From classical to quantum loss of light coherence

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Pierre Lass`egues,1Mateus Antˆonio Fernandes Biscassi,1, 2 Martial Morisse,1Andr´e Cidrim,2, 3 Pablo Gabriel Santos

Dias,2Hodei Eneriz,1Raul Celistrino Teixeira,2Robin Kaiser,1Romain Bachelard,1, 2, ∗and Mathilde Hugbart1, †

1Universit´e Cˆote d’Azur, CNRS, INPHYNI, France

2Departamento de F´ısica, Universidade Federal de S˜ao Carlos,

Rodovia Washington Lu´ıs, km 235 - SP-310, 13565-905 S˜ao Carlos, SP, Brazil

3Department of Physics, Stockholm University, 10691 Stockholm, Sweden

(Dated: October 4, 2022)

Light is a precious tool to probe matter, as it captures microscopic and macroscopic information

on the system. We here report on the transition from a thermal (classical) to a spontaneous emission

(quantum) mechanism for the loss of light coherence from a macroscopic atomic cloud. The coher-

ence is probed by intensity-intensity correlation measurements realized on the light scattered by the

atomic sample, and the transition is explored by tuning the balance between thermal coherence loss

and spontaneous emission via the pump strength. Our results illustrate the potential of cold atom

setups to investigate the classical-to-quantum transition in macroscopic systems.

Introduction.—Quantum mechanics has brought a

completely new description of a physical system, intro-

ducing the possibility of “entanglement” between its dif-

ferent states. However, this diversity in possible states

comes at the expense of a dramatic increase in complexity

as one aims at an exhaustive description of the system.

To compensate for the exponential growth of the associ-

ated Hilbert space with the number of constituents, one

may derive an eﬀective dynamics for a selected set of de-

grees of freedom, tracing over the less relevant ones. This

loss of information leads to the notion of decoherence [1],

and the partial knowledge of the system state allows for

an accurate prediction of the dynamics over a ﬁnite time

only. From a fundamental point of view, decoherence ac-

tually questions the notions of measurement, collapse of

the wavefunction [2, 3], and hidden variables in quantum

mechanics [4]. Looking toward quantum technologies,

decoherence is a major obstacle to the preservation of

quantum information, but it is also a central mechanism

behind the quantum random number generation [5].

Let us consider the prototypical example of sponta-

neous emission (with rate Γ) for a quantum emitter:

It arises from tracing over the electromagnetic modes

in which the particle excitation may be emitted. Yet,

while half of the spontaneous emission rate can be ex-

plained by the radiation reaction with a classical ap-

proach, the other half was shown to stem from the quan-

tum ﬂuctuations of the modes: “Die spontane Emis-

sion ist somit eine durch die Nullpunktsschwingungen des

leeren Raumes erzwungene Emission eines Lichtquants.”,

as wrote Weisskopf [6][7]. These zero-point ﬂuctuations

do not result from a set of unknown (or “hidden”) vari-

ables, as in classical statistical physics when microscopic

details are ignored, but rather from Heisenberg’s uncer-

tainty principle [8].

In the case of a quantum emitter, the decoherence

mechanism incarnated by spontaneous emission leaves its

mark on the radiated light, since signatures of the quan-

tum nature of the emitter, such as photon antibunch-

ing [9, 10] or Rabi oscillations [11, 12], are visible on

a time scale 1/Γ. When moving to many emitters, the

nature of the mechanism at the origin of the light coher-

ence loss can be more ambiguous, as one meets the fron-

tier between quantum physics and statistical ensembles.

For example, photon antibunching is observed in large

systems under speciﬁc conditions such as phase match-

ing [13] or conﬁnement of light in ﬁbers [14]; spontaneous

emission then sets the time scale of the light coherence.

Diﬀerently, the reduction of this coherence time due to

the particles’ motion can be understood from a classi-

cal perspective: a macroscopic information (the velocity

distribution) is extracted from the reduction of the light

coherence, without the knowledge of the microscopic tra-

jectories, and this eﬀect is at the core of the diﬀusive wave

spectroscopy technique [15–21]. This illustrates the va-

riety of phenomena which compete to set a limit to the

light coherence.

In this work, we report on the transition from a classi-

cal to a quantum mechanism for the loss of coherence in

the light scattered by a macroscopic atomic cloud of neu-

tral atoms. In the weak drive regime, the atoms scatter

light elastically, yet the ﬁnite cloud temperature, through

the Doppler eﬀect, induces a broadening of the spectrum:

the coherence loss is here a macroscopic manifestation of

the microscopic dynamics (see Fig. 1). Diﬀerently, spon-

taneous emission dominates the scattering from strongly

driven atoms, and the light coherence is then limited by

the transition rate, Γ. In this regime, the emission of

each atom is spectrally broadened [22], and the reduction

of coherence results from zero-point ﬂuctuations rather

than from unknown microscopic details [23]. Experimen-

tally, we perform intensity-intensity correlation measure-

ments to characterize the (loss of) light coherence: the as-

sociated intensity ﬂuctuations are shown to also arise, de-

pending on the regime explored, from either the Doppler

eﬀect or spontaneous emission. Alternatively, we moni-

tor ﬁeld-ﬁeld correlations, which conﬁrm that the electric

ﬁeld coherence suﬀers from the same mechanism as the

arXiv:2210.01003v1 [physics.atom-ph] 3 Oct 2022

10 210 1100101

Saturation parameter s

200

100

c(ns)

