Two-photon correlations in detuned resonance uorescence Eduardo Zubizarreta Casalengua

2025-05-06 0 0 1014.55KB 17 页 10玖币
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Two-photon correlations in detuned resonance
fluorescence
Eduardo Zubizarreta Casalengua
Departamento de F´ısica Te´orica de la Materia Condensada, Universidad Aut´onoma
de Madrid, 28049 Madrid, Spain
Faculty of Science and Engineering, University of Wolverhampton, Wulfruna St,
Wolverhampton WV1 1LY, UK
Elena del Valle
Departamento de F´ısica Te´orica de la Materia Condensada, Universidad Aut´onoma
de Madrid, 28049 Madrid, Spain
Institute for Advanced Study, Technische Universit¨at M¨unchen, 85748 Garching,
Germany
Fabrice P. Laussy
Faculty of Science and Engineering, University of Wolverhampton, Wulfruna St,
Wolverhampton WV1 1LY, UK
Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia
E-mail: fabrice.laussy@gmail.com
19 October 2022
Abstract. We discuss two-photon correlations from the side peaks that are formed
when a two-level system emitter is driven coherently, with a detuning between the
driving source and the emitter (quasi-resonance fluorescence). We do so in the context
of the theories of frequency-resolved photon correlations and homodyning, showing
that their combination leads to a neat picture compatible with perturbative two-
photon scattering that was popular in the early days of quantum electrodynamics.
This should help to control, enhance and open new regimes of multiphoton emission.
We also propose a way to evidence the quantum coherent nature of the process from
photoluminescence only, through the observation of a collapse of the symmetry of the
lineshape accompanied by a surge of its intensity of emission. We discuss several of
our results in the light of recent experimental works.
1. Introduction: historical developments of resonance fluorescence
Resonance fluorescence is the “drosophila” of quantum optics. It is the simplest yet rich
enough problem to capture many of the key considerations on light-matter interaction,
from quantization of the light field up to the riddle of measurements and observations
in quantum mechanics. It consists of driving optically a two-level system (e.g., an
arXiv:2210.03733v2 [quant-ph] 18 Oct 2022
2
atomic transition, a spin, a semiconductor exciton, a superconducting qubit, etc.) with
a coherent wave that has the same or a close frequency than that of the spontaneous
emission of the emitter. The platform is both of great fundamental interest for the
understanding of basic aspects of quantum theory as well as from a technological
perspective for its prospects as a quantum emitter, not only as a single-photon source
but also in an unsuspected regime of multiphoton emission.
The multiphoton problematic turns out to have been central to theoretical
modelling since the early days. As a basic problem of light-matter interaction, the
origin of resonance fluorescence goes back to the dawn of quantum electrodynamics
(which itself can be dated with Dirac [1]). Pioneering contributions include those of
Weisskopf [2, 3], for scattering off the ground and excited state of an atom, respectively.
A major and recurrent work still of actuality, in the low-driving regime, is that of
Heitler [4] who reported his analysis directly in the 3rd edition of his textbook “the
Quantum Theory of Radiation”, in a chapter (absent in previous editions) titled
“resonance fluorescence” (§20), where he shows that the lineshape of radiation is
provided by the driving source itself as opposed to the natural lineshape of the emitter.
His analysis follows in essence from the conservation of energy δ(ω−ωL) of the
scattering process so that each photon from the source gets scattered at the energy
with which it impinged on the atom, whence the result. The process is actually not as
trivial as it looks, with Heitler already observing that the radiation occurs “as if two
independent processes, an absorption and a subsequent emission, took place”, with the
atom “remembering” (his term) “before the emission which quantum it has absorbed”.
Seen in this way, it is less obvious why the spontaneous emission character of the emitter
plays no role. It also brings forward that a two-photon process is involved in this
scattering. In fact, as we discuss further below, it turns out to be of central importance
for the photon statistics of resonance fluorescence in this low-driving regime [5]: the
δ-shaped scattered light itself is uncorrelated and becomes antibunched only if also
detecting the weak—but at the two-photon level, essential—incoherent part of the
spectrum, that is indeed spread spectrally and originating from multiphoton events.
This peak is however very small in intensity as compared to the Rayleigh peak. This led
to some confusion in the literature [6, 7] that we hope to have clarified [8] (see also [9]):
although the incoherent peak vanishes at low intensities in one-photon observables, its
contribution rules the photon-statistics (a two-photon observable).
Multiphotons are even more prominent at higher driving. In a nonlinear quantum-
optics framework, several degenerate photons from the driving source (which we shall
from now on refer to as the “laser”) can be redistributed by the emitter at different
energies so as to produce a more complex spectral shape, known today to be a triplet
with ratio of peak heights 1:3:1 and with a splitting given by the laser intensity.
The problem was initially regarded as that of the competition between spontaneous
and stimulated emission, with a feeling shared by many theorists of the time that
spontaneous emission required quantization of the field for a correct treatment. The
exact nature of this spectral shape was the topic of some controvery, in particular it
3
took part in the debates initiated by Jaynes according to whom the light field should
not be quantized and his neoclassical theory (relying on a nonlinear feedback from the
radiation field back to the emitter) should be used instead. The neoclassicists were also
experts in solving the quantized version of a problem to provide what they assumed
were the wrong QED predictions, which is how, famously, the Jaynes–Cummings
model [10] arose. In this framework, the quantized version of resonance fluorescence
by Stroud [11] (part of Jaynes’ team) but at the one-photon level, led to incorrect
results, such as a 1:2:1 ratio of the peaks, in contrast to a semiclassical treatment by
Mollow [12] which was not quantizing the light field but obtained, for the first time,
the correct lineshape. For this reason, this characteristic result, that was originally
