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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 919 (2017) 218–237

www.elsevier.com/locate/nuclphysb

Exact solution for the inhomogeneous Dicke model

in the canonical ensemble: Thermodynamical limit and

ﬁnite-size corrections

W.V. Pogosov

a,b,c,∗

, D.S. Shapiro

a,c,d,e

, L.V. Bork

a,f

, A.I. Onishchenko

g,c,h

N.L. Dukhov All-Russia Research Institute of Automatics, Moscow, Russia

Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia

Moscow Institute of Physics and Technology, Dolgoprudny, Russia

V. A . Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia

National University of Science and Technology MISIS, Moscow, Russia

Institute for Theoretical and Experimental Physics, Moscow, Russia

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia

Received 28

September 2016; received in revised form 15 March 2017; accepted 25 March 2017

Available

online 28 March 2017

Editor: Hubert

Saleur

Abstract

consider an exactly solvable inhomogeneous Dicke model which describes an interaction between a

disordered ensemble of two-level systems with single mode boson ﬁeld. The existing method for evaluation

of Richardson–Gaudin equations in the thermodynamical limit is extended to the case of Bethe equations

in Dicke model. Using this extension, we present expressions both for the ground state and lowest excited

states energies as well as leading-order ﬁnite-size corrections to these quantities for an arbitrary distribution

of individual spin energies. We then evaluate these quantities for an equally-spaced distribution (constant

density of states). In particular, we study evolution of the spectral gap and other related quantities. We also

reveal regions on the phase diagram, where ﬁnite-size corrections are of particular importance.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

address: walter.pogosov@gmail.com (W.V. Pogosov).

http://dx.doi.org/10.1016/j.nuclphysb.2017.03.027

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

W.V. Pogosov et al. / Nuclear Physics B 919 (2017) 218–237 219

1. Introduction

Recent progress in engineering of artiﬁcial quantum systems for information technologies re-

newed an interest to Dicke (Tavis–Cummings) model [1] and its exact solution, already well

known for a long time [2–7]. Dicke model describes an interaction between a collection of

two-level systems and single mode radiation ﬁeld, while physical realizations range from su-

perconducting qubits coupled to micro

wave resonators to polaritons in quantum wells, see e.g.

Refs. [8,9] and references therein; furthermore, it can be also applied to Fermi–Bose condensates

near the Feshbach resonance [10].

The characteristic feature of macroscopic artiﬁcial quantum systems such as superconducting

qubits is a disorder in e

xcitation frequencies and inhomogeneous broadening of the density of

states. This feature is due to fundamental mechanisms: for example, an excitation energy of ﬂux

qubits depends exponentially on Josephson energies [11], which makes it extremely sensiti

to characteristics of nanometer-scale Josephson junctions. Inhomogeneous broadening appears

even in the case of microscopic two-level systems, such as NV-centers, where it is induced by

spatial ﬂuctuations of background magnetic moments [12]. In the case of NV-centers, the den-

sity of states is characterized by the q-Gaussian distribution [13]. M

oreover, there are prospect

to utilize the broadening for the construction of a multimodal quantum memory [14]. It is also

possible to engineer a density of states proﬁle by using, e.g., a so-called spectral hole burning

technique which allows to perform a signiﬁcant optimization of various characteristics of spin-

photon h

ybrid systems [13].

Inhomogeneous Dick

e model, which explicitly takes into account a disorder in excitation

energies, has been studied in Ref. [15] using a mean-ﬁeld treatment within functional-integral

representation of the partition function. This study revealed an existence of a rather rich phase

diagram. The interaction between boson and spin subsystems gives rise to a ﬁnite ga

p in the en-

ergy spectrum between the ﬁrst excited state and the ground state. It has an apparent similarity

with the superconducting gap in the Bardeen–Cooper–Schrieffer (BCS) theory of superconduc-

tivity. The mean-ﬁeld approximation for the inhomogeneous Dicke model also becomes exact

in the thermodynamical limit, as for the BCS pairing Hamiltonian. Ho

wever, this approximation

is expected to fail in the mesoscopic regime, which seems to be more relevant for near-future

technological applications with macroscopic artiﬁcial ‘atoms’, such as superconducting qubits.

