# Quantum Jump Metrology with Quantum Feedback in Cavity Networks

Kawthar Al Rasbi,1,2 Almut Beige1 and Lewis A. Clark3 1The School of Physics
and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom  2The
Department of Physics, Sultan Qaboos University, Sultanate of Oman  3Centre
for Quantum Optical Technologies, Centre of New Technologies, University of
Warsaw, Banacha 2c, 02-097 Warsaw, Poland

###### Abstract

Quantum metrology enhances measurement precision by utilising the properties
of quantum physics. In interferometry, this is typically achieved by evolving
highly-entangled quantum states before performing single-shot measurements to
reveal information about an unknown phase. While this is the theoretical
optimum approach, implementation with all but the smallest states is still
extremely challenging. An alternative approach is quantum jump metrology,
which deduces information by continuously monitoring an open quantum system
while inducing phase-dependent temporal correlations with the help of quantum
feedback. Taking this approach here, we analyse measurements of a relative
phase in an optical network of two cavities with quantum feedback in the form
of laser pulses. It is shown that the proposed scheme can exceed the standard
quantum limit without the need for complex quantum states.

##  I Introduction

Figure 1: A comparison of three general schemes to achieve an enhancement in
estimating an unknown parameter $\varphi$. (a) Classical scheme consisting of
independent and uncorrelated systems where each probe is encoded with the
parameter to be determined and then measured. In such a case the accuracy of
measurement is limited by the Standard Quantum Limit. (b) Quantum mechanical
scheme utilising entangled states give rise to higher measurement accuracy
reaching the Heisenberg Limit. (c) Quantum jump metrology scheme where the
quantum system is evolved in time inside an instantaneous feedback loop.
Sequential measurements on the system allows for accuracy enhancement beyond
the standard quantum limit due to the existence of temporal correlations.

Highly accurate measurements are important for a variety of applications,
ranging from probing biological samples lifetime1 ; lifetime2 ; lifetime3  to
the detection of gravitational waves LIGO . Often, such measurements use light
passing through interferometric devices. Classically, it is well understood
how to execute such measurements effectively kolodynski ; yang ; Loughridge ;
scott ; ataman ; blanco . One way of increasing their precision is to use
higher intensity sources. Sometimes this is not possible, for example, if the
object to be probed is fragile or has a short lifetime lifetime1 . In such
cases, precise estimations can only be made after repeating measurements many
times. More recently however, it was recognised that alternatively one could
explore the properties of quantum physics, specifically through quantum
metrology seth ; lloyd ; Maccone . These can allow for an increased precision
of measurements given the same number of probes.

Suppose an unknown parameter $\varphi$ is determined by performing exactly $N$
independent measurements, as illustrated in Fig. 1(a). In this case, the
scaling of the uncertainty $\Delta\hat{\varphi}$ of the estimator with the
number of probes $N$ is limited by the standard quantum limit, which tells us
that

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{N}\,.$ (1)

However, for correlated probes, this scaling can be improved. For maximally
correlated probes, the Heisenberg limit may be obtained, which scales with the
number of probes as

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{N^{2}}\,.$ (2)

This means that for large $N$, the parameter $\varphi$ can be estimated with
the same precision with fewer probes. Determining how to obtain and utilise
such correlated probes has thus been an active area of research.

As illustrated in Fig. 1(b), one solution is to evolve entangled quantum
states in a $\varphi$-dependent fashion followed by a collective measurement
of their state HollandB ; CavesC ; Bondurant ; DowlingNOON ; BerryW ;
Vidrighin ; Xiang ; Afek ; Nagata . However, a typical problem with this
approach is that it is difficult to implement. For example, it has been shown
that so-called N00N states, which are highly-entangled $N$ photon states, are
optimal for quantum interferometry experiments DowlingNOON ; Afek ; Nagata ;
LeeNoon ; Mitchell ; PrydeNoon ; IsraelNoon . But obtaining a reasonably large
$N$ in the laboratory to realise the above-described enhancement remains
extremely challenging Gilbert ; ThomasP . Fortunately, using entanglement is
not the only way of enhancing measurement precision BoixoN ; boixo ;
Napolitano ; Atta ; Higgins ; braun . For example, in Ref. boixo , it has been
shown that another way of overcoming the standard quantum limit in Eq. (1) is
by using non-linear or interacting systems. Taking this approach, the
uncertainty $\Delta\hat{\varphi}$ can scale such that

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{N^{k}}\,,$ (3)

where $k$ is the order of the non-linearity of the interaction. Once again
though, non-linearities are often hard to implement experimentally,
particularly when processing information with light.

In this paper we adopt an alternative approach and use quantum jump metrology
lewisstokes ; lewisbeige  to improve the precision of measurements. Quantum
jump metrology does not require the preparation of highly entangled quantum
states nor the presence of non-linear optical elements and is therefore
relatively easy to implement. Its basic idea is to deduce information about an
unknown parameter $\varphi$ by monitoring the output statistics of an open
quantum system. To ensure that the dynamics of the individual quantum
trajectories of the system depend on $\varphi$, we use quantum feedback
wiseman  which is triggered by certain measurements outcomes, as illustrated
in Fig. 1(c). Using the dynamics of open quantum systems to infer information
about an unknown parameter recently received a lot of attention in the
literature Shabani ; Beau ; Albarelli ; Haase .

Ref. lewisstokes  introduced a quantum metrology scheme for measuring the
phase difference between a coherent state prepared inside an optical cavity
and a laser providing feedback pulses. In this paper, we propose instead a
scheme which simply measures the difference $\tilde{\varphi}$ between two
phases $\varphi_{1}$ and $\varphi_{2}$ corresponding to two pathways through a
linear optics setup, which is shown in Fig. 2. The setup that we consider here
is easier to implement and much more practical than the original protocol
lewisstokes . Nevertheless, the scheme that we propose here too relies only on
coherent states, yet it is capable of producing correlated photon statistics
and thus surpassing the standard scaling due to the presence of quantum
feedback. Hence, as well as demonstrating a simple scheme with enhanced
sensing capabilities, our proposal also demonstrates the power of using
quantum feedback to obtain quantum effects, even in ‘classical-like’ states
such as coherent states.

The use of quantum feedback has found a variety of applications not only in
quantum metrology lewisstokes ; lewisbeige , but also in quantum error
correction and noise reduction qerror , quantum state stabilisation sstab ,
entanglement control fentg  and in implementing Hidden Quantum Markov Models
lewis . Moreover, it has recently been shown that quantum feedback can lead to
ergodicity breaking in quantum optical systems lewismaybee . This can again be
achieved even with coherent states and feedback in the form of displacements
of the field, thus only requiring relatively simple technology to implement.
Intuitively, it is easy to see how quantum feedback can lead to correlations
in the bath statistics of an open quantum system. Consider a quantum optical
system that emits a photon at time $t_{1}$. Then as the system is perturbed by
the feedback the emission probability for another photon is altered. Thus, the
emission at time $t_{2}$ is correlated with the emission at $t_{1}$. If this
feedback depends on an unknown parameter, these correlations can be used to
gain information for its estimation. Hence it is not surprising that quantum
feedback is a powerful tool for quantum technology tasks.

