Superfluid-Mott-insulator phase transition of light in a two-mode cavity array with ultrastrong coupling

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by  Jingtao Fan

Superfluid-Mott-insulator phase transition of light in a two-mode cavity array with
ultrastrong coupling
Jingtao Fan,1,2 Yuanwei Zhang,3 Lirong Wang,1,2 Feng Mei,1,2,∗ Gang Chen,1,2,† and Suotang Jia1,2
1State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China
2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China
In this paper we construct a new type of cavity array, in each cavity of which multiple two-level
atoms interact with two independent photon modes. This system can be totally governed by
a two-mode Dicke-lattice model, which includes all of the counter-rotating terms and therefore
7 works well in the ultrastrong coupling regime achieved in recent experiments. Attributed to its
1 special atom-photon coupling scheme, this model supports a global conserved excitation and a
0 continuous U(1) symmetry, rather than the discrete Z2 symmetry in the standard Dicke-lattice
2 model. This distinct change of symmetry via adding an extra photon mode strongly impacts the
nature of photon localization/delocalization behavior. Specifically, the atom-photon interaction
n
a featuresstable Mott-lobe structuresof photonsanda second-ordersuperfluid-Mott-insulatorphase
J transition, which share similarities with the Jaynes-Cummings-lattice and Bose-Hubbard models.
Moreinterestingly,theMott-lobestructurespredictedheredependcruciallyontheatomnumberof
8
each site. Wealso show that ourmodel can bemapped intoacontinuousXX spin model. Finally,
1
weproposeaschemetoimplementtheintroducedcavityarrayincircuitquantumelectrodynamics.
] This work broadens ourunderstandingof strongly-correlated photons.
h
p
-
t
n I. INTRODUCTION and a second-order superfluid(SF)-MI phase transition
a take place across the edge of each lobe. This Mott-lobe
u structure makes it a photonic counterpart of the Bose-
q Photons are excellent information carriers in nature,
Hubbard model [27], which simulates massive bosons in
[ and generally pass through each other without conse-
lattice and also supports a similar lobe structure. How-
quence. The realization of coherent manipulation and
1 ever, it should be noticed that a complete description
controlling of photons allows us to achieve photon quan-
v of the light-matter interaction should always incorpo-
4 tum information processing [1] as well as to explore ex- rate the counter-rotating terms, especially considering
0 otic many-body phenomena of photons [2]. Cavity array
the factthat recentexperiments ofcircuitquantumelec-
9 [3–7], in which each single-mode cavity interacts with a
trodynamics (QED) have accessed the ultrastrong cou-
4 two-level atom, is a promising platform to accomplish
0 plingregime(i.e.,theatom-photoncouplingstrengthhas
the required target and has now been considered exten-
. thesameorderofthephotonfrequency)[29–32],inwhich
1 sively [8–26]. On one hand, this platform has a novel
therotating-waveapproximationtotallybreaksdown. In
0 interplay between strong local nonlinearities and pho-
sucha case,aproperdescriptionofthe systemdynamics
7 tonhoppingofthenearest-neighborcavities,whichhasa
1 shouldresortto aRabi-lattice model. Since the counter-
phenomenologicalanalogy to those of the Bose-Hubbard
: rotating terms in the Rabi-lattice model breaks the con-
v model [27] realized, for example, by ultracold atoms in
servation of excitation number, there is, in principle, no
i optical lattices [28]. More importantly, compared with
X similar MI as that of the Bose-Hubbard model and the
thecondensed-matteroratomicphysics,cavityarrayhas
r transition between the SF and MI should be replaced
a auniquepropertythatthefundamentalmany-bodyphe-
by the coherent and incoherent type [34, 35]. These es-
nomena depend crucially on the intrinsic atom-photon
sential changes of equilibrium properties motivate us to
coupling strength [3–7].
ask a question: could the Mott-lobe structure still ex-
Forthe weakandmoderately-strongcouplingregimes, ist even though all of the counter-rotating terms of the
the counter-rotating terms of the single-site Hamilto- atom-photon coupling are taken into consideration?
nian are usually neglected by employing the rotating-
wave approximation. As a result, the property of cavity In the present paper, we try to answer this question
array is governed by a Jaynes-Cummings-lattice model by constructing a new type of cavity array, in each
[3–7]. Since this Jaynes-Cummings-lattice model pre- cavity of which multiple two-level atoms interact with
two independent photon modes. This system can be
serves a global excitation number, a series of Mott in-
totally governed by a two-mode Dicke-lattice (TMDL)
sulator (MI) phase of photons form a lobe structure
model, which includes all of the counter-rotating terms
and therefore works well in the ultrastrong coupling
regime. UnliketheRabi-latticemodel,theTMDLmodel
∗[email protected] has a global conserved excitation and a continuous
†[email protected] U(1) symmetry. This distinct change of symmetry via
2
adding an extra photon mode induces some interesting Obviously, since the rotating-wave approximation is not
many-body physics of strongly-correlated photons. employed, the TMDL model is able to completely de-
Specifically, the atom-photon interaction features stable scribe potential effects arising from the counter-rotating
Mott-lobe structures of photons and a second-order terms and is therefore reasonablein the ultrastrong cou-
SF-MI phase transition, which share similarities with pling regime, which has been achievedin current experi-
the Jaynes-Cummings-lattice [3–7] and Bose-Hubbard ments of circuit QED [29–32].
