# Broadband Quantum Memory in Atomic Ensembles

Kai<EMAIL_ADDRESS>Donny R. Pearson Jr.  Bin Fang  Elizabeth
A. Goldschmidt  Virginia O. Lorenz  Department of Physics, University of
Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA
Illinois Quantum Information Science and Technology (IQUIST) Center,
University of Illinois Urbana-Champaign, 1101 West Springfield Avenue, Urbana,
IL 61801, USA  Center for Dynamics and Control of Materials, The University of
Texas at Austin, Austin, TX 78712, USA

###### Abstract

Broadband quantum memory is critical to enabling the operation of emerging
photonic quantum technology at high speeds. Here we review a central challenge
to achieving broadband quantum memory in atomic ensembles—what we call the
‘linewidth-bandwidth mismatch’ problem—and the relative merits of various
memory protocols and hardware used for accomplishing this task. We also review
the theory underlying atomic ensemble quantum memory and its extensions to
optimizing memory efficiency and characterizing memory sensitivity. Finally,
we examine the state-of-the-art performance of broadband atomic ensemble
quantum memories with respect to three key metrics: efficiency, memory
lifetime, and noise.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Ensemble Atomic Systems
    3. 1.3 Linewidth-Bandwidth Mismatch
    4. 1.4 Metrics
  2. 2 Protocols and Hardware
    1. 2.1 Atomic Quantum Memory Protocols
    2. 2.2 Hardware
      1. 2.2.1 Hot Atoms
      2. 2.2.2 Cold Atoms
      3. 2.2.3 Solids
  3. 3 Theory
    1. 3.1 Maxwell-Bloch Equations
      1. 3.1.1 Homogeneous
      2. 3.1.2 Inhomogeneous
    2. 3.2 Efficiency Optimization
      1. 3.2.1 Signal Field Shaping
      2. 3.2.2 Control Field Shaping
      3. 3.2.3 Gaussian-Pulse-Shape Optimization
      4. 3.2.4 Inhomogeneous Profile Shaping
    3. 3.3 Sensitivity Analysis
  4. 4 State of the Art
    1. 4.1 Efficiency
    2. 4.2 Memory Lifetime
    3. 4.3 Noise
  5. 5 Conclusion

##  1 Introduction

###  1.1 Motivation

The fundamental challenges behind distributing quantum information using light
are light’s fixed speed and ubiquitous propagation losses in optical fiber.
Given the inability to copy quantum states [1], this means that, in order to
engage in quantum information processing using systems in separate locations,
one must find a way to store the quantum information contained in an optical
field for a time commensurate with the travel time between locations. This
holds for long-distance operations where ms-scale storage is required for
round-trip communication [2, 3] as well as on-chip or integrated devices where
ns-scale delays allow more complex quantum logic [4, 5, 6, 7, 8, 9]. Such a
quantum memory must store an incoming photonic quantum state, with quantum
information encoded in one (or more) of its degrees of freedom, and faithfully
retrieve that photonic state without altering its quantum information or
adding noise. The most intuitive application of this primitive operation
relates to photon synchronization: two photons arriving at, for example, a
node in a quantum network [10], may have traveled long and disparate distances
to the node and require some form of quantum memory if they are both to
impinge on a beamsplitter at the same time and affect a Bell state measurement
[11], entanglement swapping [12], quantum teleportation [13], or almost any
other quantum networking protocol. This is perhaps the most straightforward
application of quantum memory; however, quantum memory is also of critical
importance for all-optical quantum computing [14], quantum communication [15],
enhanced measurement and sensing [16], and local quantum gates [17].

There are many potential architectures for implementing quantum memory
effectively. Here we focus on the storage of quantum states of light in
collective states of atomic ensembles. This general class of schemes is widely
applicable across different wavelength and bandwidth regimes, and is only
fundamentally limited by the coherence and optical depth of the atomic system,
which can be chosen or engineered to be suitable for applications, albeit
typically at the expense of other important parameters. In this work, we focus
specifically on implementations of atomic ensemble quantum memory in the
broadband regime, which we consider to be photon bandwidths greater than 10
MHz. Broadband memory operation is of unique importance for the implementation
of quantum photonic applications at high speeds, as high clock rates and
processing speeds imply the use of photons that are short in duration and
therefore broad in bandwidth.

This chapter concerns the use of both atomic or atom-like ensemble systems for
broadband optical quantum memory. Atom-like systems include rare-earth ions
doped in solids, molecular gases, phonons in solids, and any platform in which
there exists an ensemble of particles with at least three energy levels on
which the memory interaction can be based. For ease of notation we refer to
both atomic and atom-like systems as “atomic.” Note that in this work we do
not consider “atomic-ensemble quantum memories” to include the creation of
single photons entangled with long-lived matter excitations [18], long-lived
excitations in single atoms or ions [19, 20], or other means of transducing
photonic quantum information into a material platform [21]. For more
information on these alternative mechanisms for engineered atom-photon
interactions, we refer the reader to the reviews of Refs. [2, 22].

Even within the relatively narrow scope of broadband quantum memory using
collective atomic states, there exist many physically distinct quantum memory
protocols and many distinct hardware implementations. Each protocol and
hardware implementation possesses particular advantages and disadvantages. We
provide some context for these relative advantages and disadvantages by first
introducing the fundamentals and the merits of atomic ensembles as quantum
memories in Sec. 1.2. We then discuss a critical problem for atomic ensemble
quantum memory in Sec. 1.3, the linewidth-bandwidth mismatch problem. We
present the metrics used to quantify memory performance in Sec. 1.4, before
launching into a comprehensive discussion of atomic memory protocols and
hardware implementations and their respective advantages in Sec. 2. In Sec. 3,
we review the theory of atomic ensemble quantum memory, including the various
forms of the equations of motion in the presence of homogeneous and
inhomogeneous broadening and the mathematical approximations that lead to each
physical protocol. With these theoretical foundations in mind, we then discuss
the theoretical tools developed for the optimization of memory efficiency and
the recently developed theoretical tools for investigating memory sensitivity
[23], which describes the behavior of atomic ensemble quantum memory in the
presence of experimental fluctuations and drift. Finally, in Sec. 4, we turn
to the state-of-the-art performance of broadband atomic-ensemble quantum
memories in the literature, providing empirical evidence for the advantages
and disadvantages discussed in Sec. 2. We focus on three metrics of particular
importance to broadband atomic ensemble quantum memory: efficiency (Sec. 4.1),
memory lifetime (Sec. 4.2), and noise (Sec. 4.3).

###  1.2 Ensemble Atomic Systems

Ensemble atomic systems are well-suited for quantum memory as they possess
high optical depths, controllable frequency (based in part on atomic species,
in part on the chosen detuning from atomic transitions), long-lived atomic
superposition states, low sensitivity to experimental noise, and arbitrary
storage time. As discussed below, however, atomic systems often suffer from
linewidth-bandwidth mismatch, noise from the necessary optical control fields,
and undesirable broadening mechanisms. Different atomic level structures have
been employed experimentally for atomic ensemble quantum memory, including
ladder-type [24, 25, 26, 27, 28], M-type [29, 30], and others [31, 32, 33];
however, the atomic $\Lambda$-type level structure is the most common, and all
level structures typically obey the same underlying atom-photon interaction
physics (see Fig. 1 for the $\Lambda$-type and ladder-type structures).

In the memory interaction, the ‘signal’ field of interest, which possesses
some quantum information encoded in one of its degrees of freedom, is tuned
on, near, or off resonance with the ground-to-excited-state transition
(${\left|g\right\rangle}\leftrightarrow{\left|e\right\rangle}$ in Fig. 1) of
the atoms. We note that this signal field may be a single photon, or a more
general photonic quantum state, with either discrete or continuous variable
quantum information encoded in its degrees of freedom (though typically more
care must be taken when using continuous variables [34]). For the remainder of
this chapter, we typically focus on the case of single-photon signal fields.
The classical ‘control’ field, possessing many photons, is tuned to the
excited-to-storage-state transition
(${\left|e\right\rangle}\leftrightarrow{\left|s\right\rangle}$). These two
atomic transitions are typically assumed to be dipole allowed, and the
${\left|s\right\rangle}\leftrightarrow{\left|g\right\rangle}$ transition is
assumed to be forbidden. Atoms entering the ${\left|s\right\rangle}$ state are
thus metastable. In this typical situation, the time-domain Maxwell-Bloch
equations describing the memory interaction are [35, 36, 37, 38, 39, 40]:

$\displaystyle\partial_{z}A(z,\tau)$ $\displaystyle=-\sqrt{d}P(z,\tau)$ (1)
$\displaystyle\partial_{\tau}P(z,\tau)$
$\displaystyle=-\bar{\gamma}P(z,\tau)+\sqrt{d}A(z,\tau)-i\frac{\Omega(\tau)}{2}B(z,\tau)$
(2) $\displaystyle\partial_{\tau}B(z,\tau)$
$\displaystyle=-\gamma_{B}B(z,\tau)-i\frac{\Omega^{*}(\tau)}{2}P(z,\tau),$ (3)

where $z$ represents the one-dimensional spatial dimension of the atomic
ensemble normalized to the ensemble length [i.e., $z=0$ ($z=1$) represents the
beginning (end) of the ensemble]; $\tau=t-z/c$ represents time measured in the
comoving frame of the photon ($t$ represents time in the lab frame) normalized
to the excited-state coherence decay rate $\gamma=\Gamma/2$ ($\Gamma$ is the
total excited-state population decay rate, or the linewidth of the
${\left|g\right\rangle}\leftrightarrow{\left|e\right\rangle}$ transition);
$A(z,\tau)$ is the spatially and temporally dependent signal photon field;
$P(z,\tau)$ and $B(z,\tau)$, referred to as the atomic polarization and spin
wave fields, respectively, are macroscopic field operators representing the
atomic coherences
${\left|g\right\rangle}\leftrightarrow{\left|e\right\rangle}$ and
${\left|g\right\rangle}\leftrightarrow{\left|s\right\rangle}$, which are
delocalized across the length of the medium and are shown in Fig. 1 as orange
and blue shaded regions; $d$ is the resonant optical depth of the memory;
$\bar{\gamma}=(\gamma-i\Delta)/\gamma$ is the normalized complex detuning,
where the two-photon detuning $\Delta$ is shown schematically in Fig. 1; and
$\Omega(\tau)$ is the control field Rabi frequency coupling the
${\left|e\right\rangle}$ and ${\left|s\right\rangle}$ states. All atomic
population is assumed to start in the ground state, and the metastable storage
state is assumed to have a coherence decay rate $\gamma_{B}$ that is much
smaller than the excited state decay rate ($\gamma_{B}\ll 1$, in normalized
units).

By inspection of Eqs. (1)-(3), we note that the photon field acts as a source
for the atomic polarization with coupling constant $\sqrt{d}$, which then
decays and accumulates temporal phase according to $\bar{\gamma}$. The atomic
polarization then acts as a source for the spin wave field with coupling
constant $\Omega(\tau)/2$, which then decays exponentially at a rate of
$\gamma_{B}$. Depending on the sign of each of the fields, the roles of source
and sink can be reversed, allowing for reversible mapping of population
ultimately from the photon field to the spin wave and back.

Figure 1: Schematic of the atomic level structures for (a) $\Lambda$-type and
(b) ladder-type ensemble quantum memory. Each structure possesses a ground
state ${\left|g\right\rangle}$, intermediate excited state
${\left|e\right\rangle}$ (with population decay rate $\Gamma=2\gamma$), and
metastable storage state ${\left|s\right\rangle}$. The signal field (red) and
control field (black) are typically kept in two-photon resonance with detuning
$\Delta$ from the excited state. Orange and blue shaded regions correspond to
atomic polarization and spin wave coherences, respectively.

Options other than atomic ensemble systems exist for optical quantum memory,
including notably the use of optical delay lines, free-space optical cavities
[41, 42, 43, 44, 45, 46, 47, 48], optical fibers [45, 49, 50], and optical
fiber cavities [51]. These quantum memories take the simplest approach to
photon storage by merely increasing the path length traveled by the photon
instead of transducing the photon into a material excitation. As such, these
delay-based optical quantum memories provide an important point of comparison
for atomic ensemble quantum memories. Delay-based memories possess their own
disadvantages, however, including fixed-increment delay times, unavoidable
optical losses, slow switching speed ($\sim$MHz), and high sensitivity to
thermal fluctuations and air currents. Nevertheless, delay-based quantum
memories are in principle agnostic to photon bandwidth and exhibit some of the
highest efficiencies and longest memory lifetimes in the broadband regime (see
Sec. 4), limited only by losses in the optical path. Delay-based memories
additionally typically exhibit ultra-low-noise operation. As shown in Ref.
[52], only a few atomic ensemble quantum memories can currently outperform
fiber delay lines in terms of memory efficiency and lifetime, where the total
memory efficiency for fiber delay lines is taken as
$\eta(\tau)=10^{-\varepsilon c_{n}\tau/10}$ for a fiber with loss
$\varepsilon$ (typically given in units of dB/km) and a speed of light in the
fiber of $c_{n}$ (the necessary length of fiber is therefore $L=c_{n}\tau$).
This calculation ignores coupling losses into and out of the fiber, but serves
as an important point of comparison. For particularly long-lived storage,
matter-based systems are likely the only option given the sub-ms limits of
propagation delay techniques. In the case of fiber-based delay lines, for
example, assuming state-of-the-art 0.2 dB/km loss [53], the $1/e$ lifetime of
a single fiber loop memory (without switchable delay, which introduces
additional loss) is only $\sim$100 ns. Similar constraints on memory lifetime
exist for free-space delay line memories as well, in which case the most
optimistic memory lifetime is $\sim$10 $\mu$s [41, 42, 43, 44, 45, 46, 47,
48].

Another important point of comparison between fiber-based and atomic-ensemble
quantum memory concerns the distortion of broadband pulses of light in fiber
due to group velocity dispersion. As the different frequencies of light in
broadband photons propagate in fiber at different velocities, broadband
photons experience temporal stretching and distortion described in the
frequency domain by
$A_{\text{out}}(\omega)=A_{\text{in}}(\omega)e^{i\beta(\omega-\omega_{0})^{2}L/2}$,
where $\beta$ is the second-order group velocity dispersion of the fiber
(typically given in units of fs2/mm) evaluated at the center frequency
$\omega_{0}$, and $L$ is the fiber length. By assuming an initially Gaussian
input pulse with angular $1/e^{2}$ frequency bandwidth $\sigma_{\omega}=\pi
BW/\sqrt{2\ln 2}$ (where $BW$ is the full-width at half-maximum bandwidth; see
Sec. 1.4), Fourier transforming into the temporal domain, and applying Eq.
(10), one can calculate the fidelity of a fiber-based memory as a function of
storage time $t$ and photon bandwidth. Inverting this relationship and
considering a fixed target memory fidelity $\mathcal{F}_{0}$, one can derive
the following tradeoff between storage time and fiber memory bandwidth:

$t=\frac{\sqrt{1-\mathcal{F}_{0}^{2}}}{\mathcal{F}_{0}\sigma_{\omega}^{2}\beta
c_{n}},$ (4)

which demonstrates that either high-fidelity, large-bandwidth memory is
possible at short storage times, or high-fidelity, long-storage-time memory is
possible at narrow bandwidths, but all three (high-fidelity, large-bandwidth,
long-storage-time) are not possible simultaneously in standard fiber. This
tradeoff can be alleviated if an appropriate length of dispersion-compensating
fiber is spliced onto the end of standard fiber; whether this is necessary or
not for a given storage time, bandwidth, and target fidelity can be derived
from Eq. (4).