Figure 2: Two optical cavities are monitored through a linear optics network
with photon detectors. Upon the detection of a photon, quantum feedback is
triggered and applied to the cavities, also through a linear optics network.
In this diagram and our subsequent analysis of the system, we consider a
specific case of the general dynamics presented in Sec. II, where feedback is
triggered only in mode $b_{2}$ from a photon detection in $d_{1}$ and only in
mode $b_{1}$ from a photon detection in $d_{2}$. The aim of the scheme is to
measure the phase difference $\tilde{\varphi}=\varphi_{1}-\varphi_{2}$ from
the photon statistics in the detectors.

Although Fig. 2 shows two optical cavities, the quantum metrology which we
propose here could also be implemented using only a single cavity. In this
case, the two different polarisations of light and of the states inside an
optical cavity need to be utilised. Moreover, lasers 1 and 2 would have to
differ in polarisation and each beam splitter would have to be replaced by a
single polarising beam splitter. However, for simplicity, we analyse in the
following the experimental setup shown in Fig. 2. In recent years, much
progress has been made in implementing optical cavities which confine photons
for a relatively long time. For example, strong fibre-integrated cavities FC
have already become available in laboratories worldwide. Moreover, much
progress has been made in miniaturising and increasing the efficiency of
single photon detectors PD . The proposed quantum metrology scheme is hence
feasible with current technology.

This paper is structured as follows. In Section II, we introduce the notation
and the basic theoretical models for the modelling of the linear optics cavity
network shown in Fig. 2. Afterwards, in Section III, we introduce the
measurement scheme that we propose in this paper to sense an unknown phase
shift. In Section IV, we analyse the capabilities of our quantum jump
metrology scheme and present numerical results. Finally, we summarise our
findings in Section V.

##  II A two-mode cavity network with quantum feedback

In this Section, we review the main tools for the theoretical modelling of the
experimental setup shown in Fig. 2. As we simply consider laser driving of
optical cavities, the quantised electromagnetic field inside the resonators
remains always in a coherent state. In the following we have a closer look at
the dynamics of these coherent states under the condition of no photon
emission and in case of an emission. In addition, we introduce quantum optical
master equations which can be used for the prediction of ensemble expectation
values.

###  II.1 Multi-mode Coherent States and Transformation Matrices

As mentioned already above, in this paper we consider a network of two leaky
optical cavities that are always kept in a coherent state. In Fock state
representation, the coherent state $\ket{\gamma_{i}}$ of cavity $i$ can be
solely parametrised by a complex number $\gamma_{i}$ such that

$\displaystyle\ket{\gamma_{i}}$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{|\gamma_{i}|^{2}}{2}\right)\sum\limits_{n_{i}=0}^{\infty}\frac{\gamma_{i}^{n_{i}}}{\sqrt{n_{i}\\!}}\ket{n_{i}}\,.$
(4)

Here $|n_{i}\rangle$ is the state with exactly $n_{i}$ photons in cavity $i$.
If $c_{i}$ is the annihilation operator for a single photon in cavity $i$,
then one can show that $c_{i}\ket{\gamma_{i}}=\gamma_{i}\ket{\gamma_{i}}$.
Using this notation, the state $\ket{\psi(t)}$ of both cavities at time $t$ is
always of the form

$\displaystyle\ket{\psi(t)}$ $\displaystyle=$
$\displaystyle\bigotimes_{i=1,2}\ket{\gamma_{i}(t)}$ (5)

and its dynamics can be modelled simply by tracking two complex numbers
$\gamma_{i}(t)$. Hence we can also express the state of the two cavities as

$\displaystyle{\underline{\gamma}}(t)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\gamma_{1}(t)\\\
\gamma_{2}(t)\end{array}\right)\,.$ (8)

In the following, we adopt a vector and matrix notation for convenience when
considering photon counting and quantum feedback processes.

The experimental setup in Fig. 2 contains phase shifters and beamsplitters.
Hence quantum feedback pulses do not perturb the cavities directly. Similarly,
photons arriving at a detector do not come directly from only a single cavity.
To take this into account more easily, we denote the annihilation operator of
the field mode seen by detector $i$ in the following by $a_{i}$ and the
annihilation operator of the field mode affected by laser $i$ by $b_{i}$. With
respect to these alternative modes, the state $|\psi(t)\rangle$ of the
cavities is given by the complex vectors

$\displaystyle{\underline{\alpha}(t)}=\left(\begin{array}[]{c}\alpha_{1}(t)\\\
\alpha_{2}(t)\end{array}\right)\,,~{}~{}{\underline{\beta}(t)}=\left(\begin{array}[]{c}\beta_{1}(t)\\\
\beta_{2}(t)\end{array}\right)$ (13)

with the complex numbers $\alpha_{i}(t)$ and $\beta_{i}(t)$ such that
$a_{i}|\alpha_{i}(t)\rangle=\alpha_{i}(t)\,|\alpha_{i}(t)\rangle$ and
$b_{i}|\beta_{i}(t)\rangle=\beta_{i}(t)\,|\beta_{i}(t)\rangle$. To switch from
one representation of the cavity network in Fig. 2 to another, we define
transformation matrices $M_{yx}$ with

$\displaystyle\left(\begin{array}[]{c}y_{1}\\\ y_{2}\end{array}\right)$
$\displaystyle=$ $\displaystyle M_{yx}\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)\,,$ (18)

where $x_{i},y_{i}=\alpha_{i}(t),\beta_{i}(t),\gamma_{i}(t)$ and where
$x,y=a,b,c$. Below we have a closer look at these matrices which depend on the
effect of the beamsplitters and the phase shifters shown in Fig. 2. If we
define the transformation matrices

$\displaystyle S_{\rm BS}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&{\rm i}\\\ {\rm
i}&1\end{pmatrix}\,,~{}~{}S_{\varphi_{1}}=\begin{pmatrix}1&0\\\ 0&{\rm
e}^{{\rm i}\varphi_{1}}\end{pmatrix}\,,$ $\displaystyle
S_{\varphi_{2}}=\begin{pmatrix}{\rm e}^{{\rm i}\varphi_{2}}&0\\\
0&1\end{pmatrix}\,,$ (19)

then one can now show that

$\displaystyle M_{cb}$ $\displaystyle=$ $\displaystyle S_{\varphi_{1}}S_{\rm
BS}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&{\rm i}\\\ {\rm i}\,{\rm e}^{{\rm
i}\varphi_{1}}&{\rm e}^{{\rm i}\varphi_{1}}\end{pmatrix}\,,$ $\displaystyle
M_{ac}$ $\displaystyle=$ $\displaystyle S_{\rm
BS}S_{\varphi_{2}}=\frac{1}{\sqrt{2}}\begin{pmatrix}{\rm e}^{{\rm
i}\varphi_{2}}&{\rm i}\\\ {\rm i}\,{\rm e}^{{\rm
i}\varphi_{2}}&1\end{pmatrix}\,.$ (20)

These matrices can be used to easily switch from one representation of the
cavity state into another and to easily model the detection of photons, the
no-photon time evolution and the effect of the quantum feedback pulses. For
example, $\underline{\alpha}(t)=M_{ac}\,\underline{\gamma}(t)$.

###  II.2 The effect of quantum feedback

Suppose an instantaneous strong laser pulse is applied directly to cavity $i$.
Then the effect of this operation on the coherent state
$|\gamma_{i}(t)\rangle$ of cavity $i$ can be described by a displacement
operator of the form

$\displaystyle D_{i}(\gamma)$ $\displaystyle=$
$\displaystyle\exp\left(\gamma\,c_{i}^{\dagger}-\gamma^{*}\,c_{i}\right)$ (21)

where $\gamma$ is a complex number. Taking this into account, one can show
that the result is a change such that

$\displaystyle\gamma_{i}(t)$ $\displaystyle\rightarrow$
$\displaystyle\gamma_{i}(t)+\gamma\,.$ (22)

However, in the experimental setup in Fig. 2, quantum feedback pulses do not
trigger a laser pulse that disturbs the cavities directly. Instead, because of
the presence of a beamsplitter, each laser pulse usually affects the field in
both cavities.

