# Conditional $\pi$-Phase Shift of Single-Photon-Level Pulses at Room
Temperature

Reihaneh Shahrokhshahi  Department of Physics and Astronomy, Stony Brook
University, New York 11794-3800, USA  R. Shahrokhshahi and S. Sagona-Stophel
contributed equally to this work.  Steven Sagona-Stophel  Department of
Physics and Astronomy, Stony Brook University, New York 11794-3800, USA  R.
Shahrokhshahi and S. Sagona-Stophel contributed equally to this work.  Bertus
Jordaan  Department of Physics and Astronomy, Stony Brook University, New York
11794-3800, USA  Mehdi Namazi  Department of Physics and Astronomy, Stony
Brook University, New York 11794-3800, USA  Eden Figueroa  Department of
Physics and Astronomy, Stony Brook University, New York 11794-3800, USA

###### Abstract

The development of useful photon-photon interactions can trigger numerous
breakthroughs in quantum information science, however this has remained a
considerable challenge spanning several decades. Here we demonstrate the first
room-temperature implementation of large phase shifts ($\approx\pi$) on a
single-photon level probe pulse (1.5us) triggered by a simultaneously-
propagating few-photon-level signal field. This process is mediated by
$Rb^{87}$ vapor in a double-$\Lambda$ atomic configuration. We use homodyne
tomography to obtain the quadrature statistics of the phase-shifted quantum
fields and perform maximum-likelihood estimation to reconstruct their quantum
state in the Fock state basis. For the probe field, we have observed input-
output fidelities higher than 90$\%$ for phase-shifted output states, and high
overlap (over 90%) with a theoretically perfect coherent state. Our noise-
free, four-wave-mixing-mediated photon-photon interface is a key milestone
towards developing quantum logic and nondemolition photon detection using
schemes such as coherent photon conversion.

##  I Introduction

Photons are famously useful for transmitting PhysRevLett_115_040502 ; Liao2017
and storing Wang2017 ; PhysRevLett.118.220501  quantum information due to
their typically weak interaction with the environment. Because of this, it has
been thought that photon-based quantum computers would be one of the best
architectures if an efficient photon-photon gate could be developed
PhysRevA.52.3489 . Additionally, there are a number of unique applications for
photonic quantum information processing, including utilization of qRAM
Biamonte2017  using stored qubits Lvovsky2009 , higher connectivity than
current grid-based systems Lechnere_2015  using nonlocal connections between
flying qubits, and continuous-variable quantum machine learning 1806.06871 .
111Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
There are a number of challenges towards experimentally realizing a photon-
photon phase gate: i) A system first must exhibit large enough cross-phase
modulation (XPM) per photon such that a single photon causes a $\pi$ phase
shift on a second photon state. ii) The quantum state of the phase-shifted
light must not be distorted or impure. iii) Each possibility in the “truth
table” of input combinations must be have high-fidelity operation. In
particular the state should remain the same when the phase-shifting trigger
photon is not present:

$\displaystyle(|0\rangle+|1\rangle)\otimes|1\rangle$
$\displaystyle\Rightarrow(|0\rangle-|1\rangle)\otimes|1\rangle$
$\displaystyle(|0\rangle+|1\rangle)\otimes|0\rangle$
$\displaystyle\Rightarrow(|0\rangle+|1\rangle)\otimes|0\rangle$

Recently, cavity-based and Rydberg-based systems have demonstrated large XPM
at the single-photon level Tiarks2016 ; Beck2016 ; Feizpour2015 , and have
begun investigating operation fidelities of a photon-photon gate Tiarks2019 ;
Rempe16 . However, these new schemes all either need cavities or require the
light to be stored in an atomic medium, which dramatically lowers the gate’s
efficiency. For example, the most advanced photonic processing system
currently has an efficiency ranging from 0.5% to 8%, with an entangling gate
fidelity of $63.7\%\pm 4.5\%$ Tiarks2019 . It is clear that a lot of
improvement needs to be made before these systems are ready for large problems
requiring fault tolerance PhysRevLett.117.060505 . Additionally, improvements
may need new physical architectures, as it is not clear if current gate
implementations have fundamental upper limits on fidelities similar to the
constraints present in Kerr systems PhysRevA.73.062305 ; PhysRevA.81.043823 .

Recent developments have shown that, unlike previously thought
PhysRevA.73.062305 ; PhysRevA.81.043823 , there are indeed cavity-free
nonlinear systems that are capable of large phase shifts at high-fidelity
PhysRevLett.116.023601 , and they can perform gate operations even in a full
frequency-mode framework PhysRevA.97.032314 . These systems use a paradigm
called “coherent photon conversion (CPC) Langford2011 ,” in which photons are
converted by a four-wave mixing (FWM) process. It has been shown that
universal quantum computation is possible with only a combination of these
processes and linear optics PhysRevLett.120.160502 .

