Table Of ContentPossible realization of Josephson charge qubits in two coupled Bose-Einstein
condensates
Zeng-Bing Chen1 and Yong-De Zhang2,1
1Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
2
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, P.R. China
(February 1, 2008)
2
0
We demonstrate that two coupled Bose-Einstein condensates (BEC) at zero temperature can be
0
used to realize a qubit which is the counterpart of Josephson charge qubits. The two BEC are
2
weakly coupled and confined in an asymmetric double-well trap. When the “charging energy” of
n the system is much larger than the Josephson energy and the system is biased near a degeneracy
a point, the two BEC represent a qubit with two states differing only by one atom. The realization
J
of theBEC qubitsin realistic BEC experiments is briefly discussed.
6
1
PACS numbers: 03.75.Fi, 03.67.-a, 74.50.+r
]
t
f
Bose-Einstein condensates (BEC) in dilute systems of twoBECareweaklylinkedandconfinedinanasymmet-
o
s trapped neutral atoms [1,2] have offered a new fascinat- ric double-well trap. Typically, the system may display
. ingtesting groundfor somebasicconcepts inelementary the phenomenon known as quantum coherent tunneling
t
a quantum mechanics and quantum many-body theory as of atoms, or Josephson effect [3–5,7]. However, under
m
well as for searching new macroscopic quantum coher- the conditions that the “charging energy” of the system
- ent phenomena. In particular, much interest has been is much larger than the Josephson energy and the sys-
d
focused on the possibility of coherent atomic tunneling temisbiasednearadegeneracypoint,the twoBECmay
n
o between two trapped BEC [3–8], which is analogous to represent a qubit with two states differing only by one
c the superconducting Josephson effect. atom.
[ Recent years have witnessed the rapid development of The effective many-body Hamiltonian describing
quantum information theory. One of the central results atomicBECinadouble-welltrappingpotential,V (r),
2 trap
v there is the possibility of quantum computation [9,10]. can be written in the second-quantizationform as
8 In quantum computers, an elementary building block is
¯h2
6 a set of two-level systems (“qubits”) storing quantum Hˆ = d3rψˆ† − ∇2+V ψˆ
BEC trap
3 information. Due to the superpositionof quantuminfor- Z (cid:20) 2m (cid:21)
7 mation states in qubits, a quantum computer can pro- U
0 + 0 d3rψˆ†ψˆ†ψˆψˆ, (1)
1 vide fundamentally more powerful computational abil- 2 Z
ity than its classical counterpart. So far, many systems
/0 have been proposed to perform quantum logic, includ- where m is the atomic mass, U0 = 4πa¯h2/m (with
at ingatomsintraps[11],cavityquantum-electrodynamical a denoting the s-wave scattering length) measures the
m systems[12],thenuclearmagneticresonancesystem[13], strengthofthetwo-bodyinteraction,andtheHeisenberg
and solid state systems (e.g., quantum dots [14], nu- atomicfieldoperatorsψˆ†(r,t)andψˆ(r,t)satisfythestan-
-
d clear spins of atoms in doped silicon devices [15], and dard bosonic commutation relation [ψˆ(r,t),ψˆ†(r′,t)] =
n ultrasmall Josephson junctions [16–20]). Recently it has δ(r−r′). Onecanproperlychoosethetrappingpotential
o
been demonstrated that a single Cooper pair box is a so that the two lowest states are closely spaced and well
c
macroscopictwo-levelsystemthatcanbecoherentlycon- separated from the higher energy levels, and that many-
:
v trolled [21,22], thus realizing a Josephson charge qubit body interactions only produce negligibly small modifi-
i
X [16–18,20]. Making use of superconducting quantum in- cations of the ground-state properties of the individual
terference loops containing ultrasmall Josephson junc- wells [25]. The potential has two minima at r and r ,
r 1 2
a tions, the Josephson flux qubit has also been proposed and can be expanded around each minimum as
[19,20,23]. Two recent experiments indeed realized the
quantumsuperpositionofmacroscopicpersistent-current Vtrap(r)= Vi(0)(ri)+Vi(2)(r−ri) +···
states, and demonstrated the small Josephson junction iX=1,2h i
loops as macroscopic quantum two-level systems [24]. ≡V +V +···, (2)
1 2
The implementation of qubits in a physical systems
is the first step for quantum computing. In this paper where the two constant potentials Vi(0)(ri) can be set to
we demonstrate that two coupled BEC at zero temper- zero without loss of generality.
