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Ultrastrong coupling between light and matter

TL;DR: A review of ultrastrong coupling between light and matter can be found in this paper, where the authors discuss entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models.
Abstract: Light–matter coupling with strength comparable to the bare transition frequencies of the system is called ultrastrong. This Review surveys how experiments have realized ultrastrong coupling in the past decade, the new phenomena predicted in this regime and the applications it enables. Ultrastrong coupling between light and matter has, in the past decade, transitioned from a theoretical idea to an experimental reality. It is a new regime of quantum light–matter interaction, which goes beyond weak and strong coupling to make the coupling strength comparable to the transition frequencies in the system. The achievement of weak and strong coupling has led to increased control of quantum systems and to applications such as lasers, quantum sensing, and quantum information processing. Here we review the theory of quantum systems with ultrastrong coupling, discussing entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models. We also overview the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanical systems, that have now achieved ultrastrong coupling. We conclude by discussing the many potential applications that these achievements enable in physics and chemistry.

Summary (5 min read)

INTRODUCTION

  • Being α the natural dimensionless parameter emerging in a perturbative treatment of quantum electrodynamics, its small value allows to describe most of the quantum dynamics of the electromagnetic field by only taking into account first-order (absorption, emission) or second-order processes.
  • From this crucial observation sprung a whole field of research, today called cavity quantum electrodynamics [CQED, see Fig. 1(a)], which aims to exploit different kinds of photonic resonators to modulate the coupling of light with matter.
  • It took more than two decades after the observation of SC for the CQED community to begin investigating the possibility to access a regime with larger η in which higher-order processes, which would hybridize states with different number of excitations, become observable.
  • As the light-matter coupling strength reaches the USC regime, it starts to become possible to modify the very nature of the light and matter degrees of freedom.
  • The authors then review how USC has been reached in different experimental systems.

REGIMES AND MODELS FOR LIGHT-MATTER COUPLING

  • The definitions of the WC, SC, and USC regimes compare the light-matter coupling strength g to different parameters, as shown in Fig.
  • Ultrastrong coupling is not SC with larger couplings; its definition does not involve the value of losses but instead compares g to bare energies in the system.
  • The ratio η which defines USC instead determines whether perturbation theory can be used, and to what extent approximations can be made in models for the light-matter interaction.

Models

  • Some of the most fundamental models of light-matter interaction, the quantum Rabi, Dicke, and Hopfield models, are described in Box 1.
  • In the case of the quantum Rabi model (QRM), the RWA simplifies the Hamiltonian to the standard Jaynes–Cummings model (JCM) [34] (see Table B1.I).
  • In contrast to the QRM, the spectrum of the JCM is simple and well-known [35].
  • These processes, often studied in quantum optics, are analogous to those described by Hint for the QRM.
  • For the sake of completeness, the authors here mention three other regimes of light-matter coupling which have been investigated in the literature.

PROPERTIES OF ULTRASTRONGLY COUPLED SYSTEMS

  • As η increases, several properties of coupled lightmatter systems change drastically.
  • Only the quantum Rabi model (see Box 1) gives a correct picture of the energy levels for all η; various approximate methods can be used for small or large η.
  • The Jaynes–Cummings model correctly predicts the Rabi splitting (dressed states) between neighboring pairs of energy levels, but fails when the system enters the USC regime.

Ground-state properties

  • The difference between the USC and non-USC regimes is particularly striking for the ground state of the coupled light-matter system, as shown in Fig. 2(b)-(e).
  • As η grows the coupling makes it increasingly energetically favorable to have atomic and photonic excitations in the ground state.
  • As explained below, whether or not this phase transition actually occurs depends on whether an additional term, the diamagnetic term, should be included in the Hamiltonian.
  • (f) Same as (a), but for the Hopfield model with (solid black curves) and without (dashed grey curves) the A2 term.
  • Also in this case, the ground state contains virtual light and matter excitations.

The diamagnetic term

  • In the DSC regime, the diamagnetic term can act as a potential barrier for the photonic field, localizing it away from the dipoles, leading to an effective decoupling between the light and matter degrees of freedom [63, 64].
  • A similar decoupling can occur if qubit-qubit interactions are added to the Dicke model [65].
  • (B2.4) The ratio of the coefficients of the diamagnetic and dipolar parts of the light-matter interaction, D/g, is thus at least as large as the normalized coupling η, with the equality in Eq. (B2.4) if a single transition saturates the sum rule.
  • Those claims have attracted strong criticism [69, 70] and there is for the moment no consensus on this point.
  • The historically most important role played by the dia- 8 magnetic term in CQED is linked with a series of nogo theorems [69, 73–75] seemingly demonstrating that its presence makes a system stable against superradiant phase transitions.

