Two atoms in a double well: Exact solution with a Bethe ansatz PHYSICAL REVIEW A 91, 053610 (2015) 博士生：刘彦霞 导 师：张云波 单 位:山西大学

Two Fermions in a double well: Experimental realization Outline Background Two Fermions in a double well: Experimental realization Two Atoms in a double well: An exact solution

Background

Background

Background S. Zöllner, et. al., PRL 100, 040401 (2008)

Two Fermions in a double well: Experimental realization Two ultracold fermionic atoms were deterministicly prepared in an isolated double-well. S. Murmann et.al. PRL 114, 080402 (2015)

Two Fermions in a double well: Experimental realization 2. Two-mode model is not any more a good choice for double well, especially when the interaction between two particles is strong or the excited states are involved in the tunneling dynamics. The Hamiltonian of two-mode model The basis:

Two Fermions in a double well: Experimental realization Energies of the four lowest two-particle eigenstates in a symmeric double-well potential. Particle exchange symmetric: Particle exchange anti-symmetric: Even parity: Odd parity:

Two Fermions in a double well: Experimental realization green blue

Two Atoms in a double well: An exact solution The Hamiltonian of the system

Two Atoms in a double well: An exact solution The wave function is taken as the Bethe ansatz type where For bosons the wave function of the system is symmetric under coordinates exchange

Two Atoms in a double well: An exact solution Fixed one block n The open boundary conditions are

Two Atoms in a double well: An exact solution The scattering between two particles

Two Atoms in a double well: An exact solution Connects the blocks n - 1 and n The jump condition

Two Atoms in a double well: An exact solution blocks n - 1

Two Atoms in a double well: An exact solution blocks n

Two Atoms in a double well: An exact solution The continuous condition of wave function

Two Atoms in a double well: An exact solution BA-type equation The number of the equations

Two Atoms in a double well: An exact solution The Quantum Yang-Baxter equation The number of the equations

Two Atoms in a double well: An exact solution Limiting case d=0 g=0

Two Atoms in a double well: An exact solution Two particles case BA-type equation The Quantum Yang-Baxter equation

Two Atoms in a double well: An exact solution Energy spectrum of two atoms d=0.5 d=0

Two Atoms in a double well: An exact solution g=-6.295 g=-0.5 g=1.5 g=20

Two Atoms in a double well: An exact solution g=1.5 d=0.5 d=1.5 d=10 d=50

Two Atoms in a double well: An exact solution the probabilities to find i particles in the right well The mean particle number n=1 n=0 The conservation of probability n=2 n=1 Sascha Zöllner, Phys. Rev. L 100, 040401 (2008)

Two Atoms in a double well: An exact solution Occupation probabilities of |b> state Two-mode model g=-10 g=-0.5 g=1.5 g=50

Two Atoms in a double well: An exact solution Initial state: The time evolution of the wavefunction

Two Atoms in a double well: An exact solution Initial state: the eigenstate |b> with d=300 Abruptly reduce the height of the potential barrier d=300 → d=0.5

Two Atoms in a double well: An exact solution 单粒子隧传的几率随着g的增加而增加

Two Atoms in a double well: An exact solution

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