Understanding and Improving Critical Metrology. Quenching
Superradiant Light-Matter Systems Beyond the Critical
Point
Karol Gietka, Lewis Ruks, and Thomas BuscharXiv:2110.04048v3 [quant-ph] 26 Apr 2022

Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

We carefully examine critical metrology
and present an improved critical quantum metrology protocol which relies on
quenching a system exhibiting a superradiant quantum phase transition beyond
its critical point. We show that this approach can lead to an exponential increaseof the quantum Fisher information in timewith respect to existing critical quantum
metrology protocols relying on quenchingclose to the critical point and observing
power law behaviour. We demonstratethat the Cramér-Rao bound can be saturated in our protocol through the standard
homodyne detection scheme. We explicitly show its advantage using the archetypal setting of the Dicke model and explore a quantum gas coupled to a singlemode cavity field as a potential platform.
In this case an additional exponential enhancement of the quantum Fisher information can in practice be observed with thenumber of atoms N in the cavity, even in
the absence of N -body coupling terms.

1 Introduction
Controlling and manipulating light-matter interactions is currently one of the most rapidly growing and evolving field of physics [1]. Of particular interest is the coupling of a cavity field
to a gas of atoms—quantum-gas cavity quantum electrodynamics—where the interaction between light and matter is greatly enhanced dueto the light passing through the same atomic
system many times. Whilst the back-action of
the atoms on the light field is typically neglectedin free-space configurations, its necessary inclusion in cavity quantum electrodynamics leads to
Karol Gietka: karol.gietka@oist.jp

complex nonlinear coupled dynamics [2]. The
rich ensuing dynamics in turn open up possibilities to use these systems to simulate fundamental solid-state physical systems and explore non-equilibrium many-body phenomena beyond the scope of conventional condensed matter systems [3]. Moreover, due to the presenceof non-classical correlations and the feasibilityof non-destructive monitoring of these systems
through photons leaking from the cavity mirrors,
they serve as an ideal platform for precise measurements of unknown physical parameters beyond the standard quantum limit [4–12]. Therefore, quantum-gas cavity quantum electrodynamics systems may lead to the development of newparadigms in quantum metrology.
Quantum metrology [13] is a framework inwhich quantum-mechanical effects [14], for example, non-classical correlations [15] and quantum entanglement [16], are being used to enhance
the precision of measurement beyond the standard quantum limit [14]. The figure of merit inquantum metrology is the quantum Cramér-Rao
bound [17], which sets the ultimate limit of measurement precision of an unknown parameter λusing limited resources—such as time t and number of particles—that can be potentially attained:∆2 λ ≥ 1/mIλ . Here m is the number of estimation protocol repetitions and Iλ is the quantum
Fisher information, which for pure states takes
the form [18] Iλ = 4(h∂λ ψ|∂λ ψi − h∂λ ψ|ψi2 ).
From the practical point of view, an important quantity is the classical Fisher informationP
1
Fλ = ξ p(ξ|λ)[∂λ p(ξ|λ)]2 . This quantity takes
into account the performed measurement throughthe corresponding conditional probability distribution p(ξ|λ) and can be equal to its quantum
version only for the optimal choice of measurement. However, in certain cases the optimal measurement might be beyond the scope of the exper-

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1

Figure 1: The schematic represents a toy model for aquantum phase transition (g/gc > 1 is the superradiantphase), with the black line being the effective potentialthat is felt by a quantum state (dashed-gray line). Driving the system close to the critical point (g/gc ∼ 1)
creates the correlated (squeezed) excitations at a very
slow rate (critical slowing down), since the effective potential is still of trapping form. However, if the system
is quenched beyond the critical point (g/gc > 1), thesame number of correlated excitations can be generated
much faster since the initial state will behave as if it wasplaced in an inverted harmonic oscillator potential. The
purple ellipses represent the phase space picture of thestate.

imental possibilities.
In a general metrology scheme, we can consider
a Hamiltonian of the form (h̄ = 1 in the following)
ĤQM = λĤλ + Ĥx (x),(1)

where Ĥλ is responsible for imprinting the information about the unknown parameter λ into theinitial state |ψ0 i, and Ĥx (x) describes other dynamics which in general can depend on a set ofn parameters x = x1 , x2 , . . . , xn . Provided that
one has full control over x, quantum metrology
protocols typically first employ Ĥx (x) to prepare a suitable (for example, entangled) initialstate, after which Ĥx (x) is set to 0. Subsequently the information about the unknown parameter is imprinted on the prepared initial statethrough the unitary evolution operator Û (λ, t) =exp(−itλĤλ ). By initially preparing a superposition of the minimal and the maximal eigenvalue eigenstates of the so-called local generatorĥλ = iÛ † ∂λ Û (which in this case is ĥλ = Ĥλ ),it is possible to maximize the quantum Fisher
information and reach the so-called Heisenberg
limit of the sensitivity. Note that it can be also
shown that the quantum Fisher information canbe calculated as the variance of the local generator on the initial state Iλ = 4∆2 ĥλ . For example,

in N two-mode atomic systems the Heisenberg
limit is ∆2 λ = 1/(N t)2 [19], attained by preparing the maximally entangled state. Here t is thephase acquisition time (note that the initial state
preparation time is typically not taken into account). For frequency estimation with a singlemode Gaussian field, the optimal sensitivity becomes ∆2 λ = 1/8t2 (hni2 +hni) [20] (hni being the
average number of photons) and is saturated using a squeezed vacuum state. If the imprinting
mechanism Ĥλ is additionally time-dependent,
optimal control techniques can be used to maximize the quantum Fisher information by changing x in time such that the instantaneous state
becomes the superposition of maximal and minimal eigenvalue eigenstate of the local generator [21, 22]. However, the Hamiltonian mightthen be a complicated function of time as it is
no longer composed of only Ĥλ but also involves
Ĥ(x) which in general does not commute withĤλ . Note that if the information about the unknown parameter is imprinted non-linearly, i.e.,(k)with a k-body term Ĥλ , the power-law scaling
with N in the Heisenberg limit is modified. For
example, in a system composed of N two-mode
systems with a multi-body coupling of degree k,
the Heisenberg limit becomes ∆2 λ = 1/(N k/2 t)2[17]. In principle, by exploiting N -body interactions, it could be possible to obtain an exponential enhancement of achievable precision [23],however, it has been claimed that such interactions are unphysical [17].
Although metrological schemes such as thosedescribed can in principle be used to reach theHeisenberg limit, they are often extremely demanding from an experimental point of view.
The preparation of maximally entangled statesby itself is already a challenging task—especiallyfor systems composed of many constituents—
which is complicated further by manipulation andmeasurement. Additionally, highly non-classicalstates are also extremely fragile to the effectof decoherence. Heisenberg-limited metrology istherefore currently restricted to proof-of-principle
experiments, often with systems composed ofonly a few elements such as atoms or photons [24].
However, in many cases the standard quantum
limit can be overcome with the use of less exotic states such as spin-squeezed states of atoms[19] or squeezed states of photons [20]. Finally,
the specific measurement required to saturate theAccepted in Quantum 2022-04-26, click title to verify. Published under CC-BY 4.0.