To match the requirements of a quantum photonic gate using CPC, the following
experimental benchmarks must be achieved: i) Four-wave-mixing processes can
occur for single photon inputs. ii) The fidelity of the quantum output state
in the Fock state basis is preserved. iii) The two photon inputs must be
efficiently and coherently converted into a third (initially vacuum) field and
then be coherently converted back, thereby creating a “Rabi oscillation”
between the $|1,1,0\rangle$ and $|0,0,1\rangle$ states Langford2011 .

While current work towards CPC uses nonlinear waveguides and is limited by the
efficiency of the quantum process PhysRevA.90.043808 , atomic systems can
create FWM Liu2017  at near unitary efficiencies, and are an excellent
candidate for highly efficient CPC. Additionally it has been shown that with
these same FWM systems, it is possible to achieve large XPM at the low light
level without requiring cavities, storage or Rydberg levels through a
double-$\Lambda$ system Yu2016 . Unlike Kerr-nonlinearity-based schemes, which
have increasing phase shifts per photon when the signal field coupling is
increased, double-$\Lambda$ systems can have phase shifts that are independent
of signal power, as long as the probe and signal are scaled simultaneously.

In this experiment we demonstrate the first room-temperature implementation of
a double-$\Lambda$ system in which simultaneously propagating pulses of
single-photon-level light create $\pi$ phase shifts. We perform a quantum
characterization of these phase-shifted output states using quantum state
tomography of the quadrature statistics. We show that four-wave mixing
processes at room-temperature still produce well-behaved quantum states,
observing high fidelities even for single-photon level light phase shifted by
$\pi$. Finally, we evaluate our system in the context of the first two of the
mentioned benchmarks required for implementation of the CPC protocol.

##  II Theoretical Background

A key feature of our double-$\Lambda$ system is that the output phase-shift is
phase-sensitive on the input phases of multiple fields. First, we will
highlight this sensitivity theoretically and identify a key parameter
$\Delta\phi_{FWM}$. Second, after outlining our experimental atomic system, we
will discuss how a unique pulse-sequence protocol can be used to
simultaneously extract both phase-shifts and quadrature statistics after
interaction with the medium. Third, we verify these experimental estimates of
phase-shifts are both precise and well aligned with our theoretic atomic
model. Fourth, we will use these estimates to collect phase-shift-binned
quadratures statistics for quantum state estimation. To understand the quantum
behavior of the DL process at the single-photon-level, using the results of
our quantum state tomography, we calculate characterization values including
purity, fidelity, and coherent-state overlap.

Figure 1: a) Experimental setup. A sequence of probe ($\Omega_{p}$) and signal
pulses ($\Omega_{s}$) are sent through a room-temperature ${}^{87}Rb$ atomic
ensemble. The probe output is then measured on a balanced homodyne detector
and the associated quadrature and phase values are analyzed. b) Diagram of our
sequence of input pulses. First, we send a pulsed probe field ($\Omega_{p}$),
with two CW control fields ($\Omega_{c1},\Omega_{c2})$ creating EIT. Secondly,
we add a pulsed signal $\Omega_{s}$ field (in addition to the probe
$\Omega_{p}$ pulse), creating a double-$\Lambda$ system. Thirdly, we switch
off the probe pulse (leaving the signal pulse on), creating four-wave mixing
FWM in the frequency of the probe. c) Voltage values of the balanced homodyne
detector, representing quadrature measurements of the pulse sequence. Across a
scan of the phase of the local oscillator of the homodyne, phase information
can be extracted from these pulses (as discussed in Figure 2).

###  II.1 Double-$\Lambda$ System

We use a double-$\Lambda$ system similar to the closed-loop scheme presented
in Yu2016 . While a fully quantized model of similar systems have been
constructed in Artoni2015 , we introduce a simpler semi-classical model based
on a three-level EIT system to understand how this atom-light interaction
creates a phase shift aided by four-wave mixing.

As seen in Figure 1, each individual $\Lambda$ system has a weak field
coupling states $|1\rangle$ and $|3\rangle$ (named “probe” and “signal”) and a
strong control field (labeled “c1” and “c2”) coupling states $|1\rangle$ and
$|2\rangle$, with detunings ($\Delta_{p}=\Delta_{c1}=\Delta_{1}$) and
($\Delta_{s}=\Delta_{c2}=\Delta_{2}$) such that
$\Delta_{2}-\Delta_{1}=\Delta$. Since our detunings are much smaller than our
laser frequencies ($\Delta\ll\omega$), we can make an adiabatic approximation
for the contribution of the other fields. Identifying that our additional
time-dependencies are slowly-varying relative to the field’s frequency, as
similarly treated in standard treatment of EIT RevModPhys.77.633 , the
solutions can be found via the Optical-Bloch equations:

$\displaystyle\rho_{13}$
$\displaystyle=\frac{i\Omega_{i}}{2i\Delta_{2}+\Gamma}-\frac{\Omega_{i}|\Omega_{j}|^{2}}{\gamma(2i\Delta_{2}+\Gamma)^{2}}$
(1) $\displaystyle\rho_{12}$
$\displaystyle=\frac{i\Omega_{i}\Omega_{j}^{*}}{\gamma(2i\Delta_{2}+\Gamma)^{2}}$
(2)

