Phonon-induced artificial magnetic fields

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by  Alexander Klein

epl draft
Phonon-induced artificial magnetic fields in optical lattices
Alexander Klein1,2 and Dieter Jaksch1,2
1 Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
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2 Keble College, University of Oxford, Parks Road, Oxford OX1 3PG, United Kingdom
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0
2

n PACS 37.10.Jk–Atoms in optical lattices
Ja PACS 03.75.-b–Matter waves
PACS 67.85.-d–Ultracold gases, trapped gases
2
2 Abstract. - We investigate the effect of a rotating Bose-Einstein condensate on a system of

immersed impurity atoms trapped by an optical lattice. We analytically show that for a one-
]
h dimensional, ring-shaped setup the coupling of the impurities to the Bogoliubov phonons of the
p condensate leads to a non-trivial phase in the impurity hopping. The presence of this phase can
- be tested by observing a drift in the transport properties of the impurities. These results are
t
n quantitatively confirmed by a numerically exact simulation of a two-mode Bose-Hubbard model.
a We also give analytical expressions for the occurring phase terms for a two-dimensional setup.
u The phase realises an artificial magnetic field and can for instance be used for the simulation of
q
thequantumHall effect using atoms in an optical lattice.
[

2
v
8
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8 There are several phenomena in physics such as the already been demonstratedin experiments [12]. However,
1quantum Hall effect [1] or high-temperature supercon- inordertoobservethequantumHallstatesthecentrifugal
.
8ductivity [2], that have been known for more than two term caused by the rotation has to be carefully balanced
0decades,but where a full theoreticalunderstanding ofthe byanadditionalharmonictrappingpotentialconfiningthe
8underlying mechanisms is still lacking. This is due to the gas, which is difficult to achieve in an experiment.
0
fact that a complicated many-body system consisting of In this work, we therefore propose a way of exploiting
:
velectrons in a crystal has to be described on a quantum a rotating Bose-Einstein condensate (BEC) to create an
Ximechanical level, which quickly exceeds the capability of artificial magnetic field in an optical lattice system which
modern computers. As Feynman suggested [3] it is there- is submerged into the condensate. This has the advan-
r
aforeworthwhiletosearchforalternativequantumsystems tage that the system in which the quantum Hall states
thatallowfora cleanrealisationofthe underlying models are to be observed, i.e. the optical lattice, does not need
and that are easy to manipulate. With such a quantum to be rotated and thus there is no need of employing a
simulator,we would be able to do simulations that by far balancing potential. The BEC, on the other hand, only
exceed what is currently possible on classical computers. provides the artificialmagnetic field and we are not inter-
Ultracold atoms in optical lattices are a top candidate ested in a direct observation of its states. Hence, we can
whenitcomestothe simulationofcondensedmatterphe- employa harmonictrappingpotentialwhichovercompen-
nomena [4]. It has been shown theoretically that for a sates the centrifugal force instead of exactly canceling it,
suitable experimental setup the atoms in the lattice in- which would not be possible if we wanted to observe the
deed realise typical quantum Hall states [5–7]. In a solid- quantum Hall states in the BEC directly.
state setup, these states are realisedby the application of The presenceofthe condensate cannot only be usedto
a strong magnetic field. Since the motional state of the provide an artificial magnetic field, it also allows for the
neutral atoms in the optical lattice does not couple to a simulation ofother effects which are presentin condensed
magnetic field, the effect of this field has to be simulated matter physics [13]. For example, it has been shown that
by other means. Proposals include using special lattice for suitably chosen parameters the coupling to the BEC
setups and Raman assisted hopping [8], exploiting light phononsmaychangethetransportbehaviourofthelattice
with orbital angular momentum [9,10], or simply rotat- atomsfromcoherenttoincoherent[14]andthatanattrac-
ingthe lattice [11]. Whereasthe firstproposalsarerather tiveinteractionbetweenthelatticeatomscanbemediated
complicated to implement, the rotation of the lattice has bythecondensatewhichleadstoclusteringeffects[15]. By
p-1
A. Klein and D. Jaksch
including an artificial magnetic field, the whole setup can dimensions. In this case, the deviation of φˆ(r) from φ (r)
0
thereforebeusedasawell-suitedsimulatorforphenomena is of order κ, i.e., δφˆ(r) κ, where stands for the
h i ∝ h·i
encountered in condensed matter physics. expectationvalue. Insertingφ (r)+δφˆ(r)intothe Hamil-
0
tonian Hˆ +Hˆ and keeping terms up to second order in
Theoretical description. – Inthefollowing,wewill B I
κ,weobtainthelineartermκ drχˆ†(r)χˆ(r)[φ (r)δφˆ†(r)+
derive an effective Hamiltonian for the atoms in the opti- 0
callattice. Asimilarderivationhasbeengiveninref.[13], φ∗0(r)δφˆ(r)], in addition to tRhe standard constant and
where it was assumed that the wave function describing quadratic terms in δφˆ(r) and δφˆ†(r).
