# Extracting work from coherence in a two-mode Bose-Einstein condensate

L. A. Williamson  ARC Centre of Excellence for Engineered Quantum Systems,
School of Mathematics and Physics, University of Queensland, St Lucia,
Queensland 4072, Australia  F. Cerisola  Department of Physics and Astronomy,
University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom  J. Anders
Department of Physics and Astronomy, University of Exeter, Stocker Road,
Exeter EX4 4QL, United Kingdom  Institut für Physik und Astronomie, University
of Potsdam, 14476 Potsdam, Germany  Matthew J. Davis  ARC Centre of Excellence
for Engineered Quantum Systems, School of Mathematics and Physics, University
of Queensland, St Lucia, Queensland 4072, Australia

###### Abstract

We show how work can be extracted from number-state coherence in a two-mode
Bose-Einstein condensate. With careful tuning of parameters, a sequence of
thermodynamically reversible steps transforms a Glauber coherent state into a
thermal state with the same energy probability distribution. The work
extracted during this process arises entirely from the removal of quantum
coherence. More generally, we characterise quantum (from coherence) and
classical (remaining) contributions to work output, and find that in this
system the quantum contribution can be dominant over a broad range of
parameters. The proportion of quantum work output can be further enhanced by
squeezing the initial coherent state. Due to the many-body nature of the
system, the work from coherence can equivalently be understood as work from
entanglement.

##  I Introduction

Converting disordered energy (heat) into ordered energy (work) is one of the
most fundamental processes in thermodynamics. In a classical system, disorder
arises from practical limitations on what can be known about a large system.
Observables in a quantum system may exhibit not only classical uncertainty,
but also quantum uncertainty arising from coherence [1, 2]. Quantum coherence
is defined with respect to a particular basis, and occurs when a system exists
in superpositions of eigenstates of that basis. Of particular relevance to
thermodynamics is the quantum uncertainty in a system’s energy, arising from
coherence in the energy eigenbasis [3, 4, 5].

It is now well established that coherence in the energy eigenbasis can enhance
work extraction if appropriately utilised. For a sufficiently rapid engine
cycle, coherence can enhance power output beyond that of any classical engine
operating with the same resources [6, 7, 8]. In single-shot realisations,
coherence may diminish the maximum available work [9], however, a carefully
chosen sequence of thermodynamic processes circumvents this degradation [10,
11, 12, 13]. Despite the benefits of coherence in work extraction, protocols
to realise these benefits in experimentally tractable systems are lacking.
Such expositions are timely, considering the relevance to quantum information
[14, 15, 16] and the growing field of quantum thermal machines [17, 18].

Bose-Einstein condensates (BECs) offer a pristine system in which to explore
many-body quantum physics, due to a high degree of tuneability and isolation
from the environment [19, 20]. In the field of quantum thermal machines,
bosonic particle statistics can improve engine performance [21, 22, 23, 24]
and act as a fuel in an engine-like cycle [25]. Interactions in a BEC can
enhance engine performance [26, 27] or allow access to energetic degrees of
freedom not available classically [28, 29]. However, the role of coherence in
work extraction from a BEC has not been explored. Notably, two-mode BECs can
exhibit long-lived coherence in the number state basis [30]. Incorporating
squeezing [31, 32, 33], two-mode BECs can exhibit metrologically useful
entanglement, allowing for quantum-enhanced measurement sensitivity [34, 35,
36].

Building on the generic protocol in [11], in this manuscript we show how
number-state coherence in a two-mode BEC can empower work extraction. We
introduce a formalism to quantify the work directly available from number-
state coherence and the work that would be available classically. To obtain
work that is predominantly quantum in origin, the initial state must be close
to thermal when projected onto the energy eigenbasis. This can be achieved by
using a Glauber coherent state with mean boson number less than a critical
value that depends on temperature. We show how squeezing the initial state can
substantially increase the proportion of quantum work extracted. The number-
state coherence in our protocol arises from entangling correlations between
the two modes of the BEC, and hence the work output can equivalently be
interpreted as work from entanglement. To our knowledge, our work presents the
first proposal showing how work can be extracted from coherence in a two-mode
BEC, paving the way for experimental demonstration and highlighting the
utility of two-mode BECs in thermodynamic applications.

##  II Background

###  II.1 Extracting work from coherence

To see how work can be extracted from quantum coherence generally, consider a
Hamiltonian $\hat{H}$ with thermal state
$\rho_{\mathrm{therm}}=e^{-\beta\hat{H}}/\operatorname{Tr}e^{-\beta\hat{H}}$
at inverse temperature $\beta$. Now consider a second quantum state $\rho$
that satisfies

$\braket{E_{n}}{\rho}{E_{n}}=\braket{E_{n}}{\rho_{\mathrm{therm}}}{E_{n}},$
(1)

with $\ket{E_{n}}$ the energy eigenstates of $\hat{H}$. The information
entropy associated with energy measurements of state $\rho$ (termed “energy
entropy” in [5]) is

$S^{E}=-\sum_{n}\braket{E_{n}}{\rho}{E_{n}}\ln\braket{E_{n}}{\rho}{E_{n}},$
(2)

which for an isolated system is non-decreasing under time evolution [37, 38].
When Eq. (1) is satisfied, the energy entropy of state $\rho$ is identical to
that of $\rho_{\mathrm{therm}}$. If $\rho$ is diagonal in the energy
eigenbasis then $S^{E}$ is equal to the von Neumann entropy
$S^{V}=-\operatorname{Tr}\left[\rho\ln\rho\right]$ and $\rho$ is equal to
$\rho_{\mathrm{therm}}$, which is a completely passive state [39]. If,
however, $\rho$ contains coherences (off-diagonal) terms in the energy
eigenbasis, $S^{E}$ will differ from $S^{V}$ [37, 40, 41]. The difference
between the two,

$S^{E}-S^{V}=\sum_{m,n\neq
m}\braket{E_{m}}{\rho}{E_{n}}\braket{E_{n}}{\ln\rho}{E_{m}},$ (3)

can be extracted in the form of non-classical work [11, 13, 42]

$W=\beta^{-1}(S^{E}-S^{V})=\beta^{-1}D(\rho||\rho_{\mathrm{ed}}).$ (4)

Here $D(\rho||\rho_{\mathrm{ed}})\geq 0$ is the quantum relative entropy, with
$\rho_{\mathrm{ed}}=\sum_{n}\braket{E_{n}}{\rho}{E_{n}}\ket{E_{n}}\bra{E_{n}}$
the projection of the density matrix onto the energy eigenstates (the “energy
diagonal” density matrix). The quantum relative entropy is a monotonic measure
of quantum coherence of state $\rho$ [43] and hence the work output, Eq. (4),
is quantum in origin.