Where $\Omega_{j}=E_{c1}e^{i\phi_{c1}}+E_{c2}e^{i\Delta_{1}t+\phi_{c2}}$ and
$\Omega_{i}=E_{p}e^{i\phi_{p1}}+E_{s}e^{i\Delta_{1}t+\phi_{s2}}$ are effective
Rabi frequencies containing the time dynamics of both fields. Explicitly
plugging in these time dependent relations, and solving for frequency terms
that match the probe and signal, we obtain the following relations for the
output polarizability:

$\displaystyle P(\omega_{s})$
$\displaystyle=e^{i\phi_{s}}\left(\left(\frac{i|\wp_{13}|}{2i\Delta_{2}+\Gamma}+\frac{|\wp_{13}||\wp_{23}|^{2}(|E_{c1}|^{2}+|E_{c2}|^{2})}{\gamma(2i\Delta+\Gamma)^{2}}\right)|E_{s}|+\left(\frac{|\wp_{13}||\wp_{23}|^{2}|E_{c1}||E_{c2}|}{\gamma(2i\Delta+\Gamma)^{2}}\right)|E_{p}|e^{i\Delta\phi^{s}_{FWM}}\right)$
(3) $\displaystyle P(\omega_{p})$
$\displaystyle=e^{i\phi_{p}}\left(\underbrace{\left(\frac{i|\wp_{13}|}{2i\Delta_{2}+\Gamma}+\frac{|\wp_{13}||\wp_{23}|^{2}(|E_{c1}|^{2}+|E_{c2}|^{2})}{\gamma(2i\Delta+\Gamma)^{2}}\right)|E_{p}|}_{\text{Probe-
only
Contribution}}+\underbrace{\frac{|\wp_{13}||\wp_{23}|^{2}|E_{c1}||E_{c2}|}{\gamma(2i\Delta+\Gamma)^{2}}|E_{s}|e^{i\Delta\phi_{FWM}}}_{\text{Four-
wave-mixing Contribution}}\right)$ (4)

Where P($\omega_{s}$) and P($\omega_{p}$), obtained from
$P_{out}=Tr[\rho\wp]$, are the terms proportional to $e^{i\omega_{s}t}$ and
$e^{i\omega_{p}t}$ respectively. Additionally, $\wp$ is the dipole operator,
$\Gamma$ is the decay rate from the excited state, and $\gamma$ is the
decoherence of $\rho_{12}$. We identify a critically important parameter for
our experiment: $\Delta\phi_{FWM}=\phi_{s}-\phi_{p}+\phi_{c2}-\phi_{c1}$,
which represents the relative phase between the generated four-wave-mixing and
the probe. While the first term of equation 4 represents the probe light
experiencing an effect similar to electromagnetically-induced transparency
(EIT) due to the contribution of the strong control fields, the third term in
equation 4 represents a phase-sensitive four-wave-mixing contribution, in
which the two control fields and the signal create light in the frequency of
the probe. Thus, the output polarizability can experience constructive or
destructive interference depending on the value of $\Delta\phi_{FWM}$.

These polarizabilities contribute “effectively linear” driving terms to the
slowly-varying Maxwell-Schrödinger equations and, similar to other work, can
be analytically integrated to obtain the electric field for the probe and the
signal Yu2016 ; Artoni2015 .

$\displaystyle\frac{\partial E_{p}}{\partial z}+\frac{1}{c}\frac{\partial
E_{p}}{\partial t}=i\frac{\alpha_{p}\gamma_{31}}{2L\mu_{13}}P(\omega_{p})$ (5)
$\displaystyle\frac{\partial E_{s}}{\partial z}+\frac{1}{c}\frac{\partial
E_{s}}{\partial t}=i\frac{\alpha_{s}\gamma_{31}}{2L\mu_{13}}P(\omega_{s})$ (6)

This phase-sensitivity of the FWM is an extremely critical feature of the
double-$\Lambda$ system because the output light’s amplitude
$E_{p}^{DL}(\Delta\phi_{FWM})$ and phase shift (written as $\Delta\phi_{DL}$)
are both functions of $\Delta\phi_{FWM}$:

$\displaystyle\Delta\phi_{DL}^{\omega_{p}}$
$\displaystyle=\arctan{\left(\frac{\Im[E_{p}^{DL}(\Delta\phi_{FWM})]}{\Re[E_{p}^{DL}(\Delta\phi_{FWM})]}\right)}$
(7)

Therefore, for each $\Delta\phi_{FWM}$ there is an associated double-$\Lambda$
phase-shift $\Delta\phi_{DL}$ and a particular value of $|E_{p}^{DL}|^{2}$,
which will be proportional to mean-photon number for quantum light (discussed
further in the next section and illustrated in Figure 2c and 3a).