2

Cramér-Rao bound might not be possible in certain cases; for example, the optimal measurement
might correspond to a projection onto a highly
entangled state.
A simpler metrological protocol can be constructed for situations where one does not havefull control over x. In this case, one can imprint
the information about the unknown parameter λand build up the quantum correlations simultaneously by starting from an initially uncorrelated
state [25]. This approach includes explicitly thestate preparation time in the metrological protocol in contrast to protocols in which an optimal
initial state is already prepared. However, thecost of reducing the complexity of the metrological protocol is the inability to maximize the quantum Fisher information and reach the Heisenberg
limit [26]. Nevertheless, it is still possible to beatthe standard quantum limit with moderately nonclassical states, and in certain cases reach theHeisenberg scaling. That is to say, the sensitivity follows the same power-law scaling withtime and number of particles as the Heisenberg
limit, but contains a constant prefactor f > 1,
i.e., ∆2 λ ≥ f /N 2 T 2 . In certain cases, whilst thesensitivity exhibits Heisenberg scaling, it can still
fail to beat the standard quantum limit of precision if f = Nf > N . In principle one could
then only beat it for extremely large system sizes
which might go beyond the scope of experimental
realizations [27].
A key example of simultaneously imprinting
the information about the unknown parameter
whilst creating quantum correlations is the socalled critical quantum metrology [28–42], which
has attracted much attention in recent years.
Critical quantum metrology takes advantage of
extreme sensitivity of the quantum state to perturbations when near a quantum phase transition. When changes in the unknown parameter effect such perturbations, one can use thisextreme sensitivity for high-precision measurements. If the Hamiltonian exhibits a quantum
phase transition at xc , one can use the critical quantum metrology approach to adiabatically drive the system near this critical point
and perform a suitable measurement yielding alarge Fisher information. This protocol is effective because in the thermodynamic limit of continuous quantum phase transitions, the energy
gap above the ground state closes [43]. The effectof a vanishing energy gap can be explicitly seen
if we calculate the quantum Fisher information
for the ground state |ψ0 (λ)i of the HamiltonianPĤ(λ) = n=0 En (λ)|ψn (λ)ihψn (λ)| [44]Iλ = 4X |hψn (λ)|∂λ Ĥ(λ)|ψ0 (λ)i|2n6=0[En (λ) − E0 (λ)]2.

(2)

From this expression, it is clear that the quantum
Fisher information diverges through the denominator if the energy gap above the ground statecloses. This property may lead, in principle, to anarbitrarily high estimation precision in the thermodynamic limit. Unfortunately, the adiabatic
theorem states that a quantum system remains in
its ground state only if it is varied slowly enoughsuch that no excitations occur [45]; a sufficientcondition is that the time time-scale of variation
far exceeds the inverse energy gap t  1/∆E.
Therefore, close to the critical point the metrological protocol has to be infinitely slow (critical slowing down) as the energy gap closes. Thismeans that from the viewpoint of metrology thedivergence of the quantum Fisher information is
actually due to the diverging time required for the
adiabatic protocol [37, 46]. Moreover, since theprotocol relies on creating quantum correlations
generated during the adiabatic time evolution in
the vicinity of the critical point, it also meansthat this protocol cannot reach the Heisenberg
limit [26]. Therefore, adiabatic quantum metrology protocols increase dramatically the time ofa single measurement at the cost of not reaching the Heisenberg limit. Additionally, such a
protocol might very likely operate on time scales
larger than decoherence times and will also be
unable to reach the Heisenberg limit in this respect. Recently, it has been shown that in certain
cases adiabatic time evolution is not required andmetrology that can lead to the Heisenberg scaling can be done using quenches to or close to thecritical point [47]. However, such dynamical critical quantum metrology protocols have operated
within in a single phase of the system and also
require extremely large time-scales.
In systems exhibiting a superradiant phasetransition, the exponential growth of correlated
excitations observed in a quench across the critical point [48] constitutes a macroscopic degreeof freedom with a large potential for information
storage that has not yet been explored. Quenching across a transition (see Fig. 1) further allows

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3

one to sidestep the issue of critical slowing down
encountered when preparing states near a critical point (i.e., adiabatic critical quantum metrology). Therefore, an alternative aim of quantum
metrology in systems exhibiting a superradiantphase transition could be to quench far beyond
the critical point and imprint information into
the macroscopically populated field (however, thecoupling to highly sensitive systems such as biological samples may not be suitable for such a protocol [49]). Given the prevalence of superradiantphase transitions in light-matter systems [50], we
expect that the protocol we present in the following will be applicable to a wide range of scenarios.
In the following we explore the dynamical approach to quantum metrology with light-mattersystems exhibiting superradiant quantum phasetransitions, which we show leads to an exponentially growing quantum Fisher information
caused by an exponential growth correlated photons on an arbitrarily fast time scale (scaling
inversely with the coupling strength after thequench). That said, the exponential growth islimited until the critical amount of excitations(photons) is reached as will explain in the following Sections. In contrast to the existing criticalmetrology proposals, our approach relies on creating both correlations and excitations by quenching beyond the critical point. We will demonstrate this in practice using the example of theparadigmatic Dicke model and show that quadrature measurements can saturate the Cramér-Rao
bound (extending existing results in the normal
or superradiant phase), producing an exponential growth of Fisher information in time. This is
in contrast to existing proposals in critical quantum metrology that result in a power-law scaling
with time, with a recent result obtaining a t4 scaling [37].
In addition, we propose a gas of atoms coupled
to a single-mode cavity field as a potential test
platform for our protocol. We demonstrate thatwhen the requirement of extensivity in system
quantities is dropped, the natural light-matter
coupling Hamiltonian results in a quantum
Fisher√information scaling as Iλ ∼ exp(α N t) after aquench into the superradiant regime where selforganization of the atomic gas is observed. Thisillustrates that an exponential enhancement of
Fisher information is observable not only in time,
but also in practice with N , despite the system

only having two-body interactions [17]. The exponential growth of the quantum Fisher information is also present in finite-component systems (thus finite-component quantum phase transitions).