For simplicity, we take the feedback strengths of the laser pulses as constant
in time, although it could be made time dependent for further generality. This
allows us to model the effect of the feedback by four complex numbers
$\beta_{i}^{(d)}$ which characterise the quantum feedback strength generated
by laser $i$ upon detection of a photon in detector $d$ with $d=1,2$. For
convenience we arrange these numbers into two vectors

$\displaystyle{\underline{\beta}}^{(d)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\beta_{1}^{(d)}\\\
\beta_{2}^{(d)}\end{array}\right)\,.$ (25)

Given that the feedback is triggered by the detection of a photon in detector
$d$, we observe the following effect of quantum feedback on the state
$\underline{\gamma}(t)$ of the cavities,

$\displaystyle\underline{\gamma}(t)$ $\displaystyle\rightarrow$
$\displaystyle\underline{\gamma}(t)+M_{cb}\,\underline{\beta}^{(d)}\,.$ (26)

Alternatively, in the basis of the detector modes, the state of the cavities
changes such that

$\displaystyle\underline{\alpha}(t)$ $\displaystyle\rightarrow$
$\displaystyle\underline{\alpha}(t)+M_{ab}\,\underline{\beta}^{(d)}$ (27)

with

$\displaystyle M_{ab}$ $\displaystyle=$ $\displaystyle
M_{ac}M_{cb}=\frac{1}{2}\begin{pmatrix}-{\rm e}^{{\rm i}\varphi_{1}}+{\rm
e}^{{\rm i}\varphi_{2}}&{\rm i}\left({\rm e}^{{\rm i}\varphi_{1}}+{\rm
e}^{{\rm i}\varphi_{2}}\right)\\\ {\rm i}\left({\rm e}^{{\rm
i}\varphi_{1}}+{\rm e}^{{\rm i}\varphi_{2}}\right)&{\rm e}^{{\rm
i}\varphi_{1}}-{\rm e}^{{\rm i}\varphi_{2}}\end{pmatrix}\,.$

This equation provides a complete description of the quantum feedback.

###  II.3 Master equations and Quantum Jump Approach

Next we study the effect of the possible leakage of photons through the cavity
mirrors on the state of the resonator fields. Because of the presence of
spontaneous photon emission, the calculation of expectation values for
ensemble averages requires the introduction of a density matrix $\rho$. In the
absence of quantum feedback, this density matrix evolves such that

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\sum\limits_{d=1,2}\kappa_{d}\,a_{d}\,\rho\,a_{d}^{\dagger}-\frac{1}{2}\kappa_{d}\,\left[a^{\dagger}_{d}a_{d},\rho\right]_{+}\,,$
(29)

where $\kappa_{d}$ denotes the spontaneous decay rate of a single photon in
the $a_{d}$ mode. However, using the notation introduced above, one can show
that this master equation changes into

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\sum\limits_{d=1,2}\kappa_{d}\,D_{2}(\beta^{(d)}_{2})D_{1}(\beta^{(d)}_{1})a_{d}\,\rho\,a_{d}^{\dagger}D^{\dagger}_{1}(\beta^{(d)}_{1})D^{\dagger}_{2}(\beta^{(d)}_{2})$
(30)
$\displaystyle-\frac{1}{2}\kappa_{d}\,\left[a^{\dagger}_{d}a_{d},\rho\right]_{+}$

in the presence of quantum feedback with

$\displaystyle D_{i}(\beta^{(d)}_{i})$ $\displaystyle=$
$\displaystyle\exp\left(\beta^{(d)}_{i}\,b_{i}^{\dagger}-\beta_{i}^{(d)*}\,b_{i}\right)\,.$
(31)

Unfortunately, in general, it is not possible to find analytic solutions to
this equation. In the following, we therefore have a closer look at an
unravelling of the above ensemble dynamics into individual quantum
trajectories. These are easier to study analytically, especially if the
cavities are initially prepared in a pair coherent state given by
$\underline{\gamma}(0)$.

####  II.3.1 The no-photon time evolution

To obtain the conditional no-photon evolution, we write the master equation in
Eq. (30) as

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\sum\limits_{d=1,2}\kappa_{d}\,D_{2}(\beta^{(d)}_{2})D_{1}(\beta^{(d)}_{1})a_{d}\,\rho\,a_{d}^{\dagger}D^{\dagger}_{1}(\beta^{(d)}_{1})D^{\dagger}_{2}(\beta^{(d)}_{2})$
(32) $\displaystyle-\frac{\rm i}{\hbar}\left[H_{\rm cond},\rho\right]$

with the conditional Hamiltonian $H_{\rm cond}$ given by

$\displaystyle H_{\rm cond}$ $\displaystyle=$ $\displaystyle-\frac{\rm
i}{2}\hbar\sum\limits_{d=1,2}\kappa_{d}\,a_{d}^{\dagger}a_{d}\,.$ (33)

While the terms in the first line of the above equation describe subensembles
of systems with a photon detection in output port $d$, the second line
describes the subensemble without an emission. In other words, the non-
Hermitian Hamiltonian $H_{\rm cond}$ is the generator for time evolution of
the experimental setup in Fig. 2 conditioned on no photon emission. The
corresponding time evolution operator

$\displaystyle U_{\rm cond}(t,0)$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{{\rm i}}{\hbar}H_{\rm cond}t\right)$ (34)

reduces the norm of state vectors and can be used to calculate the probability
$P_{00}(t)$ for no photon detection in both detectors in a time interval
$(0,t)$ which equals

$\displaystyle P_{00}(t)$ $\displaystyle=$ $\displaystyle\|U_{\rm
cond}(t,0)|\psi(0)\rangle\|^{2}$ (35)

given the initial state $|\psi(0)\rangle$. For example, given an initial pair
coherent state $|\psi(0)\rangle=\ket{\alpha_{1}}\ket{\alpha_{2}}$ with respect
to the modes $a_{1}$ and $a_{2}$ seen by the detector, one can show that

$\displaystyle U_{\rm cond}(t,0)\ket{\alpha_{1}}\ket{\alpha_{2}}$
$\displaystyle=$
$\displaystyle\exp\left(\frac{-|\alpha_{1}|^{2}}{2}\left(1-{\rm
e}^{-\kappa_{1}t}\right)\right)$ (36)
$\displaystyle\times\exp\left(\frac{-|\alpha_{2}|^{2}}{2}\left(1-{\rm
e}^{-\kappa_{2}t}\right)\right)~{}~{}~{}~{}$
$\displaystyle\times\ket{\alpha_{1}{\rm
e}^{-\frac{1}{2}\kappa_{1}t}}\ket{\alpha_{2}{\rm
e}^{-\frac{1}{2}\kappa_{2}t}}\,.$

Hence, using the notation introduced in Sec. II.1, we can summarise the effect
of the no-photon time evolution of the field inside the cavities as
$\underline{\alpha}(t)=M_{0}(t)\underline{\alpha}(0)$ with

$\displaystyle M_{00}(t)$ $\displaystyle=$ $\displaystyle\begin{pmatrix}{\rm
e}^{-\frac{1}{2}\kappa_{1}t}&0\\\ 0&{\rm
e}^{-\frac{1}{2}\kappa_{2}t}\end{pmatrix}\,.$ (37)

The probability of such an evolution occurring is

$\displaystyle P_{00}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-|\alpha_{1}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{1}t}\right)\right]$ (38)
$\displaystyle\times\exp\left[-|\alpha_{2}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{2}t}\right)\right]\,,$

due to Eq. (35).