referred to as the AC stark effect, became known as the “Mollow triplet” (it seems that
Zoller [13] is the first to have used this denomination). Stroud et al. mention in their
conclusions that their analysis is “incomplete in one important aspect”: the truncation
to one-photon emission. While they recognize that “in the real physical case there
will be a cascade emitting many spontaneous-emission photons”, they believed that
antibunching would make such successive emissions from a quantized model justifying
their approximation. Further support for Stroud et al.’s view came from Smithers and
Freedhoff [14] who claimed to have included multiphoton effects and yet still arrive
at the same (incorrect) result as Stroud et al., but this was disputed by Carmichael
and Walls [15] who observed that in their treatment, “Smithers and Freedhoff have not
managed to include true photon cascades but have simply followed a series of sequential
one-photon emissions”. Convincingly, by truncating their quantized version to single-
photon transitions, Carmichael and Walls showed how they downgraded Mollow’s
spectrum to one with the same attributes as Stroud’s. Mollow’s result, it must be
emphasized, although not quantizing the light-field, is not part of the neoclassical theory,
which treats spontaneous emission as a continuous process as opposed to quantum
jumps, leading to still further departures between the various models. The reason why
Mollow got the correct result is interesting: ironically, it turns out that the semiclassical
model is equivalent to a multiphoton quantized model, and that multiphoton effects
are responsible for the lineshape, although this is a single-photon observable. This has
been recognized and commented by various people at the time but the most insightful
discussion seems to be that of Mollow himself, in his 1975 follow-up paper [16]. He
carried on such a fully-quantum treatment, including multiphoton contributions of all
orders, and showed that the c-number description of the laser does not spoil the fully-
quantum nature of the problem, as long as multiphoton effects are included. These
correspond to the back-reaction in Jaynes’ neoclassical theory and to what a modern
treatment would qualify as virtual photons, i.e., the atom re-absorbing photons that
it has just emitted. In this context, the problem can be understood as a scattering
one. In the words of Mollow [16] “ the individual multiphoton scattering processes
[. . . ] are concealed from view, with only their accumulated effect exhibited”. We
will come back to this important observation later on. Another key contribution to
that approach of resonance fluorescence comes from Cohen–Tannoudji [17] and his co-
4
workers [18, 19, 20, 21], who provided both the so-called dressed-atom and a perturbative
scattering pictures. From this viewpoint, spontaneous emission is not deemed central
but is relegated to a secondary plane. Instead, dressing the atom is considered first,
yielding new eigenstates for the system with an exact (all-order) treatment of the light-
matter coupling. Then spontaneous emission is brought back to make the dressed atom
cascade down its energy diagram and in this process replacing photons from the laser
by fluorescence photons from the atom. This remains the most picturesque way to
understand the spectral shape of the Mollow triplet and can also account for a lot,
although not all, the phenomenology of correlations between the peaks.
We now turn to the detuned resonance fluorescence, i.e., when the driving laser is
close to but not right at the energy of the two-level system. In the earlier treatments,
such as from Heitler, exact resonance implied divergence and one of Heitler’s inputs
was precisely to damp the system [22] so as to arrive to a physical response for exact
resonance. In a modern quantum-optical, master equation approach, resonance is
actually simpler while off-resonance comes with additional subtleties but also with
several advantages. Not least is the fact that detuning helps the splitting of what
always remains a triplet. In fact, the first neatly resolved Mollow triplet was out of
resonance [23] and if one would stick to resonance, it would then be apparently the
improved setup of Walthers that has reported the first resonant Mollow triplet, albeit
in a conference proceedings [24] (the first report of an even better triplet in a leading
journal came however only a few months later [25]). Detuning also weakens the efficiency
of the coupling and what determines whether one is in the Heitler (low-driving) or
Mollow (high-driving) regime in this context is an interesting question that we address
elsewhere [26]. As was already described by Mollow in his magnum opus [12], when the
2LS is detuned from the laser, one gets at low driving a doublet with a peak centered on
the atom and the other peak shifted by twice the detuning, with the Rayleigh-scattered
laser sitting in between, that is to say, one always has a triplet in the non-detuned case.
This is shown in Fig. 1(e). Here it must be appreciated that the two side peaks are
vanishing with Ωσ→0 as compared to the coherent peak in the center, with a ratio
8Ω2
σ/(γ2
σ+4∆2
σ)) for their respective intensities, i.e., most of the emission comes from the
central peak, just as the case of resonance where the Lorentzian foothill is dwarfed by the
Rayleigh peak. Here too, however, two-photon observables, such as photon statistics,
are ruled by the interplay of the coherent and incoherent emission [27], regardless of their
relative intensities. The only, but striking, difference is that this central incoherent peak
has now split in two. With increasing driving, a central incoherent peak grows at the
laser position, becoming of identical height with the side peaks when Ωσ≈∆σ/√2
(exactly so in the limit γσ∆σ,Ωσ) and for higher-still driving converging towards a
resonant Mollow triplet, since detuning now becomes negligible as compared to driving.
There is therefore a smooth transition between the various cases [26].
A triplet structure comes with an obvious opportunity to correlate photons from
the various peaks. This was highlighted by Cohen–Tannoudji and Reynaud [19] and
implemented by Aspect et al. [28]. At low-enough driving but with detuning to
摘要：

Two-photoncorrelationsindetunedresonanceuorescenceEduardoZubizarretaCasalenguaDepartamentodeFsicaTeoricadelaMateriaCondensada,UniversidadAutonomadeMadrid,28049Madrid,SpainFacultyofScienceandEngineering,UniversityofWolverhampton,WulfrunaSt,WolverhamptonWV11LY,UKElenadelValleDepartamentodeFsicaT...

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