# Photocounting statistics of superconducting nanowire single-photon detectors

V. A. Uzunova  Institute of Physics, NAS of Ukraine, Prospect Nauky 46, 03028
Kyiv, Ukraine  A. A. Semenov  Bogolyubov Institute for Theoretical Physics,
NAS of Ukraine, Vul. Metrologichna 14b, 03143 Kyiv, Ukraine  Institute of
Physics, NAS of Ukraine, Prospect Nauky 46, 03028 Kyiv, Ukraine  Kyiv Academic
University, Blvd. Vernadskogo 36, 03142 Kyiv, Ukraine

###### Abstract

Superconducting nanowire single-photon detectors (SNSPDs) are efficient
measurement devices used for counting single photons. The field of their
applications covers experimental quantum-optical studies, optical quantum
computing, quantum communication, and others. After registering a photon by
such a detector, the next one cannot be registered during the dead time and
after that this ability is smoothly restored. We have included this feature
into the photodetection theory and introduced the corresponding photocounting
formula. In the regime of continuous-wave detection, the photocounting
statistics nonlinearly depends on the density operator due to a memory effect
of previous measurement time windows. The considered examples demonstrate the
strong influence of the relaxation process and the memory effect on the
resulting photocounting statistics of the SNSPDs.

##  I Introduction

Fast and efficient single-photon detectors [1, 2, 3, 4] are members of the
fundamental building blocks in modern quantum technologies. The corresponding
applications include optical universal [5] and nonuniversal [6] quantum
computers, quantum secure communication [7, 8, 9], quantum sensing and
metrology [10, 11, 12], etc. These detectors are also applied in a wider range
of research and technologies such as the design of electronic devices [13, 14]
and imaging problems in biological research and medicine [15, 16, 17].

Superconducting nanowire single-photon detectors (SNSPD) [18, 19, 3, 20, 4,
21] are highly promising measurement devices for numerous applications. This
is explained by their wide spectral sensitivity, good response speed, high
detection efficiency, low dark count rate, and high temporal resolution. The
detection process of the SNSPDs can be subdivided into a few stages [21]: (1)
an absorbed photon heats a small part of a nanowire in which a superconducting
current flows; (2) this leads to forming a normal-conducting part of the
nanowire resulting in a voltage change; (3) the detectors cannot register the
next photons during the dead-time interval $\tau_{\mathrm{d}}$; (4) the
superconducting part of the nanowire and the ability to detect another photon
are smoothly recovered during the relaxation time $\tau_{\mathrm{r}}$; (5) the
detector returns to the initial state. It is worth noting that physical models
[19, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] describing the
detection process in the SNSPDs are still developing.

The SNSPDs are applied in many fundamental quantum-optical experiments and in
different implementations of quantum-information protocols [4]. For example,
they have been recently used for demonstration of quantum supremacy in the
scheme of Gaussian boson sampling [35, 36]. These detectors have been applied
for implementations of quantum secure communication [37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] and quantum
teleportation [57, 58] protocols. The SNSPDs are key measurement elements for
implementations of strong loophole-free tests of Bell inequalities [59]. They
have also been applied as building blocks for array or time-multiplexing
detectors in various quantum-optical experiments; see, e.g., [60, 61].

Photocounting is a prominent example of quantum measurements. Its outcomes are
given by a number of clicks, which are commonly associated with the number of
photons. According to Born’s rule, the probability distribution to get $n$
clicks reads

$\displaystyle P_{n}=\operatorname{Tr}\left(\hat{\rho}\,\hat{\Pi}_{n}\right),$
(1)

where $\hat{\rho}$ is the density operator of the light mode and
$\hat{\Pi}_{n}$ is the positive operator-valued measure (POVM) [62] describing
the measurement procedure. In the idealized scenario of photon-number
resolving (PNR) detectors it is given by

$\displaystyle\hat{\Pi}_{n}\equiv\hat{F}_{n}\left[\eta\right]=:\mathrel{\frac{\left(\eta\hat{n}\right)^{n}}{n!}\exp\left(-\eta\hat{n}\right)}:,$
(2)

cf. Refs. [63, 64]. Here $\hat{n}$ is the photon-number operator,
$\eta\in[0,1]$ is the detection efficiency, and $:\mathrel{\ldots}:$ means the
normal ordering. In particular, this means that
$\hat{F}_{n}\left(1\right)=\left|n\right\rangle\left\langle n\right|$ is the
projector on the Fock state $\left|n\right\rangle$.

In realistic scenarios, photon-number resolution is not ideal, which is
described by the corresponding POVM. For example, the POVM describing array
[65, 66, 67, 68, 69] and time-multiplexing [70, 71, 72] detectors have been
derived in Ref. [73]. The corresponding measurement procedures are based on
spatial or temporal separation of modes and detecting each of them with on-off
detectors.

Another scenario of realistic detection is based on counting the photocurrent
pulses inside a measurement time window (MTW). The number of these
pulses—referred to as clicks or photocounts—is associated with the number of
photons. A problem is that each pulse is followed by a dead-time interval
during which photons cannot be detected. A classical photocounting theory for
this type of measurements has been developed in Refs. [74, 75, 76, 77, 78, 79,
80, 81, 82].

In this paper we introduce a generalization of the photocounting theory to the
scenario when the SNSPDs are used for counting pulses inside MTWs. The main
difference of the SNSPDs from other detectors consists in the effect of the
relaxation time. The common problem of the SNSPDs and the detectors
characterized by only dead time is that the corresponding time intervals from
the last registered pulse may exceed the MTW. This results in changing the
statistics of pulses for the next MTW.

We consider two detection scenarios associated with the SNSPDs. Firstly, the
scenario of independent MTWs assumes darkening the detector at the end of each
MTW. This protects statistics of pulses in the current MTW from influence of
events in the previous ones. Next, we consider the continuous-wave detection,
which assumes no darkening at the ends of the MTWs. The corresponding
photocounting statistics is affected by the previous MTWs—this influence is
referred to as a memory effect. It results in a nonlinear dependence of the
photocounting statistics on the density operator.

The rest of the paper is organized as follows. In Sec. II we give a
preliminary consideration of the photocounting formula in the Glauber-
Sudarshan representation, which is used throughout the paper. The model of
time-dependent efficiency, underlying the basis of our consideration of the
SNSPDs, is discussed in Sec. II. The POVM for the scenario of independent MTWs
is derived in Sec. IV. The scenario of continuous-wave detection involving the
memory effects from previous MTWs is considered in Sec. V. In Sec. VI we apply
the developed theory to deriving the photocounting statistics for typical
quantum states. A technique for experimental reconstruction of the time-
dependent efficiency is analyzed in Sec. VII. A summary and concluding remarks
are given in Sec. VIII.

##  II Preliminaries

Photocounting formula (1) can be conveniently rewritten as (see Refs. [83,
84])

$\displaystyle
P_{n}=\int_{\mathbb{C}}d^{2}\alpha\,P(\alpha)\,\Pi_{n}(\alpha).$ (3)

Here $P(\alpha)$ is the Glauber-Sudarshan P function [85, 86] and

$\displaystyle\Pi_{n}(\alpha)=\left\langle\alpha\right|\hat{\Pi}_{n}\left|\alpha\right\rangle$
(4)

is the $Q$ symbols of the POVM defined as the average with the coherent state
$\left|\alpha\right\rangle$. These symbols are interpreted as the
probabilities to get $n$ clicks given the coherent state
$\left|\alpha\right\rangle$. Utilizing the rule

$\displaystyle\left\langle\alpha\right|:\hat{f}\left(\hat{a},\hat{a}^{\dagger}\right):\left|\alpha\right\rangle=f\left(\alpha,\alpha^{\ast}\right),$
(5)

where $f\left(\alpha,\alpha^{\ast}\right)$ is an arbitrary function, one can
reconstruct the normal-ordering operator form of the POVM from its $Q$
symbols.

For the PNR detection, the $Q$ symbols of the POVM (2) read

$\displaystyle\Pi_{n}(\alpha)\equiv
F_{n}\left[\alpha;\eta\right]=\frac{\left(\eta|\alpha|^{2}\right)^{n}}{n!}\exp\left(-\eta|\alpha|^{2}\right).$
(6)

Here
$F_{n}\left[\alpha;\eta\right]=\left\langle\alpha\right|\hat{F}_{n}[\eta]\left|\alpha\right\rangle$
represents the $Q$ symbol of the operator $\hat{F}_{n}[\eta]$; cf. Eq. (2).
The first and second arguments of this function describe the dependence on the
phase-space complex variable $\alpha$ and the detection efficiency $\eta$,
respectively. This expression describes a well-known fact that photocounting
statistics of the coherent states is given by the Poissonian distribution.
Equation (3) has a form similar to the photocounting formula for classical
electromagnetic fields [63, 87, 88]. In the classical theory both functions,
$P(\alpha)$ and $\Pi_{n}(\alpha)$, are nonnegative. They play the roles of the
probability density of the complex amplitude $\alpha$ and the classical
response function of photocounts, respectively.

For purposes of this work, it is also useful to remind a procedure of finding
the POVM in the Fock-state basis,

$\displaystyle P_{n|m}=\left\langle
m\right|\hat{\Pi}_{n}\left|m\right\rangle,$ (7)

which can be interpreted as the probability distribution to get $n$
photocounts given $m$ photons. Since any kind of the photocounting
measurements is phase insensitive, the nondiagonal POVM elements vanish, i.e.,
$\left\langle m_{1}\right|\hat{\Pi}_{n}\left|m_{2}\right\rangle=0$ for
$m_{1}\neq m_{2}$. The probabilities $P_{n|m}$ can also be considered as
expansion coefficients of the POVM by the Fock states,

$\displaystyle\hat{\Pi}_{n}=\sum\limits_{m=0}^{+\infty}P_{n|m}\left|m\right\rangle\left\langle
m\right|.$ (8)

The same equation in terms of $Q$ symbols reads

$\displaystyle\Pi_{n}(\alpha)\exp\left(|\alpha|^{2}\right)=\sum\limits_{m=0}^{+\infty}P_{n|m}\frac{|\alpha|^{2n}}{n!}.$
(9)

Therefore, the POVM in the Fock-state basis can be obtained by expanding the
left-hand side of this expression by $|\alpha|^{2n}/n!$.

Another technique, which is used throughout the paper, is related to including
realistic values of the detection efficiency and the dark-count rate. Let the
POVM $\hat{\Pi}_{n}$ and its $Q$ symbols $\Pi_{n}(\alpha)$ describe the
idealized scenario with the unit detection efficiency and with no dark counts.
In order to include these issues in the description, one should replace

$\displaystyle\hat{n}\rightarrow\eta\hat{n}+\nu$ (10)

and

$\displaystyle|\alpha|^{2}\rightarrow\eta|\alpha|^{2}+\nu$ (11)

under the sign of normal ordering in the POVM and in the $Q$ symbols of the
POVM, respectively; cf. [89, 90, 91, 92]. Here $\eta$ and $\nu$ are the
detection efficiency and the dark-count intensity, correspondingly.

##  III Model of time-dependent efficiency

In this section we consider an idea, enabling us to study the effect of dead
and relaxation times. For a correct formulation of our model, we should take
into account two facts. Firstly, the probability to detect a photon is zero
during the dead time $\tau_{\mathrm{d}}$ after the pulse. Secondly, the
probability to detect a photon is smoothly recovered with the relaxation time
$\tau_{\mathrm{r}}$, following the dead-time interval.

