Coherent tunneling via adiabatic passage in a three-well Bose-Hubbard system

3.7 MB·English

by  C. J. Bradly

Coherent tunneling via adiabatic passage in a three-well Bose-Hubbard system
C. J. Bradly, M. Rab, A. D. Greentree, and A. M. Martin
School of Physics, The University of Melbourne, Parkville, 3010, Australia
(Dated: January 31, 2012)
We apply the Bose-Hubbard Hamiltonian to a three-well system and show analytically that co-
herent transport via adiabatic passage (CTAP) of N non-interacting particles across the chain is
possible. WeinvestigatetheeffectofdetuningthemiddlewelltorecoverCTAPwhenon-siteinter-
particleinteractionswouldotherwisedisruptthetransport. Thecaseofsmallinteractionsisrestated
usingfirst-orderperturbationtheorytodevelopcriteriaforadibaticitythatdefinetheregimewhere
CTAP is possible. Within this regime we investigate restricting the Hilbert space to the minimum
2
necessarybasisneededtodemonstrateCTAP,whichdramaticallyincreasesthenumberofparticles
1
that can be efficiently considered. Finally, we compare the results of the Bose-Hubbard model to a
0
mean-field three-mode Gross-Pitaevskii analysis for the equivalent system.
2

n PACSnumbers: 03.75.-b,05.30.Jp
a
J

0 I. INTRODUCTION ble to transport between excited states of each well [23].
3 This spatial transport is referred to as coherent tunnel-
ingviaadiabaticpassage(CTAP)andhasbeenproposed
The Bose-Hubbard model provides a useful theoreti-
] totransportsingleatoms[24,25],Cooperpairs[26],spin
s cal and experimental platform to study the properties
a states[27],electrons[23,28–31],ultracoldatomsinatom-
of strongly interacting bosonic systems. In particular,
g chip waveguides [32] and photons in three-channel opti-
considerable effort has been applied to the properties
-
cal waveguides [33]. For the case of three-channel op-
t of interacting bosons in a lattice, especially in the con-
n tical waveguides, CTAP for massless particles has been
a textofultracoldatomresearch[1,2]. TheBose-Hubbard
demonstrated [34, 35].
u model has also been extensively employed in the investi-
q gation of the properties of ultracold bosonic atoms con- Recently, considerable attention has been paid to the
t. fined in a double-well potential [3–9]. Additionally, the CTAP of Bose-Einstein condensates (BECs) in three-
a work on double-well systems has been extended to triple well systems [36–40]. These studies have primarily fo-
m
well systems [10–14]. Despite the relative simplicity of cused on a three-mode model of the system (schemat-
- the Bose-Hubbard model it provides a rich space of phe- ically shown in Fig. 1), in both the linear and nonlin-
d
nomenology to explore, much of which is inaccessible to ear regimes, to determine the robustness of CTAP. Ad-
n
o presentanalyticaltechniquesduetothegrowthinsizeof ditionally, numerical simulations, using the mean-field
c the Hilbert space for a large number of particles. Hence Gross-Pitaevskiiequation,havebeenperformedtomodel
[ there is a need to develop analytically tractable prob- CTAPofathree-dimensionalBECcontaining2000inter-

1 lems for Bose-Hubbard systems to gain insight into the acting atoms [39]. Here we take a different approach to
v analytically intractable problems. Here we utilize the studyingadiabatictransportinmany-particlesystemsby
6 Bose-Hubbard model to investigate the adiabatic trans- considering a three-site Bose-Hubbard model with N in-
0 port of ultracold bosonic atoms confined in a triple-well teracting bosons. We find that the non-interacting limit
1
potential.
6
For a single quantum particle in a three-well system
.
1 it is possible to transport the particle between the left
0
and right wells such that the probability of finding it at
2
any time in the classically accessible state in the mid-
1 1 2 3
: dle well is negligible. The ideas underpinning such a al
v transport mechanism stem from stimulated Raman adi- ti
Xi abatic passage (STIRAP) [15–19]. In general, STIRAP oten K12 Δ K23
is employed to transfer population between two atomic p
r
a states, via an intermediate state, with the occupation of
theintermediatestatestronglysuppressed,andisusedin
quantum optics for coherent internal state transfer [19–
22]. In analogy with STIRAP, it is possible to adiabat- position
ically transport a quantum particle from the left well to
the right well such that the occupation of the middle FIG. 1. Three-well system where each well is assumed to
well is exponentially suppressed. For such a system the have only one accessible energy level. Wells one and three
threestatesaretypicallythegroundstatesofeachofthe aredegenerate,whilstthemiddle-wellgroundstateisshifted
three wells and the coupling between the states is the relative to the other two by a detuning, ∆. The tunneling
rate between wells i and j is K .
tunneling rates between the wells, although it is possi- ij
2
N2 = N 0,N,0
K K
12 23
N2 = N − 1 1,N−1,0 0,N−1,1
N2 = N − 2 2,N−2,0 1,N−2,1 0,N−2,2
states where N − 3 ≤ N ≤ 1
2
N2 = 1 N−1,1,0 N−2,1,1 1,1,N−2 0,1,N−1
N2 = 0 N,0,0 N−1,0,1 states where N2 ≤ 1 and 2 ≤ N1 ≤ N − 2 1,0,N−1 0,0,N
FIG. 2. Hilbert space structure for N particles in three wells. Each layer of the Hilbert space is denoted by N and the
2
couplingbetweenlayersisviaK (blackarrows),whichcouplessites1and2,andK (redarrows)whichcouplessites2and
12 23
3.
ofthissystemmapsontoCTAPofasingleparticlealong definetheenergyscaleofthesystembythegroundstate
a chain of 2N +1 sites via an alternating coupling pro- energies of wells 1 and 3, i.e., (cid:15) =(cid:15) =0 but we include
1 3
tocol [41–43]. This mapping allows us to apply several a (positive) detuning of well 2 so that (cid:15) =∆.