####  II.3.2 Photon emission probabilities

Next we calculate the probabilities of photon emission. Having a closer look
at the factors in Eq. (38), we see that the probability for an individual
detector mode $i$ not to detect a photon equals

$\displaystyle P_{0}^{(i)}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-|\alpha_{i}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{i}t}\right)\right]\,.$ (39)

Moreover, we know that the probability to find at least one photon in detector
$i$ is given by $1-P^{(i)}_{0}(t)$. Thus the probability for no photon in
detector 1 and at least one photon in detector 2 equals

$\displaystyle P_{01}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-|\alpha_{1}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{1}t}\right)\right]$ (40)
$\displaystyle\times\left(1-\exp\left[-|\alpha_{2}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{2}t}\right)\right]\right)\,.$

Analogously,

$\displaystyle P_{10}(t)$ $\displaystyle=$
$\displaystyle\left(1-\exp\left[-|\alpha_{1}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{1}t}\right)\right]\right)$ (41)
$\displaystyle\times\exp\left[-|\alpha_{2}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{2}t}\right)\right]$

is the probability for at least one photon in detector 1 and no photon in
detector 2. To cover all possibilities, we should also consider the
probability

$\displaystyle P_{11}(t)$ $\displaystyle=$
$\displaystyle\left(1-\exp\left[-|\alpha_{1}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{1}t}\right)\right]\right)$ (42)
$\displaystyle\times\left(1-\exp\left[-|\alpha_{2}(t)|^{2}\left(1-{\rm
e}^{-\kappa_{2}t}\right)\right]\right)$

for the case in which at least one photon has been emitted into both detector
modes.

For relatively short time intervals $\Delta t$ such that the presence of two
photons in one detector becomes negligible and the probabilities
$P_{01}(\Delta t)$, $P_{10}(\Delta t)$ and $P_{11}(\Delta t)$ become the
probabilities to have exactly one photon in mode 2, exactly one photon in mode
1 and exactly one photon in each mode, respectively. In this case, the
corresponding changes of the state vector ${\underline{\alpha}}(0)$ of the
cavity fields can be described by transformation operators $M_{ij}(\Delta t)$
such that $\underline{\alpha}(t)=M_{ij}(\Delta t)\,\underline{\alpha}(0)$. To
a very good approximation, the $M_{ij}(\Delta t)$ are given by

$\displaystyle M_{01}(\Delta t)\,\underline{\alpha}(0)$ $\displaystyle=$
$\displaystyle M_{00}(\Delta
t)\left(\underline{\alpha}(0)+\underline{\beta}^{(2)}\right)$ $\displaystyle
M_{10}(\Delta t)\,\underline{\alpha}(0)$ $\displaystyle=$ $\displaystyle
M_{00}(\Delta t)\left(\underline{\alpha}(0)+\underline{\beta}^{(1)}\right)$
$\displaystyle M_{11}(\Delta t)\,\underline{\alpha}(0)$ $\displaystyle=$
$\displaystyle M_{00}(\Delta
t)\left(\underline{\alpha}(0)+\underline{\beta}^{(1)}+\underline{\beta}^{(2)}\right)~{}~{}~{}$
(43)

with $M_{00}(\Delta t)$ given in Eq. (37). We now have a complete toolbox for
modelling quantum trajectories within this system through piecewise evolution
of the system, as suggested by standard quantum jump methods Reset ; Molmer ;
Carmichael .

##  III General dynamics and temporal correlations

Figure 3: Illustration of the dynamics of the state ${\underline{\alpha}}(t)$
of the two cavities in the detector mode basis. Here we present 500 individual
trajectories for initial parameters $\gamma_{1}(0)=\gamma_{2}(0)=1$, quantum
feedback as described by Eq. (48) with $\beta_{2}^{(1)}=1$ and
$\beta_{1}^{(2)}=2$, $\kappa_{1}=\kappa_{2}=\kappa$, $\Delta
t=10^{-3}\kappa^{-1}$ and $\tilde{\varphi}=0$. We show the population
$|\alpha_{2}(t)|^{2}$ of the detector mode $a_{2}$ as a function of time in
semi-log scale. It is clear to see the difference in behaviour, as some
trajectories diverge, while others decay towards the vacuum, as signified by
the different line shades.

In this section, we study the behaviour of the experimental setup in Fig. 2 in
more detail to understand better, how it can be used to perform measurements
of the phase $\tilde{\varphi}=\varphi_{1}-\varphi_{2}$. We then consider the
fundamental limits of the measurement accuracy that we can expect for the
proposed cavity network.

###  III.1 Dynamics and Quantum trajectories

In the proceeding discussions we analyse the behaviour and the sensing
capabilities of the cavity network shown in Fig. 2 for a specific example of
quantum feedback, which can be described by

$\displaystyle\underline{\beta}^{(1)}$
$\displaystyle=\left(\begin{array}[]{c}0\\\
\beta_{2}^{(1)}\end{array}\right)\quad\underline{\beta}^{(2)}=\left(\begin{array}[]{c}\beta_{1}^{(2)}\\\
0\end{array}\right)\,.$ (48)

As pointed out already in the previous section, we treat the quantum feedback
as approximately instantaneous. For simplicity, we also consider the case of
perfect photon detection and assume that all photons are counted and trigger
feedback pulses. Losses could be incorporated but do not largely effect the
overall behaviour of the cavity network and are therefore neglected here
lewisstokes

Although the experimental setup that we analyse in this paper remains always
in a coherent state, its dynamics are nevertheless complex due to the action
of the quantum feedback. While optical cavities with continuous laser driving
smoothly evolves into a steady state, the same does not always apply in the
presence of quantum feedback. For example, when the feedback is in the form of
strong laser pulses, the free decay of the cavity field is perturbed by
‘kicks’ to the dynamics. These kicks occur the more often, the more photons
are inside the resonator and hence result into a divergence of the average
photon number inside the cavity. This highly non-linear behaviour prevents us
from obtaining a straightforward closed analytic solution to the master
equation and its statistical moments, despite the cavities always being in a
coherent state.

Considering the individual quantum trajectories of the system in Fig. 3
reveals subtle behaviour. In particular, we see the creation of two types of
dynamics. In one case we see a divergence of cavity photon numbers, with each
feedback pulse making the state even more likely to emit a photon and thus
diverging further due to subsequent feedback pulses. However, we also see
trajectories that do not follow this evolution and decay towards the vacuum
state, with only a small number of photon emissions. Generally, after a
reasonable amount of time has passed, trajectories do not swap behaviour and
clearly belong to one of two subensembles, which leads to effective ergodicity
breaking lewismaybee .

Let us emphasise here that this behaviour emerges as a result of manipulating
quantum trajectories and as such is not reproducable classically. To see this,
try to consider a “classical analogue” to he experimental setup shown in Fig.
2. For example, an optical cavity could have the output intensity monitored.
Then, based upon the measured intensity, feedback could be applied. However,
this would lead to a well defined evolution of the dynamics that would not
vary over multiple trajectories. As such, the above described generation of
individual stochastic trajectories is a truly quantum feature. Measurements
depending on this quantum feature will be capable of showing non-classical
correlations HQMM ; HQMM2 . It is this property that we exploit in this work
to develop a quantum jump metrology scheme lewisbeige ; lewisstokes . In order
to see how this quantum enhancement manifests, we now proceed to more
precisely consider the limits of measurement precision.