ature can be used to realize a qubit. This BEC qubit With the above assumptions, one can use a two-mode
is the counterpart of the Josephson charge qubits. The approximation to the many-body description of the sys-
tem [25]. Thus instead of the standard mode expansion
1
oversingle-particlestates,onecanapproximatelyexpand Using the basis |ni (n = N, N −1, ..., −N), one easily
the field operators ψˆ(r,t) in terms of two local modes obtains
[4,25]
aˆ†aˆ |ni= (N +n+1)(N −n)|n+1i. (11)
1 2
ψˆ(r,t)≈ aˆ (t)Ψ (r−r ), (3) p
i i i Consequently,aˆ†aˆ inthissubspacepermitsthefollowing
iX=1,2 1 2
representation in the basis:
where [aˆ (t),aˆ†(t)]=δ , and Ψ (r−r )≡Ψ (r) are two
i j ij i i i aˆ†aˆ = (N +n+1)(N −n)|n+1ihn|. (12)
localmodefunctions(withenergiesE0)oftheindividual 1 2
wells and satisfy i Xn p
Thus the Hamiltonian in Eq. (9) takes the form
d3rΨ∗(r)Ψ (r)≈δ . (4)
i j ij
Z Hˆ′ =E (nˆ−n )2−J (N +n+1)(N −n)
c g
Substituting Eq. (3) into the effective Hamiltonian (1) Xn p
yields the two-mode approximation of Hˆ : ×[|n+1ihn|+|nihn+1|]. (13)
BEC
To realize the Josephson charge qubits in a small
Hˆ = (E0aˆ†aˆ +λ aˆ†aˆ†aˆ aˆ )
i i i i i i i i Josephson junction system (a “single-Cooper-pair box”)
iX=1,2
[16,21], two conditions should be met: (1) The charging
−(Jaˆ†1aˆ2+J∗aˆ1aˆ†2), (5) energyofthe single-Cooper-pairbox is muchlargerthan
the Josephsonenergy;and (2)The systemis biasednear
where the parameters are estimated by [4,25] adegeneracypointso thatthe systemrepresentsa qubit
withtwostatesdifferingonlybyoneCooper-paircharge.
E0 = d3rΨ∗ −¯h2 ∇2+V Ψ , (6) Now suppose that two similar conditions are also met in
i Z i (cid:20) 2m trap(cid:21) i the present context. Then the Hamiltonian in Eq. (13)
U takes the form
λ = 0 d3r|Ψ |4, (7)
i i
2 Z
Hˆ′ =E (nˆ−n )2
¯h2 c g
J =−Z d3r(cid:20)2m∇Ψ∗1·∇Ψ2+VtrapΨ∗1Ψ2(cid:21). (8) −EJ [|n+1ihn|+|nihn+1|], (14)
2
Xn
Withoutlossofgenerality,J canberegardedarealnum-
ber. Notice that interactions between atoms in different in which EJ ≡ 2J N(N +1) ≈ NtJ is the Josephson
wells are neglected in Eq. (5) [25]. energy. InderivingpEq. (14),wehavesimplifiedEq. (12)
Defining two local number operators nˆ = aˆ†aˆ , then further by
i i i
it is easy to verify that the total number operator
aˆ†1aˆ1+aˆ†2aˆ2 =Nˆ ofatomsrepresentsaconservativequan- aˆ†1aˆ2 ≈ N(N +1) |n+1ihn| (15)
tity and thus Nˆ = N ≡ 2N = const. After neglecting p Xn
t
the constant term, the two-mode Hamiltonian Hˆ can be
if N ≫ |n| for most of the relevant number states of
rewritten as
the system. Physically, the operators |n+1ihn| and
|nihn+1| transfer a single atom between the two wells.