While today ensemble-based quantum memories compare mostly poorly to delay
lines in terms of efficiency, bandwidth, lifetime, and noise, there is
substantial room for improvement and reason to believe that further work will
lead to important advances. In particular, the extremely high losses in on-
chip waveguides make integrated photonic systems a place where ensemble
memories may be able to make a major difference in the near term. In on-chip
photonic devices, coupling on- and off-chip to fiber may introduce more loss
than even low-efficiency, evanescently coupled atomic memory, and certainly
more loss than chip-integrated rare-earth memories. Attempts to integrate
atomic ensembles on chip and in fiber have met with some success to date [54,
55, 56, 57, 58, 59, 60, 6, 7, 8, 9] and there is hope for improvement. Atomic
ensemble memories also possess the capacity for some basic quantum optical
processing that is absent among delay-based memories. These include the
capacity for built-in mode conversion, shaping, and sorting, multiphoton
quantum state preparation [61, 24], local quantum processing [17], arbitrary
spectral-temporal and polarization mode conversion, and preparation of
arbitrary superposition states that are either not possible through other
means or not possible at the same rate or bandwidth [62, 63, 64].

Mode shaping and conversion are processing tasks particularly well-researched
in atomic ensemble memories. In the broadband regime, this typically focuses
on shaping and conversion in the temporal and frequency domains, although
polarization conversion has also been demonstrated [65]. Demonstrations to
date include frequency and bandwidth conversion in a Raman memory in room-
temperature diamond [66]; frequency conversion in room-temperature molecular
hydrogen [67]; temporal and frequency multiplexing, arbitrary temporal
shaping, and temporal stretching and compression in an AFC memory in cryogenic
Tm:LiNbO3 [68]; temporal beamsplitting in a warm cesium Raman memory [62];
temporal beamsplitting and temporal stretching and compression in a laser-
cooled rubidium ATS memory [63]; and temporal compression, stretching, and
beamsplitting in a laser-cooled rubidium Raman memory [64]. Taken together,
these demonstrations show the capacity for temporal, frequency, duration, and
bandwidth conversion of atomic-ensemble memories, which may be of practical
application in developing quantum networks and quantum information processors.
We note, however, that additional work is needed to demonstrate this utility
and to extend the characterization of atomic ensembles as mode converters to
include phase as well as amplitude manipulation.

A problem is posed for broadband atomic ensembles beyond roughly MHz
bandwidths, which is the lack of sufficiently fast classical electronic
switches to effectively make use of the broad optical bandwidths demonstrated.
Fortunately, ongoing work toward GHz and THz optical switches shows promise
toward alleviating this concern [69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
80].

###  1.3 Linewidth-Bandwidth Mismatch

Figure 2: Schematic of linewidth-bandwidth mismatch. In atomic ensemble
quantum memories, the photon spectrum (bandwidth) may be (a) smaller than the
atomic lineshape (linewidth), (b) matched, or (c) mismatched.

Broad spectral bandwidth is of particular importance for quantum applications,
and we highlight this aspect of quantum memory performance in this chapter.
Quantum memory operation with broadband photons, corresponding to short photon
durations, enables the use of large clock and qubit processing rates.
Broadband operation presents a natural problem for standard atomic ensemble
quantum memory, however, as the linewidths involved in atomic ensembles are
typically narrow, and narrow storage-to-ground-state-transition linewidths are
desirable for long memory lifetimes. We refer to the problem of storing
broadband photons with high efficiency in typically narrowband atomic ensemble
quantum memories as the ‘linewidth-bandwidth mismatch’ problem. Figure 2 shows
a schematic of the different situations possible for broadband photons; high
memory efficiency is, in general, more easily achievable for the cases of
photon bandwidths smaller than the atomic excited state linewidth [Fig. 2(a)]
or when the photon bandwidth is matched to the atomic linewidth [Fig. 2(b)],
compared to the case when the two are mismatched [Fig. 2(c)]. More
specifically, we consider the linewidth-bandwidth mismatch problem to
encapsulate the empirical trend of decreasing memory efficiency with
increasing photon bandwidth; evidence for this trend is presented in Sec. 4.
This trend is physically a result of the difficulty of absorbing broadband
photons along a narrowband transition — if a broadband photon cannot be
absorbed along the ${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$
transition, it is difficult to affect the memory operation described by the
equations in Sec. 1.2. This is easiest to see in the case of the absorb-then-
transfer protocol (discussed more in Sec. 2.1), which explicitly relies on
linear absorption of the signal field. The overall memory efficiency cannot be
high without high-efficiency absorption.

We can model this phenomenon heuristically by considering the efficiency of
linear absorption in a dense atomic ensemble. Linear absorption can be modeled
by Eqs. (1)-(3) of Sec. 1.2 in the absence of a control field
[$\Omega(\tau)=0$]:

$\displaystyle\partial_{z}A(z,\tau)$ $\displaystyle=-\sqrt{d}P(z,\tau)$ (5)
$\displaystyle\partial_{\tau}P(z,\tau)$
$\displaystyle=-\bar{\gamma}P(z,\tau)+\sqrt{d}A(z,\tau),$ (6)

where we define absorption as
$\int_{-\infty}^{\infty}d\tau{\left|A_{\text{out}}(\tau)\right|}^{2}\big{/}{\left|A_{\text{in}}(\tau)\right|}^{2}$
for a Gaussian-shaped input temporal distribution $A_{\text{in}}(\tau)$ and an
output distribution $A_{\text{out}}(\tau)=A(\tau,z=1)$. This absorption is
plotted in Fig. 3 for photon bandwidths between 10 MHz and 1 THz, assuming
excited state linewidths of 1, 10, 100 MHz, and 1 GHz. Figure 3(a)-(c) show
absorption for optical depths of $d=1$, 10, 100, respectively. In general, the
larger excited state linewidths are able to absorb broadband photons with
higher efficiency as they have lesser linewidth-bandwidth mismatch. Absorption
can also be increased by increasing optical depth, but this is typically a
less effective strategy for increasing efficiency than increasing atomic
linewidth at the same optical depth. The oscillations that appear in Fig.
3(b)-(c) are due to periodic backaction of the atomic ensemble onto the photon
temporal distribution and the formation of zero-area photon pulses [81]. The
takeaway from Fig. 3 is that high efficiency at broad photon bandwidths is
difficult to achieve with (relatively) narrowband atomic transitions, and this
is due to linewidth-bandwidth mismatch. This issue introduces a trade-off
between efficiency and lifetime, since narrowband atomic transitions are
useful for preserving the coherence of collective atomic states for long
timescales, but lead to lower memory efficiency for broadband pulses.

Figure 3: Calculated upper bounds on memory efficiency based on linear
absorption, demonstrating the linewidth-bandwidth mismatch problem for photon
bandwidths between 10 MHz and 1 THz, assuming excited state linewidths of 1,
10, 100 MHz, and 1 GHz, and peak optical depths of (a) $d=1$, (b) $d=10$, and
(c) $d=100$.

One resolution to the linewidth-bandwidth mismatch problem has been
demonstrated in recent work in hot atomic barium [82, 83]. This approach
relies on large, tunable modification of the homogeneous excited state
linewidth to better match broadband photons, where the linewidth modification
is mediated by the intentional and controllable introduction of collisional
dephasing. This technique creates broad homogeneous excitation linewidths that
allow for complete absorption and storage of broadband photonic quantum states
in a system that would otherwise exhibit relatively narrowband transitions.
This approach yields a demonstrated storage efficiency of 95.6$\pm$0.3% for
ultrashort photons (500 fs), which to our knowledge is the highest measured
storage efficiency to date for any atomic-ensemble memory with bandwidth $>$10
MHz, and rivals the efficiencies demonstrated in delay-line and fiber-based
memories, as is discussed further in Sec. 4.

The controlled homogeneous broadening approach described in Refs. [82, 83]
constitutes a novel technique for alleviating the linewidth-bandwidth mismatch
problem, but it is not the only technique. In particular, the use of far-off-
resonant schemes for quantum memory in atomic ensembles is well known, and
similarly alleviates the linewidth-bandwidth mismatch problem, albeit through
a different mechanism. In far-off-resonant memory protocols, memory bandwidth
is not limited by the narrowband atomic transitions, as the large detuning
adiabatically eliminates the excited state. This elimination of the excited
state also eliminates the restriction on memory bandwidth posed by the excited
state linewidth; however, this advantage comes at the cost of large control
field power necessary to drive the off-resonant two-photon transition. This
requirement of large control field powers can be difficult to satisfy in
experiment, and also tends to lead to low efficiency at broad photon
bandwidths (see Sec. 4). By contrast, the technique of Refs. [82, 83] can be
applied on or near resonance, and therefore requires significantly less
control field power than the far-off-resonant schemes, and can achieve similar
memory bandwidths. A third technique for alleviating linewidth-bandwidth
mismatch relies on inhomogeneous broadening of the otherwise narrowband atomic
transitions, either in a controlled fashion (i.e., in the CRIB protocol; see
Sec. 2.1) or by the choice of hardware that intrinsically possesses
inhomogeneity (i.e., rare-earth ions doped in solids; see Sec. 2.2.3). This
technique can in principle be used to create inhomogeneous linewidths that
match broad photon bandwidths, and therefore allow for high-efficiency,
broadband quantum memory; however, the external fields necessary in the
controllable case tend to be large and experimentally challenging to generate,
and in the intrinsic case the inhomogeneity introduces dephasing during the
storage operation that tends to lead to lower efficiencies upon retrieval when
compared to the homogeneous case.

Regardless of efficiency, broadband atomic ensemble quantum memories tend to
possess short memory lifetimes, mostly for technical (rather than fundamental)
reasons. Narrowband atomic
${\left|s\right\rangle}\rightarrow{\left|g\right\rangle}$ transitions are
desirable and necessary for achieving long memory lifetimes, but in practice
in the broadband regime either inhomogeneous broadening (e.g., motional
dephasing) or intrinsic material constraints (e.g., phonon lifetimes) tend to
limit memory lifetime. That said, recent techniques have been developed to
alleviate inhomogeneous broadening of the storage state (see Ref. [27] and
Sec. 2.2.1), which may allow for simultaneous high-efficiency, broadband, and
long-lived quantum memory operation. Even short-lived photonic quantum
memories are useful, however, for such applications as quasi-on-demand single-
photon generation and multiphoton quantum-state preparation [61, 24], short-
timescale synchronization, mode conversion [66, 68], and local quantum
processing [17], among others. Each quantum application employing quantum
memory places different constraints on acceptable memory performance, and for
some applications short memory lifetime is not prohibitive.

###  1.4 Metrics

In what follows we give a brief description of the relevant metrics for
quantum memory performance that have been developed in the literature:

Efficiency – Quantum memory efficiency describes the integrated intensity
ratio of photons sent in to the memory, and those retrieved from it:

$\eta=\frac{\int_{-\infty}^{\infty}d\tau{\left|A_{\text{out}}(\tau)\right|}^{2}}{\int_{-\infty}^{\infty}d\tau{\left|A_{\text{in}}(\tau)\right|}^{2}}=\frac{\int_{-\infty}^{\infty}d\omega{\left|A_{\text{out}}(\omega)\right|}^{2}}{\int_{-\infty}^{\infty}d\omega{\left|A_{\text{in}}(\omega)\right|}^{2}},$
(7)

where $A_{\text{in}}(\tau)$ [$A_{\text{in}}(\omega)$] and
$A_{\text{out}}(\tau)$ [$A_{\text{out}}(\omega)$] describe the incident and
retrieved photon amplitude in the temporal (spectral) domain, respectively.

Memory efficiency in atomic ensembles is determined by two processes, storage
(or read-in) and retrieval (or read-out), which each have independent
efficiencies $\eta_{\text{stor}}$ and $\eta_{\text{ret}}$, respectively, where
$\eta=\eta_{\text{stor}}\eta_{\text{ret}}$. These efficiencies correspond to
the integrated intensity ratio of photons sent in to the memory and the
population entering the collective atomic storage state,
${\left|B_{\text{out}}(z)\right|}^{2}$, and the integrated intensity ratio of
photons retrieved from the memory and the same collective atomic state:

$\eta_{\text{stor}}=\frac{\int_{0}^{1}dz{\left|B_{\text{out}}(z)\right|}^{2}}{\int_{-\infty}^{\infty}d\tau{\left|A_{\text{in}}(\tau)\right|}^{2}},\hskip
50.00008pt\eta_{\text{ret}}=\frac{\int_{-\infty}^{\infty}d\tau{\left|A_{\text{out}}(\tau)\right|}^{2}}{\int_{0}^{1}dz{\left|B_{\text{out}}(z)\right|}^{2}},$
(8)

where $z=1$ in normalized units refers to the full length of the atomic
ensemble, and where again spectral derivatives may be substituted for the
temporal derivatives with impunity.

An important figure of comparison for memory efficiency is the optimal
efficiency at a given optical depth, $d$. Described in Refs. [40, 39, 37],
available atom number and optical depth of an atomic ensemble impose an upper
limit on the achievable memory efficiency of a given system. The optimal bound
on storage efficiency, $\eta_{\text{opt}}$, is calculated by finding the
largest eigenvalue of the antinormally ordered storage kernel

$K(z,z^{\prime})=\frac{d}{2}e^{-d(z+z^{\prime})/2}I_{0}(d\sqrt{zz^{\prime}}),$
(9)

where $I_{0}(x)$ is the zeroth order modified Bessel function of the first
kind. The optimal bound on retrieval efficiency is also $\eta_{\text{opt}}$.
Typically, optimal retrieval is only achievable for backward retrieval of the
signal field [39]; forward retrieval of the signal field typically suffers
from reabsorption loss, as the signal field is reabsorbed by the atoms as it
propagates through the ensemble after retrieval [40]. The optimal bound on
total memory efficiency is $\eta_{\text{opt}}^{2}\geq\eta$.

Bandwidth – Memory bandwidth corresponds to the full width at half maximum of
the signal photon spectral intensity stored and retrieved in the memory. For a
Gaussian-shaped signal field amplitude $A_{\text{in}}(\tau)\propto
e^{-\tau^{2}/4\sigma^{2}}$, where $\sigma=\tau_{\text{FWHM}}/(2\sqrt{2\ln 2})$
in terms of the signal temporal intensity full width at half maximum
($\tau_{\text{FWHM}}$), the Fourier-transform-limited photon bandwidth is
$BW=2\ln 2/(\pi\tau_{\text{FWHM}})$. The bandwidth of a quantum memory
determines the memory’s compatibility with short-duration pulses, which is of
critical importance to real-world quantum applications that benefit from large
clock rates and high processing speeds [84, 85].

