Thermodynamics of ultrastrongly coupled light-matter
systems
Philipp Pilar
1
, Daniele De Bernardis
1
, and Peter Rabl
1
1
Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria
We study the thermodynamic properties of
a system of two-level dipoles that are coupled
ultrastrongly to a single cavity mode. By us-
ing exact numerical and approximate analyt-
ical methods, we evaluate the free energy of
this system at arbitrary interaction strengths
and discuss strong-coupling modifications of
derivative quantities such as the specific heat
or the electric susceptibility. From this analy-
sis we identify the lowest-order cavity-induced
corrections to those quantities in the collec-
tive ultrastrong coupling regime and show that
for even stronger interactions the presence of
a single cavity mode can strongly modify ex-
tensive thermodynamic quantities of a large
ensemble of dipoles. In this non-perturbative
coupling regime we also observe a significant
shift of the ferroelectric phase transition tem-
perature and a characteristic broadening and
collapse of the black-body spectrum of the cav-
ity mode. Apart from a purely fundamen-
tal interest, these general insights will be im-
portant for identifying potential applications
of ultrastrong-coupling effects, for example, in
the field of quantum chemistry or for realizing
quantum thermal machines.
1 Introduction
Undoubtedly, the interplay between statistical physics
and the theory of electromagnetic (EM) radiation
played a very important role in the history of mod-
ern physics. Discrepancies between the predicted and
the measured spectrum of black-body radiation led
to the birth of quantum mechanics. Based on purely
thermodynamic arguments, Einstein introduced his
A-coefficient and postulated the effect of spontaneous
emission, long before it was understood microscop-
ically. Investigations of photon-photon correlations
from thermal and coherent sources of light stood at
the beginning of the field of quantum optics, and so
on. In most of these and related examples the EM
field can be treated as an independent subsystem,
which thermalizes via weak interactions with the sur-
rounding matter. This assumption breaks down in the
so-called ultrastrong coupling (USC) regime [1, 2, 3],
where the interaction energy can be comparable to the
bare energy of the photons. Such conditions can be
reached in solid-state [4, 5, 6, 7, 8, 9, 10] and molecu-
lar cavity QED experiments [11, 12, 13, 14, 15], where
modifications of chemical reactions [16, 17] or phase
transitions [18] have been observed and interpreted
as vacuum-induced changes of thermodynamic poten-
tials [19]. Together with the ability to realize even
stronger couplings between artificial superconducting
atoms and microwave photons [20, 21, 22, 23, 24],
these observations have led to a growing interest [2, 3]
in the ground and thermal states of light-matter sys-
tems under conditions where the coupling between the
individual parts can no longer be neglected.
Since an exact theoretical treatment of light-matter
systems in the USC regime is in general not pos-
sible, one usually resorts to simplified descriptions,
for example, based on the Dicke [25, 26] or the
Hopfield [27] model. However, such reduced mod-
els often do not represent the complete energy of
the system [28, 29, 30, 31, 32, 33, 34, 35] or con-
tain gauge artefacts [33, 36, 37, 38, 39] that prevent
their applicability in the USC regime. More gener-
ally, while in weakly coupled cavity QED systems
the role of static dipole-dipole interactions can of-
ten be neglected or modelled independently of the
dynamical EM mode, this is no longer the case in
the USC regime [33, 40, 41, 42, 43, 44]. An inconsis-
tent treatment of static and dynamical fields can thus
very easily lead to wrong predictions or a misinter-
pretation of results. A prominent example in this re-
spect is the superradiant phase transition of the Dicke
model [45, 46, 47], which is often described as cavity-
induced, but which can be understood as a regular
ferroelectric instability in a system of strongly attrac-
tive dipoles [33, 41]. In the past, these and other
subtle issues have led to many controversies in this
field and prevented a detailed understanding of the
ground- and thermal states of USC light-matter sys-
tems so far.
In this paper we study the thermodynamics of cav-
ity and circuit QED systems within the framework of
the extended Dicke model (EDM) [32, 33]. Although
based on several simplifications, such as the two-level
and the single-mode approximation, this model re-
mains consistent with basic electrodynamics at arbi-
trary interaction strengths and distinguishes explic-
itly between static and dynamical electric fields. It
thus allows us to evaluate the free energy of the most
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 1
arXiv:2003.11556v5 [quant-ph] 22 Sep 2020
relevant dynamical degrees of freedom in cavity QED
and to study the thermal equilibrium states of the
combined system for a macroscopic number of N 1
dipoles and in various coupling regimes.