2 Dicke modelSuperradiant quantum phase transitions are aubiquitous phenomena in quantum optics [50]which can occur in a collection of two-level systems (typically atoms) interacting with a single
harmonic oscillator (electromagnetic field). The
superradiant phase is characterised by a macroscopic number of excitations (photons) in the
twofold degenerated ground state [51, 52]. In order to illustrate how our metrology scheme exploits the superradiant phase transition, let us
consider the Dicke model. This well-known model
describes the interaction of N two-level particles
with a single-mode field and is known to exhibita superradiant phase transition. Using the collecP
tive spin operators Ŝi = N
i=1 σ̂i /2 with σ̂i being
the ith Pauli matrix, the Dicke model can be expressed asg  †ĤDM = ωâ† â + ΩŜz + √â + â Ŝx , (3)N
where ω is the frequency of the bosonic field represented by its creation and annihilation operators â† and â, Ω is the energy splitting of a
two-level particle represented by Pauli matrices
σ̂i , and g is the coupling parameter between thebosonic field√and the two-level particle. We notethat the 1/ N term is in practice introduced toguarantee extensivity of system quantities. The
Dicke model then exhibits
p a quantum phase transition in the limit of ω/ΩN → 0 for√the critical coupling strength defined as gc ≡ ωΩ. In
the thermodynamic limit N → ∞, the quantum
phasetransition occurs for an arbitrary but finitepω/Ω. For the other limiting case, i.e., N = 1(quantum Rabi model), the (finite component)pquantum phase transition occurs for ω/Ω → 0,
which becomes an equivalent of the thermodynamic limit [53]. To simplify the Dicke model wecan apply the Schrieffer-Wolff transformation[54]√†
ÛSW ĤQRM ÛSW
with ÛSW = exp{i(g/Ω N )(â† +pâ)Ŝy }. In the limit of ω/Ω → 0 the transformation is exact and yields
ĤDM ' ωâ† â + ΩŜz +Accepted in Quantum 2022-04-26, click title to verify. Published under CC-BY 4.0.

2
g2  †â + â Ŝz . (4)2ΩN4This effective model allows us to analytically calculate the spectrum including the critical groundstate and thus the quantum Fisher information.
The eigen-states of the Dicke model under the
Schrieffer-Wolff transformation become (for clarity we restrict to the low energy sector)

and for a limited time related to the numberof photons in the ground state of the quenched
Hamiltonian
1 NΩ
â â <
t
32 ω

D

†

E

g 2 gc2−
gc2 g 2

!

,

(8)

p

|ψn i = Ŝ(ξ)|ni ⊗ | ↓ i,(5)

where Ŝ(ξ) ≡ exp{(ξ/2)(â† )2 − (ξ ∗ /2)â2 } is thesqueezing operator with ξ = − 41 ln{1 − (g/gc )2 }the squeezing parameter which is real only forg < gc and | ↓ i is the collective spin down state.
The condition g < gc restricts the validity of theground state to the normal phase. Nevertheless,as has been recently shown the effective Hamiltonian can be valid dynamically given arbitrary
g for a limited time [48] which in the thermodynamic limit becomes infinity. The quantum
Fisher information with respect to λ— (which canbe set to either ω or Ω)—for the nth excited stateof the Dicke model |ψ0 i = Ŝ(ξ)|ni ⊗ | ↓ i becomes(see the Appendix A)Iλa =1 + n + n2 g 4
2 2 g4c
8λ2 1 − g

(6)

gc2

which is clearly divergent for g ' gc (superscripta implies adiabatic dynamics). However, in practice there is an implicit dependence on time andwhen this dependence is made explicit the divergence disappears [37]. This happens because thetime required to prepare a critical ground state
diverges as the reciprocal of the energy gap, [given
as the square root of δ = 4ω 2 (1 − g 2 /gc2 )], near
the critical point at the quantum phase transition.
In what follows, we will show how to eliminate
the problem of critical slowing down while still
exploiting criticality.

3 Quenching beyond the critical point
Although the effective Hamiltonian from Eq. (4)cannot be used to describe the ground state of
the system in the superradiant phase, it can be
still used to describe dynamics of a quench fromany g < gc to g > gc provided that [48]
1 g2 ω
 1,
4 gc2 Ω

(7)

which in the thermodynamic limit N Ω/ω →∞ becomes infinity (see Ref. [48] for a detailed
derivation). Therefore, for the purpose of analytical analysis, we assume now that Eq. (7) andEq. (8) are satisfied so the use of the effective
Hamiltonian is justified.
In order to shed some light on the system dynamics we rewrite the transformed Dicke Hamiltonian as (see Appendix B)!

g2
ĤDM = ω +Ŝz â† â + ΩŜzΩN
g 2  †2+â + â2 Ŝz
2ΩN

(9)

The initial spin state | ↓ i is an eigenstate of Ŝz(ground state for g/gc = 0). Since [ĤDM , Ŝz ] = 0,
we replace Ŝz → −N/2 and drop terms proportional to identity to obtain a purely photonic andquadratic Hamiltonian commonly appearing in
Gaussian metrology [55] and is essential to understand critical metrology:
!

Ĥ =


g2
g 2  †2â + â2 .
ω−â† â −
2Ω
4Ω

(10)

In fact, this Hamiltonian describes a harmonic
oscillator in quadrature space with a tunable frequencyωωĤ = P̂ 2 +
2
2

g2
1− 2
gc

!

X̂ 2 ,(11)

whose physics is easy to understand. If we quench
to g ≤ gc the vacuum starts to expand as the frequency of the harmonic oscillator is decreased.
This will happen until the frequency of the effective harmonic oscillator in quadrature spacebecomes 0 [48] which coincides with the critical
coupling g = gc . At that point, the initial state is
being constantly squeezed at a finite (quadratic)
rate in analogy to free particle expansion. By
then quenching the interaction parameter further beyond the critical coupling gc , this rate ofsqueezing increases (from quadratic to exponential), leading to a divergence of squeezing overAccepted in Quantum 2022-04-26, click title to verify. Published under CC-BY 4.0.

