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Signatures of chaos-induced mesoscopic entanglement

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2009 J. Phys. B: At. Mol. Opt. Phys. 42 031001 (http://iopscience.iop.org/0953-4075/42/3/031001) View the table of contents for this issue, or go to the journal homepage for more

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JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

doi:10.1088/0953-4075/42/3/031001

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 031001 (4pp)

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Signatures of chaos-induced mesoscopic entanglement Christoph Weiss and Niklas Teichmann Institut für Physik, Carl von Ossietzky Universität, D-26111 Oldenburg, Germany E-mail: [email protected]

Received 23 December 2008, in final form 9 January 2009 Published 27 January 2009 Online at stacks.iop.org/JPhysB/42/031001 Abstract For a Bose–Einstein condensate in a double-well potential, time-periodic shaking can lead to the emergence of non-classical mesoscopic superpositions on very short timescales. It is suggested to experimentally detect these deviations from mean-field (Gross–Pitaevskii) behaviour by investigating both the variance of the particle number and the interference pattern. The disappearance of the interference pattern followed by its reappearance a short time later could serve as an experimental signature of mesoscopic entanglement. (Some figures in this article are in colour only in the electronic version)

For a BEC in a double well, |n, N − n refers to the Fock state with n particles in the left well and N − n particles in the right well. Such NOON states have also been referred to as ‘Schrödinger-cat’ states in, e.g., the abstracts of [13] and [14]. The idea is that for a BEC of some 100 atoms the ‘cat’, rather than being alive and dead at the same time, is at two places at once or in two distinct hyperfine states at once. For such mesoscopic entangled states, the concepts of entanglement differ from what is used in, e.g., quantum communication [15]. Schemes of how target states such as the NOON state (1) can be reached can be found in [16–26] and references therein. The schemes published in the literature can be divided into cases for which the target state is reached under particularly ideal conditions and cases for which entanglement is likely to survive realistic experimental conditions [27]. A huge disadvantage of target states such as the NOON state is the fact that it is very sensitive to decoherence via particle losses. Thus, entanglement generation of highly entangled (but not perfectly entangled) states on short timescales might be preferable to trying to obtain perfect entanglement on longer timescales. The chaos-induced entanglement generation investigated in [7] could provide such a scheme. However, while it is possible to numerically identify highly entangled states, it is not at all clear if they could also be measured experimentally. Furthermore, such experimental signatures will in general rely on repeating the experiment several times. Due to experimental uncertainties, this will happen with slightly

While tunnelling control via time-periodic potential differences is already interesting on the single-particle level [1–4], Bose–Einstein condensates (BECs) in periodically shaken potentials lead to new effects. The photonassisted tunnelling predicted in [5] has recently been realized experimentally [6]. The emergence of entanglement in such systems can even be used to demonstrate the limits of the mean-field (Gross–Pitaevskii) approach to Bose– Einstein condensates [7]; limitations of the mean-field approach have recently also been discussed, e.g., in [8]. The focus in the present communication will be to predict experimentally measurable signatures of chaos-induced mesoscopic entanglement for a Bose–Einstein condensate in a double well on short timescales. Entanglement is interesting both because it is intrinsically non-classical and because many aspects are still challenges for fundamental research. While some recent research focuses on applications towards quantum computing [9], the generation of mesoscopic superpositions is another aspect pursued today [10]. In a landmark experiment, the number squeezed states of many atoms have been measured in double wells and few-well systems [11]. These states are, roughly speaking, related to the Mott-insulator states [12] of few atoms per site with very low fluctuation of the number difference of neighbouring sites. Entirely different is a particularly interesting target state for mesoscopic entanglement, the NOON state, (1) |ψNOON = √12 (|N, 0 + |0, N ). 0953-4075/09/031001+04$30.00

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© 2009 IOP Publishing Ltd Printed in the UK

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 031001

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different Hamiltonians which could lead to entirely different final states as the mean-field system is chaotic. After a brief introduction to the model used to describe a BEC in a double-well potential, interesting parameter regimes for the generation of chaos-induced mesoscopic entangled states are identified numerically. Experimentally measurable signatures are discussed and it is demonstrated that, when repeating the experiment for slightly different conditions, the signatures would still be measurable. To model the BEC in a periodically shaken double-well potential, we use the Hamiltonian [28, 29] h ¯ † Hˆ = − aˆ 1 aˆ 2 + aˆ 1† aˆ 2 + h ¯ κ aˆ 1† aˆ 1† aˆ 1 aˆ 1 + aˆ 2† aˆ 2† aˆ 2 aˆ 2 2 (2) +h ¯ (μ0 + μ1 sin(ωt)) aˆ 2† aˆ 2 − aˆ 1† aˆ 1 ,

where ( n12 )2 is the experimentally measurable [11] variance of the particle-number difference between both wells, one has an entanglement flag. For the pure states obtained by solving the N-particle Schrödinger equation, FQFI > 1