[27] models. However, in contrast to these models, the The emergence of the so-calledcounter-rotatingterms
Mott-lobe structures predicted here depend crucially on in the Dicke Hamiltonian (3) reduces the conservation
the atom number of each site, reflecting its particularity of its excitation number, Nˆ = Jˆ + aˆ† aˆ , to a
s,j z,j 1,j 1,j
among lattice models. We also show that the TMDL parity Π = exp(iπNˆ ). However, by introducing an
s,j
model can be mapped into a continuous XX spin model
extra degenerate photon mode aˆ , the Hamiltonian
2,j
under proper parameter conditions. Finally, motivated
(2) exhibits a special conserved excitation [45], Nˆ =
by recent experimental achievements of cavity array e,j
[38–40] and multimode cavity [41–43] in circuit QED, Jˆz,j+aˆ†1,jaˆ2,j+aˆ†2,jaˆ1,j,apartfromthe knownconserved
we propose a scheme to realize the TMDL model in a parity [46], even if the rotating-wave approximation is
two-mode superconducting stripline cavity array. This not applied. When the photon hopping is triggered on,
work broadens our understanding of strongly-correlated thisconservedlocalexcitationNˆe,j isreplacedbyaglobal
photons. one,
Nˆ = Nˆ = (Jˆ +aˆ† aˆ +aˆ† aˆ ), (4)
e e,j z,j 1,j 2,j 2,j 1,j
j j
X X
II. MODEL AND HAMILTONIAN
which manifests the U(1) symmetry of the Hamiltonian
(1). The conserved global excitation Nˆ and its induced
We study a photon lattice system composed by an ar- e
U(1) symmetry distinguish the TMDL model from
rayofidenticalcoupledcavities,insideeachofwhichmul-
the standard Dicke-lattice model (with a discrete Z
tipletwo-levelatomsinteractwithtwodegeneratephoton 2
symmetry and without conserved excitation). This
modes. Such a system is governedby the TMDL Hamil-
complete change of symmetry are expected to deeply
tonian
impact the behavior of strongly-correlatedphotons.
Hˆ = HˆTD t aˆ† aˆ , (1)
T j − m,j m,k
Xj hXj,kimX=1,2
where the single-site Hamiltonian III. GROUND-STATE PHASE DIAGRAM
HˆjTD = ω aˆ†m,jaˆm,j +ω0Jˆz,j + (2) Since the knowledge of the single-site limit is crucial
m=1,2 forafurtherunderstandingofmany-bodyphysics,before
X
g aˆ +aˆ† Jˆ +i aˆ aˆ† Jˆ . proceeding, we first catch some instructive insights into
1,j 1,j x,j 2,j − 2,j y,j the Hamiltonian (2). In the absence of the photon hop-
h(cid:16) (cid:17) (cid:16) (cid:17) i ping (t = 0), the excitation density Nˆ commutes with
In the Hamiltonians (1) and (2), aˆ† and aˆ are the e,j
m,j m,j the Hamiltonian (1) and each eigenstate is thus charac-
creation and annihilation operators of the mth photon
terizedbya certainexcitationnumber. With anincreas-
mode of site j, Jˆ (i = x,y,z) = N σˆl /2, with σˆl
i,j l=1 i,j i,j ing of the system parameter, the level-crossings of the
being the Paulispinoperator,is the collectivespinoper-
lowest eigenstates are expected to take place, switching
P
ator of site j, ω is the frequency of the degenerate pho-
a definite excitation density of the ground state. Armed
ton modes, ω is the atom resonant frequency, g is the
0 with this argument, we plot the ground-state mean ex-
atom-photoncouplingstrength,tisthehoppingrate,and
citation density, n = Nˆ , of the single-site Hamilto-
j,k denotes the photon hopping between the nearest- e,j
h i
neighbor sites j and k. nian (2) as a functionDof g Ein Fig. 1. The evolution of n
An intriguing feature of the Hamiltonian (2) is that reflects a conspicuous staircase, whose jump points are
the spinoperatorcouplestothetwoindependentphoton associatedwith the crossoverpoints of the lowestenergy
modes via its two orthogonalcomponents Jˆ and Jˆ, re- levels. For N = 1, n remains a constant, whereas when
x y
spectively. Without the coupling term i(aˆ aˆ† )J , increasing N, the staircase appears and becomes more
2,j − 2,j y,j and more crowded, showing that the level crossing oc-
theHamiltonian(2)reducestothestandardDickemodel
curs only for N > 2. This property is totally different
HˆD =ωaˆ† aˆ +ω Jˆ +g(aˆ +aˆ† )Jˆ , (3) from the standard Dicke model (3), where no staircase
j 1,j 1,j 0 z,j 1,j 1,j x,j
can be found for any N [see the insert part of Fig. 1],
andthe correspondingHamiltonian(1)is thus calledthe duetothenonconservationofitsexcitationdensityNˆ .
s,j
Dicke-lattice model [44] (Rabi-lattice model for N = 1 We now pay attention to the TMDL Hamiltonian (1).