Our analysis shows, first of all, that in the collective
USC regime, where G = g
√
N is comparable to the
cavity frequency ω
c
, but where the coupling g between
the cavity and each individual dipole is still small, the
coupling-induced corrections to the free energy scales
as F
g
∼ ~g
2
N/ω
c
> 0. This very generic result, which
also holds for arbitrary dipolar systems, shows that
the coupling to a cavity mode leads to a positive shift
of the free energy and when taking the limit N → ∞
for a fixed value of G, the changes in the free energy
per particle, F
g
/N, vanish. Both findings contradict
the common intuition built upon the analysis of the
Dicke model, which predicts negative corrections to
the free energy [19] and that a large collective cou-
pling to a quantized field mode can induce substantial
modifications of material properties [45, 46, 47]. Our
results are, however, consistent with similar conclu-
sions obtained in studies about molecular properties
in the ground state of cavity QED systems [48, 49, 50],
and can be intuitively explained by a simple polari-
ton picture: In the collective USC regime the cavity
field only couples to a single collective dipole mode
while the other N − 1 orthogonal excitation modes
remain unaffected. Therefore, the presence of a single
cavity mode should not have a considerable impact
on the thermodynamics of a macroscopic ensemble of
dipoles.
Surprisingly, in the regime g ∼ ω
c
this intuition
is no longer true and we find that the coupling to
the cavity can indeed influence quantities such as the
electric susceptibility or the specific heat, or even the
phase transition temperature of a ferroelectric mate-
rial. This creates a highly unusual situation where
the addition of a single degree of freedom changes the
thermodynamics of a macroscopic system. Further,
we show that the different coupling regimes of cavity
QED result in very distinct features in the black-body
spectrum of the cavity. As g in increased, the spec-
Figure 1: Sketch of a cavity QED setup where an ensemble
of dipoles is coupled to the electric field of a lumped-element
LC resonator. The system is in thermal contact with a bath
of temperature T . The black-body spectrum of the cavity
mode, S
bb
(ω), can be measured through a weak capacitive
link to a cold transmission line. See text for more details.
trum evolves from the usual polariton doublet into
a broad and disordered set of lines and, finally, col-
lapses again to a single resonance. At the same time
we find that already at moderate coupling strengths,
the light-matter interaction can either enhance or sup-
press the total radiated power. Therefore, the anal-
ysis of the EDM already provides many conceptually
important predictions, which can serve as a basis for
more detailed investigations of thermal effects in real
and artificial light-matter systems.
The remainder of the paper is structured as follows:
In Sec. 2 and Sec. 3 we first introduce the EDM and
discuss some general properties of the free energy of
a cavity QED system in different coupling regimes.
In Sec. 4 we then analyze in more detail the cavity-
induced modifications for the cases of paraelectric and
ferroelectric ensembles of dipoles. Finally, in Sec. 5
we evaluate the black-body spectrum of the cavity
mode in different coupling regimes and we conclude
our work in Sec. 6. The appendices A-D contain addi-
tional details about different approximation methods
for the free energy and the derivation of the emission
spectrum.
2 Model
We consider a generic cavity QED setup, where N
two-level dipoles are coupled to a single electromag-
netic mode. However, since we are interested in both
thermal und USC effects, we can restrict our discus-
sion to cavity and circuit QED setups in the GHz
to THz regime, where these effects are experimen-
tally most relevant. In this case the confined electro-
magnetic field can be represented by the fundamental
mode of a lumped-element LC resonator [33, 51] with
capacitance C and inductance L (see Fig. 1). This
configuration also ensures that all higher EM exci-
tations are well separated in frequency and that the
electric field is approximately homogeneous across the
ensemble of dipoles. The dipoles themselves are mod-
eled as effective two-level systems with states |0i and
|1i. The two states are separated by an energy ~ω
0
and they are coupled via an electric transition dipole
moment µ to the electric field.