In the photodetection theory, the probability to detect a single photon is
described by the detection efficiency. This means that we can consider the
detection efficiency as a function of time $t$ passed after beginning of each
pulse. This function is zero in the time-interval $[0,\tau_{\mathrm{d}}]$.
After that the detection efficiency is smoothly recovered to its initial
value. This implies that this function is given by

$\displaystyle\xi(t)=\theta(t-\tau_{\mathrm{d}})\eta_{\mathrm{r}}(t-\tau_{\mathrm{d}}),$
(12)

where $\theta(t-\tau_{\mathrm{d}})$ is the Heaviside step-function and
$\eta_{\mathrm{r}}(t)$ is the recovering efficiency. We chose the latter in
the form

$\displaystyle\eta_{\mathrm{r}}(t)=1-\exp\left({-\frac{t}{\tau_{\mathrm{r}}}}\right),$
(13)

which is used in our paper as a model of detector recovering. Although this
model may be considered as an approximation—see, e.g., Refs. [93, 94] for a
more realistic description of the time-dependent detection efficiency
$\xi(t)$—its advantage consists in possibilities of obtaining expressions
suitable for analytical study. Nevertheless, the main results of our paper are
formulated in terms of an arbitrary function $\xi(t)$, which can also be
reconstructed experimentally, as is discussed in Sec. VII and in Ref. [93]. In
a more general context, the model given by the time-dependent efficiency
$\xi(t)$ can be considered as a phenomenological description. In principle,
this model could be derived from a microscopic theory of quantum transitions
in a superconducting nanowire.

##  IV Photocounting with independent measurement time windows

In this section we consider photocounting theory with the SNSPDs in the
scenario of independent MTWs; see Fig. 1. The process of photocounting starts
at the beginning of the time window of duration $\tau_{\mathrm{m}}$. Each MTW
is followed by a time interval of darkened detector-input sufficient for the
detector to recover fully. This eliminates the influence of the dead time and
the relaxation process from the previous MTW on the measurement result in the
current one. Alternatively, this technique can be implemented by a proper
postselection of MTWs without darkening detector input.

Figure 1:  Photocounting process with the SNSPDs in the scenario of
independent MTWs is schematically depicted. The voltage pulses (solid lines)
are counted during the MTWs duration of $\tau_{\mathrm{m}}$. According to Eq.
(12), the time-dependent efficiency (dashed lines) is zero after registering
each photon during the dead time $\tau_{\mathrm{d}}$, and after that it is
smoothly recovered with the relaxation time $\tau_{\mathrm{r}}$. Detector
input is darkened between the MTWs for the time interval sufficient for full
recovering of the detector (hatched area).

###  IV.1 Positive operator-valued measure

We start with consideration of the most general situation by assuming light
being in a nonmonochromatic mode. For our purposes, such a mode can be
characterized by an intensity function $I(t)$ normalized by the condition

$\displaystyle\int\limits_{0}^{\tau_{\mathrm{m}}}dtI(t)=1.$ (14)

The function $I(t)$ should be included explicitly in the POVM. In particular,
this means that the detection efficiency in the time domain $[t,t+\Delta t]$
is given by

$\displaystyle\int\limits_{t}^{t+\Delta t}dtI(t)\xi(t-t_{\mathrm{p}}),$ (15)

where $t_{\mathrm{p}}$ is the time moment at which the previous photon was
detected. A particular example of the function $I(t)$ reads

$\displaystyle I(t)=\frac{1}{\tau_{\mathrm{m}}},$ (16)

which corresponds to a momochromatic light mode. For the sake of simplicity,
we consider the unit detection efficiency and the zero dark-count rate. The
corresponding generalization to realistic values of these parameters is
straightforwardly obtained according to Eqs. (10) and (11).

Firstly, we note that the no-count element of the POVM is simply given by

$\displaystyle\hat{\Pi}_{0}=\hat{F}_{0}[1],$ (17)

which properly describes absence of any pulses inside the MTW. Next, we derive
the POVM element corresponding to the presence of a single pulse. For this
purpose we consider the unnormalized probability density
$\pi_{1}(t_{1}|\alpha)$ to get this pulse at the time moment $t_{1}$ given the
coherent state $\left|\alpha\right\rangle$. The infinitesimal probability to
get this click during the time interval $[t_{1},t_{1}+dt_{1}]$ is obtained as
the first expansion coefficient of the expression
$1-F_{0}[\alpha;I(t_{1})dt_{1}]$, cf. Eq. (6), with respect to $dt_{1}$. This
yields $|\alpha|^{2}I(t_{1})dt_{1}$. It should be multiplied by the
probabilities to get no-counts before and after the time moment $t_{1}$, which
are given by

$\displaystyle
F_{0}\left[\alpha;\int\limits_{0}^{t_{1}}dtI(t)\right]=\exp\left(-|\alpha|^{2}\int\limits_{0}^{t_{1}}dtI(t)\right)$
(18)

and

$\displaystyle
F_{0}\left[\alpha;\int\limits_{t_{1}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{1})\right]$
(19)
$\displaystyle=\exp\left(-|\alpha|^{2}\int\limits_{t_{1}}^{\tau_{\mathrm{d}}}dtI(t)\xi(t-t_{1})\right),$

respectively. Therefore, the unnormalized probability density to get a single
pulse at the time moment $t_{1}$ given the coherent state
$\left|\alpha\right\rangle$ reads

$\displaystyle\pi_{1}\left(t_{1}|\alpha\right)={|\alpha|^{2}}I\left(t_{1}\right)\exp\left[-|\alpha|^{2}\Xi_{1}\left(\mathbf{t}\right)\right],$
(20)

where

$\displaystyle\Xi_{1}\left(t_{1}\right)=\int\limits_{0}^{t_{1}}dtI(t)+\int\limits_{t_{1}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{1}).$
(21)

The $Q$ symbol of the corresponding POVM element,

$\displaystyle\Pi_{1}(\alpha)=\int\limits_{0}^{\tau_{\mathrm{m}}}dt_{1}\pi_{1}\left(t_{1}|\alpha\right),$
(22)

is obtained via integration with all possible values of $t_{1}$.

As the last step, we derive the rest of the POVM elements, i.e. those for
$n\geq 2$. Similar to the case of $n=1$, we derive the unnormalized
probability density $\pi_{n}(\mathbf{t}|\alpha)$ to get pulses at the time
moments $\mathbf{t}=(t_{1},\ldots,t_{n})$ given the coherent state
$\left|\alpha\right\rangle$. This is composed of the following components:

  1. (i)

The probability density to get a pulse at the time moment $t_{1}$ given by
$|\alpha|^{2}I(t_{1})$;

  2. (ii)

The probability densities to get pulses at the time moments $t_{i}$ for
$i=2\ldots n$ given by $|\alpha|^{2}I(t_{i})\xi(t_{i}-t_{i-1})$;

  3. (iii)

The probability to get no pulses from the time moment $t=0$ and up to the
first pulse at the time moment $t_{1}$ given by Eq. (18);

  4. (iv)

The probability to get no pulses in the time domains between the $i$th and
$(i+1)$th pulses given by

$\displaystyle\exp\left(-|\alpha|^{2}\int\limits_{t_{i}}^{t_{i+1}}dtI(t)\xi(t-t_{i})\right);$
(23)

  5. (v)

The probability to get no pulses in the time domain between the $n$th pulse
and the time moment $t=\tau_{\mathrm{m}}$ given by

$\displaystyle\exp\left(-|\alpha|^{2}\int\limits_{t_{n}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{n})\right).$
(24)

Multiplying all these factors we arrive at the expression

$\displaystyle\pi_{n}\left(\mathbf{t}|\alpha\right)={|\alpha|^{2n}}\mathcal{I}_{n}\left(\mathbf{t}\right)\exp\left[-|\alpha|^{2}\Xi_{n}\left(\mathbf{t}\right)\right],$
(25)

where

$\displaystyle\mathcal{I}_{n}\left(\mathbf{t}\right)=I(t_{1})\prod\limits_{i=2}^{n}I(t_{i})\xi\left(t_{i}-t_{i-1}\right),$
(26)

and

$\displaystyle\Xi_{n}\left(\mathbf{t}\right)=\int\limits_{0}^{t_{1}}dtI(t)+\sum\limits_{i=1}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}dtI(t)\xi(t-t_{i})$
$\displaystyle+\int\limits_{t_{n}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{n}).$
(27)

Equation (25) can be generalized to $n=1$ by setting
$\Xi_{1}\left(t_{1}\right)$ in the form of Eq. (21) and
$\mathcal{I}(t_{1})=I(t_{1})$.

The $Q$ symbols of the POVM elements in the case of $n\geq 1$ are given by

$\displaystyle\Pi_{n}(\alpha)=\int_{T_{n}}d^{n}\mathbf{t}\,\pi_{n}\left(\mathbf{t}|\alpha\right),$
(28)

where integration is taken over the time-ordering domain $T_{n}$ such that
$0\leq t_{1}\leq t_{2}\leq\ldots t_{n}\leq\tau_{\mathrm{m}}$. Employing
property (5), one obtains the general expression for the POVM elements

$\displaystyle\hat{\Pi}_{n}=:{\hat{n}^{n}}\int_{T_{n}}d^{n}\\!\mathbf{t}\,\mathcal{I}_{n}\left(\mathbf{t}\right)\exp\left[-\hat{n}\Xi_{n}\left(\mathbf{t}\right)\right]:.$
(29)

In the case of the time-dependent detection efficiency defined by Eq. (12),
the maximal value of $n$ is restricted by the number $N+1$ or $N$, where
$N=[\tau_{\mathrm{m}}/\tau_{\mathrm{d}}]$ is the number of whole dead-time
intervals, fitting inside the MTW. The latter case is suitable only if
$N=\tau_{\mathrm{m}}/\tau_{\mathrm{d}}$. In order to generalize this POVM to
the case of a nonunit detection efficiency $\eta$ and a nonzero dark-cont
intesity $\nu$, one can use replacement described by Eqs. (10) and (11).

For practical purposes, it is convenient to change the integration variables
to $\bm{\tau}=\\{\tau_{0},\tau_{1},\tau_{2},\ldots\tau_{n}\\}$,

$\displaystyle\tau_{0}=t_{1},$ (30) $\displaystyle\tau_{i}=t_{i+1}-t_{i},$
(31) $\displaystyle\tau_{n}=\tau_{\mathrm{m}}-t_{n}.$ (32)

An inverse relation,

$\displaystyle t_{i}=\sum\limits_{j=0}^{i-1}\tau_{i}$ (33)

can be substituted in Eqs. (26) and (IV.1). The new variables correspond to
the time intervals between neighboring pulses. Herewith, $\tau_{0}$ and
$\tau_{n}$ are the intervals from the beginning of the MTW up to the first
pulse and from the last pulse up to the end of the MTW, respectively. These
nonnegative variables obey the constraint

$\displaystyle\sum\limits_{i=0}^{n}\tau_{i}=\tau_{\mathrm{m}},$ (34)

which defines an $n$-dimensional simplex of $(n{+}1)$-dimensional space.

