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Many–particle entanglement with Bose–Einstein condensates

A. Sørensen∗, L.-M. Duan†, J. I. Cirac† & P. Zoller†∗ Institute of Physics and Astronomy, University of Aarhus, DK-8000Arhus C, Denmark† Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria

We propose a method to produce entangled states of several par-ticles starting from a Bose-Einstein condensate. In the proposal, asingle fastπ/2 pulse is applied to the atoms and due to the collisionalinteraction, the subsequent free time evolution creates anentangledstate involving all atoms in the condensate. The created entangledstate is a spin-squeezed state which could be used to improvethesensitivity of atomic clocks.

The possibility of creating and manipulating entangledstates of many–particle systems has recently boosted the fieldof quantum information since it may yield new applicationswhich rely on the basic principles of Quantum Mechanics1.So far, up to four atoms have been entangled in a controlledway2,3. The experimental achievement of Bose–Einsteincondensation4–6 has also raised a lot of attention since itmay lead to applications in several fields of Science. Someof these applications are based on the fact that condensatescan be considered as pure states at the single particle level,which is a crucial requirement for the production of entangledstate. Thus, a natural question is to investigate whether Bose–Einstein condensates can also be used in some applications ofquantum information. In this paper we show that this is indeedthe case. We propose a method to obtain substantial entangle-ment of a large number of atoms with present technology. Inour proposal, a single resonant pulse is applied to all atomsin the condensate, and the collisional interaction entangles theatoms in the subsequent free evolution.

Consider a set ofN two–level atoms confined by someexternal trap. In order to describe the internal propertiesofthese atoms, it is convenient to let the internal states|a〉nand |b〉n of the n-th atom represent the two states of a fic-titious spin 1/2 particle with angular momentum operatorsj(n)z = 1/2(|a〉〈a|n−|b〉〈b|n), j(n)

x = 1/2(|b〉〈a|n + |a〉〈b|n),

and j(n)y = i/2(|b〉〈a|n − |a〉〈b|n). We consider collective

effects of the atoms which are described by total angular mo-mentum operators,~J =

∑Nn=1

~j(n). The entanglement prop-erties of the atoms can be expressed in terms of the variancesand expectation values of these operators. In the appendix weshow that if

ξ2 ≡N(∆J~n1

)2

〈J~n2〉2 + 〈J~n3

〉2< 1, (1)

whereJ~n ≡ ~n · ~J and the~n’s are mutually orthogonal unitvectors, then the state of the atoms is non–separable (i.e. en-tangled). The parameterξ2 thus characterizes the atomic en-tanglement, and states withξ2 < 1 are often referred to as“spin squeezed states”7. In the following we show how to re-duceξ2 by several orders of magnitude using the collisionalinteractions between atoms in a Bose–Einstein condensate.

We consider a two component weakly interacting

Bose-Einstein condensate, which is present in severallaboratories8,9, and we assume that the interactions do notchange the internal state of the atoms. This situation is de-scribed by the second quantized Hamiltonian

H =∑

j=a,b

∫

d3rΨ†j(r)H0,jΨj(r)

+1

2

∑

j=a,b

Ujj

∫

d3rΨ†j(r)Ψ

†j(r)Ψj(r)Ψj(r)

+ Uab

∫

d3r؆a(r)؆

b(r)Ψa(r)Ψb(r), (2)

whereH0,j is the one particle Hamiltonian for atoms in statej including the kinetic energy and the external trapping po-tentialVj(r), Ψj(r) is the field operator for atoms in the statej, Ujk = 4πh2ajk/m is the strength of the interaction be-tween particles of typej andk, parameterized by the scatter-ing lengthajk, andm is the atomic mass.