Memory Lifetime – The lifetime of an atomic ensemble quantum memory, $T$,
typically corresponds to the time it takes for the retrieved photon population
[$\langle
A_{\text{out}}\rangle\propto\int_{-\infty}^{\infty}d\tau{\left|A_{\text{out}}(\tau)\right|}^{2}$]
to reach $1/e$ of its maximum value. Assuming the retrieval control field
pulse remains unchanged aside from its arrival time in the atomic ensemble,
this is equivalent to the time it takes for the the collective atomic state
population [$\langle
B_{\text{out}}\rangle\propto\int_{0}^{1}dz{\left|B_{\text{out}}(z)\right|}^{2}$]
to reach $1/e$ of its maximum value. Occasionally, the $1/2$ lifetime of a
memory is given instead of the $1/e$ lifetime and a conversion factor
depending on the decay model must be used to compare memories (e.g., for
exponential decay, the $1/e$ lifetime $T$ and $1/2$ lifetime $T_{1/2}$ are
related by $T_{1/2}=T\ln 2$; for Gaussian decay, $T_{1/2}=T\sqrt{\ln 2}$).

Time-Bandwidth Product – A relevant figure of merit for most quantum protocols
is the time-bandwidth product, defined as the product of memory lifetime and
spectral bandwidth, $\mathrm{TBP}=T\times BW\pi/(2\ln 2)$. This figure
provides a metric for the fractional delay produced by the memory in units of
the photon duration, $\mathrm{TBP}=T/\tau_{\text{FWHM}}$. Broadband quantum
memories may have equivalent or even larger time-bandwidth product at short
storage times compared to narrowband quantum memories with long storage times.

Noise – Noise is a critical figure of merit for determining the performance of
a quantum memory. Even if the photon retrieved from a quantum memory is
retrieved in the same quantum state as the input with unit efficiency, the
presence of noise photons in addition to the retrieved signal renders such a
memory useless for almost every quantum application. This importance of noise
relates to whether a memory qualifies as ‘quantum’—true ‘quantum’ memories
must demonstrate below (ideally far below) 1 noise photon per retrieved signal
photon, such that they do not contaminate the quantum states they store.

Figure 4: Schematic of atomic ensemble quantum memory during the (a) storage
and (b) retrieval stages. The classical control field is shown in green, the
quantum signal field in red, and the noise field in blue.

The noise performance of quantum memory can be quantified several different
ways. The most popular metrics for determining noise performance are: $\langle
n_{\text{noise}}\rangle$, the average number of noise photons produced per
pulse that overlap with the retrieved signal field; signal-to-noise ratio at 1
input photon/pulse (SNR), which is the signal-to-noise-ratio of the retrieved
signal population to the average number of noise photons
[$\mathrm{SNR}=\langle A_{\text{out}}\rangle/\langle
n_{\text{noise}}\rangle=\eta\langle A_{\text{in}}\rangle/\langle
n_{\text{noise}}\rangle=\eta/\langle n_{\text{noise}}\rangle$], evaluated when
on average 1 photon per pulse is sent into the memory ($\langle
A_{\text{in}}\rangle=1$); $\mathcal{F}_{\text{noise}}$, the single-photon
fidelity of the memory, defined as
$\mathcal{F}_{\text{noise}}=1-1/(\mathrm{SNR}+1)$, which is a measure of the
fidelity of retrieving the stored signal photon rather than a noise photon,
and which may be approximated as $\mathcal{F}_{\text{noise}}=1-1/\mathrm{SNR}$
in the limit of large $\mathrm{SNR}$; and $\mu_{1}$, which is the ratio of the
average number of noise photons per pulse to the memory efficiency,
$\mu_{1}=\langle n_{\text{noise}}\rangle/\eta=1/\mathrm{SNR}$.

Some caution must be exercised in using each of these metrics, as ambiguity
exists in the definition of $\mathrm{SNR}$. As shown in Fig. 4(b), the total
photon field retrieved from a memory is made up of two contributions, which we
call the signal (shown in red) and the noise (blue). Occasionally, the total
photon field retrieved (signal and noise) is referred to as the “signal,” in
which case the signal-to-noise ratio is reported as what we would consider the
‘total-to-noise’ ratio ($\mathrm{TNR}$). The two are related by
$\mathrm{TNR}=\mathrm{SNR}+1$.

Fidelity – Independent from the noise performance of a memory, one can
consider the overlap of the retrieved photon quantum state with the quantum
state of the photon before it was sent into the memory. The definition of this
fidelity typically depends on the degree of freedom of interest where quantum
information is encoded, as in the case of quantum process tomography for
polarization-encoded photonic qubits [86], and can be performed for a single
input state or averaged over multiple input states [87]. Typically, fidelity
is defined in terms of a state overlap
$\mathcal{F}={\left|{\left\langle\Psi_{\text{out}}\right|}\Psi_{\text{in}}\rangle\right|}^{2}$
for pure states or
$\mathcal{F}={\left\langle\Psi_{\text{in}}\right|}\rho_{\text{out}}{\left|\Psi_{\text{in}}\right\rangle}$
for mixed states, which, in the temporal domain, can be expanded as

$\mathcal{F}=\frac{{\left|\int_{-\infty}^{\infty}d\tau\,A_{\text{out}}^{*}(\tau-\tau_{d})A_{\text{in}}(\tau)\right|}^{2}}{\int_{-\infty}^{\infty}d\tau\,{\left|A_{\text{in}}(\tau)\right|}^{2}\int_{-\infty}^{\infty}d\tau\,{\left|A_{\text{out}}(\tau)\right|}^{2}},$
(10)

where $\tau_{d}$ is the time delay between incident and retrieved pulses. This
definition of fidelity is sometimes called the “waveform likeness” [88, 89,
90], as it compares the likeness of the incident and outgoing temporal
waveforms, where the outgoing waveform typically encounters some distortion
[see Fig. 4(a) and (b)]. In experiment, this is typically measured via
intensity interferometry, where a reference pulse identical to the incident
signal field is interfered with the retrieved pulse. Often in this case the
interferometric visibility is reported in place of fidelity [84, 63].

Fidelity is a critical figure of merit as it relates to the degree of phase-
and entanglement-preservation of a memory. Phase- and entanglement-
preservation are critical to the use of quantum memories in quantum repeaters
[2], for example, among other applications.

Adiabaticity – The adiabaticity of a free-space atomic ensemble quantum memory
is given by $d\tau_{\text{FWHM}}\gamma$, where $d$ and $\tau_{\text{FWHM}}$
are as defined above, and $\gamma$ is the coherence decay rate of the
intermediate excited atomic state of the three-level atomic system
($\Gamma=2\gamma$ is the population decay rate). Described in detail in [37,
39, 36], quantum memories which satisfy $d\tau_{\text{FWHM}}\gamma\gg 1$ are
considered adiabatic, and memories with $d\tau_{\text{FWHM}}\gamma\sim 1$ are
typically considered non-adiabatic. The adiabatic condition
$d\tau_{\text{FWHM}}\gamma\gg 1$ ensures that for efficient memory operation
the signal field bandwidth ($\propto 1/\tau_{\text{FWHM}}$) will be smaller
than, or a similar order to, the excited state linewidth. It also ensures that
the control field used to mediate the memory operation will be weak and long
in duration relative to the timescale of evolution of the atomic states. All
of these conditions together allow for adiabatic elimination of the atomic
polarization in the equations of motion described in Sec. 1.2.

Multimode Capacity – A quantum memory capable of storing and retrieving
multiple photonic qubits independently at overlapping times is considered
multimode, as opposed to a single-mode memory, which must store and retrieve a
given photon and reinitialize before it is prepared to store and retrieve
another. Atomic ensembles may demonstrate multimode capacity in a number of
different degrees of freedom, including temporal mode [91], angular wavevector
[92], three-dimensional spatial position [93], time bin [94], and frequency
bin [95].

Multimode capacity is typically defined relative to a reference efficiency,
$\eta_{\text{ref}}$. The multimode capacity $N$ of a quantum memory
corresponds to the largest number of independent modes that can each be stored
with at least $\eta_{\text{ref}}$ efficiency.

Sensitivity – The sensitivity of atomic ensemble quantum memory is a metric
that has been discussed only recently [23, 96]. Memory sensitivity relates to
a memory’s performance in the presence of experimental noise, including
fluctuations and drift in memory parameters such as optical depth and atomic
transition linewidth as well as in the optical control field parameters used
in the memory interaction. A memory that is less sensitive is more robust to
experimental noise, and vice versa.

Memory sensitivity may be quantified in a number of different ways, and memory
sensitivity may in principle refer to the sensitivity of any of the metrics of
this section in the presence of experimental noise. The typical case, however,
concerns changes in memory efficiency in the presence of noise. When short-
timescale fluctuations lead to changes in memory efficiency, the memory
sensitivity is given in terms of the variance of memory efficiency in the
presence of fluctuations $\zeta$ in parameters $\mathcal{X}$

$V^{\text{fluc}}_{\eta}\left(\overline{\mathcal{X}}\right)=V_{\zeta}[\eta(\overline{\mathcal{X}}+\zeta)],$
(11)

where $\overline{\mathcal{X}}$ is the mean parameter value averaged over a
large number of fluctuations, and where $V_{x}[y(x)]=\int
dx\,y^{2}(x)P(x)-[\int dx\,y(x)P(x)]^{2}$ is the unconditional variance of $y$
obtained when $x$ fluctuates with some probability distribution $P(x)$. The
resulting standard deviation in memory efficiency can then be reported as
$\sigma_{\eta}^{\text{fluc}}(\overline{\mathcal{X}})=\sqrt{V^{\text{fluc}}_{\eta}(\overline{\mathcal{X}})}$.

When long timescale drift in experimental conditions lead to changes in memory
efficiency, and only one parameter of the system drifts, the memory
sensitivity can be calculated via the one-at-a-time (OAT) variance

$V_{\eta}^{\text{OAT}}(\overline{\mathcal{X}})=V_{\mathcal{X}}[\eta(\mathcal{X})],$
(12)

where $\mathcal{X}$ varies uniformly over a finite range,
$\mathcal{X}\in[\mathcal{X}^{\text{min}},\mathcal{X}^{\text{max}}]$ centered
on $\overline{\mathcal{X}}$. Again, the standard deviation
$\sigma_{\eta}^{\text{OAT}}(\overline{\mathcal{X}})=\sqrt{V_{\eta}^{\text{OAT}}(\overline{\mathcal{X}})}$
may be used to quantify the change in memory efficiency due to changes in
$\mathcal{X}$. In the case where multiple experimental parameters
$\mathcal{X}=(x_{1},...,x_{N})$ drift simultaneously, a global variance-based
sensitivity analysis is required, wherein the most prevalent sensitivity
measure is the first-order Sobol’ variance [97, 98, 99]

$V_{i}=V_{x_{i}}\\{E[\eta(\mathcal{X})|x_{i}]\\},$ (13)

where the inner expectation value, $E[\cdot]$, corresponds to the mean of
$\eta(\mathcal{X})$ when $\mathcal{X}$ is varied over all possible values in a
finite range at fixed $x_{i}$. The outer variance then measures the variance
of this mean with respect to changes in $x_{i}$. The first-order Sobol’
sensitivity index for parameter $x_{i}$ can then be calculated as

$S_{i}=V_{i}/V_{\text{tot}},$ (14)

where $V_{\text{tot}}$ is the total variance
$V_{\mathcal{X}}[\eta(\mathcal{X})]$ observed over the range of interest.

Telecom Compatibility – Quantum memories are often proposed for use before,
after, or inside networks of optical fiber. As linear absorption in standard
silica optical fiber reaches a minimum at 1310 and 1550 nm [100], the telecom
O- and C-band, respectively, a quantum memory compatible with these
wavelengths is desirable. If 1310 or 1550 nm operation is not possible, other
less-standard telecommunications bands exist in the range of 1260-1675 nm, and
wavelength compatibility within this range is more desirable than outside this
range. Telcom compatibility is therefore typically a binary metric,
designating whether or not a quantum memory is capable of storage and
retrieval of photons between 1260 and 1675 nm center wavelengths, where
special attention is given to 1310 nm and 1550 nm operation.

Latency – As with any technical device, latency describes the amount of time
required by a quantum memory for initialization and preparation before storage
and retrieval of photonic quantum states is possible. A useful figure of merit
for this aspect of quantum memory operation is the memory’s duty cycle,
defined as the ratio of the amount of time the memory is capable of storage
and retrieval to the total cycle time necessary for initialization and
subsequent memory operation [101]. In general, a memory with a larger duty
cycle (ideally 100%) is preferable to a memory with a smaller duty cycle.

Size, Weight, and Power (SWaP) – Of practical importance to the use of quantum
memories in real-world quantum applications is the size, weight, and power
consumption of the memory. Typically these metrics are not reported exactly in
the scientific literature, and may be hard to estimate, but general trends can
be intuited (such as, for example, that an atomic vapor cell held at a higher
temperature will have a higher power consumption than a cooler one, and
memories requiring high control field optical powers will have higher power
consumption than those with low powers). In general, a memory with a smaller
SWaP is preferable to a memory with a larger SWaP.

Device Lifetime – The lifetime of a quantum memory device is also of practical
importance. In the limiting case, a quantum memory that is single-use — only
capable of storing and retrieving one photonic quantum state before the device
needs to be replaced — has virtually no practical application. All quantum
memories in the literature exceed this limit; however, some platforms tend to
demonstrate longer device lifetimes than others and again general trends can
be intuited based on the required temperatures, pressures, and experimental
components.

##  2 Protocols and Hardware

###  2.1 Atomic Quantum Memory Protocols

Many protocols for quantum memory operation in atomic ensembles exist based on
distinct and quite disparate physical mechanisms. This section provides a
brief review of the ensemble memory protocols found in the literature to date.
We comment on the advantages of each memory in terms of the metrics described
above in Sec. 1.4, focusing in particular on the bandwidth limitations of each
protocol.

#### Electromagnetically Induced Transparency (EIT)

Electromagnetically induced transparency (EIT) in atomic ensembles is a
phenomenon characterized by optical transparency at wavelengths resonant with
the ${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ atomic
transition (see Fig. 1), which normally exhibits significant attenuation, and
an associated reduction in group velocity of the incident signal field [102].
This transparency and slowed group velocity is due to the presence of a
second, strong optical control field which couples the excited and metastable
atomic states (${\left|e\right\rangle}\leftrightarrow{\left|s\right\rangle}$),
where the ${\left|e\right\rangle}\rightarrow{\left|s\right\rangle}$ transition
is dipole allowed but the
${\left|s\right\rangle}\rightarrow{\left|g\right\rangle}$ tranistion is
forbidden. This is a quite general phenomenon; for the purposes of this work
we focus on EIT harnessed for quantum memory, where the ‘slow light’ of
general EIT is transformed into ‘stopped light’ via adiabatic attenuation of
the control field [103].

The pulse sequence for this protocol is as follows: A control field of
duration longer than the signal field
($\tau^{\text{ctrl}}_{\text{FWHM}}>\tau_{\text{FWHM}}$) enters the medium
ahead of the signal field in time ($\Delta\tau^{\text{ctrl}}<0$). This opens a
spectral transparency window at the signal frequency. This window is then
slowly closed after the signal field enters the medium via attenuation of the
control field. The signal field is thereby compressed and trapped in the
medium in a superposition of the metastable and ground states of the atoms.
This protocol is described in the narrowband regime in Refs. [102, 104, 103,
105, 106, 37], and in the broadband regime in Refs. [107, 90].