the condensate is real. In general, this will no longer be The quadratic terms in δφˆ(r) are diagonalised by mak-
the case if the condensate is rotated or translated, lead- ing use of the Bogoliubov transformation. The perturba-
ing to an additional phase in the hopping terms of the tion δφˆ(r) is expanded in terms of Bogoliubov phonons
lattice atoms. The Hamiltonian describing the whole sys- u and v , δφˆ(r) = ′[u (r)ˆb v∗(r)ˆb†], where ˆb† cre-
tem consists of three parts, Hˆ = HˆB +HˆI +Hˆa. Here, atqes a Boqgoliubov phoqnonq in qm−odeqq, tqhe prime aqt the
Hˆa determines the free dynamics of the atoms of species sumindicates that thPe mode correspondingto the ground
a trapped in the optical lattice, which we will call impu- state is not included, and the functions u and v solve
q q
rities in the following. In the laboratory frame, the BEC the Bogoliubov-deGennes (BdG) equations
HamiltonianHˆ describingtheatomsofspeciesbandthe
B
density-density interaction Hamiltonian HˆI are Hˆ +g φ(r)2 u (r) g(φ(r))2v (r)=E u (r), (4)
0 q q q q
| | −
HˆB = drφˆ†(r) Hˆ0+ g2φˆ†(r)φˆ(r) φˆ(r), (1) hHˆ0†+g|φ(r)|2ivq(r)−g(φ∗(r))2uq(r)=−Eqvq(r).
Z h i h i (5)
Hˆ =κ drχˆ†(r)χˆ(r)φˆ†(r)φˆ(r), (2)
I
Z Putting this expansion for the condensate field operator
where χˆ(r) is the impurity field operator,φˆ(r) is the con- into the interaction Hamiltonian and using the fact that
densate atom field operator satisfying the usual bosonic the overlaps of two different impurity modes is negligible
commutationrelations,andHˆ containsthekineticenergy allows us to rewrite the total Hamiltonian as a Hubbard-
0
term ~2 2/2m , an external trapping potential V (r), Holstein model [18,19]
b ext
− ∇
the chemical potential µ and any other terms that cor-
′
respond to rotations or translations of the BEC as will Hˆ =Hˆ + ~ω (M ˆb +M∗ ˆb†)nˆ
a q j,q q j,q q j
be detailed later. The coupling constants g >0 and κ ac-
j q
XX (6)
countfortheboson-bosonandimpurity-bosoninteraction,
+ E¯ nˆ + ~ω ˆb†ˆb ,
respectively, and m is the mass of a condensate atom. j j q q q
b
j q
In the tight-binding approximation, the impurity field X X
operator can be expanded as χˆ(r) = jηj(r)aˆj, where with ~ω = E the energies of the Bogoliubov
q q
aˆ†j creates an impurity atom in latticPe site j and ηj is excitations, the number operator nˆj = aˆ†jaˆj,
the corresponding Wannier function [16]. The probabil-
the dimensionless matrix elements M =
j,q
ity densities of these functions have a negligible overlap, (κ/~ω ) dr [φ∗(r)u (r) φ (r)v (r)] η (r)2 and the
i.e., ηj(r)2 ηj′(r)2dr 0 for j =j′, which will be im- meanqfield shift0E¯ =q κ −drn0 (r)qη (r)|2j. |
| | | | ≈ 6 j 0 j
portant later. By using the above expansion for the field R | |
R The total Hamiltonian can be brought into a more
operator χˆ the Hamiltonian for the impurities is given by R
intuitive form by applying the unitary Lang-Firsov
Hˆa =−Ja hi,jiaˆ†iaˆj+(Ua/2) jnˆj(nˆj −1)−µa jnˆj transformation [18] Hˆeff = UˆHˆUˆ†, with Uˆ =
pwihnegremµatrdiexPseclreibmeesntthbeetcwheeemnictawlPopnoetiegnhtbiaolu,rJinagissittehsPeohfothpe- exp j ′q(Mj∗,qˆb†q−Mj,qˆbq)nˆj , which yields
lattice, and Ua is the on-site interaction strength. (cid:2)P P (cid:3)
To investigate the influence of the rotating BEC on the Hˆeff =UˆHˆaUˆ†+ (E¯j Ej)nˆj Ejnˆj(nˆj 1)
− − −
impurities we employ the Bogoliubov approximation to j j
X X (7)
sdiemfipnleifφyˆ(trh)e=dyφna(rm)i+csδoφˆf(trh),ewcohnedreenφsaties.aTsooltuhtiisonenodf,twhee − 12 Vj,j′nˆjnˆj′ + ′~ωqˆb†qˆbq.