2
analytical results to the problem, which would otherwise ThereareM =(N+1)(N+2)/2waystodistributeN
becomputationallyintensive,sincethesizeoftheHilbert particlesamongstthreewellsandtheseconfigurationsare
space basis is of order N2. theFockbasisstates|N ,N ,N (cid:105)fortheHilbertspaceof
1 2 3
This paper is organized as follows: In Section II the the system. The restriction on the individual well occu-
three-site Bose-Hubbard model is introduced and the pation numbers N is (cid:80)3 N =N. Figure 2 shows the
i i=1 i
non-interacting null state is found analytically. The Hilbertspacestructureforthesystem,witheachlayerof
breakdown of CTAP in the Bose-Hubbard model is then the Hilbert space defined by N . The Fock basis states
2
investigated numerically in the presence of interactions. have a natural ordering based on the individual occupa-
In Section III, perturbative methods are used to demon- tionsofeachwell. Thisorderinggivesrisetoatriangular
strate how detuning of the middle well can maintain pattern,withlayersdefinedbythenumberofparticlesin
CTAP once interactions are introduced into the system. well 2, and the left to right ordering by the distribution
Section IV numerically investigates the fidelity of CTAP of the particles in wells 1 and 3, as depicted in Fig. 2.
as a function of the interactions and the detuning of the In general, the system can take multiple, looped paths
middle well. This is compared with an analytic determi- through the Hilbert space, giving rise to non-trivial dy-
nation of the regimes of high fidelity based on the anal- namics [45]. However, nonzero detuning on well 2 makes
ysis presented in Section III. Additionally, a compari- higher states difficult to access by the system, reducing
son between the results obtained for the Bose-Hubbard the contribution from these looped paths. The total adi-
modelandthree-modenon-linearmean-fieldcalculations abatic dynamics can, in the appropriate limit of small
is made. Finally, Section V summarizes the key results interactions, be very well described by the lowest two
obtained in this work and discusses possible avenues for layers of the Hilbert space.
further theoretical and experimental studies.
A. Non-interacting case
II. THREE WELL BOSE-HUBBARD SYSTEM
The Hamiltonian (1) cannot, in general, be solved an-
To begin the analysis we consider a three-well sys- alytically and must be investigated numerically. How-
tem, schematically shown in Fig. 1, subject to the Bose- ever, before investigating CTAP of interacting bosons
Hubbard Hamiltonian in a three-well system, we can gain significant insight
intotheproblembyconsideringthenon-interactingcase,
(cid:16) (cid:17) (cid:16) (cid:17)
Hˆ =−K aˆ†aˆ +aˆ†aˆ −K aˆ†aˆ +aˆ†aˆ which can, in the adiabatic limit, be solved analytically.
12 1 2 2 1 23 2 3 3 2
In the absence of interactions, the system closely re-
3 3
U (cid:88) (cid:88) sembles a single particle in a chain of 2N +1 sites with
+ nˆ (nˆ −1)+ (cid:15) nˆ , (1)
2 i i i i analternatingcouplingscheme[15,41–43]. Itistherefore
i=1 i=1
possible to write down the null state of the Hamiltonian
where aˆ†, aˆ and nˆ are the bosonic creation, annihila- (1)forasystemofN non-interactingparticlesinathree-
i i i
well system:
tion and number operators for site i, K (K ) is the
12 23
t3u),nnUeliisngthmeaptarritxicelele-mpaerntticlbeeitnwteeernacstiitoense1nearngdy2an(d2(cid:15)ainids |D0(cid:105)N = N1 (cid:88)N (−1)k(cid:115)k!(NN−! k)!(cid:18)KK23(cid:19)k|k,0,N−k(cid:105),
the ground state energy of site i. We only consider tun- 12
k=0
neling between nearest neighbors so we set K =0. We (2)
13
3
eigenenergies is
2
k k−2j(cid:113)
E = ∆± ∆2+4(K2 +K2 ), (4)
k,j± 2 2 12 23
K
/0
E forintegerskandj,where0≤k ≤N,0≤j ≤(cid:98)k/2(cid:99)and
E ≡ E = 0 denotes the null state energy. The corre-
0 0,0
sponding eigenstates |D (cid:105) , hereafter denoted simply
-2 k,j± N
(a) (b) as|D (cid:105),canbefoundinclosedformforanyN. Thenull
α
state (2) exists for any detuning ∆ but, as indicated by
Eq. (4), it is not unique when ∆ = 0, namely when k is
2
odd and j = 0 there are eigenenergies that are integer
multiples of ∆ which vanish when ∆ → 0. In this case
E
K 1,0- there are (N +1)/2 null states for odd N, and N/2 null
E/0 states for even N. These extra null states have contri-
butions from basis states with N > 0 so if the system
2
evolvedintooneofthemthentheCTAPtransportwould
-2
be disrupted. In addition, the adiabatic coupling (dis-
(c) (d)
cussed in Section III) between the preferred null state
0.0 0.5 1.00.0 0.5 1.0
t/τ t/τ and these extra null states is undefined, indicating that
these degenerate states are unavoidable. To prevent this
FIG. 3. Non-interacting energy eigenvalue spectrum for we can introduce a small positive detuning to lift these
N =3 using the squared-sinusoidal pulsing protocol Eq. (5). degeneracies, as shown in Fig. 3(b). This will isolate the
(a)Withnodetuning,∆/K =0,thenullstate(dashed)isde- desired null state (2) and improve the fidelity of CTAP.
generate. (b) For small detuning, ∆/K =0.2, the states are However, larger detunings, shown in Figs. 3(c,d), can in-
liftedslightly,removingthedegeneracy. (c)Moderatedetun- troduce new problems related to the adiabatic couplings
ing, ∆/K = 0.6, lifts states enough that some pass through of other eigenstates. We address this issue in Section III
thenullstateandcrossatcertaintimes,potentiallyinterrupt-
B.
ing the adiabatic transport. (d) Large detuning, ∆/K = 2,
If well 2 is not detuned, i.e. ∆=0, then CTAP is still
pushes those crossing states past the null state at the cost of
possible provided we initialize the system correctly and
bringing the E state closer to the null state at t=τ/2.
1,0−
evolve the system adiabatically, despite the extra null
states [44]. This is because when K = 1, K = 0 or
12 23
vice-versa, the extra null states are in a superposition
with normalization
of more than one basis state, whereas the preferred null
state,Eq.(2),istheonlyeigenstatethatisexactlyasin-
N =(cid:34)(cid:88)N N! (cid:18)K23(cid:19)2k(cid:35)12 =(cid:18)1+ K223(cid:19)N2 . gexlepbliacistilsystbaetleo.wF.orHtehneccea,sewoefctownoclpuadretictlheastthtihseisseshexotwrna
k!(N −k)! K K2
k=0 12 12 null states do not interfere with CTAP and demonstrate
(3)
this for the case of two particles below [43].