2.1 Hamiltonian
The Hamiltonian of the whole cavity QED system can
be written as
H
cQED
= H
em
+ H
dip
, (1)
where the two terms represent the energies of the EM
mode and the dipoles, respectively. We model the
bare dynamics of the dipoles by a spin Hamiltonian
of the form
H
dip
=
~ω
0
2
N
X
i=1
σ
i
z
+ ~
N
X
i,j=1
J
ij
4
σ
i
x
σ
j
x
, (2)
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 2
where the σ
i
k
are the usual Pauli operators for the
i-th dipole. The couplings J
ij
account for the effect
of static dipole-dipole interactions as well as possi-
ble other types of non-electromagnetic couplings be-
tween the two-level systems. For all of the explicit
calculations below we will consider the special case
of all-to-all interactions, J
ij
= J/N. In this limit
the spin Hamiltonian reduces to the Lipkin-Meshkov-
Glick (LMG) model [52]
H
dip
= ~ω
0
S
z
+
~J
N
S
2
x
≡ H
LMG
, (3)
where the S
k
= 1/2
P
i
σ
i
k
are collective spin opera-
tors. For the current purpose, this model is sufficient
to capture the qualitative features of non-interacting
(J = 0), ferroelectric (J < 0) and anti-ferroelectric
(J > 0) dipolar systems, while still being simple
enough to allow for exact numerical simulations for
moderate system sizes. However, we emphasize that
none of the general conclusions and theoretical ap-
proaches in this work depend on the assumption of
all-to-all interactions and can be extended to arbitrary
dipolar systems using more sophisticated numerical
techniques [53].
In the lumped-element limit, the energy of the EM
mode is given by
H
em
=
CV
2
2
+
Φ
2
2L
=
(Q + P/d)
2
2C
+
Φ
2
2L
, (4)
where V is the voltage difference across the capaci-
tor plates and Φ the magnetic flux. After the second
equality sign we have expressed the capacitive energy
in terms of the total charge Q, which is the variable
conjugate to Φ and obeys [Φ, Q] = i~ in the quan-
tized theory. For a sufficiently homogeneous field, the
charge is given by Q = CV −P/d, where P =
P
i
µσ
i
x
is the total polarization and d is the distance between
the capacitor plates. As usual we express Φ and Q in
terms of annihilation and creation operators a and a
†
as
Q = Q
0
(a + a
†
), Φ = iΦ
0
(a
†
− a), (5)
where Q
0
=
p
~/(2Z), Φ
0
=
p
~Z/2 and Z =
p
L/C
is the cavity impedance. Altogether, we obtain the
canonical cavity QED Hamiltonian [33]
H
cQED
= ~ω
c
a
†
a + ~g(a + a
†
)S
x
+
~g
2
ω
c
S
2
x
+ H
dip
,
(6)
where ω
c
= 1/
√
LC and g = µQ
0
/(~Cd) is the cou-
pling strength.
The form of H
cQED
given in Eq. (6) allows a clear
distinction between electrostatic and dynamical ef-
fects. Here the terms ∼ J
ij
σ
i
x
σ
j
x
represent the electro-
static energy of the ensemble with a fixed orientation
of the dipoles. This energy might be modified in the
presence of metallic cavity mirrors [33, 42, 53], but it
is independent of the frequency or the vacuum field
amplitude of the dynamical mode. The coupling to
the dynamical field is then proportional to g and in-
cludes the collective dipole-field coupling as well as
the so-called P
2
-term ∼ S
2
x
[32, 33, 51]. This dis-
tinction shows that the regular Dicke model, which
is recovered for J
ij
= −g
2
/ω
c
, describes a very spe-
cial case of a dipolar system with attractive all-to-all
dipole-dipole interactions. Although such a scenario
can be realized in circuit QED [54], the analysis of
this specific model does not provide much insights on
the behavior of more general cavity QED systems.
