${\frac{\omega_{2}-\omega_{1}}{\gamma}=\frac{3}{4}}$ (red with two maxima). In
both cases, the decays are not balanced (${\frac{\Gamma}{\gamma}=2}$) but the
couplings are chosen such that perfect transmission at some frequency is
achieved nonetheless.

We can also consider the effect of detunings on the frequency-dependent group
delay. We observe the group delay can be negative for series networks with
detuning (Fig. LABEL:DetunedN20Tau), with an asymmetric structure that depends
on the ordering of the detunings (except in the strongly coupled limit)
111111Note that a negative group delay does not imply superluminal signaling.
For a detailed discussion of negative group delays and their interpretation,
see Ref. [170]. Again, the magnitude of the group delay depends on the spacing
of discrete states, increasing when the resonant frequencies are closely
spaced.

####  3.3.3 Hybrid Networks

Figure 3.21: A hybrid network of two manifolds, each with $N_{k}$ discrete
states described by operators $c^{(k)}_{i}(\omega)$ where the index $k$ here
has only two values (${k=1,\,2}$). The discrete states are coherently coupled
at rates $g^{(1,2)}_{ij}$ and incoherently coupled to left (input, external)
and right (output, internal) continua $a$ and $b$ with decay rates
$\gamma^{(1)}_{i}$ and $\Gamma^{(2)}_{j}$, respectively. Each discrete state
is part of a manifold: a set of discrete all coupled to the same state and
continua (but not necessarily at the same rate).

We now begin to approach the case of a general two-sided quantum network. The
fully general problem is intractable analytically, but luckily there are
several simplifications we can make that correspond to the network
representing realistic photo detecting systems. To illustrate, consider the
case of two parallel networks in series: each discrete state connected to each
discrete state in the other manifold (but not necessarily at the same rate)
with each of the two manifolds of purely virtually coupled discrete states
coupled to their own continuum. One could imagine generating different
networks from this one by removing a coupling $g_{ij}$ between discrete
states, permuting which discrete states are disconnected, removing an
additional coupling, permuting, and so on. However, this is unphysical! Two
discrete states in parallel cannot be prevented from coupling to the same
discrete state except by selection rules, this would violate unitarity! But
the discrete states are also coupled to the same continuum which has the same
quantum numbers associated with it (we have removed other degrees of freedom
in going to the $1D$ problem), so they must satisfy the same selection rules.
The same argument applies in reverse: no discrete state within a manifold can
individually stop being coupled to a continuum without the rest of the
discrete states doing so as well. This argument also applies to manifolds
embedded in a larger network away from any continuum, the requirement that all
states satisfy the same selection rules is very strong. As a result, in our
analysis we can ignore partially connected networks, they are unphysical! It
also gives us a helpful way to organize discrete states, into manifolds of
purely virtually coupled discrete states (after diagonalization) that are all
coupled to the same set of discrete states and/or continua.

We can now focus on a very large class of quantum networks, hybrid systems
consisting of manifolds in series (Fig. 3.21). It will be helpful to denote
couplings between discrete states $i$ and $j$ or manifolds $k$ and $\ell$ as
$g_{ij}^{(k,\ell)}$, and denote decay rates and discrete states within a
manifold with a superscript ($\gamma_{i}^{(k)}$, $\Gamma_{i}^{(k)}$, and
$c_{i}^{(k)}(\omega)$, with $\gamma_{i}^{(k)}=0$ for $k>1$ and
$\Gamma_{i}^{(k)}=0$ for $k<M$, where $M$ is the number of manifolds). This
allows us to rewrite the general equation (3.16) in a more explicit form

$\displaystyle-i\Delta_{i}c_{i}^{(k)}(\omega)=$
$\displaystyle-\sum\limits_{j=1}^{N_{k}}\frac{\sqrt{\gamma_{i}^{(k)}\gamma_{j}^{(k)}}+\sqrt{\Gamma_{i}^{(k)}\Gamma_{j}^{(k)}}}{2}c_{j}^{(k)}(\omega)-i\sum\limits_{j=1}^{N_{k-1}}g_{ij}^{(k-1,k)}c_{j}^{(k-1)}-i\sum\limits_{j=1}^{N_{k+1}}g_{ij}^{(k,k+1)}c_{j}^{(k+1)}$
(3.28) $\displaystyle-\sqrt{\gamma_{i}^{(k)}}a_{\rm
in}(\omega)-\sqrt{\Gamma_{i}^{(k)}}b_{\rm in}(\omega)$

where the superscripts denote labels of manifolds and we’ve implicitly defined
$g_{ij}^{(0,1)}=g_{ij}^{(N,N+1)}=0\,\forall i,j$. We denote the number of
discrete states in each manifold $N_{k}$ such that
$\sum\limits_{k=1}^{M}N_{k}=N$. In general (3.28) is still only numerically
solvable but we can now note two cases that yield analytic solutions.

Figure 3.22: Transmission probability for a hybrid network consisting of $3$
manifolds in series, each with $3$ discrete states with balanced decays and
uniform critical coupling. Within each manifold, the discrete states have
frequency spacing by $2.5\gamma$. Using the same special conditions as for
series network, perfect transmission is achieved for $9$ frequencies, with the
multi-layered structure of $|T(\omega)|^{2}$ encoding the manifold structure.
Figure 3.23: Group delay for a hybrid network consisting of $3$ manifolds in
series, each with $3$ discrete states with balanced decays and uniform
critical coupling, the transmission function of which is given in Fig. 3.22.
Within each manifold, the discrete states have frequency spacing by
$2.5\gamma$.

The first is the case of critical coupling between members of each adjacent
manifold and, additionally, uniformly unbalanced decays: we first define an
effective decay rate for internal couplings within the system so that
$g_{ij}^{(k,\ell)}=\frac{\sqrt{\gamma_{i}^{(k)}\Gamma_{j}^{(\ell)}}}{2}$
(effectively specializing to the critical coupling case for series networks),
and then consider the special case of
$\Gamma_{i}^{(k)}/\gamma_{i}^{(k)}=k^{(k)}\,\forall i,k$ (inhomogeneous decays
that are uniformly unbalanced within each manifold). This leads to a
reflection coefficient of the form

$\displaystyle R(\omega)$
$\displaystyle=1-\cfrac{2h^{(1)}}{h^{(1)}-i+\cfrac{k^{(1)}h^{(1)}h^{(2)}}{-i+\cfrac{k^{(2)}h^{(2)}h^{(3)}}{\dots+\cfrac{k^{(N-1)}h^{(N-1)}h^{(N)}}{\sqrt{k^{(N)}}h^{(N)}-i}}}}$

where we have defined a new function
$h^{(k)}=\sum\limits_{i=1}^{N_{k}}\frac{\gamma_{i}^{(k)}}{2\Delta_{i}^{(k)}}$.
(The appearance of a lone $\sqrt{k^{(N)}}$ at the end of (LABEL:refseriesc1)
is due to the final purely virtual coupling to the output continuum, since
we’ve absorbed the rest of the decay rate into the function $h^{(k)}$.) This
function encodes the zeroes and singularities we found for parallel networks,
which previously gave rise to frequencies of constructive interference and
completely destructive interference.

While (LABEL:refseriesc1) is tractable, it is of limited applicability to real
systems. More relevant is the second solvable case of homogenous coupling and
decays within manifolds: $g_{ij}^{(k,\ell)}=g^{(k,\ell)}$,
$\gamma_{i}^{(k)}=\gamma^{(k)}$, and $\Gamma_{i}^{(k)}=\Gamma^{(k)}$. This
leads to a reflection coefficient of the form

$\displaystyle R(\omega)$ $\displaystyle=1-\cfrac{\gamma
f^{(1)}}{\frac{\gamma}{2}f^{(1)}-i+\cfrac{(g^{(1,2)})^{2}f^{(1)}f^{(2)}}{-i+\cfrac{(g^{(2,3)})^{2}f^{(2)}f^{(3)}}{\dots+\cfrac{(g^{(N-1,N)})^{2}f^{(N-1)}f^{(N)}}{\frac{\Gamma}{2}f^{(N)}-i}}}}$

where again we have defined a new function
$f^{(k)}=\sum\limits_{i=1}^{N_{k}}\frac{1}{\Delta_{i}^{(k)}}$. This provides a
nice model for multi-mode systems in series (for instance, a linear network of
identical multimode optical cavities; in this case, the critical coupling
strength $g=\frac{\sqrt{\gamma\Gamma}}{2}$ corresponds to optical cavities
perfectly coupled through their decays).

We see in both (LABEL:refseriesc1) and (LABEL:refseriesc2) a combination of
the structures we observed in (3.23) and (3.24) for parallel networks and
(3.25) for series networks: correlations in amplitudes between discrete states
within a given manifold manifest via a function $h^{(k)}$ or $f^{(k)}$ with
$N_{k}$ poles and $N_{k}-1$ zeroes (potentially perfectly transmitted and
reflected frequencies, depending on the hybrid network’s resonance structure
and couplings), and causal ordering of the manifolds manifests in a continued
fraction structure. This latter property makes them easily analyzable using
the Wallis-Euler recurrence relations, allowing us to find $R(\omega)$ and
from there $T(\omega)$ and the other quantities of interest.

_Perfect transmission.—_ Focusing on the case of homogenous coupling, we can
use the same trick of examining the convergence of an infinite series of
identical manifolds around one of resonant frequencies $\omega_{i}$ to find
the two critical conditions as we did for series networks, which again are
$\gamma=\Gamma$ and $g=\frac{\sqrt{\gamma\Gamma}}{2}$. We also observe that
the form of $T(\omega)$ for hybrid systems exhibits a combination of features
of series and parallel networks (Figs. 3.22, 3.24, and 3.26). This results in
layers of structure, with the small dips and peaks corresponding to intra-
manifold structure layered on top of the larger dips and peaks of the inter-
manifold structure.

When at least one discrete state from each of the manifolds have the same
resonance frequency, letting $\gamma=\Gamma$ and
$g=\frac{\sqrt{\gamma\Gamma}}{2}$ ensures perfect transmission at $M$
frequencies. For a general hybrid network, the number of peaks of unity
(perfect transmission) of $T(\omega)$ are bounded above by $M\times{\rm
Min}\\{N_{k}\\}\leq N$, with $M$ the number of manifolds and $N_{k}$ the
number of discrete states in the $k$th manifold. Here the latter equality is
reached for networks that are completely in parallel or completely in series,
with critical parameters in either case121212The bound is somewhat stronger
for critically coupled networks, with the number of perfectly transmitted
frequencies bounded by ${(M-\frac{1}{2}(1+(-1)^{M}))\times{\rm
Min}\\{N_{k}\\}}$ instead. This is due to the on-resonance broadening that
occurs for critically coupled systems, which prevents the splitting of one
peak to two when the number of manifolds $M$ is even..

_Spectral Bandwidth.—_ For hybrid networks where the first and last manifold
have the same number of discrete states $N_{1}=N_{M}$, the spectral bandwidth
is bounded above by the parallel network bandwidth for that number of discrete
states $\tilde{\Gamma}\leq\frac{N_{1}2\gamma\Gamma}{\gamma+\Gamma}$ with
equality reached in the strong coupling limit. (When the first and last
manifold have different number of discrete states $N_{1}\neq N_{M}$, the
bandwidth is bounded above by $\frac{2\gamma\Gamma}{\gamma+\Gamma}({\rm
Min}\\{N_{1},N_{M}\\}+X)$ for a homogeneously decaying network with $X$ a
network-dependent number that is always less than $1/2$.)

_Group Delay.—_ We observe the same structural properties of the group delay
(Fig. 3.23) as we did for other networks; the frequencies with the largest
delays are those where oscillations in the transmission function are most
dense. We find that networks with non-identical manifolds can give rise to
group delays that are not strictly positive (Figs. 3.25 and 3.27).

Figure 3.24: Transmission probability for a hybrid network consisting of $3$
manifolds in series, the first with $2$ discrete states and the second and
third with $3$ discrete states, with balanced decays and uniform coupling
${g=\frac{\sqrt{\gamma\Gamma}}{2}}$ (the critical value for networks without
detuning). Within the first manifold, the discrete states have frequency
spacing $5\gamma$. Within the second and third manifolds, the discrete states
are detuned by $\frac{5}{2}\gamma$, resulting in relative detuning between the
manifolds. Since the couplings are no longer tuned to the critical parameters
for a detuned system, perfect transmission is only achieved at $6$
frequencies. Figure 3.25: Group delay for a hybrid network consisting of $3$
manifolds in series, the first with $2$ discrete states and the second and
third with $3$ discrete states, with balanced decays and uniform critical
coupling. Figure 3.26: Transmission probability for a hybrid network
consisting of $3$ manifolds in series, the first two with $2$ discrete states
and the third with $3$ discrete states, with balanced decays ${\gamma=\Gamma}$
and uniform coupling ${g=\sqrt{\gamma\Gamma}}$. Within the first and second
manifolds, the discrete states are spaced by ${\frac{5}{2}\gamma}$. Within the
third manifold, the discrete states are spaced by $7\gamma$. Now no
frequencies are perfectly transmitted, and finding couplings such that perfect
transmission is achieved becomes less trivial. Figure 3.27: Group delay for a
hybrid network consisting of $3$ manifolds in series, the first two with $2$
discrete states and the third with $3$ discrete states, with balanced decays
${\gamma=\Gamma}$ and uniform coupling ${g=\sqrt{\gamma\Gamma}}$, the
transmission probability for which is shown in Fig. 3.26.