The null state is unique for ∆ (cid:54)= 0. If the system is
The time-evolution ofthe system is determinedbythe
prepared in the |D (cid:105) eigenstate and evolved adiabat-
0 N pulsing of the potential barriers K and K . If all the
12 23
ically, then the system will only occupy states in the
particles are initialized in well 1, i.e. Ψ(0) = |N,0,0(cid:105),
N = 0 layer of the Hilbert space (see Fig. 2), so there
2 then to induce the required population transfer we pulse
is vanishing probability of detecting a particle in well 2.
the left and right barriers in counter-intuitive order.
The system’s Hilbert space trajectory will also explore
The precise form of the pulsing is not as important for
the N = 1 layer, since these are necessary to couple
2 CTAPastheorder,soforconveniencewechoosesquared-
to the N = 0 states, but occupation of the N = 1
2 2 sinusoidal pulsing
states is zero in the adiabatic limit. Hence, under adi-
abatic evolution, N particles initially in state |N,0,0(cid:105) (cid:18)πt(cid:19) (cid:18)πt(cid:19)
K (t)=Ksin2 , K (t)=Kcos2 , (5)
(K12 = 0, K23 = 1) can be transported to the state 12 2τ 23 2τ
|0,0,N(cid:105) (K =1, K =0), via the states |N −k,0,k(cid:105),
12 23
with 0 < k < N. This transformation must be per- where K is the maximum tunneling rate and τ is the
formed adiabatically to maintain the system in the null total pulse time. From Eq. (2), at t = 0 (t = τ), the
state,discussedfurtherinSectionIII.Thekeyissuehere probability of being in the |N,0,0(cid:105) (|0,0,N(cid:105)) state is
is that by changing the couplings appropriately we can 1. The probability that the system is in a particular
transport the system through the Hilbert space. This is basis state |φ(cid:105) = |N ,N ,N (cid:105) as a function of time is
1 2 3
equivalent to transporting the particles from well 1 to P (t) = |(cid:104)N ,N ,N |Ψ(t)(cid:105)|2. These probabilities de-
|φ(cid:105) 1 2 3
well 3 without intermediate occupation of well 2. scribethetrajectorythesystemtakesthroughtheHilbert
In the non-interacting limit, the full spectrum of M space over the course of the pulsing. Fig. 4 shows the
4
1.0 To demonstrate CTAP in the non-interacting case we
P P
10,0,0 0,0,10 revisitthecaseofoneparticle[24,25,28–30]. Thesingle
0.8
particlebasisstatesare|0,1,0(cid:105),|1,0,0(cid:105),|0,0,1(cid:105)forwhich
P P
0.6 6,0,4 4,0,6 the Hamiltonian (1) is expressed as the matrix
P P
Pϕ 7,0,3 P 3,0,7  
0.4 5,0,5 ∆ −K12 −K23
P8,0,2 P2,0,8 Hˆ1 =−K12 0 0 , (6)
0.2 −K 0 0
P P 23
9,0,1 1,0,9
0.00.0 0.2 0.4 t/τ 0.6 0.8 1.0 and the null state, Eq. (2), is
K |1,0,0(cid:105)−K |0,0,1(cid:105)
FIG. 4. Basis state occupation probabilities P (t) for the |D (cid:105)= 23 12 , (7)
|φ(cid:105) 0 (cid:112)
non-interacting null state for ten particles. The evolution is K122+K223
for the squared-sinusoidal pulsing, with U =0 and ∆=0, in
the adiabatic limit (τ large). The basis states with N > 0 whichhasnocontributionfrom|0,1,0(cid:105). Thisisprecisely
2
are negligibly occupied throughout the pulsing. the form of the canonical null state used for CTAP [29]
and is equivalent to the more familiar dark state in dou-
bly driven three-level atoms [46]. Similarly, we may con-
population transfer, for ten particles, from the |10,0,0(cid:105) sider the two-particle system, which is spanned by the
state to the |0,0,10(cid:105) state via the null state, Eq. (2), in basisstates|0,2,0(cid:105),|1,1,0(cid:105),|0,1,1(cid:105),|2,0,0(cid:105),|1,0,1(cid:105)and
the adiabatic limit. |0,0,2(cid:105). The Hamiltonian (1) is
√ √
 
2∆ − 2K − 2K 0 0 0
√ 12 23 √
−√2K12 ∆ 0 − 2K12 −K23 √0 
 
Hˆ2 =− 2K23 √0 ∆ 0 −K12 − 2K23, (8)
 0 − 2K 0 0 0 0 
 12 
 0 −K23 −√K12 0 0 0 
0 0 − 2K 0 0 0
23
and the null state, Eq. (2), becomes B. Interacting system
√
|D (cid:105)= K223|2,0,0(cid:105)− 2K12K23|1,0,1(cid:105)+K122|0,0,2(cid:105). The value of the Bose-Hubbard model is that it can
0 K2 +K2 model the interesting physics of real interacting atomic
12 23
(9) systems. The Hamiltonian for the three-well system
Thisstatealsohasnocontributionfromstateswithoccu- Eq.(1)doesnothaveageneralnullstateforU (cid:54)=0,soto
pationofwell2(N >0). Asmentionedabove,however, findthebehavioroftheparticleswemustnumericallyin-
2
if ∆=0 there is another null state tegrate the time-dependent Schr¨odinger equation for the
wavefunction Ψ(t) of the system. Initializing the system
1 K2 as Ψ(t = 0) = |N,0,0(cid:105), we apply the CTAP pulsing se-
|D(cid:48)(cid:105)=− √ |0,2,0(cid:105)+ √ 12 |2,0,0(cid:105)
0 2 2(K2 +K2 ) quence, Eq. (5), and determine the probability P|φ(cid:105)(t)
12 23 that the system is in a particular basis state |φ(cid:105) as a
K K K2
+ 12 23 |1,0,1(cid:105)+ √ 23 |0,0,2(cid:105). function of time. If Ψ(τ) = |0,0,N(cid:105), then all the parti-
K2 +K2 2(K2 +K2 ) cles are said to have been transported across the chain,
12 23 12 23
(10) but for efficient CTAP we also require that throughout
thepulsingprotocolthecontributiontothewavefunction
from basis states with N >0 is minimized.