EXPERIMENTAL SYSTEMS WITH ULTRASTRONG COUPLING

  • The first experimental demonstration of ultrastrong (η > 0.1) light-matter coupling was reported in 2009 [15].
  • As shown in Fig. 3 and explained below, USC has since been achieved in several different systems and at different wavelengths of light.

Intersubband polaritons

  • The USC regime was first predicted [14] and demonstrated [15] exploiting intersubband polaritons in microcavity-embedded doped quantum wells.
  • In these systems, nanoscopic layers of different semiconductors create a confining potential for carriers along the growth direction, which splits electronic bands into discrete parallel subbands.
  • This appealing simple model can be spoiled in more complex devices.
  • Moreover, as the quantum-well width increases and multiple electron transitions become available, the intuitive picture in terms of single-particle states is lost.
  • Intersubband polaritons remain a scientifically and technologically interesting system to study USC phenomenology thanks to the possibility to non-adiabatically modify the coupling strength [84], making it a promising platform for quantum vacuum-emission experiments [88, 89].

Superconducting circuits

  • Superconducting circuits are a powerful platform for exploring atomic physics and quantum optics, and for QIP, since their properties (resonance frequencies, coupling strength, etc.) can be designed and even tuned in situ [7].
  • This has already been widely exploited in the SC regime to, e.g., engineer quantum states and realize quantum gates.
  • In cavity QED, the coupling scales as α3/2.

Organic molecules

  • The USC regime has also been realized at room temperature at a variety of optical frequencies, coupling cavity photons (or, in one case, plasmons [112]) to Frenkel molecular excitons [16, 17, 113–118].
  • These systems consist of thin films of organic molecules with giant dipole moments (which make it possible to reach USC) sandwiched between metal mirrors [see Fig. 3(c)] and present an interesting combination of high coupling strengths and functional capacities.
  • These devices exhibit a room-temperature dispersionless angle-resolved electroluminescence with very narrow emission lines that can be exploited to realize innovative optoelectronic devices.

Optomechanics

  • The concept of ultrastrong light-matter interaction can be extended to optomechanics.
  • Recently, the USC limit was reached in a setup where plasmonic picocavities interacted with the vibrational degrees of freedom of individual molecules [83] [see Fig. 3(a)], achieving η = g/ωm = 0.3 (ωm is the mechanical frequency).
  • The increase in coupling strength here is due to the small mode volume of the picocavity, which circumvents the diffraction limit to confine optical light in a volume measured in cubic nanometers.
  • The USC limit has also been approached in circuitoptomechanical systems by using the nonlinearity of a Josephson-junction qubit to boost η [120].

VIRTUAL EXCITATIONS

  • As shown above in Fig. 2, a clear difference between USC systems and those with lower coupling strength is the presence of light and matter excitations in the ground state.
  • These two states contain the same number of excitations.
  • Textbook quantum-optical procedures to treat open quantum systems neglect the interaction between their constituent subsystems when describing their coupling to the environment [126] (see Fig. B3).

Matter

  • Reservoir Light Matter Light-Matter system Weak and strong coupling Arbitrary coupling strengths Figure B3.
  • The operator X̂+ can be interpreted as the operator describing the annihilation of physical photons in the interacting system.
  • The photons in the ground state of a system with an atom ultrastrongly coupled to a cavity are not only unable to leave the cavity; they are tightly bound to the atom [33].
  • Another way to extract virtual photons is to use additional atomic levels.
  • The virtual photons in the USC part of such a system can also be released through stimulated emission [151], which opens up interesting prospects for experimental studies of dressed states in the USC regime [137] [Fig. 4(c)].

SIMULATING ULTRASTRONG COUPLING

  • The experimental effort required to achieve this regime is still considerable.
  • An approach that circumvents these problems is quantum simulation [152, 153], where an easy-to-control quantum system is used to simulate the properties of the quantum model of interest.
  • Several proposals for quantum simulation of USC rely on driving some part of a strongly coupled system at two frequencies.
  • Then a rotating frame can be found with renormalized parameters, set by the drives, that can be in the USC regime [157–164] (drives can also be used to set effective parameters in other ways [165, 166]).
  • Starting from a bare η below 10−3, a simulated η of above 0.6 was achieved and the dynamics of population revivals were observed.

ULTRASTRONG COUPLING TO A CONTINUUM

  • This constitutes an interesting and, so far, less explored regime of the well-known spin-boson model [173].
  • After USC to a cavity was first realized a decade ago, several theory proposals showed that superconducting circuits was a suitable platform for USC to a continuum (in this case, an open waveguide on a chip) [174–176].
  • Similar to the cavity case, the ground state contains a cloud of virtual photons (in many modes) surrounding the atom [176, 178] and the atom transition frequency experiences a strong Lamb shift [171, 173, 179].
  • Instead, similar to the nonlinear-optics-like processes [22] discussed later in this review, the counter-rotating terms allow various frequency-conversion processes [171, 172, 182] [Fig. 6(c)].