Numerous experimental implementations of the EIT protocol for quantum memory
exist (a review of most EIT quantum memory implementations can be found in
Refs. [59, 106, 108, 109, 110]). Efficiencies as high as 92% have been
demonstrated [111], as well as—separately—memory lifetimes verging on 1 minute
[112, 113]. As EIT memory requires adiabatic elimination of the excited state
however, either large optical depths or narrow bandwidths (relative to the
excited state linewidth) are required to fulfill the adiabaticity criterion
$d\tau_{\text{FWHM}}\gamma\gg 1$ [107, 114]. To date this has limited
operation of EIT-based quantum memory to bandwidths of 170 MHz [115] or less.

Noise operation in EIT systems tends to be limited by either four-wave-mixing
(FWM) or control field noise. FWM noise occurs when the strong control field
operates off-resonantly along the
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition,
generating a spontaneous Stokes or idler photon along
${\left|e\right\rangle}\rightarrow{\left|s\right\rangle}$, before operating
again along the ${\left|s\right\rangle}\rightarrow{\left|e\right\rangle}$
transition, generating an anti-Stokes or ‘spurious signal’ photon along
${\left|e\right\rangle}\rightarrow{\left|g\right\rangle}$, which overlaps with
the retrieved signal photon in all degrees of freedom. Control field noise
occurs when there is a small frequency difference between signal and control
field, and thus spectral isolation of the signal field is difficult to attain.
Compared to ATS and SR memory protocols (discussed below) at the same
bandwidth, EIT exhibits larger FWM noise due to its comparatively larger
requirements on optical depth and control field Rabi frequency [116, 117]. FWM
noise in EIT is very well characterized [118, 119, 120] and many technical
solutions to both control field and FWM noise exist [121, 65, 122, 123, 124,
125].

In ladder systems, the EIT protocol is often referred to as fast ladder memory
(FLAME), and may include a small detuning from resonance [24, 27, 28].

#### Autler-Townes Splitting (ATS)

The Autler-Townes Splitting (ATS) memory protocol is closely related to the
EIT protocol [107, 114], but typically operates most efficiently at lower
optical depths and broader photon bandwidths, placing it in the non-adiabatic
class of quantum memories ($d\tau_{\text{FWHM}}\gamma\sim 1$). The physical
mechanism of ATS quantum storage relies on dynamic control of the Autler-
Townes doublet created in the
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ absorption profile
in the presence of a strong control field. By generating a dynamic Autler-
Townes doublet that scans across the full bandwidth of the signal field (which
is typically of the same order as the excited state linewidth, $\Gamma$) as it
propagates through the atomic ensemble, uniform attenuation of the signal
field in frequency can be achieved that ensures coherent population transfer
to the spin wave state [63].

The pulse sequence used to generate the appropriate dynamical Autler-Townes
splitting consists of a control field pulse of similar duration to the signal
field ($\tau^{\text{ctrl}}_{\text{FWHM}}\sim\tau_{\text{FWHM}}$) that arrives
at the atomic ensemble contemporaneously with the signal field
($\Delta\tau^{\text{ctrl}}\sim 0$) and possesses net control field pulse area
(overlapping with the signal field) of $2\pi$.

As the bandwidth of the signal field increases, the effective optical depth of
the ${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition
decreases due to increasing necessary Autler-Townes splitting, and as the
bandwidth of the signal field decreases, population in the atomic polarization
state experiences increased decoherence due to the increased duration of the
protocol [63]. Both of these effects lead to decreased memory efficiency, and
therefore lead to an optimal photon bandwidth for a given ATS memory that is
typically of order $BW\sim\Gamma$ [63, 107, 114]. This leads to favorable
memory bandwidth compared to the EIT protocol in the same memory; however, to
date ATS memories have been limited experimentally to bandwidths of 20 MHz
[107] and below.

#### Superradiance (SR) Mediated Memory

Superradiance (SR) mediated memory is another on-resonant non-adiabatic
protocol, similar to the ATS protocol, but with distinct characteristics. The
protocol [116] relies on the effect of superradiance [126], which is
characterized by cooperative spontaneous emission in atomic ensembles. The
superradiant quantum memory protocol has three stages: absorption, writing,
and retrieval. In the absorption stage, the signal photon is absorbed linearly
along the ${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition
of the ensemble and thereby prepares the ensemble in a timed-Dicke state [127]
with a decay time shorter than the bare atomic excited state lifetime, as the
induced ordered spatial phase distribution encourages the coherent enhancement
of radiation in the direction of the absorbed photon. This short decay time
makes SR memory compatible with photons with bandwidth greater than the bare
excited state linewidth, $BW>\Gamma$. To suppress superradiant emission of the
absorbed photon and realize photon storage, the writing stage starts directly
after the absorption stage, within the superradiant decay time $T_{SR}$. This
process is accomplished by sending in a control field with $\pi$ pulse area
and duration $\tau^{\text{ctrl}}_{\text{FWHM}}\ll T_{SR}$, which maps the
generated atomic polarization to a collective spin excitation. During the
retrieval process, another $\pi$-pulse control field is applied and the photon
is superradiantly emitted.

Optimizing the absorption process in SR memory requires shaping the temporal
profile of the signal photon to match the time-reversed superradiant decay.
The optimal shape of the photon is therefore exponentially rising with time
constant inversely proportional to the linewidth $\Gamma$ and the optical
depth $d$. Currently, the first demonstration of SR memory has achieved
storage of signal photons with time constants down to $10\text{ns}$ and $3\%$
memory efficiency [116]. The lower-than-expected efficiency is due to a long
experimental control pulse duration $\tau^{\text{ctrl}}_{\text{FWHM}}$ that
does not completely satistfy the condition
$\tau^{\text{ctrl}}_{\text{FWHM}}\ll T_{SR}$. When compared with EIT and ATS
protocols, the SR protocol requires less optical depth for the same memory
efficiency at the cost of higher control power. Four-wave mixing noise in SR
memory appears to be on par with EIT but much higher than ATS memory.

#### Absorb-then-Transfer (ATT)

Similar to SR memory, the absorb-then-transfer (ATT) memory protocol [128,
129, 37, 130] occurs in three physically distinct stages. As the name
suggests, the first two stages correspond to linear absorption along the
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition in the
absence of the control field, then a short delay later the application of a
$\pi$-pulse control field along the
${\left|e\right\rangle}\rightarrow{\left|s\right\rangle}$ transition that
transfers population from the atomic polarization coherence
(${\left|g\right\rangle}\leftrightarrow{\left|e\right\rangle}$) to the spin
wave coherence
(${\left|g\right\rangle}\leftrightarrow{\left|s\right\rangle}$). These two
stages implement the storage operation. When retrieval is desired, another
control field pulse with $\pi$ pulse area is applied to the atomic ensemble,
transferring population back to the atomic polarization coherence, which then
emits the stored signal field through dipolar radiation. This memory protocol
is distinct from the photon-echo protocols discussed below as it relies on
homogeneous broadening of the intermediate excited state rather than
(reversible or structured) inhomogeneous broadening.

In order to optimize storage efficiency in the ATT protocol, the storage
control field arrival should be synchronized in time with the first zero of
the complex signal field amplitude evaluated at the middle of the ensemble
($L/2$) [130]. This protocol can implement optimal photon storage (i.e.,
$\eta_{\text{stor}}=\eta_{\text{opt}}$) for large optical depths, when the
adiabaticity criterion $d\tau_{\text{FWHM}}\gamma\gg 1$ is satisfied [129]. As
this protocol is typically employed when $\tau_{\text{FWHM}}\gamma<1$, this
implies $d\gg 1/(\tau_{\text{FWHM}}\gamma)$.

Near-off-resonant memory (NORM) operation of ATT was introduced in Ref. [82]
as a means to achieve higher efficiency than resonant ATT when control field
power is constrained. NORM balances reabsorption loss, which is worst on
resonance at large optical depths, and finite available control field power,
which leads to lower efficiency at larger detuning. NORM operation is not
necessary in the ideal case, for example, investigated in Ref. [114], when all
parameters of the control field are optimized to improve memory efficiency.
NORM operation only appears to help in the case when one parameter of the
control field is constrained, as is frequently the case experimentally. Future
work is needed to investigate this behavior theoretically.

The NORM ATT protocol is well-suited for broadband operation, as was recently
demonstrated in Refs. [82] and [83] in atomic barium vapor. A storage
efficiency of 95.6$\pm$0.3% was reached for ultrashort photons (500 fs),
which, as noted in Sec. 1.3, is the highest attained for bandwidths $>$10 MHz
(see Fig. 9 in Sec. 4.1). The total efficiency was measured to be 31$\pm$1% at
900 ∘C, limited by control field power. The lifetime of this memory was
measured to be 0.49(1) ns at 900 ∘C, which, while not long in absolute terms,
is significantly better than the $\sim$ps level lifetimes that are typical of
solid-state THz memories (see Sec. 4.2), which are usually assumed to be
population-lifetime limited; the bare atomic 0.25 sec population lifetime of
the storage state used in atomic barium leaves large room for improvement.
This lifetime corresponds to a time-bandwidth product of $\mathrm{TBP}=980\pm
20$. In terms of noise performance, a signal-to-noise ratio of
$\mathrm{SNR}=$(8.2$\pm$1.3)$\times 10^{3}$ is measured for an input of 1
photon per pulse, leading to a single photon fidelity of $F=0.99988(2)$. This
noise performance is on par with the lowest noise ladder-type atomic ensemble
memory systems [25, 24, 26, 28], as shown in Sec. 4.3. The exceptional noise
performance of this memory is explained by the ultra-large ground-storage
state splitting, which is so large ($\sim$340 THz) that it eclipses the
excited-storage state splitting ($\sim$200 THz), the transition operated along
by the control field. This means that, to first order, the control field
frequency is insufficient to excite four-wave mixing noise, which is the
dominant noise contribution in almost all other $\Lambda$-type memories. This
large ground-storage state splitting also virtually eliminates noise from
thermal population of the storage state, and furthermore allows for near
complete spectral suppression of control field leakage noise into the signal
path.

#### Off-Resonant Raman

The off-resonant Raman memory protocol [131] is well-suited for storage and
retrieval of broadband photons. In a typical $\Lambda$-type or ladder system,
the strong control field, which is off-resonant from the excited state
${\left|e\right\rangle}$ by a detuning $\Delta$ (see Fig. 1), enables a two-
photon transition between the ground state ${\left|g\right\rangle}$ and
storage state ${\left|s\right\rangle}$. In the storage process, the signal
photon, which is also detuned by $\Delta$ from the
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition, is
spatially and temporally overlapped with the control field and is thereby
mapped onto a collective spin-wave excitation. A subsequent retrieval field
(typically the same as the storage field) is then applied to the atomic
ensemble and the stored photon is deterministically released, completing the
retrieval process.

In the off-resonant Raman limit, the detuning is much greater than the excited
state linewidth, the bandwidths of signal and control fields, and the control
field Rabi frequency
${\left|\Delta\right|}\gg\Gamma,BW,{\left|\Omega(\tau)\right|}$. This allows
the off-resonant Raman protocol to operate in the adiabatic regime where the
excited state is eliminated. In principle, this may also allow for the storage
and retrieval of photons with arbitrary temporal shapes, and for arbitrary
temporal shaping upon retrieval, depending on the shape of the control field.
The memory efficiency for this protocol is proportional to the optical depth
and control pulse energy and is inversely proportional to the detuning. The
bandwidth of storage is typically only limited by the energy splitting between
ground and storage states, and has therefore exceeded 1 THz in experiment [65,
132, 133, 134]. Due to the large detuning, four-wave mixing noise is typically
dominant over control field leakage noise and other noise contributions.

In ladder systems, the off-resonant Raman protocol with few modifications is
referred to as off-resonant cascaded absorption (ORCA) [25, 26].

#### Controlled Reversible Inhomogeneous Broadening (CRIB)

In atomic ensembles that exhibit inhomogeneous broadening of the optical
transition, the atomic coherence generated by the absorption of the incoming
signal will begin to rapidly dephase. Controlled reversal of the inhomogeneous
broadening at a time $t$ following absorption leads to time-reversal of this
dephasing and the subsequent re-emission of the signal field at time 2$t$
[135, 136, 137, 128, 106, 138]. A review of the equations of motion governing
this process can be found in Sec. 3.1 of this chapter and in the proposal of
this protocol, Ref. [136]. This memory protocol is commonly referred to as
controlled reversible inhomogeneous broadening (CRIB). The bandwidth for this
protocol is set by the inhomogeneous broadening of
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition [135,
139]. A subset of these protocols are the gradient echo memories (GEM) in
which the controlled inhomogeneous broadening is given by a spatial electric
or magnetic field gradient [135].

The efficiency of CRIB is derived in Refs. [136, 138]. For a narrow absorption
feature of initial linewidth $\gamma_{0}$ that has been inhomogeneously
broadened to $\gamma$, the efficiency of CRIB is given by:

$\eta_{\text{CRIB}}=(1-e^{-d\gamma_{0}/\gamma})^{2}$ (15)

where $d$ is the optical depth of the unbroadened line. The efficiency goes as
the square of the probability of absorption, as the retrieval process is given
by time-reversed absorption [138, 140]. Additional considerations and
expressions for the efficiency are discussed in Refs. [106, 141]. To date, the
bandwidth and efficiency of experimental implementations of the CRIB protocol
have been limited to $\sim 0.7$ MHz and 69%, respectively [142].

#### Gradient Echo Memory (GEM)

Gradient echo memories (GEM) are an implementation of the CRIB protocol in
which the inhomogeneous broadening is given by a (typically spatial) electric
or magnetic field gradient [135, 106, 138]. In these schemes, the spatial
gradient gives rise to a spatially dependent detuning that can be reversed by
changing the sign of the gradient. The GEM protocol consequently does not
require backward propagation of the signal for high efficiency [135]. GEM was
first implemented at the few-photon level over a decade ago using light in the
telecommunications C-band, at a bandwidth of 5 MHz, with a storage time of
several hundred nanoseconds [143]. Implementations in cold atomic ensembles
have also been demonstrated over the last decade [144, 145, 146], however the
bandwidths demonstrated (sub-MHz) tend to be small due to the difficulty in
generating GHz-THz inhomogeneous linewidths.

#### Atomic Frequency Comb (AFC)

Figure 5: Schematic of the atomic frequency comb (AFC) memory. A periodic comb
structure of total width $\Gamma$ is formed in the inhomogeneous line. The
comb teeth have linewidth $\gamma$ and are evenly spaced by $\Delta$. More
details can be found in Ref. [94].

The atomic frequency comb (AFC) protocol is an inherently multimode photon
echo memory that utilizes many narrow, periodically spaced absorption features
created in a broad inhomogeneous
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ line by optical
pumping [147, 94, 106, 135, 59, 138]. Contrary to the CRIB and GEM protocols,
the periodic structure of this frequency comb itself leads to periodic
rephasing, without the need for external fields to reverse the inhomogeneity.
The periodic structure has total width $\Gamma$, and is composed of narrow
comb teeth of linewidth $\gamma$ evenly spaced in frequency by $\Delta$. The
finesse of the comb teeth is then $\mathscr{F}=\Delta/\gamma$. Since atoms are
selectively removed from the inhomogeneous line to form the comb structure,
the optical depth of the transition has been reduced approximately by a factor
of $\mathscr{F}$ [147, 94].

