Simulation of high-spin Heisenberg models in coupled cavities

0.14 MB·English

by  Jaeyoon Cho

Simulation of high-spin Heisenberg models in coupled cavities
Jaeyoon Cho,1 Dimitris G. Angelakis,2,3 and Sougato Bose1
1Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK
2Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117542
3Science Department, Technical University of Crete, Chania, Crete, Greece, 73100
(Dated: January 7, 2009)
WeproposeaschemetorealizetheHeisenbergmodelofanyspininanarbitraryarrayofcoupled
cavities. Our scheme is based on a fixed number of atoms confined in each cavity and collectively
applied constant laser fields, and is in a regime where both atomic and cavity excitations are sup-
pressed. It is shown that as well as optically controlling the effective spin Hamiltonian, it is also
possible to engineer the magnitude of the spin. Our scheme would open up an unprecedented way
tosimulate otherwise intractable high-spin problems in many-bodyphysics.
9
0
0
TheHeisenbergspinmodelhasplayedacrucialroleas XXX case,except for specialkinds ofmodels [15]. Addi-
2
a basic model accounting for the magnetic and thermo- tionally, going to higher spins should make perturbative
n
dynamic natures of many-body systems. Despite exten- spin-wave theory more accurate, whose predictions can
a
sive investigations, however, many aspects of the model be tested.
J
arestill largelyunexploredboth analyticallyandnumer-
7
ically, especially for the cases of higher spins. The main In this paper, we propose a scheme to realize the
] difficulty in the numerical treatment originates from the anisotropic (XXZ) or isotropic (XXX) Heisenberg spin
h factthatthe Hilbert-spacedimensionblowsupexponen- model of any spin in an arbitrary array of coupled cavi-
p tiallyasthe numberofspinsincreases. AsFeynmanfirst ties. Ourschemeisexperimentallyfeasibleinthatsimply
-
t noted [1], this difficulty would be overcome in terms of applying a small number of constant laser fields suffices
n quantum simulation based on precisely controlled quan- for our purpose. If the number of lasers is increased,
a
u tum systems. Realization of quantum simulation, ex- theindividualconstantsofthespinHamiltonianarecon-
q pected in a near future, will mark a milestone towards trolled more flexibly. Most of all, a strong advantage of
[ the realization of sophisticated quantum computation. our scheme is that the magnitude of the spin itself can

be engineered arbitrarily. This advantage contrasts with
3
In the context of quantum information processing, a all the earlier schemes mentioned above including those
v
65 pqulebmitenistaitdioennst,icasluctho aans tshe=a21rrasypsino,fanJdoseinphasofnewjuinmc-- fmorosotplytiacaslsla=tti1ce(ss, i=n w1hinichRetfh.e[5s]p)i.nsis>fix1edspiinnnchatauinres
2 2
3 tions [2] or quantum dots [3], the spin-chain Hamilto- exhibitfascinatingphysicsthats= 1 spinchainscannot
2
3 nian naturally emerges from the spin-like coupling be- have. Awell-knownexampleisHaldane’sconjecturethat
.
2 tween qubits, albeit with limited control of the coupling antiferromagneticHeisenberg integral-spinchains have a
0 constants. On the other hand, in optical lattices, per- unique disordered ground state with a finite excitation
8 turbative evolution with respect to the Mott-insulator gap,whereashalf-integral-spinchainsaregapless[16,17].