5

10
8
6

g/gc

g/gc

(a)

2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6

4
2
0
0

1

2

3

4

(b)

2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6

5

10
8
6
4
2
0
0

1

2

ωt

3

4

5

ωt

Figure 2: The logarithm of the quantum Fisher information (for initial vacuum state) normalized to λ2 as a functionof g/gc and time expressed in the units of ω −1 . Panel (a) and (b) depict the quantum Fisher information for λ = ωand λ = Ω, respectively. The dashed line illustrates the quantum Fisher information attainable by quenching thesystem to the critical point.
(a)

10

10

5

5

0
−5

−10

(b)

15

IΩ/Ω2

Iω /ω 2

15

0
−5

0

1

2

3

4

5

−10

0

1

2

3

4

5

ωt

ωt

Figure 3: Extension of Fig. 2 showing slices of the logarithm of the quantum Fisher information for different valuesof g/gc (from the bottom line 0.5, 0.68, 0.87, 1, 1.25, 1.62, 2). The dashed line represents g/gc = 1.

an arbitrarily short time-scale in the thermodynamic limit. That said, for a finite size system
the squeezing will last only until the maximum
number of excitations that a system can support
is reached [see Eq. (8)].
It is well known that squeezing Hamiltonian
leads to an exponential growth of the number ofphotons [56]. On the other hand, it is also wellknown that the quantum Fisher information is related to the number of particles [19, 20]; therefore
we can expect exponential growth of the quantum
Fisher information as well. This idea is made explicit from a heuristic extension of the analyticalexpression of the quantum Fisher information derived in the normal phase. In Ref. [47] it has beenshown that if a Hamiltonian Ĥ = Ĥ0 + λĤλ satisfies the eigenvalue equationp[Ĥ, i δ Ĉ − D̂] =

p

p



δ i δ Ĉ − D̂ ,(12)

with Ĉ = −i[Ĥ, Ĥλ ], D̂ = −i[Ĥ, Ĉ], and δ Ĉ =[Ĥ, −D̂]/i, the local generator ĥλ becomes [57]√√√cos( δ t) − 1sin( δ t) − δ t√ĥλ = Ĥλ t+
Ĉ−
D̂,
δ
δ δ
(13)

and the quantum Fisher information can be simply calculated as the variance of the local generator Iλ = 4∆2 ĥλ .
Let us now first explicitly demonstrate theshortcoming of critical metrology again for the
Dicke model [47]. It can be shown that in thenormal phase the energy gap is given as thesquare root of δ = 4ω 2 (1 − g 2 /gc2 ). Therefore,
it might naively seem that going to a criticalpoint g = gc of a quantum phase transition isindeed optimal as for δ → 0 near the criticalpoint the quantum Fisher information might diverge [47]. However, in order to see that divergence of√the quantump Fisher information the condition δ t = 2 ω 2 (1 − g 2 /ωΩ)t = O(1) has
to be satisfied. This in turn means that in order to see the√ divergence the time has to satisfy t ≈ 1/ δe which happens for t → ∞ as
1 − g 2 /ωΩ ' 0. In other words, the divergencecaused by the vanishing energy gap requires time
approaching infinity [26, 46]. However, from thediscussion above it seems that going beyond thecritical point might actually lead to a divergenceof the quantum Fisher information on exponentially shorter time scales. Let us therefore anal-

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6yse further the quantum Fisher information, or
specifically the local generator ĥλ . For the Dicke
model δ = √4ω 2 (1 − g 2 /ωΩ),
which means that√
if g > gc ≡ ωΩ, then δ is purely imaginary
and can no longer be interpreted as√an energy
gap as in the normal phase. In fact, δ can beinterpreted as the frequency of a harmonic oscillator that inverts in the superradiant phase [48][see Eq. 11]. In the superradiant regime, we thenheuristically propose the extension to the above
result aspexp( |δ |t)/2 − 1ĥλ ' Ĥ1 t +
Ĉ
|δ |
p
p
exp( |δ |t)/2 − |δ |tp−D̂,|δ |δ(14)

which means that the quantum Fisher information
grows now exponentially with increasingp|δ |t as we argued before (see Appendix C for
details). We again contrast with the recently obtained t4 scaling in the normal phase [37]. Tosummarise, at g = gc one obtains quantum Fisher
information whose growth is dictated by photons
growing quadratically in time and requires very
long protocol time, similarly to the adiabatic approach to critical quantum metrology [26, 46].
However, as the coupling parameter becomes significantly larger than gc , the mechanism responsible for the growth of quantum Fisher information
becomes the exponential growth of the number ofparticles caused by squeezing.
The analytical results showing the growth of
the quantum Fisher information as a function of
the coupling parameter and time in the superradiant regime are presented in Fig. 2 and Fig. 3. For
the sake of clarity, we take the logarithm of thenormalized quantum Fisher information. Panel(a) of Fig. 2 and Fig. 3 depict quantum Fisher
information treating ω as the unknown parameter, and (b) depict quantum Fisher informationfor Ω. One can clearly see that quenching beyond the critical point (the dashed-black line)
gives rise to much greater quantum Fisher information than driving in the vicinity of the critical point. This happens because the nonclassicalcorrelations (squeezing) and corresponding excitations (photons) are generated much faster far
beyond the critical point.
Finally, let us discuss the exponential dependence on time of the quantum Fisher informationand its relation to the standard quantum limit.

In order to derive the latter, one assumes thatthe initial state containing N particles (atoms
or photons) can be prepared instantaneously. In
this sense, the standard quantum limit does notinclude the time and energy needed to prepare
the state. Subsequently, the initial state is coherently evolved in time with a time evolution
operator that depends on the unknown parameter (unknown parameter imprint) which leads
to quadratic dependence on time of the quantum
Fisher information (and consequently of the standard quantum limit). In the protocol studied inthis work, these two stages—preparation of thestate and unknown parameter imprint—happen
at the same time. In this sense, it is difficult tomake a straightforward and meaningful comparison between the two approaches. One can thenask how to properly quantify quantum enhancement in the critical metrology approach. Fromthe classical point of view, one can clearly identify a quantum enhancement with respect to aclassical strategy since lowering the frequency ofa harmonic oscillator or even flipping the oscillator upside down does not change the state of
the system. A thorough answer to the issue of
properly quantifying critical quantum enhancement with respect to existing protocols, however,reaches beyond the scope of this work.

4 Quadrature measurements
Although the quantum Fisher information shows
exponential growth as a function of time, it does
not indicate the measurement that saturates the
Cramér-Rao bound. As discussed in the introduction, the squeezed vacuum state is the optimal state in single-mode frequency estimation forwhich the Cramér-Rao bound can be saturatedusing quadrature measurements, which can be realized by a homodyne detection scheme [58, 59].
Nevertheless, in general, the resulting state of
the considered protocol is not a perfect squeezed
vacuum state but a state only resembling thesqueezed vacuum. The perfect squeezed vac2
uum
is only√√ generated for ω = g /2Ω, i.e., g =2ωΩ = 2gc , so when the first term of the
Hamiltonian (10) vanishes. In order to determine
the optimality of the homodyne detection scheme,
we calculate the classical Fisher information
Fλ =Accepted in Quantum 2022-04-26, click title to verify. Published under CC-BY 4.0.

