††thanks: Current address is Keio University Shonan Fujisawa Campus, 5322
Endo, Fujisawa, Kanagawa 252-0882, Japan

# Phase diffusion and the small-noise approximation in linear amplifiers:
Limitations and beyond

Andy Chia  Centre for Quantum Technologies, National University of Singapore,
3 Science Drive 2, Singapore 117543<EMAIL_ADDRESS>Michal Hajdušek
Centre for Quantum Technologies, National University of Singapore, 3 Science
Drive 2, Singapore 117543  Rosario Fazio  ICTP, Strada Costiera 11, I-34151
Trieste, Italy  Dipartemento di Fisica, Università di Napoli "Federico II",
Monte S. Angelo, I-80126, Italy  Leong-Chuan Kwek  Centre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2, Singapore
117543  MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654,
Singapore  National Institute of Education, Nanyang Technological University,
1 Nanyang Walk, Singapore 637616  Vlatko Vedral  Centre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2, Singapore
117543  Department of Physics, University of Oxford, Parks Road, Oxford, OX1
3PU, UK

###### Abstract

The phase of an optical field inside a linear amplifier is widely known to
diffuse with a diffusion coefficient that is inversely proportional to the
photon number. The same process occurs in lasers which limits its intrinsic
linewidth and makes the phase uncertainty difficult to calculate. The most
commonly used simplification is to assume a narrow photon-number distribution
for the optical field (which we call the small-noise approximation). For
coherent light, this condition is determined by the average photon number. The
small-noise approximation relies on (i) the input to have a good signal-to-
noise ratio, and (ii) that such a signal-to-noise ratio can be maintained
throughout the amplification process. Here we ask: For a coherent input, how
many photons must be present in the input to a quantum linear amplifier for
the phase noise at the output to be amenable to a small-noise analysis? We
address these questions by showing how the phase uncertainty can be obtained
without recourse to the small-noise approximation. It is shown that for an
ideal linear amplifier (i.e. an amplifier most favourable to the small-noise
approximation), the small-noise approximation breaks down with only a few
photons on average. Interestingly, when the input strength is increased to
tens of photons, the small-noise approximation can be seen to perform much
better and the process of phase diffusion permits a small-noise analysis. This
demarcates the limit of the small-noise assumption in linear amplifiers as
such an assumption is less true for a nonideal amplifier.

##  1 Introduction

It was shown in the 1960s that linear amplification of light is an inherently
noisy process [1, 2, 3]. As a result, the input phase suffers from an
increased uncertainty during the course of amplification. The process by which
this occurs in a phase-insensitive linear amplifier is phase diffusion, the
same process responsible for the natural linewidth of single-mode lasers [4,
5, 6, 7, 8]. Aside from fundamental interests, phase diffusion has also been
shown to be useful for quantum random number generation [9, 10, 11, 12]. More
recently it has also been used to demonstrate the applicability of phase
squeezing as a means to reduce phase noise in linear amplifiers [13].

Phase diffusion in linear amplifiers has been studied with a variety of
methods [14, 15, 16, 17, 18, 19]. In general, one would like to characterise
phase diffusion by calculating the diffusion coefficient, or by calculating a
sensible measure of phase uncertainty (often the phase variance but not
exclusively [17, 20, 21, 22]). However, phase diffusion is inversely
proportional to the photon number which makes the diffusion coefficient or
variance difficult to calculate without approximations. The most widely used
approximation assumes the input light to be sufficiently intense such that its
photon number has an average much greater than its standard deviation (see
e.g. [15, 16]). The simplification gained from such an assumption is that one
may replace the photon number by its average. Although this approximation is
extremely useful and continues to be a standard method of analysis in quantum
optics, it restricts one to field states with a narrow photon-number
distribution. For example, weak coherent states, which may be produced by
attenuating the output of a laser may not satisfy the small-noise
approximation. These states have been shown to be useful for quantum
cryptography [23, 24, 25]. In addition, states with low number of photons are
useful in numerous fields outside of quantum information science, such as
single-molecule spectroscopy, light detection, and many others [26, 27, 28].
We note that the small-noise approximation is, in spirit, similar to what
others might call a “semiclassical” or “mean-field” approximation in quantum
optics.111The semiclassical approximation replaces the bosonic annihilation
operator $\hat{a}$ by $\langle{\hat{a}}\rangle+\delta\hat{a}$ and neglects
$(\delta\hat{a})^{2}$ and terms with higher order in $\delta\hat{a}$, where
$\delta\hat{a}$ is the deviation of $\hat{a}$ from its average value.

A primary function of linear amplifiers is to boost the strength of a weak
signal to detectable levels [29]. For quantum linear amplifiers, increasing
the signal strength comes at a cost—the amplifier must add noise if it is to
operate according to the principles of quantum mechanics. The quantum limit on
how much noise such an amplifier must add were investigated in the early days
of linear amplifiers [1, 2, 3], and has been extended recently [30]. If the
input to the amplifier is a noisy signal (which is typically the case) this
noise also gets amplified so in general the output consists of the amplified
input signal, the amplified input noise, and the noise the amplifier adds. As
a result, the signal-to-noise ratio degrades after amplification. This
suggests that the small-noise approximation tends to get worse over the course
of amplification, with the breakdown becoming more accute for weaker inputs.
One can then try to optimise the signal-to-noise ratio by using an ideal
linear amplifier—an amplifier which adds the least amount of noise required by
quantum mechanics. It is therefore particularly interesting to know how well
the small-noise approximation performs for an ideal linear amplifier as this
is the most favourable scenario for the approximation to hold. In this paper
we show that even for an ideal linear amplifier, the small-noise approximation
is still inadequate in the regime of low-intensity inputs (few photons on
average) and large amplifier gain. This is our main result which essentially
demarcates the limit of the small-noise approximation in treating phase
diffusion in linear amplifiers.

An important aspect of the small-noise approximation is that it allows one to
calculate the phase noise analytically. Naturally, one wonders if it is still
possible to make some analytical progress when the small-noise approximation
fails, or do we have to resort to numerics? In this paper we show that it is
possible to generalise the small-noise approximation and obtain the phase
uncertainty for weak inputs. In this treatment the phase uncertainty is given
by a so-called inverse-number expansion. The expansion is checked against
numerics and is also valid for nonideal linear amplifiers. We also derive a
closed-form expression for the output phase noise under the small-noise
approximation, which is then compared with the inverse-number expansion. An
appreciable difference between the phase noise obtained from the small-noise
approximation  and the inverse-number expansion can be seen for low-intensity
coherent inputs where only a few photons are present on average (and more
generally intensities that are on the order of unity). However, the
applicability of the small-noise approximation is seen to be restored by
including, on average, an additional ten photons or slightly less. This
highlights the sensitivity of the small-noise approximation to the
discreteness (or photon nature) of weak coherent inputs.

The paper is organised as follows. We first introduce our linear-amplifier
model in Sec. 2 (in terms of a master equation), and define our model of phase
diffusion. Here we also review some essential features of the standard linear
amplifier and state our measure of phase uncertainty. Here we also motivate
the relevant parameter regime to operate the amplifier in and the importance
of the coherent state as a simple model of the amplifier input. The small-
noise approximation is then explained in Sec. 3 and we shall see how this
approximation leads to a closed-form expression for the phase uncertainty. The
details are left to Appendix A. The inverse-number expansion—which is our
proposed method for going beyond the small-noise approximation—then follows
quite naturally from this. The idea behind the inverse-number expansion can be
simply explained so we leave its derivation and checks to Appendices B and C.
Using the inverse-number expansion we then present our main result in Sec. 4,
where the phase uncertainty obtained by the two approaches are compared
against each other. Here we shall see that a treatment of the phase noise
beyond the small-noise approximation is necessary for a coherent input whose
photon number is on the order of unity. This is despite the fact that such an
input gets added as little noise as possible (a condition for which the small-
noise approximation is least likely to breakdown during amplification). Of
course, our chosen path to this result is not without any drawbacks so both
the advantages and disadvantages of our approach are discussed in Appendix D.
We then conclude in Sec. 5 with a summary of our work and a discussion of the
road ahead for extending the results of this paper.

##  2 Phase diffusion in linear amplifiers

###  2.1 Linear-amplifier model

There are a number of linear-amplifier models that one can choose from. A
universal model in fact exists (a two-mode squeezer [30]) provided the noise
introduced by the amplifier is additive [31, 32]. However, for phase
diffusion, the focus of attention has been on a master-equation realisation of
the linear amplifier and we will follow suit. An important element of the
master-equation model is that it can be recast as a Fokker–Planck equation—the
quintessential equation for diffusive processes—which justifies calling the
phase evolution (of the signal being amplified) phase diffusion. The master
equation for a linear amplifier is given by

$\displaystyle\frac{d}{dt}\rho(t)=\kappa_{\downarrow}\bigg{[}\hat{a}\rho(t)\hat{a}^{\dagger}-\frac{1}{2}\hat{a}^{\dagger}\hat{a}\rho(t)-\frac{1}{2}\rho(t)\hat{a}^{\dagger}\hat{a}\bigg{]}+\kappa_{\uparrow}\bigg{[}\hat{a}^{\dagger}\rho(t)\hat{a}-\frac{1}{2}\hat{a}\hat{a}^{\dagger}\rho(t)-\frac{1}{2}\rho(t)\hat{a}\hat{a}^{\dagger}\bigg{]},$
(1)

where $\kappa_{\downarrow}$ and $\kappa_{\uparrow}$ are positive real numbers,
and $\hat{a}$ and $\hat{a}^{\dagger}$ are annihilation and creation operators
satisfying the canonical commutation relation
$[\hat{a},\hat{a}^{\dagger}]=\hat{1}$. Such a master equation may be derived
by considering the interaction of a bosonic mode with a collection of two-
level atoms. The terms proportional to $\kappa_{\downarrow}$ and
$\kappa_{\uparrow}$ collectively describe photon loss and gain respectively.
Hence the coefficients $\kappa_{\downarrow}$ and $\kappa_{\uparrow}$ can be
thought of as rates at which photons are lost or gained. An input bosonic
field prepared in the state $\rho(0)$ then experiences amplification when
there is net gain i.e. $\kappa_{\uparrow}>\kappa_{\downarrow}$, a condition
usually achieved via population inversion in the two-level atoms. The simplest
way to see this is a direct calculation of the output photon-number using (1),

$\displaystyle\big{\langle}{\hat{a}^{\dagger}\hat{a}}\big{\rangle}_{t}=G_{t}\,\big{\langle}{\hat{a}^{\dagger}\hat{a}}\big{\rangle}_{0}+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\big{(}G_{t}-1\big{)}\;,$
(2)

where we have defined the photon-number gain

$\displaystyle
G_{t}=e^{\kappa_{-}\,t}\;,\quad\kappa_{-}=\kappa_{\uparrow}-\kappa_{\downarrow}\;.$
(3)

The output can be seen to comprise one part which is simply given by the input
but now amplified by a factor $G_{t}$, and another portion, independent of the
input, interpreted as noise due to spontaneous emission. Note that a given
value of photon-number gain $G_{t}$ is determined by the product of the
amplification time $t$ and the amplification rate $\kappa_{-}$. When
$\kappa_{\downarrow}=0$, the model is said to describe an ideal linear
amplifier, so called because in this case the amplifier adds the least amount
of noise consistent with quantum mechanics [33]. From (2), the noise can be
seen to be minimised when $\kappa_{\downarrow}=0$.