###  IV.2 Regular and irregular parts of the POVM

If we consider the time-dependent efficiency in the form of Eq. (12), the POVM
can further be specified as

$\displaystyle\hat{\Pi}_{n}=\hat{\Pi}^{\mathrm{(r)}}_{n}+\hat{\Pi}^{\mathrm{(i)}}_{n},$
(35)

where $\hat{\Pi}^{\mathrm{(r)}}_{n}$ and $\hat{\Pi}^{\mathrm{(i)}}_{n}$ are
referred to as regular and irregular parts, respectively. The regular part
describes the situation with all $n$ dead-time intervals fitted inside the
MTW. The irregular part describes the situation, for which the last dead-time
interval exceeds the MTW. This representation is made by splitting the
integral for $\tau_{n}$ in two parts.

Let us derive analytical expressions for regular and irregular parts of the
POVM. Herein the presence of the Heaviside theta-function in Eq. (12) will be
accounted in the integration domain. In this case, the regular part for
$n=0\ldots N$ is given by

$\displaystyle\hat{\Pi}^{\mathrm{(r)}}_{n}=$ (36)
$\displaystyle:{\hat{n}^{n}}\int\limits_{\tau_{\mathrm{d}}}^{\tau_{\mathrm{m}}-(n-1)\tau_{\mathrm{d}}}d\tau_{n}\int_{\Omega^{n}}d^{n-1}\\!\bm{\tau}\mathcal{J}_{n}(\bm{\tau})e^{-{\hat{n}}{\Xi}^{\mathrm{(r)}}_{n}(\bm{\tau})}:.$

For $n=N+1$ this part vanishes. Here

$\displaystyle\mathcal{J}_{1}(\bm{\tau})=I(\tau_{0}),$ (37)
$\displaystyle\mathcal{J}_{n}(\bm{\tau})=I(\tau_{0})\prod\limits_{i=2}^{n}I\left(\sum\limits_{j=0}^{i-1}\tau_{i}\right)\eta_{\mathrm{r}}(\tau_{i-1}-\tau_{\mathrm{d}})$
(38)

for $n\geq 2$,

$\displaystyle{\Xi}^{\mathrm{(r)}}_{n}$
$\displaystyle(\bm{\tau})=\int\limits_{0}^{\tau_{0}}dtI(t)$ (39)
$\displaystyle+\sum\limits_{j=1}^{n}\int\limits_{\tau_{\mathrm{d}}}^{\tau_{j}}dtI\left(\sum\limits_{k=0}^{j-1}\tau_{k}+t\right)\eta_{\mathrm{r}}(t-\tau_{\mathrm{d}}).$

The integration domain $\Omega_{n}$ is defined as

$\displaystyle\int_{\Omega^{n}}d^{n-1}\\!\bm{\tau}\ldots=\int\limits_{\tau_{\mathrm{d}}}^{\Delta_{n-1}}d\tau_{n-1}\ldots\int\limits_{\tau_{\mathrm{d}}}^{\delta_{1}}d\tau_{1}\ldots,$
(40)

where

$\displaystyle\delta_{i}=\tau_{\mathrm{m}}-(i-1)\tau_{\mathrm{d}}-\sum_{j=i+1}^{n}\tau_{j}.$
(41)

As mentioned, this integration is accounted for by the zero-value domains of
the Heaviside step-functions in Eq. (12).

The irregular part of the POVM reads

$\displaystyle\hat{\Pi}^{\mathrm{(i)}}_{n}=$ (42)
$\displaystyle:{\hat{n}^{n}}\int\limits_{0}^{\tau_{\mathrm{d}}}d\tau_{n}\int_{\Omega^{n}}d^{n-1}\\!\bm{\tau}\mathcal{J}_{n}(\bm{\tau})e^{-{\hat{n}}{\Xi}^{\mathrm{(i)}}_{n}(\bm{\tau})}:,$

for $n=0\ldots N$ and

$\displaystyle\hat{\Pi}^{\mathrm{(i)}}_{N+1}$
$\displaystyle=:{\hat{n}^{N+1}}\int\limits_{0}^{\tau_{\mathrm{m}}-N\tau_{\mathrm{d}}}d\tau_{N+1}$
(43)
$\displaystyle\times\int_{\Omega^{N+1}}d^{N}\\!\bm{\tau}\mathcal{J}_{N+1}(\bm{\tau})e^{-{\hat{n}}{\Xi}^{\mathrm{(i)}}_{N+1}(\bm{\tau})}:.$

Here

$\displaystyle{\Xi}^{\mathrm{(i)}}_{n}$
$\displaystyle(\bm{\tau})=\int\limits_{0}^{\tau_{0}}dtI(t)$ (44)
$\displaystyle+\sum\limits_{j=1}^{n-1}\int\limits_{\tau_{\mathrm{d}}}^{\tau_{j}}dtI\left(\sum\limits_{k=0}^{j-1}\tau_{k}+t\right)\eta_{\mathrm{r}}(t-\tau_{\mathrm{d}}).$

The integration domain $\Omega_{n}$ and the function
$\mathcal{J}_{n}(\bm{\tau})$ are the same as for the regular part.

Consider the case of a monochromatic mode [cf. Eq. (16)] and the recovering
detector efficiency in the form of Eq. (13). This gives a possibility to
present the functions $\mathcal{J}_{n}(\bm{\tau})$,
${\Xi}^{\mathrm{(r)}}_{n}(\bm{\tau})$, and
${\Xi}^{\mathrm{(i)}}_{n}(\bm{\tau})$ in the explicit form,

$\displaystyle\mathcal{J}_{n}(\bm{\tau})=\frac{1}{\tau_{\mathrm{m}}^{n}}\prod_{j=1}^{n-1}\eta_{\mathrm{r}}(\tau_{j}-\tau_{\mathrm{d}})$
(45)

for $n\geq 2$,

$\displaystyle{\Xi}^{\mathrm{(r)}}_{n}(\bm{\tau})=\eta_{n}-\frac{\tau_{\mathrm{r}}}{\tau_{\mathrm{m}}}\sum_{j=1}^{n}\eta_{\mathrm{r}}(\tau_{j}-\tau_{\mathrm{d}}),$
(46)
$\displaystyle{\Xi}^{\mathrm{(i)}}_{n}(\bm{\tau})=\eta_{n-1}-\frac{\tau_{n}}{\tau_{\mathrm{m}}}-\frac{\tau_{\mathrm{r}}}{\tau_{\mathrm{m}}}\sum_{j=1}^{n-1}\eta_{\mathrm{r}}(\tau_{j}-\tau_{\mathrm{d}}).$
(47)

Here

$\displaystyle\eta_{n}=\frac{\tau_{\mathrm{m}}-n\tau_{\mathrm{d}}}{\tau_{\mathrm{m}}}$
(48)

is the adjusting detection efficiency describing the time free from the dead-
time interval as a ratio to the duration of the MTW.

An important example corresponds to the case of zero relaxation time,
$\tau_{\mathrm{r}}=0$, such that $\eta_{\mathrm{r}}=1$. This situation takes
place for many types of detectors, such as avalanche photodiodes and
photomultiplier tubes. The regular part of the POVM of such detectors for
$n=0\ldots N$ is given by

$\displaystyle\hat{\Pi}_{n}^{\mathrm{(r)}}=\hat{F}_{n}\big{[}\eta_{n}\big{]},$
(49)

where $\eta_{n}$ is the adjusting detection efficiency and
$\hat{F}_{n}\big{[}\eta\big{]}$ is the POVM of the PNR detectors, cf. Eqs.
(48) and (2), respectively. For $n=N+1$ this part vanishes. The irregular part
of the corresponding POVM reads

$\displaystyle\hat{\Pi}_{n}^{\mathrm{(i)}}=\sum\limits_{k=0}^{n-1}\hat{F}_{k}\left[\eta_{n}\right]-\sum\limits_{k=0}^{n-1}\hat{F}_{k}\left[\eta_{n-1}\right]$
(50)

for $n=0\ldots N$ and

$\displaystyle\hat{\Pi}_{N+1}^{\mathrm{(i)}}=1-\sum\limits_{k=0}^{N}\hat{F}_{k}\left[\eta_{N}\right].$
(51)

for $n=N+1$. Combining both parts of the POVM in Eq. (35), we arrive at the
POVM, the $Q$ symbols of which correspond to the classical photodetection
theory with the dead time [74, 75, 76, 77, 78, 79, 80, 81, 82].

We remind that all expressions presented here are given for the unit detection
efficiency $\eta$ and the zero dark-count intensity $\nu$. These equations can
be simply rewritten to the case of realistic values of these quantities. For
this purpose, one should use the replacement described by Eqs. (10) and (11).

###  IV.3 Photocounting statistics vs photon-number statistics

As follows from the previous consideration, the photocounting statistics
(statistics of pulse numbers) may significantly differ from the photon-number
statistics. Averaging both sides of Eq. (8) with the density operator
$\hat{\rho}$, we arrive at the linear expression,

$\displaystyle P_{n}=\sum\limits_{m=0}^{+\infty}P_{n|m}\mathcal{P}_{m},$ (52)

connecting the photocounting distribution $P_{n}$ with the photon-number
distribution $\mathcal{P}_{m}=\left\langle
m\right|\hat{\rho}\left|m\right\rangle$ in the case of $\eta=1$ and $\nu=0$. A
similar expression in the case of array detectors is applied for
reconstructions of $\mathcal{P}_{m}$ from $P_{n}$ via regularization of an
ill-posed problem [69]. For realistic values of $\eta$ and $\nu$, one can use
$\mathcal{P}_{m}=\operatorname{Tr}\left(\hat{\rho}\hat{F}_{m}[\eta;\nu]\right)$,
where $\hat{F}_{m}[\eta;\nu]$ is obtained from $\hat{F}_{m}[1]$ [cf. Eq. (2)]
by the replacement (10). This procedure does not result in changing the
conditional probabilities $P_{n|m}$; cf. Ref. [95]. However, it enables one to
consider the effects caused solely by the realistic resolution of photon
numbers.

We apply the technique described by Eq. (9) to the POVM (29). This yields the
expression for the probability to get $n$ pulses given $m$ photons,

$\displaystyle
P_{n|m}=\frac{m!}{(m-n)!}\int_{T_{n}}d^{n}\\!\mathbf{t}\,\mathcal{I}_{n}\left(\mathbf{t}\right)\left[1-\Xi_{n}\left(\mathbf{t}\right)\right]^{m-n}$
(53)

for $m\geq n$ and

$\displaystyle P_{n|m}=0$ (54)

for $m<n$. The latter means that we cannot get more pulses than photons at the
detector input. Integration in Eq. (53) is performed in the time-ordering
domain $T_{n}$.

Applying the same method to Eq. (36) and Eqs. (42) and (43), we can subdivide
the conditional probability $P_{n|m}$ on regular and irregular parts,
respectively. Specifically, the regular part is given by

$\displaystyle P_{n|m}^{\mathrm{(r)}}=\frac{m!}{(m-n)!}$ (55)
$\displaystyle\times\int\limits_{\tau_{\mathrm{d}}}^{\tau_{\mathrm{m}}-(n-1)\tau_{\mathrm{d}}}d\tau_{n}\int_{\Omega^{n}}d^{n-1}\\!\bm{\tau}\mathcal{J}_{n}(\bm{\tau})\left[1-{\Xi}^{\mathrm{(r)}}_{n}(\bm{\tau})\right]^{m-n}$

for $n=0\ldots N$. The irregular part reads

$\displaystyle P_{n|m}^{\mathrm{(i)}}=\frac{m!}{(m-n)!}$ (56)
$\displaystyle\times\int\limits_{0}^{\tau_{\mathrm{d}}}d\tau_{n}\int_{\Omega^{n}}d^{n-1}\\!\bm{\tau}\mathcal{J}_{n}(\bm{\tau})\left[1-{\Xi}^{\mathrm{(i)}}_{n}(\bm{\tau})\right]^{m-n}$

for $n=0\ldots N$ and

$\displaystyle
P_{N+1|m}^{\mathrm{(i)}}=\frac{m!}{(m-N-1)!}\int\limits_{0}^{\tau_{\mathrm{m}}-N\tau_{\mathrm{d}}}d\tau_{N+1}$
(57)
$\displaystyle\times\int_{\Omega^{n}}d^{N}\\!\bm{\tau}\mathcal{J}_{N+1}(\bm{\tau})\left[1-{\Xi}^{\mathrm{(i)}}_{N+1}(\bm{\tau})\right]^{m-N-1}$

for $n=N+1$. Both parts vanish for $m<n$.