2.2 Observables
Apart from H
cQED
, which determines the dynamics
and the equilibrium states of the system, it is also
important to identify the relevant measurable observ-
ables. For the dipoles, quantities of interest are the
population imbalance, hS
z
i, or the polarization along
the cavity field, hS
x
i ∼ hPi, etc. Since the operator Q
for the total charge includes the induced charges from
the dipoles, its value is typically not directly measur-
able. Therefore, the relevant observables for the cav-
ity mode are the magnetic flux Φ and the voltage drop
V (see Fig. 1) and it is convenient to introduce the
displaced photon annihilation operator
A = a +
g
ω
c
S
x
, [A, A
†
] = 1. (7)
With the help of this definition we obtain [33, 55]
V = V
0
(A + A
†
), Φ = iΦ
0
(A
†
− A), (8)
where V
0
= Q
0
/C, and the total Hamiltonian can be
written in a compact form as
H
cQED
= ~ω
c
A
†
A + H
dip
. (9)
By comparing with Eq. (1), we see that the expecta-
tion value of hA
†
Ai can be interpreted as the energy
of the dynamical cavity mode in units of ~ω
c
. This is
in contrast to the conventional photon number ha
†
ai,
which depends on the chosen gauge [33, 55] and has
no simple interpretation in a strongly coupled cavity
QED system. Note, however, while A + A
†
, A
†
A, etc.
represent physical properties of the cavity mode only,
the operators A and A
†
do not commute with all the
dipole operators and on a formal level we must still
use a and a
†
to represent the independent cavity de-
gree of freedom.
While we focus here on a lumped-element realiza-
tion of the EM mode as an explicit example, the model
in Eq. (4) and all the results discussed in this work
can be readily applied to arbitrary cavity QED sys-
tems using the replacements [33, 41, 42, 51, 55, 56]
V → E, Q → D, Φ → B. (10)
Here E, D and B are the operators for the electric
field, the displacement field and the magnetic field,
respectively. For a detailed derivation and justifica-
tion of Hamiltonian (6) in dipolar cavity QED and
circuit QED settings, see Refs. [32, 33, 36].
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 3
3 The free energy in cavity QED
By assuming that the cavity QED system is weakly
coupled to a large reservoir of temperature T , the re-
sulting equilibrium state of the system is
ρ
th
=
1
Z
e
−βH
cQED
, (11)
where β = 1/(k
B
T ) and Z = Tr{e
−βH
cQED
} is the
partition function. In this case the central quantity
of interest is the free energy,
F = −k
B
T ln(Z) = F
0
c
+ F
0
dip
+ F
g
, (12)
which we divide into the free energies F
0
c
and F
0
dip
of
the decoupled subsystems and a remaining contribu-
tion F
g
. In the following discussion we will mainly
focus on the coupling-induced part of the free energy,
F
g
, which allows us to separate the influence of light-
matter interactions from the thermodynamic proper-
ties of the bare subsystems.
For small and moderately large ensembles the par-
tition function Z and the resulting free energy can be
evaluated by diagonalizing H
cQED
numerically. For
the LMG model, which conserves the total spin s,
this can be done for each spin sector separately and
we obtain
Z =
X
s
ζ
s,N
Z
s
. (13)
Here Z
s
is the partition function of H
cQED
con-
strained to a total spin s and [45]
ζ
s,N
=
N!(2s + 1)
(
N
2
− s)!(
N
2
+ s + 1)!
(14)
accounts for the multiplicity of the respective mul-
tiplet due to permutation symmetry. For small and
moderate temperatures, we use this approach to eval-
uate the exact free energy for systems with N ≈
1 − 100 dipoles. More details about the numerical
calculations are given in Appendix A.
3.1 Mean-field theory
In the analysis of the regular Dicke model [45, 46, 47]
with N 1, a frequently applied approximation for
evaluating the free energy is based on the mean-field
decoupling of the dipole-field interaction,
(a + a
†
)S
x
→ (α + α
∗
)S
x
+ (a + a
†
)Σ
x
−(α + α
∗
)Σ
x
,
(15)
where the expectation values α = hai and Σ
x
=
hS
x
i must be determined self-consistently. Under
this mean-field approximation Hamiltonian (6) can be
written as the sum of two independent parts,
H
MF
= ~ω
c
a
†
a + ~g(a + a
†
)Σ
x
+ H
MF
dip
(α, Σ
x
), (16)
where H
MF
dip
(α, Σ
x
) = H
dip
+ ~g(α + α
∗
)(S
x
− Σ
x
) +
~g
2
S
2
x
/ω
c
. The first two terms describe the energy
0.01 0.1
1
0.01 0.1
1
10
0.01 0.1
1
10
0
0.1
0.2
0.3
0.4
0.01 0.1
1 10
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
exact
1
2
3
10
-3
10.0 1.0 10.0 1.0
10
-3
0
2
4
-3
10
10.0 1.0
0
1
3
2
-3
10
10.0 1.0
0
1
3
2
10
Figure 2: Dependence of the coupling-induced part of the
free energy, F
g
, on the cavity-dipole coupling strength, g.