###  III.2 Quantum Jump Metrology

The amount of information that can be gained from a measurement is quantified
by the Fisher information. Let ${\bf x}$ be a string of data of length $N$
with elements $x_{i}\in{\mathbb{Z}^{+}}$. The Fisher information for such data
is defined as

$\displaystyle F(P_{\varphi})$ $\displaystyle\equiv$
$\displaystyle\sum\limits_{x_{i}}P({\bf
x})\left[\partial_{\varphi}\ln\left(P_{\varphi}({\bf x})\right)\right]^{2}$
(49) $\displaystyle=$
$\displaystyle\sum\limits_{x_{i}}\frac{\left[\partial_{\varphi}P_{\varphi}({\bf
x})\right]^{2}}{P_{\varphi}({\bf x})}\,.$

Here $\varphi$ is the unknown parameter to be probed, which the probability
distribution must hence be a function of. We sum over all possible
combinations of output data ${\bf x}$. Now let $\hat{\varphi}$ be an estimator
or the unknown parameter $\varphi$. The Cramér-Rao bound tells us that the
minimum uncertainty achievable by such an estimator is bounded by the Fisher
information as

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\geq$
$\displaystyle\frac{1}{F(P_{\varphi})}\,.$ (50)

If the $N$ data points are uncorrelated, each contributes an independent
amount of information such that $\left(\Delta\hat{\varphi}\right)^{2}$ scales
like Eq. (1). If the generation of the data is repeated $M$ times, and each
repetition is statistically independent of the others, then by the same logic

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{MN}\,.$ (51)

As one would expect, obtaining more data (through repetitions or larger
sources) leads to a reduction in the uncertainty of the unknown parameter.

If however the data possesses correlations, then the information contribution
from each value may now be beyond linear with respect to the number of data
points. In particular, the correlations that may exist in quantum systems can
lead to more precise measurements when compared with a classical system with
the same number of particles seth ; lloyd ; Maccone . With maximal
correlations, such as in maximally entangled quantum states, the quantum
Cramér-Rao bound tells us

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{MN^{2}}\,.$ (52)

Different ways of obtaining such enhancements have been thoroughly explored in
recent years dowling , such as through entanglement CavesC , squeezing
Bondurant  and phase transitions Macieszczak . Unfortunately, while the theory
predicts great results, achieving such feats experimentally is more difficult,
due to the challenges in implementing quantum physics with more than a handful
of particles. Another way of improving the limit of scaling within the system
is with non-linear or interacting systems boixo . Such systems are capable of
producing scaling of

$\displaystyle\left(\Delta\hat{\varphi}\right)^{2}$ $\displaystyle\propto$
$\displaystyle\frac{1}{MN^{k}}\,,$ (53)

where $k$ is the order of the non-linearity or interaction. Such systems have
been shown to be capable of producing very good scaling boixo , but again are
typically hard to implement experimentally.

However, a quantum jump provides a non-linearity in the dynamics of a quantum
trajectory and thus a measurement that explicitly depends on the properties of
trajectories instead of depending on ensemble averages has already been shown
to be capable of enhanced measurements lewisbeige ; lewisstokes . To see this,
consider the time evolution of an open quantum system described by a density
matrix $\rho$, which can be expressed by the standard Lindbladian master
equation

$\displaystyle\dot{\rho}$ $\displaystyle=$ $\displaystyle-\frac{\rm
i}{\hbar}\left[H,\rho\right]+\Gamma\left(L\rho
L^{\dagger}-\frac{1}{2}\left[L^{\dagger}L,\rho\right]_{+}\right)\,$ (54)

with Hamiltonian $H$ and where for simplicity we have considered a single
decay channel with Lindblad operator $L$ and rate $\Gamma$. Clearly, this
equation is linear. However, the evolution of a trajectory may follow a
stochastic master equation of the form

$\displaystyle{\rm d}\rho$ $\displaystyle=$ $\displaystyle\left(-\frac{\rm
i}{\hbar}\left[H,\rho\right]-\frac{\Gamma}{2}\left[L^{\dagger}L,\rho\right]_{+}+\mbox{Tr}\left(L^{\dagger}L\rho\right)\rho\right){\rm
d}t$ (55) $\displaystyle+\left(\frac{L\rho
L^{\dagger}}{\mbox{Tr}\left(L^{\dagger}L\rho\right)}-\rho\right){\rm
d}N_{t}\,,$

with Poisson increment ${\rm d}N_{t}$ being zero for no-emission at time $t$
or one in the case of an emission Jacobs_book . From Eq. (55) it is clear that
the time evolution along an individual quantum trajectory is non-linear in
general, so long as quantum jumps exist and $L\rho L^{\dagger}\neq\rho$.
Clearly, if $\rho$ is a coherent state and $L$ is simply the annihilation
operator, this is not the case and as such there will be no quantum
enhancement. This shows why feedback is necessary and how it hence induces
quantum enhancements to sensing performance. Therefore, if the quantum jump
depends on the parameter to be measured in some way, a quantum-enhanced
measurement may be possible.

Figure 4: The probability $(P_{N>5})_{d_{i}}$ of surpassing the threshold
number of $N=5$ photon emissions within a certain time interval $(0,t)$ as a
function of the phase shift difference $\tilde{\varphi}$ for
$t=0.5\,\kappa^{-1}$ (red-solid), $t=\kappa^{-1}$ (blue-dot-dashed) and
$t=10\,\kappa^{-1}$ (green-dashed). Here we average over $10^{4}$ trajectories
with $\Delta t=10^{-3}\kappa^{-1}$, $\gamma_{1}(0)=\gamma_{2}(0)=1$ and
quantum feedback as described in Eq. (48) with $\beta_{1}^{(2)}=2$ and
$\beta_{2}^{(1)}=1$. The figure shows (a) $(P_{N>5})_{d_{2}}$, (b)
$(P_{N>5})_{d_{1}}$ and (c) $(P_{N>5})_{d_{1}}-(P_{N>5})_{d_{2}}$ as a
function of $\tilde{\varphi}$.

