1. Slow dynamics in gapless low-dimensional systems
Anatoli Polkovnikov,
Boston University
Vladimir Gritsev – Harvard
Ehud Altman - Weizmann
Eugene Demler – Harvard
Bertrand Halperin - Harvard
Misha Lukin - Harvard
CMT Seminar, Yale, 11/08/2007
AFOSR

2. (controlled and tunable Hamiltonians, isolation from environment)
Cold atoms:
(controlled and tunable Hamiltonians, isolation from environment)
1. Equilibrium thermodynamics:
Quantum simulations of equilibrium condensed matter systems
2. Quantum dynamics:
Coherent and incoherent dynamics, integrability, quantum chaos, …

3. (controlled and tunable Hamiltonians, isolation from environment)
Cold atoms:
(controlled and tunable Hamiltonians, isolation from environment)
2. Quantum dynamics:
Coherent and incoherent dynamics, integrability, quantum chaos, …

4. Qauntum Newton Craddle
Qauntum Newton Craddle. (collisions in 1D interecating Bose gas – Lieb-Liniger model)
T. Kinoshita, T. R. Wenger and D. S. Weiss, Nature 440, 900 – 903 (2006)
No thermalization during collisions of two one-dimensional clouds of interacting bosons.
Fast thermalization if the clouds are three dimensional.
Quantum analogue of the Fermi-Pasta-Ulam problem.

5. (controlled and tunable Hamiltonians, isolation from environment)
Cold atoms:
(controlled and tunable Hamiltonians, isolation from environment)
1. Equilibrium thermodynamics:
Quantum simulations of equilibrium condensed matter systems
2. Quantum dynamics:
Coherent and incoherent dynamics, integrability, quantum chaos, …
3. = 1+2 Nonequilibrium thermodynamics?

6. Adiabatic process. Assume no first order phase transitions.
Adiabatic theorem:
“Proof”:
then

7. Adiabatic theorem for integrable systems.
Density of excitations
Energy density (good both for integrable and nonintegrable systems:
EB(0) is the energy of the state adiabatically connected to the state A. For the cyclic process in isolated system this statement implies no work done at small .

8. Adiabatic theorem in quantum mechanics
Landau Zener process:
In the limit 0 transitions between different energy levels are suppressed.
This, for example, implies reversibility (no work done) in a cyclic process.

9. Adiabatic theorem in QM suggests adiabatic theorem in thermodynamics:
Transitions are unavoidable in large gapless systems.
Phase space available for these transitions decreases with d. Hence expect
Is there anything wrong with this picture?
Hint: low dimensions. Similar to Landau expansion in the order parameter.

10. More specific reason.
Equilibrium: high density of low-energy states ->
strong quantum or thermal fluctuations,
destruction of the long-range order,
breakdown of mean-field descriptions,
Dynamics ->
population of the low-energy states due to finite rate ->
breakdown of the adiabatic approximation.

11. This talk: three regimes of response to the slow ramp:
Mean field (analytic) – high dimensions:
Non-analytic – low dimensions
Non-adiabatic – lower dimensions

12. Some examples.
1. Gapless critical phase (superfluid, magnet, crystal, …).
LZ condition:

13. Second example: crossing a QCP.
tuning parameter 
gap 
   t,   0
Gap vanishes at the transition. No true adiabatic limit!
How does the number of excitations scale with  ?
A.P. 2003

14. Perturbation theory (linear response). (A.P. 2003)
Expand the wave-function in many-body basis.
Substitute into Schrödinger equation.

15. Find: Uniform system: can characterize excitations by momentum:
Use scaling relations:
Find:

16. Transverse field Ising model.
There is a phase transition at g=1.
This problem can be exactly solved using Jordan-Wigner transformation:

17. Critical exponents: z==1  d/(z +1)=1/2.
Spectrum:
Critical exponents: z==1  d/(z +1)=1/2.
Linear response (Fermi Golden Rule):
A. P., 2003
Correct result (J. Dziarmaga 2005):
Interpretation as the Kibble-Zurek mechanism:
W. H. Zurek, U. Dorner, Peter Zoller, 2005

18. Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations.
Zero temperature.