A protocol to extract work from coherence was presented in [11]. The efficacy
of this protocol depends on how closely the system’s mean energy and energy
entropy match that of the thermal state $\rho_{\mathrm{therm}}$. When these
match, any work extracted during the protocol is entirely from coherence. A
sufficient condition for this matching is that
$\rho_{\mathrm{ed}}=\rho_{\mathrm{therm}}$, Eq. (1). This can always be
satisfied in a two-level system by choosing the temperature of
$\rho_{\mathrm{therm}}$ to satisfy [44]

$k_{B}T=\frac{E_{1}-E_{0}}{\ln\frac{\braket{E_{0}}{\rho}{E_{0}}}{\braket{E_{1}}{\rho}{E_{1}}}}.$
(5)

Here $E_{0}$ and $E_{1}$ are the two energy levels with $E_{1}>E_{0}$.

In many-body systems, it is usually much harder to engineer experimentally a
non-thermal state $\rho$ satisfying
$\rho_{\mathrm{ed}}=\rho_{\mathrm{therm}}$. Achieving this requires

$r_{n}=\frac{E_{n}-E_{n-1}}{\ln\frac{\braket{E_{n-1}}{\rho}{E_{n-1}}}{\braket{E_{n}}{\rho}{E_{n}}}},$
(6)

to be insensitive to $n$, in which case $k_{B}T=r_{n}$ [44, 45]. Engineering
this for non-thermal many-body states $\rho$ is often very difficult, since
control of individual energy levels is usually not possible. As a result, the
work output from coherence from such systems will be small. As we will show
below, the energy-level structure of an interacting two-mode BEC allows
$\rho_{\mathrm{ed}}\approx\rho_{\mathrm{therm}}$ for particular experimentally
realisable initial states, hence allowing for work output with a high
contribution from coherence.

###  II.2 System setup

We consider a two-mode BEC with fixed total boson number $N$. The two modes
could be hyperfine spin states or spatial modes. Number states of the system
are denoted by $\ket{n,k}$, with $n$ and $k=N-n$ the number of bosons in the
two modes respectively. We denote the annihilation operators for the two modes
by $\hat{a}$ and $\hat{b}$, with $\hat{a}\ket{n,k}=\sqrt{n}\ket{n-1,k}$ and
$\hat{b}\ket{n,k}=\sqrt{k}\ket{n,k-1}$. The Hamiltonian for the system is
($\hbar\equiv 1$ here and throughout) [35]

$\begin{split}\hat{H}=&\omega_{1}\hat{a}^{\dagger}\hat{a}+\omega_{2}\hat{b}^{\dagger}\hat{b}+g_{11}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}+g_{22}\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}\\\
&+g_{12}\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{a}\hat{b}+E_{0}\\\
\equiv&\Delta\hat{a}^{\dagger}\hat{a}+g\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}.\end{split}$
(7)

Here $\omega_{i}$ is the single-particle energy of mode $i$ and $g_{ij}$ is
the interaction energy between particles from modes $i,j$, with
$i,j\in\\{1,2\\}$. The vacuum energy is $E_{0}$. The operator
$\hat{N}=\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}$ commutes with
$\hat{H}$ and hence the total boson number is conserved. In the second line of
Eq. (7) we replace $\hat{b}^{\dagger}\hat{b}$ by
$\hat{N}-\hat{a}^{\dagger}\hat{a}$ and define
$\Delta\equiv\omega_{1}-\omega_{2}-2(N-1)g_{22}+(N-1)g_{12}$ and $g\equiv
g_{11}+g_{22}-g_{12}$. We choose $E_{0}=-N\omega_{2}-N(N-1)g_{22}$ so that
$\hat{H}\ket{0}=0$ for convenience. The choice of $E_{0}$ will not affect the
net output of any cyclic process, for example the performance of an engine
cycle. The detuning $\Delta$ for two spatial modes can be controlled via
tuning the trapping potential [46], and for two spin states via tuning of a
Zeeman field [47, 48]. As will be seen below, we will require immiscible
interactions $g>0$ [49]. Interactions between the two modes (strength
$g_{12}$) are included for generality, but are not necessary.

Particle conservation confines the system to the subset of states
$\ket{n,N-n}$. Since number states are eigenstates of Eq. (7), the number-
state coherence realisable in a two-mode BEC [30] is equivalent to the energy
coherence enabling work extraction, Eq. (3). The thermal state of $\hat{H}$
is,

$\rho_{\mathrm{therm}}=\frac{1}{Z}e^{-\beta\hat{H}}=\sum_{n=0}^{N}p_{n}^{\mathrm{therm}}\ket{n,N-n}\bra{n,N-n},$
(8)

with

$p_{n}^{\mathrm{therm}}=\frac{1}{Z}e^{-\beta(gn(n-1)+\Delta n)}$ (9)

the thermal occupation of state $\ket{n,N-n}$ and

$Z=\sum_{n=0}^{N}e^{-\beta(gn(n-1)+\Delta n)}$ (10)

the partition function. Due to particle interactions, the boson number
distribution of a thermal state is approximately Gaussian (when $-\Delta\gg
g>0$), in contrast to the Planck distribution of non-interacting bosons. This
feature will be essential for the efficacy of our protocol as it will enable
close matching between the initial and thermal boson number distributions. We
assume $\beta g\ll 1$; noting that $gN\sim\mu$, where $\mu$ is the BEC
chemical potential, this is ensured when $\beta\mu/N\ll 1$, a condition
satisfied in dilute gaseous BECs [50, 51]. We can then approximate the sum in
Eq. (10) by an integral. We also assume $N\gg 1$ and
$\langle\hat{a}^{\dagger}\hat{a}\rangle\ll N$, i.e. most of the particles are
in the $\hat{b}$ mode throughout the protocol. We can then set
$N\rightarrow\infty$ and approximate $\ket{n,N-n}$ by a harmonic oscillator
number state, which simplifies our calculations [52, 53]. Evaluating Eq. (10)
with these two approximations gives

$Z\approx\sqrt{\frac{\pi}{\beta
g}}e^{\beta(g-\Delta)^{2}/4g}\left(\frac{1+\operatorname{erf}\left(\frac{\beta(g-\Delta)}{2\sqrt{\beta
g}}\right)}{2}\right).$ (11)

We denote the $N\rightarrow\infty$ approximation of $\ket{n,N-n}$ by $\ket{n}$
below, however it should be kept in mind that this state is an implicitly two-
mode state.

