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