Quantum metrology using time-frequency as quantum continuous variables resources sub shot-noise precision and phase space representation Eloi Descamps12 Nicolas Fabre34 Arne Keller25 and P erola Milman2

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Quantum metrology using time-frequency as quantum continuous variables: resources,

sub shot-noise precision and phase space representation

Eloi Descamps1,2, Nicolas Fabre3,4, Arne Keller2,5, and P´erola Milman2∗

1D´epartement de Physique de l’Ecole Normale Sup´erieure - PSL, 45 rue d’Ulm, 75230, Paris Cedex 05, France

2Universit´e Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France

3Departamento de ´

Optica, Facultad de F´ısica, Universidad Complutense, 28040 Madrid, Spain

4Telecom Paris, Institut Polytechnique de Paris,

19 Place Marguerite Perey, 91120 Palaiseau, France and

5Department de Physique, Universit´e Paris-Saclay, 91405 Orsay Cedex, France

We study the role of the electromagnetic ﬁeld’s frequency on the precision limits of time measure-

ments from a quantum perspective, using single photons as a paradigmatic system. We demonstrate

that a quantum enhancement of precision is possible only when combining both intensity and spec-

tral resources and, in particular, that spectral correlations enable a quadratic scaling of precision

with the number of probes. We identify the general mathematical structure of non-physical states

that achieve the Heisenberg limit and show how a ﬁnite spectral variance may cause a quantum-

to-classical-like transition in precision scaling for pure states similar to the one observed for noisy

systems. Finally, we provide a clear and consistent geometrical time-frequency phase space inter-

pretation of our results, well identifying what should be considered as spectral classical resources.

The wave nature of radiation make it a choice system

for time precision measurements: using diﬀerent interfer-

ometric techniques, precision in time is set by the inverse

of the ﬁeld’s frequency, or of the ﬁeld’s bandwidth for

non-monochromatic ﬁelds. Leaving apart the systematic

error, classical power noise is in general a limitation, but

it can be reduced to the level of the standard quantum

limit (SQL) [1] - or shot-noise -, and scales with 1/p⟨ˆn⟩,

where ⟨ˆn⟩is the average photon number, or the intensity,

of the ﬁeld. In this case, photons behave as independent

probes, which is explained by the Poissonian nature of

coherent (quasi-classical) states.

Fully exploiting quantum resources can quadratically

improve the SQL [2], and in quantum optics, a sub-shot

noise precision can be obtained from diﬀerent ﬁeld statis-

tics corresponding to squeezed [3–6], NOON [7–9] and

Schr¨odinger cat-like states [10?–12], for instance, as

well as using non-local evolutions in multi-mode states

[13].

Thus, when dealing with time precision measurements,

two key factors limiting the precision are usually set

apart: the photon number statistics - related to the parti-

cle nature of light -, which leads to the precision scaling,

and the modal properties - related to the ﬁeld’s wave

character [14] - which is treated as a classical ressource.

However, the two aforementioned aspects of radiation

are not in general independent, especially when consider-

ing intrinsically multimode non-Gaussian states. For in-

stance in [15], it was shown that entangled and squeezed

states in frequency can lead to quantum enhanced clock

synchronization and position measurement. Neverthe-

less, providing a clear picture of the interplay between

the two aspects of radiation in metrology and in quan-

tum optics in general remains an open problem, in spite

of its fundamental and practical importance.

In the present Letter we address the vast problem of

ﬁeld and mode non-separability and its consequences on

quantum metrology. By doing so, we unveil the time-

frequency phase space structure behind quantum preci-

sion limits and introduce a deﬁnition of classical ressource

which is common to both the modal and the particle as-

pects of the quantum ﬁeld. The introduced geometrical

picture provides a description of how scaling properties

with constant resources depend on modal and particle

entanglement or, in general, in the collective photonic

behavior. We study in details a paradigmatic system con-

sisting of ndistinguishable photons occupying each a dif-

ferent ancillary mode (for instance, a spatial mode) and

show how quantum metrological enhancement can be ob-

tained in such a systems. Photons in independent spatial

modes are characterized by a frequency wave-function,

and the frequency variable is treated quantum mechan-

ically, since it is directly associated to each single pho-

ton’s statistical properties. Notice that in this case, we

consider the paraxial approximation, so that the ﬁeld’s

transverse and longitudinal degrees of freedom factorize,

as in [16]. Thus, the frequency degree of freedom can

be directly associated to the longitudinal mode’s wave-

vector.