Ifthesystemisintheprimarynullstate,Eq.(9),thenit 2
canbe transformedfrom thestate |2,0,0(cid:105) when K =0 In Fig. 5 we plot P (t) for τ = 500K−1, N = 5 and
12 |φ(cid:105)
and K = 1 to the state |0,0,2(cid:105) when K = 1 and various values of U and ∆. While five particles is still in
23 12
K =0,butforbothofthesecasesweseefromEq.(10) thefew-bodyregime,ratherthanthemany-bodyregime,
23
that the secondary null state |D(cid:48)(cid:105) always has a contri- itisenoughtodemonstratetheinterestingphysicsofthe
0
bution from the |0,2,0(cid:105) state. If both particles are ini- model while still being numerically tractable with stan-
tialized in well 2 then the system will be in the preferred dard techniques. Fig. 5(a) shows transport for U = 0,
null state |D (cid:105) and will remain in that state if evolved ∆=0 and the system is confined to the lowest (N =0)
0 2
adiabatically. layer of the Hilbert space, as expected from the analysis
5
1 and 3, such that P (τ) ≈ 1. However, there is a
5,0,0 1.0 |0,0,5(cid:105)
limit to how effective this control is since increasing ∆
N2 = 0 reduces adiabaticity for constant pulsing time τ, as dis-
cussedbelow. Increasingthepulsingtimeτ (shortofthe
4,1,0
0.9
trueadiabaticlimitτ →∞)doesnotrestoreCTAPwhen
N = 1
s 2 ∆ = 0 for an interacting system and can even make it
e
at3,2,0 worse[45]. Thenon-adiabaticityinthisregimeisinstead
st N2 = 2 0.8 due to the CTAP state crossing with other eigenstates,
2,3,0
similar to Fig. 3(c) for the non-interacting case.
N = 3
2
1,4,0 (a)
0,5,0 N2 = 4 0.7
5,0,0 III. PERTURBATION ANALYSIS
N = 0
2 0.6 A. Large detuning limit
4,1,0
N = 1 We have seen that making ∆ large relative to NU re-
s 2
te3,2,0 0.5 P ϕ storesCTAPforfinitepulsingperiods,motivatinganan-
a
st N = 2 alytic perturbation investigation of the Hamiltonian. As
2
2,3,0 such, it is beneficial to rewrite the Bose-Hubbard Hamil-
N = 3 0.4 tonian (1) as H˜ =Hˆ/∆=Hˆ(0)+U˜Vˆ, where
2
1,4,0 (b)
0,5,0 N2 = 4 Hˆ(0) =−K˜ (cid:16)aˆ†aˆ +aˆ†aˆ (cid:17)−K˜ (cid:16)aˆ†aˆ +aˆ†aˆ (cid:17)+nˆ ,
5,0,0 12 1 2 2 1 23 2 3 3 2 2
0.3
(11)
N = 0
2
3
4,1,0 N = 1 0.2 Vˆ = 12(cid:88)nˆi(nˆi−1), (12)
s 2 i=1
e
t3,2,0
sta N2 = 2 0.1 K˜12 =K12/∆, K˜23 =K23/∆ and U˜ =U/∆ is the small
perturbation parameter if NU/∆ (cid:28) 1. In this Section
2,3,0
N = 3 thesuperscript(0)denotesnon-interactingquantities. It
2
1,4,0 (c) N = 4 is important to note that although the effect of interac-
0,5,0 2 0.0 tionschangeswithtimeasthedistributionofparticlesin
0.0 0.25 0.5 0.75 1.0
t/τ the system changes, Vˆ is time-independent in the basis
of Fock states.
FIG.5. HilbertspacetrajectoriesP (t)forfive-particles,us- In this limit it is possible to apply standard perturba-
|φ(cid:105)
ing thepulsing sequence defined inEq. (5), with τ =500/K. tion theory to calculate the effect of small interactions
(a) U = 0, ∆ = 0, the system evolves as expected, remain- on the null state of the non-interacting case, Eq. (2). To
ing in the N = 0 subspace. (b) NU/K = 0.03, ∆ = 0,
2 first order, the interacting null state energy is
strong interactions push the system out of the N = 0 sub-
2
space,andreducethefidelityofthetransportevenwithinthe E =E(0)+(cid:104)D(0)|Vˆ|D(0)(cid:105), (13)
N = 0 subspace. (c) NU/K = 0.03, ∆/K = 3, a large de- 0 0 0 0
2
tuningofthemiddlewellrestoresCTAPandtheparticlesare and the first-order correction to the null state wavefunc-
transported across the well with negligible occupation of the
tion is
middle well. In each plot the states are presented along the
y|1-,a4x,is0(cid:105)a;n|d0,a3r,e2(cid:105)o;rd|e1r,e3d,1f(cid:105)r;om|2,b3o,t0t(cid:105)o;m|0t,o2,t3o(cid:105)p;:||10,,25,,20(cid:105)(cid:105);; ||02,,42,,11(cid:105)(cid:105);; |D0(cid:105)=|D0(0)(cid:105)+(cid:88) (cid:104)DEα((00))|V−ˆ|ED(0(00))(cid:105)|Dα(0)(cid:105), (14)
|3,2,0(cid:105); |0,1,4(cid:105); |1,1,3(cid:105);|2,1,2(cid:105); |3,1,1(cid:105); |4,1,0(cid:105); |0,0,5(cid:105); α(cid:54)=0 0 α
|1,0,4(cid:105); |2,0,3(cid:105); |3,0,2(cid:105); |4,0,1(cid:105) and |5,0,0(cid:105). The dashed
line marks the desired final state |0,0,5(cid:105). where α represents the k,j± of Eq. (4). The eigenstates
|D(0)(cid:105) of the non-interacting Hamiltonian corresponding
α
presented in the previous section. In Fig. 5(b) the in- totheenergiesofEq.(4)canbefoundinclosedform,but
teractions are increased to NU/K = 0.03, with ∆ = 0, forlargeN itbecomesmoreefficienttoevaluateEq.(14)
and there is a significant departure from the idealized numerically. From these first order perturbative eigen-
null state transport observed in Fig. 5(a). Specifically, states it is then possible to calculate the basis state oc-
the transport is no longer confined to the lowest layer cupationprobabilitiesP (t), toinvestigatethestability
|φ(cid:105)
of the Hilbert space and P (τ) < 1. However, it is of CTAP in the presence of weak interactions. In the
|0,0,5(cid:105)
possible to recover CTAP via the introduction of detun- regime of small interactions relative to the detuning of
ing,∆. Forexample,Fig.5(c),whereNU/K =0.03and well 2 this approach is more efficient than integrating
∆/K = 3, shows full population transfer between wells the Schr¨odinger equation.