Assume that we start with a Bose–Einstein condensate instate|a〉 at very low temperature (T ≃ 0), so that all theatoms are in a single particle (motional) state|φ0〉. A fastπ/2 pulse between the states|a〉 and|b〉 prepares the atoms inthe state|φ0〉

⊗N ⊗ (|a〉+ |b〉)⊗N/2N/2 which is an eigenstateof theJx operator with eigenvalueN/2. If we choose in (1)~n1 = [0, cos(θ), sin(θ)] and~n2 along thex axis of the ficti-tious angular momentum, we haveξ2

θ = 1 at t = 0. By usingthe equations of motion of the angular momentum operatorsin the Heisenberg picture we find the time derivative ofξ2

θ att = 0

d

dtξ2θ = sin(2θ)

(N − 1)(Uaa + Ubb − 2Uab)

2h

∫

d3r|φ0|4.

(3)

This equation immediately shows that spin squeezing will beproduced for certain anglesθ if Uaa + Ubb 6= 2Uab as it is inthe Na experiments9.

To quantify the amount of squeezing which may be ob-tained, we assume identical trapping potentialsVa(r) =Vb(r) and identical coupling constants for interaction betweenatoms in the same internal stateaaa = abb > aab. Physi-cally, this could correspond to the|F = 1, MF = ±1〉 hy-perfine states of Na trapped in an optical dipole trap. Dueto the symmetry of these states their scattering lengths andtrapping potentials will be identical; moreover, due to angularmomentum conservation there are no spin exchanging colli-sions between these states as required by our model. This isexactly the experimental setup used in Ref. 9. In this exper-iment it is shown that these states have an antiferromagnetic

1

interactionaab < aaa which according to (3) enables the pro-duction of squeezed states. To avoid spin changing collisionsthat populate the state|MF = 0〉10 one has to slightly mod-ify such an experimental set–up. If theF = 1 manifold iscoupled to theF = 2 with a far off-resonant blue detunedπ-polarized microwave field, the|F = 1, MF = 0〉 state israised in energy with respect to the|F = 1, MF = ±1〉 states,since the Clebsch-Gordan coefficient for the|F = 1, MF =0〉 → |F = 2, MF = 0〉 is larger than the coefficients forthe |F = 1, MF = ±1〉 → |F = 2, MF = ±1〉 transitions,and spin exchanging collisions become energetically forbid-den. If for instance one chooses a detuningδ = (2π)25 MHzand a resonant Rabi frequency for the1 → 1 transition ofΩ = (2π)2 MHz, the energy separation is∆E = 640nK.With a typical chemical potentialµ ≈ 220 nK10, this energyseparation is much higher than the available energy in the col-lisions and the|F = 1, M = 0〉 state is completely decoupled.In Ref. 10 a much smaller energy difference is shown to ex-clude spin exchanging collision and we therefore expect thatmuch weaker fields will suffice.

The assumptionaaa = abb has several advantages. Firstly,it reduces the effect of fluctuations in the total particle number.If aaa 6= abb the mean spin performs aN dependent rotationaround thez-axis, and fluctuations in the number of particlesintroduces an uncertainty in the direction of the spin whichef-fectively reduces the average value and introduces noise intothe system. Withaaa = abb the mean spin remains in thex-direction independent of the number of atoms in the trap.Secondly, this condition ensures a large spatial overlap ofdif-ferent components of the wavefunction. After theπ/2 pulsethe spatial wavefunction is no longer in the equilibrium state.That is, due to the atomic repulsions (which are now differentthan before since the atoms are in different internal states), thespatial distribution of the atomic cloud will start oscillating.Furthermore, since the state of the system is now distributedover a range of number of particles in the|a〉 state (Na), andsince this number enters into the time evolution, theNa de-pendent wavefunctionsφa andφb are different for particles inthe states|a〉 and|b〉. With aaa = abb, φa andφb are identicalif Na equals the average numberN/2. In the limit of largeN ,the width of the distribution on differentNa’s is much smallerthanNa and all the spatial wavefunctions are approximatelyidenticalφa(Na, t) ≈ φb(Na, t) ≈ φ0(t). This relation isonly true if aaa > aab where small deviations from the aver-age wavefunction perform small oscillations. In the oppositecase the deviations grow exponentially11 resulting in a reduc-tion of the overlap of thea and theb wavefunctions and areduced squeezing. The advantages mentioned above couldalso be achieved withaaa 6= abb by using the breath–togethersolutions proposed in Ref. 11.

