Localized tendency for the superfluid and Mott insulator state in the array of dissipative cavities

0.2 MB·English

by  Lei Tan

Localized tendency for the superfluid and Mott insulator state
in the array of dissipative cavities
Lei Tan1, Ke Liu1, Chun-Hai Lv1, W. M. Liu2
1Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
2Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
(Dated: January 10, 2011)
The features of superfluid-Mott insulator phase transition in the array of dissipative cavities is
analyzed. Employing a kind of quasi-boson and a mean-filed approach, we show analytically how
dissipation and decoherence influence the critical behaviors and the time evolution of the system.
We find that there is a localized tendency, which could lead to the break of superfluidity for a
superfluidstateandsuppresstheappearanceofthelong-rangeorderformaMottstate. Eventually,
1
a collection of mixture states localized on each site will arise.
1
0
PACSnumbers: 42.50.Pq,42.60.Da,64.70.Tg,03.65.Yz
2

n
7 Ja athrreOacnyosenitosrfototllharebeairlleiitmzyeaosrfokaactbaollmeleidacpqapunliadcnaottpuiotmincsasilomsfyucslotaeutmoprlsse,.ditDccuaoveuittldyo aPnjdPHαc,okuωpkα=rj†,kαjrj,kαα,tkh(ηek∗Hαraj†m,kiαltαojni+anhf.oc.r)enthveirocnomupelnetd,
term. α = a,cPlabePls the operators and physical quanti-
be useful to attack some unclear physics and to explore tiesassociatedwithatomsandcavities,respectively. ω
ph] nInewpaprhtiecnuolmare,noovnerinthqeuapnatsutmyemarasntyh-eboedxypesryimsteemntsal[1p–r4o]-. denotesthe frequencyofenvironmentalmodes, rj†,kα ankαd
r the creationandannihilationoperatorsofquanta in
- gressesincontrollingopticalsystem[5–8]andinfabricat- j,kα
nt ing large-scalearraysof high-Q cavities [9, 10] make this tHheerekαwtehsmeto~de=l a1t.site j, and ηkα the coupling strength.
a potentialapplicationclosetoreality. However,thequan-
u The system we modelled, as depicted in Fig. 1,
tumopticalsystemistypicallydrivenbyanexternalfield
q is a two-dimensional array of resonant optical cavities,
andcoupledtothe environment[11,12],whichbringthe
[ each embedded with a two-level (artificial)atom coupled
system out of equilibrium and profoundly affected the
strongly to the cavity field. The possible realizations
1 dynamics of interest [13, 14]. New important questions
v thus arise and need to be clarified, such as whether the such as photonic bandgap cavities and superconduct-
3 linkbetweentheinitialideasofcavityarraysasquantum ing stripline resonators et al. [4]. With ωa and ωc be-
3
simulators and the realistically experimental conditions
4
1 is hold, and how dissipation and decoherence would be- (a) (b)
. have in the system. E
1 In this paper, we propose answers to these questions 2(cid:90) |(cid:14),2(cid:178)
0 c
by investigatingthe superfluid(SF)-Mott insulatorphase E
1 U
|(cid:16),2(cid:178)
1 transition in the array of dissipative cavities. Provided
: theexternaltimedependenceismuchslowerthanthein-
v E
i ternal frequencies of system, we show that there are still (cid:90) |(cid:14),1(cid:178)
X twofundamentallydifferentquantumstates,i.e. photons c
E
r localized in each cavity(Mott-like) and delocalized cross |(cid:16),1(cid:178)
a the lattices(SF-like). However,a localizedtendency does
exist for both of these two states. Specifically, the ra-
tioofthe theintercavitycouplingandon-siteinteraction 0 0
strength for the transition from Mott-like region to SF- bare dressed
like region is modified by a time dependent increment. basis basis
Moreover,thesuperfluidityofainitialSFstatewillbreak
owing to the decay of the off-diagonal long-range order.
Considerasystemconsistedofatomsandcavitiescou- FIG. 1: A type of possible topologies for two-dimensional
pled weakly to a bosonic environment at zero tempera- cavity arrays. (a) Individual cavities are coupled resonantly
ture. As the dimension of individual cavities is generally toeachotherduetotheoverlapoftheevanescentfields. Each
cavitycontainsatwo-levelsystemcoupledstronglytothecav-
muchsmaller than their spacing,we assume the photons
ity field and immerses in abosonic bath(markedby thedash
emitted from each cavity are uncorrelated. The total
line). (b) Energy eigenvalues of individual cavity-atom sys-
Hamiltonian therefore reads
temoneachsite. ωc =ωaisassumedforsimplify. Theanhar-
monicityoftheJaynes-Cummingsenergylevelscaneffectively
H =H +H +H . (1)
s bath coup provideanon-siterepulsionU toblocktheabsorptionforthe
where H is the Hamiltonian for the system, H = next photon.