###  3.4 Other Networks

####  3.4.1 General Two-Sided Networks

We now begin to extrapolate from the above analyses to a larger class of two-
sided networks. Both the series networks and hybrid networks we’ve discussed
have the key property of asymptotic irrelevance of the causal ordering of
discrete states in the strong coupling limit. This means that the
asymptotically strong-coupling behavior of these networks is entirely
described by fully parallel networks as in (3.19), and we can make several
statements that will apply to any network that has the same behavior (that is,
a network that resembles (3.19) in the strong coupling limit). From the
structure of (3.19) alone, we can bound the total number of peaks and troughs
of $|T(\omega)|^{2}$; we find there are at most $2N-1$ peaks and $2N-2$
troughs in the strong coupling limit and since the number of maxima never
decreases with increasing $g$, we can extrapolate these bounds to weakly
coupled systems as well. In transitioning from $2N-1$ peaks or constructive
interference to $N$ peaks of perfect transmission, we observe a pair-wise
merging of peaks (with one extra peak leftover when $N$ is odd). We also find
that the number of dips of zero transmission (perfect reflection) is at most
$\sum\limits_{k=1}^{M}(N_{k}-1)\leq N-1$ where the latter equality is only
reached for a completely parallel network or in the strong coupling limit.

We suspect that the above argument for upper bounds applies more generally
than just to the specific networks consisting of manifolds in series; that is,
there is a large class of networks that reduce to a parallel description in
the strong coupling limit131313It’s worth reiterating here that _all_ networks
reduce to (3.19) in the very-weak coupling limit
${2g_{ij}\ll\sqrt{\gamma_{i}\gamma_{j}}+\sqrt{\Gamma_{i}\Gamma_{j}}\,\forall
i,j}$. However, it is unproven! We cannot generalize from the above analysis
to a fully arbitrary two-sided network as there also are networks with
different topologies; for instance, we can consider networks with loops of
discrete states that exist outside the main chain of manifolds that connect
the two continua. Since it is not necessary for a photon to pass through the
loop of discrete states to make it through the network, these networks behave
differently in the strong coupling limit. We can make a further distinction
between disconnected loops (dead-ends where photons have to back track) and
connected loops (chains of manifolds that provide an alternate route to the
output continua). It is possible that connected loop networks behave more like
the loop-free networks discussed above in the strong-coupling limit, but with
their specific loop-structure encoded in $T(\omega)$ in unexpected ways.

####  3.4.2 Additional Continua

Figure 3.28: Input-output field operators for a black box network with one
additional side channel. This will give rise to losses that make perfect
transmission impossible, as well as dark counts if the side channel contains
excitations (i.e. thermal).

We have thus far only considered quantum networks with two continnua, and have
shown that perfect transmission through a general network structure is
possible. We will briefly analyze more general multi-port quantum networks
(Fig. 3.28) to illustrate how they lead to inefficiencies and dark counts.

Introducing a third continuum coupled to our network of discrete states at
rates $\mu_{i}$, we rewrite (3.16) in the form

$\displaystyle-i\Delta_{i}c_{i}(\omega)=-\sum\limits_{j}\,\left(\frac{\sqrt{\gamma_{i}\,\gamma_{j}}+\sqrt{\Gamma_{i}\,\Gamma_{j}}+\sqrt{\mu_{i}\,\mu_{j}}}{2}+ig_{i\,j}\right)c_{j}(\omega)$
$\displaystyle-\sqrt{\gamma_{i}}\,b_{in}(\omega)-\sqrt{\Gamma_{i}}\,a_{in}(\omega)-\sqrt{\mu_{i}}\,m_{in}(\omega),$
(3.31)

where we have introduced a new input field annihilation operator
$m_{in}(\omega)$ for the additional continuum (satisfying the canonical
commutation relations), satisfying the same form of input-output relations as
in (3.3)

$\displaystyle m_{out}(\omega)$
$\displaystyle=m_{in}(\omega)+\sum_{i}\,\sqrt{\mu_{i}}\,c_{i}(\omega).$ (3.32)

When there are side channels and all decay rates in the system are homogenous
($\mu_{i}=\mu$, $\gamma_{i}=\gamma$, and $\Gamma_{i}=\Gamma$), it is
impossible to achieve perfect transmission at any input frequency without
adding additional excitations (active filtering). To see this, consider taking
expectation values and imposing $m_{in}(\omega)\rightarrow 0$ over all
frequencies. Flux conservation requires that, for a perfectly transmitted
frequency $\omega^{\prime}$, we also have $m_{out}(\omega^{\prime})\rightarrow
0$. From (3.32), we see the only way to achieve this is for
$\sum_{i}c_{i}(\omega^{\prime})\rightarrow 0$ but from (3.3), we see that this
results in $a_{out}(\omega^{\prime})=a_{in}(\omega^{\prime})$ and
$b_{out}(\omega^{\prime})=b_{in}(\omega^{\prime})$—the frequency
$\omega^{\prime}$ is instead perfectly reflected by the system.

It is still possible that side channels with inhomogeneous coupling $\mu_{i}$
could yield a system that perfectly transmit some light at a frequency
$\omega^{\prime}$ satisfying both conditions
$a_{out}(\omega^{\prime})=a_{in}(\omega^{\prime})-\sum_{i}\,\sqrt{\gamma_{i}}c_{i}(\omega^{\prime})=0$
and $\sum_{i}\,\sqrt{\mu_{i}}c_{i}(\omega^{\prime})=0$. After all, the
$c_{i}(\omega^{\prime})$ with resonant frequencies above and below
$\omega^{\prime}$ generically differ in phase by $\pi$ and could in principle
cancel out. However, this would require _incredible_ fine-tuning of the
system. For a uniformly inhomogeneous parallel network such as (3.23) with a
side channel, we can easily see that this approach fails even in the low-loss
limit near resonance (where it would be most likely to succeed): we find
$m_{out}(\omega_{j})\approx\frac{2\mu_{j}}{\gamma_{j}+\Gamma_{j}}a_{in}(\omega_{j})$.
In general, output channels outside your control lead to loss.

Similarly, we can determine from (3.4.2) and (3.32) that, in general, input
channels outside your control lead to extra dark counts; photons that are in
initially populated side channel can end up in the monitored continuum. If the
side channel is at a finite temperature, there will be thermal photons that
could populate the system. The probability of this vanishes for
$k_{b}T\ll\hbar\omega_{i}\,\forall i$, but will generally be non-negligible
for high-temperature systems. The specific contribution depends on the
specific form of the full transfer matrix, which now will include a function
$D_{m}(\omega)$ that governs the probability of thermal excitations in the
side channel to end up in the monitored continuum.

Of course, the internal mode $b_{{\rm in}}$ may be occupied by thermal
excitations as well, but the contribution to dark counts by these will depend
strongly on the amplification mechanism. Furthermore, if only frequencies such
that $|T(\omega_{i})|=1$ are amplified, we find that thermal excitations
$b_{{\rm in}}$ do not contribute at all to dark counts as they always leak out
of the system, ending up in the continuum mode $a_{{\rm out}}$ populated by
reflected input photons.

###  3.5 Results

We have studied the behavior of quantum networks and have found that they
provide a diverse structure of transmission functions for modeling the first
stage of single-photon detectors: transmission of a single excitation from the
input continuum, through the system, to a monitored output continuum.
Inefficiencies and dark counts can be modeled through the incorporation of
additional continua (side channels). While we do not find fundamental limits
to transmission efficiency, spectral bandwidth, or frequency-dependent group
delay across all the studied networks (series, parallel, and hybrid), we do
observe that some networks are better suited to certain applications than
others, as we will now discuss.

_Perfect Transmission.—_ Often the most important metric for a photodetector
is photo detection efficiency, which for a quantum network is tantamount to
ensuring $|T(\omega_{i})|=1$ is achieved for some frequency or frequencies
$\omega_{i}$. The main conclusions of this chapter are as follows:

  * (i)

Ensuring the decay rates to the input and output continua are balanced
($\gamma=\Gamma$) guarantees perfect transmission for at least one frequency
in almost all quantum networks without loops, side channels, and detunings
[including all not covered by (ii)].

  * (ii)

Ensuring the couplings between manifolds are uniform and critical
($g=\frac{\sqrt{\gamma\Gamma}}{2}$) guarantees perfect transmission for at
least one frequency in almost all quantum networks without loops, side
channels, and detunings [including all not covered by (i)].

  * (iii)

For an arbitrarily detuned quantum network without loops and side channels, we
are _always_ able to find conditions for the couplings and decay rates such
that perfect transmission occurs for at least one frequency.

That finding conditions such that perfect transmission $|T(\omega_{i})|=1$ is
always possible indicates that perfect photo detection is possible in a wide
variety of physical systems.

If one additionally wants a broadened transmission spectrum at a _particular_
perfectly transmitted frequency—so that a broad range of frequencies is
detected almost perfectly—we similarly find a variety of ways to accomplish
it. One way is to use a parallel network distributed over a small range of
states. However, this will result in a number of dark states which will not be
detected at all. Instead, it is better to use a series network that meets both
the critical coupling and balanced decay conditions, resulting in a maximally
broadened on resonance transmission function as seen in Fig. 3.18.

_Spectral Bandwidth.—_ We do not find fundamental limits to the minimum or
maximum bandwidth of a network $\tilde{\Gamma}$ (or, conversely, to the
interaction time between a network and incident light $\tilde{\Gamma}^{-1}$),
but that it is generally proportional to the decay rates to both the input and
monitored continua ($\tilde{\Gamma}\propto\frac{\gamma\Gamma}{\gamma+\Gamma}$,
here assumed to be homogenous across states). We also find that some network
structures are more suited to high-bandwidth applications than others; for
networks with equivalent decay rates, we generally find that
$\tilde{\Gamma}_{\rm series}\leq\tilde{\Gamma}_{\rm
simple}\leq\tilde{\Gamma}_{\rm parallel}$. For a series network, equality with
the upper limit of $\tilde{\Gamma}_{\rm
simple}=\frac{2\gamma\Gamma}{\gamma+\Gamma}$ is reached only in the strong-
coupling limit, and the lower limit ($\tilde{\Gamma}=0$) corresponds to a
completely de-coupled or infinitely-detuned network (so that a photon can
never pass through). For a parallel network, the bandwidth is always given
$\tilde{\Gamma}_{\rm
parallel}=\sum_{i}=\frac{2\gamma_{i}\Gamma_{i}}{\gamma_{i}+\Gamma_{i}}$
regardless of detuning, so the lower limit simply corresponds to a single
discrete state (reproducing the simple model). Unlike both series and hybrid
networks where the spectral bandwidth decreases with detuning, the spectral
bandwidth of a parallel network is independent of detuning. This makes
parallel networks the better candidate for implementation of broadband single-
photon detection where the frequencies that need to be detected are distinct
(so that spectral hole-burning is a non-issue).

Considering hybrid networks, we find that their bandwidths are bounded above
and below by parallel and series networks, respectively; given a series
network with the same coupling strengths and manifold-number, and a parralel
network with the same manifold resonance structure and decay rates, the
bandwidth of a Hybrid network is bounded $\tilde{\Gamma}_{\rm
series}\leq\tilde{\Gamma}_{\rm hybrid}\leq\tilde{\Gamma}_{\rm parallel}$. The
upper limit for a hybrid network is approached only in the limit of strong
coupling, and the lower limit is approached when there is only a single
discrete state in each manifold.

_Group Delay.—_ The maximum magnitude of frequency-dependent group delay
$\tau_{g}(\omega)$ increases with both the couplings between discrete states
and the density of oscillations in the transmission function $T(\omega)$. In
particular, the bounded-box structure of a uniformly coupled series networks
yields large delays near $\omega=\pm 2g$. We observe that a negative group
delay may occur for series networks with relative detuning between the
discrete states. (For more on negative group delay, see Ref. [170].) For
general networks, the group delay can vary immensely over the range of
frequencies where $|T(\omega)|^{2}$ is non-negligible, with the effect being
strongest for hybrid networks (where the resonance structures can be most
dense). This means we can expect dispersion effects to be substantial in many
photo detection platforms. For applications such as frequency discriminating
delay-lines, we may expect hybrid networks to be the best performing candidate
as they allow for the finest control of group delay structure.

