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

2. 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

3. Background

4. Background

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

6. 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, (2015)

7. 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:

8. 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:

9. Two Fermions in a double well: Experimental realization
green
blue

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

11. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

25. 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, (2008)

26. 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

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

28. 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

29. 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

30. Two Atoms in a double well: An exact solution

31. Thank you for your attention !