[33–37], with N being the atom number of each site). Byapplyingamean-fielddecouplingapproximation[27],
3
〈〈〈
i.e., aˆ† aˆ = aˆ† aˆ + aˆ aˆ† aˆ† aˆ , 0
m,j m,k h m,ji m,k h m,ki m,j −h m,jih m,ki N=1
the many-body Hamiltonian (1) reduces to an effective
N=3
mean-field Hamiltonian N=5
−1
N=7
Hˆ = HˆTD zt ψ aˆ† +aˆ ψ 2 ,
MF j − m m,j m,j −| m| 15
Xj Xj,mh (cid:16) (cid:17) i n−2 10
(5)
s
where z denotes the number of nearest neighbors, and n 5
ψm = haˆm,ji (m = 1,2) is the variational SF order pa- −3 0
rameter, which is taken to be real for simplicity [8, 47].
−5
ψ canbedeterminedself-consistentlybyminimizingthe 0 0.5 1
m g/ω
ground-state energy E(ψ ,ψ ) of the mean-field Hamil-
1 2
tonian (5) [8]. 0 0.2 0.4 0.6 0.8
Theeffectivemean-fieldHamiltonian(5)revealsanin- g/ω
timate connections between the single-site Hamiltonian
(2) and the many-body properties. In general, even
though the global excitation Nˆ is a conserved quantity, FIG. 1: The ground-state mean excitation densities, n =
e
the excitation number Nˆe,j of each site does not con- DNˆe,jE, of the two-mode Dicke model (2) as functions of
serve, due to the photon hopping. However,as shown in
g/ω for different N. Inset: the mean excitation densities,
the Hamiltonian (5), if both ψ and ψ vanish, the sys-
tem dynamics is dominated by1 the sin2gle-site Hamilto- ns = DNˆs,jE, of the standard Dicke model. In these figures,
nian(2),andthephotonsateachsitearethuseffectively we set ω0/ω=1.
frozen and characterized by a specific excitation num-
ber n. We accordingly denote this case as a MI phase,
ψ =0. The expandedground-stateenergyinpowersof
in which the U(1) symmetry is preserved. Whereas a m
ztψ reads
U(1)symmetry-brokenphase,associatedwiththebreak-
ingoftheconservationofNˆe,j,issymbolizedbyanonzero En(ψ1,ψ2)=En(0)+En(2)+O(tzψ)4, (6)
value of ψ andcanbe anticipatedacrossa criticalhop-
m
ping rate t (g). In this condition, the photon mode m where the second-order energy correction
c
governs a macroscopic coherence over the lattice and we
E(2) = (zt+z2t2R ) ψ 2+2z2t2T ψ ψ . (7)
haveaSFphaseofthemodem. Itwasgenerallybelieved n m,n | m| n 1 2
thatthecompleteinclusionofthecounter-rotatingterms mX=1,2
would demolish the MI phase since they couple states
The coefficients R andT in Eq.(7)arederivedfrom
m,n n
withdifferentnumbersofthe dressedphotonsandthere-
the second-order perturbation theory by
fore inhibit the formation of photon blockade, which is
crucially necessary for the MI phase [33–36]. In such n (aˆ +aˆ† ) k 2
a case, the notion “SF/MI” should be replaced by “co- h | m,j m,j | i
R = , (8)
hinetrreondtu/cinecdohheerreenot”ff.erNseavesrutpheerlbesse,xcthepetTioMnDL maoltdheoluwghe m,n kX6=n(cid:12)(cid:12)(cid:12) En(0)−Ek(0) (cid:12)(cid:12)(cid:12)
still breaking the conventional conservation−o−f Nˆ , the and
s,j
counter-rotatingterms in the TMDL model preservethe
hybridized two-mode excitation Nˆe,j, attributed to the hn|(aˆ1,j +aˆ†1,j)|kihk|(aˆ2,j +aˆ†2,j)|ni+c.c
T = .
specialatom-photoncouplingschemeintheHamiltonian n h 2(E(0) E(0)) i
(2), and thus retain the possibility to form the SF-MI kX6=n n − k
(9)
phase transition.
Based on above considerations, we plot the ground- where E(0) and k arise from the eigenequation
k | i
state phase diagram in the t g plane for different N HˆTD k =E(0) k .