Signal bandwidths compatible with the AFC protocol must satisfy
$\Delta<BW\lesssim\Gamma$. Upon absorption of the signal field, the periodic
structure of the frequency comb causes the atomic coherence to rephase after a
time $2\pi/\Delta$, leading to subsequent reemission of the signal field. The
AFC protocol thus possesses a fixed, predetermined storage time set by the
spacing of the comb teeth; however, prolonged and arbitrary storage times can
be achieved by mapping the coherence to a third metastable state,
${\left|s\right\rangle}$, via optical $\pi$-pulses [94, 106, 135], as shown in
Fig. 5.

Retrieval of the signal field in the forward direction is limited to a maximum
efficiency of 54% due to reabsorption of the signal field by the medium [94,
106, 135], and is given by the closed form
$\eta_{\text{AFC}}\approx(d/\mathscr{F})^{2}e^{-7/\mathscr{F}^{2}}e^{-d/\mathscr{F}}$.
Retrieval of the signal field in the backward direction is not subject to this
constraint, and is given by
$\eta_{\text{AFC}}\approx(1-e^{-d/\mathscr{F}})^{2}e^{-7/\mathscr{F}^{2}}$
[94].

A distinct advantage of the AFC protocol is that the total number of temporal
modes that can be stored does not depend on the optical depth of the medium,
but rather the total number of comb teeth that are prepared [138]. In
contrast, the capacity of the EIT protocol to store $N$ modes scales as
$\sqrt{d}$, and for the CRIB protocol, for sufficiently large optical depth,
the number of modes $N$ scales linearly with $d$ [94, 138, 106].

Out of those protocols employing an inhomogeneously broadened
${\left|g\right\rangle}\rightarrow{\left|e\right\rangle}$ transition, the AFC
protocol tends to be the most broadband, where bandwidths up to 50 GHz at an
efficiency of 1.8% have been demonstrated in the literature [148].

#### Revival of Silenced Echo (ROSE)

Figure 6: Schematic of the ROSE protocol outlined in Ref. [149]. The rephasing
pulses (green) are counterpropagating relative to the input signal. In this
geometry, the free induction decay (blue) will be emitted in the same
direction as the rephasing beams. Spontaneous emission noise (thin red) will
still be present and pollute the retrieved echo emitted in the forward
direction.

The two-pulse photon echo is a natural candidate for quantum memory in
inhomogeneously broadened media [150]. In principle, two-pulse photon echoes
do not require any state preparation, and may utilize the entire inhomogenous
linewidth and optical depth, which may improve the multimode capacity of such
a memory compared to the AFC protocol [151, 150, 152]. Such a protocol can in
principle also be extended to a three-level system to transfer the
inhomogeneous atomic coherence to a longer-lived spin state [152]. A two-
photon echo on its own is not suitable for quantum memory, however, as
population inversion from the strong rephasing pulse creates significant and
unavoidable spontaneous emission and free induction decay (FID) noise [150,
152].

A modification to the two-pulse photon echo protocol, the revival of silenced
echo (ROSE) protocol, is able to suppress some of these inherent noise sources
and provide quantum operation. This protocol relies on phase-mismatching of
the incoming signal and the first rephasing pulse. The atomic polarization in
the medium persists regardless of the phasematching between these two fields,
and phasematching with a second rephasing pulse enables coherent emission of a
second photon in the absence of FID noise. The total storage time for this
process is $2(t_{2}-t_{1})$, where $t_{1}$ and $t_{2}$ are the arrival times
of the first and second rephasing pulses, respectively [152, 149, 153]. The
geometry of this scheme is described in Refs. [152, 149, 153] and illustrated
in Fig. 6. If both rephasing pulses are co-linear and counterpropagating
relative to the incoming signal, FID will be emitted in the direction of the
rephasing pulses and not the retrieved echo. Spontaneous emission noise cannot
be eliminated and remains a limitation to the noise performance of this
protocol.

In the limit of an infinite $T_{2}$ coherence time, and assuming forward
emission of the second echo, the efficiency of ROSE depends solely on the
optical depth as $\eta_{\text{ROSE}}=d^{2}e^{-d}$. For a finite $T_{2}$, the
efficiency also depends on $T_{2}$ and the time difference between the two
rephasing pulses as $\eta_{\text{ROSE}}=d^{2}e^{-d}e^{-4(t_{2}-t_{1})/T_{2}}$
[152, 149].

In the proposal for the ROSE scheme, the authors note that $\pi$-pulses are
not optimal for the population inversion [152]; adiabatic passage with chirped
pulses is preferable and has been used in several experimental implementations
of ROSE [151, 152, 149, 154]. To date, the ROSE protocol has been exclusively
implemented in rare-earth ion-doped solids at cryogenic temperatures: Tm3+:YAG
[153, 149, 155, 156], Er3+:YSO, [151, 154], and Tm3+:YSO [157]. Efficiencies
as high as 44% have demonstrated in 0.005% 167Er3+:YSO, as well as storage
times of up to 16 $\mu$s, though memory bandwidths are typically limited to
below 1 MHz [154, 151].

#### Hybrid Photon-Echo Rephasing (HYPER)

Hybrid photon-echo rephasing (HYPER) is a protocol that combines the concepts
of CRIB with a two-photon echo, thus avoiding the optical pumping step
required for protocols like CRIB and AFC, and reducing the noise associated
with the naïve two-pulse photon echo protocol. In HYPER, an inhomogeneously
broadened atomic ensemble is further inhomogeneously broadened in a
controllable fashion after absorption of the signal field. The controllable
broadening is then turned off, a $\pi$-pulse typical of the two-pulse photon
echo scheme is applied, and the controllable broadening is turned back on. By
introducing this controllable broadening, the standard two-pulse echo is
silenced, and after turning off the controllable broadening a second time, and
applying a second $\pi$-pulse, finally the atomic polarization rephases and
the signal field is reemitted. This scheme ultimately reduces the number of
atoms that would experience gain and undergo spontaneous emission, thus
reducing the noise inherent to two-photon echoes [139]. Analytical treatment
of the protocol, and the first experimental implementation with 0.2 MHz
bandwidth and $\sim$10% efficiency, can be found in Ref. [139].

###  2.2 Hardware

We now move on to discuss the various types of hardware used for atomic
ensemble quantum memory. Each type possesses unique advantages and
disadvantages, and may place limits on the memory protocols that are possible
to implement in each system (e.g., rare-earth ions doped in solids with
intrinsic inhomogeneous broadening).

####  2.2.1 Hot Atoms

Hot atomic ensembles have been widely used to establish different quantum
memory schemes, thanks to their advantages of room (or near-room) temperature
operation and reduced experimental complexity. Due to their relatively simple
electronic structure, high vapor pressure, and long-lived spin states, the
alkali metals rubidium and cesium have been the atomic species most widely
used for hot atomic ensemble quantum memory. The recently demonstrated barium
memory is an exception, where instead of spin states, electronic orbitals were
used for the memory interaction to enable THz-bandwidth storage [82, 83]. In
addition to hot atomic species, molecular species in room-temperature gas
cells have also been used for ensemble quantum memory [134, 67], where a
phononic storage state is used instead of a hyperfine spin level as is the
case for the alkali atoms.

Glass vapor cells have the advantage of widespread commercial availability,
and are thus one of the most frequently used hardware implementations for hot
atomic ensemble quantum memory. These cells are prepared using precision
scientific glassblowing techniques, in which the atoms are introduced to the
glass cell under vacuum conditions. One disadvantage of glass vapor cells are
the inelastic collisions between atoms inside the cell and the cell walls,
which can limit memory lifetimes in the narrowband regime [158] or when the
atoms are optically pumped into the metastable spin state [159]. These
collisions occur on the microsecond timescale for a typical, centimeter-size
cell, and can further induce spin depolarization, thus reducing the lifetime
of any collective spin state, including those used in ensemble quantum memory.
Buffer gases such as Ar, Ne or N2 are commonly loaded into the vapor cell
along with the atomic species of choice to increase the lifetime of collective
spin states and to reduce the broadening of certain transitions due to
inelastic collisions [160]. As it slows atomic diffusion and limits the
absorption of atoms onto the cell walls, the presence of a buffer gas may also
increase the device lifetime of glass vapor cells. Unfortunately, the presence
of buffer gas can also induce additional spin depolarization and quenching due
to collisions between the atoms and buffer gas [161]. An optimal buffer gas
pressure can be found that balances mitigation of wall collisions and the
increase in atom–buffer-gas collisions [162]. Paraffin-coated vapor cells have
also been studied as an alternative method to increase collective spin state
lifetimes by inducing elastic instead of inelastic wall collisions [163, 158].
A disadvantage of paraffin-coated or similar vapor cells is their complex
fabrication process, as well as the fact that the coating material may be
incompatible with elevated temperatures. Collisions between atoms and buffer
gas can also be a source of noise due to collision-induced fluorescence [161,
164]. This noise source limits the fidelity of quantum memory, particularly at
or near resonance, and is greatly reduced when the quantum memory is operated
off resonance and optical filtering is applied. Buffer-gas free vapor cells
naturally suppress collision-induced fluorescence noise [158].

The lifetime of collective spin states is not only influenced by collisions.
Broadband quantum memories using hot atomic ensembles may also suffer from
spin-wave dephasing arising from thermal atomic motion [165]. This atomic
motion may result in atoms wandering out of the effective interaction region
defined by the laser fields, or it may result in scrambling of the spin-wave
phase as the constituent atoms move within the interaction region. The rate of
this latter dephasing process is proportional to $\Delta\vec{k}\cdot\vec{v}$
with $\Delta\vec{k}=\vec{k}_{s}-\vec{k}_{c}$ representing the wavevector of
the spin wave, where $\vec{k}_{s}$ and $\vec{k}_{c}$ are the wavevectors of
signal and control fields, respectively, and $\vec{v}$ representing the
thermal velocity of the atoms. In general, it is desirable generate a spin
wave with long wavelength in order to maintain spin-wave coherence. This can
be modified somewhat through experimental design [158], but in many cases the
spin-wave wavelength is limited by the large detuning of the signal and
control fields. One scheme [27] has demonstrated elimination of motional
dephasing through dressing of the collective spin state to an auxiliary
‘sensor’ state. A sensor state corresponds to a state that, when coupled via
an optical field to the storage state, exhibits opposing energy shifts under
the same broadening mechanism. First demonstrated experimentally in a Doppler-
broadened system [27], off-resonant fields couple the storage state to the
sensor state, dressing the collective excitation such that the frequency
shifts due to the dephasing mechanism cancel out. Thus, the dephasing of the
spin wave is protected at the price of one or two additional dressing fields.
Another scheme using velocity-selective optical pumping [166] has been
proposed and proved to be effective against motional dephasing at the cost of
reduced optical depth. In this scheme the atoms are pumped out from ground
state to an auxiliary state and then pumped back to the ground state using a
narrowband laser, which selectively narrows the motional distribution of the
atoms. The rate of dephasing is therefore reduced owing to the small
$\Delta\vec{v}$ of the atoms. Other mechanisms that induce spin wave dephasing
include spatial and temporal variation of longitudinal magnetic field. Note
that this is not unique to hot atomic ensembles and experiments have been
performed to show prolonged spin-wave coherence using hot atomic vapors
through magnetic shielding [167] or applying a strong guiding magnetic field
[168, 169].

The heat-pipe oven is an alternative to glass vapor cells, and represents a
more robust, higher-temperature, customizable vapor cell [170, 171, 172].
Typically heat-pipe ovens are comprised of a stainless-steel tube covered by a
heater with water cooling at both ends. Inside the tube, the wall is covered
by mesh that acts as wick. Upon heating, atoms from liquid metal evaporate,
form a dense vapor, and diffuse toward both ends of the oven until they
condense in the water-cooled region and return to the center through the wick
due to the capillary effect. The heat-pipe oven can be used to establish a
homogeneous temperature and density distribution oven for a long period of
time. It can handle high temperatures, which is suitable for low-vapor-
pressure atomic species or high-vapor-pressure species at much higher optical
depth. The design of the oven can prevent vapor from condensing on the cell
windows, and thus the windows tend to become opaque more slowly than in
equivalent glass vapor cells. The species of atoms and buffer gas can also be
changed more readily than in sealed glass vapor cells.

Many efforts have been made to miniaturize the hot atomic vapor cell in order
to achieve chip-scale operation, enabling compact design and lower necessary
optical power [4]. Early attempts to integrate hot atoms with optical fibers
and silicon platforms include loading hot atomic vapor near tapered fibers
[173], into hollow-core fibers [174], or near waveguides [175]. These
approaches in principle allow long interaction length while maintaining tight
confinement, although to-date only quantum memory in hollow-core fibers has
been implemented experimentally [55]. The large core size of kagome fiber (26
$\mu$m) allows for a large optical depth compared to other approaches [173].
The achieved memory lifetime of 100 ns could be further extended using buffer
gas or a spin-preserving coating. Light-induced atomic desorption (LIAD) [176,
177] has been found effective in increasing optical depth without heating in
miniaturized vapor cells and in kagome fiber [178, 174, 55].

####  2.2.2 Cold Atoms

Similar to the case in hot atoms, the alkali metals are the species most
commonly used for cold atomic ensemble quantum memory. Storage times in hot
atomic vapors are limited by various mechanisms, including Doppler broadening
and transit-time broadening, which are effectively eliminated in laser-cooled
atomic ensembles. However, laser cooling tends to require large ultra-high
vacuum chambers, multiple stabilized lasers for cooling and trapping, and
typically imposes large latency and a non-unity duty cycle in order to re-cool
atoms between shots. The colder the atoms are, the denser the cloud and the
further the reduction in motion-related broadening and dephasing. However, the
maximum length of the ensemble is typically reduced for colder temperatures
due to practicalities of the various laser fields used for cooling and
trapping and the process of evaporative cooling. As a result, there continue
to be quantum memory demonstrations in cold atomic gasses in magneto-optical
traps at $\sim\mu\rm{K}$ temperatures and in Bose-Einstein condensates at
$\sim\rm{nK}$ temperatures. Optical depths as high as $\sim 1000$ can be
achieved [111]. We note that for very long-lived storage, an optical lattice
along the k-vector of the stored spin wave is required to eliminate residual
motional effects, which substantially reduces the available optical depth and
largely precludes broadband storage.

####  2.2.3 Solids

#### Optical phonon modes in diamond

Over the last 10 years the use of optical phonon modes in bulk diamond has
emerged as a platform for fast, broadband quantum memory. Such memories are
attractive due to their THz bandwidth operation across the visible and near
infrared due to the large bandgap of diamond, and room temperature operation
[133, 179, 65, 180, 132, 181]. A $\Lambda$-system consisting of the ground
state of diamond, an optical phonon mode storage state, and the conduction
band as the excited state provides a storage bandwidth of up to 40 THz,
limited only by the splitting between the ground state and the optical phonon
mode. The memory lifetime in this platform is ultimately limited to 3.5 ps by
the decay of the phonon mode [133, 181].