###  2.2 Amplifier parameter regime and input signal model

As we mentioned above, a linear amplifier serves primarily to increase a weak
signal to measurable strengths [3, 29]. The high-gain limit of linear
amplifiers ($G_{t}\longrightarrow\infty$) is thus of special interest in
research, both theoretically (such as addressing the amount of noise an
amplifier must add [3, 29]), and experimentally (such as the existing
experimental efforts to build linear amplifiers in circuit QED [29, 34, 35]).
We therefore focus on the high-gain limit here as well since this is the most
relevant regime.

Aside from a suitable amplifier model we also need a suitable model for the
amplifier input. For this we consider coherent states. Coherent states provide
a particularly good model for the input signal because their signal-to-noise
ratio is easily tunable via its average photon number. Another reason to
consider coherent states is that quantum optical experiments are typically
performed with lasers and the state of laser light can be effectively modelled
as coherent. In fact, a recent experiment demonstrating phase diffusion and
how it can be countered using squeezing was performed with a coherent state
input [13]. Aside from these advantages of coherent states, there are also
reasons for not considering other commonly used quantum-optical states: First,
for phase diffusion, states with perfect angular symmetry—the vacuum, thermal,
or number states—cannot be used as input models as they have a maximally
diffused phase. Second, for the phase-preserving amplifier model of (1), one
would not expect to observe the effects of nonclassical features on phase
diffusion such as squeezing, sub-Poissonian statistics, or macroscopic
superpositions (corresponding to the squeezed, number, and Schrödinger-cat
states). This is because such features will be completely lost even at
moderate gains ($G_{t}=2$) [33, 36, 37, 38, 39]. It has also been shown that
some nonclassicality can persist [40, 41, 42, 43]. Despite this, one should
keep in mind that one usually cares about nonclassical states because they are
useful for quantum information processing (e.g. see Refs. [44, 45]). In these
cases, most nonclassical states are interesting only when the state is
maintained (e.g. a Schrödinger-cat state remains a cat state). This is a
stronger requirement than just having any nonclassical state, a requirement
which cannot be satisfied by our amplifier model.222Generally, when
nonclassicality is important, one may choose more clever ways of
amplification. For example, a probabilistic amplifier may be used to boost a
cat state if we want to combat photon loss and at the same time and have the
state remain as a cat for whatever information processing protocol to be used.
Hence we do not consider nonclassical inputs in this paper. Our objective here
is to make clear the breakdown of the small-noise approximation in phase
diffusion.

###  2.3 Phase diffusion from the Glauber–Sudarshan distribution

Coherent states are simplest to describe using the $P$ (also called the
Glauber–Sudarshan) representation of $\rho(t)$. This is defined by a
$P(\alpha,\alpha^{*},t)$ such that the state at any time $t$ may be expanded
as

$\displaystyle\rho(t)=\int_{\mathbbm{C}}d^{2}\alpha\;P(\alpha,\alpha^{*},t)\left|{\alpha}\rangle\langle{\alpha}\right|\;,$
(4)

where the integral runs over the entire complex plane $\mathbbm{C}$. It is
well known that $P(\alpha,\alpha^{*},t)$ satisfies a Fokker–Planck equation
which can be derived from (1) [46]. It is also well known that the
Fokker–Planck equation can be converted to a set of stochastic differential
equations for $\alpha(t)$ and $\alpha^{*}(t)$ whose dynamics sample phase
space so as to be consistent with $P(\alpha,\alpha^{*},t)$ [46, 47]. In this
paper we adopt the latter approach of stochastic differential equations. That
is, instead of considering an evolving $P$ distribution, we consider $\alpha$
and $\alpha^{*}$ to be changing in time, described by

$\displaystyle d\alpha(t)={}$
$\displaystyle\frac{\kappa_{-}}{2}\;\alpha(t)\,dt+\sqrt{\kappa_{\uparrow}}\,\,dZ(t)$
(5) $\displaystyle d\alpha^{*}(t)={}$
$\displaystyle\frac{\kappa_{-}}{2}\;\alpha^{*}(t)\,dt+\sqrt{\kappa_{\uparrow}}\,\,dZ^{*}(t)\;,$
(6)

where $dZ(t)$ is a complex Wiener increment given by

$\displaystyle\big{[}dZ(t)\big{]}^{2}=\big{[}dZ^{*}(t)\big{]}^{2}$
$\displaystyle=0\;,$ (7) $\displaystyle dZ^{*}(t)\,dZ(t)$
$\displaystyle=dt\;.$ (8)

However, to describe phase diffusion we ought to change variables from
$\alpha$ and $\alpha^{*}$ to $N$ and $\Phi$, where
$\alpha=\sqrt{N}\,\exp(i\Phi)$. Our model for phase diffusion is thus given by

$\displaystyle d\Phi(t)$
$\displaystyle=\sqrt{\frac{\kappa_{\uparrow}}{2N(t)}}\,\;dV_{\Phi}(t)\;,$ (9)
$\displaystyle dN(t)$
$\displaystyle=\big{[}\kappa_{\uparrow}+\kappa_{-}\,N(t)\,\big{]}\,dt+\sqrt{2\,\kappa_{\uparrow}\,N(t)}\;dV_{N}(t)\;.$
(10)

These equations may be derived by converting (1) to its equivalent
Fokker–Planck equation in terms of the Glauber–Sudarshan distribution [46].
For phase diffusion one then needs to transform the Fokker–Planck equation
into polar coordinates [15, 16]. It is then simple to show that (9) and (10)
are equivalent to the Fokker–Planck equation in polar form. Since we are only
interested in amplification, $\kappa_{-}$ will be a positive real number.
Equations (9) and (10) are to be interpreted as Itô equations with $dV_{N}(t)$
and $dV_{\Phi}(t)$ being independent Wiener increments satisfying

$\displaystyle\big{[}dV_{N}(t)\big{]}^{2}=\big{[}dV_{\Phi}(t)\big{]}^{2}$
$\displaystyle=dt\;,$ (11) $\displaystyle dV_{N}(t)\,dV_{\Phi}(t)$
$\displaystyle=0\;.$ (12)

It is now clear from (9) that the phase $\Phi(t)$ is coupled nonlinearly to
$N(t)$. It is essentially this nonlinear coupling to $N(t)$ that makes the
phase uncertainty difficult to obtain in closed form. However, as Itô
equations, (9) and (10) can be manipulated according to the rules of Itô
calculus [48]. It is this aspect of our approach that allows us to go further
than the small-noise approximation in quantifying the amount of phase noise
appearing at the amplifier’s output. Existing treatments on the phase
uncertainty use Heisenberg equations of motion [18], the Pegg–Barnett phase
operator [15, 19], or the phase distribution $\wp_{\Phi}(\phi,t)$ obtained
from quasiprobability distributions [16, 17]. While each treatment can be
argued to have its own appeal, they all make the small-noise approximation.

In order to study the effectiveness of the small-noise approximation we will
calculate the time-dependent phase variance

$\displaystyle{\rm V}\big{[}\Phi(t)\big{]}={\rm
E}\big{[}\Phi^{2}(t)\big{]}-\big{\\{}{\rm
E}\big{[}\Phi(t)\big{]}\big{\\}}^{2}\;.$ (13)

The expectation value of any function of phase $f(\Phi)$ at time $t$ is
defined formally by

$\displaystyle{\rm
E}\big{[}f\big{(}\Phi(t)\big{)}\big{]}=\int^{\phi_{0}+2\pi}_{\phi_{0}}d\phi\;f(\phi)\,\wp_{\Phi}(\phi,t)\;,$
(14)

where the phase distribution $\wp_{\Phi}(\phi,t)$ is defined as the marginal
distribution of $\tilde{P}(n,\phi,t)$ (the $P$ function parametrised by $n$
and $\phi$) with $n$ integrated over. Note that we are following the
convention of using a capital letter to denote a random variable and the
corresponding small letter for its realisation. There are several measures of
phase uncertainty that one can choose from [21, 22]. Thus, given that phase is
a cyclic variable one may wonder why we have not used other measures which are
better at handling cyclic property of phase. The reason is that the cyclic
nature of phase never really shows up here, at least for the simple example of
a coherent-state input considered here. We will see that the phase
distribution of the amplified field is always a single-peaked function which
can always be placed in the centre of an appropriately chosen interval
$[\phi_{0},\phi_{0}+2\pi]$. In this case the usual variance as given by (13)
can be used [21].

It is widely known that $P(\alpha,\alpha^{*},t)$ can be solved in closed form
for Gaussian input states. However, this does not imply that the phase noise
can be derived in closed form. This is because on reparametrising
$P(\alpha,\alpha^{*},t)$ to get $\tilde{P}(n,\phi,t)$, the nonlinear
dependence of phase on the photon number gets introduced into the problem. The
phase variance then involves calculating the first and second moments with
respect to the marginal distribution $\wp_{\Phi}(\phi,t)$. It is shown in
Appendix D [see (D)] that even when restricted to the high-gain limit, the
phase distribution $\wp_{\Phi}(\phi,t)$ is very complicated for a coherent-
state input which makes the moments of the phase impossible to derive.

