BEC dynamics in few site systems

Maya Chuchem

Ben-Gurion University

Collaborations:

Doron Cohen [1,2]

Tsampikos Kottos (Wesleyan) [1]

Katrina Smith-Mannschott (Wesleyan) [1]

Moritz Hiller (Gottingen) [1]

Amichay Vardi (BGU) [2]

Erez Boukobza (BGU) [2] $DIP, $BSF

[1] Occupation dynamics during many body LZ transition

[2] Phase-diffusion for weakly coupled BEC systems

BHH of a dimer - the model

H =∑i=1,2

[εini +

U

2ni(ni − 1)

]− K

2(a†2a1 + a†1a2)

K - hopping

U - interaction

ε = ε2 − ε1 - bias

H = −εJz + UJ2z −KJx + Const.

N particles in a dimer →

a spin j =N

2particle

action-angle variables:

ak = eiϕk√nk , a†k =

√nke−iϕk

J+ ≡√

n1e−iϕ√

n2 (ϕ ≡ ϕ1 − ϕ2)

Jz ≡ n ≡1

2(n1 − n2)

Jx ≡1

2(a†2a1 + a

†1a2) ≈

√(N/2)2 − n2 cos(ϕ)

H ≈ Un2 − εn−K

√√√√((N2

)2

− n2

)cos(ϕ)

Josephson like,

Ec ∼ U and EJ ∼ KN

Phase space analysis

H ≈ NK

2

[12u(cos θ)2 − ε cos θ − sin θ cosϕ

]

Jz ≡ n =(

N2

)cos(θ) , u ≡ NU

K, ε ≡ ε

K

phase

space for

u > 1

and

|ε| < εc

εc =(u

23 − 1

) 32

, Ac ≈ 2πεc

u

E(θ) ∝ 12u(cos θ)2 − ε cos θ − sin θ

Wavepacket Dynamics - zero bias

[1] “θ = 0” preparation → N particles in single site

[2] “ϕ = 0” or “ϕ = π” preparations→ Coherent state

Wavepacket Dynamics - Wigner plots

[1] “θ = 0” preparation

(u = 2)

1 2 3 4 50

0.2

0.4

0.6

0.8

1

(black) time=5(red) time=20

[2] “ϕ = 0” preparation

(u = 25)

[2] “ϕ = π” preparation

(u = 25)

“ϕ = π” Wavepacket Dynamics

”Bloch vector length”

time (au)

Quasi-periodic behavior

Adiabatic population transfer

Dynamical scenarios: adiabatic/diabatic/sudden

-20 -10 0 10 20ε

-400

-200

0

200

400

600

800

Ene

rgy

N=30, k=1, U=0.27

Numerical simulation for Occupation Probability distribution

The time evolution for

the “θ = 0” minimal

Gaussian wavepacket

preparation

We look at:

〈n〉 Occupation expectation value

Var(n) Occupation variance

for different values of ε.

Occupation Statistics

Diabatic to sudden transition

〈n〉 = nf + (N − nf )(α/α0)

2

(1 + (α/α0)2)

Var(n) = (N − nf )(α/α0)

2

(1 + (α/α0)2)2

⇒

Var(n) = (〈n〉 − nf )

(1− (〈n〉 − nf )

(N − nf )

)

Driving Scenarios

NK−NK

(NK) 2

NK2

K2

K/N−K/N K−K

Q u a n t u m A D I A B A T I CU

2eff / N

S U D D E N P R O C E S SD I A B A T I C

K

SweepRate

(NU)K|NU|K

• Quantum adiabatic limit

• Diabatic approximation

• Sudden process

Thank You

References[1] M. Hiller, T. Kottos and A. Ossipov, Bifurcations in resonance widths of an open Bose-Hubbard dimer,

Phys. Rev. A 73, 063625 (2006).

[2] R. Franzosi and V. Penna, Spectral properties of coupled Bose-Einstein condensates, Phys. Rev. A 63,

043609 (2001).