CONNECTIONS TO OTHER MODELS

  • The quantum Rabi Hamiltonian (see Box 1) is closely related to a number of other fundamental models and emerging phenomena.
  • This approach enables finding topologically protected subspaces, which may help implementing decoherence-free algorithms for QIP.
  • Moreover, dark matter in cosmology may be explained through SUSY, so superconducting quantum circuits in the USC regime could in principle realize dark-matter simulations on a chip.
  • The QRM is also equivalent to a Rashba-Dresselhaus model, describing, e.g., a 2DEG with spin-orbit coupling of Rashba and Dresselhaus types interacting with a perpendicular, constant magnetic field [47].
  • This effect is analogous to the Higgs mechanism for the generation of masses of weak-force gauge bosons through gauge-symmetry breaking.

APPLICATIONS

  • The simplest answer is that USC enables more efficient interactions.
  • The coupling between a single photon and a single emitter results in significant nonlinearity, which has been used in electro-optical devices operating in the SC regime.
  • Increasing η from SC to USC results in better performance of such devices, e.g., faster con- 16 trol and response even for shorter lifetime of the device components.
  • Some quantum effects (including quantum gates) in specific realistic short-lifetime systems cannot be observed below USC.

Quantum information processing

  • Cavity- and circuit-QED systems in the USC regime are especially useful for quantum technologies like quantum metrology (e.g., novel high-resolution spectroscopy [193] utilizing smaller linewidths and improved signal-to-noise ratio) and QIP.
  • Many nonlinear-optics processes can be described in terms of higher-order perturbation theory involving virtual transitions, where the system passes from an initial state |i〉 to the final state |f〉 via a number of virtual transitions to intermediate states.
  • As discussed in the preceding section, superconducting quantum circuits with USC can also be used to simulate other fundamental models and testing their predictions, e.g., in quantum field theory and solid-state physics.
  • Possibilities and limitations of applying USC to change the electronic ground state of a molecular ensemble to control chemical reactions have also been investigated [19, 215].
  • Some of these works [213, 214] were based on the QRM as in the standard quantum-optical approach.

CONCLUSION AND OUTLOOK

  • As the authors described in this review, many intriguing physical effects have already been predicted in the USC regime of light-matter interaction.
  • Related experiments have been limited to increasing the light-matter coupling strength and verifying it by standard transmission measurements.
  • Now that USC has been reached in a broad range of systems, the authors believe that it is high time to explore experimentally the new interesting phenomena specific to USC and, finally, to find their useful applications.
  • A few decades ago, CQED in the WC and SC regimes was following the same route, which lead to important applications in modern quantum technologies.
  • The authors therefore believe that USC applications have the potential to make a profound impact.