Let us consider a special model: the monochromatic mode given by Eq. (16) and
the recovering detector efficiency in the form of Eq. (13). In this case,
integrals in Eqs. (55), (56), and (57) can be evaluated analytically for each
values of $n$ and $m$ although the general equations have a complex structure.
The corresponding expressions for $P^{\mathrm{(r)}}_{n|m}$ and
$P^{\mathrm{(i)}}_{n|m}$ can be directly used in Eq. (52) for deriving the
photocounting statistics from the photon-number statistics.

In the important case of $n=m$, the probabilities $P_{n|n}$ are reduced to a
simple analytical form,

$\displaystyle
P_{n|n}=\frac{\tau_{\mathrm{r}}^{n}}{\tau_{\mathrm{m}}^{n}}\left[\sum_{l=0}^{n}a^{l}_{n-1}(-1)^{n-l}\frac{n!(2n-2-l)!}{l!(n-l)!(n-2)!}\right.$
(58)
$\displaystyle+\left.(-1)^{n-1}e^{-a_{n-1}}\sum_{l=0}^{n-2}a^{l}_{n-1}\frac{(2n-2-l)!}{l!(n-l-2)!}\right],$

where

$\displaystyle
a_{n}=\frac{\tau_{\mathrm{m}}-n\tau_{\mathrm{d}}}{\tau_{\mathrm{r}}}.$ (59)

The quantity (58) characterizes the ability of detectors to distinguish
between photon numbers. For the PNR detectors, i.e. for
$\tau_{\mathrm{d}}=\tau_{\mathrm{r}}=0$, it takes the unit value. It tends to
zero for the detectors, for which the click number is never equal to the
number of detected photons.

##  V Continuous-wave detection

A typical photocounting technique of the continuous-wave detection assumes no
interruptions between the MTWs; see Fig. 2. In such a scenario, the detector
may not be recovered after the last pulse from the previous MTW. This affects
on the probability of events in the current MTW. Therefore, photocounting
statistics depends on quantum states in previous MTWs.

Figure 2:  Photocounting with the technique of continuous-wave detection is
schematically depicted. The voltage pulses (solid lines) are counted inside
the MTW. The time-dependent efficiency $\xi(t)$ is shown by dashed lines. The
time $\tau$ between the last pulse in the $(l-1)$th MTW and its end is shown.

Let us numerate MTWs in the order as they appear in time. The $Q$ symbols of
the POVM in the $l$th MTW, $\Lambda_{n}\left(\bm{\alpha}^{l}\right)$, depend
on the amplitudes
$\bm{\alpha}^{l}=\begin{pmatrix}\alpha_{1},\ldots\alpha_{l}\end{pmatrix}$ in
the given and all previous MTWs. Here and in the following consideration, we
use the upper indices of bold symbols in order to designate the number of
entries in the corresponding sets of numbers. Therefore, the photocounting
formula in the P representation is given by

$\displaystyle
P_{n}=\int_{\mathbb{C}^{l}}d^{2l}\bm{\alpha}^{l}\,P(\alpha_{l})P(\alpha_{l-1})\ldots
P(\alpha_{1})\,\Lambda_{n}\big{(}\bm{\alpha}^{l}\big{)}.$ (60)

This formula in the operator form reads

$\displaystyle P_{n}=\operatorname{Tr}\left(\hat{\rho}^{\otimes
l}\,\hat{\Lambda}_{n}\right).$ (61)

The most significant difference of these relations from Eqs. (1) and (3)
consists in nonlinear dependence of the photocounting distribution on the
density operator $\hat{\rho}$. In this section we will derive the POVM
$\hat{\Lambda}_{n}$ for the scenario of the continuous-wave detection in an
explicit form and consider the memory effect of the previous MTWs on the
photocounting distribution in the current one.

###  V.1 Events dependent on previous measurements

Before we start with deriving the general expression for the POVM
$\hat{\Lambda}_{n}$ in the scenario of continuous-wave detection, we consider
an important part needed for this consideration. Let us assume that the last
pulse in the $(l-1)$th MTW occurs at the time moment $\tau_{n}=\tau$ [cf. Eq.
(32)] before its end; see Fig. 2. The POVM in the $l$th MTW should now depend
on this time.

The no-count element of the POVM in the considered case differs from Eq. (17)
and is given by

$\displaystyle\hat{\Pi}_{0}(\tau)=:\exp\left[-\Xi_{0}(\tau)\hat{n}\right]:,$
(62)

where

$\displaystyle\Xi_{0}(\tau)=\int\limits_{0}^{\tau_{\mathrm{m}}}dtI(t)\xi(t+\tau)$
(63)

corresponds to the no-count efficiency. Derivation of all other elements of
the POVM resembles the points listed in Sec. IV.1 but with two modifications.
Firstly, the unnormalized probability density to get the first pulse at the
time moment $t_{1}$ is now given by $|\alpha|^{2}I(t_{1})\xi(\tau+t_{1})$.
Secondly, the probability to get no pulses before the time moment $t_{1}$ is
modified to the form

$\displaystyle F_{0}$
$\displaystyle\left[\alpha;\int\limits_{0}^{t_{1}}dtI(t)\xi(\tau+t)\right]$
(64)
$\displaystyle=\exp\left(-|\alpha|^{2}\int\limits_{0}^{t_{1}}dtI(t)\xi(\tau+t)\right).$

Other points of the derivation are not changed.

The $Q$ symbols of the POVM conditioned by the time $\tau$ for $n\geq 1$ are
given by

$\displaystyle\Pi_{n}(\alpha;\tau)=\int_{T_{n}}d^{n}\\!\mathbf{t}\,\pi_{n}\left(\mathbf{t}|\alpha;\tau\right).$
(65)

Here $\pi_{n}\left(\mathbf{t}|\alpha;\tau\right)$ is the unnormalized
probability density to get pulses at the time moments $\mathbf{t}$ given the
coherent states $\left|\alpha\right\rangle$ and the time $\tau$ between the
last pulse in the previous MTW and its end. This conditional probability
density reads

$\displaystyle\pi_{n}\left(\mathbf{t}|\alpha;\tau\right)={|\alpha|^{2n}}\mathcal{I}_{n}\left(\mathbf{t};\tau\right)\exp\left[-|\alpha|^{2}\Xi_{n}\left(\mathbf{t};\tau\right)\right].$
(66)

Here

$\displaystyle\mathcal{I}_{n}\left(\mathbf{t};\tau\right)=\mathcal{I}_{n}\left(\mathbf{t}\right)\xi(\tau+t_{1})$
(67)

[cf. Eq. (26) for $\mathcal{I}_{n}\left(\mathbf{t}\right)$],

$\displaystyle\Xi_{n}\left(\mathbf{t};\tau\right)=\int\limits_{0}^{t_{1}}dtI(t)\xi(\tau+t)$
$\displaystyle+\sum\limits_{i=1}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}dtI(t)\xi(t-t_{i})+\int\limits_{t_{n}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{n})$
(68)

for $n\geq 2$ and

$\displaystyle\Xi_{1}\left(t_{1};\tau\right)=\int\limits_{0}^{t_{1}}dtI(t)\xi(\tau+t)+\int\limits_{t_{1}}^{\tau_{\mathrm{m}}}dtI(t)\xi(t-t_{1})$
(69)

for $n=1$.

To summarize this part, we note that the POVM conditioned by the time $\tau$
is given by

$\displaystyle\hat{\Pi}_{n}(\tau)=:{\hat{n}^{n}}\int_{T_{n}}d^{n}\\!\mathbf{t}\,\mathcal{I}_{n}\left(\mathbf{t};\tau\right)\exp\left[-\hat{n}\Xi_{n}\left(\mathbf{t};\tau\right)\right]:.$
(70)

Here $\mathcal{I}_{n}\left(\mathbf{t};\tau\right)$ is given by Eq. (67) for
$n\geq 1$ and $I_{0}=1$. Similarly, $\Xi_{n}\left(\mathbf{t};\tau\right)$ is
given by Eqs. (V.1), (69), and (63) for $n\geq 2$, $n=1$, and $n=0$,
respectively.

###  V.2 POVM in the Markovian approximation

In this subsection we derive an expression for the POVM in the scenario of the
continuous-wave detection considering the measurement in the $l$th MTW; cf.
Eqs. (60) and (61). In the next sections we will show that the dependence on
the number $l$ is negligible for $l\gg 1$, and its actual value is not
important. However, for the sake of consistency, this number should be
explicitly included in our consideration. We also assume that the ratio of
dead and relaxation times to the duration of the MTW is small,

$\displaystyle\frac{\tau_{\mathrm{d}}+\tau_{\mathrm{r}}}{\tau_{\mathrm{m}}}\ll
1.$ (71)

This yields for the time-dependent efficiency

$\displaystyle\xi(t\geq\tau_{\mathrm{m}})\approx 1,$ (72)

i.e. if no pulses are registered in the $(l-1)$th MTW, then the pulses in the
former MTWs do not affect the statistics in the $l$th MTW. This is the essence
of the Markovian approximation considered here.

Similar to Sec. IV, we consider the $Q$ symbols of the POVM as the
probabilities to get $n$ pulses given the coherent state
$\left|\alpha\right\rangle$. Let $\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$
be the probability density for the time-interval [between the last pulse and
the end of the MTW] $\tau$ in the $k$th MTW given the coherent amplitudes
$\bm{\alpha}^{k}$ in this and in all previous MTWs. This function is
normalized by the condition

$\displaystyle\Lambda^{(k)}_{0}\left(\bm{\alpha}^{k}\right)+\int\limits_{0}^{\tau_{\mathrm{m}}}d\tau\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)=1,$
(73)

where $\Lambda^{(k)}_{0}\left(\bm{\alpha}^{k}\right)$ is the $0$th element of
the $Q$ symbols of the POVM, i.e. the probability to get no pulses in the
$k$th MTW given the coherent amplitudes $\bm{\alpha}^{k}$. The form of these
functions will be specified latter.

Two kinds of the outcomes in the $(l-1)$th MTW affect on the statistics in the
$l$th one:

  1. (i)

No pulses in the $(l-1)$th MTW with the probability
$\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$ occur. In this case, the
unnormalized probability density to get pulses at the time moments
$\textbf{t}^{n}$ of the $l$th MTW is $\pi_{n}(\textbf{t}^{n}|\alpha_{l})$.