s bath
2
ing the frequency of atom transition and cavity mode theprobabilityamplitudesfortheexcitationoccupiedby
respectively, in the rotating wave approximation(RWA), cavity field and environment, respectively. denotes
|∅i
such individual atom-cavity system on site j is well de- the vacuum state. Deducing the equations of these two
scribed by the Jaynes-Cummings Hamiltonian, HJC = amplitudes, one can express e in terms of e and, un-
j k c
ωaa†jaj +ωcc†jcj +β(a†jcj +h.c.). Here a†j and aj(c†j, cj) der the Born-Markov approximation, integrate out the
degrees of freedom of environment, and obtain
are atomic(photonic) rasing and lowering operators, re-
spectively, β the coupled strength. In the grand canoni-
(ω +δω iγ )e =ωe . (6)
calensemble,H isthereforegivenbycombingHJC with c c− c c c
s j
photonic hopping term and chemical potential term, δω is known as an analog to the Lamb shift in atomic
c
physics and significantly small when the coupling to en-
Hs = HjJC − κjj′c†jcj′ − µnj. (2) vironment is weak. γc is the decay rate and indicates a
Xj Xj,j′ Xj finite lifetime of cavity mode [15].
h i
Thismotivatesustointroduceaquasi-bosondescribed
κjj′ is the photonic hopping rate betweencavities. Since
byC withacomplexeigenfrequencyΩ =ω iγ ,where
the evanescent coupling between cavities decreases with j c c− c
δω hasbeenabsorbedintoω ,toeffectivelydescribethe
the distance exponentially, we restrict the summation c c
open system described by Hamiltonian (5) in terms of
j,j′ running over the nearest-neighbors. nj =a†jaj +
h i
Ptci†jocnjsisonthseitteotja.l nµumisbtehreocfhaetmomicaicl apnodtenpthioatl,onwicheerxecitthae- Hjeff|φji=ωjeff|φji, (7)
assumption µ=µj for all sites has been made. with the effective Hamiltonian Hjeff =ΩcCj†Cj and now
Due to the strongly coupling, as shown in Fig. 1(b),
theresonantfrequenciesofindividualatom-cavitysystem |φji = ecCj†|∅i. Because of loss, the system would be
nonconservative and corresponding operators would be
are split into
non-Hermitian. The communication relationof C reads
j
E|±,ni =nωc±rnβ2+ ∆42 − ∆2, (3) [Q1C,j,wCitj†h′] Q= b(1ei+ngitωγhcce)δqjju′a.liRtyecfaocgtnoirzinogf inωγcdcivisidiunaolrcdaevritoyf.
Thebosoniccommunicationrelationisthereforeapproxi-
where ,n labels the positive(negative) branch of
|± i matelysatisfiedforthehigh-Qcavity,whichismeetingin
dressed states, ∆ = ω ω is the detuning. The an-
c − a most experiments about cavity quantum electrodynam-
harmonicity of the Jaynes-Cummings energy levels can
ics(QED).
effectively provide a on-site repulsion. For instance, the
The complex eigenfrequency underlines the facts that,
resonant excitation by a photon with frequency E
,1 on one hand, dissipation is the inherent property for re-
will preventthe absorptionofa secondphotonat E|± i,
,1 alisticcavity. Whenaphotonwithcertainfrequencyhas
whichisthestrikingeffectknownasphotonblockad|e±[8i].
beeninjectedintoadissipativecavity,thecompositesys-
It is therefore feasible to realize a quantum simulator in
tem of cavity-filed plus environment cannot be charac-
terms of the system described by Eq. (2). This so called
terized merely by the frequency of the injected photon,
Jaynes-Cummings-Hubbard(JCH) mode is recently sug-
however, we must take the impacts of environment into
gested by Greentree et al. [2].
account. On the other hand, in general we are not con-
However,the situation changesdramaticallyonce tak-
cernedthe time evolutionof bath. In this way,the array
ingthecoordinatesofenvironmentintoconsideration,as
of dissipative cavities can be regardedas a configuration
describedbyHamiltonian(1). Anon-equilibriumdynam-
consisted of quasi-bosons. Quite similar operations can
ics for open quantum many-body system do arise,which
beperformedonatomtointroduceanotherkindofquasi-
is a formidable task to solve. Here we propose a treat-
bosondescribedbyA withthefrequencyΩ =ω iγ ,
j a a a
ment to eliminate those external degrees of freedom. To −
where γ is the atomic decay rate.