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Ultrastrong coupling between light and matter
Anton Frisk Kockum,
1,
Adam Miranowicz,
2, 1
Simone De Liberato,
3, 1
Salvatore Savasta,
4, 1
and Franco Nori
1, 5,
1
Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
2
Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna´n, Poland
3
School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom
4
Dipartimento di Scienze Matematiche e Informatiche,
Scienze Fisiche e Scienze della Terra, Universit`a di Messina, I-98166 Messina, Italy
5
Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
(Dated: submitted May 7, revised July 30)
Ultrastrong coupling between light and matter has, in the past decade, transi-
tioned from theoretical idea to experimental reality. It is a new regime of quantum
light-matter interaction, going beyond weak and strong coupling to make the coupling
strength comparable to the transition frequencies in the system. The achievement of
weak and strong coupling has led to increased control of quantum systems and appli-
cations like lasers, quantum sensing, and quantum information processing. Here we
review the theory of quantum systems with ultrastrong coupling, which includes en-
tangled ground states with virtual excitations, new avenues for nonlinear optics, and
connections to several important physical models. We also review the multitude of
experimental setups, including superconducting circuits, organic molecules, semicon-
ductor polaritons, and optomechanics, that now have achieved ultrastrong coupling.
We then discuss the many potential applications that these achievements enable in
physics and chemistry.
INTRODUCTION
The intuitive description of the interaction between
light and matter as a series of elementary processes, in
which a photon is absorbed, emitted, or scattered by a
distribution of charges, essentially hinges on the small
value of the fine structure constant α '
1
137
. Being α the
natural dimensionless parameter emerging in a pertur-
bative treatment of quantum electrodynamics, its small
value allows to describe most of the quantum dynamics
of the electromagnetic field by only taking into account
first-order (absorption, emission) or second-order (scat-
tering) processes.
While the value of α is fixed by nature, Purcell discov-
ered in 1946 that the intensity of the interaction of an
emitter with light can be enhanced or suppressed by en-
gineering its electromagnetic environment [1]. From this
crucial observation sprung a whole field of research, to-
day called cavity quantum electrodynamics [CQED, see
Fig. 1(a)], which aims to exploit different kinds of pho-
tonic resonators to modulate the coupling of light with
matter.
The fundamental and applied importance of control-
ling the strength of light-matter coupling g led to the
development of resonators with ever higher quality fac-
tors. In 1983, Haroche and co-workers, using a collec-
tion of Rydberg atoms in a high-Q microwave cavity,
managed to achieve a coupling strength exceeding the
losses in the system [2]. In this strong-coupling [SC, see
Fig. 1(d)] regime, it is possible to observe an oscillatory
exchange of energy quanta between the matter and the
light, called vacuum Rabi oscillations, which takes place
at a rate given by g. In the weak-coupling [WC, see
Fig. 1(c)] regime, when g is smaller than the losses, the
energy is instead lost from the system before it can be
exchanged between the light and the matter.
The SC regime was soon also reached with single atoms
coherently interacting with a microwave cavity [3] and, a
few years later, with an optical cavity [4]. In 1992, the
SC regime was demonstrated using quasi-2D electronic
excitations (Wannier excitons) embedded in a semicon-
ductor optical microcavity [5]. In this case, the resulting
system eigenstates are called cavity-polaritons. Follow-
ing these pioneering experiments, CQED has been suc-
cessfully adapted and further developed using artificial
atoms, such as quantum dots [6] and superconducting
qubits (circuit QED) [7].
In a CQED setup, the dimensionless parameter quan-
tifying the interaction is the ratio between the coupling
strength g and the bare energy of the excitations. This
quantity, the normalized coupling η, is proportional to
a positive power of α and its value in the first observa-
tions of the SC regime was smaller than 10
6
for atoms
[4] and 10
3
for Wannier excitons in semiconductor mi-
crocavities [5]. Lowest-order perturbation theory is thus
perfectly adequate to describe those experiments. The
important difference with the WC regime is that, be-
ing the coupling larger than the spectral width of the
excitations, degenerate perturbation theory needs to be
applied.
It took more than two decades after the observa-
tion of SC for the CQED community to begin investi-
gating the possibility to access a regime with larger η
in which higher-order processes, which would hybridize
states with different number of excitations, become ob-
servable. Two main paths were identified to reach such
arXiv:1807.11636v1 [cond-mat.mes-hall] 31 Jul 2018

2
Figure 1. Regimes of light-matter interaction. (a) Sketch of a CQED system with a single two-level atom (qubit; the simplest
example of a matter excitation). The parameters determining the different interaction regimes are the resonance frequency
ω
c
of the cavity mode, the transition frequency ω
q
of the qubit, the coupling strength g, and the cavity and qubit loss rates:
κ and γ. (b) Sketch of an optical resonator coupled to many quantum emitters. The light-matter coupling strength can be
enhanced by increasing the number N of emitters interacting with the resonator. The resulting collective coupling strength
scales as g
N, where g is the coupling between the light and a single emitter. (c)-(f ) Four representative CQED experiments
illustrating different light-matter interaction regimes. (c) Weak coupling: Experimental demonstration of full control of
the spontaneous-emission dynamics of single quantum dots (QDs) by a photonic-crystal nanocavity [8]. The plot shows time-
resolved micro-photoluminescence intensities of InGaAs quantum dots on resonance with the cavity (QD1), off resonance (QD2),
and in bulk without any cavity. Compared to the case without any cavity, the QDs decay faster in the presence of a resonant
cavity (which enhances the density of states that the QDs can decay to) and slower in the presence of an off-resonant cavity
(which shields the QD from the environment). This is the Purcell effect [1]. (d) Strong coupling: Data from a pioneering
experiment [9] with Rydberg atoms coupled to a superconducting microwave Fabry-Perot resonator, displaying vacuum Rabi
oscillations. An atom in the excited state |ei enters an empty resonant cavity and the excitation is exchanged back and forth
between the atom and the resonator before it decays. P
e
denotes the probability of detecting the atom in |ei as a function of
the effective interaction time. (e) Ultrastrong coupling: Microwave spectroscopy of a system with a superconducting flux
qubit coupled to a coplanar-waveguide resonator [10]. The system displays a normalized coupling strength η = g
c
= 0.12.
The plot shows the cavity transmission as a function of probe frequency ω
probe
and flux offset, which tunes the qubit frequency.
The avoided level crossing indicates a coupling between states with different numbers of excitations (one state has a single
photon in the third resonator mode; the other state has one qubit excitation and one photon in the first resonator mode). Such
a coupling requires counter-rotating terms and is not reproduced by the Jaynes-Cummings approximation (see Box 1 and the
adjacent discussion). (f ) Deep strong coupling: Magneto-THz transmission measurements on a THz metamaterial coupled
to the cyclotron resonance of a two-dimensional electron gas [11]. The splitting 2Ω
R
between the lower polariton (LP) and
upper polariton (UP) levels that emerges as the cyclotron frequency ν
c
is tuned, is a measure of the coupling strength. In
this work, a record normalized coupling of η = 1.43 was reached. (g) Phenomena and applications associated with different
strengths of light-matter interaction. Figures reproduced with permission from: (c) Ref. [8]
c
2006, APS; (d) Ref. [9],
c
2001,
APS; (e) Ref. [10]
c
2010, NPG; and (f ) Ref. [11]
c
2017, ACS.