Figure 2: Illustration of data extraction workflow for phase estimation and
quantum state estimation. a) For each “scan,” corresponding to a 5ms long
sweep of the local oscillator phase of the homodyne, our three-pulse sequence,
shown in the inset (and in Figure 1b), repeats tens of times. Peak voltage
values obtained within each of these scans provides a set of quadrature
values: $\\{X(\theta_{i})|\theta_{i}\in(0,2\pi)\\}$. These voltage values
represent a set of quadrature measurables for the cases: probe only
$\\{X(\theta)\\}^{\phi_{p}}_{p}$, double-$\Lambda$
$\\{X(\theta)\\}^{\phi_{DL}}_{DL}$, and four-wave mixing
$\\{X(\theta)\\}^{\phi_{FWM}}_{FWM}$, shown as red, teal, and blue points
respectively. Across each 5ms homodyne “scan,” phase-changes in that time
interval can be estimated. The output phase shift of the double-$\Lambda$
system, called $\Delta\phi_{DL}$, (relative to the probe-only case) is labeled
in teal. b) Quadrature values are taken continuously over multiple seconds.
The local oscillator “scans” hundreds of times, generating hundreds of sets of
quadrature values $\\{X(\theta)\\}$. Each set has a fixed associated phase-
shifts ($\Delta\phi_{FWM},\Delta\phi_{DL}$), and a range of local oscillator
phases ranging from $(0,2\pi)$. c) The relationship between DL phase shift,
$\Delta\phi_{DL}$, and FWM phase shift, $\Delta\phi_{FWM}$, for each
quadrature set are plotted as blue points, while the theoretical relationship
is plotted as a blue-dashed line. Furthermore, the binning workflow is
illustrated in the figure: Hundreds of quadrature sets associated with
particular values of $\Delta\phi_{FWM}$ are binned and collected for quantum
state estimation. For each phase-shift bin, the quantum states of these sets
$(\\{X_{P}\\},\\{X_{DL}\\},\\{X_{FWM}\\})$ are reconstructed using maximum
likelihood estimation and are illustrated as wigner functions for the EIT,
FWM, and DL cases (illustrated in further detail in Figure 4).

##  III Experimental Methods

###  III.1 Experimental Setup

As illustrated in Figure 1, we use a double-$\Lambda$ atomic system in a room-
temperature $Rb^{87}$ atomic ensemble, in which both individual $\Lambda$
subsystems exhibit EIT. The output light is sent to a homodyne detector to
extract out the quantum state. In addition to using standard techniques to
extract out phase-information and quadrature statistics lvovtecniques ;
lvovsky2009rev , we incorporate a unique pulse-sequence protocol to extract
out quadrature statistics of the double-$\Lambda$ system corresponding to
particular values of $\Delta\phi_{FWM}$.

As shown in Equation 7, because our double-$\Lambda$ system’s output light’s
amplitude $E_{p}^{DL}$ and phase shift $\Delta\phi_{DL}^{\omega_{p}}$ are
functions of the parameter $\Delta\phi_{FWM}$, in order to accurately recover
homodyne statistics describing a particular quantum state of the phase-shifted
double-$\Lambda$ system, we must either have extremely precise phase-control
of all of our input field’s phases or have a method of accurately estimating
the phases for $\Delta\phi_{FWM}$ and $\Delta\phi_{DL}$ simultaneously. We
achieve the ladder by implementing a closely-separated, three pulse sequence
with a fast local oscillator scan (as illustrated in Figure 2). With this
implementation, we can let our phases fluctuate randomly and collect
quadrature statistics of our output double-$\Lambda$ system and bin them by
the phase-shift that uniquely describes their state ($\Delta\phi_{FWM}$).
Additionally, our phase-extraction protocol also allows for additional
quadrature data to be collected describing the quantum states of the light
generated by the FWM (denoted $\rho_{FWM}$) and the probe light experiencing
EIT-like behavior without the inclusion of signal field (denoted
$\rho_{EIT}$).

Figure 3: In a,b): Comparison between the phase and amplitude of each
sinusoidal fit of individual homodyne scans and theoretical values for single-
photon-level data. a) The blue dots represents individual values of
$\Delta\phi_{DL}-\Delta\phi_{FWM}$ vs $\Delta\phi_{FWM}$ for each individual
homodyne “scan” (as discussed in Figure 2). The orange dots represent the
fitted amplitudes of each homodyne shot for the DL case. These are compared to
their theoretical values (solid orange and dashed blue lines) obtained from
equations (14) and (15). No free parameters are used in the theoretical fits.
In b) the blue points represent the phase-shift, $\Delta\phi_{DL}$ vs
$\Delta\phi_{FWM}$. Part c), to reconstruct the quantum state, the quadrature
statistics obtained from the outputs must be sorted by their associated
$\Delta\phi_{FWM}$ phase-shifts. Using the phase information illustrated in a
and b, we can collect quadrature statistics associated with different binned
phases (as discussed in Figure 2). Extracted single-photon-level quadrature
data illustrated for the double-$\Lambda$ (teal) and EIT reference (red) - for
different sets of post-selected phase shifts (for bins
$\Delta\phi_{FWM}\in.01,1.33,-.91,$ and -2.77 $\pm.37$ radians). Quadrature
values are compared to their averaged fitted values (plotted as a solid line
in red and teal).