##  III Results

Our objective is to find an experimentally realisable density matrix
containing coherence in the energy eigenbasis with the same (or almost the
same) energy and energy entropy as a thermal state Eq. (9). As we will now
argue, a Glauber coherent state (GCS) satisfies these criteria. A GCS is,

$\ket{\alpha}=\hat{D}(\alpha)\ket{0}=\sum_{n=0}^{\infty}c_{n}\ket{n},$ (12)

with $\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}$ the
displacement operator and
$c_{n}=e^{-|\alpha|^{2}/2}\frac{\alpha^{n}}{\sqrt{n!}}$ [54]. Such a state can
be realised via coherent rotation of a pure state $\ket{0,N}$ or $\ket{N,0}$
[55, 56, 46, 57, 58, 59] or via interference measurements [60, 61, 62] (see
also [63]), and gives rise to phase coherence between the two modes of the BEC
[64, 65, 66, 46, 67, 68]. Note the upper limit of the sum in Eq. (12) is set
to $\infty$ by assuming $|\alpha|^{2}\ll N$ and $N\gg 1$ (see the discussion
above Eq. (11)). A GCS is a superposition of energy eigenstates of Eq. (7) and
hence $\ket{\alpha}\bra{\alpha}$ contains off-diagonal terms
$c_{m}^{*}c_{n}\ket{n}\bra{m}$ ($m\neq n$); hence a GCS exhibits coherence
with respect to the energy eigenbasis. The GCS number-state distribution
$p_{n}^{G}=|c_{n}|^{2}$ is Poissonian with mean and variance equal to
$|\alpha|^{2}$. For large $|\alpha|^{2}$ we have,

$p_{n}^{G}=\frac{|\alpha|^{2n}e^{-|\alpha|^{2}}}{n!}\approx\frac{1}{\sqrt{2\pi|\alpha|^{2}}}e^{-(n-|\alpha|^{2})^{2}/2|\alpha|^{2}}.$
(13)

Choosing

$|\alpha|^{2}=\frac{g-\Delta}{2g},\hskip 28.45274pt\beta
g=\frac{1}{2|\alpha|^{2}},$ (14)

we obtain $p_{n}^{G}\approx p_{n}^{\mathrm{therm}}$, with
$p_{n}^{\mathrm{therm}}$ given by Eq. (9). Equivalently, it is easy to show
that Eq. (6) is independent of $n$ when $|\alpha|^{2}$ is chosen as in Eq.
(14). Hence with precise tuning of parameters, as given in Eq. (14), work can
be extracted from coherence as discussed in Sec. II.1.

For more general parameters, the work output will have both a contribution
from coherence and a “classical” contribution. Below we present an explicit
protocol to show how work can be extracted from the coherence of a GCS. We
then show how the work output depends on particle number and temperature,
identifying a regime where the work output is dominated by the contribution
from coherence. Next we show how the work output can be improved by squeezing
the initial GCS and finally we discuss the role of entanglement in the system.

###  III.1 Protocol to extract work from a GCS

Figure 1:  (a) Protocol to extract work from coherence in a two-mode BEC. Red
curves are number state distributions $\braket{n}{\rho}{n}$ and black dashed
curves are the energy spectrum $E_{n}$, both smoothed over the discrete
variable $n$. (a,i) The system is initialised in a Glauber coherent state
(GCS) at detuning $\Delta_{0}$ with mean boson number $|\alpha|^{2}\gg 1$ and
number distribution $p_{n}^{G}$, Eq. (13). The energy spectrum has a minimum
at $n_{\mathrm{min}}=\operatorname{max}(0,(g-\Delta_{0})/2)$; we choose
$\Delta_{0}<0$ in the schematic above so that $n_{\mathrm{min}}>0$. (a,ii)
Step 1: Work is exchanged as the system is rotated to the vacuum state via a
coherent coupling $\kappa$. (a,iii) Step 2: The detuning is changed to
$\Delta_{\mathrm{max}}\gg\beta^{-1}$. The new energy spectrum has a minimum at
$n=0$ with an energy spacing much larger than $\beta^{-1}$. Hence the vacuum
state is approximately thermal. (a,iv) Step 3: The system is coupled to a
reservoir at inverse temperature $\beta$ (indicated by background colouring).
(a,v) Step 4: Work is extracted isothermally as the detuning is
quasistatically adjusted to its initial value $\Delta_{0}$. The closeness of
the final thermal distribution $p_{n}^{\mathrm{therm}}$ to the initial
distribution $p_{n}^{G}$ determines the quantum and classical contributions to
the total work extracted in steps 1-4. The classical contribution can be
suppressed with careful tuning of parameters, in which case the work extracted
arises entirely from the initial energy coherence. (b) Detuning (solid line)
and the coherent coupling (dashed line) between the two modes of the BEC
during the steps in (a).

We begin in a GCS, Eq. (12), with Hamiltonian Eq. (7) and $\Delta=\Delta_{0}$.
The energy of the initial state is
$U_{i}=\braket{\alpha}{\hat{H}}{\alpha}=\Delta_{0}|\alpha|^{2}+g|\alpha|^{4}$.
To extract work from the GCS we consider the following thermodynamically
reversible steps [11] (see Fig. 1):

  1. 1.

Coherently couple the two modes of the BEC via the Hamiltonian
$\kappa(\hat{b}^{\dagger}\hat{a}+\hat{a}^{\dagger}\hat{b})$, with $\kappa>0$
the coupling rate. Evolve the system so that $\ket{\alpha}$ is displaced to
$\ket{0}$ and all particles are in the $\hat{b}$ mode. During this step,
coherence is removed and work $U_{i}$ is exchanged with the coupling field
over a time scale $\kappa^{-1}$. Turn off the coupling $\kappa$ so that the
system energy is determined by Eq. (7).