a
approach this, we rewrite Hamiltonian(1) as
We can therefore rephrase Hamiltonian (1) with the
renormalized terms,
H =Hlocal− κjj′c†jcj′ − µnj, (4)
hXj,j′i Xj H = Hjeff − κjj′Cj†Cj′ − µnj, (8)
where H = HJC +H +H . Xj <Xj,j′> Xj
local j j bath coup
a iFniirtsiatlcopnhsoitdoenrPiningttehraecctassewtihthatathbeajthth, cthaveitdyycnoanmtaicinseids withnowHjeff =ΩaA†jAj+ΩcCj†Cj+β(A†jCj+h.c.)and
governedby nj =A†jAj+Cj†Cj. OnekeyfeatureofHamiltonian(8)is
now the losses describe by leaky rates γ and γ but not
a c
Hj =ωcc†jcj + ωkcrj†,kcrj,kc + (ηk∗crj†,kccj +h.c.). by field oscillations. Without having to mention the ex-
Xk Xk ternal degrees of freedom, this effective treatment would
(5) beofgreatconceptualand,moreover,computationalad-
We denote its eigenvalue as ω and expand the eigenvec- vantage rather than the general treatment as Hamilto-
tor |φji as |φji = ecc†j|∅i+ kekrk†c|∅i. ec and ek are nian (1). A more microcosmic consideration points out
P
3
that, in cavity QED region, since the atom is dressed by
cavity field, the atom and field act as a whole subject to 0.6
a total decay rate Γ [16]. In particular, Γ = n(γ +γ )
a c
for ∆=0.
To gain insight over the role of dissipation in the SF- t) 0.4
,
Mottphasetransition,weuseameanfieldapproximation γ
κ,
which could give reliable results if the system is at least (
ψ
two-dimensional [17]. We introduce a superfluid param-
0.2
eter, ψ = RehBji = RehBj†i. In the present case, the
expected value of Bj(Bj†) is in general complex with the
formation Bj =ψ−iψγ(Bj† =ψ+iψγ). ψγ is a solvable 00 0.1 0.2 0.3
small quantity as a function of γ and γ , and vanishes
a c
zκ/β
in the limit of no loss. Using the decoupling approxima-
itniognm, Beaj†nB-fij′e=ldhHBaj†miBiltjo′n+iahnBjc′ainBjb†e−whBritj†tiehnBj†ais,athseumresouvletr- FIG. 2: The transition from Mott-like to SF-like region. In-
fluencesofdissipationonthecriticalratiodependontheleaky
single sites,
rate γ (the dot and solid line for γ = 0, 0.05, respectively)
β
HMF = {Hjeff−zκψ(Bj†+Bj)+zκ|ψ|2−µnj+O(ψλ2)}, abnludetlhineestimfoer ton=e0r,ea0c.1hγe−s1t,h0e.2trγa−n1s,itrieosnpe(ctthieverleyd),. green, and
Xj
(9)
where we have set the intercavity hopping rate κjj′ = ψ which can distinguish two slowly decayed and quali-
κ for all nearest-neighbors with z labelling the number.
tatively different regions, i.e. the SF-like and Mott-like
Corresponding to the order parameter in related closed
regions.
system, whether ψ vanishes or has a finite value can be
Inwhatfollows,wediscussthephysicsofthetransition
used to identify the Mott-like and the SF-like regions.