3
a regime. The first is to couple many dipoles to the
same cavity mode [Fig. 1(b)]. As correctly predicted by
the Dicke model [12], this leads to an enhanced cou-
pling which scales with the square root of the number
of dipoles. The second path is to use different degrees
of freedom, whose coupling is not bounded by the small
value of α [13].
In 2005, following the first path, it was predicted [14]
that this regime, which was named the ultrastrong-
coupling [USC, see Fig. 1(e)] regime, could be observed
in intersubband polaritons thanks to the large number of
electrons involved in the transitions between parallel sub-
bands in a quantum well. In 2009, the USC regime was
effectively observed for the first time in a microcavity-
embedded doped GaAs quantum well, with η = 0.11 [15].
Following this initial observation, the value of η = 0.1
has often been taken as a threshold for the USC regime.
While useful, it is important to note that the intensity of
higher-order processes depend continuously on η, and the
value of 0.1 is thus just a historical convention, without
any deeper physical meaning.
The second path has been followed in experiments with
superconducting circuits [7], in which ultrastrong cou-
pling was observed in 2010, with η = 0.12 [10]. In these
experiments, it becomes possible to explore USC of light
to a single two-level system, instead of a collective exci-
tation.
Following these experimental breakthroughs, the in-
terest in USC has blossomed, fostered by the vast phe-
nomenology which has been predicted to be observable
in this regime, including modifications of both intensity,
spectral features, and correlations of light-emitting de-
vices with USC [16, 17], as well as possible modifica-
tions of physical or chemical properties of systems ul-
trastrongly coupled to light [14, 1822]. This widespread
interest led not only to the observation of the USC regime
in a large number of physical implementations, but also
to a steady increase of the normalized coupling, whose
record is presently η = 1.43 [11].
The achievement of USC can be seen as the beginning
of a third chapter in the history of light-matter interac-
tion (see Fig. 1). Already the control of this interaction
afforded by the Purcell effect in the WC regime led to
several important applications, e.g., low-threshold solid-
state lasers [23] and efficient single- and entangled-photon
emitters [24, 25]. Cavity QED with individual atoms in
the SC regime made it possible to manipulate and control
quantum systems, enabling both tests of fundamental
physics [26] and applications [27] such as high-precision
measurements [28] and quantum information processing
(QIP) [29]. As the light-matter coupling strength reaches
the USC regime, it starts to become possible to mod-
ify the very nature of the light and matter degrees of
freedom. This opens new avenues for studying and engi-
neering non-perturbatively coupled light-matter systems,
which is likely to lead to novel applications.
In this review, we gather both theoretical insights and
experimental achievements in the field of USC. We begin
by discussing various regimes of light-matter coupling in
more detail, explaining their similarities and differences,
the models used to describe them, and their properties.
We then review how USC has been reached in different
experimental systems. This is followed by an overview of
defining characteristics of ultrastrong light-matter inter-
action such as virtual excitations and higher-order pro-
cesses, topics which affect how the interaction of an USC
system with an environment is treated. We also review
quantum simulations of the USC regime, USC to a con-
tinuum instead of a single resonator mode, and how ul-
trastrong light-matter coupling is intimately connected
to other areas of physics. We conclude with an outlook
for the field, including possible new applications and out-
standing challenges.
REGIMES AND MODELS FOR LIGHT-MATTER
COUPLING
The definitions of the WC, SC, and USC regimes com-
pare the light-matter coupling strength g to different pa-
rameters, as shown in Fig. 1. Whether the coupling is
strong or weak depends on whether g is larger or not
than the losses in the system. Ultrastrong coupling is
not SC with larger couplings; its definition does not in-
volve the value of losses but instead compares g to bare
energies in the system. It is thus possible for a system
to be in the USC regime without having SC if losses are
large [30]. The ratio η which defines USC instead deter-
mines whether perturbation theory can be used, and to
what extent approximations can be made in models for
the light-matter interaction.
Models
Some of the most fundamental models of light-matter
interaction, the quantum Rabi, Dicke, and Hopfield mod-
els, are described in Box 1. However, these models, even
though they do not approximate away some terms which
are often ignored at low light-matter coupling strengths,
still rely on some approximations, e.g., that the atoms are
two-level systems and that the light is in a single mode.
For ultrastrong light-matter coupling, these approxima-
tions may break down [3133].
As explained in Box 1, the light-matter interaction can
be divided into two parts. It is essential to note that, in
contrast to the terms in the first part (weighted by g
1
),
the terms in the second part (weighted by g
2
) do not con-
serve
ˆ
N
exc
, the total number of excitations in the system.
These latter terms are often referred to as anti-resonant
or counter-rotating. When the light and matter frequen-
cies are close to resonance, these terms can be omitted us-