[3] L. Bernstein, J. Eilbeck, and A. Scott, The quantum theory of local modes in a coupled system of nonlinear

oscillators, Nonlinearity 3, 293 (1990).

[4] S. Aubry et al., Manifestation of Classical Bifurcation in the Spectrum of the Integrable Quantum Dimer

Phys. Rev. Lett. 76, 1607 (1996).

[5] G. Kalosakas, A. R. Bishop, and V. M. Kenkre, Small-tunneling-amplitude boson-Hubbard dimer. II.

Dynamics, Phys. Rev. A 68, 023602 (2003).

[6] G. J. Milburn et al., Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential ,

ibid. 55, 4318 (1997).

[7] G. Kalosakas and A. R. Bishop, Small-tunneling-amplitude boson-Hubbard dimer: Stationary states, Phys.

Rev. A 65, 043616 (2002).

[8] G Kalosakas, A R Bishop and V M Kenkre, Multiple-timescale quantum dynamics of many interacting

bosons in a dimer J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3233-3238.

[9] J. R. Anglin and A. Vardi, Dynamics of a two-mode Bose-Einstein condensate beyond mean-field theory,

Phys. Rev. A 64, 013605 (2001).

[10] R. Franzosi, V. Penna, and R. Zecchina, Quantum Dynamics of Coupled Bosonic Wells Within the

Bose-Hubbard Picture, Int. J. Mod. Phys. B 14, 943 (2000).

[11] M. Albiez et al, Direct Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson

Junction, Phys. Rev. Lett. 95, 010402 (2005).

[12] G. P. Tsironis and V. M. Kenkre, Initial Condition Effects in the Evolution of a Nonlinear Dimer, Phys.

Lett. A 127, 209 (1988);

[13] V. M. Kenkre and G. P. Tsironis, Nonlinear effects in quasielastic neutron scattering: Exact line-shape

calculation for a dimer, Phys. Rev. B 35, 1473 (1987);

[14] V. M. Kenkre and D. K. Campbell, Self-Trapping on a Dimer: Time-dependent Solutions of a Discrete

Nonlinear Schroedinger Equation, ibid. 34, R4959 (1986);

[15] J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, The discrete self-trapping equation, Physica D 16, 318 (1985).

[16] D. Witthaut, E. M. Graefe, and H. J. Korsch, Towards a generalized Landau-Zener formula for an

interacting Bose-Einstein condensate in a two-level system, Phys. Rev. A 73, 063609 (2006) .

[17] F. Trimborn, D. Witthaut, H. J. Korsch Number-conserving phase-space dynamics of the Bose-Hubbard

dimer beyond the mean-field approximation, arXiv:0802.1142v2.

[18] E. M. Graefe, H. J. Korsch, and D. Witthaut, Mean-field dynamics of a Bose-Einstein condensate in a

time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman

adiabatic passage, Phys. Rev. A 73, 013617 (2006).

[19] B. Wu and J. Liu, Commutability between the Semiclassical and Adiabatic Limits, Phys. Rev. Lett. 96,

020405 (2006).

[20] J. Liu, B. Wu, Q. Niu, Nonlinear Evolution of Quantum States in the Adiabatic Regime Phys. Rev. Lett.

90, 170404 (2003).

[21] Jie Liu et al, Theory of nonlinear Landau-Zener tunneling, Phys. Rev. A 66, 023404 (2002).

[22] B. Wu and Q. Niu, Nonlinear Landau-Zener tunneling, Phys. Rev. A 61, 023402 (2000).

[23] J. P. Dowling, G. S. Agarwal,W. P Schleich, W igner distribution of a general angular-momentum state:

Applications to a collection of two-level atoms, Phys. Rev. A 49, 4101 (1994).

[24] G. S. Agarwal, Relation between atomic coherent-state representation, state multipoles, and generalized

phase-space distributions, Phys. Rev. A. 24, 2889 (1981).