Xξ1p(ξ|λ)

∂p(ξ|λ)
∂λ

2

,

(15)

7

(a)

3π
4
π
2

0
0.0

0.80

0.80

0.60

π
2

0.60

0.20
0.5

1.0

1.5

2.0
ωt

2.5

3.0

3.5

4.0

0.99

3π
4

0.40

π
4

(b)

π

0.99

φ

φ

π

0.00

0.40

π
4

0.20

0
0.0

0.5

1.0

1.5

2.0
ωt

2.5

3.0

3.5

4.0

0.00

Figure 4: Classical Fisher information for quadrature measurement as a function of quadrature direction φ and time√
expressed in units of ω normalized to the quantum Fisher information. The calculations are performed for g/gc ≈ 2which corresponds to the generation of a perfectly squeezed vacuum in Eq. (10). We plot 0 < φ < π as theinformation is invariant under rotation by π. Panel (a) treats ω as the unknown parameter and panel (b) treats Ω as
the unknown parameter. In both cases the optimal quadrature direction converges to π/4.

for the distribution obtained by quadrature measurement, where p(ξ|λ) is the probability of measuring an outcome labeled by ξ for a given valueof an unknown parameter λ. The homodyne
detection boils down to the measurement of aiφgeneralized√ quadrature operator Q̂(φ) = (âe +†−iφâ e )/ 2 which requires additional optimization over the direction φ; this optimal direction
will, in general, depend on the unknown parameter itself (see Appendix E). The dependence ofthe optimal quadrature direction on time andother parameters of the system is a complication with respect to adiabatic protocols where
the optimal direction is always fixed [37]. In
order to saturate the Cramér-Rao bound one
could therefore use adaptive real-time feedback
techniques [60]. On the other hand, one could
first estimate an approximate value of an unknown parameter with some alternative techniques, and subsequently use an adaptive metrological scheme to maximize the sensitivity [61–64]. In the adaptive approach, the couplingstrength g could be increased from relatively low
value (even below the critical value) to estimate
and update the information about the unknown
parameter in every round which should still leadto the possibility of exploiting enhanced rate ofcreating excitations.
To simplify the cumbersome expression for theclassical Fisher information, we first note that theresulting state is Gaussian as the Hamiltonian isquadratic in â and â† . This means that all theinformation about the state is contained in themean and variance of quadrature measurementson the wavefunction. As the initial wavefunction
has a mean value equal to 0 and is subsequently

being squeezed without any displacement, all theinformation is then stored in the variance [65, 66].
As a consequence, measuring the second momentof the quadrature and using it as an estimatorallows us to replace the classical Fisher information formula from Eq. (15) by the inverse of
the error propagation formula (signal-to-noise ratio) [55, 58]
Fλ =|∂λ Q̂2 (φ)|2,∆2 Q̂2 (φ)(16)

which can be calculated analytically (see Appendix D).
The analytical results comparing the quantum
and classical Fisher information for quadrature
measurement as a √function of quadrature angle φand time for g = 2gc (calculation for different
values of g are given in Appendix E) are shown in
Fig. 4. As the state is being more squeezed andmore photons are generated, the optimal quadrature direction converges exponentially with timeto π/4 which is the direction in which the state is
being antisqueezed (for other choices of g the direction of squeezing will be different). This can be
explicitly seen if we calculate the classical Fisher
information (treating ω as the unknown parameter)|∂ω Q̂2 (φ)|2∆2 Q̂2 (φ)=

2 cos2 (2φ) sinh4 (ωt)

2,
ω 2 (cosh (2ωt) − sin(2φ) sinh (2ωt))(17)

and assuming ωt  1 we obtain to lowest order|∂ω Q̂2 (φ)|2
cos2 (2φ)',
2ω 2 (sin(2φ) − 1)2∆2 Q̂2 (φ)(18)

which can be expanded again to lowest order

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8around φ = π/4:|∂ω Q̂2 (φ)|2
1
'
2 .
2
2
∆ Q̂ (φ)
2ω 2 φ − π4(19)This profound result indicates that in order toobserve an arbitrary large sensitivity the quadrature direction (phase of the local oscillator) has tobe precisely controlled. This can be interpreted
as an inherent feature of precision measurements;
namely, arbitrarily large sensitives also require an
arbitrary precise measuring apparatus. Any uncertainty of such an apparatus—in this case, apotential instability of the measured quadrature
angle—will directly influence the precision of estimation (see Fig. 5).
3π
8

6
5

φ

5π
16

4

π
4

3
2

3π
16
π
8

0.5

1
0
1.0

1.5

2.0

2.5

3.0

3.5

4.0

ωt

Figure 5: Logarithm of the classical Fisher informationfrom Eq. (19) (multiplied by ω 2 ). As the classical Fisher
information grows, the angle-window for the optimal
quadrature direction becomes smaller and√converges toa single point (here π/4). Result for g = 2gc .

5 Dynamical Dicke superradiance inthe cavity
In this section, we briefly discuss the possible
implementation of the proposed protocol by exploiting Dicke superradiance in atoms interactingwith single-mode cavity field. It is a well-known
fact that the self-organization phase transition,
which can be observed in systems of thermal [67]and ultracold atoms [68] trapped and interactingwith a cavity mode, can be mapped to the Dicke
model [69]. For simplicity, we consider a onedimensional zero-temperature Bose-Einstein condensate composed of N atoms each with mass m
trapped inside a high-Q single-mode cavity withfrequency ωc and cavity mode function cos kx
with x being the cavity axis and k = 2πωc /c. The
atoms are illuminated from a transverse direction
to the cavity axis by a pump laser with frequency

ω far red-detuned from the atomic frequency ωA .
This laser changes the effective photon energy inthe cavity (detuning) to ∆c = ω − ωc . We assume now that the detuning ∆A = ω − ωA ismuch larger than the rate of spontaneous emission such that the atomic excited state can be
adiabatically eliminated. The dispersive atomfield interaction strength is U0 = g02 /∆A with asingle photon Rabi frequency g0 , and the effective
cavity pump coupling strength describing atomphoton coupling is η = Ωg0 /∆A , where Ω is the
Rabi frequency of the coupling to the transverse
driving field. Assuming the positional wavefunction is spanned by two Fourier modes ĉ0 and ĉ1
obeying the standard bosonic commutation relations, the Hamiltonian can be written as [69]iyĤcav = − δc â† â + ωR Ŝz + √ (â† − â)Ŝx
N
!
(20)
1 Ŝz
†
+ uâ â+,2N
where δc =√∆c − 2u, ωR = h̄k 2 /2m, u = N U0 /4,
and y = 2N η. In the first line of the above
Equation we identify the Dicke model, while thesecond line represents the additional cavity frequency shift inherent to the Bose-Einstein condensate cavity system. The latter, however, does
not change the critical coupling [69] and can besafely neglected in typical experimental situations
when |u| < |δc |. Applying the Wolff-Schrieffer√transformation Û = exp(y(â† − â)Ŝy /ωR N )to the above Equation and projecting onto theground state for y = 0 yields (see Appendix F)!

y 2  †2â + â2 ,4ωR(21)
which describes apharmonic oscillator with frequency ν √= 4δc 1 − (y/yc )2 and critical coupling yc = −ωR δc . Since the frequency depends√
on the number of atoms N through y = 2N η,
the quantum Fisher
information picks up an ad√
ditional power N -enhancement with respect to
Eq. (13), i.e., [assuming (y/yc )2  1]

Ĥcav '

y2
−δc +
2ωR



â† â −√Iλ ∝ exp 8 2s

N |δC | 
ηt .
ωR

(22)

This happens because more atoms in the cavity
increase the probability of photon scattering into
the cavity mode and the number of scattered photons is proportional to N 2 [70].