The first proof-of-principle experiment to demonstrate the suitability of
diamond as a quantum memory yielded a noise floor well below 1 photon per
pulse [133]. The primary noise source was FWM noise intrinsic to most
$\Lambda$-type memory schemes; however, the authors highlight that FWM is
suppressed due to the large dispersion in diamond [133]. Other sources of
noise, such as resonant fluorescence, are greatly suppressed by operating with
a detuning of $\sim$950 THz from the conduction band [133, 65, 132].

True single-photon storage was first demonstrated with THz-bandwidth (260 fs)
photons produced from a heralded spontaneous parametric down conversion (SPDC)
source [65]. The preservation of nonclassicality in the retrieved photons was
determined by measuring the second-order correlation function using Hanbury-
Brown Twiss interferometry [133, 65]. Storage of polarization qubits and of a
single photon from a polarization-entangled pair has been demonstrated with a
fidelity of 76% for picosecond-long storage [132].

#### Rare-earth ions

Trivalent rare-earth ions doped in solids at cryogenic temperatures have many
properties that make them a promising platform for quantum memory devices,
including long-lived coherence, compatibility with integrated photonics, lack
of motional dephasing, high density of ions, moderate optical depths, and
telecommunications wavelength compatibility [182, 183, 184, 185, 186, 187,
188, 189, 190]. Optical coherence times can exceed milliseconds [191, 192]
and, to date, the longest spin coherence time observed is 6 hours [193].

The long optical and spin coherence properties afforded to the rare earths is
due to their unique electronic structure. The optical transitions of interest
are between their 4f-4f valence electronic levels, which are dipole-forbidden
in the free ion. The 4f valence shell is electronically shielded by filled 5s
and 5p orbitals, which renders the 4f levels largely insensitive to their
environment. Incorporation of the rare-earth ion in a solid introduces a
crystal electric field which acts as a perturbation to the free-ion states.
For sufficiently low site-symmetry of the rare-earth ion, the crystal electric
field mixes in additional states, which weakly allows electronic transitions
[194, 189].

The electronic states of the free ion are labeled by Russell-Saunders term
symbols where the electronic spin $J$ and projection $m_{j}$ are considered
good quantum numbers [189, 194, 195]. These states are $(2J+1)$-degenerate,
known as Kramer’s degeneracy. The addition of the crystal electric field
either partially or completely lifts the degeneracy of these states. For
sufficiently low site symmetry and ions with an even number of electrons—the
non-Kramer’s ions—the degeneracy can be lifted completely. The species with an
odd number of electrons—the Kramer’s ions—require an external field to fully
break this degeneracy. Rare-earth ions can have further splittings of their
crystal field levels from hyperfine structure, the Zeeman effect, and higher
order moments [189, 194].

While these electronic levels are largely insensitive to their environment, it
is residual environmental effects that broaden the optical and spin
transitions. Homogeneous broadening is largely due to dynamical processes
within the solid and can be sorted broadly into four categories: ion-phonon
interactions, ion-ion interactions, ion-host lattice interactions, and pure
dephasing. Optical phonons are suppressed at or below 4 K [194, 189, 196].
Microwave phonons can couple the spin transitions and primarily interact with
the ion via multiphonon processes that scale nonlinearly with temperature
[194, 191, 197]. Ion-ion interactions account for a number of interactions
such as instantaneous spectral diffusion, in which the excitation of one ion
shifts the electric dipole of a neighboring ion [194, 198, 189, 196, 199]. The
dominant source of dephasing below 4 K is interactions between the ion and the
host lattice, which acts as a spin bath [189]. For example, fluctuating
magnetic fields due to nuclear spin flips of Y3+ are the dominant source of
dephasing in YSO [200]. This dephasing can be suppressed to extend the
coherence times by application of a moderate external magnetic field to
polarize the spin bath, as well as finding spin transitions that have zero
effective first-order Zeeman shift [201, 202, 203, 193], and employing
rephasing techniques like dynamical decoupling [204, 205, 206, 207, 208].

Defects and strain in the crystalline host result in inhomogeneous broadening
of both the optical and spin transitions. The local crystal field seen by the
ions varies site-to-site, resulting in a static inhomogeneous shift to their
transition energy. In commercially doped rare-earth materials, the
inhomogeneous broadening of the optical transition can be on the order of 10’s
of MHz to 100’s of GHz depending on factors such as (but not limited to) the
host material, doping concentration, and isotopic purity. The inhomogeneous
broadening is always many orders of magnitude larger than the homogeneous
linewidth [209, 210, 211, 212, 213, 214, 194, 189, 196, 215]. In the case of
rare-earth doped solids, where the ions themselves are substitutional dopants,
they may be the primary source of this inhomogeneous broadening [195, 216,
189, 135].

Ions with an even number of electrons—the non-Kramer’s ions (Eu3+, Pr3+,
Tm3+)—have a quenched electronic spin and consequently are more insensitive to
their environment and have longer coherence properties. The nuclear spin state
splittings in these ions are between 10-100 MHz—too narrow to be optically
resolved in many host materials due to the large inhomogeneous linewidth [217,
214, 218, 219, 220, 221, 202, 222]. The spin coherence properties of the non-
Kramer’s ions can be exploited for long storage times by utilizing techniques
such as spectral holeburning to resolve the nuclear spin states; however, this
comes at the expense of sacrificing optical depth—and therefore the storage
efficiency—by only using a small subset of the atoms in the inhomogeneous line
[135].

In the Kramer’s ions (Er3+,Yb3+, Nd3+)—the paramagnetic species—application of
an external magnetic field can yield Zeeman splittings large enough for the
Zeeman sublevels to to be resolvable in the inhomogeneous line. These species
are of particular interest due to their larger bandwidths, on the order of
hundreds of MHz to GHz, and potential for interfacing with platforms such as
superconducting qubits [223, 224, 225, 226]. While their Zeeman states can be
accessible and have the potential for larger memory bandwidths than the non-
Kramer’s ions, with state splittings of several hundred MHz to GHz, their
electronic spin has stronger coupling to the environment due to its larger
magnetic moment and thus is more susceptible to fluctuations and exhibits
shorter coherence times. A challenge of utilizing the Kramers species is the
ability to efficiently optically pump them [227, 228], which is a prerequisite
of many quantum memory protocols such as AFC and CRIB [94, 106, 135]. Erbium
is one of the most attractive rare-earths due to its telecom C-band transition
at 1.5 $\mu$m; however erbium can be difficult to optically pump since the
lifetime of its Zeeman sublevels is only about an order of magnitude slower
than the excited state lifetime [229, 197]. However, using Kramers isotopes
with nuclear spin helps substantially and spin coherence times as long as 1
second have been achieved in 167Er3+:YSO [230]. Progress has been made
recently to overcome the limitations of optical pumping in both Yb3+ and Nd3+
[227, 231, 190].

There have been many proof-of-principle light storage experiments with both
classical light and weak coherent states that show these systems can operate
with the proper SNR to store single photons [232, 233, 234, 235]. Storing
single-photon states in a rare-earth-ion-based quantum memory has also been
achieved [236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248].

To date, the longest storage time of a bright classical pulse was 53 minutes
and for weak coherent states exceeding 1.0 second [249, 205]; however, both
were acheived with low storage efficiencies. On-chip storage of
telecommunications C-band light at the few-photon level has also been
demonstrated in a photonic nanoresonator with isotopically purified Er-167:YSO
[250]. Rare-earth ions have demonstrated quantum memory protocols with
bandwidths exceeding a GHz [236, 244, 251, 252].

##  3 Theory

###  3.1 Maxwell-Bloch Equations

In this section, we expand upon the three-level Maxwell-Bloch equations
described in Sec. 1.2. Section 3.1.1 provides several approximations to the
Maxwell-Bloch equations relevant for the memory protocols discussed in Sec.
2.1 that rely on homogeneous broadening, while Sec. 3.1.2 provides more
information on the Maxwell-Bloch equations in the presence of inhomogeneous
broadening.

Each of the protocols in Sec. 2.1 requires or permits a certain type of line
broadening for the intermediate excited state shown in Fig. 1, and the type of
broadening present in a given system limits the possible memory protocols that
can be implemented in that system. The types of excited-state line broadening
are broadly categorized into those with homogeneous and inhomogeneous
mechanisms. Homogeneous broadening mechanisms affect each atom in the system
equally, whereas inhomogeneous mechanisms affect different classes of atoms
differently. These classes of atoms may be distinguished in frequency, spatial
position, momentum, or any other degree of freedom. Homogeneous broadening
mechanisms, which typically apply to cold atoms—but, importantly for broadband
operation, are not restricted to cold atoms—include effects such as natural or
lifetime broadening, resonance broadening, collisional broadening (which is
separated into impact and quasi-static regimes, depending on the timescale of
collisions [253]), and phonon broadening in solids. Inhomogeneous broadening
mechanisms include Zeeman and Stark broadening (these are typically considered
inhomogeneous due to spatial or temporal non-uniformity in the magnetic and
electric fields, respectively), Doppler broadening, and crystal-field or
impurity-based broadening in solids [194].

####  3.1.1 Homogeneous

Several approximations to Eqs. (1)-(3) from Sec. 1.2 lead to the mathematical
descriptions of EIT, ATS, ATT, SR, and Raman protocols, which we compile here.
In addition, we provide representations of the Maxwell-Bloch equations in the
spectral domain and in the temporal domain split into amplitude and phase.
These different forms of the Maxwell-Bloch equations are useful for both
analytical and numerical investigation of quantum memory behavior.

In the case of EIT, the approximations $\Delta=0$ (i.e., $\bar{\gamma}=1$) and
$BW\ll\gamma$ (or $\tau_{\text{FWHM}}\gamma\gg 1$) for both the signal and
control fields implies that the time evolution of the atomic polarization in
Eq. (2) may be adiabatically eliminated, and a reduced set of partial
differential equations can be found:

$\displaystyle\partial_{z}A(z,\tau)$
$\displaystyle=-dA(z,\tau)+i\sqrt{d}\frac{\Omega(\tau)}{2}B(z,\tau)$ (16)
$\displaystyle\partial_{\tau}B(z,\tau)$
$\displaystyle=-i\sqrt{d}\frac{\Omega^{*}(\tau)}{2}A(z,\tau)-\frac{{\left|\Omega(\tau)\right|}^{2}}{4}B(z,\tau),$
(17)

which are significantly less computationally expensive to simulate.

In the case of ATS, the assumptions $\Delta=0$,
${\left|\Omega\right|}\gg\bar{\gamma},\sqrt{d}$ imply the elimination of the
first two terms on the right-hand-side of Eq. (2), and in the case of a
constant control field (which without loss of generality can be assumed to be
purely imaginary), the simplified coupled partial differential equations

$\displaystyle\partial_{\tau}P(z,\tau)$
$\displaystyle=-i\frac{\Omega(\tau)}{2}B(z,\tau)$ (18)
$\displaystyle\partial_{\tau}B(z,\tau)$
$\displaystyle=-i\frac{\Omega^{*}(\tau)}{2}P(z,\tau),$ (19)

lead to periodically oscillating atomic polarization and spin wave populations
that are $\pi/2$ out of phase:

$\displaystyle A(z,\tau)\propto P(z,\tau)$
$\displaystyle=\sin\left(\frac{{\left|\Omega(\tau)\right|}}{2}t\right)$ (20)
$\displaystyle B(z,\tau)$
$\displaystyle=\cos\left(\frac{{\left|\Omega(\tau)\right|}}{2}t\right).$ (21)

For ATT, the assumptions $\Delta=0$, $BW\geq\gamma$ (or
$\tau_{\text{FWHM}}\gamma\leq 1$), and $d\gg 1$ imply that we can drop the
$\bar{\gamma}P(z,\tau)$ term in Eq. (2). Further assuming a $\pi$-pulse
control field that arrives after absorption of the signal field implies the
terms involving $\Omega(\tau)$ are eliminated during the initial absorption
stage, leaving only the simplified equations:

$\displaystyle\partial_{z}A(z,\tau)$ $\displaystyle=-\sqrt{d}P(z,\tau)$ (22)
$\displaystyle\partial_{\tau}P(z,\tau)$ $\displaystyle=\sqrt{d}A(z,\tau),$
(23)

which has exact solution

$\displaystyle A(z,\tau)$
$\displaystyle=e^{t-dz}[1-\Theta(\Delta\tau^{\text{ctrl}})]$ (24)
$\displaystyle P(z,\tau)$
$\displaystyle=\sqrt{d}e^{t-dz}[1-\Theta(\Delta\tau^{\text{ctrl}})]$ (25)

during the absorption stage (until the arrival of the control field,
$\tau<\Delta\tau^{\text{ctrl}}$), where $\Theta(\Delta\tau^{\text{ctrl}})$ is
the Heaviside step function. The $\pi$-pulse control field then in principle
transfers $P\rightarrow B$ exactly. Applying the definition of storage
efficiency, we find $\eta_{\text{stor}}=1-e^{-2d}$. The treatment of the SR
memory protocol is nearly identical, but operates under the assumption of
$BW\gg\gamma$ (or $\tau_{\text{FWHM}}\gamma\ll 1$) and superradiant coupling
between atoms in the ensemble. The approximations leading to the Raman
protocol are similar to EIT, but in the far-off-resonant limit, i.e.,
$\Delta\gg\gamma$ (i.e., $\bar{\gamma}\approx-i\Delta/\gamma=-i\bar{\Delta}$)
and $BW\ll\Delta$. These conditions also imply the adiabatic elimination of
the atomic polarization, leading to the simplified system:

$\displaystyle\partial_{z}A(z,\tau)$
$\displaystyle=-i\frac{d}{\bar{\Delta}}A(z,\tau)-\frac{\sqrt{d}}{\bar{\Delta}}\frac{\Omega(\tau)}{2}B(z,\tau)$
(26) $\displaystyle\partial_{\tau}B(z,\tau)$
$\displaystyle=\frac{\sqrt{d}}{\bar{\Delta}}\frac{\Omega^{*}(\tau)}{2}A(z,\tau)-i\frac{{\left|\Omega(\tau)\right|}^{2}}{4\bar{\Delta}}B(z,\tau).$
(27)

Several other forms of the Maxwell-Bloch equations are useful both
analytically and numerically. In the spectral domain, we can rewrite the
Maxwell-Bloch equations via application of the Fourier transform as:

$\displaystyle\partial_{z}\tilde{A}(z,\omega)$
$\displaystyle=-\sqrt{d}\tilde{P}(z,\omega)$ (28) $\displaystyle
i\omega\tilde{P}(z,\omega)$
$\displaystyle=-\bar{\gamma}\tilde{P}(z,\omega)+\sqrt{d}\tilde{A}(z,\omega)-\frac{i}{\sqrt{2\pi}}\frac{\tilde{\Omega}(\omega)}{2}*\tilde{B}(z,\omega)$
(29) $\displaystyle i\omega\tilde{B}(z,\omega)$
$\displaystyle=-\gamma_{B}\tilde{B}(z,\omega)-\frac{i}{\sqrt{2\pi}}\frac{\tilde{\Omega}^{*}(-\omega)}{2}*\tilde{P}(z,\omega).$
(30)

Here we have made use of the convolution theorem for Fourier transforms and
the fact that the Fourier transform of a conjugated function is the conjugate
of the function’s Fourier transform reflected about $\omega=0$. We have used
“$*$” to express the convolution of two functions, e.g.
$\tilde{\Omega}(\omega)*\tilde{B}(z,\omega)=\int_{-\infty}^{\infty}du\,\tilde{\Omega}(\omega-u)\tilde{B}(z,u)$.
Here $\omega$ represents the difference from each field’s center frequency.
This form of the Maxwell-Bloch equations can be particularly useful for
simulation of broadband quantum memory. As usual, application of the Fourier
transform has converted equations involving temporal derivatives to simple
algebraic equations.