6
1.0 50,0,0 1.0 mizedatt=τ/2forthepulsingschemegivenbyEq.(5).
(a) P5,0,0 P0,0,5 (b) For ∆/K (cid:29)1 at t=τ/2
0.8
0.75 N2 = 0 √ √ √
Pϕ0.5 P3,0,2 P2,0,3 states0,0,50 00..46Pϕ A1,0− = 2πτKN2∆ + π2τ∆N + π16Nτ∆K32 +O(∆−5). (16)
0.25 P4,0,1 P1,0,4 N2 = 1 0.2 The leading order of Eq. (16) is proportional to ∆,
0.0 49,1,0 0.0 so there is a limit on the detuning before the adia-
0.0 0.25 0.5 0.75 1.0 0.0 0.25 0.5 0.75 1.0
t/τ t/τ batic coupling between the null state and the close state
|D(0) (cid:105) becomes too large and the transport becomes
FIG. 6. (a) P (t) numerically calculated for the complete 1,0−
|φ(cid:105) non-adiabatic. To first order in ∆, the requirement
Hilbert space (red solid curve) and via perturbation theory √
A (cid:28) 1 therefore becomes τK/(2π N) (cid:29) ∆/K for
(dashed black curve) for N = 5, ∆/K = 1, τK = 500 and 1,0−
NU/∆ = 0.01. In both cases the occupation of states with the non-interacting system. Note that in the true adia-
N2 >0istoosmalltobevisible. (b)Hilbertspacetrajectory batic limit, τ →∞, the adiabaticity parameters Aα will
foratwo-layerHilbertspaceforN =50,∆/K =1,τK =500 always vanish.
and NU/∆=0.01. In the limit of small interactions relative to the detun-
ing ∆ (cid:29) NU, we again use perturbation theory to ap-
Fig. 6(a) compares the perturbation theory method to
proximatetheadiabaticityparameterbyfindingthefirst-
a complete Hilbert space calculation for five particles.
ordercorrectiontoalltheeigenstates. InsertingEqs.(13)
Forsmallinteractionsrelativetothedetuning,transport
and (14) in Eq. (15) we obtain
from the state |5,0,0(cid:105) to |0,0,5(cid:105) is dominated by the
N2 =0 states, just as in the non-interacting case. (cid:12)(cid:12)(cid:104)D(0)|D˙(0)(cid:105) (cid:104)D(1)|D˙(0)(cid:105)+(cid:104)D(0)|D˙(1)(cid:105)
A =(cid:12) α 0 + α 0 α 0
α (cid:12)(cid:12)Eα(0)−E0(0) Eα(0)−E0(0)
B. Adiabaticity (cid:16) (cid:17) (cid:34)(cid:18)NU(cid:19)2(cid:35)(cid:12)(cid:12)
−(cid:104)D(0)|D˙(0)(cid:105) E(1)−E(1) +O (cid:12),
α 0 α 0 ∆ (cid:12)
(cid:12)
To determine when it is appropriate to describe trans-
port of the particles from well 1 to well 3 via CTAP, the (17)
adiabaticityoftheevolutionneedstobeverified. Increas-
where
ing the pulsing time τ will in general make the trans-
port more adiabatic, assuming no eigenstate crossings,
but otherwise the adiabaticity of the transport within |D(1)(cid:105)=(cid:88) (cid:104)Dβ(0)|Vˆ|Dµ(0)(cid:105)|D(0)(cid:105), (18)
the null state |D (cid:105) depends on the coupling to the other µ E(0)−E(0) β
0 β(cid:54)=µ µ β
energy eigenstates |D (cid:105) and their energies E . To quan-
α α E(1) =(cid:104)D(0)|Vˆ|D(0)(cid:105), (19)
tify the adiabaticity of the transport we introduce the µ µ µ
parameter
which is valid for any pulsing protocol. As with the non-
|(cid:104)D |D˙ (cid:105)| interacting case, the adiabaticity of the transport in the
A = α 0 , (15) interacting case can be improved by increasing the puls-
α |E −E |
α 0 ingtimeτ,butforfinitetime,asrequiredinexperiment,
Eq. (17), can be used to show where the CTAP protocol
and say that the evolution is adiabatic if A (cid:28) 1 for
α
becomes non-adiabatic and is used in the next section in
all α (cid:54)= 0. Eq. (15) may be generalized to any pair of
conjunction with numerical results.
eigenstates but here we are interested in the coupling
of the CTAP null state to other nearby eigenstates. It
is also clear from Eq. (15) that removing degeneracies
IV. RESTRICTED HILBERT SPACE
vianon-zerodetuning∆improvestheadiabaticityofthe
transport. In particular, we focus on the large detuning
limit ∆/K (cid:29)1. For the case of small interactions relative to the de-
In the absence of interactions, the eigenenergy that tuning, if the system is evolved adiabatically with all
is closest to E (for any N) is E(0) = (∆ − particles initially in the state |N,0,0(cid:105), the CTAP pro-
0 1,0−
(cid:112)4(K2 +K2 )+∆2)/2, from Eq. (4). It is then pos- tocol only probes those states which have zero (N2 =0)
12 23 or singular occupation (N = 1) of well 2, i.e. the bot-
sible to determine |D1(0,0)−(cid:105) and calculate the adiabatic tom two layers of the Hilb2ert space structure, shown in
coupling A1,0− for the squared-sinusoidal pulsing. Re- Fig. 2. If the system is not evolved adiabatically, or the
quiring A1,0− (cid:28) 1 will suppress the coupling between interactions are large, then the full Hilbert space con-
thiseigenstateandthenullstate. Sincethisistheeigen- tributes to the dynamical evolution of the system, as
state closest to the null state, Aα will most likely be less seen in Fig. 5(b). From this observation it is possible
than A1,0− for all other eigenstates α. From Fig. 3(d) to consider the following question. If the Hilbert space
we can also see that E(0) , and therefore A , is maxi- is restricted to the lower layers (N = 0,1,...), does
1,0− 1,0− 2
7
thesolutioninthislimitrepresentavalidapproximation
1.0
for the adiabatic transport between states |N,0,0(cid:105) and
|0,0,N(cid:105), via CTAP? Hilbert space restrictions, if valid,
0.75
will facilitate the investigation of systems for large N.