##  3 The small-noise approximation and beyond

The small-noise approximation is first explained in Sec. 3.1 and we show how
an expression for the phase variance can be obtained under this approximation.
This naturally sets the stage for the main method of analysis—the inverse-
number expansion in Sec. 3.2. The expansion can be stated quite simply and the
idea behind it is in fact quite straightforward. Therefore we leave the
details of its derivation to Appendix B.

###  3.1 Phase variance within the small-noise approximation

To obtain the phase variance we need its second moment. Using the Itô chain
rule and (9) we obtain

$\displaystyle\frac{d}{dt}\;{\rm
E}\big{[}\Phi^{2}(t)\big{]}=\frac{\kappa_{\uparrow}}{2}\;{\rm
E}\bigg{[}\frac{1}{N(t)}\bigg{]}\;.$ (15)

The second moment of phase thus depends on the expectation value of ${\rm
E}[1/N(t)]$. As alluded to in Sec. 1, the small-noise approximation assumes
the field being amplified has a narrow photon-number distribution so that one
may regard $N(t)$ as essentially a sure variable (a deterministic quantity)
whose value is equal to ${\rm E}[N(t)]$. Under this assumption, one may thus
replace (15) by

$\displaystyle\frac{d}{dt}\;{\rm
E}\big{[}\Phi^{2}(t)\big{]}=\frac{\kappa_{\uparrow}}{2\,{\rm E}[N(t)]}\;.$
(16)

We can then derive ${\rm E}[N(t)]$ directly from (10) to be [or take it from
(2) with the appropriate change of notation since the photon number is already
a normally ordered quantity]

$\displaystyle{\rm E}\big{[}N(t)\big{]}=G_{t}\,{\rm
E}\big{[}N(0)\big{]}+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\big{(}G_{t}-1\big{)}\;,$
(17)

Here we see explicitly that under the small-noise approximation the
fluctuations in $N(t)$ are ignored as it is only the mean of $N(t)$ that is
used to determine the phase variance. This is clearly quite different to ${\rm
E}[1/N(t)]$ where both a deterministic part and a noisy part contribute to the
denominator inside the expectation value. As we shall see, the inverse-number
expansion on the other hand attempts to deal with the $1/N(t)$ dependence
wholly which allows the fluctuations in $N(t)$ as given in (10) to influence
the phase variance. We can now derive the phase uncertainty under the small-
noise approximation by substituting (17) into (16). The derivation is a little
tedious so we leave the details to Appendix A. The result is simply

$\displaystyle{\rm V}[\Phi(t)]={}$ $\displaystyle{\rm
V}[\Phi(0)]-\frac{1}{2}\,\ln\\!\Bigg{[}\frac{{\rm E}[N(0)]}{{\rm
E}[N(0)]+\kappa_{\uparrow}/\kappa_{-}}\Bigg{]}$
$\displaystyle-\frac{1}{2}\ln\Bigg{[}\frac{\big{\\{}{\rm
E}[N(0)]+\kappa_{\uparrow}/\kappa_{-}\big{\\}}G_{t}}{\big{\\{}{\rm
E}[N(0)]+\kappa_{\uparrow}/\kappa_{-}\big{\\}}G_{t}-\kappa_{\uparrow}/\kappa_{-}}\Bigg{]}.$
(18)

Later we will compare the phase variance obtained from the inverse-number
expansion to (3.1).

###  3.2 Inverse-number expansion

In order to go further than the small-noise approximation we must try to deal
with the ${\rm E}[1/N(t)]$ appearing in (15). To this end, let us work out how
${\rm E}[1/N(t)]$ changes in time. Using (10) and Itô calculus we find

$\displaystyle\frac{d}{dt}\;{\rm
E}\bigg{[}\frac{1}{N(t)}\bigg{]}=-\,\kappa_{-}\,{\rm
E}\bigg{[}\frac{1}{N(t)}\bigg{]}+\kappa_{\uparrow}\;{\rm
E}\bigg{[}\frac{1}{N^{2}(t)}\bigg{]}\;.$ (19)

Hence we see that ${\rm E}[1/N(t)]$ depends on ${\rm E}[1/N^{2}(t)]$. However,
since amplification increases the photon number we expect that on average
$N(t)$ to increase, and hence $1/N(t)$ to decrease. This is illustrated in
Fig. 1 where different realisations of $N(t)$ corresponding to an ideal linear
amplifier are shown.

Figure 1:  (a) Fifty realisations of $N(t)$ obtained from (10) with
$\kappa_{\downarrow}=0$ and $\kappa_{\uparrow}=2$ for a coherent-state input
$|{\alpha}\rangle$ with $|\alpha|^{2}=N(0)=3$. (b) Ensemble average of $N(t)$
obtained from averaging over the realisations in (a) (blue solid curve), and
analytically from (10) (red dashed curve).

If at time $t$ we find that on average $1/N^{2}(t)\ll 1/N(t)$, then we can
neglect the second term in (19) and obtain, as a first-order approximation,

$\displaystyle{\rm
E}\bigg{[}\frac{1}{N(t)}\bigg{]}\approx\frac{1}{G_{t}}\;{\rm
E}\bigg{[}\frac{1}{N(0)}\bigg{]}\;.$ (20)

Clearly, one can continue in a similar fashion and improve on (20) by taking
into account the effect of ${\rm E}[1/N^{2}(t)]$. The evolution of ${\rm
E}[1/N^{2}(t)]$ is simple to obtain and is given by

$\displaystyle\frac{d}{dt}\;{\rm E}\bigg{[}\frac{1}{N^{2}(t)}\bigg{]}=$
$\displaystyle-2\,\kappa_{-}\,{\rm
E}\bigg{[}\frac{1}{N^{2}(t)}\bigg{]}+4\,\kappa_{\uparrow}\,{\rm
E}\bigg{[}\frac{1}{N^{3}(t)}\bigg{]}\;.$ (21)

On neglecting ${\rm E}[1/N^{3}(t)]$ we can then solve (21) straightforwardly.
The second-order approximation to ${\rm E}[1/N(t)]$ is then obtained by first
solving (21) with ${\rm E}[1/N^{3}(t)]$ neglected and then using the resulting
expression for ${\rm E}[1/N^{2}(t)]$ to solve (19). Performing these steps we
arrive at

$\displaystyle{\rm E}\bigg{[}\frac{1}{N(t)}\bigg{]}$
$\displaystyle\approx\frac{1}{G_{t}}\;{\rm
E}\bigg{[}\frac{1}{N(0)}\bigg{]}+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\,\frac{1}{G_{t}}\>\bigg{(}\,1-\frac{1}{G_{t}}\bigg{)}\,{\rm
E}\bigg{[}\frac{1}{N^{2}(0)}\bigg{]}\;.$ (22)

From this we can now appreciate the essential simplification made by the
small-noise approximation—namely that it avoids the coupling of ${\rm
E}[1/N(t)]$ to higher order moments in $1/N(t)$. In general we will find that
$E[1/N^{k}(t)]$ couples to $E[1/N^{k+1}(t)]$ so that in principle, solving for
${\rm E}[1/N(t)]$ requires us to solve an infinite set of coupled differential
equations involving all the moments of $1/N(t)$. In practice however, we would
expect (and in fact we shall find) that truncating the set of coupled
differential equations at ${\rm E}[1/N^{K}(t)]$ by neglecting ${\rm
E}[1/N^{K+1}(t)]$ for a finite $K$ captures the dynamics of ${\rm E}[1/N(t)]$
sufficiently accurately. Our objective is thus to first obtain a general
expression for ${\rm E}[1/N(t)]$ for any $K$ (i.e. any truncation) and
subsequently an expression for the phase variance ${\rm V}[\Phi(t)]$. It will
be convenient to define

$\displaystyle\Upsilon(t)=\frac{1}{N(t)}\;,$ (23)

where using Itô’s lemma, one can show that $\Upsilon(t)$ is the stochastic
process defined by

$\displaystyle d\Upsilon(t)=$
$\displaystyle-\big{[}\kappa_{-}\Upsilon(t)-\kappa_{\uparrow}\,\Upsilon^{2}(t)\big{]}\,dt-\Upsilon^{2}(t)\,\sqrt{2\,\kappa_{\uparrow}\Upsilon^{-1}(t)}\,\;dV_{N}(t)\;.$
(24)

Based on (20) and (22) we propose the following expansion for ${\rm
E}[\Upsilon(t)]$ ,

$\displaystyle{\rm E}\big{[}\Upsilon(t)\big{]}=\sum_{n=1}^{K}\,g_{n}(t)\,{\rm
E}\big{[}\Upsilon^{n}(0)\big{]}.$ (25)

We call (25) the inverse-number expansion up to $K$th order. Equations (20)
and (22) are thus examples of the inverse-number expansion up to order one,
and order two respectively. The first two expansion coefficients can thus be
read off from these equations,

$\displaystyle g_{1}(t)$ $\displaystyle=\frac{1}{G_{t}}\;,$ (26)
$\displaystyle g_{2}(t)$
$\displaystyle=\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\,\frac{1}{G_{t}}\>\bigg{(}\,1-\frac{1}{G_{t}}\bigg{)}\;.$
(27)

Note that to avoid convergence issues in (25) the realisations of $N(0)$ need
to be greater than one. The problem of obtaining the phase variance now
requires one to find the expansion coefficients $\\{g_{n}(t)\\}_{n=1}^{K}$ up
to any order $K$, and for arbitrary parameter values $\kappa_{\uparrow}$ and
$\kappa_{\downarrow}$. In essence, the important step is the application of
Itô’s lemma to the $n$th power of $\Upsilon(t)$, giving

$\displaystyle\frac{d}{dt}\;{\rm E}\big{[}\Upsilon^{n}(t)\big{]}=b_{n}\,{\rm
E}\big{[}\Upsilon^{n}(t)\big{]}+c_{n}\;{\rm
E}\big{[}\Upsilon^{n+1}(t)\big{]}\;,$ (28)

where we have defined for ease of writing

$\displaystyle b_{n}=-n\,\kappa_{-}\;,\quad c_{n}=n^{2}\,\kappa_{\uparrow}\;.$
(29)