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9

From the viewpoint of experimental realizations of the Dicke model in the cavity systems [68, 71–74], even though the condition from
Eq. (7) may not be always satisfied, and the observation of squeezing might not be possible dueto the spontaneous symmetry breaking, the exponential growth of the Fisher information shouldstill be an observable effect. This is due to thefact that the exponential growth of the numberof photons following the dynamical phase transition (superradiance) occurs for a broad spectrumof parameters even in the mean-field approach(see the experimental results from Ref. [71]). Ina general case, the analytical description of thesystem might be more obscure than the elegant
formalism with an inverted harmonic oscillator
as the spin dynamics will not be frozen; nevertheless, the information about the unknown parameter should be still imprinted in the numberof photons that grows exponentially in time [71].
It is worth noting that we have not taken into
account the effect of photon losses. Nevertheless,as was shown in Ref. [48], photon losses will only
affect the rate at which the photons are being
created and will not destroy metrologically useful correlations stored in a squeezed state. This
happens because the squeezed vacuum state is arelatively robust quantum resource. Therefore,
even in the presence of decoherence, one can still
expect exponential growth of the quantum andclassical Fisher information in time.

6 Conclusions & Outlook
In this work, in order to eliminate the critical slowing down that hinders protocols of critical quantum metrology, we have proposed toquench the system far beyond the critical point
and thus increase the rate at which correlations
are created. The presented idea is significantly
faster (i.e., exponential versus power-law scaling
in time) than existing adiabatic and dynamical
protocols, which can be of fundamental importance in realistic scenarios where deleterious effects of decoherence and experimental noise are
inevitable. The price being paid for the enhancement is an additional dependence of the optimal
quadrature angle on the parameters of the system. This however could be overcome by using
an adaptive strategies.
The protocol can be understood as a non-linear type of interferometry where the information about the unknown parameter is not only
stored in the phase of the wave function but also
in the number of excitations (photons) and thenonclassicality of them itself. The results are fully
valid close to the thermodynamic limit but canalso be applied to finite size systems under additional constraints on time [48]. They can also
be extended to a more general case in which theinverted harmonic oscillator description may not
be fully valid. In this case the spin degree of freedom will also experience squeezing and lead tolight-matter entanglement.
As quenching the system into the superradiantphase generates a Gaussian state, we have argued
and shown analytically that the quadrature measurement can saturate the quantum Cramér-Rao
bound for an optimal angle outside of the normal
phase. The optimal quadrature angle becomes
effectively a point in the limit of ωt  1 which
can be understood as a condition that a measuring apparatus has to satisfy in order to perform
a very precise measurement.
Our protocol can be readily tested in quantum simulators which realize the (optical) quantum Rabi model [75–77], and quantum simulators which can realize the (optical) Dicke model[68, 72], i.e., a macroscopic number of spins coupled to a single-mode field. As discussed in thiswork, a promising candidate is a gas of atoms inan optical cavity. Moreover, we have shown thatthese systems can exhibit sensitivity that additionally scales exponentially with the square rootof the number of atoms N confirming that exponentially scaling Fisher information can be aphysical and observable effect [23] and does notnecessarily require a N -body term.
The presented idea paves a way towards
metrology protocols which can take benefit fromstoring the information about the unknown parameter in the number of excitations. An intriguing possibility is testing the proposed protocol
in quantum simulators which realize the Dicke(Rabi) model without photonic degrees of freedom [78–81]. Our analysis shows that measuringsimply excitations of the system—such as centerof-mass excitations in a spin-orbit coupled BoseEinstein condensate [78]—could lead to enhanced
metrology.

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10

Acknowledgments

for his support. Simulations were performed using the open-source QuantumOptics.jl frameworkin Julia [82]. This work was supported by the Okinawa Institute of Science and Technology Graduate University. K.G. acknowledges support fromthe Japanese Society for the Promotion of Science(P19792).

The authors are pleased to acknowledge Farokh
Mivehvar, Friederieke Metz, Tim Keller, and Jing
Li for inspiring discussions and thank the anonymous Referees for their suggestions and comments. K.G. would like to thank Szemek Keppe

A Quantum Fisher information for the eigen-states of the Dicke model in thenormal phase
The quantum Fisher information can be calculated by using the following formulaIλ = 4(h∂λ ψ|∂λ ψi − h∂λ ψ|ψi2 ),(23)

where λ is an unknown parameter. The nth eigenstate |ψn i for the Dicke model in the thermodynamic
limit below the critical point is|ψn i = Ŝ(ξ)|ni ⊗ | ↓ i,(24)
where Ŝ(ξ) ≡ exp{(ξ/2)(â† )2 − (ξ ∗ /2)â2 } is the squeezing operator with ξ = − 14 ln{1 − (g/gc )2 } and√gc = ωΩ. We begin by calculating the derivative of the instantaneous eigenstate |ψn i with respect
to an unknown parameter which we assume to be ω. This yields|∂ω ψn i =


 
2
g2
†
2
â
−
â
Ŝ(ξ)|ni ⊗ | ↓ i,
8ω(g 2 − ωΩ)(25)

and allows to calculate the overlap h∂ω ψn |ψn i
 2
g2
†
â† − â2 Ŝ(ξ)|nih∂λ ψn |ψn i =
hn|
Ŝ
(ξ)
8ω(g 2 − ωΩ)

2 
2 
g2
†
†
|ni
=
hn| â cosh(|ξ|) + â sinh(|ξ|) − â cosh(|ξ|) + â sinh(|ξ|)
8ω(g 2 − ωΩ)g2=cosh(|ξ|) sinh(|ξ|)hn| â† â + ââ† − â† â − ââ† |ni = 0,2
8ω(g − ωΩ)




(26)

where we have usedŜ † (ξ)âŜ(ξ) = â cosh(|ξ|) − ei arg(ξ) â† sinh(|ξ|) = â cosh(|ξ|) + â† sinh(|ξ|).

