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# "Squeezed States in Bose-Einstein Condensate"

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Research talk given at Colgate University back in 2002

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### "Squeezed States in Bose-Einstein Condensate"

1. 1. Ari Tuchman Matt Fenselau Mark Kasevich Squeezed States in a Bose-Einstein Condensate Yale University Brian Anderson (JILA) Masami Yasuda (Tokyo) Chad Orzel \$\$ - NSF, ONR (Now at Union College)
2. 2. Bose-Einstein Condensate 2001 NOBEL PRIZE in PHYSICS Eric Cornell Carl Wieman Wolfgang Ketterle “ For the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates&quot;.
3. 3. Uncertainty Principle  x  p   / 2 Best known form: Fundamental limit on knowledge Improve measurement of position Lose information about momentum Position - Momentum Uncertainty  x  0  p   Important on microscopic scale ~ 10 -34 Minimum Uncertainty Wavepacket  x  p = / 2  p = / 2  x  x h h h h
4. 4. Uncertainty and Light Light Wave: Uncertainty:  E  t  / 2 Energy- Time Uncertainty Energy: Amplitude of wave Time: Phase of wave h
5. 5. Squeezed States  N   N Number-phase uncertainty  N   1/2 Coherent State: Minimum Uncertainty State  N  = 1/2 Squeezed State: Smaller  N Larger  Still  N  = 1/2 Studied with light -> Do same thing with atoms  N 
6. 6. Michelson Interferometer Laser Beam Splitter Mirror  L Light from two arms overlaps, interferes Can measure changes in path length difference (  L)  Can measure phase shifts (  ) Detector
7. 7. http://www.ligo.caltech.edu/ Laser Interferometer Gravitational Wave Observatory
8. 8. Interference of Molecules M. Arndt et al., Nature 401, 680-682, 14.October 1999 Source Grating Detector
9. 9. Atom Interferometry N 1 N 2 General Scheme: Detectors for Rotation, Acceleration, Gravity Gradients, etc. Beam splitters/ gratings Atom Beam Improve by using Squeezed States?
10. 10. Bose-Einstein Condensation High Temperature Like classical particles BEC Low Temp. Quantum wavepackets T < T c First Rb BEC, JILA, 1995
11. 11. Interfering BEC (Ketterle group, MIT) Two BEC's created in trap Let fall, overlap, interfere Fringes in overlap region M.R. Andrews et al ., Science 275, 637 (1997)
12. 12. Path to BEC Laser Cooling Cool atoms to ~ 100  K Trap ~ 10 8 - 10 9 Atoms Room Temperature Rb vapor cell Magnetic Trap (TOP) Evaporative Cooling Remove hot atoms from trap Remaining Atoms get colder BEC ~ 30,000 atoms T < 100nK
13. 13. Absorptive Imaging CCD Illuminate sample with collimated laser Atoms absorb light => Image “shadow” on camera BEC Probe Only Subtracted Image 50  m BEC
14. 14. Optical Lattice Laser shifts energy levels Lower energy of ground state |g> |e> Standing Wave Periodic Potential Atoms trapped in high intensity
15. 15. Optical Lattice U o 1-D Optical Lattice  840 nm (  = 60 nm) Focus to 60  m, retro-reflect <0.04 photons/sec Neglect scattering Atoms localized at anti-nodes of standing wave Array of traps spaced by  /2 BEC
16. 16. Atomic Tunnel Array Output Array Output: Measured pulse period of ~1.1 msec is in excellent agreement with calculated  J = mg  z/  (  z=  / 2) .
17. 17. Tunnel Array <ul><li>Tunnel array: </li></ul><ul><li>Under appropriate conditions, atoms tunnel from lattice sites to the continuum. </li></ul><ul><li>Waves interfere to form pulses in region A. </li></ul>
18. 18. Tunnel Array <ul><li>Wavefunction of atoms at q th lattice site: </li></ul><ul><li>Each site has a probability of tunneling out of lattice, into continuum: </li></ul><ul><li> Emission of deBroglie waves. </li></ul><ul><li>Relative macroscopic phase  q ( t ) depends on initial phase at t =0 and on g . </li></ul>Macroscopic Quantum State Phase
19. 19. Double-Well Potential Tunneling : Atoms hop between wells Tunneling Energy:  Mean Field Interaction: Collisions between atoms in same well Collision Energy: Ng  Ratio Ng  /  Determines Character of Ground State H =  (a L + a R + a R + a L ) + g  (N L 2 + N R 2 )
20. 20. Ground states Assume |  =  c n |n, N-n  Left trap Right trap Ng  non-interacting) |  = { (a L + + a R + )/  2} N | vac  Ng  For Ng    |  = { a L + } N/2 { a R + } N/2 | vac  Note: Squeezed solutions can not be obtained with Gross-Pitaevskii Eq., which assumes a coherent state and large N. Squeezed
