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我们考虑一个完全可解的非均匀Dicke模型,该模型描述了具有单模玻色子场的两级系统的无序集合之间的相互作用。 在热力学极限中评估Richardson-Gaudin方程的现有方法已扩展到Dicke模型中Bethe方程的情况。 使用这种扩展,我们给出了基态和最低激发态能量的表达式,以及针对这些量的自旋能量的任意分布的前导有限大小校正。 然后,我们评估这些数量的等距分布(状态的恒定密度)。 特别是,我们研究了光谱间隙和其他相关量的演变。 我们还揭示了相位图上的区域,其中有限尺寸的校正尤为重要。
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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
a
N.L. Dukhov All-Russia Research Institute of Automatics, Moscow, Russia
b
Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia
c
Moscow Institute of Physics and Technology, Dolgoprudny, Russia
d
V. A . Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, Russia
e
National University of Science and Technology MISIS, Moscow, Russia
f
Institute for Theoretical and Experimental Physics, Moscow, Russia
g
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
h
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
We
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.
© 2017 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
3
.
*
Corresponding author.
E-mail
address: walter.pogosov@gmail.com (W.V. Pogosov).
http://dx.doi.org/10.1016/j.nuclphysb.2017.03.027
0550-3213/© 2017
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
3
.
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
ve
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 =
L
n
2
n
s
z
n
+ωb
†
b + g
L
n
(b
†
s
−
n
+bs
+
n
), (1)
where b
†
and b correspond to the boson degree of freedom:
[b, b
†
]=1, (2)
while s
z
n
and s
±
n
correspond to the paulion degrees of freedom and describe a set of L two-level
systems:
[s
+
n
,s
−
n
]=2s
z
n
, (3)
[s
z
n
,s
±
n
]=±s
±
n
. (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
n
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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