  2. (ii)

Pulses in the $(l-1)$th MTW occur such that the probability density for the
time-interval $\tau$ is $\rho^{(l-1)}\left(\tau|\bm{\alpha}^{l-1}\right)$. In
this case, the unnormalized probability density to get pulses at the time
moments $\textbf{t}^{n}$ of the $l$th MTW is
$\pi_{n}(\textbf{t}^{n}|\alpha_{l};\tau)$.

Employing the law of total probability, we get the unnormalized probability
density to get pulses at the time moments $\textbf{t}^{n}$ of the $l$th MTW
given the coherent amplitudes $\bm{\alpha}^{l}$ in current and all previous
MTWs in the form

$\displaystyle\lambda^{(l)}_{n}(\textbf{t}^{n}$
$\displaystyle|\bm{\alpha}^{l})=\pi_{n}(\textbf{t}^{n}|\alpha_{l})\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$
(74)
$\displaystyle+\int\limits_{0}^{\tau_{\mathrm{m}}}d\tau\pi_{n}(\textbf{t}^{n}|\alpha_{l};\tau)\rho^{(l-1)}\left(\tau|\bm{\alpha}^{l-1}\right).$

Integrating this relation with respect to $\textbf{t}^{n}$ in the time-
ordering domain $T_{n}$, we get for the $Q$ symbols of the POVM

$\displaystyle\Lambda^{(l)}_{n}(\bm{\alpha}^{l})$
$\displaystyle=\Pi_{n}(\alpha_{l})\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$
(75)
$\displaystyle+\int\limits_{0}^{\tau_{\mathrm{m}}}d\tau\Pi_{n}(\alpha_{l};\tau)\rho^{(l-1)}\left(\tau|\bm{\alpha}^{l-1}\right).$

This equation connects the POVM $\hat{\Lambda}^{(l)}_{n}$ for the $l$th MTW in
the scenario of continuous-wave detection with the POVM $\hat{\Pi}_{n}$ in the
scenario of independent MTW and the conditional POVM $\hat{\Pi}_{n}(\tau)$.

Let us now derive a technique for obtaining the functions
$\rho^{(l-1)}\left(\tau|\bm{\alpha}^{l-1}\right)$ and
$\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$. For this purpose, we
explicitly separate the $n$th time in Eqs. (74) and (75) as
$\textbf{t}^{n}=(\textbf{t}^{n-1},t_{n})$. The function
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ can be expressed via the
probability density
$\lambda^{(k)}_{n}(\textbf{t}^{n-1},t_{n}|\bm{\alpha}^{k})$ as

$\displaystyle\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ (76)
$\displaystyle=\sum\limits_{n=1}^{+\infty}\int_{T_{\tau}^{n-1}}d^{n-1}\textbf{t}^{n-1}\lambda^{(k)}_{n}(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau|\bm{\alpha}^{k}),$

where the term with $n=1$ assumes no integration and the integration domain
$T_{\tau}^{n-1}$ is defined as $0\leq t_{1}\leq t_{2}\leq\ldots
t_{n-1}\leq\tau_{\mathrm{m}}-\tau$. Indeed, integration with the first $(n-1)$
components of $\textbf{t}^{n}$ gives the probability density for the time
$t_{n}=\tau_{\mathrm{m}}-\tau$ in the case of $n$ pulses and sum is taken over
all $n\neq 0$.

Employing Eq. (76) to Eq. (74) and setting $n=0$ in Eq. (75), we arrive at a
system of recurrence relations for
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ and
$\Lambda_{0}\left(\bm{\alpha}^{k}\right)$,

$\displaystyle\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$
$\displaystyle=G(\tau|\alpha_{k})\Lambda^{(k-1)}_{0}\left(\bm{\alpha}^{k-1}\right)$
(77)
$\displaystyle+\int\limits_{0}^{\tau_{\mathrm{m}}}d\tau^{\prime}H(\tau|\alpha_{k};\tau^{\prime})\rho^{(k-1)}\left(\tau^{\prime}|\bm{\alpha}^{k-1}\right).$
$\displaystyle\Lambda^{(k)}_{0}(\bm{\alpha}^{k})$
$\displaystyle=\Pi_{0}(\alpha_{k})\Lambda^{(k-1)}_{0}\left(\bm{\alpha}^{k-1}\right)$
(78)
$\displaystyle+\int\limits_{0}^{\tau_{\mathrm{m}}}d\tau\Pi_{0}(\alpha_{k};\tau)\rho^{(k-1)}\left(\tau|\bm{\alpha}^{k-1}\right).$

Here we use the notations

$\displaystyle G(\tau|\alpha_{k})$ (79)
$\displaystyle=\sum\limits_{n=1}^{+\infty}\int_{T_{\tau}^{n-1}}d^{n-1}\textbf{t}^{n-1}\pi_{n}(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau|\alpha_{k}),$

and

$\displaystyle H(\tau|\alpha_{k};\tau^{\prime})$ (80)
$\displaystyle=\sum\limits_{n=1}^{+\infty}\int_{T_{\tau}^{n-1}}d^{n-1}\textbf{t}^{n-1}\pi_{n}(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau|\alpha_{k};\tau^{\prime}).$

Resolving the recurrence relations (77) and (78) with the initial conditions

$\displaystyle\rho^{(0)}=0\textrm{ and }\Lambda^{(0)}_{0}=1$ (81)

we may get explicit expressions for
$\rho^{(l-1)}\left(\tau|\bm{\alpha}^{l-1}\right)$ and
$\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$.

As a summary of this part we note that the POVM $\hat{\Lambda}_{n}$ in the
photocounting formula (60) and (61) is given by Eq. (75). Constituents of this
expression can be obtained as a solution to the system of recurrence relations
given by Eqs. (77) and (78). The POVM element
$\Lambda^{(l-1)}_{0}\left(\bm{\alpha}^{l-1}\right)$ can in principle be
excluded from Eq. (75) by employing the normalization condition (73). One can
also formulate the recurrence relation solely for the function
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ by excluding the POVM element
$\Lambda^{(k)}_{0}\left(\bm{\alpha}^{l-1}\right)$ in Eq. (77). Similarly to
the case of independent MTWs, the detection efficiency $\eta$ and dark-count
intensity $\nu$ can be easily included by applying the rules (10) and (11) to
all coherent amplitudes or the photon-number operators.

###  V.3 Approximation by the uniform distribution

In real practical applications, resolving the system of recurrence relations
(77) and (78) may be an involved numerical task. In this subsection we
introduce a reasonable approximation for this problem. The main condition for
its applicability is stronger than the condition (72) for the Markovian
approximation. Namely, we assume that there exists a time parameter
$\Delta\ll\tau_{\mathrm{m}}$ such that

$\displaystyle\xi(t\geq\Delta)\approx 1.$ (82)

This yields

$\displaystyle\pi_{n}\left(\mathbf{t}|\alpha;\tau>\Delta\right)\approx\pi_{n}\left(\mathbf{t}|\alpha\right),$
(83)
$\displaystyle\Pi_{n}\left(\alpha;\tau>\Delta\right)\approx\Pi_{n}\left(\alpha\right).$
(84)

Particularly, in the case of $\tau_{\mathrm{r}}=0$ the parameter $\Delta$
should be chosen as $\Delta=\tau_{\mathrm{d}}$. Another condition for the
discussed approximation assumes that the number of time bins free from dead-
and relaxation-time intervals must significantly exceed $\Delta$, i.e.
$\Delta\ll\tau_{\mathrm{m}}-|\alpha|^{2}(\tau_{\mathrm{d}}+\tau_{\mathrm{r}})$.
This implies that in the domain $\tau\in[0,\Delta]$ the probability density
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ can be modeled by the uniform
distribution,

$\displaystyle\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)=\frac{1-Q^{(k)}(\bm{\alpha}^{k})}{\Delta}.$
(85)

Here $Q^{(k)}(\bm{\alpha}^{k})$ is the probability to get no pulses inside the
time interval $\tau\in[0,\Delta]$ for the $k$th MTW.

In the approximation by the uniform distribution, the POVM (75) is
significantly simplified. Firstly, we split the integration domain in the
right-hand side of this equation into two parts: The first and the second ones
are $[0,\Delta]$ and $[\Delta,\tau_{\mathrm{m}}]$, respectively. Next, we
apply the simplification (84) to the second part. We also use the fact that

$\displaystyle\int\limits_{\Delta}^{\tau_{\mathrm{m}}}d\tau\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)=Q^{(k)}\left(\bm{\alpha}^{k}\right)-\Lambda^{(k)}_{0}\left(\bm{\alpha}^{k}\right),$
(86)

which directly follows from the normalization condition (73) and from the
expression

$\displaystyle
1-Q^{(k)}\left(\bm{\alpha}^{k}\right)=\int\limits_{0}^{\Delta}d\tau\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$
(87)

for the probability to get the last pulse in the domain $\tau\in[0,\Delta]$.
Finally, we apply the approximation (85) for the probability density
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$. As a result, the POVM (75) is
reduced to the form

$\displaystyle\Lambda^{(l)}_{n}(\bm{\alpha}^{l})$
$\displaystyle=Q^{(l-1)}\left(\bm{\alpha}^{l-1}\right)\Pi_{n}(\alpha_{l})$
(88)
$\displaystyle+\frac{1-Q^{(l-1)}\left(\bm{\alpha}^{l-1}\right)}{\Delta}\int\limits_{0}^{\Delta}d\tau\Pi_{n}(\alpha_{l};\tau).$

The information about the memory effect caused by events in the previous MTWs
is encoded in this relation by the probability
$Q^{(l-1)}\left(\bm{\alpha}^{l-1}\right)$.

The advantage of this approximation is that the probability density
$\rho^{(k)}\left(\tau|\bm{\alpha}^{k}\right)$ is replaced by the probability
$Q^{(k)}\left(\bm{\alpha}^{k}\right)$, which is a number depending on the
coherent amplitudes in the previous MTWs. In order to find the corresponding
recurrence relations, we integrate Eq. (77) in the domain $[0,\Delta]$ and
apply the technique used for derivation of Eq. (88). Thus we get

$\displaystyle
Q^{(l)}\left(\bm{\alpha}^{l}\right)=C\left(\alpha_{l}\right)Q^{(l-1)}\left(\bm{\alpha}^{l-1}\right)+B\left(\alpha_{l}\right),$
(89)

where

$\displaystyle
B\left(\alpha_{l}\right)=1-\frac{1}{\Delta}\int\limits_{0}^{\Delta}d\tau\int\limits_{0}^{\Delta}d\tau^{\prime}H(\tau|\alpha_{l};\tau^{\prime})$
(90) $\displaystyle
C\left(\alpha_{l}\right)=A\left(\alpha_{l}\right)-B\left(\alpha_{l}\right),$
(91) $\displaystyle A\left(\alpha_{l}\right)=1-\int\limits_{0}^{\Delta}d\tau
G(\tau|\alpha_{k}).$ (92)

The recurrence relation (89) should be resolved with the initial condition

$\displaystyle Q^{(0)}=1,$ (93)

which implies that the first MTW is not affected by the previous MTWs.

The solution to the relation (89) with the initial condition (93) reads

$\displaystyle\mathcal{Q}_{l-1}\big{(}\bm{\alpha}^{l-1}\big{)}=$
$\displaystyle B(\alpha_{l-1})+B(\alpha_{l-2})C(\alpha_{l-1})$
$\displaystyle+B(\alpha_{l-3})C(\alpha_{l-2})C(\alpha_{l-1})+\ldots.$ (94)

The first term of this relation describes the effect of the previous MTW. The
second term describes the effect of two previous MTWs, and so on. Therefore,
the memory effect from the previous MTWs is conveniently encoded by the
probability $\mathcal{Q}_{l-1}\big{(}\bm{\alpha}^{l-1}\big{)}$ in the POVM
(88).