###  III.2 Atomic Scheme

In our experiment, two hyperfine ground states are coupled to an excited state
by weak probe fields called probe and signal ($\Omega_{p}$ and $\Omega_{s}$)
and two strong control fields ($\Omega_{c1}$ and $\Omega_{c2}$), as shown in
Figure 1. We utilize two external cavity diode lasers phase-locked at 6.8 GHz
to correspond to the 87Rb ground state splitting. For the first $\Lambda$
system, the probe and control field are 400 MHz red detuned from the
$|5S_{1/2},F=1\rangle$ $\Rightarrow$ $|5P_{1/2},F=1\rangle$ and the
$|5S_{1/2},F=2\rangle$ $\Rightarrow$ $|5P_{1/2},F=1\rangle$ transition
respectively. The signal and second control field for the second $\Lambda$
system are 80 MHz blue detuned from the probe and first control field
respectively. Both probe and signal fields are fixed to have the same linear
polarization, while the control fields will be set to have the same orthogonal
polarization to allow for convenient separation of the control fields after
atomic interaction. The intensity of the probe and signal fields are
temporally modulated by means of acousto-optical modulators (AOM). For our
medium we use an 5 cm long glass cell with anti-reflection coated windows
containing isotopically pure 87Rb with 10 Torr of Ne buffer gas kept at a
temperature of 336 K. In this medium, 1.5 us long single-photon pulses can be
transmitted through the EIT medium with 80$\%$ efficiency in the presence of
the first control field $\Omega_{c1}=45mW$. Additionally, with the addition of
the second control field $\Omega_{c2}=5mW$, this drops to 50$\%$.

###  III.3 Homodyne Detection of Pulse Sequence

As illustrated in Figure 1 b, we send a pulse sequence consisting of three
cases: probe pulse only, probe and signal pulse on (creating the
double-$\Lambda$ system), and signal only (generating FWM). Each pulse is
$1.5\mu s$ long, each case is separated with a 20$\mu s$ interpulse delay, and
this is repeated every $60\mu s$. Control fields 1 and 2 are on during the
whole cycle.

We use a standard balanced homodyne detector to obtain accurate phase and
quadrature information to estimate the quantum state of the phase shifted
light lvovsky2009rev ; lvovtecniques . A homodyne detector can obtain phase
information of cw light through interference with a strong coherent source,
producing an output voltage of the form:

$\displaystyle V_{homo}$ $\displaystyle=E_{p}E_{LO}\cos(\phi_{LO}-\phi_{p})$
(8) $\displaystyle=E_{p}E_{LO}\cos(\omega_{pz}t-\phi_{p})$ (9)

where $E_{p}$ ($E_{LO}$) and $\phi_{p}$ ($\phi_{LO}$) are the amplitude and
phase respectively of the probe (local oscillator). The phase of the probe,
$\phi_{p}$, relative to the local oscillator can be obtained by scanning the
local oscillator phase (with a frequency of $\omega_{pz}$) and fitting the
resultant voltage $V(t)\propto\cos(\omega_{pz}t-\phi_{p})$ to a cosine curve
to extract the offset, $\phi_{p}$.

Using our pulse sequence, we can extract out this sinusoidal behavior by
extracting the voltage values associated with the peaks of the pulses. The
peaks of each case in the pulse sequence forms its own sinusoidal function per
scan of the local oscillator, and can be seen as the red, teal and blue fits
in Figure 2a for $\phi^{p}$, $\phi^{p}_{DL}$, and $\phi^{p}_{FWM}$.

As a result, we can obtain the phase-shifts $\Delta\phi_{DL}$ and
$\Delta\phi_{FWM}$:

$\displaystyle\Delta\phi_{DL}$
$\displaystyle=(\phi^{p}_{DL}-\phi_{LO})-(\phi_{p}-\phi_{LO})=\phi^{p}_{DL}-\phi_{p}$
(10) $\displaystyle\Delta\phi_{FWM}$
$\displaystyle=(\phi^{p}_{FWM}-\phi_{LO})-(\phi_{p}-\phi_{LO})=\phi^{p}_{FWM}-\phi_{p}$
(11)

By alternating between the three cases and quickly scanning the local
oscillator phase, we can obtain a phase estimate faster than the timescale of
the phase-fluctations of our experiment (typically due to slight optical path
length changes due to air fluctations). Using a Spectrum M2i.2031 digitizer
card with 2 GSamples of onboard memory, we store continuous homodyne data in
bursts of 1.3 seconds, with an internal sampling clock of 100MHz. We acquire
these large bursts of data, split the data into individual 5ms “shots”
(corresponding to the time to make a complete $2\pi$ cycle of the local
oscillator phase), and obtain the phase-shift per shot from each fit
(illustrated in Figure 2b). Because the frequency of our local oscillator scan
is 200 Hz (a frequency faster than typical phase fluctuations in our
experiment), each fit provides an accurate shot-by-shot estimation of the
phases of the light pulses within the 5ms time window.