− j | i k | i
in Fig. 2. These results show two typical phases: the The critical hopping rate t can be obtained by the
c
U(1) symmetry-preserved MI with ψ1 =ψ2 = 0 and the following procedure. (i) We first write a 2 2 Hessian
×
symmetry-broken SF with nonzero ψ1 and ψ2. A fur- matrix in terms of Eq. (7), i.e., ij = ∂2En(2)/∂ψi∂ψj,
ther analysis of ψ near the critical point demonstrates M
m and then derive its two eigenvalues ε and ε . (ii) These
1 2
thatthe transitionbetweenthesetwophasesisofsecond
two eigenvalues generate two equations, ε =0 and ε =
1 2
order. According to the Landau’s theory [48, 49], the
0, with respect to t. Each of these equations, say ε =
m
phase boundary of such a continuous transition can be 0, supports a trivial solution tT = 0 and a nontrivial
m
obtainedbyaperturbationmethod,inwhichtheground- solution tN = 0. (iii) The critical transition point is
state energy E (ψ ,ψ ) is expanded up to second order m 6
n 1 2 finally given by
in ψ [14, 35]. We expand E (ψ ,ψ ) of the nth MI
m n 1 2
phase around the critical value of the order parameter t =min(tN,tN). (10)
c 1 2
4

(a)1.0 (b)1.0 (a)0.7 (b)5
MI: n=3.5
0.6 MI: n=2.5 4
0.8 0.8 MI: n=1.5
0.5 3
SF
g/00..46 MI:n=-1.5 SF /g00..46MI: n=-0.5 SF /00..34 MI: n=0.5 n12
0.2
0.2 0.2MI: n=-1.5 0.1 MI: n=-0.5 0
0.0 10-1t/ 100 0.0 10-3 10t-2/ 10-1 100 0.010-4 10-3 10-2 10-1 -10.0 0.2 0.4 0.6
t/ /

(c)1.0 (d)1.0 FIG. 3: (a) Ground-state phase diagram of the Hamiltonian
(11)inthet−µplaneand(b)thecorrespondingmeanexcita-
0.8 0.8
tiondensity,n=DNˆe,jE,ofthesingle-sitelimitasafunction
0.6 SF 0.6 SF
g/ MI: n=-0.5 g/ MI: n=-0.5 of µ/ω. In these figures, we set g/ω=ω0/ω=1 and N =1.
0.4 0.4
MI: n=-1.5 MI: n=-1.5 MI: n=-2.5
0.2MI: n=-2.5 0.2MI: n=-3.5
0.0 -5 -4 -3 -2 -1 0 0.0 -5 -4 -3 -2 -1 0 coupling instead [34, 35]. This is in sharp contrast
10 10 10t/ 10 10 10 10 10 10t/10 10 10
to both the cases of the Jaynes-Cummings-lattice [3–7]
and Bose-Hubbard [27] models, which are often studied
withintheframeworkofgrandcanonicalensemblewhere
FIG.2: Ground-statephasediagrams oftheHamiltonian (5)
in the t−g plane, when (a) N = 1, (b) N = 3, (c) N = 5, a chemical potential is introduced to fix the (conserved)
and (d) N = 7. The MI phase is characterized by its lobes, numberofexcitationsonthelattice[8,14]. Ontheother
each of which supports a constant mean excitation density hand,thestandardDicke-orRabi-latticemodeldoesnot
n = DNˆe,jE. For a comparison, the phase boundaries of the support any conserved excitations, due to the inclusion
of the counter-rotating terms. This makes the descrip-
Dicke-latticemodel are also shown bythered-dashedcurves.
In these figures, we set ω0/ω=1. tion of grand canonical ensemble irrelevant to some ex-
tent and no well-defined chemical potential thus exists
[48,49]. However,theconservedexcitationintheTMDL
model motivates us to introduce a chemical potential µ
The obtained boundaries are shown by the black solid
and access a theory of grand canonical ensemble. We
curves in Fig. 2. The most important finding, as ex-
nowextendEq.(1)tothefollowingHamiltonianingrand
pected, is that the missing Mott lobes in the stan-
canonical ensemble:
dard Dicke-lattice model [33, 34] (see the red dashed
curve in Fig. 2) reappear. More interestingly, our pre-
dictedMott-lobestructuredependscruciallyontheatom Hˆ = Hˆ µNˆ (11)
G C e
number N, which has no counterpart in the Jaynes- −
= HˆGTD t aˆ† aˆ ,
Cummings-lattice [3–7] and Bose-Hubbard [27] models j − m,j m,k
(NotethattheN-dependentphasediagramfortheTavis- Xj hXj,kimX=1,2
Cummings-lattice,whichisnothingbuttheDicke-lattice
after the rotating-wave approximation, has been inves-
where the on-site two-mode Dicke Hamiltonian becomes
tigated previously [47, 50, 51]. In that case, the atom
HˆGTD = HˆTD µNˆ . Following the same mean-field
numberN onlyslightlyshiftsthephaseboundaryofeach j j − e,j
theory, we plot the phase diagram in the t µ plane in
lobe, rather than its total structure). Specifically, when −
Fig. 3. As shown in Fig. 3(a), the engineered chemical
N = 1, the atom-photon coupling features only a single
potentialµstillfeaturestheMottlobes,whichisadirect
Mott lobe, as shown in Fig. 2(a). With the increasing of
analog of those of the Bose-Hubbard model [27]. Once
N,however,moreandmoreMottlobesemerge,asshown
again, a clear interpretation of this lobe structure is still
in Figs. 2(b)-2(d). This N-dependent behavior of the
based on the dynamics of the single-site limit, which is
Mott lobes is a direct legacy of the N-dependent stair-
governedbytheHamiltonianHˆGTD. Asthechemicalpo-
case of n governedby the single-site Hamiltonian (2). In j
fact, since in the MI phase, the mean-field Hamiltonian tentialcouplestoaconservedquantityNˆe,j intheHamil-
(5)equals to the single-siteHamiltonian(2), there exists tonian HˆGTD, the eigenstates are independent of µ, due
j
a one-to-one correspondence between Fig. 1 and Fig. 2. to the simultaneous diagonalization of HˆTD and Nˆ .