An important feature of the solution (V.3) is that contributions of higher
terms quickly vanish with growing their number. In many practical situations,
it is sufficient to consider only a few first terms. Therefore, only several
previous MTWs affect the photocounting statistics in the given one. Hence, the
considered measurement process is ergodic, i.e. the dependence on $l$ vanishes
for $l\gg 1$ and statistics in different MTWs is almost the same. This implies
that the photocounting statistics of the $l$th MTW is equal to the statistics
obtained from events sampled from the MTWs with consecutive numbers. This
result justifies a technique of data processing widely used with the
continuous-wave detection. Indeed, in typical experimental applications one
averages events sampled from different MTWs without repeating the whole
measurement procedure and averaging data from the MTWs with the same number.

###  V.4 Photocounting statistics vs photon-number statistics

An important consequence from the nonlinear dependence of the photocounting
distribution on the density operator $\hat{\rho}$ in Eq. (61) is that the
expression (52) connecting photon-number and photocounting statistics does not
hold anymore. Evidently, in the considered situation it should be replaced by

$\displaystyle
P_{n}=\sum\limits_{m_{1},\ldots,m_{l}=0}^{+\infty}\Lambda_{n|m_{l},\ldots,m_{1}}\mathcal{P}_{m_{l}}\ldots\mathcal{P}_{m_{1}}.$
(95)

Here $P_{n}$ and $\mathcal{P}_{k}$ are photocounting and photon-number
distributions, respectively, and

$\displaystyle\Lambda_{n|m_{l},\ldots,m_{1}}=\left\langle
m_{l},\ldots,m_{1}\right|\hat{\Lambda}_{n}\left|m_{l},\ldots,m_{1}\right\rangle$
(96)

is the POVM (88) in the Fock representation, where we have omitted the MTW-
number $l$ for the sake of simplicity.

Converting the POVM (88) in the Fock representation, we get

$\displaystyle\Lambda_{n|m_{l},\ldots,m_{1}}$ (97)
$\displaystyle=\left[P_{n|m_{l}}-{D}_{n|m_{l}}\right]Q_{m_{l-1}\ldots
m_{1}}+{D}_{n|m_{l}}.$

Here we have introduced operators in the Fock representation. The probability
$P_{n|m}$ to get $n$ pulses given $m$ photons in the current MTW and no pulses
in the time-interval $\tau\in[0,\Delta]$ of the previous MTW is presented by
Eq. (53). The conditional probability ${D}_{n|m}=\left\langle
m\right|\hat{D}_{n}\left|m\right\rangle$ describes the situation with the
presence of last pulse in the time-interval $\tau\in[0,\Delta]$ of the
previous MTW. This probability is uniformly averaged with the time $\tau$ and
corresponds to the operator

$\displaystyle\hat{D}_{n}=\frac{1}{\Delta}\int\limits_{0}^{\Delta}d\tau\hat{\Pi}_{n}\left(\tau\right)$
(98)

[cf. Eq. (65) for the conditional POVM $\hat{\Pi}_{n}\left(\tau\right)$]. The
symbol $Q_{m_{l-1}\ldots m_{1}}$ is decomposed according to Eq. (V.3) as

$\displaystyle Q_{m_{l-1}\ldots m_{1}}=B_{m_{l-1}}$
$\displaystyle+B_{m_{l-2}}C_{m_{l-1}}$
$\displaystyle+B_{m_{l-3}}C_{m_{l-2}}C_{m_{l-1}}+\ldots.$ (99)

Here $B_{m}=\left\langle m\right|\hat{B}\left|m\right\rangle$ and
$C_{m}=\left\langle m\right|\hat{C}\left|m\right\rangle$ correspond to the
operators with the $Q$ symbols given by Eq. (90) and Eq. (91), respectively.

The conditional probabilities $D_{n|m}$ can be obtained from Eqs. (65), (66),
and (98) by applying the technique described in Sec. II. This leads to the
expression

$\displaystyle D_{n|m}=\frac{m!}{\Delta(m-n)!}$ (100)
$\displaystyle\times\int\limits_{0}^{\Delta}d\tau\int_{T_{n}}d^{n}\\!\mathbf{t}\,\mathcal{I}_{n}\left(\mathbf{t};\tau\right)\left[1-\Xi_{n}\left(\mathbf{t};\tau\right)\right]^{m-n}$

for $m\geq n$ and

$\displaystyle D_{n|m}=0$ (101)

for $m<n$. The coefficients $A_{m}$, $B_{m}$, and $C_{m}$ are expressed via
the operators $\hat{G}\left(\tau\right)$ and
$\hat{H}\left(\tau|\tau^{\prime}\right)$ [cf. Eqs. (90), (91), and (92)] in
the Fock representation as

$\displaystyle A_{m}=1-\int\limits_{0}^{\Delta}d\tau G_{m}\left(\tau\right),$
(102) $\displaystyle
B_{m}=1-\frac{1}{\Delta}\int\limits_{0}^{\Delta}d\tau\int\limits_{0}^{\Delta}d\tau^{\prime}H_{m}\left(\tau|\tau^{\prime}\right),$
(103) $\displaystyle C_{m}=A_{m}-B_{m}.$ (104)

Here the functions

$\displaystyle G_{m}\left(\tau\right)$
$\displaystyle=\sum\limits_{n=1}^{m}\frac{m!}{(m-n)!}$ (105)
$\displaystyle\times\int_{T_{\tau}^{n-1}}d^{n-1}\textbf{t}^{n-1}\mathcal{I}_{n}\left(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau\right)$
$\displaystyle\times\left[1-\Xi_{n}\left(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau\right)\right]^{m-n}$

and

$\displaystyle H_{m}$
$\displaystyle\left(\tau|\tau^{\prime}\right)=\sum\limits_{n=1}^{m}\frac{m!}{(m-n)!}$
(106)
$\displaystyle\times\int_{T_{\tau}^{n-1}}d^{n-1}\textbf{t}^{n-1}\mathcal{I}_{n}\left(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau;\tau^{\prime}\right)$
$\displaystyle\times\left[1-\Xi_{n}\left(\textbf{t}^{n-1},\tau_{\mathrm{m}}-\tau;\tau^{\prime}\right)\right]^{m-n}$

are obtained from Eqs. (79) and (80) with the technique described in Sec. II.

##  VI Examples for typical quantum states

In this section, we consider applications of the obtained theory for the
SNSPDs to derivation of photocounting statistics for typical quantum states.
We will concentrate on understanding roles of two factors: (1) the relaxation
time $\tau_{\mathrm{r}}$ in the scenario of independent MTWs and (2) the
memory effect of the previous MTWs in the scenario of continuous-wave
detection. For this purpose, we compare the statistic obtained from the PNR
detectors with three different situations. Firstly, we model the relaxation in
the scenario of independent MTWs by replacing the dead time
$\tau_{\mathrm{d}}$ with $\tau_{\mathrm{d}}+\tau_{\mathrm{r}}$ in the POVM
(49), (50), (51). Next, we directly apply the derived POVM (29) with the model
of smooth relaxation (13). Finally, we consider the memory effect of the
previous MTWs; cf. the POVM (88) and the photocounting formula (60). For all
cases we use a monochromatic mode described by Eq. (16).

The first example is the coherent state $\left|\alpha_{0}\right\rangle$. Since
its $P$ function is given by the Dirac delta-function,

$\displaystyle P(\alpha)=\delta(\alpha-\alpha_{0}),$ (107)

the corresponding statistics is obtained by replacing $\alpha$ or $\alpha_{k}$
for all $k$ with $\alpha_{0}$ in the $Q$ symbols of the POVM for the scenario
of independent MTWs or in continuous-wave detection, respectively. The result
is shown in Fig. 3. It is clear that the presence of the dead time alone
strictly changes the statistics in comparison with the case of the PNR
detectors. The smooth relaxation described by Eq. (13) and the memory effect
of the previous MTWs result in significant modifications of the statistics as
well.

Figure 3:  Photocounting statistics for the coherent state
$\left|\alpha_{0}\right\rangle$ with $\alpha_{0}=2$ is shown. Four cases are
presented: (i) PNR detectors with the statistics determined from the POVM (2);
(ii) the simple model described by the POVM (49), (50), (51) with the dead
time $\tau_{\mathrm{d}}$ replaced by $\tau_{\mathrm{d}}+\tau_{\mathrm{r}}$;
(iii) the model given by the POVM (29) in the scenario of independent MTWs;
(iv) the model of continuous-wave detection accounting the memory effect of
the previous MTWs [cf. Eq. (88)] for $l=3$ and $\Delta=0.3\tau_{\mathrm{m}}$.
In all appropriate cases we chose $\tau_{\mathrm{d}}=0.05\tau_{\mathrm{m}}$
and $\tau_{\mathrm{r}}=0.2\tau_{\mathrm{m}}$.

As the next example, we consider the Fock state $\left|k\right\rangle$
attenuated with the efficiency $\eta$, which can also be considered as the
detection efficiency. The corresponding density operator is given by

$\displaystyle\hat{\rho}=\sum\limits_{l=0}^{k}\mathcal{P}_{l}\left|l\right\rangle\left\langle
l\right|,$ (108)

where

$\displaystyle\mathcal{P}_{l}=\binom{k}{l}\eta^{l}(1-\eta)^{k-l}$ (109)

is the photon-number distribution. The photocounting distribution is directly
obtained from Eq. (52) and Eq. (95) in the scenarios of independent MTWs and
continuous-wave detection, respectively. It is shown in Fig. 4 for $k=4$.
Firstly, it is clearly seen that even the dead-time alone results in a
significant increase of events with $n<k$. Secondly, the consideration of a
smooth model for the relaxation crucially changes the probability distribution
as well. Finally, we note the significant impact of the previous MTWs.

Figure 4: Photocounting statistics as in Fig. 3 is shown but for the Fock
state $\left|4\right\rangle$. The upper and lower plots correspond to $\eta=1$
and $\eta=0.8$, respectively.

