BEC dynamics in few site systems - BGUphysics.bgu.ac.il/~dcohen/ARCHIVE/cst_TLK.pdf · [18]E. M....
Transcript of BEC dynamics in few site systems - BGUphysics.bgu.ac.il/~dcohen/ARCHIVE/cst_TLK.pdf · [18]E. M....
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)
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.
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).