Hˆ′ =E (nˆ−n )2−J(aˆ†aˆ +aˆ aˆ†),
c g 1 2 1 2 TheformofHˆ′ inEq. (14)resemblestheHamiltonian
nˆ ≡(nˆ −nˆ )/2, E =λ +λ ,
1 2 c 1 2 thatisusedtorealizetheJosephsonchargequbits[16,21].
n = 1 [(E0−E0)+(N −1)(λ −λ )], (9) Similarly,onecanintroducethephase-differenceoperator
g 2E 2 1 t 2 1 ϕˆthatis conjugate withnˆ: 1[|n+1ihn|+|nihn+1|]≡
c 2
cosϕˆ. In this case, we recover the Hamiltonian widely
where nˆ is the number difference operator, E is the
c used in the literature [20]:
“charging energy” and n acts as a control parameter
g
and is known as the “gate charge” in other context [20]. Hˆ′ =E (nˆ−n )2−E cosϕˆ.
c g J
Due to the conservationoftotalatomnumber, the prob-
lem can be restricted to the subspace of definite N, i.e., However, it should be noted that there are some sub-
tleties in doing so, mainly in the context of elementary
|n ,n i =|N +n,N −ni≡|ni,
1 2 quantum mechanics, as is well known [26,7]. Yet, iden-
Nˆ|ni =2N|ni, nˆ|ni=n|ni, (10) tifying nˆ and ϕˆ as a canonically conjugate pair is very
popular and seems to be well justified in the context of
where|n ,n i=|n i |n i ,with|n i denotingtheusual condensed matter physics, typically dealing with a large
1 2 1 1 2 2 i i
numberstatesofthetwolocalmodes(nˆ |n i =n |n i ). number of particles [27,8].
i i i i i i
2
The first condition stated above implies E ≫E . In Finally,itisworthpointingoutthatthesame(orsim-
c J
this case, the energies of the system’s states are domi- ilar) Hamiltonian as in (5) also arises in other context,
natedby thechargingpartofthe Hamiltonian(14). The e.g., in a two-mode nonlinear optical directional coupler
second condition means that when n is approximately [30] and in two-species BEC [31]. The latter has been
g
half-integer (say n + 1), the charging energies of two proposed to create macroscopic quantum superpositions
deg 2
adjacent states (|n i and |n +1i) are nearly degen- (the “Schr¨odinger cat states”). Thus the present work
deg deg
erate, and as such the Josephson term mixes the two may be viewed in a wider context.
adjacent states strongly. As a result, the eigenstates of In conclusion, we have suggested that two coupled
the Hamiltonian (14) are now superpositions of |n i BECatzerotemperaturecanbeusedtorealizetheBEC
deg
and |n +1i with a minimum energy gap E between counterpart of the superconducting Josephson charge
deg J
them. In this case, the two coupled BEC form a BEC qubits under the conditions that the charging energy of
qubitthatisanalogoustotheJosephsonchargequbitsof thesystemismuchlargerthantheJosephsonenergyand
ultrasmall Josephson junctions. Thus the system under the systemis biasednear a degeneracypoint. Ifthe con-
study actsasa“single-atombox”,aBECcounterpartof ditions canbe met, the twoBEC mayrepresenta single-
the single-Cooper-pair box [22]. Similarly to the case of atom box, with two states differing only by one atom.
ultrasmall superconducting Josephson junctions [16,20], It remains to be seen how non-zero temperatures affect
|n i≡|↑i and |n +1i≡|↓i. The effective Hamilto- the predicted BEC qubits. Due to extremely good co-
deg deg
nian in the spin-1 notion is correspondingly herence of BEC observed so far, we expect decoherence
2
wouldnotrendertheBECqubitsunobservable. Another
H=−Ec′σ − EJσ . (16) interesting issue is how to entangle two BEC qubits.
2 z 2 x Note added–Yu Shi recently brought our attention to
his related work [32].
Here
We aregratefulto the Referee for valuable suggestion.
ThisworkwassupportedbytheNationalNaturalScience
σ =|↑ih↑|−|↓ih↓|, σ =|↑ih↓|+|↓ih↑|,
z x Foundation of China, the Chinese Academy of Sciences,
Ec′ =Ec[1−2(ng−ndeg)], (ng−ndeg)∼1/2, (17) andby the LaboratoryofQuantumCommunicationand
Quantum Computation of USTC.
from which the dependence of H (or E′) on the gate
c
charge n and the degenerate point is evident. The ex-
g
pressionofE′ showsthatthoughE ≫E ,the E -term
c c J J
may dominate the E′-term near the degenerate point.
c
With the Hamiltonian (16) at hand, the single-qubit op-
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