We introduce the basic principles of quantum metrol-

ogy considering the example of phase estimation. For

such, we deﬁne a probe, whose evolution depends on a

parameter θto be estimated. The probe is then mea-

sured and an estimator infers the value of the parameter

from the measurement results. For an unbiased estima-

tor, the average value of θ,⟨θ⟩=θr, where θris the real

value of the parameter. The diﬀerent outcomes xare ob-

tained with probability p(x|θ), and precision is limited

by the Cram´er-Rao bound [17]: δθ ≥1/pνF (θ), where

F(θ) = Rdx 1

p(x|θ)∂p(x|θ)

∂θ 2is the Fisher information

(FI) and νthe number of independent repetitions of this

arXiv:2210.05511v5 [quant-ph] 19 Dec 2023

procedure. In quantum metrology, the probe is a quan-

tum state, and we can consider that it evolves by the ac-

tion of a unitary operator ˆ

U(θ) depending on the param-

eter to be estimated θ. The optimization of the FI over

all possible measurements leads to the quantum Fisher

information (QFI) FQ(θ) [18] that sets a more general

bound for the precision of estimating the parameter θ, the

quantum Cram´er-Rao (QCR) bound δθ ≥1/pνFQ(θ).

The QFI for pure states is proportional to the overlap

between the initial state and the displaced one, FQ(θ) =

8(1 − |⟨ψθ|ψθ+dθ⟩|)/dθ2and |ψθ+dθ⟩=ˆ

U(dθ)|ψθ⟩=

eiˆ

Hdθ/ℏ|ψθ⟩. Expanding the unitary operator up to sec-

ond order in dθ, we obtain the well known expression for

the QFI, FQ(θ) = 4(∆ ˆ

H)2[19], i.e., it is proportional to

the variance of the Hamiltonian computed in the state

used as a probe (the initial state). We have then an in-

equality that will be central to this contribution:

δθ ≥1/(2√ν∆ˆ

H).(1)

Beating the SQL involves obtaining a scaling of the

QFI better than ⟨ˆn⟩, which quantiﬁes the amount of

available resources (in quantum optics, the average ﬁeld’s

intensity, in general). Quantum mechanical scaling can

be as good as the Heisenberg limit, where the QFI is pro-

portional to ⟨ˆn2⟩(a scaling shown to be optimal [20]).

The quadrature phase space was show to provide a

clear geometrical picture of (1)[10?, 11], since ˆ

Hgen-

erates a phase space trajectory. The maximal precision

can be seen as the minimum displacement of a Wigner

function so as it becomes distinguishable from the initial

one. In particular, sub-Planck structures are associated

to sub-shot noise precision.

In quantum optics, phase estimation is often linked to

time (or delay) estimation for single mode ﬁelds: the free

evolution on diﬀerent optical paths results in a phase gain

proportional to the frequency of the ﬁeld (which is con-

stant for monochromatic ﬁelds). This evolution can also

be visualized as translations in the time frequency phase

space (TFPS), where exotic spectral properties have also

been observed but not yet associated to any quantum ef-

fect [21–23]: although it was shown that the right choice

of modes for non-monochromatic Gaussian single-mode

states is essential for optimizing precision measurements

[3, 24], the frequency related statistical properties of the

ﬁeld can be disregarded in this case and the ﬁeld’s spec-

tral properties become mere quantities that do not play

a role in the scaling of the QFI. Rather, they simply de-

termine the units in which the scaling is computed (this

can be seen from Eq. (2), for instance). However, for

many quantum states, such as intrinsically multi-mode

non-Gaussian states, it may not be possible to separate

the modal and the ﬁeld statistics. This is the case of

frequency entangled single photon states - which are the

main subject of this Letter - that are used as a resource

in various quantum optical protocols [25–29], including

metrological ones [15, 30–34]. For these reasons it is cru-

cial for quantum optical based metrological protocols to

establish a consistent formalism that demonstrates how

various optical resources, as modes and the ﬁeld statis-

tics, interplay and contribute to the establishment of pre-

cision limits in parameter estimation.

In order to do so, we will study free evolution as the

generator of the probe state’s dynamics. This will en-

able the deﬁnition of a common classical reference with a

clear interpretation both in the quadrature phase space

and in the frequency-time representation using coherent

states. Then, we’ll show how quantum metrological ad-

vantage can appear from frequency correlation properties

and interpreted in the time-frequency phase space. Since

throughout this Letter we’ll mostly consider evolutions

generated by Hamiltonians, we’ll restrict our discussion

to the variance of this operator.