j e,j
Asaresult,eachMottlobeisspecifiedbyadefinitemean Thus, the ground-state competition leads to a staircase
excitation density n. behavior of the excitation density Nˆ when varying µ,
e,j
We emphasize that in the TMDL model, on the one asshowninFig.3(b). Andaccordingly,eachMottlobein
hand, no chemical potential is needed to engineer the Fig. 3(a)is characterizedby the correspondingplateaux.
Mott lobes, which is here stabilized by the atom-photon
5
IV. EFFECTIVE SPIN MODEL: THE (a)
CONTINUOUS XX MODEL Resonator A
t t t t
It has been well established that the Jaynes-
Cummings-lattice model, respecting a U(1) symmetry, atoms atoms atoms
can be mapped to a continuous XX spin model (the
isotropic XY spin model) [11, 14], whereas the Rabi-
t t t t
lattice model with the counter-rotating terms has been Resonator B
demonstrated to be in the Ising universality class,owing
to its discrete Z symmetry [34, 35, 37]. As revealed in
2
Circuit QED element
thispaper,however,theinclusionofthecounter-rotating
terms does not always break the continuous symmetry.
ResonatorB
(cid:11)(cid:69)(cid:12)
Especially,forourTMDL model,the U(1)symmetryas- L L L
b b b
sociatedwiththeconservedexcitationnumberisasigna-
ture of its intimate connection with the continuous spin fs-1 C fs C fi C
b b b b b b
model. To confirm this argument, we focus on the sys- ResonatorA
tem dynamics in the t g plane, which is governed by L L L
theHamiltonian(1). We−firstconsiderthecaseofN >2, a (cid:68)(cid:87)(cid:82)(cid:80) a a
which supports a multi-lobe structure in the phase dia- fs-1 C fs C fi C
b a a a a a
gram.
When parameters are tuned close to the degenerate
f
point in the MI phase with t g, i.e., the boundary f J C
≪ f J
between two nearest Mott lobes, we can truncate the
Hilbertspacetotwooftheexcitationnumbereigenstates (cid:68)(cid:87)(cid:82)(cid:80) L1
n and n+1 , where n denotes the eigenstate of the
e|xicitatio|ndensiity Nˆe,j w| iith eigenvalue n (as verifiednu- L2 Cg
mericallybelow,n variesonlybyoneacrossthe degener-
ate point). Utilizing the commutation relations between
thephotonannihilationoperatoraˆ andthe excitation
m,j FIG.4: (a)Schematicdiagramofourproposedtwo-modecou-
number Nˆ , we can map aˆ in the reduced Hilbert pled circuit QED elements (black dashed line), one of which
e,j m,j
space n , n+1 into containsacoupleofsuperconductingstriplineresonatorsand
{| i | i} finiteJosephsonjunctionsactingasartificialtwo-levelatoms.
aˆ αΣˆ−+βΣˆ+, aˆ αΣˆ− βΣˆ+, (12) The nearest two elements are coupled through the series ca-
1,j → j j 2,j → j − j pacitanceoftheresonatorswithaphotonhoppingratet. (b)
where Σˆ+ = n n+1 and Σˆ− = n+1 n are the Theeffectivecircuitdiagram ofeachelement. Thefabricated
j | ih | j | ih | artificial atom (black dashedline) isassumed tobeplaced at
redefinedPaulispinladderoperators,andthecoefficients
a point, which is labeled by thesuperscript s of theflux.
αandβ canbedeterminednumerically(seeAppendix A
for details). Therefore, the effective spin Hamiltonian of
the TMDL model reads
regime,wecanstillobtaintheeffectiveHamiltonian(13)
∆ in the subspace spanned by the two lowestenergy levels.