_Tradeoffs.—_ For an arbitrary series or hybrid network with arbitrary decay
rates, perfect transmission requires a particular choice of the couplings.
This critical value is generically of order $\sqrt{\frac{\gamma\Gamma}{2}}$.
However, the spectral bandwidth is maximized when
$g\gg\sqrt{\frac{\gamma\Gamma}{2}}$. Furthermore, the magnitude and location
of the maximum group delays depends strongly on the coupling $g$ (and
especially so in the high-$N$ limit). So for series and hybrid networks, there
is a clear tradeoff between efficient transmission and the spectral bandwidth
(which saturates in the high-$g$ limit)141414Of course, this assumes a large
spectral bandwidth is a desired. There is always an obvious tradeoff between
the spectral bandwidth and the amount one learns about the incoming light; the
more frequencies that can be detected efficiently, the less a single click
tells you about the incoming light. Whether a high bandwidth or a low
bandwidth is preferred will ultimately depend on the application of a SPD.,
with the group delay changing as well. We contrast this with the case of
parallel networks where the three quantities are completely independent:
perfect detection requires balanced decays ($\gamma=\Gamma$), the bandwidth
can be directly scaled by scaling both decay rates together, and the
frequency-dependent group delay $\tau_{g}(\omega)$ alone depends on the
detuning between discrete states, which is an independent quantity. So for
photodetectors where all three quantities must be determinable independently,
parallel networks are the best choice. The exception to this are situations
where a negative group delay is required, which parallel networks never
exhibit. Then the use of a hybrid or series network is unavoidable, as are
their accompanying tradeoffs.

_Final Remarks.—_ We have studied a variety of quantum networks to uncover
tradeoffs and limits fundamental to single-photon detection that may arise in
the first step of photo detection (transmission). The spectral bandwidth is
the only main quantity of interest where there appear to be fundamental limits
for particular classes of networks (parallel, series, and hybrid), but even
these scale with the relevant decay rates for the networks. Given the freedom
to adjust coupling strengths and decay rates, arbitrary group delay,
transmission efficiency, and spectral bandwidth are attainable for any network
so long as they are attained individually (together there may be tradeoffs for
series and hybrid networks).

We do not find fundamental limitations to dark counts and losses in this
analysis of transmission: in general uncontrolled input channels lead to dark
counts and uncontrolled output channels lead to loss. These can always be
mitigated by cooling the system ($k_{b}T\ll\hbar\omega_{i}\,\forall i$) and
reducing coupling to side channels
($\mu_{i},\,\nu_{i}\ll\gamma_{i}+\Gamma_{i}\,\forall i$). These side channels
can be as simple as material absorption, where a photon cab be absorbed by the
material of the photodetector and contributes to a heating process and where
the material of the photodetector acts as a thermal bath, generating potential
dark counts.

Other sources of noise, such as from signal amplification (for more on this
specifically, see Ref. [5]), classical parameter fluctuations, as well as the
noise inherent in a non-ideal quantum measurement (as described by an
arbitrary photo detection POVM) can also be included in our model to give a
fully quantum description of the entire single-photon detection process, as
will be done in the coming sections.

###  3.6 An Additional Toy Model POVM

In this chapter, we have not yet discussed in detail what happens to the
excitation after it ends up in the output mode $b_{{\rm out}}$, except that it
can give rise to a macroscopic photodetector click. Following Ref. [143], we
can construct an extremely simplified POVM from $T(\omega)$ by assuming _any_
excitation in the output continuum will lead to a click (that is, that the
continuum is monitored). This single-step POVM will not be realistic151515In
particular, the inclusion of a monitored continuum of states leads to a
divergent dark count rate., but it has pedagogical value in highlighting the
role of $T(\omega)$ in our work.

Assume that we finalize the photo detection process by ascertaining at some
time $t$ whether an excitation is indeed in the continuum $b_{{\rm out}}$ and
also that the amplification is ideal and lossless so that, if a photon makes
it to the monitored continuum, it will be detected. We can then define
normalized filtered photon states (following Ref. [143])

$\displaystyle\left|T\,\phi_{t}\right\rangle=\frac{1}{\sqrt{\pi\,\tilde{\Gamma}}}\int_{0}^{\infty}\,d\omega\,T^{*}(\omega)\,e^{i\,\omega\,t}\,\hat{a}^{\dagger}(\omega)\,\left|\textnormal{vac}\right\rangle.$
(3.33)

From the quantum jump method [171, 89], we know that a quantum jump from the
final manifold of discrete states to the monitored output continuum will occur
in an infinitesimal time $dt$ with conditional probability
$\frac{\tilde{\Gamma}\,dt}{2}$. We then infer the POVM element for detecting a
photon at a particular time $t$ after the photodetector has been on for a time
$dt$

$\displaystyle\hat{\Pi}_{t}=\frac{\tilde{\Gamma}\,dt}{2}\,\left|T\,\phi_{t}\right\rangle\left\langle
T\,\phi_{t}\right|.$ (3.34)

(To reiterate, here $t$ refers to a possible time of detection analogous to
the time $T$ from the previous chapter, and does not refer to the time
evolution of the input state.) The probability of getting a click for a
normalized input photon $\hat{\rho}$ is
$\textnormal{Tr}\left(\hat{\Pi}_{t}\hat{\rho}\right)\leq 1$. So the input
state that will be detected with maximum probability is
$\hat{\rho}=\left|T\,\phi_{t}\right\rangle\left\langle T\,\phi_{t}\right|$,
yielding an infinitesimal probability of detection of
$\frac{\tilde{\Gamma}\,dt}{2}$. This is because this assumes that the
detection events at times separated by a time $dt$ correspond to different
measurement outcomes which is highly idealized; any realistic photo detection
outcome corresponds to detection within an integrated time-window.
Nonetheless, (3.34) provides a POVM description of a continuous measurement
process, and can be used to construct a partition of unity.

To take a finite time-window into account, we consider a time-integrated POVM
element $\hat{\Pi}_{\tau}$, where a click corresponds to an excitation
entering the final output continuum sometime between $t=T_{0}$ when we first
turn on the detector and $t=T_{0}+\tau$. Then the time-integrated POVM element
is

$\displaystyle\hat{\Pi}_{\tau}=\int_{T_{0}}^{T_{0}+\tau}\,dt\,\frac{\tilde{\Gamma}}{2}\,\left|T\,\phi_{t}\right\rangle\left\langle
T\,\phi_{t}\right|$
$\displaystyle\approx\int_{-\infty}^{\infty}\,d\omega\,|T(\omega)|^{2}\left|\omega\right\rangle\left\langle\omega\right|\,\,\,\,\,\,(\tau\gg\tilde{\Gamma}^{-1})$
(3.35)

where, for $\tau\rightarrow\infty$, the projectors
$\left|\omega\right\rangle\left\langle\omega\right|$ are truly monochromatic
because no timing information is obtained. With the time-integrated form in
(3.6), we can see that the conditions for perfect transmission discussed above
correspond to perfect detection in the limit of $\tau\gg\tilde{\Gamma}^{-1}$
as we used previously in (3.6). Physically, a non-infinitesimal integration
time $\tau$ means a photon incident on the photodetector at a time $t=-\tau$
has had sufficient time to propagate through the network before we end the
integration at time $t=0$ and check whether there has been a click. This is in
agreement with our observation that, for the simple model in (3.8),
$\tau_{g}(\omega_{0})=\tilde{\Gamma}^{-1}$: simply, we must at least allow
enough time for an on-resonance photon to travel through the network (interact
with the device) to achieve perfect detection of a monochromatic on-resonance
photon. (3.6) is also illustrative of what the POVM element represents: the
information a detector click reveals about what led up to it.

We can also use (3.6) to construct the POVM element for not getting a click in
the finite time $\tau$, which is simply
$\hat{\Pi}_{0}=\hat{1}-\hat{\Pi}_{\tau}$, such that the full POVM
$\\{\hat{\Pi}_{0},\,\hat{\Pi}_{\tau}\\}$ forms a partition of unity for the
relevant Hilbert space; that is, the Hilbert space spanned by single-photon
states and the vacuum. (See Ref. [98] for inclusion of general photon Fock
states in the photo detection POVM.) This POVM corresponds to a photodetector
that is reset after each integration time $\tau$. Similarly, one can divide up
the the full detection window into $N$ time intervals $\tau_{i}$ without
resetting the device in between so that the full POVM is
$\\{\hat{\Pi}_{0},\,\hat{\Pi}_{\tau_{i}}\\}_{i=1...N}$, reproducing a
continuous measurement POVM in the limit $\tau_{i}\rightarrow 0$. In this
case, the measurement outcomes are not orthogonal [143]; even if we know a
photon is incident at a definite time $t^{\prime}$ we cannot predict with
certainty when it will be detected.

##  Chapter 4 Amplification

In this chapter, we show that detection of single photons is not subject to
the fundamental limitations that accompany quantum linear amplification of
bosonic mode amplitudes, even though a photodetector does amplify a few-photon
input signal to a macroscopic output signal. Alternative limits are derived
for _nonlinear_ photon-number amplification schemes with optimistic
implications for single-photon detection. Four commutator-preserving
transformations are presented: one idealized (which is optimal) and three more
realistic (less than optimal). Our description makes clear that nonlinear
amplification takes place, in general, at a different frequency
$\omega^{\prime}$ than the frequency $\omega$ of the input photons. This can
be exploited to suppress thermal noise and dark counts past what is possible
with linear amplification up to a fundamental limit imposed by nonlinear
amplification into a single bosonic mode.

###  4.1 Quantum amplification and noise

The fundamental relations between quantum noise and quantum amplification are
most straightforwardly derived in the Heisenberg picture. Thus, the standard
way [172] to describe linear phase-preserving quantum amplification of a
bosonic mode amplitude $a$ is through Caves’ relation for the annihilation
operator $\hat{a}$,

$\displaystyle\hat{a}_{{\rm out}}=\sqrt{G}\hat{a}_{{\rm
in}}+\sqrt{G-1}\hat{b}^{\dagger}_{{\rm in}},$ (4.1)

where $\hat{b}^{\dagger}$ is the creation operator corresponding to an
independent auxiliary bosonic mode $b$. Here the input field amplitude of mode
$a$ is amplified by a factor of $\sqrt{G}$, but there is a cost: extra noise
arising from the additional mode $b$111This could be a mode internal to the
detector.. If this mode contains (thermal) excitations, mode $a$ after
amplification will contain excitations, too, and their number is multiplied by
$G-1$. Even if mode $b$ is in the vacuum state, it still adds noise [172]. It
is clear that this extra noise is due to the additional creation operator term
proportional to $\sqrt{G-1}$ in Eq. (4.1), but since that term is necessary so
as to preserve the standard bosonic commutation relation $[\hat{a}_{{\rm
out}},\hat{a}^{\dagger}_{{\rm out}}]=\mathds{1}$ this tradeoff between linear
amplification and added noise is fundamental. Indeed, phase-preserving linear
amplification in proposed number resolving platforms using superconducting
qubits have noise bounded by the Caves limit [173, 52] and similar bounds are
found for linear amplification used in quantum circuits [174].

Recently, there has been some effort to describe all parts of the
photodetection process, including amplification [175], fully quantum
mechanically [176, 26, 98, 24]. One conclusion that may be drawn from that
research is that there is no severe amplification-driven tradeoff between
efficiency and (thermally induced) dark counts. In particular, even though a
few-photon signal must be amplified to a macroscopic level [forcing us to
consider $G\gg 1$], thermal fluctuations in internal detector modes apparently
do not get amplified by the same factor of $G$. Experiments [22, 23] on
superconducting nanowires demonstrate that over a wide range of detected
wavelengths dark count rates can indeed be extremely low (on the order of one
dark count per day). How can we reconcile these results with that of the
previous paragraph?

The answer, as we will show, is that amplification is not necessarily linear.
That is, in the Heisenberg picture, the transformation of the bosonic
annihilation operator can be nonlinear while still preserving the bosonic
commutation relation. And, perhaps surprisingly, that way of amplifying can
decrease the amount of noise added. Previous models of nonlinear amplification
were developed in a different context, that of amplifying superpositions of
different number states [177, 178, 179, 180, 181, 182]. The relevant figure-
of-merit there was the signal to noise ratio of the output relative to that of
the input, and noise from the auxiliary modes was ignored. Here, in contrast,
we are interested exclusively in the noise in the output and especially in the
contributions arising from thermal fluctuations in the auxiliary modes, as
functions of the gain factor $G$. In order to explore this issue, we construct
physical transformations that implement nonlinear amplification satisfying the
correct mathematical requirements which, to the authors’ knowledge, have not
appeared in the literature.