This dependence is shown in the individual plots for different
temperatures and dipole-dipole coupling strengths, J, and for
ω
c
= ω
0
. In each plot the exact numerical results for N = 20
dipoles are compared with approximate results obtained from
mean-field theory (F
MF
g
), second-order perturbation theory
(F
(2)
g
) and from a variational calculation (F
V
g
).
of a displaced oscillator, which is minimized for α =
−(g/ω
c
)Σ
x
. With the help of this relation between
α and Σ
x
, the total partition function in mean-field
approximation is given by
Z
MF
(Σ
x
) = Z
0
c
×
¯
Z
MF
dip
(Σ
x
). (17)
Here, the first factor is the partition function of the
bare cavity and
¯
Z
MF
dip
(Σ
x
) = Tr{exp[−β
¯
H
MF
dip
(Σ
x
)]} is
the partition function of an ensemble of dipoles with
effective Hamiltonian (which includes the constant en-
ergy shift from the displaced oscillator)
¯
H
MF
dip
(Σ
x
) = H
dip
+
~g
2
ω
c
(S
x
− Σ
x
)
2
. (18)
The free energy for the whole system in mean-field
approximation is then given by
F
MF
= min
Σ
x
{−k
B
T ln [Z
MF
(Σ
x
)]}, (19)
and F
MF
g
= F
MF
− F
0
c
− F
0
dip
are the correspond-
ing coupling-induced corrections. Note that the min-
imization of the free energy also ensures that the self-
consistency condition hS
x
i = Σ
x
is satisfied.
Equation (18) shows that cavity-induced correc-
tions to the thermodynamic properties of a dipolar
system are only affected by fluctuations, but not by
the mean orientation of the dipoles. Therefore, by ap-
plying a second mean-field decoupling for the dipoles
(see Appendix B), the effect of the cavity vanishes
completely and F
MF
g
= 0. We conclude that a full
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 4
mean-field treatment, as frequently employed to study
the ground states and thermal phases of the Dicke
model or of collective spin models [57, 58, 59], can-
not be used to analyze the thermodynamics of actual
cavity QED systems.
To take fluctuations of the dipoles into account we
can evaluate the partition function of the spin sys-
tem,
¯
Z
MF
dip
(Σ
x
), exactly. In this case we find that
in the paraelectric phase, where Σ
x
= 0, the cav-
ity induces a renormalization of the interaction term,
J → J + g
2
N/ω
c
. This renormalization becomes a
substantial modification of the dipolar system already
in the collective USC regime, g
√
N ∼
√
ω
0
ω
c
, and
could, at first sight, even prevent the ferroelectric in-
stability for J < 0. While such a shift of the phase
transition point is not observed when a proper min-
imization over Σ
x
is carried out, a comparison with
the exact free energy in Fig. 2 shows that mean-field
theory systematically overestimates the influence of
the cavity mode, in particular at higher temperatures.
This somewhat counterintuitive trend can be traced
back to the fact that the mean-field decoupling in
Eq. (15) neglects contributions which are second order
in ~g(a + a
†
)S
x
and scale approximately as
H
(2)
g
∼ −
~g
2
ω
c
S
2
x
. (20)
Therefore, the mean-field decoupling neglects an es-
sential contribution from the light-matter interaction
and the approximation becomes uncontrolled.
As shown in the examples in Fig. 2, at larger cou-
pling parameters g/ω
c
and low temperatures, the
mean-field predictions agree reasonably well with the
exact results. However, this agreement seems to
be accidental since at higher temperatures there are
again substantial deviations and the limit g/ω
c
1
is not reproduced correctly. Although not shown ex-
plicitly, a very similar trend is also found in the anti-
ferroelectric case, J > 0. In summary, these results
for small and large couplings indicate that also on a
qualitative level Hamiltonian
¯
H
MF
dip
(Σ
x
) does not cor-
rectly capture the influence of the cavity mode.