21. 21. Lattice potential Use variational method to find ground-state: Example solution: “ Soft” Bose-Hubbard model 30 lattice sites Ng  ~50 atoms/site (center) Lattice plus harmonic potential Vary n 0 ,  Ansatz, |  =    |  i  ( i indexes lattice site) where, |  i  ~  exp -{(n-n 0 ) 2/    } |n 
22. 22. Lattice potential vs. double well Ng  / 
23. 23. Quantum Optics and BEC Coherent state: Undefined phase, fixed amplitude Number-phase uncertainty  N   1/2 (from Loudon, Quant. Theory of Light) Recent work by Javanainen, Castin and Dalibard, 1996 State of system when tunneling fast, interactions weak State when mean-field large, tunneling slow Fock state:
24. 24. Tunnel Array as Phase Probe ~12 wells occupied Release atoms from lattice Atom clouds expand, overlap, interfere Like multiple-slit diffraction Coherent State: Well-defined phase Sharp interference Fock State: Large phase variance Interference washes out Atoms held in lattice
25. 25. Squeezed State Formation (a) (b) (c) (d) (e) (f) 8 E r 18 E r 44 E r ramp = 200 msec Lattice strength Harmonic trap off Density image
26. 26. 3-D Picture
27. 27. Squeezed State Formation Peak Contrast vs. Well Depth <ul><li>Fit gaussians to cross sections; Peak width determines contrast </li></ul><ul><li>Vary condensate density by changing TOP gradient </li></ul>
28. 28. Simple Theory Comparison Convert B ’ q , U o to Ng  Compare to model to extract phase variance   2 = S   o 2 ~ S (1/N) Fit (Ng  /  ) C Theory: C = ½ Fit: C = 0.54(9) 0 10 20
29. 29. Fock  Coherent 200 ms 150 ms 44 E r 11 E r 13 E r 41 E r 44 -> 11 E r Adiabatically ramp up to make squeezed state Ramp down to return to coherent state Non-Adiabatic (2ms ramp up, 10ms dephasing):
30. 30. Quantum state dynamics Adiabatically ramp lattice depth to prepare number squeezed states Suddenly drop lattice depth to allow tunneling (Drop slow compared to vibration frequency in well) Time dependent variational estimate for phase variance per lattice well Experimental signature: breathing in interference contrast Number squeezed state Time Variance time Lattice depth
31. 31. Quantum State Dynamics 1 ms 5 ms 9 ms 13 ms 17 ms 21 ms 25 ms 29 ms
32. 32. Conclusion Can make number-squeezed states with a BEC in an optical lattice Use interference of atoms to probe phase state Observed factor of ~30 reduction in  N N= 2500 ± 50  2500 ± 2 Future: Look at transition between coherent/ squeezed Quantum State Dynamics Ultimate Goal: Squeezed State Atom Interferometry Have Shown: Quantum Phase Transition
33. 34. Quantum State Dynamics 1ms 4ms 6ms 8ms 10ms 12ms 15ms 19ms 23ms
34. 35. Dephasing Mechanisms 1) Ensemble phase dispersion (inhomogeneous broadening) 2) Coherent-state (self) phase dispersion 3) Squeezing Mean number (thus phase) varies trap-to-trap. Mean-field interaction + initial number variance  phase dispersion at each trap control parameter Trap i External control parameter used to control quantum many-body state at each trap Trap 2 Trap N ··· (Phasor diagrams) Trap 1  n Trap i Trap i time evolution Trap i
35. 36. Inhomogeneous Phase Broadening 2ms Hold t Ramp up in 2ms, hold for variable time Wells evolve independently ~23 E r Dephasing Time ~ (B q ) -2 => Harmonic trap
36. 37. Quantum State Dynamics: Exp’t Vary Low Lattice Level:  ~ (Ng  ) 1/2
37. 38. Quantum State Dynamics: Exp’t 3ms 7ms 13ms 19ms Max Value: 42 E r Hold at: 11 E r
38. 40. BEC Apparatus 87 Rb F = 2 m = 2 state Single Vapor Cell MOT ~ 10 4 atoms in condensate TOP and RF evaporation 1-D Optical Lattice  850 nm (  = 70 nm) Focus to 60  m Absorptive Imaging
39. 41. Double-well system Left trap Right trap H =  (a L + a R + a R + a L ) - g  a L + a L a R + a R Hamiltonian tunneling mean field Literature A. Imamoglu, M. Lewenstein, and L. You, PRL 78 2511 (1997). R. Spekkens and J. Sipe, PRA 59 , 3868 (1999). A. Smerzi and S. Raghavan, cond-mat/9905059. J. Javanainen, preprint, 1998. What is the many-body ground state of this system (assume N atoms are partitioned between the two traps)? Adiabatically manipulate tunnel barrier height
40. 42. Hold and Release 200 msec ramp 200 msec ramp + 100 msec hold 200 msec ramp + 500 msec hold ramp hold Lattice strength Harmonic trap off ~6 E r depth Density image High atomic density