0 0 j6=j′ q
Gross-Pitaevskiiequation (GPE) [17] for κ=0, X X
ThepresenceoftheBECmediatesanon-retardedinterac-
0=(Hˆ +g φ (r)2)φ (r). (3)
0 | 0 | 0 tion Vj,j′ = ′q~ωq Mj,qMj∗′,q+Mj∗,qMj′,q between im-
In contrast to [13] we here do not assume that the wave purities in different lattice sites j and j′. The charac-
P (cid:0) (cid:1)
functionφ isreal. Toproceedwerequireaweakimpurity- teristic potential energy of an impurity in the deformed
0
ξbho(sro)n=co~u/plimngbgκn0s(urc)hthtehhatea|lκin|/gglnen0(grt)hξ,hDn(0r()r)≪= 1φ,0(wri)th2 con′qd~eωnqs|aMtej,qis|2d.escribed by the polaronic level shift Ej =
thedensityofthecondensateandD thenumberof|spatia|l The transformation of the impurity Hamiltonian Hˆ
p P a
p-2
Phonon-induced artificial magnetic fields
yields
(a) (b)
0.03
UˆHˆaUˆ† =−Ja (Xˆiaˆi)†Xˆjaˆj v=RΩ J~ae2πiαa
hXi,ji αa 0
(8)
U
+ a nˆ (nˆ 1) µ nˆ , −0.03
2 j j − − j 0.02 300
Xj Xj 0.015 150
g / 2aER 0.01 0 Ω [rad/s]
where Xˆ† =exp (M∗ ˆb† M ˆb ) is a Glauber dis-
j q j,q q− j,q q
placement operator that creates a coherent phonon cloud Fig. 1: (a) Setup of the scheme. The ring confining the BEC
around the impu(cid:2)rPity. Thus, the Ham(cid:3)iltonian Hˆ de- is rotated with angular velocity Ω. This inducesa phase twist
eff
scribes the behaviour of polarons according to an ex- αa whenanimpurityhopsfromonelatticesite(bluedots)toa
tended Hubbardmodel [4]providedthat c aJ /~, with neighbouringone. (b)Inducedphasetwistαa versustheangu-
a
≫ lar rotation speed Ω and the interaction within the BEC. We
c∼ gn0/mb the phonon velocity and a the lattice spac- assumed a BEC of 87Rb with linear density n0 =5 106m−1
ing [13]. ×
p in a ring with a circumference of L = 12µm. The impurities
In ref. [13] it has been shown that for low tempera- are21Naatomsinalatticewith30sites,whichcorrespondsto
tures kBT Ej the behaviour of the polarons is essen- a lattice constant a=400nm. The coupling strength between
≪
tially coherent. Here, kB is Boltzmann’s constant. If also theimpurityandtheBEC isgiven byκ/2aER =0.035, where
Ja Ej, we can treat the hopping term in eq. (8) as ER =(2π~)2/2ma(2a)2 is the recoil energy with ma the mass
≪
a perturbation and trace over the condensate degrees of of theimpurities.