0 state can be described by an effective spin-chain Hamil- Our scheme could be used to prepare a groundstate, for
:
v tonian[4,5]. Thisapproachhasitsownmeritinthatthe example,throughanadiabaticevolution,andmeasureits
i spin-coupling constants can be optically controlled to a excitationgapandspincorrelationfunctions[5]. Thead-
X
greatextent. Analternativeapproach,recentlyunderac- vantage of our scheme is, however, more apparent when
r tive investigation,is to use the arrayof coupled cavities, we consider higher-spin problems intractable with any
a
which are ideally suited to addressing individual spins previous method. One of the intriguing examples is the
[6, 7, 8, 9, 10, 11, 12, 13, 14]. In this approach, a spin quantum spin dynamics of the ferric wheels such as Fe
6
is represented by either polaritons or hyperfine ground and Fe composed of s = 5 spins [18]. These systems
10 2
levels. The former, proposed in Refs. [8] and [14], allows would be simulable with a relatively small number (6 or
a stronger spin-spin coupling than the latter, but lacks 10)ofcavities. Spinchainsalsoplayanimportantroleas
the optical control of the coupling. On the other hand, a quantum channel for short-distance quantum commu-
the latter, proposed in Ref. [9], retains the optical con- nication[19]. Thepropertyofans=1antiferromagnetic
trollability,butreliesonrapidswitchingofopticalpulses spinchainasa quantumchannelstronglydepends onits
and the consequent Trotter expansion, which unavoid- phase [20]. In some phases, it provides an efficient chan-
ably involves additional errors and makes error-free im- nel, outperforming that of a ferromagnetic chain. It has
plementationmoredifficult. Moreimportantly,theques- been also shown that a ground state with an excitation
tionofsimulatingchainsofhigherspins,whichmayhave gap, as is the case for the spin-1 chain, can serve as a
a completely different phase diagram, remains open. In moreefficientquantumchannel[21]. Thegroundstateof
somesense,thesearemoreimportanttosimulatebecause theantiferromagneticspin-1chainalsoestablishestheso-
unlike spin-1 chains, they do not have exact analytical called localizable entanglementbetween two ends, which
2
solutions for a wide range of parameters including the can be extracted by measuring every intermediate spins
2
Rabi frequencies of the classical fields, g is the corre-
j
sponding atom-cavity coupling rate, and J is the inter-
cavity hopping rate of photons. Both the transitions are
coupledtothesamecavitymode. Thetransitionbetween
a and b is induced by two-photon Raman transition
| i | i
using far-detuned lasers. Here, j,k represent nearest
h i
neighbor pairs. For now, we ignore the spontaneous de-
cay rate γ of the atom.
Before we proceed, it is instructive to write down our
parameter regime:
FIG. 1: Involved atomic levels and transitions. Both transi- g2 g2
tions |ai↔ |ei and |bi ↔ |ei are coupled to the same cavity 1 = 2 , (2)
∆ ∆
mode with coupling rates g1 and g2 and with detunings ∆1 1 2
and∆2,respectively. TwolaserfieldswithRabifrequencyΩ1 M
tainvdelyΩ,2anadrethalesotraanpspiltiieodnwbietthwedeentugnrionugnsd∆le1vaelnsd|a∆i2a,nrdes|pbiecis- ∆j,∆1−∆2 ≫r 2 gj ≫J >∼|Ωj|, (3)
g2 g2
driven with Rabi frequency ω/2 by Raman lasers. γ denotes M 1 M 1 ω ω 2J. (4)
theatomic spontaneous decay rate. ∆ ∼(cid:12) ∆ ± (cid:12)∼| |≫
1 (cid:12) 1 (cid:12)
(cid:12) (cid:12)
The condition (2) c(cid:12)an be fulfi(cid:12)lled with conventionally
used alkali-metal atoms, such as rubidium and caesium.
[22] and then used for quantum communication. It is
For example, one may choose ground hyperfine lev-
hardtodemonstratetheseschemesinopticallatticesow-
els F =1,m = 1 and F =2,m = 1 of a 87Rb
ing to the difficulty of addressing individual sites [23]. F F
atom| to represent−ai and |b , respectively−, aind use σ+-
Although the idea of communicating using spin chains
| i | i
polarized light, for which g > g . The detuning ∆
is ultimately meant for solid-state applications, our sys- 1 2 j
is then comparable to the hyperfine splitting between
tem can serve as a preliminary test for comparing and
the two levels. Although there are multiple excited lev-
contrasting the performance of various spin-s chains.