Here we study the bath statistics of an open quantum system in order to
extract information about an unknown internal system parameter, such as a
phase $\varphi$. If the emission of quanta from the system to the bath creates
a quantum jump within the dynamics of the system, temporal correlations may
exist in the bath, which may allow for enhanced measurement sensitivity
lewisbeige ; lewisstokes . It is straightforward to see that such correlations
exist. Consider an open quantum system whose dynamics can be described by a
set of Kraus operators $K_{x}$ with

$\displaystyle\sum\limits_{x}K^{\dagger}_{x}K_{x}$ $\displaystyle=$
$\displaystyle\mathbb{1}\,.$ (56)

Let $\rho$ be a density matrix describing the internal state of an open
quantum system. Then upon the measurement of $x$ in the bath, the internal
state of the system evolves to $\rho\rightarrow K_{x}\rho K^{\dagger}_{x}$.
This state is unnormalised, occurring with probability
$\mbox{Tr}\left[K_{x}\rho K^{\dagger}_{x}\right]$. Now consider a series of
$N$ measurements of $\left\\{x_{i}\right\\}_{i=1}^{i=N}$. The resulting
density matrix occurs with probability

$\displaystyle p(x_{1},\dots,x_{N})$ $\displaystyle=$
$\displaystyle\mbox{Tr}\left[\left(\prod\limits_{i=1}^{N}K_{x_{N+1-i}}\right)\rho\left(\prod\limits_{i=1}^{N}K_{x_{i}}^{\dagger}\right)\right]\,.$

The Kraus operators $K_{x}$ do not necessarily commute, thus the events and
measurements are not independent from one another. This can be seen more
clearly by comparing the probabilities

$\displaystyle p(x_{N}|x_{N-1})$ $\displaystyle=$
$\displaystyle\frac{\mbox{Tr}\left[K_{N}K_{N-1}\mathcal{T}_{1}^{N-2}\left(\rho\right)K_{N-1}^{\dagger}K_{N}^{\dagger}\right]}{\mbox{Tr}\left[K_{N-1}\mathcal{T}_{1}^{N-2}\left(\rho\right)K_{N-1}^{\dagger}\right]}\,,$

which is the probability of measuring $x_{N}$ after $x_{N-1}$ in the previous
time step with the probability of measuring $X_{N}$ after $x_{N-1}$ and
$x_{N-2}$

$\displaystyle p(x_{N}|x_{N-1}x_{N-2})$ $\displaystyle=$
$\displaystyle\frac{\mbox{Tr}\left[K_{N}K_{N-1}K_{N-2}\mathcal{T}_{1}^{N-3}\left(\rho\right)K_{N-2}^{\dagger}K_{N-1}^{\dagger}K_{N}^{\dagger}\right]}{\mbox{Tr}\left[K_{N-1}K_{N-2}\mathcal{T}_{1}^{N-3}\left(\rho\right)K_{N-2}^{\dagger}K_{N-1}^{\dagger}\right]}\,.$

The superoperator $\mathcal{T}_{i}^{j}$ describes a Markovian evolution from
time-step $i$ to $j$. In general, Eqs. (III.2) and (III.2) differ, meaning the
measurement statistics do not form a Markov chain and thus possess
correlations lewisbeige ; lewisstokes . If the Kraus operators are dependent
on the unknown parameter, these correlations may lead to a Fisher information
growing faster than linearly and thus allows for enhanced precision. As we
shall see below, in the case of the cavity network with quantum feedback that
we consider here, this is the case and thus the potential for obtaining
quantum enhanced measurements exists.

##  IV A simple measurement scheme for estimating phase difference

In this section, we finally discuss, how the phase difference
$\tilde{\varphi}=\varphi_{1}-\varphi_{2}$ between two pathways of the network
shown in Fig. 2 can be measured. We propose a simple experimental scheme and
analyse its performance by calculating the uncertainty of $\tilde{\varphi}$
based on an ensemble of simulated quantum trajectories.

###  IV.1 The basic protocol

Figure 5: The uncertainty in the phase shift measurement for phase differences
$\tilde{\varphi}=\pi/10$ as function of time is shown for the solid line. The
probabilities shown in Fig. 4 are generated with $10^{3}$ trajectories and 10
subensembles of these are generated to obtain the variance of the estimator.
The dashed line shows the extrapolated value of the reciprocal of the Fisher
information, providing an estimated lower bound on potential the sensitivity
of the data. The Fisher information is obtained exactly for 12 time steps
$\Delta t$ and the subsequent fit is extended for all time. We plot the data
of our results only for later times, where the trend is clearer to see. All
parameters are as in Fig. 4.

As Fig. 3 shows, there are two classes of trajectory. Moreover, it shows that
the relative size of the subensemble associated with each class has a
relatively strong dependence on the phase difference $\tilde{\varphi}$ that we
want to estimate. This applies since the only way that energy is put into the
system is through quantum feedback, which is triggered upon photon emission.
The number of detected photons therefore acts as a reliable signal for the
type of trajectory being observed. For example, a convergent trajectory will
have emitted no or few photons until a given time $t$, whereas an eventually
divergent one is likely to have emitted many photons due to the repeated
pumping of energy into the system. In the following, we therefore take the
probability of the total number of emitted photons until a given time $t$
surpassing a certain threshold $N$ as our measurement signal. In particular,
we take the probability of the number of photons surpassing 5 as our signal,
denoted as $P_{N>5}$. This signal is a good indicator of which trajectory
class the system is following. However, it is important to choose the
parameters carefully in order to ensure the signal reveals useful information.
For example, choosing the threshold number of photons to be too low won’t
faithfully distinguish between the trajectory classes, depending also on the
strength of the feedback and size of the initial state. Therefore, finding a
balance between these is essential when determining the phase difference
$\tilde{\varphi}$.

As mentioned in the previous section, an analytic calculation for the
behaviour of the ensemble is straightforward due to the quantum feedback
giving rise to non-linear dynamics. Hence, to determine the uncertainty of the
estimator of $\tilde{\varphi}$, we numerically simulate a large number of
trajectories over a coarse-grained timescale. By sampling these trajectories,
we are able to estimate the probability of the system emitting a number of
photons surpassing the threshold we set. From this, we then infer the
uncertainty of measuring $\tilde{\varphi}$ from this data as a function of the
time $t$.

###  IV.2 Measurement of phase difference

Here we consider a specific set of parameters to demonstrate the utility of
our system for estimating a phase in an optical cavity network. As with the
threshold value, it is important to choose feedback parameters that are both
not too small and not too big. For example, for very weak feedback, we are
unlikely to deduce information about $\tilde{\varphi}$ over reasonable
timescales. Moreover, for very strong feedback, the dynamics of the cavity
network becomes dominated by the feedback, almost all trajectories diverge and
the measurement outcomes are essentially independent of the unknown phase that
we want to identify. Finally, in order to avoid starting in the vacuum state
and for the practicality of implementation, we start each trajectory with a
feedback pulse to prepare a non-trivial initial state. With all of these
factors in mind, Fig. 4 shows three different measurement signals obtained
after three different times $t$. Three different observables are considered,
namely, the probabilities $(P_{N>5})_{d_{1}}$ and $(P_{N>5})_{d_{2}}$ and
their difference $(P_{N>5})_{d_{1}}-(P_{N>5})_{d_{2}}$. All three observables
can be used to infer the value of $\tilde{\varphi}$.

As we shall see below, considering only the output of either of the detectors,
i.e. considering only $(P_{N>5})_{d_{1}}$ or $(P_{N>5})_{d_{2}}$, does not
give the best sensitivity since some of the available information from the
dynamics of the system is omitted in this case. It is better to take
information from both detectors into account and to use instead
$(P_{N>5})_{d_{1}}-(P_{N>5})_{d_{2}}$ as the estimator for $\tilde{\varphi}$.
To show that this is indeed the case, we now determine the measurement
uncertainty $\delta\hat{\varphi}$ with the help of the standard error
propagation formula

$\displaystyle\left(\Delta\varphi\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{\left(\Delta
O\right)^{2}}{\left|\frac{\partial\left\langle
O\right\rangle}{\partial\varphi}\right|^{2}}\,,$ (60)

where $O$ denotes the relevant observable. The variance in the numerator of
this equation is obtained by sampling over a number of subensembles of
trajectories, while the visibility in the denominator is obtained numerically
from Fig. 4. The result of this calculation is shown in Fig. 5. As in Fig. 4,
three different observables are considered. Our results show that as time
increases we obtain better accuracy for measuring the phase shift
$\tilde{\varphi}$ in either detector mode. However, we obtain our best results
when we consider the difference between the two threshold probabilities
$(P_{N>5})_{d_{1}}$ and $(P_{N>5})_{d_{2}}$.