K
Rather than look at complete Hilbert space trajecto- U/0.5 0.07
ries for every different value of ∆ and U to judge the N 0.005.06
validity of a restriction, we can consider two criteria 0.25
to identify successful CTAP. Firstly, to determine if all
(a) (b)
the particles have been transported across the chain we 0.0
1.0
calculate the probability that at the end of the pulsing
P (τ) ≈ 1. Secondly, to ensure that occupation of
|0,0,N(cid:105) 0.75
well 2 has been suppressed during the protocol we cal-
culate the maximum occupation of a state with N =1, K
2 U/0.5 0.07
maxP|N1,N2=1,N3(cid:105)(t). The latter can be thought of as a N 0.005.06
lower bound on occupation of well 2, since, if occupa-
0.25
tion of a N = 1 state becomes too significant at any
2
(c) (d)
point, then the system may access higher basis states,
0.0
which reduces the fidelity of the CTAP protocol. These 1.0
two conditions are not mutually exclusive but in tandem
they can be used to determine the fidelity of the popula- 0.75
tion transfer. Figs. 7(c-f) plots these two quantities for K
atwo-andfour-layerHilbertspacerestriction,compared U/0.5 0.07
N 0.005.06
to the full Hilbert space calculation, Fig. 7(a,b) for five
particles. 0.25
The strongest restriction is to consider only the two (e) (f)
0.0
bottom layers of the Hilbert space structure, states with 1.0
N = 0,1. There are only 2N +1 basis states so the
2
problem scales linearly with N, rather than with N2 for 0.75
the full system, and the Hamiltonian can be written in a
K
tridiagonal form. The restricted Hamiltonian is still not U/0.5
invertible analytically for general U but the Schr¨odinger N
equation can easily be integrated numerically for a sys- 0.25
tem of several hundreds of particles, since the problem (g) (h)
is then linear in N (there are 2N +1 basis states in the 0.0
0.0 0.75 1.5 2.25 3.0 0.0 0.75 1.5 2.25 3.0
two bottom layers, of the Hilbert space structure). Fur- Δ/K Δ/K
thermore, the null state of this restricted Hamiltonian
in the non-interacting limit is exactly the null state for FIG. 7. (a,c,e) [white=1, black=0] P|0,0,N(cid:105)(t = τ) for: (a)
a full Hilbert space calculation; (c) a restricted two-layer
CTAP of a single particle along a chain of 2N +1 sites
(N = 0,1) Hilbert space calculation and (e) a restricted
[41, 42]. Fig. 6(b) shows the Hilbert space trajectory for 2
four-layer (N = 0,1,2,3) Hilbert space calculation. (b,d,f)
N =50andNU/∆=0.01usingthetwo-layerrestriction 2
[white=0] maxP (t) for: (b) a full Hilbert space
(N =0 and N =1). In this figure the only states with |N1,N2=1,N3(cid:105)
2 2 calculation [black=0.38], with N = 1; (d) a restricted two-
2
significant occupation are those with N2 = 0, with es- layer(N2 =0,1)Hilbertspacecalculation[black=0.35],with
sentiallyzeroprobabilityofthestateswithN2 =1being N2 =1and(f)arestrictedfour-layer(N2 =0,1,2,3)Hilbert
occupied. SinceonlythestateswithN =1coupletothe space calculation [black=0.45], with N = 1. In (a,c,e) the
2 2
rest of the Hilbert space structure (see Fig. 2), it could soliddashedlineisgivenbytheenergylevelcriterion,Eq.(23)
be expected that in this case a two layer Hilbert space and solid curves are for A1,0− =0.05,0.06,0.07 as calculated
is a good representation of the transport that would be from Eq. (17). (g) [white=1, black=0] Meanfield calculation
ofthefinalpopulationofwellthree: N (t=τ). (h)[white=0,
observed in a full Hilbert space calculation. 3
black=1]Meanfieldcalculationofthemaximumpopulationof
The two-layer restriction, while powerful, is only valid
well two: maxN (t)/N. For the above plots τK = 500 and
2
incertainregimes. Inparticular,Fig.7(c)showsthatthe
for (a-f) N =5.
two-layerrestrictionallowsfullpopulationtransferinthe
case of high interaction strength but low detuning, while of computation for a larger Hilbert space. A weaker
Fig. 7(d) shows that at higher interaction strengths well restriction is to consider the four bottom layers of the
2 is being occupied, regardless of detuning, in contrast Hilbert space tree (N = 0,1,2,3). The basis contains
2
to the full Hilbert space calculation, Figs. 7(a,b). Away 4N −2 states, so it still scales linearly with N, but the
from the small interaction limit, these results show that Hamiltonian is not tridiagonal. The benefit is being able
the two-layer restriction does not model CTAP well. to investigate couplings to higher states of the Hilbert
Toincorporatelargerinteractionswecansacrificeease space in the presence of strong interactions. Figs. 7(e,f)
8
showthecalculationsforthefour-layerrestrictedsystem. }
Weimmediatelyseethattheinclusionofbasisstateswith (c)
N2 =2,3 is necessary to show that CTAP is unstable in Δ Δ
the high interaction, low detuning regime, as shown, for rgy }(b)
example, in Fig. 5(b). This concept of expanding the ne
Hilbert space from the bottom up, rather than consid- e 3U Δ 3U
2U 2U 2U 2U
ering the whole space, is analogous to the way matrix
(a)
product states minimize entanglement by exploring the 3,0,0 2,1,0 2,0,1 1,1,1 1,0,2 2,1,0 0,0,3
Hilbert space as required [47]. To determine the point basis states
at which this instability occurs we need to consider the
energiesofthebasisstatestofindlimitsintheparameter FIG. 8. Relative energies of basis states in the N = 0 and
2
space for when CTAP works. N =1subspacesforasystemofthreeparticles. (a)Withno
2
interactions(U =0)andnodetuning(∆=0)thesestatesare
alldegenerate. (b)Intheinteractingsystemwithnodetuning
(cid:80)
(∆ = 0), the interaction energy ( UN (N −1)/2) shifts
i i i
each state differently (red shifts). Note that this includes
A. Limiting the CTAP parameter space
|1,1,1(cid:105), which has zero interaction energy. (c) Detuning the
middlewellintheinteractingsystemshiftstheN =1states
2
All the models show that CTAP is not possible in the bythesameamount(bluearrows). Thedashedlineindicates
that there exists a minimum detuning for which all N = 1
high detuning, high interaction regime. Although we 2
states will have higher energy than all N =0 states.