Another useful form of the Maxwell-Bloch equations arises from separating the
amplitude and phase of each field, leading to the following 6 equations of
motion:

$\displaystyle\partial_{z}{\left|A(z,\tau)\right|}$
$\displaystyle=-\sqrt{d}{\left|P(z,\tau)\right|}\cos[\phi_{P}(z,\tau)-\phi_{A}(z,\tau)]$
(31) $\displaystyle\partial_{z}\phi_{A}(z,\tau)$
$\displaystyle=-\sqrt{d}\frac{{\left|P(z,\tau)\right|}}{{\left|A(z,\tau)\right|}}\sin[\phi_{P}(z,\tau)-\phi_{A}(z,\tau)]$
(32) $\displaystyle\partial_{\tau}{\left|P(z,\tau)\right|}$
$\displaystyle=-\bar{\gamma}{\left|P(z,\tau)\right|}+\sqrt{d}{\left|A(z,\tau)\right|}\cos[\phi_{A}(z,\tau)-\phi_{P}(z,\tau)]$
$\displaystyle\hskip
20.00003pt+\frac{1}{2}\real[\Omega(\tau)]{\left|B(z,\tau)\right|}\sin[\phi_{B}(z,\tau)-\phi_{P}(z,\tau)]$
(33) $\displaystyle\hskip
30.00005pt+\frac{1}{2}\imaginary[\Omega(\tau)]{\left|B(z,\tau)\right|}\cos[\phi_{B}(z,\tau)-\phi_{P}(z,\tau)]$
$\displaystyle\partial_{\tau}\phi_{P}(z,\tau)$
$\displaystyle=\Delta+\sqrt{d}\frac{{\left|A(z,\tau)\right|}}{{\left|P(z,\tau)\right|}}\sin[\phi_{A}(z,\tau)-\phi_{P}(z,\tau)]$
$\displaystyle\hskip
20.00003pt-\frac{1}{2}\real[\Omega(\tau)]\frac{{\left|B(z,\tau)\right|}}{{\left|P(z,\tau)\right|}}\cos[\phi_{B}(z,\tau)-\phi_{P}(z,\tau)]$
(34) $\displaystyle\hskip
30.00005pt+\frac{1}{2}\imaginary[\Omega(\tau)]\frac{{\left|B(z,\tau)\right|}}{{\left|P(z,\tau)\right|}}\sin[\phi_{B}(z,\tau)-\phi_{P}(z,\tau)]$
$\displaystyle\partial_{\tau}{\left|B(z,\tau)\right|}$
$\displaystyle=-\gamma_{B}{\left|B(z,\tau)\right|}+\frac{1}{2}\real[\Omega(\tau)]{\left|P(z,\tau)\right|}\sin[\phi_{P}(z,\tau)-\phi_{B}(z,\tau)]$
$\displaystyle\hskip
30.00005pt-\frac{1}{2}\imaginary[\Omega(\tau)]{\left|P(z,\tau)\right|}\cos[\phi_{P}(z,\tau)-\phi_{B}(z,\tau)]$
(35) $\displaystyle\partial_{\tau}\phi_{B}(z,\tau)$
$\displaystyle=-\frac{1}{2}\real[\Omega(\tau)]\frac{{\left|P(z,\tau)\right|}}{{\left|B(z,\tau)\right|}}\cos[\phi_{P}(z,\tau)-\phi_{B}(z,\tau)]$
$\displaystyle\hskip
30.00005pt-\frac{1}{2}\imaginary[\Omega(\tau)]\frac{{\left|P(z,\tau)\right|}}{{\left|B(z,\tau)\right|}}\sin[\phi_{P}(z,\tau)-\phi_{B}(z,\tau)],$
(36)

where each field is written
$X(z,\tau)={\left|X(z,\tau)\right|}e^{i\phi_{X}(z,\tau)}$ for $X=A,P,$ and
$B$, and where $\real(\cdot)$ and $\imaginary(\cdot)$ designate the real and
imaginary parts of the enclosed function, respectively. This form of the
equations of motion is particularly useful for analysis and simulations
involving chirped optical fields or fields with non-trivial temporal/spectral
phase.

####  3.1.2 Inhomogeneous

Inhomogeneous broadening of the intermediate excited state is treated as a
coherent sum of homogeneous distributions separated in frequency [38], leading
to the following equations of motion:

$\displaystyle\partial_{z}A(z,\tau)$ $\displaystyle=-\sqrt{d}P(z,\tau)$ (37)
$\displaystyle\partial_{\tau}P_{\Delta}(z,\tau)$
$\displaystyle=-(\bar{\gamma}-i\Delta)P_{\Delta}(z,\tau)+\sqrt{d}\sqrt{p_{\Delta}}A(z,\tau)-i\frac{\Omega(\tau)}{2}B_{\Delta}(z,\tau)$
(38) $\displaystyle\partial_{\tau}B_{\Delta}(z,\tau)$
$\displaystyle=-\gamma_{B}B_{\Delta}(z,\tau)-i\frac{\Omega^{*}(\tau)}{2}P_{\Delta}(z,\tau),$
(39)

where $P(z,\tau)=\int d\Delta\sqrt{p_{\Delta}}P_{\Delta}(z,\tau)$ in Eq. (37)
and $p_{\Delta}$ represents the normalized fraction of atoms with
inhomogeneous detuning $\Delta$ (i.e., $\int d\Delta\,p_{\Delta}=1$). Here we
have redefined $\bar{\gamma}=(\gamma-i\Delta_{0})/\gamma$ to designate the
detuning $\Delta_{0}$ of the optical fields relative to the atoms with no
inhomogeneous shift (i.e., for $\Delta=0$).

Analytic expressions for memory efficiency and other metrics for each of the
protocols discussed in Sec. 2.1 may be derived from Eqs. (37)-(39) by assuming
the appropriate inhomogeneous spectral profile $p_{\Delta}$ and the
appropriate pulse sequence [i.e., for most protocols, during storage the
control field is off and terms involving $\Omega(\tau)$ in Eqs. (37)-(39) may
be neglected].

Again, these equations may be Fourier transformed into the spectral domain or
split into amplitude and phase for ease of computation, just as in the
homogeneous case described above.

###  3.2 Efficiency Optimization

The primary focus of most existing theoretical work dealing with atomic
ensemble quantum memory revolves around the goal of increasing memory
efficiency. In principle, the same or similar tools that have been developed
for efficiency optimization may also be used to improve other metrics, but
efficiency is uniquely important for quantum memory applications and has
accordingly received the majority of theoretical attention. The techniques
described in the following Secs. 3.2.1 and 3.2.2 are most fully described in
Refs. [36, 37, 38, 39, 40].

####  3.2.1 Signal Field Shaping

One particularly powerful tool for increasing memory efficiency is
optimization of the temporal profile of the signal field in both amplitude and
phase. The idea behind this optimization technique is that every stage of the
memory process can be treated as a linear integral mapping with a unique
integral kernel fully describing the map. Each kernel can be decomposed into
singular vectors and associated singular values that act as a prefactor
describing the efficiency for each mapping between singular vectors. For
example, the storage process is described by the map

$B_{\text{out}}(z)=\int_{-\infty}^{\infty}d\tau\,K_{\text{stor}}(z,\tau)A_{\text{in}}(\tau),$
(40)

where $A_{\text{in}}(\tau)$ is the incident signal field temporal profile,
$K_{\text{stor}}(z,\tau)$ is the integral kernel for the storage operation,
and $B_{\text{out}}(z)$ is the long-lived spatially dependent spin wave
resulting from the storage process. Singular value decomposition of this
storage kernel takes the form

$K_{\text{stor}}(z,\tau)=\sum_{j}\lambda_{j}B_{j}(z)A^{*}_{j}(\tau),$ (41)

where $B_{j}(z)$ and $A_{j}(\tau)$ are the left-singular and right-singular
vectors (sometimes called the ‘optimal modes’) of $K_{\text{stor}}(z,\tau)$
and $\lambda_{j}$ are its singular values. The exact form of the storage
kernel $K_{\text{stor}}(z,\tau)$ depends on parameters of the memory such as
the optical depth, linewidth, control field shape, etc. Given Eqs. (40) and
(41), it is clear to see that the largest storage efficiency is achieved when
$A_{\text{in}}(\tau)$ matches the right-singular vector of the storage kernel
with the largest singular value, $A_{\text{opt}}(\tau)$, as any admixture of
$A_{\text{in}}(\tau)$ with other modes will imply a component of the mapping
with non-optimal $\lambda_{j}$, or non-optimal efficiency. Thus if it is
possible to arbitrarily shape the signal field to be stored, one can optimize
memory efficiency by shaping $A_{\text{in}}(\tau)$ to match
$A_{\text{opt}}(\tau)$. One can determine the exact form of
$A_{\text{opt}}(\tau)$ by numerically or analytically constructing
$K_{\text{stor}}(z,\tau)$ and computing its singular value decomposition based
on the experimental parameters at hand, or by an iterative experimental
process based on time-reversal described in Refs. [110, 37, 254, 255].

####  3.2.2 Control Field Shaping

Shaping of incident single-photon light pulses that contain sensitive encoded
quantum information can introduce loss, which is undesirable for almost every
application of quantum memory. It is generally more desirable to shape the
strong, many-photon control field, where loss can be easily compensated for by
increasing control field power. The approach for optimizing memory efficiency
through the temporal shape of the control field then takes on a similar form
to the signal-field shaping discussed above. In the case of quantum storage,
instead of matching the signal field to the right-singular vector of the
storage kernel with the largest singular value, the task becomes to shape the
storage kernel itself and thereby shape its right-singular vector with largest
eigenvalue to match the input signal field. This can be accomplished
analytically in several limiting cases, or, in the general case, this can be
accomplished numerically through an iterative process: First, the storage
kernel is constructed numerically for an initial guess of the control field
shape (amplitude and phase), $\Omega_{1}(\tau)$, which is discretized along a
series of spline points. The optimal mode $A^{1}_{\text{opt}}(\tau)$ of this
initial storage kernel is constructed and compared to the target signal field
$A_{\text{in}}(\tau)$. A series of $N$ numeric interpolations between
$A^{1}_{\text{opt}}(\tau)$ and $A_{\text{in}}(\tau)$ are constructed,
$A_{k}(\tau)$, where $A_{1}(\tau)=A^{1}_{\text{opt}}(\tau)$ and
$A_{N}(\tau)=A_{\text{in}}(\tau)$. For each $A_{k}(\tau)$, the control field
spline points are optimized, leading to an optimal control field shape
$\Omega_{k}(\tau)$ that maximizes storage efficiency for each $A_{k}(\tau)$
and provides the initial guess for $\Omega_{k+1}(\tau)$. This optimization at
each step can either be implemented by constructing the storage kernel for
each guess and comparing the optimal mode from the SVD of the kernel with
$A_{k}(\tau)$, or by numerically integrating the single instance of the
Maxwell-Bloch equations with input $A_{k}(\tau)$ and the control field guess.
Assuming a large enough number of interpolations $N$, this iterative process
slowly transforms the storage kernel such that its optimal mode overlaps with
the final $A_{N}(\tau)$ (i.e.,
$A^{N}_{\text{opt}}(\tau)=A_{N}(\tau)=A_{\text{in}}(\tau)$, where the
superscript $N$ is just a label, not a multiplicative power).

In practice, this optimization technique tends to produce similar efficiencies
compared to the signal-field shaping technique, however the relative strength
of each technique for optimizing memory efficiency remains an open question.
It is possible that regions of the memory parameter space exist where control
field shaping either outperforms or underperforms signal-field shaping, and it
is possible that optimization of both fields together may yield higher
efficiencies in some regions than optimization of either field shape alone.

Figure 7: (a) Comparison of signal-field shape-based and Gaussian-pulse-shape
optimization techniques for an optical depth of $d=50$. The dotted line marks
the optimal storage efficiency, $\eta_{\text{opt}}\approx 95\%$. (b) Gaussian-
optimized memory efficiency and (c)-(e) optimized control field parameters in
the broadband regime. Adapted from Ref. [114].

####  3.2.3 Gaussian-Pulse-Shape Optimization

In contrast to the two techniques described above, each of which requires
arbitrary shaping of the respective light field temporal envelope, recent work
reported in Ref. [114] addresses the question of how far memory efficiency can
be optimized using a fixed shape of the temporal envelope, one which is a good
approximation for the shape that is natively produced by modelocked lasers and
single-photon sources. The shape chosen in Ref. [114] is a Gaussian, such that
the signal and control field shapes are

$\displaystyle A_{\text{in}}(\tau)$ $\displaystyle=e^{-\tau^{2}/4\sigma^{2}}$
(42) $\displaystyle\Omega(\tau)$
$\displaystyle=\Omega_{0}\,e^{-[(\tau-\Delta\tau^{\text{ctrl}})/2\sigma^{\text{ctrl}}]^{2}},$
(43)

where again all timescales are normalized by $1/\gamma$, and
$\sigma=\tau_{\text{FWHM}}/(2\sqrt{2\ln{2}})$,
$\Omega_{0}=\theta/(2\sqrt{\pi}\sigma^{\text{ctrl}})$, the control-field pulse
area is $\theta=\int_{-\infty}^{\infty}d\tau\,\Omega(\tau)$, the temporal
delay of the control field relative to the maximum of the signal field is
$\Delta\tau^{\text{ctrl}}$, and the duration of the control field is
$\tau^{\text{ctrl}}_{\text{FWHM}}=2\sqrt{2\ln{2}}\sigma^{\text{ctrl}}$.

The approach of Gaussian-pulse-shape optimization has three main benefits:
First, related to the motivation of using the pulse shapes that are native to
many modelocked lasers and single-photon sources, it demonstrates that over a
large range of memory parameter space (for a large range of optical depths $d$
and signal durations/linewidths $\tau_{\text{FWHM}}\gamma$) very high storage
efficiencies very close to the optimal bound discussed in Sec. 1.4 can be
achieved with only these native Gaussian shaped pulses. This eliminates the
need in most quantum memories for experimentally complex and costly pulse
shaping methods. Fig. 7(a) shows a comparison of the storage efficiencies
achieved using both signal-field, arbitrary shape-based optimization and the
optimization of Gaussian-shaped control fields for $d=50$. Over a wide range
of signal-field pulse durations $\tau_{\text{FWHM}}\gamma$, the Gaussian-
pulse-shape optimization technique produces the same efficiency as the shape-
based technique, within numerical error. For the most broadband photons
(smallest $\tau_{\text{FWHM}}\gamma$), the shape-based technique outperforms
Gaussian-pulse-shape optimization, but the storage efficiency is far from the
optimal bound in both cases. Additionally, the Gaussian-pulse-shape
optimization technique is significantly less computationally expensive,
allowing for a continuous calculation of optimized storage efficiency as a
function of $\tau_{\text{FWHM}}\gamma$, whereas the shape-based optimization
method is only computationally tractable in a point-wise fashion.