(27)

An analogous (but much more tedious) calculation for the h∂ω ψn |∂ω ψn i yieldsh∂ω ψn |∂ω ψn i =g 4 (1 + n + n2 ).
32ω 2 (g 2 − ωΩ)2(28)

Substituting the h∂ω ψn |ψn i and h∂ω ψn |∂ω ψn i into the quantum Fisher information formula givesIω =g 4 (1 + n + n2 )(1 + n + n2 ) g 4
=
.


2 2 4
8ω 2 (g 2 − ωΩ)2
8ω 2 1 − g2 gc

(29)

gc

Treating Ω as an unknown parameter one obtainsIΩ =(1 + n + n2 ) g 4.

g 2 2 gc4
2
8Ω 1 −(30)

gc2

This extends the result of n = 0 [37] to an arbitrary n.
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11

B Schrieffer-Wolff transformation of the Dicke model
In the case of the Dicke modelg  †ĤDM = ωâ† â + ΩŜz + √â + â Ŝx ,N(31)

the Schrieffer-Wolff transformation is a simultaneous rotation of the (collective) spin around the y-axis
and displacement of the harmonic oscillator
√ 

 g
ω
ÛSW = exp i √â† + â Ŝy .
(32)
gc ΩN
As a result of the transformation, one obtains
!
!
√
√
ω
ω
g
g
†
ĤDM = ω â + i √Ŝyâ − i √
Ŝy
gc ΩN
gc ΩN
!
! !
√
√
g
ω
g
ω
†
†
√
√
+ Ω cos(â + â) Ŝz − sin(â + â) Ŝx
gc ΩN
gc ΩN
!
! !
√
√

g  †
g
g
ω
ω
†
†
√
√
+√
â + â cos(â + â) Ŝx + sin(â + â) Ŝz .
gc ΩN
gc ΩN
N

(33)

Expanding the above Hamiltonian in the Taylor series assuming ω/Ω  1 gives Eq. (4) from the main
text
!

g2
g 2  †2
ĤDM = ω +â + â2 Ŝz .
Ŝz â† â + ΩŜz +
(34)
ΩN
2ΩN
If the initial state is a collective spin down state, i.e. Ŝz | ↓ i = −N/2| ↓ i, then the Hamiltonian can be
simplified to!

g2
g 2  †2
ĤDM = ω −â† â +â + â2 ,
(35)
Ω2
4Ω
which can be rewritten in the form of a harmonic oscillator with a tunable frequencyωωĤ = P̂ 2 +
2
2

g2
1− 2
gc

!

X̂ 2 .

(36)

The above Hamiltonian can be used to calculate the ground state but only in the normal phase. Fromthe viewpoint of metrology, however, it is more interesting to explore the dynamics of this Hamiltonian.
Following Ref. [48], it can be shown that this Hamiltonian gives rise to proper (squeezing) dynamics
until certain number of excitations is reached. To be more precise this number can be calculated to beD

1 NΩâ â <
t
32 ω
†

E

g 2 gc2−
gc2 g 2

!

,

(37)

which can be further elevated by increasing N .

C Local generator for the Dicke model under the Schrieffer-Wolff transformation
Following the derivation from Ref. [47], it can be shown that the transformed local generator ĥλ for a
Hamiltonian Ĥ = Ĥ0 + λĤλ can be expressed asĥλ = ie

iĤt



∂λ e

−iĤt



Z t

e

=
0

iĤs

−iĤs

Ĥλ e

ds =

Z tX
∞

∞
i
i
X
(is)n h(it)n+1 h
Ĥ, Ĥλ ds = −i
Ĥ, Ĥλ ,nn(n + 1)!
0 n=0 n!
n=0(38)

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12where [Ĥ, Ĥλ ]n+1 = [Ĥ, [Ĥ, Ĥλ ]]n and [Ĥ, Ĥλ ]0 = Ĥλ . Now it is possible to show that [Ĥ, Ĥλ ] = iĈ,
[Ĥ, iĈ] = −D̂, and [Ĥ, −D̂] = iδ Ĉ, which results in the following commutation relations[Ĥ, Ĥλ ]2n+1 = iδn Ĉ

and [Ĥ, Ĥλ ]2n+2 = −δn D̂.

(39)

Therefore, one can express the local generator ĥλ asĥλ = Ĥλ t − i∞X(it)2n+2
n=0(2n + 2)!

iδn Ĉ + i∞
X(it)2n+3
n=0(2n + 3)!

cos(
δn D̂ = Ĥλ t +√√√δ t) − 1sin( δ t) − δ t
√
D̂.
Ĉ −
δ
δ δ
(40)

In the low-energy sector we can rewrite the Dicke model as

ω 2
g2 2
P̂ + X̂ 2 −
X̂ ,
(41)
2
2Ω
√
√
with X̂ = (â + â† )/ 2 and P̂ = (â − â† )/ 2i. Assuming that ω is unknown, by choosing Ĥω =
g2
1
2
2
2
2 (X̂ + P̂ ) and Ĥ0 = − 2Ω X̂ , one can easily show that

Ĥ =


g2 
Ĉ = −X̂ P̂ + P̂ X̂ ,
2Ω

g2ω 2
g4
D̂ = −(X̂ − P̂ 2 ) + 2 X̂ 2 ,ΩΩ!

and δ = 4ω 2

g2
1−
,
ωΩ

(42)

leading to
ĥω =

t
2



X̂ 2 + P̂ 2 −g 2 sin
+ 2
gc ω
2

√

cos

g2
2gc2 ω


δωt −
√
δδ

g
) and gc =where δ = δ /ω 2 = 4(1 − ωΩan unknown parameter yielding

√

√

√



δωt − 1 δδωt X̂ 2 − P̂ 2

X̂ P̂ + P̂ X̂
g 4 sin
− 4
gc ω



√



δωt −
√
δδ

√

(43)
δωt

X̂ 2 ,

ωΩ. Analogous calculations can be performed treating Ω as√δωt − 1 
g 2 ωt 2g 2 cosĥΩ = 2 X̂ + 2
X̂ P̂ + P̂ X̂
2gc Ω
2g Ω
δ
√ c  √
√
 √
4 sin

δωt − δωt 
δωt − δωt
g 2 sing√√X̂ 2 .
+ 2
X̂ 2 − P̂ 2 − 4
gc Ω
gc Ω
δδ
δδ

(44)

Similar local generator can be found for g as has been shown in Ref. [47]√√δωt − 1 g ωt 2g cos
ĥg = − √ X̂ − 2
X̂ P̂ + P̂ X̂
gc
δ
gc Ω
√
 √
√
 √
 2g 3 sinδωt − δωt 
δωt − δωt
2g sin√√− 2
X̂ 2 − P̂ 2 + 4
X̂ 2 .
gc
g
δδ
δδ
c

(45)

D Error propagation formula for the second moment of Quadrature estimation
In order to calculate the error propagation formula, we first move to the Heisenberg picture andevaluate how â and â† evolve in time. Using matrix notation it can be easily shown that a Bogoliubovtransformation is obtained:
!
!