Hˆ = Σˆz J ΣˆxΣˆx+ΣˆyΣˆy , (13)
2 i − i j i j
Xi hXi,ji(cid:16) (cid:17)
where ∆ is the energy gap between the two states n
| i
and n+1 and acts as a longitudinal field, and J = V. POSSIBLE EXPERIMENTAL
2t(α|2 + βi2) is the isotropic exchange interaction. As IMPLEMENTATION
| | | |
expected, we reproduce the continuous XX model even
taking the counter-rotating terms into account. Havingrevealedsomestrikingfeaturesofthetwo-mode
We now turn to the special case of N =1, where only cavity array, we now turn to the experimental imple-
a single Mott lobe exists. In this case, the mapping pro- mentation of the Hamiltonian (1). Motivated by recent
cedure of N >2 can not be employed directly. However, experimental achievements of cavity array [38–40] and
similar to Ref. [34], the energy gap between the two multimode cavity [41–43] in circuit QED, we propose
lowest energy levels is of higher-order small, compared a scheme, depicted in Fig. 4, to implement the TMDL
with the gap to the next energy level in the ultrastrong model. As shown in Fig. 4(a), the structure we con-
coupling regime, and the numerical calculation verifies siderisaseriesofidenticalcircuitQEDelementscoupled
thatthesetwolowestlevelsarestillcharacterizedbytwo throughcapacities. Eachofthese elements simulates the
nearest excitation numbers n and n+1 (see Appendix single-site two-mode Dicke model (2) and the capacitive
B). Based on these facts, in the ultrastrong coupling coupling gives rise to the photon hopping of different el-
6
ements. The effective circuit diagram of each element Fig. 4(a) with N =1. According to the theory of circuit
is shown in Fig. 4(b). A Josephson junction, acting as QED, we can regard the flux φ and the charge Q as the
an artificial two-level atom, is coupled to two different canonical coordinate and momentum, respectively. In
superconducting stripline resonators. this sense, the Lagrangian of a circuit QED element in
We first focus on the circuit QED element labeled in Fig. 4(b) is written as
(φ˙i)2 (φ˙i)2 (φ˙ )2 (φ˙s+φ˙ )2 (φi−1 φi)2
= C b + C a +C J +C˜ b J b − b (14)
b a J g
L 2 " 2 2 2 #− 2Lb
i i6=s i
X X X
(φi−1 φi)2 (φ )2 (φ φ )2 (φs−1 φs φ +φ )2 φ +φ
a − a f f − J a − a− f J E cos J ext ,
J
− 2L − 2L − 2L − 2L − φ
i6=s(cid:20) a 1 2 a (cid:21) (cid:18) 0 (cid:19)
X
where C˜ = C + C and φ is the external flux of neglected in the continuous limit, where the number of
g g a ext
theJosephsonjunction. NoticethatinderivingEq.(14), the sites becomes infinite. Based on this consideration,
the relation φ˙s = φ˙s +φ˙ has been used. Moreover, in we obtain
a b J
terms of the Kirchoff’s law at the point, there exists an
extra constraint relation φ = (L L +L L )φ /L + (Qi)2 (φi φi−1)2 (Qi)2
L1ULs2inφgas−Q1j−=φ∂sa //L∂Σφ˙,j,wwheefreobLtΣai=n1tLha2eLeax+p1Lre12sLsiaoJ+nLo1fΣLφ˙2j. Hres = Xi " 2Cbb + b−2Lbb + 2Caa (19)
(cid:0) k (cid:1)L k k (φi φi−1)2
in terms of Qj, i.e., + a− a .
k 2L
a (cid:21)
φ˙ 1 C +C˜ , C˜ Q
φ˙Js = C b C˜ ,g C−+gC˜ QJs (15) The Hamiltonian of the artificial atom reads
(cid:18) b (cid:19) Σ (cid:18) − g J g (cid:19)(cid:18) b (cid:19)
C˜ +C L˜
and H = g b(Q )2+ J (φ )2 (20)
at 2C J 2L2L L J
Σ Σ 2 a
1
φ˙im6=s = CmQim6=s (m=1,2), (16) −EJcos φJ +φφext .
(cid:18) 0 (cid:19)
where C =C˜ C +C˜ C +C C .
Σ g b g J J b
The interaction between the artificial atom and the res-
By means of Eqs.(14)-(16), together with the relation
onator is governed by the Hamiltonian
betweentheLagrangianandtheHamiltonian,weexpand
the Hamiltonian of the circuit QED element as a sum of
three contributions, i.e., H = L˜c φ (φs φs−1) C˜gQ Qs. (21)
int 2L2L L J a− a − C J b
Σ 2 a Σ
H =H +H +H . (17)
s res at int
In Eqs. (18)-(21), L˜ = L2L3+3L2L2L +3L2L2L +
In Eq. (17), the Hamiltonian of the stripline resonatoris J 1 2 1 2 a 1 2 a
3L2L L2 + L2L3 + L L3L + 2L L2L2 + L L L3
given by 1 2 a 1 a 1 2 a 1 2 a 1 2 a −
2L L L2 4L L L L 2L L L2 + L2L + L2L ,
Σ 1 2 − Σ 1 2 a − Σ 1 a Σ 2 Σ a
L˜ = L2L3 + L2L2L + L L3L 2L L L2 + L2L ,
(Qi)2 (φi φi−1)2(Qi)2 (φi φi−1)2 s 1 2 1 2 a 1 2 a − Σ 1 2 Σ 2
H = b + b− b a + a− a andL˜ =4L L L2+4L L L L 2L2L3 4L2L2L
res Xi " 2Cb 2Lb 2Ca 2La # 2L21L2cL2a−2ΣL1L132L2a−2ΣL1L122L22aa−−2L2Σ1L22.− 1 2 a−
C˜ +C 1 (Qs)2 We thus take the continuous limit of the canonical
+ g2CΣ J − 2Cb!(Qsb)2− 2Caa pi.ea.r,amφieters iφn t(hxe)suapnedrcQoniductiQng s(txri)p,liannedretshoennatporros-,
m → m i m → m i
L˜ 1 mote them to quantum operators obeying the canonical
+ 2L2ΣLs2La − 2La!(φsa−φas−1)2. (18) commutation relation φˆm(x),Qˆn(y) = iδ(x−y)δm,n.
