量子計算的進展及展望課件

1、Quantum phase transition of Bose-Einstein condensates on a ring with periodic scattering lengthDalian, Aug. 3, 2010 Zheng-Wei Zhou(周正威） Key Lab of Quantum Information , CAS, USTCIn collaboration with:Univ. of Sci. & Tech. of ChinaS.-L. Zhang(張少良）X.-F. Zhou (周祥發(fā)) X. Zhou (周幸祥) G.-C. Guo (郭光燦)Rice Uni

2、v.Han Pu (浦晗)Lisa 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

3、waveguidesGupta, et al. PRL (2005)4 coaxial circular electromagnetsBECs in a ring shaped magnetic waveguide. Optical dipole trap using Laguerre-Gaussian beamsBackground: Ring potential for cold atomsAtom-Atom InteractionsUltracold collision governed by s-wave scattering length, a.a0: repulsive inter

4、actionsa0: attractive interactionsControl with external magnetic or optical fieldsCornish, et al. PRL (2000)Feshbach resonanceBackground: Bosons on a ringBackground: Bosons on a ringToroidal system with sufficient transverse confinement:Weakly interacting particlesGP EquationL. D.Carr, et. al., PRA

5、62, 063211 (2000)Background: Bosons on a ringKanamoto, PRA 67,013608 (2003)Phase transition at = -0.5ground statePeriodically modulated scattering length (2 periods)Bosons on a ring with modulated interaction- Many bosons: Mean field analysisMFT solutions 2-fold degeneracyin symmetry breaking regime

6、Symmetry breaking occurs at The original symmetry manifest itself in the 2-fold degeneracy of GS.densityEnergy vs. | Phase transition0.52一個成功的經(jīng)驗：標準的Bogoliubov方法求解均勻調制1. Full many-body Hamiltonian2. Decompose into plane waves (Fourier decomposition)3. Rewrite Hamiltonian asWhen =3，凝聚穩(wěn)態(tài)的能級交叉導致量子相變。動力學

7、非穩(wěn)點Bosons on a ring with modulated interaction- A few bosons: Quantum mechanical analysis2. Decompose into plane waves (Fourier decomposition)1. Full many-body Hamiltonian3. Rewrite Hamiltonian asBosons on a ring with modulated interaction- A few bosons: Quantum mechanical analysis4. Basis states ar

8、e Fock states (angular momentum e-states)5. Diagonalize Hamiltonian in the span of this 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

9、 statesCorrelation and entanglementLeft-right spatial correlation function for N=2, 4, and 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

10、wave function defined asThe rapid vanishing of the energy gap for large means that the ground 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 sup

11、erpositions ofthe quantum ground state and its first excited state.Energy gap between the quantum mechanical ground state and the first excited state as a function of particle number N.the mean-field states are “selected” states另外一種求解該問題的途徑 Time evolving block decimation algorithm We first compute t

12、he SD of according to the bipartite splitting of the system into qubit 1 and the 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

13、 function for n-qubit system: Repeat these steps, we can express state as: coefficients 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

14、shows that the Schmidt coefficients of the state of decay exponentially with : Initialization 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:

15、i) by extracting it from the solution of the DMRG method; ii) by considering any product state, and 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 as

16、sume that does not depend on time. After a time interval T, the evolved state is given by The can be decomposed 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 gat

17、es and to a number of times, and by updating decomposition at each step. Errors and computational cost The 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 unit

18、ary evolution. ii) The order-p Trotter expansion error scale as Lemma 2 implies that updating after a two-body 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 discretiza

19、tion scheme單粒子能量（d2）單粒子能量（d3）歸一化的凝聚粒子數(shù)（d2）歸一化的凝聚粒子數(shù)（d3）ConclusionWe use the exact 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, 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 rin