4
ing the rotating-wave approximation (RWA). In the case
of the quantum Rabi model (QRM), the RWA simpli-
fies the Hamiltonian to the standard Jaynes–Cummings
model (JCM) [34] (see Table B1.I). The JCM, which has
been a workhorse of quantum optics in the WC and SC
regimes, conserves
ˆ
N
exc
ˆa
ˆa + ˆσ
+
ˆσ
(symbols defined
in Box 1) and restricts the resulting light-matter dynam-
ics to two-dimensional Hilbert subspaces [35]. However,
the RWA is not justified in the USC regime, when all
terms in the light-matter interaction come into play.
BOX 1 MODELS FOR LIGHT-MATTER COUPLING
The quantum Rabi model [36] (QRM) is a paradigm of quantum physics as one of the simplest and most fundamental
models of light-matter interaction. In the QRM, the interaction between a single-mode bosonic field (e.g., a cavity
mode with frequency ω
c
) and a generic two-level system (TLS, or a qubit, with level splitting ω
q
) is described by the
quantum Rabi Hamiltonian (~ = 1)
ˆ
H
Rabi
= ω
c
ˆa
ˆa +
1
2
ω
q
ˆσ
z
+
ˆ
H
int
, (B1.1)
ˆ
H
int
= g
ˆ
X ˆσ
x
= g
1
ˆaˆσ
+
+ ˆa
ˆσ
+ g
2
ˆaˆσ
+ ˆa
ˆσ
+
, (B1.2)
where ˆa (ˆa
) is the annihilation (creation) operator of the cavity mode, ˆσ
= |gihe| (ˆσ
+
= |eihg|) is the lowering
(raising) operator between the ground (|gi) and excited (|ei) states of a given TLS, ˆσ
x
= ˆσ
+ˆσ
+
and ˆσ
z
= |eihe|−|gihg|
are Pauli operators, and
ˆ
X = ˆa + ˆa
is the canonical position operator of the electric field of the cavity mode. For
simplicity, we ignore the vacuum-field energy in the free Hamiltonian in Eq. (B1.1). Moreover, g, g
1
, g
2
denote light-
matter coupling strengths. In the QRM, g = g
1
= g
2
, but this condition can be relaxed.
A Rabi-type model can also be applied to describe the interaction between two coupled harmonic oscillators. This
is an effective description of many systems, where the light is coupled not to a single atom or molecule, but to an
ensemble of these. For example, the standard fermion-boson QRM can be generalized to a purely bosonic multi-mode
Hopfield model [37], which describes the interaction between photons and collective excitations (e.g., plasmons or
phonons) of a matter system. A simplified two-mode version of the Hopfield model is
ˆ
H
Hopfield
= ω
c
ˆa
ˆa +
1
2
ω
b
ˆ
b
ˆ
b +
ˆ
H
0
int
+ H
dia
, (B1.3)
ˆ
H
0
int
= g
ˆ
X
ˆ
Y
0
= ig
1
ˆa
ˆ
b
ˆa
ˆ
b
+ ig
2
ˆa
ˆ
b
ˆa
ˆ
b
, (B1.4)
where
ˆ
b (
ˆ
b
) is the annihilation (creation) operator for collective excitations of a matter system of frequency ω
b
,
and
ˆ
Y
0
= i(
ˆ
b
ˆ
b) is the quadrature corresponding to the canonical momentum operator of the matter mode. The
Hamiltonian H
dia
describes the diamagnetic term (also referred to as the A
2
term), which is proportional to
ˆ
X
2
. This
term is also sometimes added to the standard QRM. The physical meaning of H
dia
, and the conditions under which
this term can be omitted, are explained in Box 2.
1 atom N atoms
no RWA Quantum Rabi model [36] Dicke model [12], Hopfield model [37]
with RWA Jaynes–Cummings model [34] Tavis–Cummings model [38]
Table B1.I. The models used to describe various regimes of light-matter interaction.
Although the QRM does not conserve
ˆ
N
exc
, it does
conserve the parity
ˆ
P = exp(
ˆ
N
exc
). A generalized
QRM, which is obtained by replacing the term g
ˆ
X ˆσ
x
by
g
ˆ
X(ˆσ
x
cos θ + ˆσ
z
sin θ) (with a parameter θ 6= 0, π) does
not conserve even
ˆ
P ; this Hamiltonian features in exper-
iments with superconducting circuits [10, 39]. Note that
the JCM conserves both
ˆ
N
exc
and
ˆ
P .
An analytical approach to find the spectrum of the
QRM was discovered only in 2011 [40] (and has since
been extended to multiple TLSs [41, 42] and multiple
bosonic modes [43]). But this solution is still based on
conjectures and numerical calculations of transcendental
(non-analytic) functions. A particular difficulty is to find
exceptional eigenvalues of
ˆ
H
Rabi
with no definite parity
(doubly degenerate) [40]. In contrast to the QRM, the
spectrum of the JCM is simple and well-known [35].
The QRM can be simulated with the standard JCM in
experiments using various tricks, as discussed later in this
review. Also, the coupling g can be enhanced in various
ways, e.g., by increasing the number of TLSs or cavity