3.2 Collective USC regime
Many cavity QED experiments are operated in the
regime G = g
√
N . ω
c
and N 1, where the collec-
tive coupling G can become comparable to the photon
frequency, but the coupling of each individual dipole
to the cavity mode is still very small, g ω
c
. In this
regime, we can treat the dipole-field interaction,
H
g
= ~g(a + a
†
)S
x
+
~g
2
ω
c
S
2
x
, (21)
as a small perturbation and expand the free energy
in powers of g. As a result of this calculation, which
is detailed in Appendix C, we obtain the lowest-order
correction to the bare free energy. It can be written
in the form
F
(2)
g
= N
~g
2
4ω
c
f
g
. (22)
The dimensionless function f
g
∼ O(1) contains two
contributions, one arising from the average value of
the S
2
x
term and a second-order contribution from
the linear coupling term, ~g(a
†
+ a)S
x
. The result-
ing expression for f
g
still involves non-trivial correla-
tion functions of spin operators, which for interact-
ing dipoles must be evaluated numerically. For non-
interacting dipoles this calculation can be carried out
analytically and we obtain the explicit result
f
g
(J = 0) =
ω
2
0
− ω
0
ω
c
tanh
~ω
0
2k
B
T
coth
~ω
c
2k
B
T
(ω
2
0
− ω
2
c
)
.
(23)
In Fig. 2, this prediction from perturbation theory is
compared to the exact free energy and we find that
the cavity-induced corrections to the free energy are
very accurately reproduced by F
(2)
g
at low and high
temperatures, even for collective coupling strengths
as large as G ≈ ω
c
.
By taking the limit T → 0, Eq. (23) provides us
directly with the lowest-order correction to the ground
state energy of a cavity QED system [1],
E
(2)
0
= F
(2)
g
(T → 0, J = 0) = N
~g
2
4ω
c
ω
0
ω
0
+ ω
c
, (24)
which agrees with Hamiltonian perturbation theory.
In the opposite high-temperature limit we obtain
F
(2)
g
(T → ∞) ' N
~
3
g
2
ω
2
0
48ω
c
k
2
B
T
2
. (25)
Therefore, the cavity-induced corrections to the free
energy vanish quadratically with increasing tempera-
ture. Importantly, we find that for all temperatures
F
(2)
g
≥ 0, which is in stark contrast to the negative
correction terms obtained within the framework of the
regular Dicke model [19]. More generally, one can
show that also for an arbitrary system of interacting
two-level dipoles (see Appendix C)
0 ≤ f
g
<
4(∆S
x
)
2
N
, (26)
where (∆S
x
)
2
= hS
2
x
i
0
− hS
x
i
2
0
. For a regular dipolar
system away from a critical point, the spatial extent of
the individual correlations, hσ
i
x
σ
j
x
i−hσ
i
x
ihσ
j
x
i, is finite
and the collective fluctuations scale as (∆S
x
)
2
∼ N.
Therefore, under very generic conditions, by taking
the limit N → ∞ with G kept fixed, we obtain
lim
N→∞
F
(2)
g
N
= 0. (27)
This result confirms our basic intuition that the cou-
pling of many dipoles to a single mode should not af-
fect extensive thermodynamic properties. Note that
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 5
this conclusion does not necessarily hold for a stronger
scaling of fluctuations, i. e. (∆S
x
)
2
∼ N
2
. This scal-
ing can be found, for example, in the LMG model for
J < ω
0
when the symmetry of the ferroelectric phase
is not explicitly broken. However, even in this special
case our exact numerical calculations confirm that the
bound f
g
< 1 still holds.