Finally, we can extract the homodyne quadrature statistics for any of the
outputs of the three cases in the sequence. For single-photon level light, the
voltage output associated with the peak of the pulses (in each sequence) of
the homodyne is quantum mechanically a measurement of the generalized
quadrature:

$X_{\theta}=ae^{i\theta}+a^{\dagger}e^{i\theta}$

Figure 4: Reconstructed density matrices represented as Wigner functions are
plotted, illustrating the relationship between inputs and outputs for
different post-selected phase-shifts. The red and purple plots are the input
states of the probe and signal. The teal plot is the phase-shifted probe
output for a) 0.3 radians phase shift, b) $\pi/2$ phase shift, and c) $\pi$
phase shift.

These quadrature statistics can be collected and binned by the assocated phase
shift $\Delta\phi_{FWM}$, where the data across multiple local oscillator
scans can be evaluated using Maximum-likelihood estimation to uncover their
quantum states in the Fock state basis lvovsky2009rev . This workflow is
illustrated in Figure 2c and the resultant binned data is shown in Figure 3c.

##  IV Results

###  IV.1 Analysis of Input-phase vs. Phase-shift

As our local oscillator sweeps through its phase, the output voltage
associated with the peaks of the output pulses in the sequence vary
sinusoidally (illustrated in Figure 2a and 2b). To compare our experimental
results with our semi-classical theory, we can predict this behavior as a
linear inteference of the “probe-only” contribution (the first term in
equation 3) and “FWM” contribution (second term):

$\displaystyle E_{DL}(t)$
$\displaystyle=E_{E}\sin(w_{pz}t)+E_{F}\sin(w_{pz}t+\Delta\phi_{FWM})$ (12)
$\displaystyle=A(E_{F},E_{E},\Delta\phi_{FWM})\sin(w_{pz}t+\Delta\phi_{FWM})$
(13)

Where $E_{E}$ and $E_{F}$ correspond to amplitudes obtained from sinusoidal
fits of homodyne shots extracted from the first and third pulse in the
sequence respectively. We solve for the phase $\Delta\phi_{DL}$ and amplitude
$A(E_{F},E_{E},\Delta\phi_{FWM})$, obtaining:

$\displaystyle\Delta\phi_{DL}=\arctan\left(\frac{1+\frac{E_{F}}{E_{E}}\cos(\Delta\phi_{FWM})}{\frac{E_{F}}{E_{E}}\sin(\Delta\phi_{FWM})}\right)$
(14) $\displaystyle
A=\sqrt{\left(\frac{E_{E}^{2}+E_{F}^{2}}{E_{E}^{2}}+2\frac{E_{F}}{E_{E}}\cos(\Delta\phi_{FWM})\right)}$
(15)

$E_{E}$ and $E_{F}$ can be obtained accurately by averaging fits of shots of
homodyne data from pulses 1 and 3 in our pulse sequence. One of each
individual shot to be averaged is illustrated as red and blue sinusoidal fits
in Figure 2a. Thus our simple model matches the experimental data without any
free parameters at the single-photon level. The theoretical amplitude vs
$\Delta\phi_{FWM}$ in equation 15 is visualized as the solid orange line in
Figure 2a, while the dashed blue line in Figure 2b represents amplitude A vs
$\Delta\phi_{FWM}$ from equation 14. Additionally, the small degree of
uncertainty in our experimental estimation of our phases ($\Delta\phi_{FWM}$
and $\Delta\phi_{DL}$) implies we can accurately associate quadrature
statistics with a particular phase-shifted ouput state, thereby allowing
accurate quantum state estimation through binning.

###  IV.2 Quantum State Reconstruction for Binned Phases

For each sweep of the homodyne local oscillator phase, an accurate value of
the output phase-shift $\Delta\phi_{DL}$ and the four-wave mixing phase-shift
$\Delta\phi_{FWM}$ can be extracted from the peaks of the pulses (as
illustrated in Figure 2b). Therefore, for every output phase-shift
(represented by a point in Figure 2c), we obtain an entire set of homodyne
statistics $\\{X(\theta)\\}$, all associated with a particular FWM phase shift
$\Delta\phi_{FWM}$. We then organize the output phase-shifts into 10 bins and
then combine quadrature sets $\\{X(\theta)\\}$ with associated FWM phase-
shifts in the same the bin region. A selection of these combined sets are
plotted in Figure 3c.

Figure 5:  a) Maximum values of the Wigner functions of each post-selected
phase are plotted, illustrating the motion in phase-space of the probe output
as the global input phase is changed. These maximums are illustrated for the
double-$\Lambda$ states (in teal), the four-wave mixing (in purple), and the
probe-only case (in red) is labeled as a reference. b) Plots of mean photon
number from reconstructed states, as a function of post-selected phase shifts.
Photon numbers are plotted as dashed lines for double-$\Lambda$ (dashed teal),
four-wave-mixing (dashed purple), EIT (dashed red), and probe input (solid
red).

For each combined set (binned by $\Delta\phi_{FWM}$), we perform a quantum
state reconstruction for each case: probe-only, double-$\Lambda$, and FWM.
Additionally we also measure and reconstruct the density matrix for the probe
and signal fields without the cell.