Considering Fig. 4, we see that the optimal phase to conduct an estimation at
is around $\tilde{\varphi}=0$ due to the sharpness of the gradient at this
point. This corresponds to a crucial point in the dynamics. When
$\tilde{\varphi}=0$, only one detector mode is ever occupied. However, moving
away from this point, the other detector mode begins to be occupied too. Thus,
taking advantage of this distinction in the signal allows for the best
measurement. Due to numerical instabilities evaluating at this point however,
we take our data at a nearby value of $\tilde{\varphi}=\pi/10$, where the
gradient is still large for reasonable amounts of time.

###  IV.3 Fisher information for the photon statistics of an optical cavity
network

Finally, we calculate the Fisher information of the photon statistics using
the methods outlined in Section III.2 in order to obtain a bound on the
optimum precision our measurement scheme can achieve. The probability of a
certain trajectory with a given number of time steps can be calculated using
Eq. (III.2). To do so, it has to be taken into account that every individual
time step has one of four different possible event types (no-photon, photon in
a single detector or photons in both detectors). The theoretical model to
describe each of these events has been introduced in Section II. A drawback of
this however is that an exact calculation of the Fisher information for a
trajectory of length $N$ requires summing over $4^{N}$ trajectories. Hence
this approach becomes computationally very challenging for large $N$.
Nevertheless, as illustrated in Fig. 5, for small $N$, we find scaling beyond
linear, of the form $F(N)\propto N^{2}-N$. This result is in agreement with
the scaling behaviour observed in Ref. lewisbeige , thus demonstrating the
presence of correlations in the photon statistics that depend on
$\tilde{\varphi}$.

The limitation of only having exact results for the Fisher information for a
small number of time steps means we do not always have a strict bound for the
system. Instead, we extrapolate the scaling shown for short times. We expect
that this will be an upper bound on the Fisher information, as it is likely
that at large times the scaling will reduce rather than increase due to a
breakdown in correlations between far away time steps. This estimated bound
can nevertheless be useful for comparing to the uncertainty of our proposed
measurement scheme though, as illustrated above in Fig. 4. In particular, the
sensing performance in Fig. 5(c) approaches our extrapolated bound, thus
suggesting that our fit result is reasonable.

##  V Conclusions

In this paper we have demonstrated how the phase difference between the “arms”
of an optical cavity network (c.f. Fig. 2) can be inferred requiring only
single photon detectors and quantum feedback in the form of strong but
approximately instantaneous laser pulses. Despite the simplicity of the
proposed measurement scheme, we have shown that it is capable of measuring the
phase difference with a sensitivity beyond the standard quantum limit. As in
previous work lewisbeige ; lewisstokes , the presence of quantum jumps in the
form of photon emissions is continuously monitored. Subsequent driving of the
cavities creates non-linearities in the system dynamics, thereby inducing
correlations in the photon statistics observed by the detectors.

In the experimental setup that we consider here, the cavities remain always in
a coherent state. Although photon emission does not alter the state of the
cavities, it reveals information about the state of the resonators. Similarly,
not observing photons reveals information. The dependence of the quantum
feedback induced dynamics on the parameter that we want to measure leads to
effective ergodicity-breaking in the dynamics of the system, similar to the
ergodicity breaking discussed in Ref. lewismaybee  and results in two
different classes of trajectory. In this paper, we have shown that measuring
the probability for these two classes of dynamics to occur reveal information
about the phase difference between the arms of the cavity network beyond the
standard quantum limit. Even better scaling might be achieved by preparing the
cavities in a more complex initial state than a coherent state. However, the
use of coherent states offers experimental simplicity and as such the proposed
scheme can be operated more straightforwardly and for longer periods of time.

A limitation of our work is that the results are only attainable numerically,
and the Fisher information is only calculable for small times. However, the
expected behaviour has been predicted with approximations previously and the
observed behaviour here is in line with this lewismaybee . Moreover, it may be
possible to obtain estimates for the Fisher information at large times by
using sampling techniques. Even in this case however, the number of possible
trajectories is extremely large even for reasonable values of $N$, thus
meaning the number of trajectories needed to sample over may also need to be
large. Due to how well the predicted bound matches the uncertainty predicted
from our measurement signal though, we believe that our fitting of the Fisher
information to be reasonable.

An alternative measurement protocol for gaining information about the phase
would be to utilise all information gained in the continuous monitoring of the
cavity network and follow a Bayesian inference procedure. In an open system
where the photon statistics are observed this is perhaps the most natural way
to envisage the measurement of an unknown parameter pstat . This could be
further supplemented by a strong quantum measurement of the cavity state at
the end of the observation which supplements the information gained from
monitoring the photon statistics. For the purpose of this work though, we
choose to just consider the Fisher information as a proof-of-principle that a
quantum enhancement exists. Because of this property, cavity networks like the
one shown in Fig. 2 are very likely to attract more attention for applications
in quantum sensing and might play a crucial role in the development of quantum
machine learning devices.

### Acknowledgements

LAC acknowledges support from the Foundation for Polish Science within the
”Quantum Optical Technologies” project carried out within the International
Research Agendas programme co-financed by the European Union under the
European Regional Development Fund. KAR acknowledges the support from the
Ministry of Higher Education, Research and Innovation in the Sultanate of Oman
funded by The National Postgraduate Scholarship Programme.