We refer the reader to Appendix B for the proof of (28) and the subsequent
derivation of the time-dependent expansion coefficients. The result is

$\displaystyle g_{n}(t)=\sum_{k=1}^{n}\,\beta_{k,n}\,e^{b_{k}\;\\!t},$ (30)

where $\beta_{1,1}=1$, otherwise

$\displaystyle\beta_{k,n}=\left.c_{1}\,c_{2}\,\cdots\,c_{n-1}\middle/\underset{j\neq
k}{\prod_{j=1}^{n}}\,(b_{k}-b_{j})\quad n\geq 2\right.\;.$ (31)

Note from (30) that $g_{n}(t)\longrightarrow 0$ as $t\longrightarrow\infty$.
This means that the right-hand side of (15) eventually goes to zero and the
second moment of phase reaches a constant value. We can write out, as an
example, the first three coefficients of the inverse-number expansion:

$\displaystyle g_{1}(t)={}$ $\displaystyle e^{b_{1}\;\\!t}\;,$ (32)
$\displaystyle g_{2}(t)={}$
$\displaystyle\frac{c_{1}}{(b_{1}-b_{2})}\;e^{b_{1}\;\\!t}+\frac{c_{1}}{(b_{2}-b_{1})}\;e^{b_{2}\;\\!t}\;,$
(33) $\displaystyle g_{3}(t)={}$
$\displaystyle\frac{c_{1}\,c_{2}}{(b_{1}-b_{2})(b_{1}-b_{3})}\,e^{b_{1}\;\\!t}+\frac{c_{1}\,c_{2}}{(b_{2}-b_{1})(b_{2}-b_{3})}\;e^{b_{2}\;\\!t}+\frac{c_{1}\,c_{2}}{(b_{3}-b_{1})(b_{3}-b_{2})}\;e^{b_{3}\;\\!t}\;.$
(34)

It is simple to see that (26) and (27) are reproduced on substituting (29)
into (32) and (33). The advantage of the inverse-number expansion can also be
seen in the complexity of $g_{3}(t)$, where it would have been somewhat
tedious to derive by solving a set of three coupled differential equations. Of
course, after obtaining (25) one still has to calculate ${\rm E}[\Phi^{2}(t)]$
according to (15), and then the variance defined by (13). The expression for
${\rm E}[\Phi^{2}(t)]$ thus entails an integral of $g_{n}(t)$. We provide a
discussion on the validity of the inverse-number expansion in Appendix C.

##  4 Phase uncertainty

We now have everything to present our main result. It is shown that for
relatively weak inputs (whose number average is a few photons), the small-
noise approximation inadequately captures the high-gain output phase noise
even for an ideal linear amplifier. Such a linear amplifier is optimal with
respect to the small-noise approximation, so if the approximation fails here,
one can expect it to fail more severely for either nonideal linear amplifiers,
or ideal amplifiers but with less gain. We have also provided a discussion on
the use of the Glauber–Sudarshan distribution in describing the phase of the
amplified field in Appendix D. In essence this validates our result in the
high-gain regime.

###  4.1 General expression

Substituting (25) back into (15) and integrating we have

$\displaystyle{\rm E}\big{[}\Phi^{2}(t)\big{]}={\rm
E}\big{[}\Phi^{2}(0)\big{]}+\frac{\kappa_{\uparrow}}{2}\,\sum_{n=1}^{K}\,{\rm
E}\big{[}\Upsilon^{n}(0)\big{]}\int^{t}_{0}dt^{\prime}\,g_{n}(t^{\prime})\quad.$
(35)

To obtain the variance of $\Phi(t)$ we also require its mean. This is simple
to get from (9) as the phase is entirely driven by a Wiener process so that on
averaging (9) we have

$\displaystyle{\rm E}\big{[}\Phi(t)\big{]}={\rm E}\big{[}\Phi(0)\big{]}\;.$
(36)

Squaring (36) and recalling the definition of the variance from (13) we arrive
at

$\displaystyle{\rm V}\big{[}\Phi(t)\big{]}={\rm
V}\big{[}\Phi(0)\big{]}+\sum_{n=1}^{K}\,\chi_{n}(t)\,{\rm
E}\big{[}\Upsilon^{n}(0)\big{]}\;.$ (37)

where we have defined, on using (30),

$\displaystyle\chi_{n}(t)=\frac{\kappa_{\uparrow}}{2}\,\int^{t}_{0}dt^{\prime}\,g_{n}(t^{\prime})=\frac{\kappa_{\uparrow}}{2}\,\sum_{k=1}^{n}\;\frac{\beta_{k,n}}{b_{k}}\;\big{(}e^{b_{k}\;\\!t}-1\big{)}\;.$
(38)

###  4.2 Ideal linear amplifier

As explained earlier, an ideal linear amplifier is an amplifier that adds the
least amount of noise. Such an amplifier corresponds to the model in Sec. 2.1
with $\kappa_{\downarrow}=0$ so that

$\displaystyle\kappa_{-}=\kappa_{\uparrow}\;.$ (39)

Thus the ideal linear amplifier is entirely characterised by a single
parameter, $\kappa_{\uparrow}$ (assuming a fixed $t$). This sufficiently
simplifies the expression for the phase variance so that we may write it as an
explicit function of $\kappa_{\uparrow}$. From (29) we have

$\displaystyle b_{n}$ $\displaystyle=-n\,\kappa_{\uparrow},$ (40)
$\displaystyle c_{1}\,c_{2}\,\cdots\,c_{n-1}$
$\displaystyle=\big{[}(n-1)!\big{]}^{2}\;(\kappa_{\uparrow})^{n-1}\;,$ (41)
$\displaystyle\underset{j\neq k}{\prod_{j=1}^{n}}\,(b_{k}-b_{j})$
$\displaystyle=(\kappa_{\uparrow})^{n-1}\,\underset{j\neq
k}{\prod_{j=1}^{n}}\,(j-k)\;.$ (42)

It then follows that $\beta_{k,n}$ is independent of $\kappa_{\uparrow}$.
Recall that we defined $\beta_{1,1}=1$ while

$\displaystyle\beta_{k,n}=\left.\big{[}\,(n-1)!\,\big{]}^{2}\middle/\underset{j\neq
k}{\prod_{j=1}^{n}}\,(j-k)\;,\quad n\geq 2\right.\;,$ (43)

and the phase variance then simplifies to

$\displaystyle{\rm V}\big{[}\Phi(t)\big{]}={\rm
V}\big{[}\Phi(0)\big{]}+\sum_{n=1}^{K}\,\chi_{n}(t)\,{\rm
E}\big{[}\Upsilon^{n}(0)\big{]}\;,$ (44)

where $\chi_{n}(t)$ is now given by

$\displaystyle\chi_{n}(t)=\frac{\big{[}\,(n-1)!\,\big{]}^{2}}{2}\,\sum_{k=1}^{n}\,\frac{1-G_{t}^{-k}}{k\,\underset{j\neq
k}{\prod_{j=1}^{n}}\,(j-k)}.$ (45)

Note that the product in the denominator of (45) skips over terms for which
$j=k$ so that for $k=1$ we have $\prod_{j=1,j\neq 1}^{1}\,(j-1)=1$. The first
time-dependent coefficient for the ideal linear amplifier is therefore

$\displaystyle\chi_{1}(t)=\frac{1}{2}\;\bigg{(}1-\frac{1}{G_{t}}\bigg{)}\;.$
(46)

Figure 2:  Phase variance for an ideal linear amplifier with
$\kappa_{\uparrow}=1$ and $\Phi(0)=\pi$. Comparison between the ‘exact’ phase
variance derived using the inverse-number expansion for $K=4$ versus the
small-noise approximation and a first-order inverse-number expansion. The
analytic expression for phase variance is shown to be consistent with the
sample variance corresponding to 500 realisations of $\Phi(t)$. (a) and (b):
$N(0)$=2.25. (c) and (d): $N(0)$=13.

Let us now compare the phase variance obtained from integrating the inverse-
number expansion to the variance obtained with the small-noise approximation
[recall (3.1) from Sec. 3.1]. This is illustrated in Fig. 2 for an ideal
linear amplifier which we now explain. First, in Fig. 2(a) we show 500 sample
paths of the phase which gives us a visualisation of how the phase of an input
signal diffuses as it gets amplified. In Fig. 2(b) we plot the phase variance
from the 500 phase realisations (blue jiggly curve) along with the variance
obtained from (44) for $K=4$ (red, solid line), $K=1$ (cyan, dotted line), and
the small-noise approximation (black dashed line). Recall that we defined the
small-noise approximation in (16) back in Sec. 3.1. The same is true for Fig.
2(c) and (d) with the only difference being the value of $N(0)$. For Fig. 2(a)
and (b) we took $N(0)=2.25$, while for (c) and (d) we used $N(0)=13$. This
allows us to see how the small-noise approximation performs when applied to
the amplification of two inputs of different strengths. Clearly, the inverse-
number expansion does a much better job of estimating the output phase noise
for relatively weak inputs. Another interesting observation from Fig. 2 is
that even the lowest-order approximation ($K=1$ in the inverse-number
expansion) as explained in Sec. 3.2 performs visibly better than the small-
noise approximation.

With the comparison shown in Fig. 2 at hand, let us now try to understand some
of its essential features: First note how in both Fig. 2(b) and (d) the small-
noise approximation appears to underestimate ${\rm V}[\Phi(t)]$. It is in fact
a general property that the small-noise approximated phase variance provides a
lower bound for ${\rm V}[\Phi(t)]$. We can show this by considering the
evolution of the variance given by

$\displaystyle\frac{d}{dt}\;{\rm V}[\Phi(t)]=\frac{d}{dt}\;{\rm
E}[\Phi^{2}(t)]=\frac{\kappa_{\uparrow}}{2}\;{\rm
E}\bigg{[}\frac{1}{N(t)}\bigg{]}\;,$ (47)

where we have used the fact that ${\rm E}[\Phi(t)]$ is constant. Now we note
that Jensen’s inequality states that ${\rm E}[f(X)]\geq f({\rm E}[X])$ where
$f(X)$ is a convex function of the random variable $X$. It can then be shown
that $1/N$ is convex and hence it follows from Jensen’s inequality that

$\displaystyle\frac{d}{dt}\;{\rm
V}[\Phi(t)]\geq\frac{\kappa_{\uparrow}}{2}\;\frac{1}{{\rm E}[N(t)]}\;.$ (48)

The right-hand side of the inequality (48) is exactly the rate of change of
the phase variance in the small-noise approximation. Thus the exact phase
variance must always rise at least as fast, and hence will be greater than or
equal to, the phase variance in the small-noise approximation. Note that (47)
also says that as $N(t)$ increases without bound,

$\displaystyle\lim_{t\to\infty}\;\frac{d}{dt}\;{\rm V}[\Phi(t)]=0\;.$ (49)

This means that the phase variance approaches to some constant value and is
therefore upper bounded. This precisely is what we see in both Fig. 2(b) and
(d) where the phase variance can be seen to level off after some transient
dynamics.