[25] C.S. Chuu, F. Schreck, T.P. Meyrath, J.L. Hanssen, G.N. Price, and M.G. Raizen, Direct Observation of

Sub-Poissonian Number Statistics in a Degenerate Bose Gas, Phys. Rev. Lett. 95, 260403 (2005).

[26] A. M. Dudarev, M. G. Raizen and Q. Niu, Quantum Many-Body Culling: Production of a Definite Number

of Ground-State Atoms in a Bose-Einstein Condensate, Phys. Rev. Lett. 98, 063001 (2007).

[27] J. R. Anglin, Second-quantized Landau-Zener theory for dynamical instabilities, Phys. Rev. A 67, 051601

(2003).

[28] P. Solinas, P. Ribeiro, R. Mosseri, Dynamical properties across a quantum phase transition in the

Lipkin-Meshkov-Glick model, arXiv:0807.0703.

[29] A. Altland, and V. Gurarie, Many body generalization of the Landau Zener problem, Phys. Rev. Lett. 100,

063602 (2008)

[30] G. Ferrini, A. Minguzzi, and F. W. J. Hekking, Number squeezing, quantum fluctuations, and oscillations in

mesoscopic Bose Josephson junctions, Phys. Rev. A 78, 023606 (2008).

[31] C.-S. Chuu, et. al, Direct Observation of Sub-Poissonian Number Statistics in a Degenerate Bose Gas, Phys.

Rev. Lett. 95, 260403 (2005).

[32] A. M. Dudarev, M. G. Raizen, and Q. Niu, Quantum Many-Body Culling: Production of a Definite Number

of Ground-State Atoms in a Bose-Einstein Condensate, Phys. Rev. Lett. 98, 063001 (2007).

[33] S. Foelling, S. Trotzky, P. Cheinet, M. Feld, R. Saers, A. Widera, T. Mueller, I. Bloch, Direct Observation

of Second Order Atom Tunneling, Nature 448, 1029 (2007)

[34] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner, S. Foelling, I. Bloch, Counting

atoms using interaction blockade in an optical superlattice, arXiv:0804.3372

[35] I. Bloch, J. Dalibard, W. Zwerger, Many-Body Physics with Ultracold Gases, Rev. Mod. Phys. 80, 885

(2008).

[36] I. Bloch, U ltracold quantum gases in optical lattices, Nature Phys. 1, 23-30 (2005).

[37] D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983).

[38] Q. Niu,and D.J. Thouless, Quantized adiabatic charge transport in the presence of substrate disorder and

many-body interaction, J. Phys. A 17 2453 (1984).

[39] M.V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).

[40] J.E. Avron, A. Raveh and B. Zur, Adiabatic quantum transport in multiply connected systems, Rev. Mod.

Phys. 60, 873 (1988).

[41] J.M. Robbins and M.V. Berry, Discordance between quantum and classical correlation moments for chaotic

systems, J. Phys. A 25, L961 (1992).

[42] * M.V. Berry and J.M. Robbins, Chaotic classical and half-classical adiabatic reactions: geometric

magnetism and deterministic friction, Proc. R. Soc. Lond. A 442, 659 (1993).

[43] M. L. Polianski, M. G. Vavilov, and P. W. Brouwer, Noise through quantum pumps, Phys. Rev. B 65,

245314 (2002)

[44] M. Moskalets and M. Bttiker, Dissipation and noise in adiabatic quantum pumps, Phys. Rev. B 66, 035306

(2002)

[45] A. Andreev and A. Kamenev Counting Statistics of an Adiabatic Pump, Phys. Rev. Lett. 85, 1294 (2000).

[46] Y. Makhlin and A.D. Mirlin, Counting Statistics for Arbitrary Cycles in Quantum Pumps, Phys. Rev. Lett.

87, 276803 (2001).