  2. 2.

Tune the detuning to $\Delta_{\mathrm{max}}>0$. This does not change the
system energy since the system is in state $\ket{0}$.

  3. 3.

Couple the system to a reservoir at inverse temperature $\beta$ and allow to
thermalise. The detuning $\Delta_{\mathrm{max}}$ and $\beta$ are chosen to
satisfy $\beta\Delta_{\mathrm{max}}\gg 1$. Occupation of modes $\ket{n}$ with
$n>0$ are therefore thermally suppressed (see Eq. (10)) and hence $\ket{0}$ is
approximately the thermal state. The energy increase during thermalisation is
$\approx\Delta_{\mathrm{max}}e^{-\beta\Delta_{\mathrm{max}}}$ and is hence
negligible for $\beta\Delta_{\mathrm{max}}\gg 1$.

  4. 4.

Quasistatically change the detuning back to $\Delta_{0}$ such that the system
remains in thermal equilibrium with the reservoir throughout. Work is
extracted and heat is absorbed in this step. The final state is thermal with
energy $U_{f}=-\partial\ln Z/\partial\beta$ evaluated from Eq. (11) with
$\Delta=\Delta_{0}$.

The coherent coupling in the first step could be achieved using Josephson
oscillations for two spatial modes [46, 57] or via a Rabi pulse for two spin
states [58, 59]. Complete conversion is possible when $\kappa$ is much larger
than the other energy scales in the problem, which ensures the dynamics is in
the Josephson regime, rather than the self-trapping regime [55, 56, 69, 70].
Note the effect of changing $\Delta$ in steps 2-4 could also be achieved by
manipulating $g$ using a Feshbach resonance [71, 72, 73] with $\Delta$ fixed:
$g$ would be increased to a value $\beta g_{\mathrm{max}}\gg 1$ in steps 2 and
3 and then quasistatically reduced back to its initial value in step 4. This
alternative protocol would not change the results below.

The total mean work output from the protocol in Fig. 1 is [11]

$W=U_{i}-U_{f}-\beta^{-1}(S^{V}_{i}-S^{V}_{f}),$ (15)

with $S_{i}^{V}(S_{f}^{V})$ the initial (final) von Neumann entropies of the
system. We decompose $W$ into two parts: A classical part $W_{\mathrm{class}}$
and a quantum part $W_{\mathrm{quant}}$,

$\begin{split}W_{\mathrm{class}}=&U_{i}-U_{f}-\beta^{-1}(S^{E}_{i}-S^{E}_{f}),\\\
W_{\mathrm{quant}}=&\beta^{-1}(S^{E}_{i}-S^{V}_{i}).\end{split}$ (16)

Here $S^{E}_{i}(S^{E}_{f})$ is the initial (final) energy entropy, see Eq.
(2). Equation (16) decomposes the total entropic change $S^{V}_{f}-S^{V}_{i}$
into a classical contribution $S^{E}_{f}-S^{E}_{i}$ and a quantum contribution
$S^{E}_{i}-S^{V}_{i}=D(\rho||\rho_{\mathrm{ed}})$ arising from coherence, Eq.
(3). For the protocol in Fig. 1 we have $S^{V}_{i}=0$ since the initial state
is pure and $S^{E}_{f}=S^{V}_{f}$ since the final state is thermal. Note that
a projection of the initial state Eq. (12) onto the energy eigenbasis prior to
work extraction would result in $W_{\mathrm{quant}}=0$, but would not change
$W_{\mathrm{class}}$. The protocol in Fig. 1 avoids this measurement, allowing
$W_{\mathrm{quant}}$ to be extracted. A decomposition similar to Eq. (16) was
applied to ergotropy in [13].

###  III.2 Classical and quantum contributions to the work extracted from a
GCS

With an initial state (12), the outputs $W_{\mathrm{quant}}$ and
$W_{\mathrm{class}}$ in Eq. (16) can be calculated analytically. We choose
$|\alpha|$ so that the GCS has mean boson number equal to the thermal state,
$|\alpha|^{2}=\langle\hat{n}\rangle_{\mathrm{therm}}\equiv\langle\hat{n}\rangle$.
Here $\hat{n}=\hat{a}^{\dagger}\hat{a}$ and a “therm” subscript denotes
expectation values for state $\rho_{\mathrm{therm}}$, Eq. (8). We have checked
that this choice of $|\alpha|$ minimises $W_{\mathrm{class}}$ to a very good
approximation over the parameters explored. The energetic and entropic terms
in Eq. (16) are

$\begin{split}U_{i}-U_{f}=&-g\left(\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}-\langle\hat{n}\rangle\right),\\\
S^{E}_{i}=&\frac{1}{2}+\frac{1}{2}\ln
2\pi\langle\hat{n}\rangle+O\left(\frac{1}{\langle\hat{n}\rangle}\right),\\\
S^{E}_{f}=&\frac{1}{2}+\frac{1}{2}\ln
2\pi\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}+S_{\mathrm{NG}},\end{split}$
(17)

where
$\langle\delta\hat{n}^{2}\rangle=\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}$.
We have also introduced

$S_{\mathrm{NG}}=\ln\frac{1+\operatorname{erf}u}{2}-uG(u)-\ln\left[1-2uG(u)-2G(u)^{2}\right],$
(18)

with $u=\frac{\beta(g-\Delta_{0})}{2\sqrt{\beta g}}$ and
$G(u)=\frac{e^{-u^{2}}}{\sqrt{\pi}(1+\operatorname{erf}u)}=O(e^{-\beta\Delta_{0}^{2}/g})$.
The entropic correction $S_{\mathrm{NG}}$ arises since the thermal
distribution is not a perfect Gaussian, and is only important for $u\lesssim
1$.