We can also try to understand why the small-noise approximation performs worse
for smaller inputs. Recall from Sec. 3.1 that we expect the small-noise
approximation to be good when the photon-number distribution is narrow. If the
phase variance obtained from making such an approximation is to be accurate
for all times then this condition must be met for all times as well. The
condition of a narrow number distribution can also be understood as having a
strong signal-to-noise ratio $\sigma(t)$, defined as

$\displaystyle\sigma(t)=\frac{\big{(}\,{\rm E}[N(t)]\,\big{)}^{2}}{{\rm
V}[N(t)]}\;.$ (50)

In fact, a Taylor expansion around the mean of $N(t)$ up to second order gives

$\displaystyle{\rm E}\bigg{[}\frac{1}{N(t)}\bigg{]}\approx\frac{1}{{\rm
E}[N(t)]}\;\Bigg{[}1+\frac{1}{\sigma(t)}\Bigg{]}\;.$ (51)

Therefore to understand why the discrepancy between the ${\rm V}[\Phi(t)]$
obtained from the small-noise approximation and inverse-number expansion is
larger in Fig. 2(b) compared to Fig. 2(d), we should consider how
$1/\sigma(t)$ evolves for different $N(0)$. From (51) we expect the larger
$1/\sigma(t)$ is, the worse the small-noise approximation performs. A closed-
form expression for $1/\sigma(t)$ can be derived. Remember from (17) of Sec.
3.1 we already have (for any $\kappa_{\uparrow}$ and $\kappa_{\downarrow}$)

$\displaystyle{\rm E}\big{[}N(t)\big{]}=G_{t}\,{\rm
E}\big{[}N(0)\big{]}+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\big{(}G_{t}-1\big{)}\;.$
(52)

So to obtain $\sigma(t)$ we only need ${\rm E}[N^{2}(t)]$. It is not too
difficult to show that in general (i.e. arbitrary $\kappa_{\uparrow}$ and
$\kappa_{\downarrow}$),

$\displaystyle{\rm E}[N^{2}(t)]=G^{2}_{t}\,{\rm
E}[N^{2}(0)]+4\kappa_{\uparrow}\,G^{2}_{t}\,\bigg{\\{}\bigg{[}\frac{\kappa_{-}\,{\rm
E}[N(0)]+\kappa_{\uparrow}}{\kappa_{-}^{2}}\bigg{]}\bigg{(}1-\frac{1}{G_{t}}\bigg{)}-\frac{\kappa_{\uparrow}}{2\kappa_{-}^{2}}\bigg{(}1-\frac{1}{G^{2}_{t}}\bigg{)}\bigg{\\}}.$
(53)

The number variance ${\rm V}[N(t)]={\rm E}[N^{2}(t)]-({\rm E}[N(t)])^{2}$ is
thus given explicitly by (52) and (53), and hence $\sigma(t)$. The analytic
result for $1/\sigma(t)$ is plotted in Fig. 3 to illustrate its dependence on
$N(0)$. In particular, the $N(0)=2$ and $N(0)=13$ curves correspond to the two
situations shown in Fig. 2 (ignoring the small difference of 0.25 for the
weaker input). We have also plotted $1/\sigma(t)$ for intermediate values of
$N(0)$ to illustrate how the evolution of the signal-to-noise ratio depends on
$N(0)$.

Figure 3:  Inverse of the signal-to-noise ratio of an ideal amplifier for
different the input strengths: $N(0)=2$ (red dashed curve), $N(0)=3$ (black
dotted curve), $N(0)=6$ (purple, dash-dot curve), and $N(0)=13$ (blue solid
curve).

As can be seen from Fig. 3, $1/\sigma(t)$ rises much faster for $N(0)=2$ than
$N(0)=13$. This is consistent with the phase variance observed in Fig. 2(b)
where the small-noise phase variance diverges from the inverse-number
expansion result early on during transience and severely underestimates the
stationary phase variance thereafter. On the other hand, the $1/\sigma(t)$
curve for $N(0)=13$ rises much more slowly and remains at a much lower level
compared to the $N(0)=2$ case. Thus we now understand why in Fig. 2(d) the
small-noise phase variance stays close to the inverse-number formula during
transience and continues to do so at later times when compared to Fig. 2(b).

It is also interesting to consider the case of a fixed input strength for
different levels of non-ideality. This is shown in Fig. 4(a) for $N(0)=3$. For
nonideal amplifiers, $1/\sigma(t)$ takes longer to reach steady state value so
we have plotted the curves for a longer time. At long times (i.e. large
$G_{t}$), the more ideal the amplifier is, the lower its $1/\sigma(t)$ value.
This ordering is in fact preserved for all input strengths for large $G_{t}$.
With the help of (52) and (53), we obtain a simple expression for
$1/\sigma(t)$ when $G_{t}\gg 1$ (assuming a coherent-state input with
amplitude $\alpha$)

$\displaystyle\frac{1}{\sigma}=\Sigma\big{(}|\alpha|^{2}\big{)}=\frac{\big{(}|\alpha|^{2}+\kappa_{\uparrow}/\kappa_{-}\big{)}^{2}-|\alpha|^{4}}{\big{(}|\alpha|^{2}+\kappa_{\uparrow}/\kappa_{-}\big{)}^{2}}.$
(54)

From this we see that $\Sigma(|\alpha|^{2})\leq 1$ (with equality when
$|\alpha|^{2}=0$ i.e. no input), and that $\Sigma(|\alpha|^{2})\longrightarrow
0$ for $|\alpha|^{2}\longrightarrow\infty$, so that nonideal and ideal
amplifiers become indistinguishable in so far as $\sigma(t)$ is concerned. For
all nonzero and finite $|\alpha|^{2}$ the behaviour of $\Sigma(|\alpha|^{2})$
is shown in Fig. 4(b). We plot $\Sigma(|\alpha|^{2})$ for the same nonideal
amplifiers as in Fig. 4(a). We find the more ideal the amplifier is, the
better it performs for any input strength. This again corroborates with
discrepancy seen in Fig. 2 between the phase variance obtained from the small-
noise approximation and the inverse-number expansion.

Figure 4: (a) Inverse of the signal-to-noise ratio for $N(0)=3$ but different
levels of non-ideality: $\kappa_{\uparrow}=0.6$, $\kappa_{\downarrow}=0.4$
(blue solid curve), $\kappa_{\uparrow}=0.7$, $\kappa_{\downarrow}=0.3$
(purple, dash-dot curve), $\kappa_{\uparrow}=0.8$, $\kappa_{\downarrow}=0.2$
(black dotted curve), $\kappa_{\uparrow}=1$, $\kappa_{\downarrow}=0$ (red
dashed curve). Note that for short times ($t\lesssim 1.5$) i.e. low $G_{t}$,
the order of performance is actually reversed (not clearly visible from the
plot). (b) Inverse of the signal-to-noise ratio for large gain as a function
of the input strength. The colour coding corresponding to different
$\kappa_{\uparrow}$ and $\kappa_{\downarrow}$ values is the same as in (a).

##  5 Conclusion

In this paper we have shown the often-used small-noise approximation to be
inadequate for capturing phase diffusion in linear amplifiers when the input
contains a few photons on average. However, the phase noise of a ‘slightly’
stronger input (say ten photons or more) may be described reasonably well
within the small-noise approximation. We have demonstrated this even under
conditions most favourable to the small-noise approximation—amplification of a
single-mode field by an ideal linear amplifier with large gain (equivalent to
amplifying an input for a long time with a fixed amplification rate). This is
also the regime where one would ideally like to operate a linear amplifier. We
obtained such a result by proposing the inverse-number expansion using the $P$
(or Glauber–Sudarshan) function. From our examples, we saw that the inverse-
number expansion with a reasonably small set of terms was sufficient to
capture the phase uncertainty beyond transience. In terms of the $P$ function,
a coherent state is a completely noise-free point in phase space which makes
such inputs extremely easy to handle. In this case, all of noise seen at the
amplifier’s output is interpreted as added noise due to the amplifier being
quantum-mechanical. The disadvantage, is that it cannot predict the correct
transient phase noise (though this is actually the less interesting case). The
extended discussion of Appendix D implies that the $W$ (i.e. Wigner) function
is a better choice for capturing phase diffusion because it has the special
property of giving true marginals. One possible extension which can be
explored is to recast the inverse-number expansion in terms of the $W$
function. This would then allow us to obtain a valid estimate for the phase
noise even at small gain, or short times. The cost to this extension is that
the initial inverse-number statistics become much more difficult to calculate
even for a simple input such as the coherent state.

## Acknowledgements

This research is supported by the MOE grant number RG 127/14, and the National
Research Foundation, Prime Minister’s Office, Singapore under its Competitive
Research Programme (CRP Award No. NRF-CRP-14-2014-02). We would also like to
thank Ranjith Nair for useful discussions.

##  Appendix A Phase variance within the small-noise approximation

We have already said in Sec. 3.1 that the small-noise approximation replaces
the field intensity by its mean value so that we have for the second moment of
$\Phi(t)$,

$\displaystyle\frac{d}{dt}\;{\rm
E}\big{[}\Phi^{2}(t)\big{]}=\frac{\kappa_{\uparrow}}{2\,{\rm E}[N(t)]}\;,$
(55)

where ${\rm E}[N(t)]$ is given by

$\displaystyle{\rm E}\big{[}N(t)\big{]}={\rm
E}\big{[}N(0)\big{]}\;e^{\kappa_{-}\,t}+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;\big{(}e^{\kappa_{-}\,t}-1\big{)}\;.$
(56)

This is a good approximation provided that $({\rm E}[N(t)])^{2}\gg V[N(t)]$.
Here we shall derive the phase variance in closed form within the small-noise
approximation.