freedom [13]. This allows us to derive the Hamiltonian
1 causes persistent currents [20]. This is similar to the be-
Hˆ(1) = J˜ aˆ†aˆ + U˜ nˆ (nˆ 1)
− i,j i j 2 j j j − haviour of an electron in a superconducting ring subject
hXi,ji Xj (9) to a magnetic flux. In a two-dimensional setup a suitable
1
phase factorrealisesanartificialmagneticfield, whichen-
µ˜ nˆ V nˆ nˆ ,
j j i,j i j
− − 2 ables for instance the investigation of quantum Hall ef-
j i6=j
X X fects [6]. In the remainder of this paper, we will charac-
with µ˜ = µ κn + E , U˜ = U 2E , and J˜ = terise the occurring phase term for different setups and
j 0 j j a j i,j
J Xˆ†Xˆ . H−ere denotes the ave−rageover the ther- discuss ways of detecting its influence on the dynamics of
ahh i jii hh·ii the impurity atoms.
mal phonon distribution. After some tedious algebra we
find
BEC in a ring. – We first consider the case where
the BEC is loaded into a quasi one-dimensional trap of
1 ′
Xˆ†Xˆ = exp M M 2(2N (T)+1) length L with periodic boundary conditions, which cor-
hh i jii −2 | i,q− j,q| q ! responds to a ring of radius R = L/2π and has already
q
X
been achieved in experiments [21,22]. The impurities
1 ′
exp M M∗ M∗ M , (10) are correspondingly trapped by a one-dimensional opti-
× 2 q i,q j,q− i,q j,q! cal lattice which is ring-shaped, see fig. 1(a). We assume
X
that the ring trapping the condensate is rotated with an
where Nq(T) = [exp(~ωq/kBT)−1]−1 is the number oc- angular speed Ω. The Hamiltonian Hˆ0 describing the
cupationofmodeq. As wasdiscussedinref.[13],the first interaction-free part of the BEC is then given by Hˆ =
0
exponential term leads to a suppression of the coherent (~2/2m )∂2/∂x2+i~v∂/∂x µ,wherev =RΩ. Thefull
b
− −
hoppingJa. Thesecondexponential,whichdoesnotoccur GPE is solved by the wave function φ0 = √n0exp(iq0x),
for the cases presented in [13]since there φ0 was assumed where q0 = 2πj/L for some integer j such that q0 =
to be real, has always an absolute value of 1 and causes a m v/~ ∆q with ∆q [ π/L,π/L]. This choice of q
b 0
− ∈ −
phase whenever an atom hops from one lattice site j to a ensures that the BEC is in its ground state with a chemi-
neighbouringonei. Bydefining the reducedhoppingcon- calpotentialµ=~2q2/2m +gn ~vq andthatthephase
0 b 0− 0
stant J˜ = J exp ′ M M 2(2N (T)+1)/2 of the BEC fulfills the periodic boundary conditions.
a a − q| i,q− j,q| q
tthene hinoptphiengfotremrm−Jo(cid:16)˜fathPehi,Hjiaemxpil(t2oπniiαain,j)Hˆaˆ†i(1aˆ)jcwanithbeαiw,jri=t(cid:17)- uqq)T(xxh])/e√=BLdA,Gqwehexeqpru[eia(ttqhio+encsqo0ae)rxffie]/csi√eonlLvtesadanrdbeyvgqtiv(hxeen)m=boydBtehqeefuxrnpecl[ait(itqoion−ns
1/(4πi) ′(M M∗ M∗ M ). 0
This phqase di,qepejn,qds−Ponit,qhe jp,qroperties of the order pa- Aq±Bq =(EqB/ε0q)±1/2,andthe quasi-momentafulfilthe
P condition q = 2πj/L for some integer j. The energies of
rameterφ0 anddrasticallyaffectsthe dynamicsoftheim- theBogoliubovmodesaregivenby~ω =EB ~2q∆q/m
purities. For example, for a quasi one-dimensional setup q q − b
the presence of such a phase twist in nearest neighbour with EqB = ε0q(ε0q+2gn0) and ε0q = ~2q2/2mb. In or-
hopping corresponds to a rotation of the ring and thus der to calculaqte the coupling matrix elements Mj,q we as-
p-3
A. Klein and D. Jaksch
sumethatthelatticetrappingtheimpuritiesissufficiently sites in each direction.