The thermal mean and variance are

$\langle\hat{n}\rangle_{\mathrm{therm}}=\frac{1}{\sqrt{\beta
g}}\left(u+G(u)\right),$ (19)
$\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}=\frac{1}{2\beta
g}\left(1-2uG(u)-2G(u)^{2}\right).$ (20)

Again, the $G(u)$ terms are non-Gaussian corrections, which are only important
for $u\lesssim 1$. Note for $u\gg 1$ we have
$u^{2}\approx\langle\hat{n}\rangle_{\mathrm{therm}}^{2}/2\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$
and hence $u^{-1}$ is essentially the normalized second-order correlation
function of the thermal bosonic field [74]. The thermal mean, variance and
skewness
$\langle\delta\hat{n}^{3}\rangle=\langle(\hat{n}-\langle\hat{n}\rangle\rangle$
are shown in the inset to Fig. 2(a). We focus on cases
$\langle\hat{n}\rangle_{\mathrm{therm}}\gtrsim 1$, which is ensured when the
initial detuning is less than the reservoir temperature,

$-\infty<\beta\Delta_{0}\lesssim 1.$ (21)

We hence allow $\beta\Delta_{0}$ to be negative. This is easy to engineer in a
two-mode BEC but difficult to engineer in other bosonic systems that do not
have a constrained particle number, e.g. photons.

Figure 2:  (a) Work extracted from a two-mode BEC undergoing the protocol in
Fig. 1 for different mean bosonic numbers $\langle\hat{n}\rangle$. Work
extracted for an initial GCS (blue curves) divided into quantum $\beta
W_{\mathrm{quant}}$ (blue solid curve) and classical $\beta
W_{\mathrm{class}}$ (blue dotted curve) contributions, see Eq. (16). The work
output is dominated by $W_{\mathrm{quant}}$ for $\langle\hat{n}\rangle\lesssim
n_{G}$, with $n_{G}$ given by Eq. (24) and marked by a vertical dashed line.
The number state distributions for the GCS and thermal states coincide almost
perfectly when $\langle\hat{n}\rangle=1/2\beta g$, at which point
$W_{\mathrm{class}}\approx 0$ is minimised (dip in blue dotted curve). Inset:
First three cumulants of the thermal boson number distribution, with
$\langle\delta\hat{n}^{k}\rangle=\langle(\hat{n}-\langle\hat{n}\rangle)^{k}\rangle$
for $k=2,3$. The distribution is Gaussian for $u\gg 1$, in which case
$\langle\hat{n}\rangle_{\mathrm{therm}}\approx u/\sqrt{\beta g}$,
$\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}\approx 1/2\beta g$ and
higher order cumulants are zero. (b) (i)-(iv) Number state distributions for
different mean bosonic numbers (four panel values are marked by crosses in
(a)) for a GCS (blue curve) and a thermal state (black dashed curve). The GCS
distribution is narrower (wider) than the thermal distribution for
$\langle\hat{n}\rangle$ smaller (larger) than
$\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}\approx 1/2\beta g\approx
501$, which results in non-zero $W_{\mathrm{class}}$. Pale red curves in (a)
and (b): as for the GCS but for a squeezed coherent state. These are included
here for comparison and will be discussed in Sec. III.3. The GCS and squeezed
distributions coincide in (ii). All results are for $\beta g=10^{-3}$.

Analytic expressions for $W_{\mathrm{class}}$ and $W_{\mathrm{quant}}$ follow
from Eq. (16)–(20) and are plotted in Fig. 2(a). The minimum of the classical
work occurs when $p_{n}^{G}$ and $p_{n}^{\mathrm{therm}}$ are as close as
possible, see Fig. 2(b). For large $\langle\hat{n}\rangle$ both distributions
are approximately Gaussian and hence will be equal when their respective means
and variances are equal. Since $p_{n}^{G}$ has equal mean and variance, this
requires

$|\alpha|^{2}=\langle\hat{n}\rangle_{\mathrm{therm}}=\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}.$
(22)

This occurs when $\beta\Delta_{0}\approx-1$, at which point
$\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}\approx 1/2\beta g$. At this
optimal point,
$W_{\mathrm{class}}=O(\sqrt{\langle\hat{n}\rangle}e^{-\langle\hat{n}\rangle})$
and the work output almost exclusively consists of $W_{\mathrm{quant}}$. (Note
our choice of units: plotting work in units of $\beta^{-1}$ is equivalent to
plotting the information lost if the work was dissipated as heat, according to
Landauer’s principle [75].)

For
$\langle\hat{n}\rangle\neq\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$,
the protocol is suboptimal in the sense that $W_{\mathrm{class}}$ is no longer
approximately zero. Here the thermal bosons are either bunched or anti-
bunched, caused by the boson interactions [76]. For
$\langle\hat{n}\rangle>\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$
(thermal anti-bunching), large number fluctuations in the initial state result
in a large $U_{i}-U_{f}$, see Fig. 2(b). In this regime, $u\gg 1$ and Eq.
(17)–(20) simplify to give

$\begin{split}W_{\mathrm{class}}&\approx
g\langle\hat{n}\rangle-\frac{1}{2\beta}\left[1+\ln\left(2\beta
g\langle\hat{n}\rangle\right)\right],\\\
W_{\mathrm{quant}}&\approx\frac{1}{2\beta}\left[1+\ln\left(2\pi\langle\hat{n}\rangle\right)\right].\end{split}\hskip
28.45274pt(u\gg 1)$ (23)

Using Eq. (23), we find that $W_{\mathrm{quant}}>W_{\mathrm{class}}$ for

$\langle\hat{n}\rangle\lesssim\frac{1}{\beta
g}\left[-\Omega_{-1}\left(-\sqrt{\frac{\beta
g}{4\pi\exp(2)}}\right)\right]\equiv n_{G},$ (24)

see Fig. 2(a). Here $\Omega_{k}$ is the $k$th branch of the omega function
111Also known as the Lambert $W$-function, denoted by $W_{k}$. We avoid this
notation to avoid confusion with work output.. The function $-\Omega_{-1}(-x)$
satisfies $1\leq-\Omega_{-1}(-x)<\infty$ for $x\in(0,\exp(-1)]$ and
monotonically decreases with increasing $x$, with
$-\Omega_{-1}(-x)\sim\ln\left(\frac{1}{x}\ln\frac{1}{x}\right)$ for small $x$
[78]. For $\langle\hat{n}\rangle>n_{G}$, the classical work output grows
extensively due to the extensive growth of $U_{i}-U_{f}$, see Eq. (23). The
quantum work also increases with $\langle\hat{n}\rangle$ due to increased
coherence in the initial state. The increase, however, is subextensive, with
$W_{\mathrm{quant}}\sim\frac{1}{2\beta}\ln\langle\hat{n}\rangle$. For
$\langle\hat{n}\rangle<\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$
(thermal bunching), the number fluctuations of the initial state are less than
the final state, see Fig. 2(b). Although this increases $W_{\mathrm{class}}$,
we still have $W_{\mathrm{quant}}>W_{\mathrm{class}}$ as long as
$\langle\hat{n}\rangle\gg 1$, see Fig. 2(a).