To start, we note that the phase is coupled to $N(t)$ but it is purely noisy
(i.e. it has no deterministic evolution) driven by a Wiener process. The
average of $\Phi(t)$ is therefore equal to its initial value ${\rm
E}[\Phi(0)]$. Integrating (55) then gives us

$\displaystyle{\rm V}[\Phi(t)]={\rm E}[\Phi^{2}(t)]-\big{(}{\rm
E}[\Phi(t)]\big{)}^{2}={\rm
V}[\Phi(0)]+\frac{\kappa_{\uparrow}}{2}\int^{t}_{0}dt^{\prime}\;\frac{1}{{\rm
E}[N(t^{\prime})]}\;.$ (57)

Using (56), we write the integral of $1/{\rm E}[N(t)]$ in a more compact form

$\displaystyle\int^{t}_{0}dt^{\prime}\;\frac{1}{{\rm
E}[N(t^{\prime})]}=\int^{t}_{0}dt^{\prime}\;\frac{1}{a\,e^{\kappa_{-}t^{\prime}}+b}\;,$
(58)

where $a$ and $b$ are constants defined by

$\displaystyle a={}$ $\displaystyle{\rm
E}[N(0)]+\frac{\kappa_{\uparrow}}{\kappa_{-}}\;,$ (59) $\displaystyle b={}$
$\displaystyle-\frac{\kappa_{\uparrow}}{\kappa_{-}}\;.$ (60)

The exponential in the denominator can be dealt with if we make the
substitution

$\displaystyle u=a\,e^{\kappa_{-}\,t}\;.$ (61)

The corresponding indefinite integral can then be written entirely in terms of
$u$:

$\displaystyle\int dt\;\frac{1}{a\,e^{\lambda t}+b}={}$
$\displaystyle\frac{1}{\kappa_{-}}\int du\;\frac{1}{u(u+b)}$ (62)
$\displaystyle={}$ $\displaystyle\frac{1}{\kappa_{-}\,b}\;\bigg{[}\int
du\;\frac{1}{u}-\int du\;\frac{1}{(u+b)}\bigg{]}$ (63) $\displaystyle={}$
$\displaystyle\frac{1}{\kappa_{-}\,b}\;\Big{[}\ln u-\ln(u+b)\Big{]}\;.$ (64)

The definite integral then follows on plugging in the definition of $u$ and
then evaluating the limits. The result is simply

$\displaystyle\int^{t}_{0}dt\;\frac{1}{a\,e^{\lambda t}+b}={}$
$\displaystyle\frac{1}{\kappa_{-}\,b}\;\Big{[}\ln\\!\big{(}a\,e^{\kappa_{-}\,t}\big{)}-\ln\\!\big{(}a\,e^{\kappa_{-}\,t}+b\big{)}\Big{]}-\frac{1}{\kappa_{-}\,b}\;\Big{[}\ln\\!\big{(}a\big{)}-\ln\\!\big{(}a+b\big{)}\Big{]}\;.$
(65)

Substituting this back into (57) and using (59) and (60) for $a$ and $b$ we
arrive at (3.1) in the main text.

##  Appendix B Derivation of the inverse-number expansion

###  B.1 Main proof

Recall from (24) of Sec. 3.2 that the stochastic differential equation for
$\Upsilon(t)$ is

$\displaystyle
d\Upsilon(t)=-\big{[}\kappa_{-}\Upsilon(t)-\kappa_{\uparrow}\,\Upsilon^{2}(t)\big{]}\,dt-\Upsilon^{2}(t)\,\sqrt{2\,\kappa_{\uparrow}\Upsilon^{-1}(t)}\,\;dV_{N}(t)\;.$
(66)

Applying Itô’s lemma to $\Upsilon^{n}(t)$ we have

$\displaystyle d\Upsilon^{n}(t)={}$
$\displaystyle\frac{d\Upsilon^{n}}{d\Upsilon}\;d\Upsilon(t)+\frac{1}{2}\,\frac{d^{2}\Upsilon^{n}}{d\Upsilon^{2}}\;\big{[}d\Upsilon(t)\big{]}^{2}$
(67) $\displaystyle={}$
$\displaystyle-n\Upsilon^{n-1}(t)\,\Big{\\{}\big{[}\kappa_{-}\Upsilon(t)-\kappa_{\uparrow}\,\Upsilon^{2}(t)\big{]}\,dt+\Upsilon^{2}\,\sqrt{2\,\kappa_{\uparrow}\Upsilon^{-1}(t)}\,\;dV_{N}(t)\Big{\\}}$
$\displaystyle+n(n-1)\,\kappa_{\uparrow}\,\Upsilon^{n+1}(t)\,dt$ (68)
$\displaystyle={}$
$\displaystyle-n\,\kappa_{-}\,\Upsilon^{n}(t)\,dt+n^{2}\,\kappa_{\uparrow}\Upsilon^{n+1}(t)\,dt-\Upsilon^{2}\,\sqrt{2\,\kappa_{\uparrow}\Upsilon^{-1}(t)}\,dV_{N}(t)\;.$
(69)

It then follows on taking the average of (69),

$\displaystyle\frac{d}{dt}\;{\rm E}\big{[}\Upsilon^{n}(t)\big{]}=b_{n}\,{\rm
E}\big{[}\Upsilon^{n}(t)\big{]}+c_{n}\;{\rm
E}\big{[}\Upsilon^{n+1}(t)\big{]}\;.$ (70)

where

$\displaystyle b_{n}=-n\,\kappa_{-}\;,\quad c_{n}=n^{2}\,\kappa_{\uparrow}\;.$
(71)

This proves that ${\rm E}[\Upsilon^{n}]$ couples only to ${\rm
E}[\Upsilon^{n+1}]$. Let us now define a $K\times 1$ vector $\bm{x}(t)$ by

$\displaystyle\bm{x}(t)=\left[\begin{array}[]{c}{x_{1}}\\\ {x_{2}}\\\
{\vdots}\\\ {x_{K}}\end{array}\right]=\left[\begin{array}[]{c}{{\rm
E}[\Upsilon(t)]}\\\ {{\rm E}[\Upsilon^{2}(t)]}\\\ {\vdots}\\\ {{\rm
E}[\Upsilon^{K}(t)]}\end{array}\right]\;.$ (80)

This defines the mean of the inverse-number $\Upsilon$ as the first component
of $\bm{x}(t)$. We can then write (70) in the form of a linear matrix
differential equation for $\bm{x}$:

$\displaystyle\frac{d}{dt}\,\bm{x}(t)=A\,\bm{x}(t)\;,$ (81)

where $A$ is the $K\times K$ matrix

$\displaystyle A=\left[\begin{array}[]{ccccc}b_{1}&c_{1}&0&\cdots&0\\\
0&b_{2}&c_{2}&&\\\ \vdots&&\ddots&\ddots&\\\ &&&b_{K-1}&c_{K-1}\\\
0&&&&b_{K}\end{array}\right],$ (87)

Furthermore, $A$ is a triangular matrix with $K$ distinct eigenvalues given
directly by $\\{b_{n}\\}_{n=1}^{K}$. Letting $D={\rm diag}(\bm{b})$, an
eigendecomposition of $A$ then gives

$\displaystyle A=SDS^{-1}\;,$ (88)

where the $n$th column of $S$ is the eigenvector corresponding to $b_{n}$:

$\displaystyle
S=\Big{[}\bm{v}^{(1)}\;\bm{v}^{(2)}\;\cdots\;\bm{v}^{(K)}\Big{]}\;,\quad
A\,\bm{v}^{(n)}=b_{n}\,\bm{v}^{(n)}\;,\quad n=1,2,\ldots,K\;.$ (89)

The solution to (81) can then be written in component form as

$\displaystyle x_{r}(t)={}$
$\displaystyle\sum_{j=1}^{K}\sum_{m=1}^{K}\sum_{n=1}^{K}\,S_{r,j}\,\big{(}e^{Dt}\big{)}_{j,m}\,\big{(}S^{-1}\big{)}_{m,n}\,x_{n}(0)$
(90) $\displaystyle={}$
$\displaystyle\sum_{m=1}^{K}\sum_{n=1}^{K}\,S_{r,j}\,\big{(}S^{-1}\big{)}_{m,n}\,e^{b_{m}\,t}\,\bm{x}_{n}(0)$
(91) $\displaystyle={}$
$\displaystyle\sum_{n=1}^{K}\Bigg{[}\sum_{m=1}^{K}\,S_{r,m}\,\big{(}S^{-1}\big{)}_{m,n}\,e^{b_{m}\,t}\Bigg{]}\,x_{n}(0)\;.$
(92)

The inverse-number expansion is then given by (92) for $r=1$. This gives us a
general expression for the time-dependent coefficients in terms of $S$ and its
inverse on comparison to (25):

$\displaystyle
g_{n}(t)=\sum_{k=1}^{K}\,\beta_{k,n}\,e^{b_{k}\,t}\;,\quad\beta_{k,n}=S_{1,k}\,\big{(}S^{-1}\big{)}_{k,n}\;.$
(93)

It can be shown that $S$ is given by

$\displaystyle
S_{m,n}=\begin{cases}\text{\Large$\frac{\prod_{j=m}^{n-1}c_{j}}{\prod_{j=m}^{n-1}(b_{n}-b_{j})}$}\;,\quad
m<n\;,\\\ 1\;,\quad m=n\;,\\\ 0\;,\quad m>n\;.\end{cases}$ (94)

Note that $S$ is not an orthogonal matrix so that its transpose does not equal
its inverse. Nevertheless, because $S$ is an upper-triangular matrix it is
fairly straightforward to analytically obtain its inverse using standard
symbolic software such as Mathematica. The inverse of $S$ is again upper
triangular, given by