The reconstructed density matrices can additionally be mapped to a Wigner
function representation for easier visualization. Figure 4 illustrates an
input-output representation for different phase-shifted output states, while
Figure 5a shows how both the four-wave mixing and the double-$\Lambda$ system
traverse phase-space. In Figure 5a, we observe an unshifted circle (in dark
blue) representing the max-values of each Wigner function of the four-wave
mixing in phase space, indicating the mean photon number of the quantum state
of the FWM does not change with phase. This is unlike the double-$\Lambda$
system which has its maximum values in phase space represented by a shifted
circle (illustrated in teal), as expected by our semiclassical theory.

###  IV.3 High fidelity of output quantum state

By comparing the quantum state of the phase-shifted output of the
double-$\Lambda$, $\rho_{DL_{\theta}}$, with the quantum state of the light
without the cell, $\rho_{in_{\theta}}$, the fidelity can be calculated.

$F_{\theta}=\left(tr\sqrt{\sqrt{\rho_{in_{\theta}}}\rho_{DL_{\theta}}\sqrt{\rho_{in_{\theta}}}}\right)^{2}$
(16)

This input-output fidelity is plotted as a function of output phase shift
(plotted as orange dashed lines in Figure 6a). This fidelity reaches its
highest value of $94.4\%\pm.5\%$ for the bin $\Delta\phi_{FWM}\in(1.69,2.43$)
radians. The fidelity is also observed to reach $F=91.9\%\pm 0.4\%$ for the
bin $\Delta\phi_{FWM}\in(2.42,3.14$) radians.

Most of the behavior of the input-output fidelity can be explained with our
semi-classical model. When the FWM is in-phase with the probe (ie,
$\Delta\phi_{FWM}=0$), the fidelity is lower because the quantum interference
is constructive, producing a probe output is that larger than its input. When
out-of-phase ($\Delta\phi_{FWM}=\pi$), the amplitude of the output flips in
sign because of the unbalanced size of the probe and the four-wave-mixing
generated by the signal field. These different cases can be seen in the
homodyne sweeps illustrated in Figure 2b.

To be sure that our phase-shifted states do not obtain some additional quantum
noise within our room-temperature system (such as Langevin noise
PhysRevA.77.012323 ), we measured the fidelity between our state and a
theoretical coherent state of the same mean photon number (obtained from the
reconstruction). As shown in Figure 6b, we find that our phase-shifted states
retain remarkably high fidelity at the single-photon level - remaining above
90$\%$. Additionally, the purity of these quantum states were evaluated by
both calculating the purity operator, $\rho^{2}$. For the DL system exhibiting
large phase-shifts (within the bin $\Delta\phi_{FWM}\in(2.42,3.14$) radians),
we observe a purity of $0.92\pm 0.02$, as compared to a purity of $0.93\pm
0.01$ for the EIT-like case and $0.96\pm 0.02$ for the FWM-only case. This
high level of purity indicates that we are neither measuring significant
parasitic thermal light nor decoherence in our phase-shift process.

Figure 6: a) Plots of fidelity from reconstructed states, as a function of
post-selected phase shifts. The “input-output” fidelity (plotted as a dashed
red line) is found by comparing the density matrix overlap between a reference
in which the probe is reconstructed without the Rb cell and the phase-shifted
double-$\Lambda$ output state. The overlap between reconstructed states
obtained between the pulse sequence 1 (probe-only case) and the pulse sequence
2 (double-$\Lambda$ case) is plotted as a dashed green line. The expected
fidelity plots for perfect coherent states are plotted for comparison (as
dotted red and green lines). b) Illustrating how similar the post-selected
reconstructed states are to a perfect coherent state. For each post-selected
phase-shift, the associated reconstructed state is compared to a theoretical,
perfect coherent state of the same mean photon number. This is done for
quadrature statistics for each pulse sequence (Probe-only, double-$\Lambda$
system, and FWM in red, teal, and purple respectively)

##  V Discussion

In summary, we have shown that a simple room-temperature atomic system can
create large phase-shifts in the quantum state of simultaneously propagating
single-photon-level pulses; we have performed a quantum characterization of
these phase-shifted output states using quantum state tomography; and we have
demonstrated that four-wave mixing processes at room-temperature still have
well-behaved quantum states with high fidelities, even for single-photon level
light phase shifted by $\pi$. This bodes well for the future construction of
FWM-mediated quantum nonlinear systems.

###  V.1 Phase and Quantum State Estimation

While a number of groups have characterized fwm_dressed  or implemented four-
wave mixing PhysRevA.91.013843 , to our knowledge we are the first to
reconstruct its Fock-state decomposition. In particular, we note the utility
of our pulse-sequence protocol, which enables a “shot-by-shot” determination
of the phase. While similar experiments require careful designs of the optical
setup to reduce any potential change in path between the fields, our “shot-by-
shot” analysis allows us to accurately measure the phase with significantly
less errors due to phase fluctuations (typically due to air fluctuations),
thereby allowing for accurate quantum state estimation. This protocol could be
useful for future experiments including: quasi-real-time characterzation of
quantum light PhysRevLett.116.233602  and nondemolition measurements of single
photons PhysRevLett.77.2352 .