###  4.2 Nonlinear amplification

The idea is that for detecting single photons it is sufficient to have an
output field whose total number of excitations is given by $N_{\rm out}=N_{\rm
in}+Gn_{a}$ with $n_{a}$ the number of input photons we would like to detect,
and $N_{\rm in}$ the (fluctuating) number of excitations present in the output
mode prior to amplification, which is itself not amplified. A physically
allowed but highly idealized unitary transformation that accomplishes this is
easiest written down in the Schrödinger picture as

$\displaystyle\left|n\right\rangle_{a}\left|M\right\rangle_{1}\left|N\right\rangle_{2}\longmapsto\left|n\right\rangle_{a}\left|M-Gn\right\rangle_{1}\left|N+Gn\right\rangle_{2}.$
(4.2)

This is valid for any number $n$ of input photons222The transformation in Eq.
(4.2) can be realized only when ${M\geq Gn}$. There is always such a
restriction on amplification relations; the energy transferred to reservoir 2
must come from somewhere., even though in practice we will be interested
mainly in small values of $n$, say, $n=0,1,2$).

All states here are number (Fock) states of bosonic modes and as written this
transformation is superposition-preserving. The transformation involves two
energy reservoirs: energy is transferred from the first reservoir to the
second with the amount of energy transferred determined by the number $n$ of
input photons in mode $a$ (with nothing happening at all when $n=0$). The
assumption is that excitations of the two reservoirs have identical energies,
$\hbar\omega^{\prime}$, such that energy is conserved. The input mode can have
any frequency $\omega$. The second reservoir ideally starts out with $N=0$
excitations—corresponding to the zero temperature limit—such that in the end
it would contain exactly $Gn$ excitations if the input field contained $n$
photons. Clearly, this ideal transformation would represent perfect
(noiseless) amplification of a photon number state (and $G$ will have to be an
integer for this to work).

Figure 4.1: An input photon with frequency $\omega$ undergoes amplification
into a macroscopic signal via electron-shelving [31, 183, 184]: when an on-
resonance photon is absorbed, an atom (modeled here as a three-level system)
enters the first excited state. This is followed by a laser beam tuned to the
second transition frequency ${\omega_{L}\approx\omega}^{\prime}$, which
induces fluorescence. If there are multiple input photons, they are absorbed
by multiple atoms and the fluorescence signal is increased proportionally. The
number of fluorescence modes may be reduced by placing a high-Q cavity around
the atom so that amplification moves towards the ideal transformation given in
Eq. 4.2.

Now we wish to describe this ideal process in the Heisenberg picture so as to
make a direct comparison with Eq. (4.1). In that picture, the ideal
transformation that is linear in the number operator for the excitations in
the second reservoir should be333The construction of physically implementable
transformations such as Eq. (4.3) (and transformations including higher powers
of the input photon number operator) are highly constrained by two conditions:
that the spectrum of the operator-representation be $\mathbb{N}^{0}$ (the
natural numbers including zero) and that the commutator be preserved [178].
From these conditions, in Eq. (4.3) we need _at least_ one term with a number
operator on the right with a prefactor of one. Additional reservoirs with
arbitrary prefactors are allowed but they will carry additional noise and
decrease the SNR.

$\hat{b}^{\dagger}_{\rm out}\hat{b}_{\rm out}=\hat{b}^{\dagger}_{\rm
in}\hat{b}_{\rm in}+G\hat{a}^{\dagger}_{\rm in}\hat{a}_{\rm in}.$ (4.3)

We are now going to do three things: (A) we will construct expressions for
$\hat{b}^{\dagger}_{\rm out}$ and $\hat{b}_{\rm out}$ such that Eq. (4.3) is
reproduced and such that their commutator $[\hat{b}^{\dagger}_{\rm
out},\hat{b}_{\rm out}]=\mathds{1}$; (B) we will add non-ideal features that
make the model more realistic, and (C) we will include fluctuations in the
initial number of excitations in the $b$ mode and calculate the signal-to-
noise ratio (SNR) for the final number of excitations in the $b$ mode, both
for the ideal limit and the more realistic models. A comparison with linear
amplification will then show how nonlinear amplification improves upon the
former.

To start with part of task (B), we adjust the idealized Schrödinger picture to
get rid of two features that make the process in Eq. (4.2) obviously
inapplicable to real detectors, but such that the Heisenberg picture Eq. (4.3)
is still valid. First note that the $n$ photons in the process in Eq. (4.2)
are not destroyed, whereas in a standard detector they are. We fix that by
introducing another quantum system $S$ with a continuum of energies $E$ that
can absorb the energy $n\hbar\omega$ of the $n$ photons. This modifies Eq.
(4.2) by adding a step (potentially at a later time)

$\left|n\right\rangle_{a}\left|E\right\rangle_{S}\longmapsto\left|0\right\rangle_{a}\left|E+n\hbar\omega\right\rangle_{S}.$
(4.4)

Since this extra step does not affect the state of reservoir 2, the crucial
expression Eq. (4.3) is unchanged when the continuum of energies absorbing the
energy is traced over. The second change concerns phase: in the Schrödinger
picture we can insert random phase factors $\exp(i\phi^{\mbox{$\clubsuit$}})$
on the right-hand side of Eq. (4.2). Critically, this is not a global phase
but a local phase factor, that will be different even for the same Fock state.
Thus, the amplification process destroys superpositions of different number
states (e.g., coherent states will not be coherently amplified) and the
process is now also irreversible (as any amplification process in a real
detector is). It destroys any entanglement between the different modes as
well444Phase randomization is necessary for optimal amplification and
measurement of photon number due to number-phase uncertainty. Indeed,
amplification of photon number deamplifies phase and vice versa, see [181]..

For task (A) we would like to use the polar decompositions of the creation and
annihilation operators. That is, in analogy to the polar decomposition of a
complex number, $z=\exp(i\phi)\sqrt{|z|^{2}}$, we would like to write

$\displaystyle\hat{b}_{{\rm out}}$
$\displaystyle=\hat{S}\,\sqrt{(\hat{b}^{\dagger}\hat{b})_{{\rm
in}}+G(\hat{a}^{\dagger}\hat{a})_{{\rm in}}},$ (4.5)

where $\hat{S}$ is a unitary operator which could be written in the suggestive
form $\exp(i\hat{\phi})$ for some hermitian operator $\hat{\phi}$. In a
finite-dimensional Hilbert space of dimension $s+1$ there is no problem
defining $\hat{S}$: it is a shift operator that acts on number states
$\left|N\right\rangle$ of the bosonic mode as

$\displaystyle\hat{S}\left|N\right\rangle=e^{i\phi^{\mbox{$\clubsuit$}}}\left|N-1\right\rangle\,\,\,{\rm
for}\,\,s\geq N>0,$ (4.6)

with $\hat{S}\left|0\right\rangle=\left|s\right\rangle$ and
$\phi^{\mbox{$\clubsuit$}}$ the random phase we introduced earlier. Since Fock
space is infinite-dimensional, we use the Pegg-Barnett trick [185] of
truncating the Hilbert space at a high excitation number $s$ and only in the
end (when calculating physical quantities) taking the limit
$s\rightarrow\infty$. It is easy to verify that the relation Eq. (4.5) yields
the commutator $[\hat{b}_{{\rm out}},\hat{b}^{\dagger}_{{\rm
out}}]=\mathds{1}_{{\rm in}}-(s+1)\left|s\right\rangle\left\langle s\right|$,
in which the extra Pegg-Barnett term won’t contribute to any physical
quantity, while ensuring a traceless commutator, necessary in finite
dimensions555Note the dimension of both input and output mode Hilbert spaces
is ${s+1}$; they necessarily match in the Heisenberg picture..

Construction of Hamiltonians that implement nonlinear photon number
amplification was done elsewhere (see [180, 182]), but they are far more
complicated than the Hamiltonians implementing linear amplification; in the
interaction picture phase-preserving linear amplification is implemented with
the simple interaction Hamiltonian

$\displaystyle\hat{H}=i\hbar\kappa(\hat{a}\hat{b}-\hat{a}^{\dagger}\hat{b}^{\dagger})$
(4.7)

which amplifies via two-mode quadrature squeezing (as introduced in Ref.
[186]) with a (non-integer) time-dependent gain factor $G=\cosh(\kappa t)$
with $\kappa$ the coupling strength and $t$ the interaction time [187]. In
contrast, an interaction Hamiltonian implementing nonlinear amplification even
for a modest gain of $G=5$ has $40$ terms (fewer terms for smaller $G$, more
terms for larger $G$) [182]. Since we are only interested in the output signal
and not the details of the evolution of other operators, we will not construct
a system Hamiltonian; indeed, the same transformation on the output signal
could be given by different Hamiltonians corresponding to different physical
implementations of photon number amplification that give the same overall gain
factor.

The nonlinear expression Eq. (4.5) does not seem to have appeared in the large
literature on bosonic amplification (for a review, see, e.g., [52]). [177,
178, 179] did discuss photon-number amplifiers (especially in the high-photon
number limit) decades ago, but no attempt was made there to find commutator-
preserving operator equations.

###  4.3 More realistic models for amplification

Continuing with task (B), in a more realistic description the reservoirs
consist of many modes. So, instead of having just one bosonic output mode $b$
we really should describe many output reservoir modes. For example, we may
have $G$ modes $b_{k}$ [recall that for non-linear amplification $G$ must be
an integer] each one of which satisfies

$\displaystyle\hat{b}_{k\,{\rm out}}$
$\displaystyle=\hat{S}_{k}\,\sqrt{(\hat{b}^{\dagger}\hat{b})_{k\,{\rm
in}}+(\hat{a}^{\dagger}\hat{a})_{{\rm in}}},\,\,\,\,\,k=1\ldots G.$ (4.8)

Here the macroscopic signal monitored and analyzed consists of the sum of all
detected excitations (since each mode by itself contains just a microscopic
number of excitations we cannot simply assume to be able to count those
individual numbers: then we would not need amplification at all!). That is, we
consider as our macroscopic output signal

$\displaystyle\hat{I}_{{\rm
out}}=\sum_{k=1}^{G}(\hat{b}^{\dagger}\hat{b})_{k\,{\rm
out}}=\sum_{k=1}^{G}(\hat{b}^{\dagger}\hat{b})_{k\,{\rm
in}}+G(\hat{a}^{\dagger}\hat{a})_{{\rm in}}.$ (4.9)

Another extension is to “avalanche” photodetection where one small-scale
amplification event triggers the next and the process repeats, giving rise to
a macroscopic signal. Iterating the transformation Eq. (4.3) of single mode
amplification $N$ times with a gain factor $g$ in each step gives a total gain
factor $G=g^{N}$ and an input-output relation

$(\hat{b}^{\dagger}\hat{b})_{\rm
N\,out}=\sum_{k=1}^{N}g^{N-k}(\hat{b}^{\dagger}\hat{b})_{k\,{\rm
in}}+G(\hat{a}^{\dagger}\hat{a})_{{\rm in}}$ (4.10)

where mode $\hat{b}_{k}$ here contains the output of the $k$th amplification
step, and the last mode $b_{N}$ contains the signal.

Another extension, relevant for $n>1$, describes multiplexing: the idea is
that $n$ photons are most conveniently detected by $n$ detectors that each
detect one (and only one) photon, along the lines of [188, 189]. We will not
describe this model in any detail, except to state that amplification would in
that case be described by $Gn$ modes, each containing exactly one extra
excitation.

Lastly, we combine both multi-mode and multi-step extensions above by
repeating the process in Eqs. (4.8) and (4.9) of amplification into several
($g$) modes $N$ times, again with a total gain factor defined $G=g^{n}$ and an
input-output relation for the macroscopic signal

(4.11) $\displaystyle\hat{I}_{{\rm
out}}=\sum_{k_{N}=1}^{G}(\hat{b}^{\dagger}\hat{b})_{k_{N}\,{\rm
out}}=\sum\limits_{n=1}^{N}\sum\limits_{k_{n}=1}^{g^{n}}(\hat{b}^{\dagger}\hat{b})_{k_{n}\,{\rm
in}}+G(\hat{a}^{\dagger}\hat{a})_{{\rm in}}$

where the mode $\hat{b}_{k_{n}}$ is the $k_{n}$th mode in the $n$th step.

Note that in our nonlinear amplification models the amplified signal ends up
in a different bosonic mode or modes: indeed, a photodetector typically
converts the input signal (light) to an output signal of a physically
different type, e.g., electron-hole pairs (which may sometimes be approximated
as composite bosons; see also [190, 191, 192, 193]).

###  4.4 Number fluctuations

We turn to task (C) and calculate the noise in photon number introduced by the
amplification process and by the coupling to reservoirs. For the reservoir we
monitor, we write

$\displaystyle\left\langle(\hat{b}^{\dagger}\hat{b})_{\rm{in}}\right\rangle=\overline{n}_{b};\,\left\langle(\hat{b}^{\dagger}\hat{b})^{2}_{\rm{in}}\right\rangle=\overline{n}^{2}_{b}+\Delta
n_{b}^{2}$ (4.12)

and make no further assumptions about its initial state.