deep such that the Wannier functions ηj are well approx- Drift of the impurities. The presence of the phase
imated by Gaussians of width σ centered at lattice site j,
twist α can be measured by observing its influence on
a
i.e., η (x) exp[ (x x )2/2σ2]/ √πσ, where x is the
j j j thetransportpropertiesoftheimpurities. Weconsiderthe
≈ − −
position of lattice site j. We furthermore note that due
following case: A single impurity is initially trapped in a
p
to the translationalinvariance the phase α =α does
i,i+1 a finitespatialregionofthelattice. Thiscanbeachievedby
not depend on the index i and we thus find
applyinganadditionaltrappingpotentialV (j)=V (j
ini 0
−
j )2, in which case the single impurity starts in the state
α = 1 κ2n0 ε0q 1 e−q2σ2/2sin(qa) . (11) ψ0 exp (j j )2/2w2 aˆ† vac , where w is
a 2π L EB (~ω )2 | iini ∝ j − − 0 ini j| i ini
qX6=q0 q q the widthPof the(cid:2)initial state. Wi(cid:3)thout a phase twist αa
afterturningtheinitialpotentialoff,theimpurityexpands
We see that the finite width of the Wannier functions σ
symmetrically around the lattice site j and no overall
0
leads to an effective cut-off of the sum for large quasi-
drift to either higher or lower lattice sites occurs.
momenta q and thus large energies ~ω . Furthermore,
q Thisbehaviourchangesifafinitephasetwistα istaken
a
the expression for α contains the Bogolibov excitation
a into account as shown in fig. 2(a). For the chosen pa-
energies~ωq oftherotatingcondensateaswellasthestatic rameters and a relatively short time of t = 5~/J , which
structure factor ε0/EB of the non-rotating BEC [23,24]. a
q q under standard experimental conditions corresponds to a
Thevalueofα dependscriticallyontherotationspeed
a few miliseconds, a drift towards higher lattice site num-
Ω. For zero rotation it can be shown from eq. (11) that
bers is clearly visible. Such drifts can be measured in
α = 0. If the rotation speed is increased, the phase
a experiments [21] and thus allow to probe the presence of
twist α initially increases linearly with Ω, as shown in
a an induced phase twist. The mean position of the im-
fig. 1(b). However, for a critical rotation speed a jump purity after an evolution time of t = 5~/J is shown in
a
occursand the value of the phase twist changesfrompos-
fig. 2(b). It exhibits a periodic behaviour in α with a
a
itive to negative. This behaviour can be explained as
period of 1, which is caused simply by the fact that the
follows: Due to the finite size of the ring, the possible
integer part of α only contributes a trivial phase. In to-
a
quasi-momentumstatesoftheBECarerestrictedtovalues
tal, the behaviour is akin to an electron in a ring subject
q = 2πj/L for some integer j. For rotation speeds close
0 to a magnetic flux. The electron will also drift towards
to zero, the ground state of the BEC exhibits zero quasi-
a preferential direction and thus will cause a current as
momentum. Only if the rotation speed is above a critical
known from superconducting rings.
value Ω = ~/2m R2 will the ground state change to
crit b
Numericalsimulationsofthefullringsystem. Wenow
a non-zero quasi-momentum. Further quasi-momentum
compare our prediction to a system where a numerically
jumpsoccuratoddmultiplesofthiscriticalangularspeed.
exactsolutionis possible,namely asmalltwo-modeBose-
Together with the change in the quasi-momentum of the
Hubbard model. We assume that the condensate is rep-
BEC,alsothe jump inthe inducedphase twistα occurs.
a
resented by a mode c that experiences a rotation whereas
Tosomeextendonecouldthussaythattheinducedphase
the other mode a representing a single impurity does not.
is caused by a mismatch of the angular rotation speed Ω
The system is described by the Hamiltonian
andthequasi-momentumq oftheBEC,whichisenforced
0
to be quantised by the boundary conditions. It should be
Hˆ = J aˆ†aˆ +U aˆ†aˆ cˆ†cˆ (12)
noted that at the critical points where the BEC changes BH − a i j I j j j j
its quasi-momentum, the ground state of the condensate hXi,ji Xj
is degenerate, which leads to a failure of the Bogoliubov J e2πiαccˆ†cˆ +h.c. + Uc cˆ†cˆ†cˆ cˆ .