Second, Gaussian-pulse-shape optimization provides three physically intuitive
optimization parameters for the control field — the control-field pulse area,
duration, and delay relative to the signal field — whereas the shape-based
technique relies on arbitrary pulse shaping that is arguably less physically
intuitive. The resulting optimized control field parameters are shown in Fig.
7(c)-(e). In addition to providing a convenient lookup table for the optimized
control parameters to use experimentally for a quantum memory with a given
optical depth, signal duration, and linewidth, Fig. 7(c)-(e) also agrees with
and builds upon the physical intuition for each quantum memory protocol. In
the EIT region, Fig. 7(c)-(e) shows that control fields with large duration
and pulse area should be used that arrive before the signal field; control
fields of this type open the well-known transparency window of EIT and
adiabatically close this window as the signal field enters the medium and is
compressed and stored. Similarly the physical intuition holds in the ATS and
ATT regions, where control fields of $2\pi$ and $\pi$ pulse area, control
durations of $\tau_{\text{FWHM}}$ and less than $\tau_{\text{FWHM}}$, and
delays of 0 and positive delays implying a control field that arrives after
the signal field are used, respectively, as predicted for both of these
protocols. These physically intuitive control field parameters are practically
useful when dealing with tunable experiments, but also allow for more
sophisticated sensitivity analysis, as detailed in Ref. [23] and Sec. 3.3.

Third, Gaussian-pulse-shape optimization reveals a continuous mathematical
transformation between the three resonant protocols of EIT, ATS, and ATT. As
shown in Fig. 7(b) and (c)-(e), memory efficiency remains at the optimal bound
across a continuous range of memory parameters
$\mathcal{M}=(d,\tau_{\text{FWHM}}\gamma)$ where the memory protocol behavior
changes continuously from EIT to ATS to ATT. As $\mathcal{M}$ varies, the
control field parameters
$\mathcal{G}=\left(\theta,\Delta\tau^{\text{ctrl}},\tau^{\text{ctrl}}_{\text{FWHM}}\right)$
also vary continuously and monotonically. Naïvely, one might predict
discontinuities in $\mathcal{G}$ as one protocol region changes into another,
but Fig. 7 shows this is not the case, and a continuous transformation between
memory protocols exists and can be exploited to achieve optimal memory
efficiency in the regions of memory parameter space between protocols, or in
the regions of “mixed” memory behavior.

####  3.2.4 Inhomogeneous Profile Shaping

While in principle the three optimization techniques above (Sec. 3.2.1-3.2.3)
can be applied to both homogeneous and inhomogeneous systems, they have
primarily been applied in the homogeneous case. In the inhomogeneous case
another unique degree of freedom is available for optimization, namely the
inhomogeneous broadening profile, which can be straightforwardly manipulated
via spectral hole burning and related experimental techniques.

In Ref. [38], the authors consider several inhomogeneous profile shapes,
including a delta function (identical to the homogeneous case), a Lorentzian
distributed inhomogeneous profile, and a Gaussian distributed inhomogeneous
profile typical of Doppler-broadened gases. The efficiencies resulting from
the three inhomogeneous are compared (where, unsurprisingly, the delta
function profile leads to the highest efficiency), and a general procedure for
calculating the efficiency for different inhomogeneous profile shapes is
developed. In Ref. [256], a similar analysis is carried out assuming a
periodic inhomogeneous frequency profile characteristic of the AFC protocol.
The authors show that in this case, the optimal inhomogeneous profile shape is
a square-toothed frequency comb with an analytically calculable tooth width
depending on the achievable peak optical depth. In addition to these
techniques, one can imagine implementing a similar procedure to Sec. 3.2.3 to
calculate the optimal inhomogeneous profile shape in the absence of the
assumptions made in the AFC protocol. Ref. [222], for example, has examined
the EIT visibilities resulting from different inhomogeneous shapes, which may
lead to the discovery of an optimal inhomogeneous shape for the EIT protocol
in inhomogeneously broadened media.

###  3.3 Sensitivity Analysis

Although efficiency optimization has received the lion’s share of theoretical
attention related to atomic ensemble quantum memory, recent work reported in
Ref. [23] introduces theoretical tools aimed at the distinct task of
characterizing and comparing the sensitivity of different quantum memory
protocols to external perturbations.

Figure 8: Select results from sensitivity analysis of $\Lambda$-type quantum
memory in the presence of (a)-(d) short-timescale fluctuations of the memory
parameters (optical depth, $d$, and linewidth, $\gamma$), and (e)-(g) long-
timescale drift in control field parameters.

Figure 8 shows a few selected results from Ref. [23], considering the case of
short-timescale, shot-to-shot fluctuations $\zeta=(\zeta_{d},\zeta_{g})$ in
memory parameters $\mathcal{M}$ drawn from a normal distribution $P(\zeta)\sim
e^{-(\zeta_{d}^{2}g^{2}+\zeta_{g}^{2}d^{2})/[2(\epsilon_{M}dg)^{2}]}$, where
$g=\tau_{\text{FWHM}}\gamma$ [Fig. 8(a)-(d)], and long-timescale drift in
control field parameters $\mathcal{G}_{s}$ for a control field with arbitrary
profile defined by spline points $\mathcal{G}_{s}=(\xi_{1},...,\xi_{N})$ [Fig.
8(e)-(g)]. The situation in mind addressed by Fig. 8(a)-(d) is frequently
encountered in atomic-ensemble quantum memory using hot atoms, wherein the
average optical depth and excited state linewidth are fixed with minimal long-
timescale drift (typically through setting the vapor cell temperature) but
there may be non-negligible shot-to-shot fluctuations in optical depth and
linewidth arising from density fluctuations in the interaction volume defined
by the optical modes. With this situation in mind, Fig. 8(a) shows average
memory efficiency in the presence of memory parameter fluctuations
$\epsilon_{M}=5\%$, and Fig. 8(b) shows the magnitude of the fluctuations in
memory efficiency arising from $\epsilon_{M}=5\%$. First, the average memory
efficiency drops slightly from the case without fluctuations [e.g., compared
to Fig. 7(b)], as expected. Second, we find that the absorb-then-transfer
(ATT) region is most sensitive to these memory parameter fluctuations, as it
is this region that possesses the largest fluctuations in memory efficiency
(for the same fluctuations in memory parameters) shown in Fig. 8(b). The
relatively large sensitivity of the ATT protocol compared to the other
resonant memory protocols (i.e., ATS and EIT) is explained by the overlap
fidelity of optimal control fields neighboring any given point in
$\mathcal{M}=(d,\tau_{\text{FWHM}}\gamma)$, shown in Fig. 8(c). Where the
overlap fidelity is low, one expects a significant mismatch between the
control field used (which corresponds to the average optical depth and
linewidth) and the optimal control field (corresponding to the optical depth
and linewidth modified by a given shot-to-shot fluctuation), which should
result in lower average efficiency and larger efficiency fluctuations. When
the overlap fidelity is high, one expects (and we observe) high average
efficiency and lower efficiency fluctuations. Across several points in the
memory parameter space shown by the markers in Fig. 8(b), sampling over all
three resonant memory protocols, we measure efficiency fluctuations as a
function of the memory parameter fluctuation magnitude, $\epsilon_{M}$, and
plot the results in Fig. 8(d). First, we observe efficiency fluctuations that
are smaller, as a percentage, than the memory parameter fluctuations that
cause them (i.e., memory parameter fluctuations of magnitude
$\epsilon_{M}=5\%$ cause at most memory efficiency fluctuations of 2.5%). This
behavior persists over all reasonable magnitudes of memory parameter
fluctuations, shown in Fig. 8(d), which implies that all three resonant memory
protocols are ‘stable.’ We also note that the EIT and ATS protocols are
significantly less sensitive to memory parameter fluctuations than the ATT
protocol (by about a factor of 3), across all memory parameter fluctuation
magnitudes.

In addition to deviations in memory parameters, we can also consider the
effect of deviations in control field parameters within the same general
mathematical framework. Typically memory parameters fluctuate over a short
timescale around a fixed center point; control field parameters may also
fluctuate in this manner, but a more common problem experimentally is long
timescale drift from an initially optimal setpoint. Investigating the
sensitivity of a memory to this long timescale drift is, coincidentally,
equivalent to probing the difficulty of finding the optimal setpoint for that
memory. Memory protocols that allow for a larger region of control field
parameter space with optimal efficiency, and which therefore lead to less
difficulty in finding the optimal control field setpoint, are less sensitive
to control field drift out of that optimal region. The converse is also
true—memory protocols that permit only a small region of control field
parameter space with optimal efficiency lead to more difficulty in finding the
optimal control field setpoint and are more sensitive to control field drift.
This situation is explored in Fig. 8(e)-(g) for optimal arbitrary control
field shapes. Fig. 8(e) shows the general approach; the optimal control field
shape for a certain memory protocol is found by the procedure described in
Sec. 3.2.2, and each spline point parameterizing the control field shape is
allowed to drift by $\epsilon_{G}=5\%$ of its optimum. The resulting change in
memory efficiency is recorded and is used to compute a normalized sensitivity
relative to the rest of the points along the control field shape. The results
of this procedure are shown for two optimal control fields in Fig. 8(f) and
(g), in the adiabatic [EIT-like, $\mathcal{M}=(50,1.5)$] and non-adiabatic
[ATT-like, $\mathcal{M}=(50,0.01)$] regimes, respectively. Fig. 8(f) shows
that EIT-like control field shapes are most sensitive to drift along the
trailing edge of the control field shape, which overlaps with the signal field
in time and adiabatically closes the transparency window. Importantly, this
result is in agreement with the experimental determination of Ref. [159]. For
ATT-like control field shapes, Fig. 8(g) shows that the most sensitive regions
of the control field are those with the largest Rabi frequency. This agrees
with physical intuition for the ATT protocol (see Sec. 2.1 for a review), as
it is these regions of the control field that have the largest effect on its
net pulse area, and therefore have the largest effect on the efficiency of the
atomic-polarization–to–spin-wave transfer process. The same one-at-a-time
sensitivity analysis is performed for Gaussian control fields in Ref. [23], as
well as a more sophisticated and computationally expensive Sobol’ analysis
that probes correlations between Gaussian control field parameters.

We stress here that the sensitivity analysis performed in Ref. [23] represents
only a small subset of all sensitivity calculations that can be performed, and
which may be relevant and useful for quantum memory experiments. The general
framework developed in Ref. [23] may also be applied to characterize the
sensitivity of off-resonant memory protocols, or those protocols that make use
of inhomogeneously broadened ensembles. Moreover, Ref. [23] considers only the
case of ‘efficiency sensitivity,’ or how fluctuations and drift in certain
parameters affect memory efficiency. The same approach may straightforwardly
be repurposed to evaluate such metrics as fidelity sensitivity, bandwidth
sensitivity, or lifetime sensitivity, to name only a few. This may be the
subject of future work.

##  4 State of the Art

Figure 9: State of the art storage efficiencies for (a) atomic ensemble and
(b) all quantum memories (including delay-based) in the broadband regime.
Delay-line and fiber memories tend to have high efficiency independent of
bandwidth. Citation numbers appear in Ref. [257]. Adapted from Ref. [82].

Figure 10: State of the art memory lifetimes for (a) atomic ensemble and (b)
all quantum memories (including delay-based) in the broadband regime. Citation
numbers appear in Ref. [257]. Adapted from Ref. [83].

###  4.1 Efficiency

With the foundation for the quantum memory protocols and theory laid above, in
this section we move on to discuss the state of the art for experimental
quantum memory performance in the broadband regime, which we consider to be
memory bandwidths greater than 10 MHz (defined as the full width at half-
maximum of the signal frequency intensity distribution, see Sec. 1.4).

Figure 9(a) shows the state-of-the-art storage efficiencies for atomic
ensemble quantum memories in the broadband regime, as of the writing of this
chapter. Each memory protocol is given a different marker type. As can be
seen, Raman and AFC memories are the most popular types in the broadband
regime, and Raman memories are the only type used in the THz-bandwidth regime
except for Ref. [30], which most closely resembles the absorb-then-transfer
protocol. Resonant, homogeneous atomic ensemble protocols (e.g., EIT, ATS, SR,
FLAME) tend to be limited to bandwidths below 1 GHz, with the exception of
Ref. [30], due to linewidth-bandwidth mismatch. More details on how Ref. [30]
alleviates the linewidth-bandwidth mismatch problem can be found in Sec. 1.3.
Fig. 9(b) shows the performance of all quantum memories in the broadband
regime, as of the writing of this chapter, including delay-line and fiber
memories, which tend to have high efficiency independent of bandwidth.

In Fig. 9, we show state-of-the-art storage efficiencies; total efficiency
follows a similar trend, but is always significantly lower, and the trend is
complicated by non-optimal phasematching and reabsorption loss during
retrieval, as well as spin wave decay, which vary depending on the reference
and reduce total efficiency non-uniformly across the bandwidths shown.

###  4.2 Memory Lifetime

Figure 10 shows the state-of-the-art memory lifetimes for atomic ensemble
quantum memories [Fig. 10(a)] and all quantum memories [Fig. 10(b)] in the
broadband regime. An empirical tradeoff exists between memory lifetime and
bandwidth. We believe this trend primarily arises from technical
considerations, rather than fundamental physical ones — in the THz-bandwidth
regime, for example, the Raman memories shown suffer from low memory lifetime
not due to a fundamental aspect of the Raman protocol, but because the
hardware employed (solids and molecular gases) have inherently limited storage
state lifetimes. Ref. [30] represents a unique case in this regard, where the
lifetime is instead limited by atomic motion (sometimes called Doppler
broadening) due to the high temperatures used, yet the inherent storage state
lifetime is long, of order 0.1 seconds [258]. The delay-based memories shown
in Fig. 10(b) break this trend, and all tend to have nanosecond to microsecond
delay times.

###  4.3 Noise

We plot the state-of-the-art noise performance for broadband atomic ensemble
quantum memories in Figure 11. We plot the signal-to-noise ratio as defined in
Sec. 1.4 for each broadband memory found in the literature. The dark, medium,
and light red regions represent the signal to noise ratios for $<$90%, $<$99%,
and $<$99.9% single-photon fidelities. Across all bandwidths shown, only a few
quantum memories exceed an $\mathrm{SNR}$ of 103; most memories in this region
employ a ladder-type energy level system (ORCA and FLAME memories,
specifically) that is in principle noise-free. Ref. [30] is the only atomic
ensemble memory in this regime that employs a $\Lambda$-type level structure,
and is also in principle noise-free to first order due to a large ground-
state–storage-state splitting compared to the storage-state–excited-state
splitting.

Figure 11: State-of-the-art noise performance for atomic ensemble and delay-
based quantum memories in the broadband regime. Dark, medium, and light red
regions correspond to the signal to noise ratios necessary for $<$90%, $<$99%,
and $<$99.9% single-photon fidelities. Note that many fiber and delay-line
memories do not report a figure of merit for noise performance, and are
therefore not plotted here, as the added noise is virtually zero; some of