have seen that detuning can recover CTAP in the pres- 2
ence of interactions, too much detuning will still break
the adiabaticity of the transport. The solid curves in
the Hilbert space is
Figs. 7(a,c,e) are contours of constant adiabaticity, cal-
culated from Eq. (17), for A1,0− = 0.05,0.06,0.07. For U (cid:18)N −1(cid:19)(cid:18)N −3(cid:19)
a given finite pulsing time τ these curves accurately pre- E|N2−1,1,N2−1(cid:105) =2× 2 2 2 +∆, (21)
dict the limit in the parameter space where adiabatic
E =E
transport is possible. Increasing τ will push this bound- |N−2,1,N(cid:105) |N,1,N−2(cid:105)
2 2 2 2
ary further out and allow stronger interactions relative U (cid:18)N −2(cid:19)(cid:18)N −4(cid:19)
=
to the detuning of well 2.
2 2 2
The other region of interest is the low detuning, high (cid:18) (cid:19)(cid:18) (cid:19)
U N N −2
interactionregime,∆(cid:46)NU. Inthisregimethetwo-layer + +∆, (22)
2 2 2
Hilbert space calculation, shown in Figs. 7(c,d), signifi-
cantlydeviatesfromthefullandfour-layerHilbertspace for an odd and even number of particles respectively.
calculations,showninFigs.7(a,b)and(e,f),respectively. Hence,thecriterionthatthehighestenergyoftheN =0
2
Inanalogytothenon-interactingcaseinFig.3(c),thisis states is less than the lowest energy of the N =1 states
2
wherethedetuningislowenoughforsomeeigenstatesto is given by
cross the interacting CTAP state; the adiabaticity of the
transport is broken by these crossing states in the small (cid:40)NU(cid:0)N + 1 − 3 (cid:1) odd N,
detuning regime, and not necessarily at t = τ/2, which ∆> NU(cid:0)N4 + 21 − 43N(cid:1) even N. (23)
is typically the time when the adiabaticity parameter is 4 2 4N
maximized. To understand the breakdown of CTAP in
These inequalities give the minimum detuning needed to
this region we need to consider the energies of the ba-
efficiently transport N interacting particles across the
sis states in the presence of interactions. Empirically,
three wells. The region to the right of the dashed line
we find that efficient CTAP only occurs when the maxi-
plotted in Figs. 7(a,c,e) satisfies Eq. (23) for N = 5,
mum energy of the N =0 states of the Hilbert space is
2 while transport in the region to the left is unstable.
lessthantheminimumenergyoftheN =1statesofthe
2
Hilbertspace,asshowninFig.8. Thatis,thereisamini-
mumdetuningwhichmakesN =1states–whichcouple
2 B. Comparison with Gross-Pitaevskii analysis
to higher states – energetically unfavourable, therefore
resistinganyattemptbythesystemtoevolveintohigher
We finally turn our attention to previous studies of
Hilbert space layers. For N particles the highest energy
CTAP in dilute gas BECs [36–40], which have focused
basis state of the N =0 layer of the Hilbert space is
2 onaGross-Pitaevskiimean-fieldthree-modemodel. Fol-
lowing [39] we consider a one dimensional system, with
E =U (cid:0)N2−N(cid:1) (20) each of the three wells containing a single mode, equiv-
|N,0,0(cid:105) 2 alent to the Bose-Hubbard system schematically shown
in Fig. 1. The energy of each mode is the ground state
and the lowest energy basis state of the N = 1 layer of energy of each well, U is again the interaction strength
2
9
(equivalent to g in [39]), ∆ is the detuning of the sec- so much as to break the adiabaticity of the transport.
ond mode relative to the left and right modes, and K Byconsideringtheweaklyinteractingcaseasapertur-
ij
describe the wavefunction overlap, and hence tunneling bation of the non-interacting case we have defined useful
ratebetweenmodesiandj. IntheGross-Pitaevskiianal- limits for when adiabatic transport is possible using a
ysis the quantities of interest are the populations N (t) squared-sinusoidal pulsing. These limits stem from min-
i
of mode i. Using the same squared-sinusoidal pulsing imizingtheadiabaticityparameters,i.e.thecouplingbe-
of Eq. (5), Figs. 7(g,h) plot the final population of mode tween the CTAP state and the closest eigenstate, and
three,N (t=τ)/N,andthemaximumpopulationofwell give good agreement to the limit of detuning and inter-
3
two, maxN (t)/N, respectively. The regimes of high fi- action strength for which adiabatic transport is possible
2
delityCTAPinthethree-modeGross-Pitaevskiianalysis for a finite pulsing time.
qualitatively agree with the Bose-Hubbard models. The Havingestablishedthedetuning-interactionregimefor
two models are typically applied to very different limits, which CTAP is possible we have investigated restricting
the former being a semi-classical mean-field model for a theHilbertspacetotheminimumnumberofbasisstates
large number of particles, and the latter a linear quan- necessary to allow CTAP. For zero or very small interac-
tummodelthatallowsforentanglement. Theagreement tions relative to the detuning only states with N =0 or
2
of the two models demonstrates the robustness of the 1 need to be included in the basis. For interaction ener-
CTAP protocol. giescomparabletoorgreaterthanthedetuning,wemust
include states with N > 1, but we can still restrict the
2
Hilbert space to the lowest layers to allow CTAP. Apart
V. CONCLUSIONS from the adiabaticity criterion Eq. (17), which holds for
the restricted and unrestricted Hilbert spaces, we have
We have considered adiabatic transport of N particles obtained a limit on what defines this strongly interact-
across a three-well Bose-Hubbard system and solved the ing regime by comparing the energy of individual basis
non-interacting case analytically for which we obtained states. In this regime we have seen that adiabatic trans-
the null space and energy eigenspectrum in closed form. port is not stable.