els, their contributions can be summed up and denoted
We use two ground levels of a three-levelatom to rep-
resent an s = 1 spin (in a rotated basis, as will be seen by single parameters in what follows. The other condi-
2 tions (3) and (4) can be satisfied simultaneously when
later). We start by recalling that in terms of two states
and of one atom, the s = 1 spin is described in Mg /∆ J/ Mg . Forexample,ourschemeworks
t|↓eirms of|↑oiperators sZ = 1( 2 ), s+ = , q 2 j j ≫ q 2 j
tahnadtsif−th=er|e↓iahr↑e|.MOiduernstti2caar|lt↑iianthgo↑m|p−osi,n|o↓tniiehs↓c|aannostbrsaeirg|v↑haittfhio↓or|n- wbyelsltirnocnagseat∆omj/-1c0a0v0it∼y cqouMp2lginjg/.100∼J,whichisallowed
wardly define total spin operator SZ = M sZ with Ourregimeischosensothattheexcitationoftheatom
j=1 j
or the cavity photon is suppressed (condition (3)), while
S± = M s± (j is the index for the atomPs), by which
j=1 j thecommunicationbetweenatomsismediatedbyvirtual
the atoPms represent S = M, M 1, and so on. Keeping cavity photons. The first step is to adiabatically elimi-
2 2 −
this in mind, let us consider a coupled array of identical natethe excitedstate usingtheconventionalmethod,by
cavities in anarbitrarygeometry,eachof whichcontains which the effective Hamiltonian is given by
M identical single atoms. We employ the Dicke-type
model, in which every atom in a cavity interacts with H = g12 Λaa+Λbb a†a
the cavity mode with the same coupling strength [24]. − ∆ j j j j
Xj 1 (cid:0) (cid:1)
Let us first consider a simple case, as depicted in Fig. 1.
Let us denote by |ψijk the state |ψi of the kth atom in − µ1Λbja+µ2Λajb aj +h.c. (5)
the jth cavity. In the rotating frame, the Hamiltonian Xj (cid:2)(cid:0) (cid:1) (cid:3)
reads + ω(Λab+Λba) J(a†a +a a†),
2 j j − j k j k
H = ei∆1tΩ Λeb+ei∆2tΩ Λea+h.c. Xj hXj,ki
1 j 2 j
Xj (cid:2) (cid:3) where µ = gjΩ∗j. Now let us introduce spin operatorsin
+ (ei∆1tg Λea+ei∆2tg Λeb)a +h.c. j ∆j
1 j 2 j j (1) a rotated basis
Xj (cid:2) (cid:3)
ω 1 1
+ (Λab+Λba) J(a†a +a a†), = (a + b ), = (a b ) (6)
2 j j − j k j k (cid:26)|↑i √2 | i | i |↓i √2 | i−| i (cid:27)
Xj hXj,ki
to represent an s = 1 spin. Note that these are the
2
where Λxy = M (x y ) (x,y = a,b,e), a is the eigenstates of ω(a b + b a). The underlying idea is
annihilatjion opPerka=to1r|foirhth|ejkjth cavity mode, ∆jj is the to apply the R2am|ainhl|aser|siwhit|h Rabi frequency ω2 con-
correspondingdetuning,Ω and ω arethecorresponding stantly, introducing a fixed amount of energy splitting
j 2
3
ω between the two spin states. The total spin is then
| |
defined in terms of the operators
M M
SZ = sZ and S± = s±, (7)
j jk j jk
Xk=1 kX=1
where sZ = 1( ) , S+ = ( ) , and
jk 2 |↑ih↑|−|↓ih↓| jk jk |↑ih↓| jk
s− =( ) . ThetotalspinisgivenbyS2 =(SZ)2+
jk |↓ih↑| jk j j
1(S+S−+S−S+). If M is even (odd), the atoms repre-
2 j j j J
sent integral (half-integral) spins up to M2 . The atomic FIG.2: AdditionallaserstogetfullcontrolofthespinHamil-
operatorsarenowwrittenasΛ↓↓ = s−s+ = M SZ, tonian. These lasers are applied in addition to the set up of
Λ↑↑ = s+s− = M +SZ, Λj↑↓ =PSk+,jkanjdk Λ↓↑2=−Sj−. Fig. 1.