We assume that there is some (unknown) number of photons in the input mode $a$
that we want to measure. We thus consider input states that are diagonal in
the photon number basis, with some nonzero photon number fluctuations $\Delta
n_{a}$. (Thanks to the randomized phase assumption we can use this assumption
without loss of generality for our nonlinear models.) So, we write

$\displaystyle\left\langle(\hat{a}^{\dagger}\hat{a})_{\rm{in}}\right\rangle=\overline{n}_{a};\,\left\langle(\hat{a}^{\dagger}\hat{a})^{2}_{\rm{in}}\right\rangle=\overline{n}^{2}_{a}+\Delta
n_{a}^{2}.$ (4.13)

In the following we always assume the initial states of modes $a$ and $b$ to
be independent, such that

$\left\langle
f(\hat{a},\hat{a}^{\dagger})g(\hat{b},\hat{b}^{\dagger})\right\rangle=\left\langle
f(\hat{a},\hat{a}^{\dagger})\right\rangle\left\langle
g(\hat{b},\hat{b}^{\dagger})\right\rangle$ (4.14)

for any functions $f$ and $g$.

For linear phase-insensitive amplification Eq. (4.1), we find the following
variance in the number of excitations in the amplified signal:

$\displaystyle\sigma_{(\hat{a}^{\dagger}\hat{a})_{\rm out}}^{2}$
$\displaystyle=$ $\displaystyle G^{2}\Delta n_{a}^{2}+(G-1)^{2}\Delta
n_{b}^{2}+$ (4.15) $\displaystyle
G(G-1)(2\overline{n}_{a}\,\overline{n}_{b}+\overline{n}_{a}+\overline{n}_{b}+1).$

We observe that in addition to amplification of the thermal fluctuations in
the auxiliary mode $b$ [the second term in Eq. (4.15)], even at zero
temperature where $\Delta n_{b}^{2}=0$ there is inherent noise from the
amplification process itself [the second line is strictly positive for $G>1$].

We should also consider linear phase-sensitive amplification [172], described
by

$\displaystyle\hat{a}_{{\rm out}}=\sqrt{G}\hat{a}_{{\rm
in}}+\sqrt{G-1}\hat{a}^{\dagger}_{{\rm in}}.$ (4.16)

Here, compared to Eq. (4.1) the $\hat{b}^{\dagger}$ term is replaced by the
$\hat{a}^{\dagger}$ term, such that the commutator $[\hat{a}_{{\rm
out}},\hat{a}^{\dagger}_{{\rm out}}]$ is still preserved. This gives a
variance

$\displaystyle\\!\\!\sigma_{(\hat{a}^{\dagger}\hat{a})_{\rm out}}^{2}$
$\displaystyle=$ $\displaystyle(6G(G-1)+1)\Delta n_{a}^{2}+$ (4.17)
$\displaystyle 2G(G-1)(\overline{n}^{2}_{a}+\overline{n}_{a}+1).$

There is again extra amplification noise for $G>1$ [the second line], much
like what we found for phase-insensitive amplification.

We compare these two results for linear amplification to the result for the
nonlinear amplification process described by Eq. (4.5). The variance in
excitation number is

$\displaystyle\sigma_{(\hat{b}^{\dagger}\hat{b})_{\rm out}}^{2}$
$\displaystyle=\Delta n_{b}^{2}+G^{2}\Delta n^{2}_{a}.$ (4.18)

Here the number fluctuations in the auxiliary mode are not amplified and there
is no additional amplification noise either; if the input field is in a number
state such that $\Delta n_{a}=0$, the gain factor $G$ will not effect the
noise at all. This is a result of the reservoir and auxiliary mode being
coupled conditionally on the presence of the input field; the transformation
Eq. (4.2) _always_ transfers exactly $Gn$ excitations to the auxiliary mode,
leaving the noise in the auxiliary mode unchanged when $\Delta n_{a}=0$. In
contrast, linear amplification _never_ adds a definite number of excitations
to the auxiliary mode because the input mode and the auxiliary mode are
directly coupled; even at zero temperature there is inherent noise from the
amplification of the auxiliary mode amplitude by a factor $\sqrt{G-1}$ [see
Eq. (4.1)]. For linear amplification this is necessary so as to preserve the
spectrum of the number operator [which non-linear amplification through Eq.
(4.3) does by construction].

Already we are able to see that the scheme of amplification into a single mode
is optimal; any transformation that would reduce the prefactor of $\Delta
n_{b}^{2}$ in Eq. (4.18) below unity would fail to realize a well-behaved
annihilation operator (for details, see again33footnotemark: 3)!

For nonlinear amplification into many modes described by the more realistic
model in Eqs. (4.8) and (4.9), we find

$\displaystyle\sigma_{\hat{I}_{\rm out}}^{2}$ $\displaystyle=G\Delta
n_{b}^{2}+G^{2}\Delta n^{2}_{a},$ (4.19)

where for simplicity we assumed all reservoir modes to be independent with the
same number fluctuations and we have defined $\hat{I}_{\rm out}$ as the
integrated signal in Eq. (4.11). This shows amplifying according to Eq. (4.8)
is suboptimal; even though it still beats both linear amplification limits
Eqs. (4.15) and (4.17) the noise in the reservoir modes is still amplified.
Similarly, amplification using multiple fermionic degrees of freedom will be
sub-optimal; a similar multi-mode description will be necessary666One way
around this limitation is for the incident photons to only interact with a
single symmetrized collective degree of freedom of many fermions, on which a
measurement is then made. In this idealized case, this collective degree of
freedom plays the role of a single bosonic mode and amplification could still
be described by Eq. (4.5) and photon number amplification is improved past the
limit for linear fermionic amplification [174]..

Defining the total gain $g^{N}=G$ with $N$ the number of steps, we find for
our multi-step models that

$\displaystyle\sigma_{(\hat{b}^{\dagger}\hat{b})_{\rm
out}}^{2}=\frac{G^{2}-1}{g^{2}-1}\Delta n^{2}_{b}+G^{2}\Delta n^{2}_{a}$
(4.20)

for amplification of $g$ excitations into a single mode and

$\displaystyle\sigma_{\hat{I}_{\rm out}}^{2}=G\frac{G-1}{g-1}\Delta
n^{2}_{b}+G^{2}\Delta n^{2}_{a}$ (4.21)

for amplification of a single excitation into $g$ modes. (Note Eqs. (4.20) and
(4.21) reduce to Eqs. (4.18) and (4.19) for $g=G$.)

###  4.5 Signal-to-noise ratios

We can now write down explicit tradeoff relations between amplification and
number fluctuations in terms of signal-to-noise ratios for all types of
amplification discussed here, for the case where the number of input photons
is fixed to be $n_{a}$ (and so $\Delta n_{a}=0$). Using the standard signal-
to-noise ratio as the number of excitations in the amplified mode minus the
background, divided by the standard deviation in the number of excitations, we
get

$\displaystyle\textnormal{SNR}_{{\rm PhaseInsensitive}}$ $\displaystyle\leq$
$\displaystyle\frac{G}{G-1}\frac{n_{a}}{\Delta n_{b}}$ (4.22)
$\displaystyle\textnormal{SNR}_{{\rm PhaseSensitive}}$ $\displaystyle\leq$
$\displaystyle\frac{2G-1}{\sqrt{2G(G-1)}}n_{a}$ (4.23)
$\displaystyle\textnormal{SNR}_{{\rm SingleMode}}$ $\displaystyle=$
$\displaystyle\frac{Gn_{a}}{\Delta n_{b}}$ (4.24)
$\displaystyle\textnormal{SNR}_{{\rm GModes}}$ $\displaystyle=$
$\displaystyle\frac{Gn_{a}}{\sqrt{G}\Delta n_{b}}=\frac{\sqrt{G}n_{a}}{\Delta
n_{b}}.$ (4.25)

The linear amplification mechanisms have increasingly worse signal-to-noise
ratios as $G$ increases777The signal-to-noise ratios Eqs. (4.22) and (4.23)
for linear amplification become infinite at ${G=1}$ simply because there is no
noise when both ${G=1}$ and ${\Delta n_{a}=0}$., albeit saturating in the
limit $G\rightarrow\infty$. In contrast, the signal-to-noise ratios for the
nonlinear amplification mechanisms improve with increasing $G$, with
amplification into a single-mode performing best, with an improvement by a
factor of $G$ over an unamplified signal888We find the linear dependence on
$G$ resulting from single-shot single-mode amplification holds for
transformations describing higher-order amplification of photon number
operator, again subject to the constraints of [23].. Amplification into $G$
modes performs worse than single mode amplification, showing an improvement
over an unamplified signal by a factor of $\sqrt{G}$. In the large $G$ limit,
this can be considered the well-known $1/\sqrt{G}$ reduction in the standard
error; while there are $G$ noisy modes, they do each carry the signal we wish
to measure so the total noise is reduced by a factor of $\sqrt{G}$.

Figure 4.2: The signal-to-noise ratio results we have derived are plotted
logarithmically. Here, the linear amplification scheme plotted is the phase
insensitive one (Eq. 4.22). The dashed lines correspond to the multi-step
amplification schemes in Eq. 4.26 and Eq. 4.27 and perform worse than their
single-step counterparts. The fundamental limit is reached only by the single
mode amplification scheme (Eq. 4.24) implementing ideal amplification. The
trade-space is further filled in by considering additional noise modes that
are not occupied by excitations deterministically.

Similarly, we consider multi-step amplification models

$\displaystyle\textnormal{SNR}_{{\rm MultiStepSingleMode}}$ $\displaystyle=$
$\displaystyle\frac{G\sqrt{g^{2}-1}n_{a}}{\sqrt{G^{2}-1}\Delta n_{b}}$ (4.26)
$\displaystyle\textnormal{SNR}_{{\rm MultiStepMultiMode}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{G(g-1)}n_{a}}{\sqrt{G-1}\Delta n_{b}}.$ (4.27)

These intermediate noise limits fill in the space between the optimal SNR in
Eq. (4.24) and linear amplification as shown in Fig. 4.2999The space is
further filled in by considering nonlinear amplification where $G$ excitations
are distributed into ${G^{\prime}>G}$ modes so that ancillary modes contribute
only to the noise and not to the signal. In this case, we find that the SNR
goes to $0$ as ${G^{\prime}\rightarrow\infty}$; the effect of additional noise
modes is always to reduce the SNR and move us away from the optimal SNR in Eq.
(4.24).. (Indeed, multi-step multi-mode amplification performs slightly
_worse_ than both linear mechanisms for $g=2$!)

In general, nonlinear amplification enables SNRs that, as functions of the
total gain factor $G$, do not saturate with the gain $G$ and outperform what
is possible with linear amplification. This is most apparent in the low
temperature limit where the lack of zero-temperature noise for non-linear
amplification (due to amplification taking place at a different frequency) is
most prominent.

###  4.6 Single-photon pre-amplification

While this chapter focuses on the amplification part of the photodetection
process, we now very briefly consider the pre-amplification process. We
certainly cannot treat that part in full generality here and we adopt several
simplifications in order to arrive at an important result concerning the
suppression of thermal noise. First, we assume that we can decouple the
amplification stage from the pre-amplification filtering [by having an
irreversible step in between the two] such that filtering does not interfere
negatively with the absorption/transduction part [26]. We then focus on just
the time/frequency degree of one incoming photon. A single absorber with some
resonance frequency $\omega_{0}$ able to absorb that single photon will act as
a frequency filter. If the pre-amplification filtering is passive (easy to
implement, but we certainly can go beyond this101010See, for example, [194].
The result is that, instead of certain frequencies, it is certain spectral
“Schmidt modes” that are detected perfectly.) and unitary (i.e., lossless: we
consider this because we are interested in the fundamental limits of
photodetection. Internal losses only degrade performance.), then frequency
filtering is described by the linear transformation

$\hat{a}_{{\rm out}}(\omega)=T(\omega)\hat{a}_{{\rm
in}}(\omega)+R(\omega)\hat{c}_{{\rm in}}(\omega)$ (4.28)

where $c_{{\rm in}}(\omega)$ is yet another internal bosonic detector mode at
the same frequency as the input mode111111 Photon-number resolved
photodetection can be achieved by multiplexing an $n$-photon signal to many
(${N\gg n}$) single-photon detectors [188], each satisfying Eq. (4.28)
independently. However, this means an additional noise mode will be added with
each splitting of the signal, decreasing the integrated signal-to-noise ratio.
To avoid added noise a nonlinear multi-photon filtering process could be used,
but for this a full S-matrix treatment must be used, see [135, 136, 137]..
Here $T(\omega)$ and $R(\omega)$ are “transmission” and “reflection”
coefficients which satisfy $|T(\omega)|^{2}+|R(\omega)|^{2}=1$ and which are
determined by the resonance structures internal to the photodetector. The
amplification process that follows the initial absorption of the photon energy
is applied to the operator $\hat{a}_{\rm out}(\omega)$ of Eq. (4.28), so that
(explicitly displaying the different frequencies of the modes now) ideal
amplification (single-mode and single-shot) is described by

$\\!\hat{b}^{\dagger}_{\rm out}(\omega^{\prime})\hat{b}_{\rm
out}(\omega^{\prime})=\hat{b}^{\dagger}_{\rm in}(\omega^{\prime})\hat{b}_{\rm
in}(\omega^{\prime})+G\hat{a}^{\dagger}_{\rm out}(\omega)\hat{a}_{\rm
out}(\omega).$ (4.29)

This makes rigorous the idea that one can amplify at any frequency, enabling
the mantra that one should amplify at high (optical) frequencies [195].
Namely, thermal fluctuations at a frequency $\omega^{\prime}$ may be
suppressed by choosing the reservoir mode frequency $\omega^{\prime}$ such
that $\hbar\omega^{\prime}\gg kT$. This suppression is exponential: $\Delta
n_{b}^{2}\propto\bar{n}_{b}\propto\exp(-\hbar\omega^{\prime}/kT)$. Note that
number fluctuations in the internal mode $c_{{\rm in}}(\omega)$ at the input
frequency will be amplified by the subsequent amplification process. However,
one can in principle construct ideal detectors for light with a particular
frequency $\omega_{0}$ [26], such that $|T(\omega_{0})|=1$ and hence
$R(\omega_{0})=0$, avoiding internally generated dark counts at that
particular frequency. If instead of a single monochromatic mode a small range
of frequencies is amplified with differing probabilities (that is, a
particular temporal mode), matching the amplification spectrum to the
filtering spectrum is sufficient for reducing internally generated dark
counts, as we will discuss in more detail in the next chapter.