$\displaystyle\big{(}S^{-1}\big{)}_{m,n}=\begin{cases}\text{\Large$\frac{\prod_{j=m}^{n-1}c_{j}}{\prod_{j=m+1}^{n}(b_{m}-b_{j})}$}\;,\quad
m<n\;,\\\ 1\;,\quad m=n\;,\\\ 0\;,\quad m>n\;.\end{cases}$ (95)

We prove in Appendix B.2 that $S$ is given by (94). It is interesting to note
that $S$ has a structure such that for any $K$, its inverse can be constructed
directly by taking matrix powers. This is due to $S$ being an upper-triangular
matrix with its main diagonal given by only ones. We can then write it in
terms of a nilpotent matrix $N$ as $S=I+N$ where $I$ is the $K\times K$
identity matrix and $N$ is strictly upper triangular such that $N^{p}=0$ for
all $p\geq K$. The inverse of $S$ can then be obtained as a finite (geometric)
series in $N$

$\displaystyle S^{-1}=\sum_{q=0}^{K-1}\,N^{q}\;.$ (96)

This provides an alternative way to compute $S^{-1}$. Since $N$ is strictly
upper triangular, its powers are not too difficult to compute for small values
of $K$. Note that even though we need only the first row of $S$ for
$g_{n}(t)$, the remaining elements of $S$ are still required to obtain
$S^{-1}$. Taking the first row of $S$ we have

$\displaystyle S_{1,1}={}$ $\displaystyle 1\;,$ (97) $\displaystyle
S_{1,n}={}$ $\displaystyle\frac{c_{1}\,c_{2}\cdots
c_{n-1}}{(b_{n}-b_{1})(b_{n}-b_{2})\cdots(b_{n}-b_{n-1})}\;,\quad n\geq 2\;.$
(98)

It is not too difficult to show, by substituting (95)–(98) into the definition
of $\beta_{k,n}$ in (93) that

$\displaystyle g_{n}(t)=\sum_{k=1}^{n}\,\beta_{k,n}\,e^{b_{k}\;\\!t}\;,$ (99)

where $\beta_{1,1}=1$, and

$\displaystyle\beta_{k,n}=\left.c_{1}\,c_{2}\,\cdots\,c_{n-1}\middle/\underset{j\neq
k}{\prod_{j=1}^{n}}\,(b_{k}-b_{j})\;,\quad n\geq 2\right.\;.$ (100)

Note the top limit of the sum in (99) is now $n$ rather than $K$ as a result
of (95) for $m>n$.

###  B.2 Eigenvectors of $A$

The goal here is to prove that $S$ has the form given in (94). This is
equivalent to showing that the $n$th column of $S$, which we denote by
$\bm{v}^{(n)}$, satisfies

$\displaystyle A\,\bm{v}^{(n)}=b_{n}\,\bm{v}^{(n)}\;,\quad n=1,2,\ldots,K\;.$
(101)

In component form, the left-hand side of (101) is

$\displaystyle\big{[}A\,\bm{v}^{(n)}\big{]}_{m}={}$
$\displaystyle\sum_{j=1}^{K}\;\big{(}\,b_{m}\,\delta_{m,j}+c_{m}\,\delta_{m,j-1}\,\big{)}\,v^{(n)}_{j}$
(102) $\displaystyle={}$
$\displaystyle\sum_{j=1}^{K}\;b_{m}\,\delta_{m,j}\,S_{j,n}+\sum_{j=1}^{K}\;c_{m}\,\delta_{m,j-1}\,S_{j,n}=b_{m}\,S_{m,n}+c_{m}\,S_{m+1,n}$
(103)

where we have noted that $v^{(n)}_{j}=S_{j,n}$ in the second equality. The
eigenvalue equation (101), when written in terms of the components of $S$ is
therefore

$\displaystyle b_{m}\,S_{m,n}+c_{m}\,S_{m+1,n}=b_{n}\,S_{m,n}\;.$ (104)

We first show that $S$ satisfies (104) for all $m$ and $n$ such that $m<n$.
Using (94) we have

$\displaystyle
b_{m}\,S_{m,n}+c_{m}\,S_{m+1,n}=b_{m}\;\frac{\prod_{k=m}^{n-1}c_{k}}{\prod_{k=m}^{n-1}\,(b_{n}-b_{k})}+c_{m}\;\frac{\prod_{k=m+1}^{n-1}c_{k}}{\prod_{k=m+1}^{n-1}\,(b_{n}-b_{k})}\;.$
(105)

Now we note that

$\displaystyle
c_{m}\;\prod_{k=m+1}^{n-1}c_{k}=\prod_{k=m}^{n-1}c_{k}\;,\quad\prod_{k=m+1}^{n-1}\,(b_{n}-b_{k})=\frac{1}{b_{n}-b_{m}}\;\prod_{k=m}^{n-1}\,(b_{n}-b_{k})\;.$
(106)

Equation (105) thus becomes

$\displaystyle b_{m}\,S_{m,n}+c_{m}\,S_{m+1,n}={}$ $\displaystyle
b_{m}\;\frac{\prod_{k=m}^{n-1}c_{k}}{\prod_{k=m}^{n-1}\,(b_{n}-b_{k})}+(b_{n}-b_{m})\;\frac{\prod_{k=m}^{n-1}c_{k}}{\prod_{k=m}^{n-1}\,(b_{n}-b_{k})}$
(107) $\displaystyle={}$ $\displaystyle
b_{n}\;\frac{\prod_{k=m}^{n-1}c_{k}}{\prod_{k=m}^{n-1}\,(b_{n}-b_{k})}$ (108)
$\displaystyle={}$ $\displaystyle b_{n}\,S_{m,n}\;.$ (109)

For $m>n$ it is clear that (94) gives zero for the left-hand of (104) and also
zero for the right-hand side. Finally, for $m=n$ it is also trivial to see
that (94) gives $b_{m}$ for both sides of (104).

##  Appendix C Validity of the inverse-number expansion

###  C.1 An iterative derivation of the inverse-number expansion and

To show that (25), (30), and (31) of Sec. 3.2 are indeed correct we consider
an alternative way to derive the inverse-number expansion. Let us return to
(28) and note that it could have been solved iteratively. That is, it has the
formal solution

$\displaystyle{\rm E}\big{[}\Upsilon^{n}(t)\big{]}={\rm
E}\big{[}\Upsilon^{n}(0)\big{]}\,e^{b_{n}\;\\!t}+c_{n}\int^{t}_{0}\,dt^{\prime}\;{\rm
E}\big{[}\Upsilon^{n+1}(t^{\prime})\big{]}\,e^{b_{n}(t-t^{\prime})}\;.$ (110)

This now gives us a recursive relation between any two consecutive moments of
$\Upsilon(t)$ so we may write ${\rm E}[\Upsilon(t)]$ up to any desired order
in moments of $\Upsilon(0)$ by repeated application of (110). For example,
application of (110) for $n=1,2,3$ gives

$\displaystyle{\rm E}\big{[}\Upsilon(t)\big{]}={\rm
E}\big{[}\Upsilon(0)\big{]}\,e^{b_{1}\;\\!t}$
$\displaystyle+c_{1}\int^{t}_{0}\,dt_{1}\;e^{b_{2}\;\\!t_{1}}\;{\rm
E}\big{[}\Upsilon^{2}(0)\big{]}$
$\displaystyle+c_{1}\,c_{2}\int^{t}_{0}\,dt_{1}\int^{t_{1}}_{0}\;dt_{2}\;e^{b_{3}\;\\!t_{2}}\,e^{b_{2}(t_{1}-t_{2})}\;{\rm
E}\big{[}\Upsilon^{3}(0)\big{]}$ (111)
$\displaystyle+c_{1}\,c_{2}\,c_{3}\int^{t}_{0}\,dt_{1}\int^{t_{1}}_{0}\;dt_{2}\int^{t_{2}}_{0}\;dt_{3}\;e^{b_{3}\;\\!(t_{2}-t_{3})}\,e^{b_{2}(t_{1}-t_{2})}\;{\rm
E}\big{[}\Upsilon^{4}(t_{3})\big{]}.$

Using the definition of the expansion coefficients in (25), we find on
repeated application of (C.1) the general result

$\displaystyle g_{1}(t)={}$ $\displaystyle e^{b_{1}\;\\!t}\;,$ (112)
$\displaystyle g_{n}(t_{0})={}$ $\displaystyle c_{1}\,c_{2}\cdots
c_{n-1}\;e^{b_{1}\;\\!t}\int^{t_{0}}_{0}dt_{1}\;e^{(b_{2}-b_{1})t_{1}}\int^{t_{1}}_{0}dt_{2}\;e^{(b_{3}-b_{2})t_{2}}\cdots$
$\displaystyle\cdots\int^{t_{n-2}}_{0}dt_{n-1}\;e^{(b_{n}-b_{n-1})t_{n-1}}\;,\quad
n\geq 2\;.$ (113)

Equations (112) and (C.1) may also be seen as an alternative form for the
time-dependent coefficients of the inverse-number expansion. Compared to (30)
and (31) they are rather clumsy to use for large $K$ as (C.1) requires one to
compute an $n$-fold integral for $g_{n-1}(t)$. However, the integrals only
ever involve simple exponentials so they are not too difficult to evaluate for
small $n$. We can at least show that they match the results in (33) and (34).
Considering $n=2$ and $n=3$,

$\displaystyle g_{2}(t)={}$ $\displaystyle
c_{1}\,e^{b_{1}\;\\!t}\int^{t}_{0}dt_{1}\;e^{(b_{2}-b_{1})\,t_{1}}$
$\displaystyle={}$
$\displaystyle\frac{c_{1}}{b_{2}-b_{1}}\;\big{(}\,e^{b_{2}\,t}-e^{b_{1}\,t}\big{)}\;,$
(114) $\displaystyle g_{3}(t)={}$ $\displaystyle
c_{1}\,c_{2}\,e^{b_{1}\;\\!t}\int^{t}_{0}dt_{1}\;e^{(b_{2}-b_{1})t_{1}}\int^{t_{1}}_{0}dt_{2}\;e^{(b_{3}-b_{2})t_{2}}$
$\displaystyle={}$
$\displaystyle\frac{c_{1}\,c_{2}}{(b_{3}-b_{2})(b_{3}-b_{1})}\;\big{(}e^{b_{3}\,t}-e^{b_{1}\;\\!t}\big{)}-\frac{c_{1}\,c_{2}}{(b_{3}-b_{2})(b_{2}-b_{1})}\;\big{(}e^{b_{2}\,t}-e^{b_{1}\;\\!t}\big{)}\;.$
(115)

It is clear that (114) is identical to (33). To see that (115) is consistent
with (34) we only have to add the coefficients of $\exp(b_{1}\;\\!t)$ in
(115).