clearly identifies entanglement [32]. Defining entangled states as states which cannot be written as product states [15], this statement is supported by the fact that for a single atomic coherent state (5) one has FQFI 1. For the NOON state (1), one has FQFI = N ; we thus define highly entangled states as being characterized by N (10) FQFI > . 2 However, simply measuring the variance of the particlenumber difference will not at all prove entanglement. Statistical mixtures could also lead to such results; performing in the shaken double well under conditions similar to [11] will not lead to a pure state but rather to a mixture of highly entangled states. Another experimental signature will be provided by interference patterns2 after switching off the potential and letting the wavefunction expand for some time. A single atomic coherent state |ϑ, ϕ leads to an interference pattern used to experimentally detect the phase between the condensates [35]

where the operator aˆ j(†) creates (annihilates) a boson in well j , the tilt is described by μ0 , μ1 is the amplitude of the periodic shaking, is the single-particle tunnelling frequency and 2¯hκ is the pair interaction energy. The corresponding Gross–Pitaevskii dynamics can be mapped onto that of a non-rigid pendulum [30] which displays a co-existence of chaotic and regular dynamics [31]. To derive the Gross–Pitaevskii equation, one usually assumes that all N particles occupy the same single-particle state. Within the two-mode approximation, to identify a singleparticle state, two complex amplitudes c1 and c2 are sufficient (|ci |2 1, i = 1, 2). Discarding an overall phase factor, the population imbalance cos ϑ ≡ |c1 |2 − |c2 |2 ,

0 ϑ π,

and the phase between the two amplitudes c2 c1 , c1 , c2 = 0, exp(iϕ) ≡ c c 1

I = [1 + sin(ϑ) cos(X − φ)],

(4)

are sufficient to uniquely identify a single-particle state. On the N-particle level, putting all atoms into this single-particle state leads to the wavefunction N N 1/2 |ϑ, ϕ = cosn (ϑ/2) sinN−n (ϑ/2) n n=0 (5)

These bimodal phase states [15] are sometimes called atomic coherent states. Superpositions of two such states |ψsuperpos ≡ N (|ϑ1 , ϕ1 + |ϑ2 , ϕ2 )

Visibility 1,

(7)

For θ1 = 0 and θ2 = π , the wavefunction (6) becomes the NOON state (1). While numerical projections on such highly entangled states can be easily calculated to identify the emergence of mesoscopic entanglement [7], to predict experimental signatures requires greater care. The quantum Fisher information [32, 33] provides a first clue into that direction. Defining FQFI ≡ 1

( n12 )2 , N

(13)

which might simply mean sin(ϑ) 0 (cf equation (11)). However, when combined with equation (10), it will be the signature for the superposition of two (or more) distinct atomic coherent states with sin(ϑ1 ) sin(ϑ2 ) and cos(X − ϕ1 ) + cos(X − ϕ2 ) = 0. While a combination of the experimentally measurable entanglement signatures, equations (10) and (13), provides strong indications for entanglement, similar effects could be obtained by, e.g., heating. As for the case of the Mott-insulator transition [12], we will thus have to show that a clear interference pattern can reappear after the system was in the state believed to be (and numerically shown to be) a highly entangled mesoscopic state. Figure 1 shows both the quantum Fisher information (8) and the visibility for a small BEC of 100 atoms (cf [36]); to

(6)

are highly entangled if1 |ϑ1 , ϕ1 |ϑ2 , ϕ2 |2 1.

(11)

which still has to be multiplied by a Gaussian envelope; X essentially is the spatial variable in the direction connecting the two wells of the double well. To characterize the quality of interference fringes, Imax − Imin , (12) Visibility ≡ Imax + Imin where Imax is the highest and Imin is the lowest value of I. Given the fact that even a single atomic coherent state can produce an impressive interference pattern, interference cannot be used in a straightforward manner as an entanglement flag as in [10, 26]. Let us thus consider the disappearance of interference patterns,

(3)

2

× ei(N−n)ϕ |n, N − n.

(9)

(8)

2

In [34], it was suggested to distinguish NOON states (1) from the corresponding statistical mixtures via centre-of-mass measurements.

The atomic coherent states become orthogonal in the limit N → ∞.

2

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 031001


(a)

(b)

Figure 1. Two-dimensional projection of both the visibility (left, equation (12)) and the quantum Fisher information (right, equation (8)) as a function of the initial atomic coherent state |ϑ0 , ϕ0 (equation (5)); z0 = cos(ϑ0 ). For each initial state, the final value after a time evolution during t1 = 10/ is shown for ω = 3, μ0 = 1.5, N = 100, N κ/ = 0.8 and 2μ1 /ω = 1.841 183 781, such that the J1 Bessel function reaches its first maximum, equation (16). (This figure is in colour only in the electronic version)

choose much larger condensates would make the entangled states too short-lived. In order to numerically calculate the visibility, one needs [37] ˆ ˆ † (x) (x)|ψ (14) ψ|