Before analyzing quantitatively the complete system, weestimate the amount of spin squeezing we can reach with ourproposal by using a simple model. Assuming the same wave-functionφ0 for both |a〉 and |b〉 atoms is constant and inde-pendent of the numberNa, the spin dependent part of theHamiltonian (2) may be written asHspin = hχJ2

z , whereχdepends on the scattering lengths and the wavefunctionφ0.The spin squeezing produced by this Hamiltonian can be cal-

culated exactly7. In the limit of largeN , the minimum obtain-

able squeezing parameter isξ2θ = 1

2

(

3N

)2/3, which indicates

that our proposal might produce a reduction ofξ2 by a factorof ∼ N2/3 which would be more than three orders of magni-tude with105 atoms in the condensate.

In contrast to the simplified HamiltonianHspin the realHamiltonian (2) will also entangle the internal and motionalstates of the atoms which is a source of decoherence for thespin squeezing. To quantify this effect we have performeda direct numerical integration following the procedure devel-oped in Ref. 11. We split the whole Hilbert space into orthog-onal subspaces containing a fixed number of particlesNa andNb = N − Na in each of the internal states, respectively. Ineach subspace we make a Hartree-Fock variational ansatz interms of three-dimensional spatial wavefunctionsφa(Na, t)andφb(Nb, t), which are evolved according to the time de-pendent coupled Gross-Pitaevskii equations. This is an ap-proximation to the full problem which is valid in the limitof low temperatures and short times, where the population ofthe Bogoliubov modes is small. Particularly it is a good ap-proximation in our case withaab < aaa, abb where there areno demixing instabilities. With this procedure, the decoher-ence induced by the entanglement with the motional state iseffectively taken into account. Together with the predictionfrom the simple HamiltonianHspin the result of the simula-tion is shown in Fig. 1. The two curves are roughly in agree-ment confirming that the system is able to approximate the re-sults of the HamiltonianHspin. The numerical solution showsfluctuations due to the oscillations of the spatial wavefunc-tion. The large dips atωt ≈ 4, 9, 13, and 18 are the pointswhere the atomic cloud reach the initial width. At these in-stants the overlap of the wavefunctions is maximal and thetwo curves are very close (up to a factor of two). The smalldips atωt =2, 7, 11, and 16 corresponds to the points of max-imum compression. With the realistic parameters used in thefigure, our simulation suggests that three orders of magnitudesqueezing is possible. Also, note the time scale in the fig-ure. The maximally squeezed state is reached after approx-imately two oscillation periods in the trap. For a fixed ratioof the scattering lengthsaab/aaa, the optimal time scales as(aaa/d0)

−2/5N−1/15, whered0 =√

h/(mω) is the width ofthe ground state of the harmonic potential.

2

1

10-1

10-2

10-3

10-4

0 4 8 12 16 20

wt

x2

q

FIG. 1. Reduction in the squeezing parameterξ2. A fast π/2pulse between two internal states is applied to all atoms in the con-densate. The subsequent free time evolution results in a strongsqueezing of the total spin. The angleθ is chosen such thatξ2

θ isminimal. The solid line is the results of a numerical integration(see text). For numerical convenience we have assumed a spheri-cally symmetric potentialV (r) = mω2r2/2. The parameters areaaa/d0 = 6 ∗ 10−3, abb = 2aab = aaa, andN = 105. Thedashed curve shows the squeezing obtained from the HamiltonianHspin = hχJ2

z . The parameterχ is chosen such that the reductionof Jx obtained from the solution in Ref. 7 is consistent with the re-sults of Ref. 11.