###  III.3 Improved protocol using squeezing

The boson number distribution of a GCS is Poissonian, and hence the mean and
variance are always equal. For sufficiently large mean boson number, Eq. (24),
the work output becomes dominated by $W_{\mathrm{class}}$. The classical work
output can be suppressed much more effectively by squeezing the initial
coherent state. Squeezing of a two-mode BEC is possible using multiple
techniques [79, 35], including time evolution with a one-axis twisting
Hamiltonian [80, 81, 33] and parametric downconversion via spin-changing
collisions [82, 83, 84]. A squeezed coherent state is [85, 86, 87],

$\ket{\alpha,\zeta}=\hat{D}(\alpha)\hat{S}(\zeta)\ket{0},$ (25)

where
$\hat{S}(\zeta)=e^{(\zeta^{*}\hat{a}\hat{a}-\zeta\hat{a}^{\dagger}\hat{a}^{\dagger})/2}$
is the squeezing operator and $\zeta=|\zeta|e^{i\theta}$ parameterises the
squeezing. The distribution $p_{n}^{S}=|\braket{n}{\alpha,\zeta}|^{2}$ is [88,
89]

$p_{n}^{S}=\frac{(\frac{1}{2}\tanh|\zeta|)^{n}}{n!\cosh|\zeta|}\left|H_{n}(\lambda\alpha)\right|^{2}e^{-\alpha^{2}(1+\cos\theta\tanh|\zeta|)},$
(26)

with $H_{n}$ the Hermite polynomials ($H_{0}(x)=1$, $H_{1}(x)=2x$, etc.) and

$\lambda=\frac{e^{-i\theta/2}\cosh|\zeta|+e^{i\theta/2}\sinh|\zeta|}{\sqrt{\sinh
2|\zeta|}}.$ (27)

Figure 3:  (a) Work extracted from an initial squeezed coherent state, divided
into quantum $\beta W_{\mathrm{quant}}$ (solid curves) and classical $\beta
W_{\mathrm{class}}$ (dotted curves) contributions. Colours indicate different
values of $\beta g$. The work output is dominated by $W_{\mathrm{quant}}$ for
$\langle\hat{n}\rangle\lesssim n_{S}$, with $n_{S}$ given by Eq. (32) and
marked by vertical dashed lines. Lighter-coloured thicker lines give the large
$\langle\hat{n}\rangle$ result for $W_{\mathrm{class}}$, Eq. (30),
demonstrating subextensive scaling of $W_{\mathrm{class}}$. Grey dashed line
gives the large $\langle\hat{n}\rangle$ result for $W_{\mathrm{quant}}$, Eq.
(31). (b) Optimised squeezing parameters $e^{-2|\zeta|}$ (solid curves, left
axis) and $\cos\theta$ (dotted curves, right axis) for the squeezed coherent
state (see text for details). For $\langle\hat{n}\rangle\leq 2(3\beta
g)^{-3/2}$ the squeezed variance can be matched to the thermal variance with
appropriate choice of $e^{-2|\zeta|}$. For larger $\langle\hat{n}\rangle$
matching is no longer possible, see Eq. (29); minimising the squeezed variance
then gives $e^{-2|\zeta|}=(4\langle\hat{n}\rangle)^{-1/3}$ (gray line),
independent of $\beta g$. This results in a discontinuity in the slope of
$e^{-2|\zeta|}$ and $W_{\mathrm{class}}$ at $\langle\hat{n}\rangle=2(3\beta
g)^{-3/2}$ (marked by vertical arrows for $\beta g=10^{-5}$).

The boson number distribution for a squeezed state can have a variance smaller
or larger than its mean, controllable via the squeezing parameter $\zeta$.
This additional control allows the initial number distribution to be matched
more closely to $p_{n}^{\mathrm{therm}}$, see Fig. 2(b). The work output from
an initial squeezed state is compared with that of a GCS in Fig. 2(a).
Squeezing suppresses $W_{\mathrm{class}}$ over a much larger range of
$\langle\hat{n}\rangle$ compared to the GCS. Note the small but finite minimum
of $W_{\mathrm{class}}/\langle\hat{n}\rangle$ for a GCS, which arises due to
the approximation in Eq. (13), is reduced even further by squeezing, see Fig.
2(a). Further squeezed results are shown in Fig. 3(a) for different values of
$\beta g$. For the squeezed state, we choose $|\alpha|^{2}$ so that the mean
boson number is equal to that of the thermal state and choose $\zeta$ to
minimise differences in higher order moments. Details are discussed in
Appendix A and the values of $|\zeta|$ and $\cos\theta$ obtained are plotted
in Fig. 3(b). Numerical details to obtain $p_{n}^{S}$ are described in
Appendix B.

Although squeezing gives much greater control over the initial boson number
distribution, its variance is still constrained by $\langle\hat{n}\rangle$
[90]. For large $\langle\hat{n}\rangle$ the minimum variance possible for a
squeezed state is (see Appendix C)

$\operatorname{min}\langle\delta\hat{n}^{2}\rangle=\left(\frac{27}{32}\right)^{1/3}\langle\hat{n}\rangle^{2/3}$
(28)

Achieving
$\langle\delta\hat{n}^{2}\rangle=\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$
is hence only possible when the mean boson number is not too large,

$\langle\hat{n}\rangle\leq\sqrt{\frac{32\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}^{3}}{27}}\approx\sqrt{\frac{4}{27(\beta
g)^{3}}}.$ (29)

The constraint (29) should be contrasted with the strict constraint for a GCS,
Eq. (22). Matching the mean and variances of $p_{n}^{S}$ and
$p_{n}^{\mathrm{therm}}$ gives $U_{i}=U_{f}$ and hence
$W_{\mathrm{class}}=\beta^{-1}(S_{f}^{E}-S_{i}^{E})$, see Eq. (16).