###  4.7 Connections

The models for amplification considered here apply to other types of quantum
measurement as well. For example, electron-shelving [31, 183, 184] is a well-
known method to perform atomic state measurements. Here one particular atomic
state (e.g., one of the hyperfine ground states of an ion) is coupled
resonantly to a higher-lying excited state which can then decay back by
fluorescence only to that same ground state. A laser tuned to that transition
can then induce the atom to emit a macroscopic amount (visible by eye) of
fluorescent light. In the language accompanying Eq. (4.2), the laser beam
forms the first reservoir, while the second reservoir consists of vacuum modes
that are filled with fluorescent light as described by Eq. (4.8). The gain
factor $G$ (the number of fluorescence photons) is determined by the ratio of
Einstein’s coefficients for spontaneous and stimulated emission and the total
integration time. By placing the atom/ion inside a high-Q optical resonator
(with resonant frequency $\omega^{\prime}$) we would reduce the number of
output modes and thereby get closer to the optimum. The idea of placing a
detector inside a resonant cavity is, of course, not new [196], but that idea
is usually associated with increasing the coupling to light. Although we do
have that effect as well, the main purpose here is to reduce the number of
output modes, and thereby increase the SNR (Fig. 4.1).

A method similar to electron shelving (but using a collective metastable state
of an atomic gas) was proposed in [197] for implementing high-efficiency
photon counting with an, in principle, quantum nondemolition measurement. Our
model describes that detector too and clarifies why such a measurement is
possible; nonlinear amplification enables low-noise photon counting and does
not destroy the input photons without an additional step as in Eq. (4.4).

In [177], a transformation similar to Eq. (4.2) is given: $n$ photons are
directly converted to $Gn$ photons in a single mode. Though this
transformation is unphysical (there is no way to preserve the commutator), a
SNR is calculated that increases linearly with $G$ like our Eq. (4.24).
However, the SNR found in [177] diverges for a photodetector with unit
efficiency, which is not the case once fluctuations in the reservoir mode are
properly taken into account as our results clarify.

In [175], Yang and Jacob propose an interesting model for amplification that
makes use of a first-order phase transition for a collection of $N$
interacting spin-1/2 particles. These spins are coupled both to an input
photon and to an output bosonic mode. The SNR (as we define it here) for that
model scales as $\sqrt{N}$ while the gain $G$ of that model is linear in $N$.
Thus, the SNR scales with $\sqrt{G}$ just as our Eq. (4.25): the number of
spins in [175]’s model plays a similar role as our number of amplification
modes.

###  4.8 Remarks

We discussed various linear and nonlinear amplification schemes for bosonic
modes. For detecting few photons, we found that the latter add considerably
less noise, leading to better signal-to-noise ratios, as exemplified in Eqs.
(4.22)–(4.27). Unlike for linear amplification, number fluctuations in
internal detector modes are not amplified, while the number of photons that we
want to detect is amplified. All amplification schemes explicitly preserve the
bosonic commutation relations.

While amplification into a single-mode may not be feasible in practice, it
provides the fundamental lower limit to noise in photon-number measurements
across amplification mechanisms. In practice, one may have many output modes
and thus may find a SNR closer to Eq. (4.25), which is worse by a factor of
$\sqrt{G}$ than the fundamental limit (but still better by a factor of
$\sqrt{G}$ than linear amplification), or one may have multiple amplification
steps as in Eq. (4.26), or both as in Eq. (4.27). To test this, we suggest
that measurement of the gain dependence of the SNR for a given photodetector
should provide a rough but useful indication of the underlying amplification
mechanism.

##  Chapter 5 Measurement and Applications

The model of a SPD as an isolated two-level system is highly idealized. In a
more realistic system, photodetection is an extended process wherein a photon
is transmitted into the detector, interacting with the system and triggering a
macroscopic change of the photodetector state (amplification) which can then
be measured classically. Many theories of single-photon detection have been
developed over the past century, [44, 47, 48, 49, 126, 50, 51, 127, 52, 26,
24, 25] and indeed there are numerous implementations of SPD technology [128,
129, 77, 22, 66, 23]. Across all systems, we have identified the three stages
of transmission, amplification, and measurement as universal. In the previous
two chapters, we have studied amplification and transmission in isolation,
discussing the tradeoffs and limits that emerge for each stage individually.
In this chapter, we will combine our results with a final quantum measurement
and derive a POVM that incorporates all three stages quantum mechanically,
extending our model to include fluctuations of system parameters. Then, in the
remainder of this chapter we will discuss applications enabled by such a photo
detecting system.

###  5.1 Realistic SPD POVMs

The time-dependent two-level system from the previous section enabling
arbitrary wavepacket projection is incorporated into the three-stage model as
the trigger for the amplification mechanism. This will enable us to still
project onto arbitrary single-photon wavepackets, a result we will prove at
the end of this section. We will assume in this analysis that the system is
left on for a sufficient time such that the subnormalization of $\Psi(t)$ is
minimal and ${\cal W}\approx 1$ (2.13).

Figure 5.1: A POVM description of the three-stage model of photodetection,
where the chain of inference (left to right) moves opposite the arrow of time,
connecting a macroscopic “click” outcome to the state of the input field. A
“click” outcome represented by the POVM element $\hat{\Pi}_{T}$ indicates
${k\in K_{\rm click}}$ excitations were measured in the final measurement with
detection efficiency $\eta$. This retrodicts that ${n\approx k/\eta}$ (and,
strictly, ${n\geq k}$) excitations were present post-amplification, either
from thermal fluctuations or an amplification process. In turn, this indicates
that ${m\approx(n-\bar{n}_{\rm th})/G\leq n}$ excitations were likely incident
to the amplification process trigger, with $\bar{n}_{\rm th}$ the expected
number of thermal excitations already in the amplification target mode. If $m$
is larger than the expected number of thermal excitations in the amplification
trigger mode $\bar{m}_{\rm th}$, we conclude that one input photon of the form
$\left|T\Psi_{T}\right\rangle\left\langle T\Psi_{T}\right|$ was likely
present. In addition to the states written explicitly, there are other
possible states where $k$, $n$, and $m$ deviate from their most likely values.
These states (denoted by parallelograms) contribute to the POVM element, as
the internal state of the photodetector is in general highly mixed.
Nonetheless, in the end a SPD POVM element $\hat{\Pi}_{T}$ only projects onto
the two input states $\left|T\Psi_{T}\right\rangle\left\langle
T\Psi_{T}\right|$ and $\left|\rm vac\right\rangle\left\langle\rm vac\right|$,
remaining relatively pure.

In the spirit of what a POVM does (connect present outcomes to probabilistic
statements about quantum states in the past), we will begin our endeavor at
the very end of the photodetection process following Fig. 5.1. Consider a
macroscopic measurement performed at time $T$ with a binary response triggered
by $k$ excitations measured in the amplified signal111There is of course
latency in any detector, but this is not interesting: one merely shifts
${T\rightarrow T+\delta T}$ in each step so that the final POVM matches the
timing of a detection event to the quantum state projected onto.. Such a POVM
can be written as a projector onto Fock states in the Hilbert space internal
to the system

$\displaystyle\hat{\Pi}_{T}=\sum\limits_{k\in K_{\rm
click}}\left|k,\,T\right\rangle\left\langle k,\,T\right|.$ (5.1)

Here, we have defined a set $K_{\rm click}$ that sets the threshold for how
many amplification excitations must be measured to trigger a macroscopic
detection event. At this stage, we can already see that the internal state the
POVM projects onto is highly mixed, but this will not directly translate to an
impure measurement on the Hilbert space of input photons. Indeed, this is what
we would expect; we do not need to know _precisely_ the internal state of the
photodetector in order to use it to efficiently detect the presence of a
single photon.

The macroscopic measurement performed on the amplified signal will, in
general, be inefficient. We model this in a standard way [198], using a
beamsplitter with frequency independent transmission amplitude $\sqrt{\eta}$.
We can then rewrite the POVM element (5.1) so that it projects onto Fock
states in the amplification target mode prior to the measurement

$\displaystyle\hat{\Pi}_{T}=\sum\limits_{k\in K_{\rm
click}}\sum\limits_{n=k}^{\infty}\Pr(n|k)\left|n,\,T\right\rangle\left\langle
n\,T\right|$ (5.2)

where we have defined

$\displaystyle\Pr(n|k)={n\choose k}\eta^{k}(1-\eta)^{n-k}$ (5.3)

the probability to detect $k$ excitations given that there were $n$
excitations in the output mode of the amplification process. An inefficiency
$1-\eta$ affects photodetection by changing which post-amplification Fock
states are projected onto: the larger $1-\eta$ is, the more post-amplification
states $\left|n,\,T\right\rangle\left\langle n,\,T\right|$ will be projected
onto intermediately by the same POVM element $\hat{\Pi}_{T}$, making it harder
to distinguish between signal and noise.

We now move one step further back in the chain of inference (Fig. 5.1) so the
POVM element $\hat{\Pi}_{T}$ projects onto the number of excitations $m$ input
to the amplification trigger. Amplification is a generic feature of
photodetection; without a macroscopic change in the internal state of a
photodetector, there is no way to correlate detector outcomes with the
presence of a single photon [199, 175, 200] (that is, without invoking
additional single-excitation detectors in an argument _circulus in probando_).
There are many interesting methods for implementing amplification [172, 197,
174, 52, 173, 199, 175, 200], but the fundamental quantum limit to
amplification of any bosonic Fock state is achieved by a Schrödinger picture
transformation [5]

$\displaystyle\left|m\right\rangle_{\rm trig}\left|M\right\rangle_{\rm
res}\left|N\right\rangle_{\rm targ}\longmapsto\left|m\right\rangle_{\rm
trig}\left|M-Gm\right\rangle_{\rm res}\left|N+Gm\right\rangle_{\rm targ}$

such that exactly $G$ excitations are transferred from the reservoir mode to
the target mode for each excitation in the trigger mode. In using this
expression we do impose a restriction that there must be $M>Gn$ excitations in
the reservoir mode, but restrictions of this type are to be expected (the
energy for amplification must come from somewhere) and we will be most
interested in few photons ($n=0,1,2$) in this analysis. In most physical
platforms $G$ will fluctuate222For instance, electron shelving [31] is exactly
described by (5.1) in the high-Q cavity limit when at most a single excitation
is present with the laser acting as reservoir. Here the fluctuations in $G$
will be sub-Poissonian due to photon bunching., as will other (classical)
system parameters which we will return to at the end of this section.
(Exceptions do exist; for Hamiltonians that implement deterministic
amplification schemes [with small integer values for $G$] see Ref. [182].)
However, even with a definite gain factor $G$ and number of input excitations
$m$, we will still not end up with exactly $n=N+Gm$ excitations if the target
mode is initially in a thermal state with mean occupation number $\bar{N}$ (as
opposed to a Fock state with exactly $N$ excitations). We now assume this,
writing the state of the target mode in the Fock basis

$\displaystyle\hat{\rho}_{\rm targ}^{(th)}=\sum\limits_{N=0}^{\infty}P^{\rm
th}_{N,\,T}|N,\,T\rangle\\!\langle N,\,T|,$ (5.5)

with the probability for $N$ thermal excitations given by

$\displaystyle P^{\rm
th}_{N}=\frac{1}{1+\bar{N}}\left(\frac{\bar{N}}{1+\bar{N}}\right)^{N};\,\bar{N}=\frac{1}{e^{\frac{\hbar\omega^{\prime}}{k_{b}T}}-1},$
(5.6)

where $\omega^{\prime}$ and $k_{B}T$ are the frequency and the thermal energy
of the target mode. Assuming the ideal amplification scheme in (5.1), we now
write the POVM element $\hat{\Pi}_{k}$ in terms of the number $m$ excitations
that trigger the application mechanism

$\displaystyle\\!\hat{\Pi}_{T}=\sum\limits_{k\in K_{\rm
click}}\sum\limits_{n=k}^{\infty}\Pr(n|k)\sum\limits_{m=0}^{{\rm
Int}_{-}[\frac{n}{G}]}P^{\rm
th}_{n-Gm}\left|\Psi_{T}\right\rangle\left\langle\Psi_{T}\right|^{\otimes m}$

where we define
$\left|\Psi_{T}\right\rangle\left\langle\Psi_{T}\right|^{\otimes 0}=|{\rm
vac}\rangle\langle{\rm vac}|$, $P^{\rm th}_{N}=0$ for $N<0$, and the function
${\rm Int}_{-}$ rounding down to the nearest integer. We can now see the
benefit of having a large gain factor G; it shifts the probability
distribution over $n$ that corresponds to non-zero excitations in the trigger
mode, minimizing its overlap with the probability distribution for zero
excitations. In this way, one can dramatically reduce the background noise
(dark counts) without decreasing signal by changing the minimum value of the
set $K_{\rm click}$333The set $K_{\rm click}$ need not have a maximum.. In
(5.1) we have reintroduced the state $\left|\Psi_{T}\right\rangle$ defined in
(2.2) as the state projected onto by the trigger mechanism. As in the previous
section, we will assume a time-dependent resonance frequency $\Delta(t)$ and
decay rate $\kappa(t)$ so that arbitrary pulse-shaping is possible.