19. Most divergent regime: k0 = 0
Agrees with the linear response.
Assuming the system thermalizes

20. Finite temperatures.
Instead of wave function use density matrix (Wigner form).

21. Artifact of the quadratic approximation or the real result?
Results
d=1,2
Non-adiabatic regime!
d=1;
d=2;
d=3
Artifact of the quadratic approximation or the real result?

22. Numerical verification (bosons on a lattice).
Use the fact that quantum fluctuations are weak and expand dynamics in the effective Planck’s constant (saddle point parameter)

23. Classical limit – use Gross-Pitaevskii equations with initial conditions distributed according to the thermal density matrix.

24. How do we add quantum corrections?
Idea: expand quantum evolution in powers of .
Take an arbitrary observable
We have two fields propagating in time forward and backward .
Treat  exactly, while expand in powers of .

25. Results:
Leading order in : start from random initial conditions distributed according to the Wigner transform of the density matrix and propagate them classically (truncated Wigner approximation):
Expectation value is substituted by the average over the initial conditions.
Exact for harmonic theories!
Not limited by low temperatures!
Asymptotically exact at short times.
Subsequent orders: quantum scattering events (quantum jumps)

26. Results (1d, L=128)
Predictions:
finite temperature
zero temperature

27. T=0.02

28. Thermalization at long times.

29. 2D, T=0.2

30. Conclusions. Mean field (analytic): Non-analytic Non-adiabatic
Three generic regimes of a system response to a slow ramp:
Mean field (analytic):
Non-analytic
Non-adiabatic
Open questions: general fate of linear response at low dimensions, non-uniform perturbations,…

31. M. Greiner et. al., Nature (02)
Adiabatic increase of lattice potential
Superfluid
Mott insulator
What happens if there is a current in the superfluid?

32. ??? U/J Drive a slowly moving superfluid towards MI. p U/J p/2 p SF MI
possible experimental sequence:
~lattice potential
??? p
U/J
p/2
Stable
Unstable SF MI

33. Include quantum depletion.
Equilibrium: 
Current state:  p
With quantum depletion the current state is unstable at

34. Meanfield (Gutzwiller ansatzt) phase diagram
Is there current decay below the instability?

35. Role of fluctuations Phase slip
Below the mean field transition superfluid current can decay via quantum tunneling or thermal decay .

36. 1D System. – variational result
semiclassical parameter (plays the role of 1/ )
N~1
Large N~
C.D. Fertig et. al., 2004
Fallani et. al., 2004

37. Higher dimensions.
Longitudinal stiffness is much smaller than the transverse. r
Need to excite many chains in order to create a phase slip.

38. Phase slip tunneling is more expensive in higher dimensions:
Stability phase diagram
Crossover
Stable
Unstable

39. Current decay in the vicinity of the superfluid-insulator transition

40. Use the same steps as before to obtain the asymptotics:
Discontinuous change of the decay rate across the meanfield transition. Phase diagram is well defined in 3D!
Large broadening in one and two dimensions.

41. Detecting equilibrium SF-IN transition boundary in 3D.
U/J
p/2
Superfluid MI
Extrapolate p
Easy to detect nonequilibrium irreversible transition!!
At nonzero current the SF-IN transition is irreversible: no restoration of current and partial restoration of phase coherence in a cyclic ramp.

42. J. Mun, P. Medley, G. K. Campbell, L. G. Marcassa, D. E. Pritchard, W
J. Mun, P. Medley, G. K. Campbell, L. G. Marcassa, D. E. Pritchard, W. Ketterle, 2007

43. Conclusions.
Three generic regimes of a system response to a slow ramp:
Mean field (analytic):
Non-analytic
Non-adiabatic
Smooth connection between the classical dynamical instability and the quantum superfluid-insulator transition.
Quantum fluctuations
Depletion of the condensate. Reduction of the critical current. All spatial dimensions.
mean field
beyond mean field
Broadening of the mean field transition. Low dimensions

44. Density of excitations:
Energy density:
Agrees with the linear response.