approximation[25]. − c j j+1 2 j j j j
The parameters presented in fig.1(b) fulfill the condi- Xj (cid:16) (cid:17) Xj
tions for our derivation to be valid and can be achieved Here, J is the hopping matrix element of the atoms in
c
usingpresentexperimentaltechniques[13–15,26,27]. The mode c, U their interaction constant, U the interaction
c I
precise value of the coupling constant g can be changed betweenatomsinmodecanda,andα =m Ω N a2/2π~,
c c c s
via a Feshbach resonance. The temperature chosen for where m is the mass of the atoms, Ω the speed with
c c
our calculations is zero, but the discussed effects will be which the ring for atoms in mode c is rotated, N the
s
observableaslongasthetemperatureislowerthanthepo- numberoflatticesites,athelatticeconstant,andperiodic
laronenergy, which is typically on the order of a few tens boundary conditions have been assumed [25].
of nanokelvin [13]. This also ensures that the first factor Wefirstcalculatethegroundstateoftheatomsinmode
ineq. (10)is close to one leading to sufficiently largehop- c for different values of α and vanishing coupling to the
c
ping. Inthe aboveexamples the size ofthe inducedphase impurity atom. We assume that the impurity is in state
twist can reach values as large as α 0.03. This will ψ at time t = 0 when the interaction U is turned on
a ini I
≈ | i
significantly affect the phase of the atomic wave function and let the system evolve for a finite evolution time. For
and make effects of the induced artificial magnetic field two different Ω α the density distributions of the im-
c c
∝
observableevenif the lattice only consistsofa few tens of purity after this time are shown in fig. 3(a), using typical
p-4
Phonon-induced artificial magnetic fields
30 30 11.2
(a) (b) 0.1 (a) (b)
25 n n
20 ositio20 0.08 ositio11.1
j1105 ean end p10 〈〉 n j00..0046 ean end p101.91
5 m 0.02 m
0 2 td 4 6 00 0α.a5 1 0 5 10j 15 20 10.80 0.02 αc 0.04 0.06
Fig.2: (a) Plot oftheimpuritydistribution versusdimension- Fig. 3: Expansion of a single impurity atom initially in state
less time td =tJa/~. A darker colour corresponds to a higher ψ ini with j0 = 11 and width wini = √2. (a) Density distri-
density. (b) Mean position of the impurity after an evolution |buition after a time td = tJa/~ = 4 for αc = 0.0227 (dashed
time of td =6. We havechosen an initial state ψ ini of width line)andαc =0.0251(solidline). (b)Meanpositionoftheim-
| i
w = √2 centered at lattice site j = 15. In (a), the chosen purity after the evolution time t = 4. The two crosses mark
ini 0 d
phasetwist was αa =0.03. thepositions of thedensities in (a). Theother parameters are
Uc = Jc = Ja, UI = 2Jc with three atoms in mode c and a
lattice consisting of Ns =21 sites.
experimental parameters. A small drift to either the left
or the right is clearly visible. We expect this difference
always given by (n,l)=(0,0).
to be by far more pronounced for a larger lattice with a
For brevity, let us introduce the multi-index ν =(n,l).