As the last example, we consider the squeezed vacuum state,

$\displaystyle\left|r\right\rangle=\frac{1}{\sqrt{\cosh
r}}\sum_{n=0}^{\infty}\binom{2n}{n}^{1/2}\left(\frac{\tanh
r}{2}\right)^{n}\left|2n\right\rangle,$ (110)

where $r$ is the squeezing parameter. The corresponding photocounting
distribution can be obtained with two different methods. The first one is
based on the fact that the photon-number distribution for this state is given
by

$\displaystyle\mathcal{P}_{n}$ $\displaystyle=\frac{(i\sinh
r)^{n}}{[1+(2-\eta)\eta\sinh^{2}r]^{\frac{n+1}{2}}}$ (111)
$\displaystyle\times\operatorname{P}_{n}\left[\frac{(1-\eta)\eta\sinh
r}{i\sqrt{1+(2-\eta)\eta\sinh^{2}r}}\right],$

where $\operatorname{P}_{n}(x)$ are the Legendre polynomials. These
probabilities can be substituted in Eqs. (52) and (95) in order to obtain the
photocounting distribution for the SNSPDs in the scenarios of independent MTWs
and continuous-wave detection, respectively. Alternatively, for the scenario
of independent MTWs one can directly use the photocounting formula (1) with
the POVM (29) and get the photocounting distribution in the form

$\displaystyle P_{n}=\int_{T^{n}}d^{n}\textbf{t}p_{n}(\textbf{t}),$ (112)

where

$\displaystyle
p_{n}(\textbf{t})=\operatorname{Tr}\left[\hat{\pi}_{n}(\textbf{t})\hat{\rho}\right]$
(113)

is the unnormalized probability density to get pulses at the time moments t,
$\hat{\rho}=\left|r\right\rangle\left\langle r\right|$ is the density
operator, and $\hat{\pi}_{n}(\textbf{t})$ is the operator, the $Q$ symbol of
which is given by $\pi_{n}(\textbf{t}|\alpha)$, cf. Eq. (25). The function
$p_{n}(\textbf{t})$ can be obtained explicitly as

$\displaystyle p_{n}(\textbf{t})$
$\displaystyle=\frac{n!i^{n}\mathcal{I}_{n}(\textbf{t})\sinh^{n}r}{[S_{n}(\textbf{t})]^{n+1}}$
(114) $\displaystyle\times\operatorname{P}_{n}\left(\frac{i\sinh
r[\Xi_{n}(\textbf{t})-1]}{S_{n}(\textbf{t})}\right),$

where

$\displaystyle
S_{n}(\textbf{t})={\sqrt{1-\sinh^{2}r\Xi_{n}(\textbf{t})\left[\Xi_{n}(\textbf{t})-2\right]}}.$
(115)

With these expressions, the integral in Eq. (112) can be evaluated
numerically. Similar calculations can be directly conducted for the scenario
of continuous-wave detection. The result is presented in Fig. 5. Evidently,
dead and relaxation times as well as the memory effect of the previous MTWs
play a crucial role in the photocounting statistics for the squeezed vacuum
states.

Figure 5:  Photocounting statistics as in Fig. 3 is shown but for the squeezed
vacuum state $\left|r\right\rangle$. The squeezing parameter and the detection
efficiency are chosen as $r=1.5$ and $\eta=0.8$, respectively.

##  VII Reconstruction of the time-dependent efficiency

For our purposes, we have used the time-dependent efficiency in the form of a
decaying exponent as is given by Eqs. (12) and (13). However, the realistic
function may differ from this simple model. In this section we analyze a
technique reported in Ref. [93], enabling reconstruction of the efficiency
$\xi(t)$ from the experimental data.

Let us consider a monochromatic wave in the coherent state
$\left|\alpha_{0}\right\rangle$ in the MTW of duration $\tau_{\mathrm{m}}$.
The intensity of this wave (the mean energy per time unit) is given by the
value $\lambda=|\alpha_{0}|^{2}/\tau_{\mathrm{m}}$. We will derive the
probability density for the time between pulses in such a scenario. It is
composed of two parts:

  * •

The probability density to get the second pulse at the time moment $t$ given
the first pulse registered at the time moment $t=0$ in the form
$\lambda\xi(t)$.

  * •

The probability to get no pulses in the time domain $(0,t)$ given by

$\displaystyle\exp\left[-\lambda\int\limits_{0}^{t}d\tau\xi(\tau)\right].$
(116)

The resulting probability density for the time between pulses is given by

$\displaystyle
P(t)=\lambda\xi(t)\exp\left[-\lambda\int\limits_{0}^{t}d\tau\xi(\tau)\right],$
(117)

which is the product of two mentioned components.

Similar to Eq. (82) we assume the existence of time $\Delta$, for which the
relaxation processes are finished. Let the intensity be chosen such that the
condition

$\displaystyle\lambda\ll\left[\int\limits_{0}^{\Delta}d\tau\xi(\tau)\right]^{-1}$
(118)

is satisfied. For times $t\leq\Delta$ the probability density (117) can be
approximated as

$\displaystyle P(t)\approx\lambda\xi(t).$ (119)

Therefore, the probability density of the time between pulses for
$t\leq\Delta$ is proportional to the time-dependent efficiency $\xi(t)$. This
fact can be used for its reconstruction in experiments.

##  VIII Summary and conclusions

To summarize, we have proposed a photodetection theory of the SNSPDs for the
technique of counting voltage pulses, which occur during the MTWs. We have
concentrated with three issues: dead time, relaxation time, and the memory
effect from the previous MTWs. The latter can be eliminated with the
considered here technique of independent MTWs assuming either darkening
detector input after each MTW or a proper postselection of the MTWs.

Our idea is based on modeling the dead time and the relaxation process by the
time-dependent efficiency. This efficiency becomes zero after each pulse
during the dead-time interval and then it is smoothly recovered. For a better
understanding of the role of dead and relaxation times, the recovering part of
this efficiency has been approximated by the exponential function. However,
the realistic dependence may have a different form, which can also be applied
in our theory. We have analyzed an experimental technique of its
reconstruction from experimental data.

The measurement technique of continuous-wave detection assuming no
interruptions between the MTWs is considered in the Markovian approximation.
In this case, the well-known photodetection formula significantly differs from
its standard form. Indeed, the memory effect of the previous MTWs results in
nonlinear dependence of the photocounting distribution on the density
operator.

In the most general case, photocounting statistics in each MTW depends on its
number. We have shown that in the Markovian scenario this dependence is weak
such that the measurement process can be considered as ergodic. This result
justifies a widely-used experimental technique of averaging results from the
events sampled in all MTWs.

The SNSPDs are widely used in many experimental works in quantum optics. They
also successfully applied for practical implementations of quantum-information
technologies. We hope that our results will be useful for a proper analysis of
corresponding experimental data and for further development of modern quantum
technologies.

###### Acknowledgements.

The authors acknowledge support from the National Research Foundation of
Ukraine through the project 2020.02/0111 “Nonclassical and hybrid correlations
of quantum systems under realistic conditions”. They also thank B. Hage and J.
Kröger for enlightening discussions.