## References

  * (1) A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame and J. L. O’Brien, Measuring protein concentration with entangled photons, Appl. Phys. Lett. 100, 233704 (2012).
  * (2) F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout and M. W. Mitchell, Entanglement-enhanced probing of a delicate material system, Nature Photon. 7, 28 (2013).
  * (3) M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor and W. P. Bowen, Biological measurement beyond the quantum limit, Nature Photon. 7, 229 (2013).
  * (4) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).
  * (5) S. Aaronson and A. Arkhipov, The computational complexity of linear optics. In Proceedings of the forty-third annual ACM symposium on Theory of computing (pp. 333-342). ACM. (2011).
  * (6) R. Loughridge and D. Y. Abramovitch, A tutorial on laser interferometry for precision measurements, 2013 American Control Conference, pp. 3686-3703 (2013). doi: 10.1109/ACC.2013.6580402.
  * (7) R. Demkowicz-Dobrzanski, M. Jarzyna and J. Kolodynski, Quantum limits in optical interferometry, Progress in Optics 60, 345 (2015).
  * (8) Z. Blanco-Garcia, Quantum Control of the States of Light in a Mach-Zehnder Interferometer, J. Phys.: Conf. Ser. 839, 012021 (2017).
  * (9) S. Yang and G. Zhang, A review of interferometry for geometric measurement, Meas. Sci. Technol. 29, 102001 (2018).
  * (10) S. Ataman, A. Preda and R. Ionicioiu, Phase sensitivity of a Mach-Zehnder interferometer with single-intensity and difference-intensity detection, Phys. Rev. A 98, 043856 (2018).
  * (11) V. Giovannetti, S. Lloyd and L. Maccone, Quantum-Enhanced Measurements: Beating the Standard Quantum Limit. Science 306, 1330 (2004).
  * (12) V. Giovannetti, S. Lloyd and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  * (13) V. Giovannetti, S. Lloyd and L. Maccone, Advances in quantum metrology, Nat. Photon. 5, 222 (2011).
  * (14) C. M. Caves,  Quantum-mechanical noise in an interferometer, Phys. Rev. D 23 1693 (1981).
  * (15) R. S. Bondurant and J. H. Shapiro, Squeezed states in phase-sensing interferometers. Phys. Rev. D 30, 2548 (1984).
  * (16) M. J. Holland and K. Burnett,  Interferometric detection of optical phase shifts at the Heisenberg limit, Phys. Rev. Lett. 71 1355 (1993).
  * (17) D. W. Berry and H. M. Wiseman,  Optimal states and almost optimal adaptive measurements for quantum interferometry, Phys. Rev. Lett. 85, 5098 (2000).
  * (18) T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki and S. Takeuchi,  Beating the standard quantum limit with four-entangled photons, Science 316, 726 (2007).
  * (19) J. P. Dowling, Quantum optical metrology —The lowdown on high-NOON states, Contemp. Phys. 49, 125 (2008).
  * (20) I. Afek, O. Ambar, and Y. Silberberg, High-NOON states by mixing quantum and classical light, Science 328, 879 (2010).
  * (21) G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman and G. J. Pryde, Entanglement-enhanced measurement of a completely unknown optical phase. Nat. Photon. 5, 43 (2011).
  * (22) M. D. Vidrighin, G. Donati, M. G. Genoni, X.M. Jin, W. S. Kolthammer, M. S. Kim and I. A. Walmsley, Joint estimation of phase and phase diffusion for quantum metrology, Nat. Comm. 5, 1 (2014).
  * (23) H. Lee, P. Kok and J. P. Dowling,  A quantum Rosetta stone for interferometry, J. Mod. Opt. 49, 2325 (2002).
  * (24) M. W. Mitchell, J. S. Lundeen and A. M. Steinberg, Super-resolving phase measurements with a multiphoton entangled state, Nature 429 161 (2004).
  * (25) G. J. Pryde and A. G. White, Creation of maximally entangled photon-number states using optical fiber multiports, Phys. Rev. A 68, 052315 (2003).
  * (26) Y. Israel, I. Afek, S. Rosen, O. Ambar and Y. Silberberg,  Experimental tomography of NOON states with large photon numbers, Phys. Rev. A 85, 022115 (2012).
  * (27) G. Gilbert, M. Hamrick and Y. S. Weinstein,  Practical quantum interferometry using photonic N00N states, In Quantum Information and Computation V (Vol. 6573, p. 65730K), International Society for Optics and Photonics (2007).
  * (28) N. Thomas-Peter, B. J. Smith, A. Datta, L. Zhang, U. Dorner and I. A. Walmsley, Real-world quantum sensors: evaluating resources for precision measurement, Phys. Rev. Lett. 107, 113603 (2011).
  * (29) B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman and G. J. Pryde, Entanglement-free Heisenberg-limited phase estimation, Nature 450, 393 (2007).
  * (30) S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan and C. M. Caves, Quantum-limited metrology with product states, Phys. Rev. A 77, 012317 (2008).
  * (31) S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji and C. M. Caves, Quantum metrology: dynamics versus entanglement. Phys. Rev. Lett. 101 040403 (2008).
  * (32) M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell and M. W. Mitchell,  Interaction-based quantum metrology showing scaling beyond the Heisenberg limit, Nature 471, 486 (2011).
  * (33) A. Datta and A. Shaji, Quantum metrology without quantum entanglement, Mod. Phys. Lett. B 26, 1230010 (2012).
  * (34) D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell and S. Pirandola, Quantum-enhanced measurements without entanglement, Rev. Mod. Phys. 90, 035006 (2018).
  * (35) L. A. Clark, A. Stokes and A. Beige, Quantum-enhanced metrology with the single-mode coherent states of an optical cavity inside a quantum feedback loop, Phys. Rev. A 94, 023840 (2016).
  * (36) L. A. Clark, A. Stokes and A. Beige, Quantum jump metrology, Phys. Rev. A 99, 022102 (2019).
  * (37) H. M. Wiseman and G. J. Milburn, Quantum measurement and control, Cambridge University Press (2009).
  * (38) A. Shabani, J. Roden and K. B. Whaley,  Continuous measurement of a non-Markovian open quantum system, Phys. Rev. Lett. 112, 113601 (2014).
  * (39) J. F. Haase, A. Smirne, S. F. Huelga, J. Kolodynski and R. Demkowicz-Dobrzanski, Precision limits in quantum metrology with open quantum systems, Quantum Meas. Quantum Metrol. 5, 13 (2016).
  * (40) M. Beau and A. del Campo, Nonlinear quantum metrology of many-body open systems, Phys. Rev. Lett. 119, 010403 (2017).
  * (41) F. Albarelli, M. A. Rossi, D. Tamascelli and M. G. Genoni, Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment, Quantum 2, 110 (2018).
  * (42) C. Ahn, A. C. Doherty and A. J. Landahl, Continuous quantum error correction via quantum feedback control. Phys. Rev. A 65, 042301 (2002).
  * (43) J. Wang and H. M. Wiseman, Feedback-stabilization of an arbitrary pure state of a two-level atom, Phys. Rev. A 64, 063810 (2001).
  * (44) J. Wang, H. M. Wiseman and G. J. Milburn, Dynamical creation of entanglement by homodyne-mediated feedback, Phys. Rev. A 71, 042309 (2005).
  * (45) L. A. Clark, W. Huang, T. Barlow and A. Beige, Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback. In ISCS 2014: Interdisciplinary Symposium on Complex Systems (pp. 143-151). Springer, Cham. (2015).
  * (46) L. A. Clark, F. Torzewska, B. Maybee and A. Beige, Non-ergodicity in open quantum systems through quantum feedback, EPL 130, 54002 (2020).
  * (47) G. K. Gulati, H. Takahashi, N. Podoliak, P. Horak and M. Keller, Fiber cavities with integrated mode matching optics, Sci. Rep. 7, 5556 (2017).
  * (48) I. E. Zadeh et al., Efficient single-photon detection with 7.7 ps time resolution for photon-correlation measurements, ACS Photonics 7, 1780 (2020).
  * (49) G. C. Hegerfeldt, How to reset an atom after a photon detection: Applications to photon-counting processes, Phys. Rev. A 47, 449 (1993).
  * (50) J. Dalibard, Y. Castin and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  * (51) H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, Volume 18 (Springer, Berlin, 1993).
  * (52) A. Monras, A. Beige and K. Wiesner, Hidden Quantum Markov Models and non-adaptive read-out of many-body states, Appl. Math. and Comp. Sciences 3, 93 (2011); arXiv:1002.2337.
  * (53) L. A. Clark, W. Huang, T. M. Barlow and A. Beige, Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback, ISCS 2014: Interdisciplinary Symposium on Complex Systems, Emergence, Complexity and Computation 14, p. 143, Springer (2015). arXiv:1406.5847.
  * (54) J. Dowling and K. Seshadreesan, Quantum Optical Technologies for Metrology, Sensing, and Imaging, J. Light. Technol. 33, 2359 (2015).
  * (55) K. Macieszczak, M. Guta, I. Lesanovsky and J. P. Garrahan, Dynamical phase transitions as a resource for quantum enhanced metrology. Phys. Rev. A 93, 022103 (2016).
  * (56) K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).
  * (57) A. Lumino, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe and F. Sciarrino, Experimental phase estimation enhanced by machine learning, Phys. Rev. Applied 10, 044033 (2018).