If the mean boson number is too large such that Eq. (29) is not satisfied, the
variance of the squeezed distribution cannot be made small enough to match
$\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}$. We then choose
$\langle\delta\hat{n}^{2}\rangle$ to take on its minimum value Eq. (28) by
choosing $e^{-2|\zeta|}=(4\langle\hat{n}\rangle)^{-1/3}$ (see Appendix C).
This results in subextensive scaling of $W_{\mathrm{class}}$ for large
$\langle\hat{n}\rangle$,

$W_{\mathrm{class}}\approx
g\left(\frac{27}{32}\right)^{1/3}\langle\hat{n}\rangle^{2/3},$ (30)

see Fig. 3(a). Notably, this contrasts the extensive scaling of
$W_{\mathrm{class}}$ for a GCS, which arises from the strict constraint
$\langle\hat{n}\rangle=\langle\delta\hat{n}^{2}\rangle$, see Fig. 2(a) and Eq.
(23). For large $\langle\hat{n}\rangle$, we find $W_{\mathrm{quant}}$ is well
approximated by

$W_{\mathrm{quant}}\approx\frac{1}{2\beta}\left[1+\frac{1}{3}\ln\left(\frac{27\langle\hat{n}\rangle^{2}}{32}\right)\right],$
(31)

see Fig. 3(a). This follows from approximating $p_{n}^{S}$ by a Gaussian
distribution and using Eq. (28). Equations (30) and (31) give the following
criteria for $W_{\mathrm{quant}}>W_{\mathrm{class}}$,

$\langle\hat{n}\rangle\lesssim\sqrt{\frac{32}{27}n_{G}^{3}}\equiv n_{S},$ (32)

with $n_{G}$ the equivalent bound for a GCS given by Eq. (24). The estimate
Eq. (32) agrees well with our numerics, see Fig. 3(a). Notably, $n_{S}\gg
n_{G}$ for large $n_{G}$.

For small $\langle\hat{n}\rangle$, we observe an abrupt drop in
$W_{\mathrm{class}}$, see Fig. 3(a). At this point, $S_{i}^{E}$ increases
above $S_{f}^{E}$ and hence $W_{\mathrm{class}}$ becomes negative
($W_{\mathrm{class}}=S_{f}^{E}-S_{i}^{E}$ for small $\langle\hat{n}\rangle$,
see Eq. (29)). We expect that the increase in $S_{i}^{E}$ above $S_{f}^{E}$ is
due to oscillations that arise in the squeezed distribution for small
$\langle\hat{n}\rangle$ [91, 92], see Fig. 2(b) 222The change of sign of
$W_{\mathrm{class}}$ means that there is value of $\langle\hat{n}\rangle$
where $W_{\mathrm{class}}=0$ despite differences in the squeezed and thermal
distributions. The choice of squeezing parameters can possibly be optimised to
find similar “serendipitous” points of zero or small $W_{\mathrm{class}}$ for
other values of $\langle\hat{n}\rangle$..

###  III.4 Relation with entanglement

Finally, we show how the work from coherence in this system can equivalently
be interpreted as work from entanglement [94, 95, 96, 97]. To see this, we
evaluate the entanglement entropy $\mathcal{S}$ between the two modes of the
BEC. A sufficient condition for entanglement is then $\mathcal{S}>0$ [98]. The
entanglement entropy of a pure state $\rho$ is

$\mathcal{S}=-\operatorname{Tr}_{1}\rho_{1}\log\rho_{1},$ (33)

where $\rho_{1}=\operatorname{Tr}_{2}\rho$ and $\operatorname{Tr}_{i}$ is a
partial trace over boson states of modes $i=1,2$. The trace
$\operatorname{Tr}_{2}$ can be evaluated in the number state basis for both
modes. Due to conservation of total particle number, this projects $\rho$ onto
the number state basis for the first mode,

$\rho_{1}=\sum_{n=0}^{N}\braket{n,N-n}{\rho}{n,N-n}\ket{n}_{1}\prescript{}{1}{\bra{n}}$
(34)

with
$\ket{n}_{1}\prescript{}{1}{\bra{n}}=\operatorname{Tr}_{2}\ket{n,N-n}\bra{n,N-n}$.
The “$1$” subscript distinguishes $\ket{n}_{1}$ from the implicitly two-mode
state $\ket{n}$. Equations (33) and (34) give

$\mathcal{S}=S_{i}^{E}=\beta W_{\mathrm{quant}}.$ (35)

Hence the coherence in this system arises due to entangling correlations
between the two modes, which is subsequently extracted as
$W_{\mathrm{quant}}$. The final thermal state is a classical mixture of
product states, and hence does not possess any entanglement. The quantum work
output from coherence can therefore equivalently be interpreted as work from
entanglement. Squeezing results in additional entanglement according to the
criteria in [32, 99].

##  IV Discussion and conclusion

We have shown how work can be extracted from coherence in a two-mode BEC. Work
almost entirely from coherence can be extracted from a GCS with precisely
tuned mean boson number. The protocol can be substantially improved using
squeezing, enabling work extraction predominantly from coherence over a much
broader range of parameters. Due to the many-body nature of the system, the
work from coherence can equivalently be interpreted as work from entanglement.
A challenging step in our protocol is the controlled transfer of heat to the
BEC alongside particle exchange between the two modes to reach equilibrium. A
possible heat source is a separate atomic gas in analogy with sympathetic
cooling [100]. Particle exchange could occur via tunnelling for two trapped
modes [101, 102] or via spin-changing collisions with the reservoir for two
spin states [103, 104].

The Hamiltonian (7) also describes a large, fully-connected spin chain [105]
and hence could be realised in other atomic systems [35], including atomic
ensembles coupled to light [106, 107], Rydberg arrays [108, 109] and trapped
ions [110, 111]. The Hamiltonian also resembles that of photons in a non-
linear medium [88], with the important difference that we allow the free-
photon spectrum to be negative, $\Delta<0$. To realise this, photon number
would need to be conserved [112, 113].

We have limited our analysis to Glauber coherent and squeezed coherent states.
Improved operation may be possible using self-phase modulation of the squeezed
state [114] or by using non-Gaussian states [115] at the expense of more
involved state generation. We have also limited our analysis to initial states
with zero von Neumann entropy; relaxing this provides an interesting area for
future investigation. Extracting work from coherence can enhance power output
compared to classical systems [6]. Our results provide a starting point to
explore this in a two-mode BEC, and may result in a many-body quantum
advantage when incorporated into an engine cycle.