###  C.2 Number and phase as stochastic processes

Figure 5:  Inverse number for a coherent-state input $|{\alpha}\rangle$ with
$|\alpha|^{2}=\Upsilon(0)=3$ and $\kappa_{\downarrow}=0$ and
$\kappa_{\uparrow}=2$ (ideal linear amplifier). (a) Five samples of the
process $\Upsilon(t)$ from simulating (24). (b) Analytic expressions for the
inverse-number expansion [(25), (30) and (31)] corresponding to $K=1$ (green
dashed curve), $K=2$ (blue solid curve), and $K=3$ (black dotted curve). (c)
Sample average for ${\rm E}[\Upsilon(t)]$ obtained from the five realisations
in (a) (noisy blue curve) and the analytic formula corresponding to $K=3$ (red
dashed curve). (d) ${\rm E}[\Upsilon(t)]$ obtained from averaging over 200
realisations of $\Upsilon(t)$ (solid blue curve), and from the inverse-number
expansion for $K=3$ (red dashed curve).

We can illustrate how the inverse-number expansion works by comparing it to
the average of $\Upsilon(t)$ over many realisations obtained from simulating
(24). This comparison is made in Fig. 5 for the simple case of an ideal linear
amplifier. We first show how different realisations of $\Upsilon(t)$ look like
in Fig. 5(a). For visual clarity only five realisations of $\Upsilon(t)$ are
shown. We then illustrate the analytic expression of ${\rm E}[\Upsilon(t)]$
for a few values of $K$. If the claimed time-dependent coefficients given in
(30) and (31) are correct then we ought to find a value of $K$ for which there
is a good agreement between the analytic expression for ${\rm E}[\Upsilon(t)]$
and that obtained from a large sample of $\Upsilon(t)$. In Fig. 5(b) we show
how the sum in (25) changes for $K=1$ (green dashed curve), $K=2$ (blue solid
curve), and $K=3$ (black dotted curve). In Fig. 5(c) we compare the inverse-
number expansion with $K=3$ with the average of the five realisations of
$\Upsilon(t)$ shown in (a). The same comparison is made with 200 realisations
in Fig. 5(d), and as can be seen, the sample average (solid blue line) is much
smoother due to the large sample size and is in good agreement with the
analytic result (red dashed line). We note that in practice there is some
threshold value of $K$ beyond which the inverse-number expansion starts to
deviate from the ${\rm E}[\Upsilon(t)]$ calculated by averaging over many
sample paths though in theory we expect this $K$ to be arbitrarily large. We
find that for a coherent-state input $|{\alpha}\rangle$, this threshold tend
to be larger for larger $\alpha$. As long as $K$ is less than the threshold we
find good agreement between the analytic result given by the inverse-number
expansion and the stochastic simulations.

##  Appendix D Phase in quantum optics and the use of the $P$ distribution

Here we comment on the nature of our treatment of phase and its relation to
other work on phase diffusion in linear amplifiers. We have chosen to work
with the stochastic differential equations that are consistent with the
Glauber–Sudarshan function as opposed to other quasiprobability distributions.
It has been shown explicitly for linear amplifiers that it is the
Fokker–Planck equation for the Wigner function (the $W$) rather than the
Husimi (the $Q$), or the Glauber–Sudarshan (the $P$) which gives the correct
phase diffusion coefficient in the small-noise approximation [15, 16]. This is
perhaps not hard to accept given that $W(q,p)$, i.e. when parametrised by the
canonical position and momentum, guarantees true marginal distributions when
either $q$ or $p$. The idea of using the Wigner function is that one would
expect this to still be true if the Wigner function was reparametrised by
number and phase (a more complete discussion of this point is given in Sec. 5
in connection to possible future directions). Then in what sense does the
Glauber–Sudarshan distribution provide a valid description of the phase noise
in linear amplifiers? If we use the $P$ rather than the $W$ then we forgo the
transient dynamics of phase during amplification. The stationary statistics of
phase however remain correct. It is already known that $P$, $W$, and $Q$ all
give the same limiting phase distribution for an ideal linear amplifier in the
high-gain regime [17, 49] (though there is no reason to expect this to
breakdown for nonideal amplifiers). For a fixed amplification rate
$\kappa_{\uparrow}$, the high-gain regime simply means that we amplify the
field for sufficiently long. With the assistance of Ref. [17] we can actually
compute the output phase probability density from the $P$ distribution
explicitly for a coherent-state input $|{\alpha}\rangle$ in the high-gain
limit. The result, for $\alpha=|\alpha|\exp(i\theta)$, is

Figure 6:  Plots of $\wp_{\Phi}(\phi,t)$ as given by the Glauber–Sudarshan
function (red solid line), and the Pegg–Barnett distribution (blue dashed
line). The parameters are $\kappa_{\uparrow}=1$, $\kappa_{\downarrow}=0$ and
$\alpha=-1.5$. The two phase distributions are shown after (a) a short time,
where $\epsilon=0.1$; and (b) after transience, where $T=4$.

$\displaystyle\wp_{\Phi}(\phi,t)={}$
$\displaystyle\frac{\cos\big{(}\phi-\theta\big{)}}{2\pi}\;e^{-\eta(t)}\Big{\\{}\sqrt{\pi\,\eta(t)}\,\exp\\!\big{[}\,\eta(t)\,\cos^{2}\\!\big{(}\phi-\theta\big{)}\big{]}$
$\displaystyle\times\left(1+\text{erf}\Big{[}\sqrt{\eta(t)}\,\cos\big{(}\phi-\theta\big{)}\Big{]}\right)+\sec\\!\big{(}\phi-\theta\big{)}\Big{\\}}\;,$
(116)

where ${\rm erf}(x)$ is the error function given by

$\displaystyle{\rm
erf}(x)=\frac{2}{\sqrt{\pi}\,}\;\int^{x}_{0}dy\;e^{-y^{2}}\;,$ (117)

and we have defined $\eta(t)$ to be the (noise-free) output signal normalised
by the spontaneous-emission noise [see (17) from Sec. 3.1]

$\displaystyle\eta(t)=\frac{G_{t}\,|\alpha|^{2}}{n_{\rm sp}(t)}\;,\quad n_{\rm
sp}(t)=G_{t}-1\;.$ (118)

Note that here $G_{t}=\exp(\kappa_{\uparrow}\,t)$ since an ideal linear
amplifier is assumed. Using (D) the phase statistics beyond transience can now
be seen explicitly. To show that it converges to the correct phase
distribution we use the Pegg–Barnett probability density as a reference. The
Pegg–Barnett phase probability corresponding to some arbitrary state $\rho(t)$
is defined on a truncated $s$-dimensional Hilbert space spanned by the the
Fock basis $\\{|{n}\rangle\\}_{n=0}^{s}$ [50]:

$\displaystyle\wp_{\Phi}(\phi_{m},t)=\frac{s+1}{2\,\pi}\;\langle{\phi_{m}}|\,\rho(t)\,|{\phi_{m}}\rangle\;,$
(119)

where for each $m$, the phase state $|{\phi_{m}}\rangle$ is defined by

$\displaystyle|{\phi_{m}}\rangle=\frac{1}{\sqrt{s+1}\,}\;\sum_{n=0}^{s}\,e^{in\phi_{m}}\,|{n}\rangle\;,\quad\phi_{m}=\phi_{0}+\frac{2\pi}{s+1}\;m\;.$
(120)

Figure 7:  Phase variance as a function of time as obtained from the $P$
function (red solid line) and the Pegg–Barnett distribution (blue dashed
line). The parameter values are the same as those in Fig. 6.

Generally, all phase statistics obtained using the Pegg–Barnett operator must
be calculated with a finite value of $s$, i.e. on the truncated space first.
To obtain the correct statistics we must then take the
$s\longrightarrow\infty$ limit at the end (and only in the end). An exception
to this limiting procedure applies for physical states which Pegg and Barnett
defined to be states with finite average photon number. This is often true and
is also true here. We obtained the Pegg–Barnett phase probability distribution
numerically from the exact model given by the master equation (1) in Sec. 2.1.
We show the result of (D) with the Pegg–Barnett distribution in Fig. 6 at two
times. In both Fig. 6(a) and (b) the Pegg–Barnett phase probability is shown
as the blue dashed curve while (D) is shown as the red solid curve. At $t=0$
the phase variance from the $P$ distribution would be zero. This is because
the coherent state has a delta-function representation in terms of $P$. The
Pegg–Barnett distribution on the other hand would correspond to the actual
phase distribution which would have some finite width at $t=0$. After a short
time $\epsilon$ the two distributions then broaden a little as shown in Fig.
6(a). The $P$-function phase density given by (D) can be seen to be much more
sharply peaked than the Pegg-Barnett phase density. This makes sense since it
diffused from an initial delta function $\delta(\phi-\theta)$ at $t=0$. As
time progresses we find the $P$-function phase density to diffuse further,
eventually converging to the Pegg–Barnett distribution at time $T$ (the value
of $T$ is taken to be four). We also show how the phase variance evolves in
the interval $[\epsilon,T]$ in Fig. 7. It can be seen that the phase diffuses
much more quickly when described in terms of the $P$ distribution (red solid
curve). In relation to this we note that previous analysis have shown the
Pegg–Barnett phase diffusion coefficient to be smaller than that from the $P$
function, but under the small-noise assumption [15]. In the end, it is clear
that at $t=T=4$ the two phase variances differ only slightly, suggesting that
the Glauber–Sudarshan distribution provides a good estimate for the phase
variance after transience, or equivalently, in the high-gain regime (which is
the limit in which one typically would like to operate the amplifier in [29]).
The $P$ distribution then has the advantage that coherent-state inputs are
easily handled as it amounts to specifying deterministic initial conditions
for the number and phase. In this case, all of the output noise is interpreted
as added noise due to amplification as the input is completely noise-free.

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