j k jk jk 2 j j j j j
SubstitPuting these operators, the Hamiltonian in the ro-
tating frame reads where µ± = µ µ , µ = g1Ω∗3( 1 + 1 ), and µ =
H =−Xj (cid:20)(cid:26)eiλtµ+12SjZ +ei(λ+ω)tµ2−12Sj+ agn22Ωo∗4th(e∆1r23f+a4r-∆d2e1+t3uδ±n).edT4lhaeseS3rtfiareklds2(huifsti∆n−g1µazd|∆biffi1e+hrbδe|nintdleuvceelda4nbdy
polarization)resultsinadding µz(eiωtS++e−iωtS−)
µ− j 2 j j
ei(λ−ω)t 12S− a +h.c. (8) in Hamiltonian (8). Recall thaPt in our previous deriva-
− 2 j (cid:27) j (cid:21)
tion, we have adjusted 0,λ,λ ω so that they are
J(a†a +a a†), distinct in frequency with{ the sim±ilar}frequency spacing
− j k j k
X (condition (4)), thereby causing each summation in the
hj,ki
Hamiltonian (8) to contribute independently to the con-
where λ = M g12 and µ± = µ µ . Note that in view stantsinthefinalHamiltonian(9). Weadjustω,λ,λ ω,
∆1 12 1± 2 λ δ,andλ δ ωinthesamespirit. Fortheeaseof±pre-
of conditions (3) and (4), the effective Rabi frequency − − ±
sentation,let us take a particularsituation whereω >0,
(cid:12)qM2 µ±12(cid:12) is much smaller than λ, |λ±ω|, and |ω|. This λ = 3ω, and λ−δ = −6ω, although this is not a neces-
(cid:12)allows us(cid:12)to make use of the adiabatic elimination once sarycondition. WethenobtainthesameHamiltonian(9)
(cid:12)more. W(cid:12)e extend the method in Ref. [25] to keep up to with parameters given by A = 1( 9 µ− 2 9 µ− 2),
thethirdordertermsandtakeonlythesubspacewithno B = 1( µ+ 2 9 µ− 2 1 µλ+ 126(cid:12)+129(cid:12) µ−−325)(cid:12), 3C4(cid:12) =
cavity photon. Simple algebra as in Ref. [25] yields the λ 12 − 16 12 − 2 34 (cid:12) 35(cid:12) 34 (cid:12) (cid:12)
final Heisenberg spin Hamiltonian λ1(23|µz|2(cid:12)(cid:12)− 13(cid:12)(cid:12)6 µ−12 2(cid:12)(cid:12)− 7(cid:12)(cid:12)30 µ−34 2(cid:12)(cid:12)), D(cid:12)(cid:12) = λJ(cid:12)(cid:12)2(3425(cid:12)(cid:12) µ−12 2 +
333 µ− 2),an(cid:12)dE(cid:12)= J (2(cid:12)µ+(cid:12)2+1 µ+ 2). Not(cid:12)eth(cid:12)atC
H =Xj (cid:2)A(Sj)2+B(SjZ)2+CSjZ(cid:3) wi1s2h2di5elet(cid:12)(cid:12)eor3tm4h(cid:12)(cid:12)einretderi(cid:12)mndseap(cid:12)reenaλd2lesnot(cid:12)ld(cid:12)(cid:12)yet1te2h(cid:12)r(cid:12)(cid:12)amnikn2se(cid:12)(cid:12)dtof3rt4eh(cid:12)(cid:12)eelyt.erHme(cid:12)n|µcez,|(cid:12)2t/hλis,
(9)
D(SXSX +SYSY)+ESZSZ , parametersetcoversanyanisotropicorisotropicHeisen-
−hXj,ki(cid:2) j k j k j k(cid:3) berg spin models, with the single-ion anisotropy turned
on or off.
where A = λ |µ−12|2, B = |µ+12|2 λ |µ−12|2, C = ThegroundstateofthespinHamiltoniancouldbepre-
λ2−ω2 2 λ − λ2−ω2 2 pared by the adiabatic method, as described in Ref. [5].
ω |µ−12|2, D = J( µ−12 2 + µ−12 2), E = 2J µ+12 2. Although the Hamiltonian (9) looks similar to a ferro-
−λ2−ω2 2 2 |λ+ω| |λ−ω| | λ |
This Hamiltonian already covers a wide range of para- magneticone (D,E >0), one canalsosimulate the anti-
metric regimes for the Heisenberg spin model, although ferromagnetic spin Hamiltonian, since other parameters
individual control of the parameters is limited owing to A, B, and C can be adjusted to have any sign. This can
their mutual dependency. Interestingly,the Hamiltonian beeasilyseenbynotingthatiftheparametersA,B,and
alsocontainsthe single-ionanisotropy(SZ)2, whichis of Careadjustedtobe 1timesthoseofadesiredantiferro-
j −
essentialimportanceinhigh-spincases[26],whereasit is magnetic Hamiltonian, the Hamiltonian is equivalent to
merely a meaningless constant in the spin-1 case. the antiferromagnetic one up to global factor 1, hence
2 −
Full controlof the individual parameters is allowedby with an inverted energy spectrum. Consequently, the
bringing in more lasers, shown in FIG. 2, in addition to adiabatic preparation, starting from an antiparallel spin
the set up of FIG. 1. The classical fields with Rabi fre- configuration [5], ends up with the highest energy state,
quency Ω and Ω , which are applied with an additional which in fact is the ground state of the corresponding
3 4
detuning δ λ, make a similarcontributionto the effec- antiferromagnetic Hamiltonian.