Our results demonstrate that quantum protocols utilizing effective four-wave
mixing processes in Rb cells are achievable, despite their room-temperature
operation. The reconstructed states of the phase-shifted light resemble near-
perfect coherent states. These states’ fidelities (as compared to perfect
coherent states) are as high as $94.4\%\pm 0.5\%$ for the bin
$\Delta\phi_{FWM}\in(1.69,2.43)$ radians, a surprising result considering that
thermal noise typically affects output fidelities in storage of light in
similar systems. Additionally, for particular input-phases, we find the
reconstructed quantum state of light’s input-output fidelity is over 90$\%$.

###  V.2 Truth-Table Elements for Gate Operation

As discussed in the introduction, the traditional truth-table element of
interest for gate operation is of the form:

$(|0\rangle+|1\rangle)\otimes|1\rangle\Rightarrow(|0\rangle-|1\rangle)\otimes|1\rangle$

where the triggering field is no longer a single-photon-level coherent state,
but a single photon Fock state. While a complete measurement of this fidelity
of this truth table element would require either Rubidium-tuned single photon
sources or a dual-mode quantum process analysis Fedorov2015 , we believe our
results allow us to answer many questions pertaining to the double-$\Lambda$
system’s potential functionality for gate operation.

Our analysis of one of the two modes is equivalent to a reduction in the
combined Hilbert space $H_{A}=H_{P}\otimes H_{S}$ Fedorov2015  to partial
trace of the form: $\rho^{P}=Tr_{S}\rho_{A}$. Although our probe mode has been
reconstructed as a reduced density matrix $\rho^{P}$, because of the high
level of purity of the reconstructed states, we can approximate our probe
density matrix as a pure state. Therefore, for this case we can explicitly
expand out the Fock-state elements of the probe mode after quantum process:

$\displaystyle(|0\rangle+|\alpha_{p}||1\rangle+...)$
$\displaystyle\Rightarrow(|0\rangle+e^{i\Delta\phi}|\alpha_{p}||1\rangle+...)$
$\displaystyle|\alpha_{s}\rangle$
$\displaystyle\Rightarrow|\tilde{\alpha_{s}}\rangle$

where $\tilde{\alpha_{p}}$ and $\tilde{\alpha_{s}}$ represent coherent states
that have modified phases and amplitudes for the probe and signal field
respectively. Extrapolating from Figure 5b, we approximate these values to be
$|\alpha_{p}|=\sqrt{\langle N\rangle}\approx 0.71$ and
$\Delta\phi^{p}_{FWM}\approx 1.86$ radians.

While these results appear to be immediately promising for quantum logic, some
of the elements of the truth table necessary for quantum-phase gate operation
are fundamentally limited. In this experiment, the primary mechanism for the
phase-shift is due to phase-controllable frequency conversion via the FWM
nonlinear process. While quantum interference in our system is generated by a
$\chi^{3}$ nonlinearity, the quantum coherence terms in $\rho_{12}$ generating
four-wave mixing interfere linearly with respect to the input fields. This
means that

$\displaystyle|0\rangle_{p}\otimes|1\rangle_{s}$
$\displaystyle\not\to|0\rangle_{p}\otimes|1\rangle_{s}$ (17)
$\displaystyle\to((1-\epsilon)|0\rangle_{p}+\epsilon|1\rangle_{p})\otimes|1\rangle_{s}$
(18)

The “phase-triggering” photon $|1\rangle_{s}$ is partially converted into the
other frequency mode, and conversion lowers the fidelity of the desired output
state.

##  VI Conclusion

We conclude that while the current double-$\Lambda$ system cannot form a full
“truth table” necessary for quantum logic, the well-behaved nature of the
analysed quantum states demonstrates that near-resonant atomic systems are a
viable candidate for coherent-photon conversion even for bulk-glass cells
operating at room-temperature.

Since the primary mechanism creating the phase-shifted light in this
double-$\Lambda$ system are two interfering four-wave-mixing channels, it
evaluates the first two benchmarks for implementation of the CPC protocol, as
outlined in the introduction. For example, if parasitic nonlinear terms,
spectral entanglement, or thermal effects dominate the process at the single
photon level, then such a system will not be experimentally viable for a
future CPC gate.

Observing no detrimental effects on fidelity of the quantum state of these
1-to-1 photon processes indicates that this architecture is ready to explore
1-to-2 photon conversion necessary for CPC Langford2011 . This would give our
system the potential to achieve 2-qubit gate operations and quantum
nondemolition measurements of single photons.

##  VII Acknowledgements

The authors kindly thank Julio Gea Banacloche, Bing He, Balakrishnan
Viswanathan, and Aephraim Steinberg for enlightening discussions. The work was
supported by the US-Navy Office of Naval Research, grant number N00141410801,
the National Science Foundation, grant numbers PHY-1404398 and PHY-1707919 and
the Simons Foundation, grant number SBF241180. B. J. acknowledges financial
assistance of the National Research Foundation (NRF) of South Africa. S. S.
acknowledges financial assistance of the United States Department of Education
through a GAANN Fellowship P200A150027-17.

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