Indeed, structures, which consist of tens or hundreds of superconducting qubits and show signa-

tures of global coherence, have been successfully f

abricated and explored very recently [8,16].

Such structures are often refereed to as superconducting metamaterials.

The mesoscopic re

gime of the Dicke model as well as the emergence of the macroscopic limit

can be properly described only using approaches based on a canonical ensemble, which takes

into account that the ‘particle’ number is ﬁxed. This circumstance makes it difﬁcult to apply

standard mean-ﬁeld methods. By ‘particle’ number one should understand the total number of

bosons and e

xcited two-level systems (spins), so they can be referred to as pseudo-particles. The

limiting validity of grand canonical description is well known in the case of pairing correlations

in ultrasmall metallic systems at low temperatures and in nuclei, for which usual mean-ﬁeld

approximation can gi

ve results inadequate even on a qualitative level [17]. For example, it pre-

dicts vanishing of superconducting correlations below certain mean interlevel distance, while

more advanced approaches show that they do not disappear. One of such approaches is to turn

to Richardson–Gaudin solution of BCS pairing Hamiltonian [2,18] via Bethe ansatz technique,

which wa

s utilized to evaluate various characteristics along the crossover from few-particle sys-

tems to the macroscopic regime, see, e.g., Refs. [19,20]. In particular, ﬁnite-size corrections can

220 W.V. Pogosov et al. / Nuclear Physics B 919 (2017) 218–237

be elaborated iteratively using the electrostatic analogy for Bethe equations in the thermodynam-

ical limit [21,22]. For an example of a recent application of this exact solution for the evaluation

of form factors, see Ref. [23], while the extension of this approach to other pairing models was

reported in Ref. [24].

Dicke model belongs to the same class of e

xactly solvable quantum models as Richardson

and Gaudin models [2,25–27]. To a certain extent, it can be viewed as Richardson model in

which interaction between spins is mediated by a bosonic degree of freedom. However, in con-

trast to the Richardson model, Dicke model supports arbitrarily lar

ge number of pseudo-particles

through this degree of freedom. This fact sometimes makes it not so straightforward to apply

ideas relevant for Richardson–Gaudin models to the Dicke model, see, e.g., recent developments

on the particle-hole duality [29–31]. The phase diagram of the inhomogeneous Dicke model in

the thermodynamical limit is much richer

, since it contains larger number of controlling parame-

ters which include pseudo-particle density, mean detuning between the spin and boson energies,

as well as spin-boson coupling energy [15].

The aim of the present paper is to construct a solution of Bethe equations for inhomogeneous

Dick

e model in the thermodynamical limit in the spirit of the approach developed by Richard-

son, as well as to evaluate, within the canonical ensemble, leading ﬁnite-size corrections both

to the ground state energy and low-energy e

xcited states. Although we basically follow the ap-

proach of Richardson [21], many aspects of the derivation are different, since Bethe equations for

Dicke model include a term of a new (divergent) type, which signiﬁcantly modiﬁes the multipole

expansion encoded into the approach and also alters a nonlinear equation for the electrostatic

ﬁeld.

We then apply the deri

ved formulas for the simplest case of an equally-spaced distribution of

spin energies. We reveal an existence of the rich phase diagram and ﬁnd regions of parameters,

where ﬁnite-size corrections are of a particular importance.

2. Preliminaries

We consider a Hamiltonian of the form

H =



2

+ωb

†

b + g



†

−

+bs

), (1)

where b

†

and b correspond to the boson degree of freedom:

[b, b

†

]=1, (2)

while s

and s

correspond to the paulion degrees of freedom and describe a set of L two-level

systems:

−

]=2s

, (3)

]=±s

. (4)

This Hamiltonian commutes with the operator of the total pseudo-particle number, i.e., the

number of bosons plus the number of excited two-level systems. Let us denote this number as M.

We then consider a limit L →∞, while M behaves in the same way, so that M/L is constant.

Let us also assume that the density of ener

gies 

grows at L →∞, whereas ω is independent

on L and g scales as 1/

√

L. For any ﬁxed M, there are different eigenstates of the Hamiltonian.

The lowest energy state is the ground state at given M, while others represent excited states.

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