The POVM element in (5.1) now projects onto quantum states _internal_ to the
photodetector. We need to connect the internal continuum of states coupled to
the amplification trigger to the external continuum containing the photons we
wish to detect (the transmission stage in Fig. 5.1). This is accomplished by
introducing a two-sided quantum network to serve as the transmission stage
[108]. Any such network is completely described by a single complex frequency-
dependent transmission coefficient $T(\omega)$ (related to a reflection
coefficient at each frequency via $|T(\omega)|^{2}+|R(\omega)|^{2}=1$ and
$R(\omega)T^{*}(\omega)+R^{*}\omega)T(\omega)=0$). We now invoke the single-
photon assumption so that there is at most a single excitation input to the
quantum network. Any other excitations present in the internal continuum will
be from internally-generated thermal fluctuations reflected by the quantum
network back to the trigger mechanism. In this way, we can construct a POVM
element that projects onto product-states $\left|\psi_{\rm
ex}\right\rangle\left\langle\psi_{\rm ex}\right|\otimes\left|\psi_{\rm
in}\right\rangle\left\langle\psi_{\rm in}\right|$ of the external and internal
continua

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\hat{\Pi}_{T}$
$\displaystyle=$ $\displaystyle\sum\limits_{k\in K_{\rm
click}}\sum\limits_{n=k}^{\infty}\Pr(n|k)\left(P^{\rm th}_{n}|{\rm
vac}\rangle\langle{\rm vac}|\otimes|{\rm vac}\rangle\langle{\rm
vac}|\phantom{\sum\limits_{m=1}^{{\rm Int}_{-}[\frac{n}{G}]}}\right.$ (5.8)
$\displaystyle+$ $\displaystyle\sum\limits_{m=1}^{{\rm
Int}_{-}[\frac{n}{G}]}P^{\rm th}_{n-Gm}\rho^{2m}|{\rm vac}\rangle\langle{\rm
vac}|\otimes\left|R\Psi_{T}\right\rangle\left\langle R\Psi_{T}\right|^{\otimes
m}\phantom{\sum\limits_{m=1}^{{\rm Int}_{-}[\frac{n}{G}]}}$ $\displaystyle+$
$\displaystyle\left.\sum\limits_{m=1}^{{\rm Int}_{-}[\frac{n}{G}]}mP^{\rm
th}_{n-Gm}\tau^{2}\rho^{2(m-1)}\left|T\Psi_{T}\right\rangle\left\langle
T\Psi_{T}\right|\otimes\left|R\Psi_{T}\right\rangle\left\langle
R\Psi_{T}\right|^{\otimes m-1}\right).$

The first line corresponds to dark counts generated from thermal excitations
post-amplification and the second line corresponds to dark counts generated by
thermal excitations that then trigger the amplification mechanism. Only the
third line contains a projection onto a photon to be detected. (The
multiplicative factor $m$ in the third line is combinatorial in origin: $m$
total excitations in the trigger mode with $m-1$ generated from thermal
fluctuations.) In writing (5.8) we have defined transmitted and reflected
normalized single-photon states and coefficients

$\displaystyle\left|T\Psi_{T}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\int_{-\infty}^{\infty}\,d\omega\tilde{\Psi}(\omega)T^{*}(\omega)e^{i\,\omega\,T}\hat{a}^{\dagger}(\omega)\left|\textnormal{vac}\right\rangle$
$\displaystyle\left|R\Psi_{T}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\int_{-\infty}^{\infty}\,d\omega\tilde{\Psi}(\omega)R^{*}(\omega)e^{i\,\omega\,T}\hat{b}^{\dagger}(\omega)\left|\textnormal{vac}\right\rangle$
$\displaystyle\tau$ $\displaystyle=$ $\displaystyle\sqrt{\int
d\omega|\tilde{\Psi}(\omega)|^{2}|T(\omega)|^{2}}$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle\sqrt{\int
d\omega|\tilde{\Psi}(\omega)|^{2}|R(\omega)|^{2}}$ (5.9)

where $\hat{a}^{\dagger}$ and $\hat{b}^{\dagger}$ are the creation operators
for the external and internal continua and we have defined a Fourier-
transformed wavepacket for the amplification trigger mode
$\tilde{\Psi}(\omega)={\rm FT}[\sqrt{\kappa(t)}\Psi(t)]$. We can now see how
pre-amplification dark counts (the second line of (5.8)) can be suppressed: by
reducing the overlap of $|\tilde{\Psi}(\omega)|^{2}$ and $|R(\omega)|^{2}$,
that is, by only amplifying the frequencies we wish to detect so that
$\rho^{2}\ll 1$. In this case, the POVM element (5.8) will be dominated by the
$m=1$ term of the third line (the signal to be detected with no thermal
excitations), as well as potentially the first line. (To reiterate, these are
dark counts post-amplification, but these can be reduced by amplifying at a
high frequency such that $\hbar\omega^{\prime}\gg k_{B}T$, where
$\omega^{\prime}$ and $k_{B}T$ are the frequency and thermal energy of the
target mode.)

Finally, we trace over the internal continuum, which we assume is in a thermal
state with thermal energy $k_{B}T^{\prime}$444At my advisor’s request, I would
like to apologize for the overuse of the letter “T” in this dissertation. so
the POVM projects onto the external continua only

$\displaystyle\hat{\Pi}_{T}$ $\displaystyle=$ $\displaystyle\sum\limits_{k\in
K_{\rm click}}\sum\limits_{n=k}^{\infty}\Pr(n|k)\left(\sum\limits_{m=0}^{{\rm
Int}_{-}[\frac{n}{G}]}P^{\rm th}_{n-Gm}P^{{}^{\prime}\rm
th}_{m}\rho^{2m}\right.|{\rm vac}\rangle\langle{\rm vac}|$ (5.10)
$\displaystyle+$ $\displaystyle\left.\sum\limits_{m=1}^{{\rm
Int}_{-}[\frac{n}{G}]}mP^{\rm th}_{n-Gm}P^{{}^{\prime}\rm
th}_{m-1}\tau^{2}\rho^{2(m-1)}\left|T\Psi_{T}\right\rangle\left\langle
T\Psi_{T}\right|\right)$ $\displaystyle\equiv$ $\displaystyle w_{0}|{\rm
vac}\rangle\langle{\rm vac}|+w_{T}\left|T\Psi_{T}\right\rangle\left\langle
T\Psi_{T}\right|$

where in the last line we have absorbed the sums in front of the two
projectors into weights so that the POVM element has the form of (1.9) and
with $P^{{}^{\prime}\rm th}_{j}$ the probability to have $j$ excitations (now
in the non-monochromatic reflected mode defined in (5.1)). For a finite
detector on-time $T_{0}>-\infty$, the weights $w_{0}$ and $w_{T}$ will be
slightly less than in (5.10) due to wavepacket sub-normalization (2.13).
However, this deviation is negligible provided the detector is left on for a
time comparable to the temporal mode’s width.

We now reconsider the question of projecting onto an arbitrary wavepacket,
including the full quantum description. We find that this is possible to do in
principle, _provisio_ $T(\omega)$ is nowhere zero (except at infinity), a
result we will now prove. That is, we can ensure that the single-photon
wavepacket $\left|T\Psi_{T}\right\rangle$ has any desired (smooth) shape and
will be projected onto with a high-efficiency and high-purity measurement.

_Proof.—_ Consider a photon with complex normalized spectral wavepacket
$\tilde{f}(\omega)$. If detection is achieved with a time-dependent two-level
system preceded by a quantum network with filtering transmission function
$T(\omega)$, the system will project onto a state
$\left|T\Psi_{T}\right\rangle$ as defined in (5.1). In the low-noise limit
this will be the only state projected onto by the (pure) POVM element. From
the Born rule, the probability of detection will be

$\displaystyle
P_{T}=w_{T}\frac{1}{\tau^{2}}\left|\int_{-\infty}^{\infty}d\omega\tilde{f}(\omega)\tilde{\Psi}(\omega)T^{*}(\omega)\right|^{2}$
(5.11)

with $w_{T}$ the overall weight given by (5.10) and maximum possible detection
efficiency, which can be arbitrarily close to unity. It is possible to achieve
$P_{T}=w_{T}$ (mode-matched detection) in (5.11) if and only if

$\displaystyle\tilde{\Psi}(\omega)=\frac{\tilde{f}^{*}(\omega)}{T(\omega)}e^{i\omega
T}.$ (5.12)

From (2.12), we know that it is possible to generate an arbitrary temporal
wavepacket ${\rm FT}^{-1}[\tilde{\Psi}(\omega)]=\sqrt{\kappa(t)}\Psi(t)$ from
a time-dependent two-level system. The Fourier transform of a continuous
smooth function is itself smooth and continuous. Thus, if the right hand side
of (5.12) is a well-defined spectral wavepacket (smooth and continuous), one
can find functions $\kappa(t)$ and $\Delta(t)$ such that
$\tilde{\Psi}(\omega)$ has the form of (5.12). ∎

_Remark.—_ Arbitrary wavepacket detection (and thus Fourier-limited
simultaneous measurements of time and frequency) is in principle possible only
when there are no photonic band gaps induced by the filter; if
$T(\omega^{\prime})=0$ for some frequency $\omega^{\prime}$, there is simply
no way to compensate for the lost information about $\omega^{\prime}$.
Photonic band gaps are a generic feature of parallel (and hybrid) quantum
networks [108] as well as certain non-Markovian systems [201].
Network/reservoir engineering must be employed to ensure any $\omega^{\prime}$
where $T(\omega^{\prime})=0$ is not a frequency of interest.

The POVM $\\{\hat{\Pi}_{T},\,\hat{\Pi}_{0}\\}$ with $\hat{\Pi}_{T}$ defined in
(5.10) and $\hat{\Pi}_{0}=\hat{\mathds{1}}-\hat{\Pi}_{T}$ provides a complete
description of the single-photon detection process that is fully quantum from
beginning to end (Fig. 5.1). However, there is a final element that must be
considered to make the description applicable to labratory systems: classical
parameter fluctuations. For continuous parameter fluctuations over any system
parameter or set of system parameters $X$, these are naturally incorporated

Symbol | System Parameter | Fluctuations
---|---|---
$K_{\rm click}$ | Macroscopic Detection Threshold | $w_{0},\,w_{T}$
$\eta$ | Macroscopic Detection Efficiency | $w_{0},\,w_{T}$
$k_{B}T$ | Target Mode Thermal Energy | $w_{0},\,w_{T}$
$\omega^{\prime}$ | Target Mode Frequency | $w_{0},\,w_{T}$
$G$ | Amplification Gain | $w_{0},\,w_{T}$
$\tilde{\Psi}(\omega)$ | Amplification Trigger Mode | $w_{0},\,w_{T},\,\left|T\Psi_{T}\right\rangle$
$k_{B}T^{\prime}$ | Internal Continuum Thermal Energy | $w_{0},\,w_{T}$
$T(\omega)$ | Transmission Coefficient | $w_{0},\,w_{T}\,\left|T\Psi_{T}\right\rangle$
Table 5.1: The effects of classical fluctuations in system parameters on the
final POVM: either the weights and states are changed, or only the weights are
changed. The fluctuations over the functions $\tilde{\Psi}(\omega)$ and
$T(\omega)$ (and thus $R(\omega)$ by unitarity) could be caused by
fluctuations in other system parameters (decay rates, resonances) internal to
those functions. A subset of fluctuations in $\tilde{\Psi}(\omega)$ are
fluctuations in the time of detection $T$. Importantly, these shift the
wavepacket $\Psi(t)$ projected onto, resulting in a mixed measurement with
larger temporal uncertainty (jitter) that depends on the ratio of the
fluctuations in $T$ to the width of the temporal wavepacket.