3.3 Non-perturbative regime
The physics of cavity QED changes drastically once
g ∼ ω
c
and the light-matter interactions become non-
perturbative at the level of individual dipoles. To
analyze this regime, it is usually more convenient to
change to a polaron frame,
˜
H
cQED
= UH
cQED
U
†
, via
the unitary transformation U = e
g
ω
c
S
x
(a
†
−a)
[60, 61,
62]. In this frame the cavity QED Hamiltonian can
be written as [32, 33, 38, 63]
˜
H
cQED
= ~ω
c
a
†
a + H
dip
+ H
int
, (28)
where the interaction part now takes the form
H
int
= ~ω
0
US
z
U
†
− S
z
. (29)
An immediate benefit of the polaron representation
is that the interaction is proportional to ω
0
. This
shows that for ω
0
→ 0 the coupling to the dynamical
mode vanishes and we recover the electrostatic limit,
˜
H
cQED
(ω
0
→ 0) = ~ω
c
a
†
a +
P
i,j
~J
ij
4
σ
i
x
σ
j
x
.
For finite ω
0
the effects of H
int
are more involved.
For T = J = 0 it can be shown that up to second order
in H
int
and for g/ω
c
& 2, the low-energy behavior of
the dipolar system is well-described by the effective
spin Hamiltonian [32]
H
eff
= ~ω
0
e
−
g
2
2ω
2
c
− 1
S
z
+
~ω
2
0
ω
c
2g
2
S
2
x
−
~
S
2
.
(30)
This effective model captures two important signa-
tures of non-perturbative light-matter interactions,
which will be relevant for the discussions below.
Firstly, there is a strong suppression of the dipole
oscillation frequency when g/ω
c
& 1. Secondly, the
cavity mediates an all-to-all anti-ferroelectric coupling
∼ ω
2
0
.
In principle, we can again apply perturbation the-
ory to evaluate F
g
up to second order in H
int
and ex-
tend the results from Sec. 3.2 into the strong-coupling
regime. However, such an approach is only reliable
when ω
0
ω
c
and the resulting expressions are much
more involved. Therefore, this method is only briefly
summarized in Appendix C. As a less accurate, but
more intuitive approach we can use the variational
principle of Bogoliubov to derive an upper bound F
V
for the free energy,
F ≤ F
∗
+ h
˜
H
cQED
− H
∗
i
ρ
∗
≡ F
V
. (31)
Here ρ
∗
is the thermal state and F
∗
the corresponding
free energy for the trial Hamiltonian H
∗
. Based on
(a) (b)
10
0
10
-2
10
-4
0.1
1
10
3
0
0.1
0.2
exact
variational
0 1 2
Figure 3: (a) Plot of the zero-field susceptibility χ
z
(solid
lines) for different coupling parameters g/ω
c
. The dashed
lines indicate the predictions from the approximate formula
given in Eqs. (37)-(39). The x-markers show the results ob-
tained from the perturbation theory discussed in Appendix
C.3. (b) Dependence of the Curie constant α
C
(g) on the
dipole-field coupling strength. The exact numerical results
are in perfect agreement with the analytic scaling derived in
Eq. (38) from the variational free energy, F
V
. For all plots
N = 20, ω
c
= ω
0
and J = 0 have been assumed.
the discussion above we choose
H
∗
= ~ω
c
a
†
a + ~˜ω
0
S
z
+
~J
N
S
2
x
, (32)
which describes a non-interacting cavity QED sys-
tem, but with a variable frequency ˜ω
0
. By minimizing
F
V
with respect to ˜ω
0
for each g we obtain (see Ap-
pendix D)
˜ω
0
(g) = ω
0
e
−
g
2
2ω
2
c
(1+2N
th
)
, (33)
where N
th
= 1/(e
β~ω
c
−1). While from the compari-
son in Fig. 2 we see that overall F
V
g
= F
V
−F
0
c
−F
0
dip
does not reproduce the quantitative behavior of F
g
very accurately, we will see in the following that there
are still many cavity-induced modifications that can
be directly explained by this simple renormalization
of the dipole frequency.
4 Para- and ferroelectricity in the USC
regime
While the free energy contains all the relevant infor-
mation about the cavity QED system, we are usually
interested in derivative quantities such the suscepti-
bility, the specific heat, etc., or the existence of dif-
ferent phases and the transitions between them. To
understand in which way the coupling to a quantized
cavity mode can influence such quantities, we discuss
in this section three elementary examples.
4.1 USC modifications of the Curie law
As a first example it is instructive to consider the
simplest case of non-interacting dipoles, where
H
dip
= ~ω
0
S
z
. (34)
Accepted in Quantum 2020-09-15, click title to verify. Published under CC-BY 4.0. 6