The analysis so far has left out a number of possible im-perfections. Specifically, we have assumed that all scatteringlength are real so that no atoms are lost from the trap and wehave not considered the role of thermal particles. To estimatethe effect particle losses we have performed a Monte Carlosimulation12 of the evolution of squeezing from the Hamil-tonianHspin. The particle loss is phenomenologically takeninto account by introducing a loss rateΓ which is identicalfor atoms in the|a〉 and the|b〉 state. In Fig. 2 we show theobtainable squeezing in the presence of loss. Approximately10 % of the atoms are lost at the timeχt ≈ 6 ∗ 10−4 wherethe squeezing is maximally without loss. With the parame-ters of Fig. 1 this time corresponds to roughly two trappingperiods. Such a large loss is an exaggeration of the loss com-pared to current experiments and the simulation indicate thateven under these conditions, squeezing of nearly two ordersofmagnitude may be obtained. On the other hand, the effects ofthermal particles can be suppressed at sufficiently low temper-atures but due to the robustness with respect to particle losesshown in Fig. 2, we expect to obtain high squeezing even atsome finite temperatures.

In conclusion we believe that we have presented a sim-ple and robust method to produce entangled states of alarge number of atoms with present technology. The pro-duced entangled state are interesting in fundamental physicsand they also have possible technological applications inatomic clocks13,14, where the projection noise(∆Jθ)

2 is cur-rently the main source of noise15. In particular Boyer andKasevich14 have shown how to use an entangled state withhalf of the particlesN/2 in one internal statea, and the other

half in the other stateb to obtain the Heisenberg limit in atomicinterferometric measurements. On the other hand, it has beenshown that if the atoms are prepared in a state withξ2 < 1before they are injected into an atomic clock, one can reducethe frequency noise (variance in the frequency measurements)or the measuring time to obtain a desired precision by a factorξ2 as compared to the case in which one uses atoms in an un-correlated state13. If the squeezing produced by our proposalis transferred into a suitable clock transition, and the atomiccloud is allowed to expand so that the role of collisions is re-duced during the frequency measurement, the states could bedirectly applied in the current set–up of atomic clocks. Othertheoretical proposals for noise reduction in neutral atom sys-tems have been made16, and a weak squeezing of the spinhas recently been produced experimentally17,18. However,our proposal has the considerable advantage that it offers avery strong noise reduction, and it is directly applicable to ex-isting experimental set–ups. In future experiments with neg-ligible particle loss even for very long interaction times,theHamiltonianHspin could also be used to produce maximallyentangled state of any number of atoms2,19, which could re-duce the frequency noise to the fundamental limit of quantummechanics20.

1

10-1

10-2

10-3

10-4

0 4 8 12 16 20

x2

q

ct 10X-4

FIG. 2. Quantum Monte Carlo simulation of squeezing in thepresence of loss. The full line shows squeezing obtained from theHamiltonianHspin with particle loss described by a constant lossrateΓ = 200χ. The dashed curve shows squeezing without particleloss.

AppendixHere we present the derivation of Eq. (1) as a criterion forentanglement. AnN -particle density matrixρ is defined to beseparable (non-entangled) if it can be decomposed into

ρ =∑

k

Pkρ(1)k ⊗ ρ

(2)k ⊗ ... ⊗ ρ

(N)k , (4)

where the coefficientsPk are positive real numbers fulfilling∑

k Pk = 1, andρ(i)k is a density matrix for thei’th parti-

cle. The variance ofJz may be described as(∆Jz)2 = N

4 −∑

k Pk

∑

i〈j(i)z 〉2k +

∑

k Pk〈Jz〉2k −〈Jz〉

2, and using Cauchy-

Schwarz’s inequality and〈j(i)x 〉2k + 〈j

(i)y 〉2k + 〈j

(i)z 〉2k ≤ 1/4

3

we find three inequalities for separable states∑

k Pk〈Jz〉2k ≥

〈Jz〉2, −

∑

k Pk

∑

i〈j(i)z 〉2k ≥ −N

4 +∑

k Pk

∑

i〈j(i)x 〉2k +

〈j(i)y 〉2k, and 〈Jx〉

2 ≤ N∑

k Pk

∑

i〈j(i)x 〉2k. From these in-

equalities we immediately find that any separable state obeysξ2 ≥ 1 and hence any state withξ2 < 1 is non-separable.AcknowledgmentsThis work was supported by the Austrian Science Founda-tion, the European Union project EQUIP, the TMR europeannetwork, the ESF under the PESC program ”Quantum Infor-mation”, the Institute for Quantum Information GmbH, andThomas B. Thriges Center for Kvanteinformatik. A. S. ac-knowledges the hospitality of the University of Innsbruck.