Followingthe standardhquantizationpirocedure incircuit
Since the last three terms in the Hamiltonian (18) do QED [52], the quantized canonical parameters are ex-
not involve a sum over sites, their contributions can be pressed as
7
φˆ (x ) = ωm,noLmD cos(noπxi)(aˆ +aˆ† ) (22)
m i n π D m,no m,no
nXo=1p o
+ ωm,neLmD sin(neπxi)(aˆ +aˆ† ),
n π D m,ne m,ne
nXe=2p e
Qˆ (x ) = i ωm,noCmD cos(noπxi)(aˆ aˆ† ) (23)
m i − n π D m,no − m,no
nXo=1p o
i ωm,neCmD sin(neπxi)(aˆ aˆ† ),
− n π D m,ne − m,ne
nXe=2p e
π
where ωm,n = nπ/(D√LmCm) is the eigenfrequency, D ω2 = . (30)
isthelengthoftheresonator,andno andne areoddand D√LbCb
even integers, respectively.
When the external flux is set to φ /φ =π, the two-
ext 0 The tunability of the inductance and the capacitance
levelapproximationof the Josephsonjunction givesthat
of the two superconducting stipline resonators allows us
[53, 54]
to set ω = ω = ω and g = g = g , under which
1 2 1 2 0
theHamiltonian(26)reducestothesingle-sitetwo-mode
φˆ φˆ σˆ (24)
J ⇔h↓| J|↑i x Rabi model. Using the same procedure, the Hamilto-
nian (26) can be extended straightforwardly to the case
and
withseveraltwo-levelartificialatoms,i.e., the single-site
Qˆ ω0 φˆ σˆ , (25) two-mode Dicke Hamiltonian (2). When a series of such
J J y
⇔ 4eE φ h↓| |↑i circuit QED elements are coupled capacitively with the
Q 0
hopping rate t [see Fig. 4(a)], the TMDL Hamiltonian
with E = (C˜ +C )/(2C ), where ω is the resonant (1) can be achieved.
Q g b Σ 0
frequencyofthetwo-levelsystem, and arethetwo
|↓i |↑i We emphasize that the improvement of current ex-
lowestmacroscopicstatesoftheHamiltonianH ,andσˆ
at i perimentaltechniques inthe ultrastrong-couplingcircuit
(i = x,y,z) is the Pauli spin operator spanned by these
QED [29–32] makes our proposal a promising candidate
two macroscopic states.
to exhibit relevant physics of the TMDL model.
At low temperature, we only keep the mode resonate
with the artificial atom (i.e., n = 1) and neglect other
non-resonate terms. Under this single-mode approxima-
tion of the resonator and the two-level approximationof
theartificialatom,theHamiltonianoftheconsideredcir-
cuit QED element is finally expressed as
VI. DISCUSSIONS
1
Hˆ = ω aˆ†aˆ +ω aˆ†aˆ + ω σˆ (26)
s 1 1 1 2 2 2 2 0 z
+g (aˆ +aˆ†)σˆ +ig (aˆ aˆ†)σˆ , Up to now, our discussions are restricted to the case
1 1 1 x 2 2− 2 y of the degenerate photon modes (ω = ω = ω) and the
1 2
equal atom-photon coupling strengths (g = g = g).
where 1 2
If these conditions are not fulfilled, there would not
L˜c ω1/LaDsin(πxs/D) φˆJ be a strict conservation law of Nˆe, and an instructive
g = h↓| |↑i, (27)
1 − 2L2L question is whether the Mott-lobe structure still exists
p Σ 2 in such a case or not. To briefly show the influence of a
slightdeviationof these two equalities, ω =ω =ω and
1 2
g2 = C˜gω0√ω2Cb4DπecEos(φπxCs/D)h↓|φˆJ|↑i, (28) fgo1r=digff2er=engt,ωw2e/pωl1ot[Fthige.p5h(aas)e] odriagg2r/agm1s[Finigt.he5(tb−)]g, pwlhaenne
Q 0 Σ N = 3. It can be seen clearly from these figures that a
slightdeviationofthe idealconditiondoes notbreakthe
π Mott-lobestructurebutmerelyshiftthephaseboundary.