## Data Availability

The code used to produce the data shown in Figs. 2 and 3 is available upon
reasonable request to LW<EMAIL_ADDRESS>
## Acknowledgements

This research was supported by the Australian Research Council Centre of
Excellence for Engineered Quantum Systems (EQUS, CE170100009) and the
Australian government Department of Industry, Science, and Resources via the
Australia-India Strategic Research Fund (AIRXIV000025). FC and JA gratefully
acknowledge funding from the Foundational Questions Institute Fund (FQXi-
IAF19-01) and EPSRC (EP/R045577/1). JA thanks the Royal Society for support.

##  Appendix A Choosing squeezing parameters

We choose $\alpha$ and $\zeta$ for the initial squeezed coherent state as
follows. The first three cumulants of $p_{n}^{S}$ are evaluated from the
generating function $K(s)=\ln\sum_{n}p_{n}^{S}e^{sn}$, which can be written in
closed form using Mehler’s formula [116]

$\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{n!}\left(\frac{u}{2}\right)^{n}=\frac{1}{\sqrt{1-u^{2}}}e^{\frac{2xyu-(x^{2}+y^{2})u^{2}}{1-u^{2}}}.$
(36)

This gives

$\begin{split}\langle\hat{n}\rangle=&\sinh^{2}|\zeta|+|\alpha|^{2},\\\
\langle\delta\hat{n}^{2}\rangle=&2\cosh^{2}|\zeta|\sinh^{2}|\zeta|+|\alpha|^{2}\left(\cosh
2|\zeta|-\sinh 2|\zeta|\cos\theta\right),\\\ \langle\delta
n^{3}\rangle=&\frac{1}{2}\sinh 2|\zeta|\sinh 4|\zeta|\\\
&+\frac{3|\alpha|^{2}}{2}\left(\cosh 4|\zeta|-\frac{1}{3}-\sinh
4|\zeta|\cos\theta\right),\end{split}$ (37)

with $\langle\delta n^{k}\rangle=\langle(n-\langle\hat{n}\rangle)^{k}\rangle$
for $k=2,3$. We set
$\langle\hat{n}\rangle=\langle\hat{n}\rangle_{\mathrm{therm}}$ by choosing
$|\alpha|^{2}=\langle\hat{n}\rangle_{\mathrm{therm}}-\sinh^{2}|\zeta|$. We
solve for $|\zeta|$ by minimising $|\langle\delta
n^{2}\rangle-\langle\delta\hat{n}^{2}\rangle_{\mathrm{therm}}|$. Since the
Hamiltonian (7) depends only on $\langle\hat{n}\rangle$ and
$\langle\delta\hat{n}^{2}\rangle$, this choice minimises $|U_{f}-U_{i}|$.
Additionally, for not-too-large squeezing this minimises
$|S^{E}_{f}-S^{E}_{i}|$. We then choose $\cos\theta$ to minimise
$|\langle\delta n^{3}\rangle|$.

##  Appendix B Numerical details for extracting squeezed state statistics

Numerical computation of $p_{n}^{S}$ requires care for large $n$ due to
numerical overflow and underflow [117]. We evaluate $p_{n}^{S}$ using the
underlying recursion relation [88, 89]
$c_{n}=\braket{n}{\alpha,\zeta}=\frac{\alpha(1+e^{i\theta}\tanh|\zeta|)}{\sqrt{n}}c_{n-1}-e^{i\theta}\tanh|\zeta|\sqrt{\frac{n-1}{n}}c_{n-2}$.
After each iteration we normalise the distribution. Defining $\Delta
n=\operatorname{ceil}(50\sqrt{\langle\delta\hat{n}^{2}\rangle})$, we begin the
recursion at
$n^{*}=\operatorname{max}(0,\operatorname{round}(\langle\hat{n}\rangle-\Delta
n/2))$ with $c_{n^{*}}=1$ and $c_{n<n^{*}}=0$, and evaluate terms up to
$n^{*}+\Delta n$. In cases where $n^{*}>0$, we find $|\alpha|\gg 1$ and hence
$c_{n^{*}+1}\approx\frac{\alpha(1+e^{i\theta}\tanh|\zeta|)}{\sqrt{n}}c_{n^{*}}$
(this is exact for $n^{*}=0$). Hence why the values of $c_{n<n^{*}}$ are not
needed.

##  Appendix C Minimum variance of a squeezed state

To a very good approximation $\cos\theta=\pm 1$, see Fig. 3(b). Defining
$x=e^{2|\zeta|\cos\theta}\left(=e^{\pm 2|\zeta|}\right)$, the equation for
$\langle\delta\hat{n}^{2}\rangle$ is (see Eq. (37)),

$\langle\delta\hat{n}^{2}\rangle=\frac{1}{8x^{2}}\left(x^{4}-4x^{2}+8\left(\langle\hat{n}\rangle+\frac{1}{2}\right)x-1\right).$
(38)

Stationary points of $d\langle\delta\hat{n}^{2}\rangle/dx$ satisfy the
polynomial

$x^{4}-4\left(\langle\hat{n}\rangle+\frac{1}{2}\right)x+1=0.$ (39)

Equation (39) can be solved for $x$ using standard methods. For large
$\langle\hat{n}\rangle$ this gives two real roots and two complex roots. The
two real roots are

$\begin{split}x_{1}\approx&\frac{1}{4\langle\hat{n}\rangle+2},\\\
x_{2}\approx&(4\langle\hat{n}\rangle)^{1/3},\end{split}$ (40)

with $x_{1}$ corresponding to a local maximum of Eq. (38) and $x_{2}>x_{1}$ a
local minimum. Substituting $x<x_{1}$ into the expression for
$\langle\hat{n}\rangle$ (Eq. (37)) would give a negative value for
$|\alpha|^{2}$, hence $x\geq x_{1}$. Therefore $x_{2}$ gives the global
minimum of $\langle\delta\hat{n}^{2}\rangle$ for valid values of $x$.
Substituting $x_{2}$ into Eq. (38) gives Eq. (28) for large
$\langle\hat{n}\rangle$.

For the purpose of open access, the authors have applied a ‘Creative Commons
Attribution’ (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

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