5
fields, or by applying classical (single-photon) drives to
a single TLS or a cavity field. Recently, an exponential
enhancement of the coupling g was predicted with a two-
photon drive (i.e., squeezing) of the cavity field [44, 45].
A generalization of the QRM to N TLSs (which can
correspond to a single multi-level system or a large spin)
is known as the Dicke model [12]. Under the RWA, the
Dicke model reduces to the Tavis–Cummings model [38]
(see Table B1.I). Another generalized version of the quan-
tum Rabi model, with g
1
6= g
2
, enables studying super-
symmetry (SUSY), which exists if g
2
1
g
2
2
= ω
c
ω
q
(i.e.,
when the Bloch–Siegert shift [46] is zero) [47]. Note that
g
1
= g
2
if the Rabi model is derived from first principles.
In Box 1, we give the Hamiltonian for the Hopfield
model. In this case, the g
1
terms describe parametric
frequency conversion, which conserves the total number
of excitations
ˆ
N
0
exc
ˆa
ˆa +
ˆ
b
ˆ
b, while the g
2
terms de-
scribe parametric amplification, which does not conserve
ˆ
N
0
exc
. These processes, often studied in quantum optics,
are analogous to those described by H
int
for the QRM.
This simplified Hopfield model has been applied to de-
scribe experimental data of a two-dimensional electron
gas interacting with terahertz cavity photons in the USC
regime [4850].
Other regimes of light-matter coupling
For the sake of completeness, we here mention three
other regimes of light-matter coupling which have been
investigated in the literature. The first is the deep-
strong-coupling [DSC, see Fig. 1(f)] regime, in which η
becomes larger than one and higher-order perturbative
processes are not only observable, but can become dom-
inant. Theoretically investigated for the first time in
2010 [51], this regime was finally demonstrated exper-
imentally in 2017 using different physical implementa-
tions [11, 39].
The second, the very-strong-coupling (VSC) regime, is
achieved when g becomes comparable with the spacing
between the excited levels of the quantum emitter. In
this regime, although the number of excitations is con-
served and first-order perturbation gives an adequate de-
scription of the system, the coupling is large enough to
hybridize different excited states of the emitter, modify-
ing its properties. This regime was initially predicted by
Khurgin in 2001 [52], and observed in microcavity polari-
tons in 2017 [53].
The third is the multi-mode-strong-coupling (MMSC),
where g exceeds the free spectral range of the resonator
that the matter couples to. This regime has recently
been reached with superconducting qubits coupled to ei-
ther microwave photons in a long transmission-line res-
onator [54] or phonons in a surface-acoustic-wave res-
onator [55].
In the rest of this review, we will largely speak of USC,
with the implicit understanding that, according to the
value of η and other energy scales, the system under in-
vestigation could also be in the WC, SC, VSC, MMSC,
or DSC regimes.
PROPERTIES OF ULTRASTRONGLY COUPLED
SYSTEMS
As η increases, several properties of coupled light-
matter systems change drastically. In Fig. 2(a), we plot,
as a function of η, the lowest energy levels of a light-
matter system with a single atom on resonance with a
cavity mode. Only the quantum Rabi model (see Box 1)
gives a correct picture of the energy levels for all η; vari-
ous approximate methods can be used for small or large
η. The Jaynes–Cummings model correctly predicts the
Rabi splitting (dressed states) between neighboring pairs
of energy levels, but fails when the system enters the USC
regime.
Ground-state properties
The difference between the USC and non-USC regimes
is particularly striking for the ground state of the cou-
pled light-matter system, as shown in Fig. 2(b)-(e). For
small η, the lowest-energy state of the system is simply an
empty cavity with the atom in its ground state. However,
as η grows the coupling makes it increasingly energeti-
cally favorable to have atomic and photonic excitations
in the ground state. The exact nature of these excita-
tions is discussed later in this review, in the section on
virtual excitations. Here we only note that for very large
η, in the DSC regime, as shown in Fig. 2(e), the ground
state of the QRM consists of photonic Schr
¨
odinger-cat
states entangled with the atom and exhibits nonclassical
properties such as squeezing [18, 56].
As shown in the inset of Fig. 2(b), the mean num-
ber of photons in the ground state starts to increase
rapidly when η approaches and passes one. In the case
of many atoms coupled to the light, as described by the
Dicke model (see Box 1), it is predicted that a quantum
phase transition, known as the superradiant phase transi-
tion [5961] takes place at a critical value of η, separating
phases with and without photons in the ground state of
the system. However, as explained below, whether or not
this phase transition actually occurs depends on whether
an additional term, the diamagnetic term, should be in-
cluded in the Hamiltonian.
In Fig. 2(f), we plot the energy levels for the case where
the matter instead consists of many atoms and is de-
scribed as bosonic collective excitations in the Hopfield
model (see Box 1). The impact of the diamagnetic term
is clearly seen here. In Fig. 2(g), we plot the ground-
state population in the same way as in Fig. 2(b) with the