ω = , (29)
1
D√L C
a a
8
(a) 1 g2/g1=1 (b) 1 ω2/ω1=1
0.8 g2/g1=1.02 0.8 ω2/ω1=1.02
g2/g1=1.05 ω2/ω1=1.05
ωg/100..46 MI:n=−0.5 g2S/gF1=1.1 ωg/100..46 MI:n=−0.5 ωS2/Fω1=1.1 Appendix A: Mapping aˆ1,j and aˆ2,j to the spin
operators
0.2 0.2
MI:n=−1.5 MI:n=−1.5
100 −4 10−2 100 100 −4 10−2 100 We first notice that the commutation relations be-
t/ω t/ω
1 tween the photon annihilation operator and the exci-
tation number operator satisfy aˆ ,Nˆ = aˆ and
1,j e,j 2,j
FIG. 5: (a) Phase boundaries for g2/g1 = 1 (black solid aˆ ,Nˆ = aˆ . Taking thesehtwo equaitions into ac-
2,j e,j 1,j
curve), g2/g1 = 1.02 (red dashed curve), g2/g1 = 1.05
(blue dotted-dashed curve), and g2/g1 = 1.1 (green dotted hcount, theimatrix elements of aˆ1,j +aˆ2,j and aˆ1,j −aˆ2,j
in the basis of the excitation eigenstates n and m are
curve), when ω1/ω = ω2/ω = ω0/ω = 1. (b) Phase bound- | i | i
aries for ω2/ω1 = 1 (black solid curve), ω2/ω1 = 1.02 (red expressed respectively as
dashedcurve),ω2/ω1 =1.05(bluedotted-dashedcurve),and
ω2/ω1 = 1.1 (green dotted curve), when g1/g = g2/g = n aˆ +aˆ m = n aˆ +aˆ ,Nˆ m (A1)
1,j 2,j 1,j 2,j e,j
ω0/ω1 =1. In these figures, we set N =3. h | | i h | | i
= (mh n) n aˆ +aˆi m ,
1,j 2,j
− h | | i
n a a m = n [aˆ aˆ ,N ] m (A2)
1,j 2,j 1,j 2,j e,j
VII. CONCLUSION h | − | i −h | − | i
= (n m) n aˆ aˆ m .
1,j 2,j
− h | − | i
Insummary,we haveconstructedanew type ofcavity
Toobtainanonzerovalueof n aˆ +aˆ m ( n aˆ
array system, which is governed by the TMDL model. h | 1,j 2,j| i h | 1,j−
aˆ m ), we should have m = n+1 (m = n 1), and
This model incorporates all of the counter-rotating 2,j| i −
inthereducedHilbertspace n , n+1 ,theoperators
terms of the atom-photon coupling and therefore works {| i | i}
aˆ +aˆ and aˆ aˆ thus read
well in the ultrastrong coupling regime achieved in 1,j 2,j 1,j − 2,j
recent experiments. Unlike the standard Dicke-lattice
0 0
model, the TMDL has a global conserved excitation aˆ +aˆ 2α 2αΣˆ−, (A3)
and a continuous U(1) symmetry. This distinct change 1,j 2,j → 1 0 ⇐⇒ j
(cid:18) (cid:19)
of symmetry strongly impacts the nature of photon 0 1
aˆ aˆ 2β 2βΣˆ+, (A4)
localization/delocalization behavior. Specifically, the 1,j − 2,j → 0 0 ⇐⇒ j
(cid:18) (cid:19)
atom-photon interaction features Mott-lobe structures
of photons and a second-order SF-MI phase transition, from which we can straightforwardly obtain aˆ
1,j
which share similarities with the Jaynes-Cummings- αΣˆ− +βΣˆ+ and a αΣˆ− βΣˆ+, i.e., Eq. (12)→of
lattice and Bose-Hubbard models. However, the j j 2,j → j − j
the main text. The coefficients α and β can be deter-
Mott-lobe structures predicted here depend crucially on
mined numerically.
the atom number of each site, reflecting its particularity
among lattice models. We have also shown that the
TMDL model can be mapped into a continuous XX
spin model under proper parameter conditions. Finally, Appendix B: Numerical demonstration of the two
we have proposed an experimentally-feasible scheme to state subspace {|ni,|n+1i} in the ultrastrong
realizetheTMDL modelinatwo-modesuperconducting coupling regime for N =1
stripline cavity array.
Figure ?? shows the low-lying spectrum of the Hamil-
tonian(2)withN =1,fromwhichwecanseeclearlythat
the two lowest energy levels become quasi-degenerate in
VIII. ACKNOWLEDGEMENTS the ultrastrong coupling regime. Moreover, as shown in
the inset of this figure, both of these two levels support
This work is supported partly by the NSFC under the well-definedexcitationnumbers, whose difference re-
Grants No. 11422433, No. 11674200, No. 11604392, mainsone. Thisguaranteesthevalidityofthetruncation
No. 11434007, and No. 61378049; the FANEDD under of the Hilbert space to an effective two state subspace
Grant No. 201316;SFSSSP; OYTPSP; and SSCC. n , n+1 for a large g/ω.
{| i | i}
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