1、Quantum phase transition of Bose-Einstein condensates on a ring with periodic scattering lengthDalian,Aug.3,2010 Zheng-Wei Zhou(周正威）周正威）Key Lab of Quantum Information,CAS,USTCIn collaboration with:Univ.of Sci.&Tech.of ChinaS.-L.Zhang(张少良）X.-F.Zhou(周祥发)X.Zhou(周幸祥)G.-C.Guo (郭光灿)Rice Univ.Han Pu (浦晗)Li
2、sa C.QianMichael L.WallOutlineBackground:Bosons on a ringBosons on a ring with modulated interactionMany bosons:Mean field analysisA few bosons:Quantum mechanical analysis;Entanglement and correlationConclusionOctober,2009KITPCBackground:Ring potential for cold atoms Magnetic waveguidesGupta,et al.P
3、RL(2005)4 coaxial circular electromagnetsBECs in a ring shaped magnetic waveguide.Optical dipole trap using Laguerre-Gaussian beamsBackground:Ring potential for cold atomsAtom-Atom Interactions Ultracold collision governed by s-wave scattering length,a.a0:repulsive interactions a0:attractive interac
4、tions Control with external magnetic or optical fieldsCornish,et al.PRL(2000)Feshbach resonanceBackground:Bosons on a ringBackground:Bosons on a ringrR22222(,)(,)(,)2itgtttmR Toroidal system with sufficient transverse confinement:Weakly interacting particlesGP Equation2gNL.D.Carr,et.al.,PRA 62,06321
5、1(2000)Background:Bosons on a ring2gNKanamoto,PRA 67,013608(2003)0.5:uniform amplitute0.5:soliton state(symmetry breaking)Phase transition at =-0.5ground state sin(2)gPeriodically modulated scattering length(2 periods)222(,)sin(2)2,tittt 22222(,)(,)(,)2itgtttmR Bosons on a ring with modulated intera
6、ction-Many bosons:Mean field analysisMFT solutions 60.02-fold degeneracyin symmetry breaking regime60.054.025.0Symmetry breaking occurs at 0.52The original symmetry manifest itself in the 2-fold degeneracy of GS.densityEnergy vs.|Phase transition0.52一个成功的经验：标准的Bogoliubov方法求解均匀调制1.Full many-body Hami
7、ltonian2.Decompose into plane waves(Fourier decomposition)3.Rewrite Hamiltonian asWhen=3，凝聚稳态的能级交叉导致量子相变。，凝聚稳态的能级交叉导致量子相变。动力学非稳点动力学非稳点Bosons on a ring with modulated interaction-A few bosons:Quantum mechanical analysis2.Decompose into plane waves(Fourier decomposition)1()2LilllLa e 2220(sin 2)HdN 1.
8、Full many-body Hamiltonian3.Rewrite Hamiltonian as2224llklmklmklmklmNilklmHl a aa a a aa a a a Bosons on a ring with modulated interaction-A few bosons:Quantum mechanical analysis4.Basis states are Fock states(angular momentum e-states)1101,.,.,LLLnnnn nn 5.Diagonalize Hamiltonian in the span of thi
9、s basisground-state energy per particlesDensity profile of quantum mechanical ground states with N=6.No spontaneous symmetry breaking happens in quantum mechanical ground states!Energy and density profile of ground statesCorrelation and entanglementLeft-right spatial correlation function for N=2,4,a
10、nd 6.This implies that the quantum ground state is a Schrdinger cat state for large !Correlation and entanglementground state.Entanglement of ground state for N=2(N=2)we calculate the overlap of the ground-state wave function defined asThe rapid vanishing of the energy gap for large means that the g
11、round state and the first excited state essentially become degenerate,a result in accordance with the MFT analysis.The two degenerate solitonlike states found in MFT are just the symmetric and antisymmetric superpositions ofthe quantum ground state and its first excited state.Energy gap between the
12、quantum mechanical ground state and the first excited state as a function of particle number N.the mean-field states are“selected”states另外一种求解该问题的途径另外一种求解该问题的途径 Time evolving block decimation algorithm We first compute the SD of according to the bipartite splitting of the system into qubit 1 and the
13、 n-1 remaining qubits.where ,we expand each Schmidt vector in a local basis for qubit 2,then we write each in terms of at most Schmidt vectors a and the corresponding Schmidt coefficients ,finally we can obtain A wave function for n-qubit system:Repeat these steps,we can express state as:coefficient
14、s In a generic case grows exponentially with n.However,in one-dimensional settings it is sometimes possible to obtain a good approximation to by considering only the first terms,with Problem:Numerical analysis shows that the Schmidt coefficients of the state of decay exponentially with :Initializati
15、on We consider only Hamiltonians made of arbitrary single-body and two-body terms.With the interactions restricted to nearest neighbors,The ground state can be obtained through one of the following methods:i)by extracting it from the solution of the DMRG method;ii)by considering any product state,an
16、d by using the present scheme to simulate an evolution in imaginary time according to ,The second method rely on simulating a Hamiltonian evolution from a product state.Evolution For simplicity,we assume that does not depend on time.After a time interval T,the evolved state is given by The can be de
17、composed as The Trotter expansion of order p for reads where and where a for first and second order expansions.The simulation of the time evolution is then accomplished by iteratively applying gates and to a number of times,and by updating decomposition at each step.Errors and computational cost The
18、 main source of errors in the algorithm are the truncation and the Trotter expansion.i)The truncation error is Truncation errors accumulate additively with time during the simulation of a unitary evolution.ii)The order-p Trotter expansion error scale as Lemma 2 implies that updating after a two-body
19、 gate requires basic operations.Gates and are applied times and each of them decomposes into about n two body gates.Therefore operations are required on .The finite-differerence discretization scheme单粒子能量（单粒子能量（d2）单粒子能量（单粒子能量（d3）归一化的凝聚粒子数（归一化的凝聚粒子数（d2）归一化的凝聚粒子数（归一化的凝聚粒子数（d3）ConclusionWe use the exac
20、t diagonalization and TEBD to study the behavior of few particles systems,which reveals that the degeneracy found in the soliton phase of the MFT is lifted.Instead,the ground state is comprised of a strongly anti-correlated macroscopic superposition of solitons peaked at different spatial locations,
21、and can be regarded as a Schrdinger cat state,which becomes increasingly fragile as the total number of atoms increase.We studied the ground states of 1D BECs in a ring trap with d spatial periods of modulated scattering length,within and beyond the Gross-Pitaevskii mean-field theory.In the MFT,the
22、ground state undergoes a quantum phase transition between a sinusoidal state matching the spatial symmetry of the modulated interaction strength and a bright solitonlike state that breaks such a symmetry.the dfold ground state degeneracy was found in the symmetry-breaking regime.Reference:Lisa C.Qian,Michael L.Wall,Shaoliang Zhang,Zhengwei Zhou,and Han Pu,Phys.Rev.A 77,013611(2008).Zheng-Wei Zhou,Shao-Liang Zhang,Xiang-Fa Zhou,Xingxiang Zhou,Guang-Can Guo,Han Pu,in preparation.