The ﬁrst studied system consists of states that are sep-

arable in n(orthogonal) auxiliary mode basis (as spatial

modes, for instance). The free evolution Hamiltonian is

given by ˆ

H=ℏˆ

Ω, where ˆ

Ω = Pn

i=1 αiˆωi,αi=±1, is a

collective mode operator and i= 1, ..., n denotes the spa-

tial modes. The operators ˆωiare the frequency operators

(see Supplementary Material A) acting on each mode i.

The QFI for pure states is proportional to the variance

of ˆ

Ω, which can be expressed, for mode separable states

as (see Supplementary Material F):

(∆ˆ

Ω)2=

i=1

(⟨ˆni⟩(∆ωi)2+ (∆ˆni)2¯ω2

i),(2)

where ˆniis the photon number operator in spatial

mode iand ∆ˆniits root mean square (RMS). ¯ω2

(Rω|Si(ω)|2dω)2is the average frequency squared in

mode i. The function Si(ω) is a complex function, or the

ﬁeld’s spectrum in the i-th mode, with R|Si(ω)|2dω = 1.

Thus, |Si(ω)|2behaves as a classical density probability

distribution. Finally, ∆ωiis the frequency RMS where,

again, ωiis considered as a continuous random variable

with density probability distribution |Si(ω)|2. A ﬁrst re-

mark is that Eq. (2) explicits two types of contributions

to the QFI: one coming from the photon number vari-

ance and another from the frequency variance. While

the mechanisms by which the ﬁrst one can be associated

with a quantum metrological advantage have been ex-

tensively studied [35] and are related to the quadrature

phase space structure, the other is often associated to a

mere free classical resource, since it depends only linearly

on the average photon number.

In order to gain insight we analyze Eq.(2) for

a coherent state of amplitude βand spectrum

S(ω) in a single spatial mode [36, 37], |β⟩=

e(RβS(ω)ˆa†(ω)−β∗S∗(ω)ˆa(ω)dω)|0⟩. In this case, (2) becomes

(∆ˆ

Ωc)2=|β|2Rω2|S(ω)|2dω =|β|2ω2. This result can

be interpreted from diﬀerent perspectives. In ﬁrst place,

it is proportional to the ﬁeld’s intensity |β|2, and corre-

sponds to the shot-noise limit, as expected. In second

place, it does not depend on the total energy of the sys-

tem |β|2¯ω, usually considered as a resource, but rather

to the spectral’s ﬂuctuations ω2[38]. Thus, for a given

ﬁxed ﬁeld’s energy, one can freely engineer its spectrum

so as to deﬁne diﬀerent time precision scales using ∆ω

while keeping the same shot-noise scaling (as done in [21–

23], for instance). This suggests that the ﬁeld’s energy

should not be considered as the classical ressource, and

spectral properties should play a role. We’ll study this

issue by considering intrinsically multimode states where

the frequency variance takes a more complex form. In

this case, frequency and intensity properties are not in-

dependent and the frequency variance can be used to

modify the scaling of the QFI even in a situation where

the photon number variance vanishes.

To demonstrate this, we examine a system compris-

ing nsingle photons, each occupying a distinct ancillary

mode. Each photon has a given frequency proﬁle (spec-

trum), and this system forms a subspace denoted Sn(see

[29] and Supplementary Material A). Hence, if photons

are prepared in a separable state, (∆ ˆ

Ωs)2=Pn

i=1(∆ωi)2,

where (∆ωi)2=Rω2|Si(ω)|2dω −(Rω|Si(ω)|2dω)2,

and Si(ω) is the spectrum of the i-th photon. We’ll sup-

pose, for simplicity, that all the single photons have the

same frequency RMS ∆ω- also called the frequency RMS

per photon -, and that the considered state is pure (our

results can be easily generalized for non-pure states and

arbitrary RMS per photon). Thus, (∆ˆ

Ωs)2=n(∆ω)2,

since we have the equivalent to nindependent probes.

This is the same scaling as the shot-noise. By com-

paring it to the coherent state scaling, we can identify

n(∆ω)2=|β|2ω2. Both expressions are proportional to

the number of photons: not surprisingly, a coherent state

represents the same resource as nindependent photons

[15, 38, 39]. Nevertheless, it is noteworthy that while for

a coherent state the scaling on the average photon num-

ber is due to the fact that (∆ˆn)2=|β|2, ∆ˆn= 0 in Sn.