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TL;DR: In this article, the authors considered a radiating gas as a single quantum-mechanical system, and the energy levels corresponding to certain correlations between individual molecules were described, where spontaneous emission of radiation in a transition between two such levels leads to the emission of coherent radiation.
Abstract: By considering a radiating gas as a single quantum-mechanical system, energy levels corresponding to certain correlations between individual molecules are described. Spontaneous emission of radiation in a transition between two such levels leads to the emission of coherent radiation. The discussion is limited first to a gas of dimension small compared with a wavelength. Spontaneous radiation rates and natural line breadths are calculated. For a gas of large extent the effect of photon recoil momentum on coherence is calculated. The effect of a radiation pulse in exciting "super-radiant" states is discussed. The angular correlation between successive photons spontaneously emitted by a gas initially in thermal equilibrium is calculated.

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"Ultrastrong coupling between light ..." refers methods in this paper

  • ...with RWA Jaynes–Cummings model [34] Tavis–Cummings model [38]...

    [...]

  • ...of the quantum Rabi model (QRM), the RWA simplifies the Hamiltonian to the standard Jaynes–Cummings model (JCM) [34] (see Table B1....

    [...]

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Frequently Asked Questions (15)
Q1. What is the effect of the Purcell effect on the light-matter coupling?

As the light-matter coupling strength reaches the USC regime, it starts to become possible to modify the very nature of the light and matter degrees of freedom. 

Ultrastrong coupling modifies the physics of an atom in a waveguide dramatically compared to when the coupling is low enough for the RWA to be applicable. 

an exponential enhancement of the coupling g was predicted with a twophoton drive (i.e., squeezing) of the cavity field [44, 45]. 

The diamagnetic term can in fact be removed from the Hamiltonian by performing a Bogoliubov rotation in the space of the photon operator, at the cost of a renormalization of the cavity frequency: ωc → √ ω2c + 4ωcD. 

While the value of α is fixed by nature, Purcell discovered in 1946 that the intensity of the interaction of an emitter with light can be enhanced or suppressed by engineering its electromagnetic environment [1]. 

In 1983, Haroche and co-workers, using a collection of Rydberg atoms in a high-Q microwave cavity, managed to achieve a coupling strength exceeding the losses in the system [2]. 

The list of emerging applications of USC goes on much longer: QIP, quantum metrology, nonlinear optics, quantum optomechanics, quantum plasmonics, superconductivity, metamaterials, quantum field theory, quantum thermodynamics, and even chemistry QED and materials science. 

The first is the deepstrong-coupling [DSC, see Fig. 1(f)] regime, in which η becomes larger than one and higher-order perturbative processes are not only observable, but can become dominant. 

As discussed in the preceding section, superconducting quantum circuits with USC can also be used to simulate other fundamental models and testing their predictions, e.g., in quantum field theory and solid-state physics. 

if the cavity is ultrastrongly coupled to an electronic two-level system, yet another way to release photons from |E0〉 is through electroluminescence [21] [Fig. 4(d)]. 

In particular, a better control of chemical reactions can be realized via polaron decoupling, induced by SC or USC, of electronic and nuclear degrees of freedom in a molecular ensemble [20]. 

The third is the multi-mode-strong-coupling (MMSC), where g exceeds the free spectral range of the resonator that the matter couples to. 

It took more than two decades after the observation of SC for the CQED community to begin investigating the possibility to access a regime with larger η in which higher-order processes, which would hybridize states with different number of excitations, become observable. 

In 2005, following the first path, it was predicted [14] that this regime, which was named the ultrastrongcoupling [USC, see Fig. 1(e)] regime, could be observed in intersubband polaritons thanks to the large number of electrons involved in the transitions between parallel subbands in a quantum well. 

The second, the very-strong-coupling (VSC) regime, is achieved when g becomes comparable with the spacing between the excited levels of the quantum emitter.