## References

  * Hadfield [2009] R. H. Hadfield, Nat. Photonics 3, 696 (2009).
  * Eisaman  _et al._ [2011] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011).
  * Natarajan  _et al._ [2012] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, Supercond. Sci. Technol. 25, 063001 (2012).
  * You [2020] L. You, Nanophotonics 9, 2673 (2020).
  * Kok  _et al._ [2007] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
  * Aaronson and Arkhipov [2013] S. Aaronson and A. Arkhipov, Theory Comput. 9, 143 (2013).
  * Gisin and Thew [2007] N. Gisin and R. Thew, Nat. Photonics 1, 165 (2007).
  * Xu  _et al._ [2020] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, Rev. Mod. Phys. 92, 025002 (2020).
  * Pirandola  _et al._ [2020] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, B. D., R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Adv. Opt. Photon. 12, 1012 (2020).
  * Giovannetti  _et al._ [2011] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011).
  * Degen  _et al._ [2017] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).
  * Polino  _et al._ [2020] E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, AVS Quantum Science 2, 024703 (2020).
  * Somani  _et al._ [2001] S. Somani, S. Kasapi, K. Wilsher, W. Lo, R. Sobolewski, and G. Gol’tsman, J. Vac. Sci. Technol. B. 19, 2766 (2001).
  * Zhang  _et al._ [2003] J. Zhang _et al._ , Electron. Lett. 39, 1086 (2003).
  * Puszka  _et al._ [2013] A. Puszka, L. D. Sieno, A. D. Mora, A. Pifferi, D. Contini, G. Boso, A. Tosi, L. Hervé, A. Planat-Chrétien, A. Koenig, and J.-M. Dinten, Biomed. Opt. Express. 4, 1351 (2013).
  * Zysk  _et al._ [2007] A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart M.D., J. of Biomedical Optics 12, 051403 (2007).
  * Fujimoto  _et al._ [1995] J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, Nat. Med. 1, 970 (1995).
  * Semenov  _et al._ [2001] A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, Phys. C: Supercond. 351, 349 (2001).
  * Gol’tsman  _et al._ [2001] G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, Appl. Phys. Lett. 79, 705 (2001).
  * Zhang  _et al._ [2020] H. Zhang, L. Xiao, B. Luo, J. Guo, L. Zhang, and J. Xie, J. Phys. D: Appl. Phys. 53, 013001 (2020).
  * Esmaeil Zadeh  _et al._ [2021] I. Esmaeil Zadeh, J. Chang, J. W. N. Los, S. Gyger, A. W. Elshaari, S. Steinhauer, S. N. Dorenbos, and V. Zwiller, Appl. Phys. Lett. 118, 190502 (2021).
  * Kerman  _et al._ [2006] A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, Appl. Phys. Lett. 88, 111116 (2006).
  * Yang  _et al._ [2007] J. K. W. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M. Rosfjord, and K. K. Berggren, IEEE Trans. Appl. Supercond. 17, 581 (2007).
  * Semenov  _et al._ [2009] A. Semenov, P. Haas, H.-W. Hübers, K. Ilin, M. Siegel, A. Kirste, D. Drung, T. Schurig, and A. Engel, J. Mod. Opt. 56, 345 (2009).
  * Kerman  _et al._ [2009] A. J. Kerman, J. K. W. Yang, R. J. Molnar, E. A. Dauler, and K. K. Berggren, Phys. Rev. B 79, 100509(R) (2009).
  * Bulaevskii  _et al._ [2012] L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, Phys. Rev. B 85, 014505 (2012).
  * Zotova and Vodolazov [2012] A. N. Zotova and D. Y. Vodolazov, Phys. Rev. B 85, 024509 (2012).
  * Engel and Schilling [2013] A. Engel and A. Schilling, J. Appl. Phys. 114, 214501 (2013).
  * Renema  _et al._ [2014] J. J. Renema, R. Gaudio, Q. Wang, Z. Zhou, A. Gaggero, F. Mattioli, R. Leoni, D. Sahin, M. J. A. de Dood, A. Fiore, and M. P. van Exter, Phys. Rev. Lett. 112, 117604 (2014).
  * Engel  _et al._ [2015] A. Engel, J. J. Renema, K. Il’in, and A. Semenov, Supercond. Sci. Technol. 28, 114003 (2015).
  * Vodolazov [2017] D. Y. Vodolazov, Phys. Rev. Applied 7, 034014 (2017).
  * Sunter and Berggren [2018] K. A. Sunter and K. K. Berggren, Appl. Opt. 57, 4872 (2018).
  * Zhao  _et al._ [2018] Q.-Y. Zhao, D. F. Santavicca, D. Zhu, B. Noble, and K. K. Berggren, Appl. Phys. Lett. 113, 082601 (2018).
  * Allmaras  _et al._ [2019] J. P. Allmaras, A. G. Kozorezov, B. A. Korzh, K. K. Berggren, and M. D. Shaw, Phys. Rev. Applied 11, 034062 (2019).
  * Zhong  _et al._ [2020] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Science 370, 1460 (2020).
  * Zhong  _et al._ [2021] H.-S. Zhong, Y.-H. Deng, J. Qin, H. Wang, M.-C. Chen, L.-C. Peng, Y.-H. Luo, D. Wu, S.-Q. Gong, H. Su, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, J. J. Renema, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett. 127, 180502 (2021).
  * Hadfield  _et al._ [2006] R. H. Hadfield, J. L. Habif, J. Schlafer, R. E. Schwall, and S. W. Nam, Appl. Phys. Lett. 89, 241129 (2006).
  * Takesue  _et al._ [2007] H. Takesue, S. W. Nam, Q. Zhang, R. H. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, Nat. Photonics 1, 343 (2007).
  * Rosenberg  _et al._ [2009] D. Rosenberg, C. G. Peterson, J. W. Harrington, P. R. Rice, N. Dallmann, K. T. Tyagi, K. P. McCabe, S. Nam, B. Baek, R. H. Hadfield, R. J. Hughes, and J. E. Nordholt, New J. Phys. 11, 045009 (2009).
  * Liu  _et al._ [2010] Y. Liu, T.-Y. Chen, J. Wang, W.-Q. Cai, X. Wan, L.-K. Chen, J.-H. Wang, S.-B. Liu, H. Liang, L. Yang, C.-Z. Peng, K. Chen, Z.-B. Chen, and J.-W. Pan, Opt. Express 18, 8587 (2010).
  * Sasaki  _et al._ [2011] M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, Opt. Express 19, 10387 (2011).
  * Wang  _et al._ [2012] S. Wang, W. Chen, J.-F. Guo, Z.-Q. Yin, H.-W. Li, Z. Zhou, G.-C. Guo, and Z.-F. Han, Opt. Lett. 37, 1008 (2012).
  * Tang  _et al._ [2015] Y.-L. Tang, H.-L. Yin, S.-J. Chen, Y. Liu, W.-J. Zhang, X. Jiang, L. Zhang, J. Wang, L.-X. You, J.-Y. Guan, D.-X. Yang, Z. Wang, H. Liang, Z. Zhang, N. Zhou, X. Ma, T.-Y. Chen, Q. Zhang, and J.-W. Pan, IEEE J. Sel. Top. Quantum Electron. 21, 116 (2015).
  * Tang  _et al._ [2014] Y.-L. Tang, H.-L. Yin, S.-J. Chen, Y. Liu, W.-J. Zhang, X. Jiang, L. Zhang, J. Wang, L.-X. You, J.-Y. Guan, D.-X. Yang, Z. Wang, H. Liang, Z. Zhang, N. Zhou, X. Ma, T.-Y. Chen, Q. Zhang, and J.-W. Pan, Phys. Rev. Lett. 113, 190501 (2014).
  * Takemoto  _et al._ [2015] K. Takemoto, Y. Nambu, T. Miyazawa, Y. Sakuma, T. Yamamoto, S. Yorozu, and Y. Arakawa, Sci. Rep. 5, 14383 (2015).
  * Valivarthi  _et al._ [2015] R. Valivarthi, I. Lucio-Martinez, P. Chan, A. Rubenok, C. John, D. Korchinski, C. Duffin, F. Marsili, V. Verma, M. D. Shaw, J. A. Stern, S. W. Nam, D. Oblak, Q. Zhou, J. A. Slater, and W. Tittel, J. Mod. Opt. 62, 1141 (2015).
  * Tang  _et al._ [2016] Y.-L. Tang, H.-L. Yin, Q. Zhao, H. Liu, X.-X. Sun, M.-Q. Huang, W.-J. Zhang, S.-J. Chen, L. Zhang, L.-X. You, Z. Wang, Y. Liu, C.-Y. Lu, X. Jiang, X. Ma, Q. Zhang, T.-Y. Chen, and J.-W. Pan, Phys. Rev. X 6, 011024 (2016).
  * Yin  _et al._ [2016] H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, Phys. Rev. Lett. 117, 190501 (2016).
  * Boaron  _et al._ [2018] A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, Phys. Rev. Lett. 121, 190502 (2018).
  * Liu  _et al._ [2019a] H. Liu, W. Wang, K. Wei, X.-T. Fang, L. Li, N.-L. Liu, H. Liang, S.-J. Zhang, W. Zhang, H. Li, L. You, Z. Wang, H.-K. Lo, T.-Y. Chen, F. Xu, and J.-W. Pan, Phys. Rev. Lett. 122, 160501 (2019a).
  * Liu  _et al._ [2019b] Y. Liu, Z.-W. Yu, W. Zhang, J.-Y. Guan, J.-P. Chen, C. Zhang, X.-L. Hu, H. Li, C. Jiang, J. Lin, T.-Y. Chen, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, Phys. Rev. Lett. 123, 100505 (2019b).
  * Wang  _et al._ [2019] S. Wang, D.-Y. He, Z.-Q. Yin, F.-Y. Lu, C.-H. Cui, W. Chen, Z. Zhou, G.-C. Guo, and Z.-F. Han, Phys. Rev. X 9, 021046 (2019).
  * Minder  _et al._ [2019] M. Minder, M. Pittaluga, G. L. Roberts, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, Nat. Photonics 13, 334 (2019).
  * Fang  _et al._ [2020] X.-T. Fang, P. Zeng, H. Liu, M. Zou, W. Wu, Y.-L. Tang, Y.-J. Sheng, Y. Xiang, W. Zhang, H. Li, Z. Wang, L. You, M.-J. Li, H. Chen, Y.-A. Chen, Q. Zhang, C.-Z. Peng, X. Ma, T.-Y. Chen, and J.-W. Pan, Nat. Photonics 14, 422 (2020).
  * Chen  _et al._ [2020] J.-P. Chen, C. Zhang, Y. Liu, C. Jiang, W. Zhang, X.-L. Hu, J.-Y. Guan, Z.-W. Yu, H. Xu, J. Lin, M.-J. Li, H. Chen, H. Li, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, Phys. Rev. Lett. 124, 070501 (2020).
  * Wei  _et al._ [2020] K. Wei, W. Li, H. Tan, Y. Li, H. Min, W.-J. Zhang, H. Li, L. You, Z. Wang, X. Jiang, T.-Y. Chen, S.-K. Liao, C.-Z. Peng, F. Xu, and J.-W. Pan, Phys. Rev. X 10, 031030 (2020).
  * Sun  _et al._ [2016] Q.-C. Sun, Y.-L. Mao, S.-J. Chen, W. Zhang, Y.-F. Jiang, Y.-B. Zhang, W.-J. Zhang, S. Miki, T. Yamashita, H. Terai, X. Jiang, T.-Y. Chen, L.-X. You, X.-F. Chen, Z. Wang, J.-Y. Fan, Q. Zhang, and J.-W. Pan, Nat. Photonics 10, 671 (2016).
  * Valivarthi  _et al._ [2016] R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, Nat. Photonics 10, 676 (2016).
  * Shalm  _et al._ [2015] L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, Phys. Rev. Lett. 115, 250402 (2015).
  * Bohmann  _et al._ [2017] M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, Phys. Rev. A 95, 033806 (2017).
  * Bohmann  _et al._ [2018] M. Bohmann, J. Tiedau, T. Bartley, J. Sperling, C. Silberhorn, and W. Vogel, Phys. Rev. Lett. 120, 063607 (2018).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, Cambridge, 2010).
  * Mandel and Wol [1995] L. Mandel and E. Wol,  _Optical Coherence and Quantum Optics_ (Cambridge University Press, Cambridge, 1995).
  * Kelley and Kleiner [1964] P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
  * Paul  _et al._ [1996] H. Paul, P. Törmä, T. Kiss, and I. Jex, Phys. Rev. Lett. 76, 2464 (1996).
  * Castelletto  _et al._ [2007] S. A. Castelletto, I. P. Degiovanni, V. Schettini, and A. L. Migdall, J. Mod. Opt. 54, 337 (2007).
  * Schettini  _et al._ [2007] V. Schettini, S. V. Polyakov, I. P. Degiovanni, G. Brida, S. Castelletto, and A. L. Migdall, IEEE J. Sel. Top. Quantum Electron. 13, 978 (2007).
  * Blanchet  _et al._ [2008] J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, Phys. Rev. Lett. 101, 233604 (2008).
  * Hloušek  _et al._ [2019] J. Hloušek, M. Dudka, I. Straka, and M. Ježek, Phys. Rev. Lett. 123, 153604 (2019).
  * Achilles  _et al._ [2003] D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, Opt. Lett. 28, 2387 (2003).
  * Fitch  _et al._ [2003] M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, Phys. Rev. A 68, 043814 (2003).
  * Řeháček  _et al._ [2003] J. Řeháček, Z. Hradil, O. Haderka, J. Peřina, and M. Hamar, Phys. Rev. A 67, 061801(R) (2003).
  * Sperling  _et al._ [2012] J. Sperling, W. Vogel, and G. S. Agarwal, Phys. Rev. A 85, 023820 (2012).
  * Ricciardi and Esposito [1966] L. M. Ricciardi and F. Esposito, Kybernetik 3, 148 (1966).
  * Müller [1973] J. W. Müller, Nucl. Instrum. Methods 112, 47 (1973).
  * Müller [1974] J. W. Müller, Nucl. Instrum. Methods 117, 401 (1974).
  * Cantor and Teich [1975] B. I. Cantor and M. C. Teich, J. Opt. Soc. Am. 65, 786 (1975).
  * Teich  _et al._ [1978] M. C. Teich, L. Matin, and B. I. Cantor, J. Opt. Soc. Am. 68, 386 (1978).
  * Vannucci and Teich [1978] G. Vannucci and M. C. Teich, Opt. Commun. 25, 267 (1978).
  * Saleh [1978] B. Saleh,  _Photoelectron Statistics_ (Springer, Berlin, Heidelberg, 1978).
  * Rapp  _et al._ [2019] J. Rapp, Y. Ma, R. M. A. Dawson, and V. K. Goyal, IEEE Trans. Signal Process. 67, 3471 (2019).
  * Straka  _et al._ [2020] I. Straka, J. Grygar, J. Hloušek, and M. Ježek, J. Lightwave Technol. 38, 4765 (2020).
  * Cahill and Glauber [1969a] K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1882 (1969a).
  * Cahill and Glauber [1969b] K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969b).
  * Glauber [1963] R. J. Glauber, Phys. Rev. 131, 2766 (1963).
  * Sudarshan [1963] E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
  * Mandel  _et al._ [1964] L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. 84, 435 (1964).
  * Lamb, Jr. and Scully [1969] W. Lamb, Jr. and M. O. Scully, in  _Polarization: Matiere et Rayonnement_ , edited by H. Bederson (Presses Universitaires de France, Paris, 1969) p. 363.
  * Pratt [1969] W. K. Pratt,  _Laser Communication Systems_ (Wiley, New York, 1969).
  * Karp  _et al._ [1970] S. Karp, E. L. O’Neill, and R. M. Gagliardi, Proc. IEEE 58, 1611 (1970).
  * Lee  _et al._ [2004] H. Lee, U. Yurtsever, P. Kok, G. M. Hockney, C. Adami, S. L. Braunstein, and J. P. Dowling, J. Mod. Opt. 51, 1517 (2004).
  * Semenov  _et al._ [2008] A. A. Semenov, A. V. Turchin, and H. V. Gomonay, Phys. Rev. A 78, 055803 (2008).
  * Autebert  _et al._ [2020] C. Autebert, G. Gras, E. Amri, M. Perrenoud, M. Caloz, H. Zbinden, and F. Bussières, J. Appl. Phys. 128, 074504 (2020).
  * Caloz  _et al._ [2017] M. Caloz, B. Korzh, N. Timoney, M. Weiss, S. Gariglio, R. J. Warburton, C. Schönenberger, J. Renema, H. Zbinden, and F. Bussières, Appl. Phys. Lett. 110, 083106 (2017).
  * Kovalenko  _et al._ [2018] O. P. Kovalenko, J. Sperling, W. Vogel, and A. A. Semenov, Phys. Rev. A 97, 023845 (2018).