In Fig. 1 (a) and (b) we show the Joint Spectral Intensity

(JSI) of separable states of two independent (separable)

photons (n= 2). Also, we can identify the frequency

dependency of the coherent state scaling to a frequency

variance centered at ω= 0.

We can now calculate the variance of the operator ˆ

Ω

for a pure non-separable state in Sn, which gives:

(∆ˆ

Ω)2=

i=1

(∆ωi)2+

i=1,i̸=j

αiαj(⟨ˆωiˆωj⟩−⟨ˆωi⟩⟨ˆωj⟩).(3)

This variance is bounded, and in the case where all the

variances of the single photons are the same we can

show that (∆ˆ

Ω)2≤n2(∆ω)2, which corresponds pre-

cisely to the Heisenberg limit. We now compare this

result to the usual computations of precision limits in

phase measurements in quantum optics, where the role

of mode variance is disregarded as a quantum resource

and the quantum metrological advantage is exclusively

due to the photon number variance: for NOON [40, 41]

or Schr¨odinger cat states [42], which saturate the Heisen-

berg limit, (∆ˆn)2∝ ⟨ˆn⟩2, where ⟨ˆn⟩is the average photon

number. For states in Sn, however, the photon number

variance is always equal to zero and the variance in the

global evolution generator ˆ

Ω explicitly depends on modal

properties only. Thus, the Heisenberg limit can only

be reached by exploiting mode (frequency) entanglement

and the associated mode/particle statistical properties

of an intrinsically multi-mode state. Interestingly, in or-

der to reach the Heisenberg limit, these variables must

behave as maximally correlated classical ones. However,

this is by no means a paradox: since the considered states

are pure and because of the single photon statistics, these

correlations eﬀectively contribute to the QFI, leading to

the possibility to attain the Heisenberg limit.

We now discuss in detail the type of states that satu-

rate the Heisenberg limit for (3) and provide a geometri-

cal intuitive picture of the observed scaling, pointing out

the analogies and diﬀerences with respect to the quadra-

ture phase space [10, 11]. These states, also discussed in

[15], are maximally entangled in the (local) variables ωi,

and their general mathematical expression (see Supple-

mentary Material B) for αi= 1 ∀ireads:

|ψ⟩=ZdΩf(Ω) 

Ω + ω0

1

Ω + ω0

2... 

Ω + ω0

n,(4)

where ω0

iare constants. The spectral function thus only

depends on one variable (Ω), and |ψ⟩has a (non-physical)

spectrum that is inﬁnitely localized in all collective vari-

ables except for Ω = Pn

i=1 ωi, the one associated to the

operator ˆ

Ω. This means that all the photons display a

collective behavior associated to a re-scaled de Broglie

wavelength λ=c/Ω [39]. As can be seen from the JSI

shown in Fig. 1(c) (for n= 2), these states are repre-

sented by diagonals, and the variance of each mode iis

the projection of these diagonals on the corresponding

frequency axis. This geometrically illustrates the role

of correlations in the scaling. This type of states with

diﬀerent spectral functions is currently produced in ex-

periments for n= 2 (see [25–27, 30, 43], for instance

and Sup. Mat. sections D and E). In addition, from

Fig. 1 (c) and (d) we see that entanglement, even if nec-

essary, is not suﬃcient to obtain sub-shot-noise scaling,

and that the symmetry of the spectral variance plays an

important role on the state’s metrological precision. In

the case where we have nphotons in nmodes, the same

type of geometrical picture can be built, and the states

saturating the Heisenberg limit are diagonals of a ndi-

mensional hypercube.

These scaling eﬀects can also be observed in the time

frequency phase space (TFPS). For this, we deﬁne the

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Quantummetrologyusingtime-frequencyasquantumcontinuousvariables:resources,subshot-noiseprecisionandphasespacerepresentationEloiDescamps1,2,NicolasFabre3,4,ArneKeller2,5,andP´erolaMilman2∗1D´epartementdePhysiquedel’EcoleNormaleSup´erieure-PSL,45rued’Ulm,75230,ParisCedex05,France2Universit´eParisCit´e...

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Quantum metrology using time-frequency as quantum continuous variables resources sub shot-noise precision and phase space representation Eloi Descamps12 Nicolas Fabre34 Arne Keller25 and P erola Milman2

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