Download Quantum dynamics with ~10 6 - Weizmann Institute of Science

Quantum dynamics with ultra cold atoms
Nir Davidson
Weizmann Institute of Science
I. Grunzweig, Y. Hertzberg, A. Ridinger (M. Andersen, A. Kaplan)
Billiards
 n  10  10
6
8
0.1
BEC
 n  1
0.2
0.3
0.4
0.5
0.6
0.7
Eitan Rowen, Tuesday
Dynamics inside a molecule:
quantum dynamics on nm scale
E
1 nm
Fsec laser pulse
R
Is there quantum chaos?
• Classical chaos: distances between close points grow exponentially
• Quantum chaos: distance between close states remains constant
 H
n2 exp i
 
   H
  exp  i
    
 
  n1  n2 | n1
 
Asher Peres (1984): distance between same state evolved by close
Hamiltonians grows faster for (underlying) classical chaotic dynamics ???
 H  
 H  
n exp i 2   exp  i 1   n
   
    
 n |n
Answer: yes….but also depends on many other things !!!
One thing with many names: survival probability = fidelity = Loschmidt echo
R. Jalabert and H. Pastawski, PRL 86, 2490 (2001)
Atom-optics billiards:
decay of classical time-correlations
Fraction of Surviving Atoms
10
0
-1
10
-2
10
-3
10
-4
10
0.00
0.05
0.10
0.15
Time [sec]
…and effects of soft walls, gravity, curved manifolds, collisions…..
PRL 86, 1518 (2001), PRL 87, 274101(2001), PRL 90 023001 (2003)
Wedge billiards: chaotic and mixed phase space
Criteria for “quantum” to “classical” transition
Old: large state number
n  1
New: “mixing” to many states by small perturbation
But “no mixing”
is hard to get
n n'  n  1
n n'   nn'  E / E  n  106
Quantum dynamics with <n>~106: challenges and solutions:
• Very weak (and controlled) perturbation –optical traps + very
strong selection rules
• No perturbation from environment - ultra cold atoms
• Measure n  106  106  1 mixing – microwave spectroscopy
• Pure state preparation? - echo
Pulsed microwave spectroscopy
Prepare Atomic Sample → MW-pulse Sequence → Detect Populations
On
• cooling and trapping
~106 rubidium atoms
• optical pumping to 1
Off
π-pulse:
1 i 2
1
i
1
2
2
2
π/2-pulse: 1 
3
optical
transition
MW “clock”
transition
2 (5S1/ 2 , F  3, mF  0)
1 (5S1/ 2 , F  2, mF  0)
Ramsey spectroscopy of free atoms
MW Power
H = Hint + Hext → Spectroscopy of two-level Atoms
1
π/2
 1  ie
 1  i 2 /
2
π/2
T
iT
2 / 2
Time
1 e 1  1 e  2 / 2
iT
1
P2  1  cos ΔT 
2
iT
Ramsey spectroscopy of trapped atoms
H  H1 1 1  H2 2 2


 H int  H ext
2
EHF
General case: Nightmare
Short strong pulses: OK (Projection)
1
Vopt  I /( L   A )
Microwave
pulse
|2,Ψ>
|1,Ψ>
|1,Ψ>
H2
H1
e-iH2t|2,Ψ>
Microwave
pulse
<Ψ| eiH1te-iH2t|Ψ>…
e-iH1t|1,Ψ>
MW Power
Ramsey spectroscopy of single eigenstate
T
π/2
π/2
Time
P2  n



1
1  n(t  0) n(t  T ) cos nT 
2
For small Perturbation: n n'   nn'  n(t  0) n(t  T )  1
MW Power
Ramsey spectroscopy of thermal ensemble
T
π/2
π/2
Time
P2  n



1
1  n(t  0) n(t  T ) cos nT 
2
For small Perturbation: n n'   nn'  n(t  0) n(t  T )  1
Averaging over the thermal
ensemble destroys the
Ramsey fringes
MW Power
Echo spectroscopy (Han 1950)
π/2
T
π
T
π/2
Time
t=T
t=2T
NOTE: classically echo should not always work for dynamical system !!!!
MW Power
Echo spectroscopy
π/2
T
π
T
π/2
Time
De-Coherence
Ramsey
Echo
Coherence
BUT: it works here !!!!
Echo vs. Ramsey spectroscopy
  H1  
  H2  
n exp  i
  exp   i
  n
   
    
Ramsey
H2
H1
 H  
 H  
 H  
 H  
n exp  i 1   exp  i 2   exp   i 1   exp   i 2   n
   
   
    
    
H2
H1
H1
H2
Echo
Quantum dynamics in Gaussian trap
Tosc/2
De-Coherence
Tosc
Calculation for H.O.
=1.5 nm
n' n  δn,n'
=3.2 nm
Coherence
=10 nm
2
EHF
1
Long-time echo signal
De-Coherence
n n'   nn'  E    E / E  n
1
P2   1  n' n n
2
n n'   nn'  E    E / E  n
Coherence
n n'   nn'  E / E  1/ n
•2-D: E / E  10
6
•1-D:E / E  10
3
4



Observation of “sidebands”
Π-pulse
4π-pulse
Quantum stability in atom-optic billiards
<n>~104
Quantum stability in atom-optic billiards
<n>~104
D. Cohen, A. Barnett and E. J. Heller, PRE 63, 046207 (2001)
Avoid Avoided Crossings
Quantum dynamics in mixed and chaotic phase-space
Incoherent
Coherent
Perturbation-independent decay
Perturbation strength
Quantum dynamics in perturbation-independent regime
0,4
P2
Chaotic
Mixed
0,2
0,0
0,000
0,005
Time between pulses (s)
0,010
Shape of perturbation is also important
… and even it’s position
No perturbation-independence
Finally: back to Ramsey (=Loschmidt)
Conclusions
•Quantum dynamics of extremely high-lying states in billiards:
survival probability = Loschmidt echo = fidelity=dephasing?
• Quantum stability depends on: classical dynamics, type and
strength of perturbation, state considered and….
• “Applications”: precision spectroscopy (“clocks”)
quantum information
Can many-body quantum dynamics be reversed as well?
(“Magic” echo, Pines 1970’s, “polarization” echo, Ernst 1992)
Atom Optics Billiards
•Control classical dynamics (regular, chaotic, mixed…)
•Quantum dynamics with <n>~106 ????
Tzahi
Ariel
Nir
Atom Optics Billiards

Positive (repulsive) laser potentials of various shapes.
Standing
Wave
Trap
Beam
• Low density  collisions
• Z direction frozen by a standing wave
• “Hole” in the wall  probe time-correlation function