Coherence time

Spontaneous

Emission

Thermal

FIG. 1. Coherence time of the light radiated by a macroscopic

cloud, as a function of the saturation parameter (dashed: the-

ory; dots: experiment with error bars, see main text). Left

inset: For a weak monochromatic drive on the atoms, the

elastically scattered light acquires a frequency shift due to the

Doppler eﬀect. The ﬁnite temperature of the cloud broadens

the light spectrum, setting the coherence time. Right inset:

Strongly driven atoms each emits a broadened spectrum (Mol-

low triplet).

intensity (Siegert relation), both in the classical and in

the quantum regime of coherence loss.

Thermal coherence loss versus spontaneous

emission.—Our experimental setup, see Fig. 2(a)

and [24], allows us to measure simultaneously the ﬁrst-

order (ﬁeld-ﬁeld) and second-order (intensity-intensity)

correlation functions of the scattered light [25]:

g(1)(τ) = hˆ

E−(t)ˆ

E+(t+τ)i

hˆ

E−(t)ˆ

E+(t)i,(1)

g(2)(τ) = hˆ

E−(t)ˆ

E−(t+τ)ˆ

E+(t+τ)ˆ

E+(t)i

hˆ

E−(t)ˆ

E+(t)i2.(2)

E+refers to the positive frequency component of the

electric ﬁeld in the measured mode, h.ieither to the av-

erage over time or to the expectation value [26], and we

here consider the steady-state limit, t→ ∞. In all cases,

we also average over conﬁgurations.

Two examples of g(2)(τ) correlation functions taken

from the experiment are presented in Fig. 2(b). For a

weak pump, with saturation parameter s1, most light

is scattered elastically by each atom in its own (moving)

frame. In the laboratory frame, this motion translates

into a change in frequency of the light (Doppler eﬀect).

The interference between the ﬁelds scattered by the dis-

ordered ensemble of moving atoms leads to a Doppler-

broadened spectrum, with a coherence time τT

c∝1/√T

(Tthe temperature). Our temperature-induced intensity

coherence time τT

c≈260 ns, extrapolated for s= 0 from

measurements at several low-svalues, corresponds to a

temperature of about 200 µK and a cloud with an optical

depth of 6 [21]. This statistical analysis is the basis, for

example, of diﬀusive wave spectroscopy technique [15–

20], and it is a purely classical mechanism of coherence

loss.

101102103

(ns)

1.0

1.2

1.4

1.6

1.8

s≈24

s≈0.004

Intensity correlation g(2)( )

(b)

Rabi osc.

(a)

-10

Spectral

(c)

FIG. 2. (a) Schematic setup of the experiment, see [24] for

details. (b) Temporal evolution of the second-order coherence

g(2)(τ), in a low-saturation temperature-dominated regime

[s= (4.0±0.8) ×10−3] and in a high-saturation spontaneous

emission-dominated regime (s= 24 ±5). Dashed lines: ﬁts

of the decay capturing the coherence time (τcis computed as

the half-width at half-maximum); the solid thicker line is a

ﬁt containing the (coherent) Rabi oscillation of the saturated

regime. (c) Observation of the Siegert relation, which writes

˜g(2)(ω) = δ(ω) + ˜g(1)(ω)~˜g(1)∗(ω) in the frequency space (~

the convolution), for the large-sregime, s≈60. The elastic

component is broadened by the temperature; the spectra are

normalized to one.

To enter the regime where coherence loss is based on

quantum randomness, we use a strong resonant pump

(s1). Each atom then presents a spectrally broadened

ﬂuorescence, the so-called Mollow triplet [22, 27], which

is characterized by a peak at resonance and two side-

bands shifted by ±Ω from the carrier. The beating be-

tween these peaks of inelastic scattering is manifested as

Rabi oscillations in the g(2)(τ), whereas the peak widths

(∆ω≈Γ) set the coherence time of this spontaneously

emitted light to τSE

c≈16 ns, see Fig. 2(b). This broad-

ening mechanism does not rely on the microscopic state

of the system (and macroscopically captured by temper-

ature, for example) or any “hidden variable”, but rather

on zero-point ﬂuctuations [23].

In our experiment, the transition between the classical

and quantum regimes occurs when the ratio of sponta-

neously emitted to elastically scattered power PSE/PESL

is inverted. This ratio here corresponds to the saturation

parameter s[22, 28], tuned via the pump Rabi frequency

Ω. In Fig. 1, we present the evolution of the coherence

time of the light when crossing from the classical to the

quantum regime of coherence loss. The coherence time

was extracted from the second-order correlation g(2)(τ)

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FromclassicaltoquantumlossoflightcoherencePierreLassegues,1MateusAnt^onioFernandesBiscassi,1,2MartialMorisse,1AndreCidrim,2,3PabloGabrielSantosDias,2HodeiEneriz,1RaulCelistrinoTeixeira,2RobinKaiser,1RomainBachelard,1,2,andMathildeHugbart1,y1UniversiteC^oted'Azur,CNRS,INPHYNI,France2Departamentod...

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