∼
tive Hamiltonian as those with Ω and Ω . This can be Although atomic excitation is heavily suppressed, the
1 2
reflected in Hamiltonian (8) by adding the same terms main source of decoherence in our system is the sponta-
with λ and µ± replaced by λ δ and µ±, respectively, neous decay of atoms. In relation to the effective spin
12 − 34
4
model, the atomic spontaneous decay results in depolar- ous assumption of strong atom-cavity coupling. Note,
ization of the spins. This effect can be accounted for however, that testing Haldane’s conjecture for higher-
by considering a conditional Hamiltonian H = H spin chains is more demanding, since the lowest exci-
C
i (γ′ Λaa + γ′ Λbb), where the effective decay rate−s tation gap is expected, from its asymptotic behavior, to
2 j A j B j
arPe approximately given by γ′ = γ |Ω1|2 + |Ω3|2 decrease rapidly with increasing M, while the spin cor-
A (cid:16) ∆21 (∆1+δ)2(cid:17) relation length increases rapidly [16]. There are vari-
and γ′ =γ |Ω2|2 + |Ω4|2 , assuming other lasers are ous micro-cavity technologies under active development
B (cid:16) ∆22 (∆2+δ)2(cid:17) whichareexpectedtofallintoourregimeofstrongatom-
sufficiently detuned and thus make a negligible contri-
cavity coupling [27], such as superconducting microwave
bution to the decay. In particular, if Ω s are chosen in
j
cavities [28], photonic bandgap microcavities [29], and
such a way that the two contributions are balanced, i.e.,
γ′ = γ′ = γ′, the depolarization is nearly independent microtoroidal cavities [30]. These models would be also
A B suited to having a fixed number of atoms in a cavity, by
of the spin state. In this case, the conditional Hamil-
tonian is approximately given by H = H iNMγ′, virtueoftheprogressinthemicro-fabricationtechniques.
C − 2 For example, coupling two superconducting qubits with
where N is the number of cavities. Consequently, the
a single cavity mode in a well-controlled way has been
state of the system at time t may be written as ρ(t) =
demonstrated recently [28], which suggests the viability
e−NMγ′tρ (t)+(1 e−NMγ′t)ρ ,whereρ (t)isthe de-
Q − M Q ofthe proposedwayofengineeringspins [31]. Animpor-
siredquantumstateevolvedbythespinHamiltonianand
tant point is that contrary to the system-specific ideas,
ρ is the fully mixed state. This property is useful for
M our scheme relies on a general model, which would be
testing condensed-matter theories, since even under de-
available in a wide range of current or future systems.
polarization, the quantum nature retained in the coher-
ent portion ρ (t) could be observed over a time scale
Q
1/NMγ′. Onerequirementisthatthespin-spincouplin∼g ThisworkwassupportedbytheKoreaResearchFoun-
ratemultipliedby(M/2)2shouldbemuchlargerthanthe dationGrant(KRF-2007-357-C00016)fundedbytheKo-
global decoherence rate. Reminding that the coupling reanGovernment (MOEHRD). SB thanks the Engineer-
rate is given by 2J µ 2/λ2 (J/M)(Ω / M/2g )2, ing and Physical Sciences Research Council (EPSRC)
j j j
we require γ ∼J | ∆|j 2∼. Since ∆ p M/2g UK for an Advanced Research Fellowship and for sup-
≪ 2N(cid:0)√M/2gj(cid:1) j ≫ p j port through the Quantum Information Processing IRC
from condition (3), this requirement can be met for a (GR/S82176/01)andthe RoyalSociety and the Wolfson
moderate N if J > γ is satisfied along with our previ- Foundation.
∼
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