Detuning well 2 relative to wells 1 and 3 makes the null Finally, we have compared CTAP in the quantum
state unique and it is this state in which N particles can Bose-Hubbard model to CTAP in a three-mode mean-
be adiabatically transported from well 1 to 3, or vice- fieldGross-Pitaevskiiapproximation. Theregimeofhigh
versa, without occupation of well 2. Even without de- fidelity of CTAP in both models matches and clearly
tuning, CTAP is possible if the system is initialized cor- demonstrates the robustness of the CTAP protocol in
rectly. The non-interacting N-particle three-well system both the quantum and semi-classical mean-field lim-
exhibitssimilardynamicstoadiabatictransportofsingle its. This mapping between the mean-field approach and
particles along longer chains of 2N +1 sites. the Bose-Hubbard model reinforces the use of the semi-
The interacting Bose-Hubbard three-well system can- analytic constraints we have developed to determine the
not be solved analytically and there is no general null validity of CTAP.
state but the time-dependent Schr¨odinger equation can
be integrated numerically for few particles. We have
shown that for stronger interactions relative to the mid- ACKNOWLEDGMENTS
dle well detuning, which disrupt the transport, CTAP
can be maintained by detuning well 2 enough to isolate A.D.G.acknowledgestheAustralianResearchCouncil
the CTAP state from other energy eigenstates, but not for financial support (Project No. DP0880466).
[1] M. Greiner et al., Nature 415, 39 (2002). (2011).
[2] M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989). [14] T. F. Viscondi, K. Furuya and M. C. de Oliveira, EPL
[3] A. Smerzi et al., Phys. Rev. Lett. 79, 4950 (1997). 90, 10014 (2010).
[4] G. J. Milburn et al., Phys. Rev. A 55, 4318 (1997). [15] B. W. Shore et al., Phys. Rev. A 44, 7442 (1991).
[5] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). [16] J. Oreg, F. T. Hioe and J. H. Eberly, Phys. Rev. A 29,
[6] M. Albiez et al., Phys. Rev. Lett. 95, 010402(2005). 690 (1984).
[7] T. Schumm et al., Nat. Phys. 1, 57 (2005). [17] J. R. Kuklinski et al., Phys. Rev. A 40, 6741 (1989).
[8] R. Gati et al., Phys. Rev. Lett. 96, 130404 (2006). [18] U. Gaubatz et al., J. Chem. Phys. 92, 5363 (1990).
[9] S. Fo¨lling et al., Nature (London) 448, 1029 (2007). [19] K. Bergmann, H. Theuer and B. W. Shore, Rev. Mod.
[10] S. Mossmann and C. Jung, Phys. Rev. A 74, 033601 Phys. 70, 1003 (1998).
(2006). [20] M.Weitz,B.C.YoungandS.Chu,Phys.Rev.Lett.73,
[11] S. Mossmann and C. Jung, Phys. Lett. A 372, 817 2563 (1994).
(2008). [21] R. Wynar et al., Science 287, 1016 (2000).
[12] A. I. Streltsov et al., Phys. Rev. A 83, 043604 (2011). [22] K. Winkler et al., Phys. Rev. Lett. 98, 043201 (2007).
[13] T. F. Viscondi and K. Furuya, J. Phys. A 44, 175301 [23] J. H. Cole et al., Phys Rev. B 77, 235418 (2008).
10
[24] K. Eckert, M. Lewenstein, R. Corbala´n, G. Birkl, W. [36] E. M. Graefe, H. J. Korsh and D. Witthaut, Phys. Rev.
ErtmerandJ.Mompart,Phys.Rev.A70,023606(2004). A 73, 013617 (2006).
[25] K. Deasy et al., preprint, arXiv:quant-ph/0611174 [37] G.-F. Wang et al., Phys. Rev. A 74, 033414 (2006).
(2006). [38] B. Liu et al., Phys. Rev. A 75, 033601 (2007).
[26] J. Siewert and T. Brandes, Adv. Solid State Phys. 44, [39] M. Rab et al., Phys. Rev. A 77, 061602(R) (2008).
181 (2004). [40] V. O. Nesterenko et al., Laser Physics 19, 616 (2009).
[27] T. Ohshima et al., e-print arXiv:quant-ph/0702019. [41] D.PetrosyanandP.Lambropoulos,OpticsCommunica-
[28] P. Zhang et al., Phys. Rev. A 69, 042307 (2004). tions 264, 419 (2006).
[29] A. D. Greentree et al., Phys. Rev. B 70, 235317 (2004). [42] L. M. Jong et al., Nanotech. 20, 405402 (2009).
[30] J. Fabian and U. Hohenester, Phys. Rev. B 72, [43] C. Hill, A.D. Greentree and L.C.L. Hollenberg, New J.
201304(R) (2005). Phys 13, 125002 (2011).
[31] L. C. L. Hollenberg et al., Phys. Rev. B 74, 045311 [44] V. Pyragas and G. Juzeliunas, arXiv:1110.6379.
(2006). [45] L.M.JongandA.D.Greentree,Phys.Rev.B81,035311
[32] S. O’Sullivan et al., Phys. Scr. T140, 014029 (2010). (2010).
[33] S. Longhi, J. Phys. B 40, F189 (2007). [46] E. Arimondo and G. Orriols, Lettere Al Nuovo Cimento
[34] S. Longhi et al., Phys. Rev. B 76, 201101(R) (2007). 17, 333 (1976).
[35] G.DellaValleetal.,Appl.Phys.Lett.92,011106(2008). [47] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).