 â
â(t)
= exp itÂ,(46)â† (t)â†
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13

where

2

Â =

g
− 2Ω
g2
ω − 2Ω

g2
2Ω − ω
g2
2Ω

!

.

(47)

Upon diagonalization of the above matrix, one can findâ(t) = â cos√

2i sinδωt/2 − â

√

√



δωt/2

δ

+ (â − â† )ig 2 sin

√



δωt/2
√
,
gc2 δ

(48)

and similarly for â† (t). The second moment of the quadrature operator for initial vacuum state can
now be calculated to give
√

√

g 2 sin(2φ) sinδωtg 2 cos(2φ) cos
δωt1
g 2 cos(2φ)√
h0|Q̂2 (t)|0i = −
+
−
2
gc2 δ
2gc2 δ
2g 2 δ√ c√g 4 sin2
δωt/2
δωt/2
g 4 cos(2φ) sin2
+
+
,
gc4 δ
2gc4 δ

(49)

and all the higher moments can be calculated from this expression as the distribution
is Gaussian. In√
order to get a further insight, we can now focus on the special case of g = 2gc , which boils down topure squeezing, and set ω as the unknown√parameter (without the loss of generality). Evaluating thesignal-to-noise ratio and substituting g = 2gc into the expression yields|∂ω Q̂2 (φ)|22 cos2 (2φ) sinh4 (ωt)=.
ω 2 (cosh (2ωt) − sin(2φ) sinh (2ωt))2∆2 Q̂2 (φ)(50)

Assuming now ωt  1 such that cosh(ωt) ' sinh(ωt) ' exp(ωt)/2, we finally obtaincos2 (2φ)|∂ω Q̂2 (φ)|2',2ω 2 (sin(2φ) − 1)2∆2 Q̂2 (φ)(51)

which diverges as φ approaches π/4 which can be explicitly seen by expanding in Taylor series aroundφ = π/4|∂ω Q̂2 (φ)|2
1
'
(52)
2 .
2
2
∆ Q̂ (φ)
2ω 2 φ − π4

E Classical Fisher information for an arbitrary g/gc in the superradiant phase√
In the main text, we have presented calculations for a special case of g = 2gc which corresponds topure squeezing. The natural question that arises is whether the Cramér-Rao bound can be saturated
for other choices of g/gc . To this end, let us calculate the optimal angle of the quadrature direction.
√
Assuming g/gc = r, the optimal direction is given bys

φ = cos−1 

f (r) 
,
2

(53)

with
√

2r2 sinh2 r − 1ωt + 8r − 8√
f (r) = 2
r cosh 2 r − 1ωt − r2 + 8r − 8
√
√
√
√
q


 (54)22csch
r − 1ωt (r − 2)2 sinh6 r − 1ωt r2 cosh 2 r − 1ωt − r2 + 8r − 8√+.
r2 cosh 2 r − 1ωt − r2 + 8r − 8
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14

In the limit of√r − 1ωt → ∞ the optimal angle becomes
s



g2
lim φ = cos−1  1 − c2  ,
t→∞
g

(55)

which is the direction perpendicular to the squeezing direction as discussed in the main text. In order toshow the saturation of the Cramér-Rao bound for other values of g/gc we performed simulations whose
results are presented in Fig. 6. The figure depicts classical Fisher information normalized to quantum
Fisher information as a function of φ and time. As can be seen, in every case the saturation of the
Cramér-Rao bound is possible. We interpret this result as a potential confirmation of the saturation of
the Cramér-Rao bound by the homodyne detection scheme. For g/gc > 1.25, the simulations cannot
resolve the peak of the classical Fisher information as the optimal quadrature angle converges to asingle point given by Eq. (55).

1

3

4

0.00

π
2

0.80
0.60

π
2

π
4

0.20
1

2
ωt

3

4

0.00

0

1

2
ωt
(f)

3

4

0.00

0

2
ωt

3

4

1

2

3

4

0.00

0

(g)

π

0.80

3π
4

0.60

π
2

0

0.00

0.80

3π
4

0.60

π
2

0.40

π
4

1

2
ωt

3

4

0.40
0.20
0

1

2
ωt
(h)

3

4

0

0.00

0.00

0.99

g/gc ≈ 1.62

0.80
0.60
0.40

π
4

0.20
0

0.80
0.60

π

0.99

g/gc ≈ 1.55

0.99

g/gc ≈ 1.32

π
4

ωt

0.20
1

π
2

0.20

0

0.40

0

0.60
0.40

π
4

0.99

g/gc ≈ 1.47

0.80

3π
4

φ

φ

0.40

π
4

0

π
2

0.20

π
3π
4

0.40

0

0

0.99

g/gc ≈ 1.40

3π
4

φ

2
ωt
(e)

0.80
0.60

g/gc ≈ 1.25

3π
4

(d)

π

0.99

φ

0

g/gc ≈ 1.17

(c)

π

0.99

π
4

0.20

π

0

π
2

0.40

π
4

0

0.80
0.60

φ

φ

π
2

3π
4

φ

g/gc ≈ 1.10

3π
4

(b)

π

0.99

φ

(a)

π

0.20
0

1

2
ωt

3

4

0.00

Figure 6: Classical Fisher information normalized to the quantum Fisher information as a function of φ and time forvarious values of g/gc . In all the cases the homodyne detection in the optimal direction saturates the Cramér-Rao
bound.

F Schrieffer-Wolff transformation for cavity superradianceFollowing Ref. [69] a quantum gas of atoms interacting dispersively with a cavity mode can be mapped
to the following Hamiltonianiy
1 ŜzĤcav = −δC â â + ωR Ŝz + √ (â† − â)Ŝx + uâ† â
+
2
N
N

!

†

.

(56)

Using the Schreiffer-Wolff transformation of the formÛ = exp

y
√

ωR N



†





â − â Ŝy ,and assuming |δC |/ωR N  1, |u| < |δC |, neglecting terms proportional to(57)pδc /ωR N gives

y2
1 Ŝz
Ĥcav ≈ −δC â† â + ωR Ŝz +(â† − â)2 Ŝz + uâ† â
+
2ωR N
2
N
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!

.

(58)

15

The above Hamiltonian commutes with the Ŝz operator which means that hŜz i is a constant of motion.
Therefore, if the initial state is a collective spin-down state, the operator can be replaced by the
eigenvalue −N/2 to givey2 †Ĥcav ≈ −δC â† â −(â − â)2 .
(59)
4ωR
The considered transformation can be understood as an elimination (time-dependent) of the spin
degree of freedom.

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