1 Special issue on quantum information.Phys. World11 No. 2, 33-57 (1998).

2 Sackett, C.A.et al. Experimental entanglement of four particles.Nature404, 256-259 (2000).

3 Rauschenbeutel, A.et al. Step-by-step engineered multiparticleentanglement.Science288, 2024-2028 (2000).

4 Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E.& Cornell, E.A. Observation of Bose-Einstein condensationin adilute atomic vapor.Science269, 198-201 (1995).

5 Davis, K.B.et al.Bose-Einstein condensation in a gas of sodiumatoms.Phys. Rev. Lett.75, 3969-3973 (1995).

6 Parkins, A.S. & Walls, D.F. The physics of trapped dilute-gasBose-Einstein condensates.Physics Reports303, 1-80 (1998).

7 Kitagawa, M. & Ueda, M. Squeezed spin states.Phys. Rev. A47,5138-5143 (1993).

8 Hall, D.S., Matthews, M.R., Ensher, J.R., Wieman, C.E. & Cor-nell, E.A. Dynamics of component separation in a binary mix-ture of Bose-Einstein condensates.Phys. Rev. Lett81, 1539-1542(1998).

9 Stenger, J.et al.Spin domains in ground-state Bose-Einstein con-densates.Nature396, 345-348 (1998).

10 Miesner, H.-J.et al. Observation of metastable states in spinorBose-Einstein condensates.Phys. Rev. Lett.82, 2228-2231(1999).

11 Sinatra, A. & Castin, Y. Binary mixtures of Bose-Einstein con-densates: Phase dynamics and spatial dynamics.Eur. Phys. J. D8319-332 (2000).

12 Mølmer, K., Castin, Y. & Dalibard J. Monte Carlo wave-functionmethod in quantum optics.J. Opt. Soc. Am. B10, 524-538 (1993).

13 Wineland, D.J., Bollinger, J.J., Itano, W.M. & Heinzen, D.J.Squeezed atomic states and projection noise in spectroscopy.Phys. Rev. A50, 67-88 (1994).

14 Boyer, P. & Kasevich, M. A. Heisenberg-limited sprectroscopywith degenerate Bose-Einstein gases.Phys. Rev. A56, R1083-1086 (1997).

15 Santarelli, G.et al. Quantum projection noise in an atomic foun-tain: A high stability cesium frequency standard.Phys. Rev. Lett.82, 4619-4622 (1999).

16 Sørensen, A. & Mølmer, K. Spin-spin interaction and spin squeez-ing in an optical lattice.Phys. Rev. Lett.83, 2274-2277 (1999).

17 Hald, J., Sørensen, J.L., Schori, C. & Polzik, E.S. Spin squeezed

atoms: A macroscopic entangled ensemble created by light.Phys.Rev. Lett.83, 1319-1322 (1999).

18 Kuzmich, A., Mandel, L. & Bigelow, N.P. Generation of SpinSqueezing via Continuous Quantum Nondemolition Measure-ment,Phys. Rev. Lett.85 1594-1597 (2000).

19 Mølmer, K. & Sørensen, A. Multiparticle entanglement of hottrapped ions.Phys. Rev. Lett.82, 1835-1838 (1999).

20 Bollinger, J.J., Itano, W.M., Wineland, D.J. & Heinzen, D.J. Op-timal frequency measurements with maximally correlated states.Phys. Rev. A54, 4649-4652 (1996).

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