1. Images of Bose-Einstein condensates
Jacek Dziarmaga
& Krzysztof Sacha
PRA 67, (2003)
cond-mat/
Kraków, Poland

2. Condensate interference
Bogoliubov theory
Diagonal t-dependent vacuum
Measurements on the vacuum

3. Condensate interference
Javanainen & Yoo, PRL 76, 161 (1996)
Fock state
Single density measurement

4. Condensate interference
In experiment
In computer experiment
annihilation of atoms

5. Condensate interference
Phase ``collapse’’ by
density measurement

6. Quantum Bogoliubov theory
Hamiltonian
Small fluctuations
Expansion
Fig.Jav

7. Bogoliubov transformation
particles <-> quasiparticles
Bogoliubov Hamiltonian
Bogoliubov vacuum

8. Time-dependent Bogoliubov theory
Condensate in the ground state
Condensate in an excited state

9. Time-dependent Bogoliubov theory
Excited state =
t-dependent vacuum
Ansatz
Solution

10. N-conserving theory N-conserving operator Vacuum is N-particle state
Girardeau & Arnowitt, PR 113, 755 (1959)
Castin & Dum, PRA 57, 3008 (1998)
N-conserving operator
Vacuum is N-particle state

11. Diagonal dynamical vacuum
We claim
D & Sacha, PRA 67, (2003)
cond-mat/

12. Proof (constructive)

13. q - representation
Real coordinates q

14. q - representation
Probability for q

15. Density measurement

16. Dark soliton Fig:imprinting Phase imprinting Organizers
Burger et al., PRL 83, 5198 (1999)

17. Phase imprinting in Bogoliubov theory
Condesate
Density
after 2ms
Total
density
after 2ms
In focus
Mode 1

18. Density measurement
Truncated vacuum

19. Conclusion

20. Condensate in thermal state
Condensate in ``excited thermal state’’