(5.13) $\displaystyle\\!\hat{\Pi}_{T}=\int dX{\rm Pr}(X)\left(w_{0}|{\rm
vac}\rangle\langle{\rm vac}|+w_{T}\left|T\Psi_{T}\right\rangle\left\langle
T\Psi_{T}\right|\right)$

where we have assumed a (known) probability distribution ${\rm Pr}(X)$. In
(5.13), the system parameter(s) $X$ could be such that only the weights
$w_{0}$ and $w_{T}$ depend on $X$, or $X$ could be such that the state
$\left|T\Psi_{T}\right\rangle$ depends on $X$ as well (for a summary, see Tbl.
5.1). In the case of the latter, the POVM will become less pure and will need
rediagonalization to determine which states are projected onto555By varying
certain key parameters, it is possible to induce an exceptional-point
structure in (5.13), for instance, by introducing a discrete probability
distribution over resonance frequencies corresponding to classical ignorance
about a discrete set of detector settings. Here the exceptional point occurs
when the frequencies are made degenerate, which (since the discrete states
have identical quantum numbers) is forbidden by unitarity. Here the range over
the resonances are distributed is the exceptional-point parameter [202].. This
final POVM not only includes ignorance about the internal state of the
photodetector as was depicted in Fig. 5.1, but also classical ignorance about
the state of the photodetector due to system-lab interactions.

###  5.2 Applications

Using the time-dependent two-level system, we are able to project onto
orthogonal quantum states (Fig. 5.2). This enables efficient detection of
photonic qubits, an essential component of any quantum internet [203, 204].
More generally, temporal modes provide a complete framework for quantum
information science [205], with efficient detection of orthogonal modes (and
their superpositions to create mutually unbiased bases) a key ingredient.
Fully manipulable temporal modes also play a key role in error-corrected
quantum transduction [206], where a time-reversed temporal mode can restore an
unknown superposition in a qubit. Here, efficient detection of arbitrary
temporal modes is essential so that quantum jumps out of the dark state are
efficiently heralded.

Figure 5.2: The time-dependent coupling $\kappa(t)$ that generates a
wavepacket exactly orthogonal to the minimum uncertainty Gaussian wavepacket
in Fig. 2.3 are plotted for times of detection ${T=2\sigma}$ (${{\cal
W}=0.72}$) and ${T=7\sigma}$ (${{\cal W}=1-10^{-5}}$) and detector on-time
${T_{0}=-\infty}$. The wavepacket’s time-dependent amplitude $A(t)$ (depicted
in the rotating frame) is an approximate first-order Hermite-Gaussian pulse,
where the singular region of zero as been expanded with half-width of
${z=0.5\sigma}$. In this way, the wavepacket’s phase (solid black line, not to
scale) can go from $0$ to $\pi$ in a finite time, whereas the Heaviside phase-
flip in the exact Hermite-Gaussian pulse requires an unphysical delta-function
detuning (second line of (2.2). Both the phase and amplitude have been
convolved with a triangular smoothing function with full-width
${s=0.5\sigma}$, ensuring $A(t)$ is continuously differentiable to first order
[207, 208], which is pre-requisite for solving the Bernoulli equation (2.21).
For any finite ${z\geq s>0}$ the phase flip [here implemented with a
triangular detuning $\Delta(t)$] occurs while the amplitude is zero. This
ensures exact orthogonality of the approximate first-order Hermite-Gaussian to
the gaussian pulse. In the limit ${s,z\rightarrow 0}$, an exact first-order
Hermite-Gaussian is recovered. This smoothing procedure generalizes to higher
order Hermite-Gaussian pulses, forming a mutually unbiased basis for efficient
detection of higher dimensional qudits [205].

High-purity measurements that project onto orthogonal single-photon
wavepackets also enable super-resolved measurements [209]. Suppose we have two
single-photon sources emitting almost identical pure states differing slightly
in either emission time or central frequency

$\displaystyle\left|\tilde{\phi}_{1}\right\rangle$
$\displaystyle=\frac{\left|\phi_{1}\right\rangle+\sqrt{\epsilon}\left|\phi_{2}\right\rangle}{\sqrt{1+\epsilon}}$
$\displaystyle\left|\tilde{\phi}_{2}\right\rangle$
$\displaystyle=\frac{\left|\phi_{1}\right\rangle-\sqrt{\epsilon}\left|\phi_{2}\right\rangle}{\sqrt{1+\epsilon}}$
(5.14)

with $\left\langle\tilde{\phi}_{1}|\tilde{\phi}_{2}\right\rangle$ real,
$\epsilon\ll 1$, and $\left\langle\phi_{1}|\phi_{2}\right\rangle=0$.
Alternatively, we may imagine a single source of light but the light we
receive may have either been slightly Doppler-shifted or it may have been
slightly delayed.

Suppose now that we receive one photon that could equally likely be from
either source so that our input state is

$\displaystyle\hat{\rho}=\frac{1}{2}\left|\tilde{\phi}_{1}\right\rangle\left\langle\tilde{\phi}_{1}\right|+\frac{1}{2}\left|\tilde{\phi}_{2}\right\rangle\left\langle\tilde{\phi}_{2}\right|.$
(5.15)

If we can measure both
$\hat{\Pi}_{1}=\eta\left|\phi_{1}\right\rangle\left\langle\phi_{1}\right|$ and
$\hat{\Pi}_{2}=\eta\left|\phi_{2}\right\rangle\left\langle\phi_{2}\right|$
(that is, if we have separate photodetectors with these (pure) POVM elements,
or a single non-binary-outcome photodetector), then we find the probability of
clicks

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\left[\hat{\Pi}_{1}\hat{\rho}\right]=\eta\frac{1}{1+\epsilon}$
$\displaystyle P_{2}$ $\displaystyle=$ $\displaystyle{\rm
Tr}\left[\hat{\Pi}_{2}\hat{\rho}\right]=\eta\frac{\epsilon}{1+\epsilon}$
(5.16)

so that the ratio of clicks gives a direct estimate of $\epsilon$, even for
low efficiency $\eta$. Here all that is needed for time-frequency domain
super-resolved measurement of $\epsilon$ are SPDs with time-dependent
couplings and resonance frequencies as opposed to nonlinear optics [210].

In traditional quantum key distribution (QKD) schemes (that is, _not_
measurement device independent (MDI)-QKD), specification of the measurement
POVM is essential to robust security proofs [211, 212, 213]. Here, we have
verified several assumptions about an eves-dropper’s capabilities common in
security proofs: that high-purity measurements are possible, that high
efficiency measurements are possible, and (for continuous-variable (CV)-QKD
proofs) minimum time-frequency uncertainty measurements are possible. In
particular for CV-QKD, an eavesdropper can perform measurements that project
onto variable-width spectral modes, disrupting temporal correlations between
Alice and Bob (who are assumed to use fixed time-frequency bins) [214]. Here,
the capacity to adjust the width of the spectral mode $\tilde{\Psi}(\omega)$
provides Alice and Bob a new strategy to mitigate Eve’s attack and extract a
secure key.

More generally, detector tomography [215, 95, 94, undeen2009, 94, 216] is an
important tool across implementations of single-photon and number-resolved
photodetection. Real-time tomography could be useful in QKD protocols
resistant to “trojan-horse attacks” [217] or any SPD platform subject to time-
dependent environmental parameter fluctuations: for instance, atmospheric
turbulence in MDI-QKD [218] or interplanetary medium in deep space classical
communications [219]. Recently tomography speed-ups have been achieved using
machine learning assisted tomography protocols [220]. The POVMs derived in
this thesis provide priors which can further speed up detector tomography
[221]. These include approximate effects of environmental fluctuations as
outlined in Fig. 3.1 and a global optimum POVM for single-photon detection
(5.10) which can be used to incorporate detector calibration and optimization
into _in situ_ tomographic protocols.

##  Chapter 6 Conclusions

Having constructed a fully quantum model incorporating the three stages of
photodetection (transmission, amplification, and measurement), we can now
discuss the fundamental limits and tradeoffs inherent to photodetection as a
whole.

We have constructed single-photon measurements that are fundamentally limited
in two ways simultaneously: the first is that they can project onto Fourier-
limited (Gaussian) time-frequency states as illustrated in Fig. 2.3, and the
second is that the amplification scheme reaches a Heisenberg-limited (linear
in the gain $G$) signal-to-noise ratio, surpassing the standard quantum limit
(a signal-to-noise ratio going like of $\sqrt{G}$). Achieving these
simultaneously is possible in principle with no drawback. Indeed, the only
stringent tradeoff we encounter in this analysis is between efficiency and
photon counting rate, which becomes substantial when an SPD is reset at a
faster rate than $\sim 1/\Delta t$. (The photon does not have sufficient time
to excite the two-level system with high probability before the system is
reset.) For other figures of merit, we find that they are either independent,
or deteriorate together111For instance, inefficiency and dark counts both
increase with the coefficient $\rho$ in (5.1) when onwae considers an
amplification scheme like electron shelving, where the absorption of one
excitation precludes the absorption of a second.. While it does appear from
(5.10) that improving efficiency also increases dark counts, these are
decoupled by ensuring the coefficient $\rho\ll 1$—that is, by making
$T(\omega)$ broader than $\tilde{\Psi}(\omega)$. While it is commonsense that
one should only amplify the frequencies they wish to detect, our work
clarifies how enormously important this is. The dark counts produced in this
way are insuperable; they cannot be removed post-amplification without
removing the single-photon signal as well.

One optimistic conclusion from our work is that, while there are emergent
tradeoffs to consider when constructing SPDs due to their multi-stage nature,
these are generally avoidable with foresight and engineering. For instance,
using a parallel network allows one to simultaneously optimize transmission
efficiency $|T(\omega)|^{2}$, spectral bandwidth (3.1), and frequency
dependent group delay (3.3) despite the individual tradeoffs present for other
network types. Using a time-dependent amplification trigger, it is possible to
still achieve Fourier-limited simultaneous measurements of time and frequency
provided no frequencies of interest have been removed from the signal (that
is, destructive interference has not resulted in $T(\omega)=0$) as
demonstrated in (5.12). While some noise is unavoidable, minimal post-
amplification arbitrarily high signal can be reached by amplifying into fewer
modes even at room temperature by amplifying at higher frequencies.
Furthermore, detection of a particular arbitrary wavepacket changes pre-
amplification dark counts; the functions $T(\omega)$ and
$\tilde{\Psi}(\omega)$ both determine the projected wavepacket
$\tilde{f}(\omega)=T(\omega)\tilde{\Psi}(\omega)e^{-i\omega T}$ and are
interchangeable up to complex conjugation. However, they have different roles
in determining dark counts. One can significantly reduce pre-amplification
dark counts by ensuring that the transmission stage $T(\omega)$ has a broader
spectrum than the trigger mechanism $\tilde{\Psi}(\omega)$; pre-amplification
dark counts go to zero as the reflection function $R(\omega)$ and trigger
mechanism spectrum $\tilde{\Psi}(\omega)$ become orthogonal.

Another conclusion from this work is also rather optimistic. Here we have
given a quantum description of an entire single-photon detection process
projecting onto arbitrary single-photon states and the only fundamental
limitations encountered are Heisenberg and Fourier limits. Incorporating
realistic descriptions of amplification and a final measurement reduce
efficiency and increase dark counts, but even so a Heisenberg-limited
measurement is still achievable in principle. Similarly, incorporating the
filtering of a first irreversible step does not impede implementation of
Fourier-limited measurements provided no frequencies are completely blocked
from entering the trigger mechanism. Even considering parameter fluctuations
(5.13) in the two internal thermal energies $k_{B}T$ and $k_{B}T^{\prime}$,
amplification frequency $\omega^{\prime}$, and amplification gain factor
$G$—which are unavoidable in any realistic system—Fourier limited time-
frequency measurements are achieved. To our knowledge, this works contains the
first proposed quantum procedure for reaching Fourier-limited time-frequency
measurements in a realistic quantum system. This is a fundamental limit to SPD
performance enforced by foundational quantum theory, and provides a limit to
any precision measurements involving time and frequency. Furthermore, probing
Heisenberg limits paves the way for future experimental tests of quantum
theory.

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