longer evolution time. However, for computational rea-
If we assume that the condensate is in a state ν , we can
sons we had to restrict ourselves to a small system size. expand the field operator of the condensate as ψˆ0(r,ϕ)
The behaviour of the mean end position versus the phase →
√N ψ (r,ϕ) + ψ (r,ϕ)ˆb , where ˆb† ˆb = N
twistαc is showninfig.3(b). As wesee this driftexhibits is th0eνn0umber of atνo6=mν0s inν stateνν , and wheνa0ssνu0mi e tha0t
a jump at a value of αc = 1/2Ns. The reason for this ˆb†ˆb N forPν = ν . As we co0nsider the interaction-
jump is similar to the saw-toothbehaviour ofthe induced h ν νi ≪ 0 6 0
freecasethisisexactlythe sameasanexpansioninterms
phase α observedinfig. 1(b). Due to the quantisationof
a of Bogolibov phonons, which would yield v = 0 and
the BEC quasi-momenta there exist critical rotation ve- ν
u = ψ (r,ϕ). The chemical potential is determined by
locitiesΩcrit =2π2(2j+1)~/m N2a2 atwhichtheground ν ν
c b s solving the Gross-Pitaevskii equation for the correspond-
stateoftheBECchangesfromonequasi-momentumstate
ing state ν , which gives µ = E . This allowsus to
to another one [25]. These critical velocities correspond 0 ν0 − n0,l0
derivethe phasetwist α inthe samewayasfor the ring
to phase twists α = (2j + 1)/2N for integer j. One i,j
c s optical lattice. In order to solve the integral in the ex-
can show that due to these jumps the induced phase α
a pressionforthecouplingmatrixelements M weassume
changes analogously to the behaviour shown in fig. 1(b). j,ν
that the Wannier functions are strongly localised and can
This change is then also reflected in the drifts that the
be approximated by a Dirac delta function. If the con-
impurity atoms experience.
densate is in the ground state ν = (0,0), we find for the
0
induced phase twist
Thetwo-dimensionalcondensate.– Aphasetwist
αini,jaitsraalpsorointadtuicnegdwifitwheacnognusliadrervaelotwciot-ydiΩm.enFsoiornsaimlBpElicC- α(i,0j,0) =2κπ23Na04 (~ω1 )2(nn+!l)!e−ξi2−ξj2
ity we assume an interaction free condensate, i.e., g = 0. 0 νX6=ν0 ν (13)
In the rotating frame, the (single particle) Hamiltonian ξlξlLl(ξ2)Ll(ξ2)sin(l(ϕ ϕ )),
× i j n i n j i− j
describing the condensate subject to a harmonic trap-
ping potential ωtr thus reads Hˆ0 = (~2/2mb)( ′)2 + where (ξi,ϕi) and (ξj,ϕj) are the coordinates of the two
m ω2(r′)2/2 µ Ωe r′ pˆ′, wh−ere e is th∇e unit latticesitesbeforeandafterthejumpoftheimpurity. We
vebctotrr in z di−recti−on anzd·pˆ′ ×= i~ ′. It czan be shown immediately see that if ϕi = ϕj, which corresponds to
that the solutions for this Ha−milt∇onian are given by a hopping in radial direction, the induced phase is zero.
ψn,l(r,ϕ) = n!/π(n+l)!eilϕe−ξ2/2ξlLln(ξ2)/a0 with en- Furthermore, the induced phase vanishes as ξi or ξj go to
ergiesE +µ=(2n+l+1)~ω ~Ωl. Here,ξ =r/a with infinity, which can be explained by the vanishing density
a = ~n/,lmpω the harmonictro−scillator length and0(r,ϕ) of the condensate for large distances.
0 b tr
Ourtheoryisalsoapplicableifthecondensateisinitially
are polar coordinates. The quantum numbers n and l are
p in an excited state, for example in the state ν = (0,1)
restricted to n = 0,1,2,... and l = n, n+1,.... From 0
− − that exhibits a vortex. In this case, we derive for the
this we see that for high rotation speeds Ω > ω the
tr
| | phase twist
system is unstable since for suitable (n,l) the energy can
bbeeinmgaddreivinenfininitfienlyiteslmyaflalrwahwiachyfcroormrestphoenodrsigtino dthueeatototmhes α(i,0j,1) =2κπ23Na04 (~ω1 )2(nn+!l)!e−ξi2−ξj2(ξiξj)l+1
centrifugal term. We thus restrict the rotation speed to 0 νX6=ν0 ν (14)
moderate values |Ω| < ωtr, for which the ground state is ×Lln(ξi2)Lln(ξj2)sin[(ϕi−ϕj)(l+1)].
p-5
A. Klein and D. Jaksch
0.5 0.08 IRC(GrantNo.GR/S82176/01)andEuroQUAMProject
(a) (b)
No. EP/E041612/1, the EU through the STREP project
0.4
0.06
OLAQUI, and the Keble Association (A.K.).
0.3
αred0.2 αred0.04
0.1 0.02 REFERENCES
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supported by the United Kingdom EPSRC through QIP 88 (2001) 010405.
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