0.5 0

with ˆ † (x) = 1 (x)aˆ 1 + 2 (x)aˆ 2 ,

(15)

0

τ

10

50 0

0

τ

10

Figure 2. Time evolution of a typical initial state (ϑ = π/2, ϑ = −0.93π ; cf figure 1) which evolves into having both low visibility (left, equation (12)) and high quantum Fisher information (right, equation (8)) at t1 = 10/ . All other parameters can be found in figure 1. At t1 13.5/ (right border of the plots), both the interference pattern has reappeared and the quantum Fisher information is considerably reduced which offers the possibility of distinguishing the vanishing of an interference pattern from heating.

where i is the mode which was localized in well i with i = 1, 2 before switching off the well. The parameters were chosen such that they correspond to the one-photon resonance in photon-assisted tunnelling [5]. Many aspects of photon-assisted tunnelling in this case can be understood by replacing the time-dependent Hamiltonian (2) with a timeindependent untilted one with renormalized effective singleparticle tunnelling frequency : eff ≡ J1 (2μ1 /ω),

100 FQFI

Visibility

1

(16) Figure 2 shows the time dependence of both visibility and quantum Fisher information for a typical initial state which fulfils both entanglement requirements (equations (10) and (13)): for times which are of the order of 1.35t1 , both the interference patterns reappear and the quantum Fisher information (while still indicating entanglement) is reduced considerably. Both the visibility and the quantum Fisher information could only be measured by repeating the experiment several times. Taking similar initial conditions and identical Hamiltonians, as usual, does not pose any problem (cf figure 1). However, the repetitions will involve Hamiltonians which differ at least slightly in each run. Figure 3 shows that even for slightly different Hamiltonians, both visibility and quantum Fisher information would depend only weakly on variations of the driving amplitude and the driving frequency. However, the plots show that the requirements on the accuracy of the driving frequency are stronger than on the driving amplitude (the latter should not be too surprising as a photon-assisted-tunnelling system is known to have a comparatively narrow dependence on the frequency [6]). Changing the interaction constant κ

where J1 is the Bessel function of an integer order 1. In figure 1, the shaking amplitude μ1 was chosen such that Experimentally, J1 (2μ1 /ω) reaches its first maximum. interactions with N κ ≈ can be achieved [35] which are necessary for the mean-field system to become chaotic (cf [7]). In order to really find typical behaviour, the initial conditions were changed systematically to map the entire space of initial conditions for the Gross–Pitaevskii equation. The map also contains experimentally realistic initial conditions with all particles being in one well [35]. The entanglement flags are shown after letting the system evolve for 10 , (17) t1 = which indeed is a small timescale for entanglement generation as it is suggested to use collapse and revival to produce perfect superpositions on timescales of [15] π Trevival = ; (18) 2κ for N κ = 0.8 and N = 100, one has 20N κ/ t1 = ≈ 0.05. (19) Trevival Nπ 3

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 031001 60

(a) FQFI

Visbility

1

0

1.8 1.9 2μ1/ω

FQFI 0

2.8

3 ω/Ω

3.2

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20

60

(c)

References (b)

40

0

2

Visbility

1

1.7


1.7

1.8 1.9 2μ1/ω

2

(d)

40 20 0

2.8

3 ω/Ω

3.2

Figure 3. Visibility (12) and quantum Fisher information (8) at times t = t1 = 10/ (solid lines) and t = 1.35t1 (dotted lines) for the initial state of figure 2 under slightly different experimental conditions. If the driving amplitude is varied, both the values for the visibility (a) and for the quantum Fisher information (b) only depend weakly on the value of the driving amplitude. Varying the driving frequency gives a stronger dependence of both visibility (c) and quantum Fisher information (d) on the driving frequency; however, the dependence is still weak enough to allow experiments with very similar final signatures (all parameters not shown here can be found in figures 1 and 2).

and thus mimicking the particle-number dependence lead to curves which are as flat as for the driving amplitude. To conclude, it was shown that vanishing of interference fringes after releasing the BEC from the double well could serve as an experimental signature of chaos-induced entanglement generation. To achieve this, it should be accompanied by a high variance of the particle-number difference between both wells which corresponds, for pure states, to having high quantum Fisher information. Generation of mesoscopic entanglement and reappearance of interference fringes take place on timescales, an order of magnitude shorter than for generation of mesoscopic entanglement via collapse and revival [15]. Thus, although the final states would not be perfect entangled states, chaos-induced mesoscopic entanglement will be interesting experimentally beyond demonstrating that the mean-field (Gross–Pitaevskii) approach is not valid for these conditions [7].

Acknowledgments We thank M Holthaus for his continuous support and S Arlinghaus, Y Castin, J Dalibard, J Estève, B Gertjerenken, M Oberthaler and A Ridinger for insightful discussions. Computer power was obtained from the GOLEM I cluster of the Universität Oldenburg. NT acknowledges funding by the Studienstiftung des deutschen Volkes.

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