betweentheseregionsfromtwoaspects. First,westartin
ψ can be examined analytically in terms of second-
theMottphaseanddiscusstheimpactsofdissipationon
order perturbation theory, with respect to the dressed
transitionratio(zκ) . Considertheinitialstateisdeepin
basis. For energetically favorable we assume each site is β c
the Mott phase, zκ =0, and we continually increase the
preparedinthenegativebranchofdressedstate. Butbe- β
causethedressedbasisisdefinedonn 1,agroundstate intercavity coupled rate. For the related ideal case, we
0 with the energy E0 =0 need to b≥e supplemented, will reach the SF phase at zβκ = (zβκ)′c ≃ 0.16. However,
| i | i in the presence of dissipation, owing to the continuous
χ leakage of coherence the effective tunnelling energy will
ψ =e−Γt . (10) be lower than what we set. Stated in another way, the
r−zκΘ
photonhopping ratenotonlydependent onthe coupling
χ and Θ are functions of all the parametersof the whole parameterbetweencavities,butalsoonthecoherencein-
system. Since the evanescent parameter κ is a typical side cavities. As shown in Fig. 2, one cannot expect the
small quantity in systems of coupled cavities, the per- appearance of photon hopping at zκ 0.16. We must
β ≃
turbation theory gives good qualitative and quantitative continuously increase κ and the rate of increase must
descriptions comparing to the numerically results given fasterthanthedecayofcoherence. Onlyinsuchwaycan
by explicitly diagonalizing [2, 18]. the transition occur before the system reach the steady
Arguably the most interestingsituation is the photon- state. However, even we start exactly at zβκ = (zβκ)′c,
photon interactions are maximized, namely, cavities on we also cannot expect we are in the SF state. As the
resonantwith atoms and with one initial excitations per dissipation is the inherent nature of open quantum sys-
site [19]. In addition, Γ = γa + γc = γ. With F1 = tem, the critical ration (zβκ)c, even at t = 0, is modified
ωΘc=−β−1µ an+d F32+2=√2−ω>c+0,(a√nd2−1)β+µ, in eq. (10), by an increment ≃ 2βγ22. This can be understood in a
2F12+2γ2 4F22+4γ2 simplified picture. If there was no excitation in system,
i.e. wearein ,the decaywouldnothappen. However,
|∅i
F1 (3+2√2)F2 1 theopenquantumsystemisinherentlydifferentfromthe
χ= + + . (11)
2F2+2γ2 4F2+4γ2 zκe 2γt correspondingly perfected system.
1 2 −
In contrast, we start with the SF phase in the vicinity
In the absence of loss, one can recognize χ = 0 is the of transition ratio and track the time evolution of sys-
well known self-consistent equation and therefore distin- tem. The exponential term in Eq.(10), e Γt, indicates
−
guish the SF phase and Mott phase. Nevertheless, the the expected decay of ψ and related physics quantities.
coupling to environment inducing a non-equilibrium dy- However,the more important impacts of the presence of
namics, and there is thus no strict phase exist. How- external environment are revealed by χ. As illustrated
ever,there remainsa time-dependent boundarygivenby in Fig. 3, for t β 1, ψ has a slightly decrease scaled
−
≪
4
rate zκ given initially in SF phase will cross the critical
β
point at a time t 1 ln κ . Before t , a non-localre-
0.6 (t) 0.5 gionisstillrecogcni≃ze−da2γsnonκ′c-local. Thecdissipationdoes
n
) ∆ not change the fundamental nature of the system, albeit
t
(
n with the decrease of long-range coherence. Nevertheless,
p 0.4 0
/ 0 tc 50 100 beyond tc, the superfluidity breaks down and gives rise
t) to alocalizedwavefunction,i.e. atransitionto Mott-like
(
ψ regiondooccur. Ananalogouseffectisrecentlydescribed
0.2
inopticallatticesystem,whereatime-dependenttunnel-
ingratecontrolledbyexternalfiledleadstoasweepfrom
0 the SF to the Mott phase [20]. Differently, in our case
0 20 40 60 80 100 the decay of long-range order is resulted from the leak-
t(β−1) age of coherence, and the total number of excitations is
not conservative. As a result, there remains a statistical
fluctuation acted on each site.
FIG. 3: The temporal decrease of the superfluidity and the
photon number fluctuation on each site for a certain initial In conclusion, we have shownanalytically the features
state(inset). For a initial SF state (the red line for zβκ =0.3 of SF-Mott phase transition in the coupled cavity arrays
and blue line for zβκ = 0.2), before tc the long-range order in the presence of dissipation. Our analysis fully takes
decays continuously and the fluctuation on each site is a to- into account the intrinsically dissipative nature of open
tal effect of photon hopping and photon leakage (the solid
quantum many-body system, and identifies how dissipa-
line for βγ = 0.01 and dash line for βγ = 0.02). Beyond tc, tion and decoherence would come into play. For further
the superfluidity breaks down and the related photon num-
experimentrealization,wepredictthereisalocalizedten-
berfluctuationbehavesasthefluctuationofaMott-likestate
dency characterized by the suppression to the superflu-
(dot-dash line).
idity.
This work was supported by NSFC under grants
by βγ22. For t > β−1, the leader term is zκe−2γt. It is Nos. 10704031, 10874235, 10934010 and 60978019,
this term pronounces the decrease of effective tunnelling the NKBRSFC under grants Nos. 2009CB930701,
energy which, mathematically, originates from the non- 2010CB922904and2011CB921500,andtheFundamental
diagonalelementinHamiltonian(9)respectingtotheoc- Research Funds for the Central Universities under grant
cupationnumber basis. Consequently,a photon hopping No. lzujbky-2010-75.
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