1. The document discusses the formation of squeezed quantum states in Bose-Einstein condensates trapped in optical lattices. By slowly ramping up the depth of the optical lattice, the atoms can be prepared in a number-squeezed state. 2. Releasing the atoms from the lattice allows their wavefunctions to overlap and interfere, providing a way to probe the quantum phase state of the atoms. Number-squeezed states are observed to produce interference patterns with higher contrast than coherent states. 3. Variational calculations of the quantum state dynamics during lattice ramping and dephasing agree qualitatively with experimental observations of the transition between coherent and squeezed states.

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. 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".

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. Uncertainty and Light Light Wave: Uncertainty:  E  t  / 2 Energy- Time Uncertainty Energy: Amplitude of wave Time: Phase of wave h

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. 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. http://www.ligo.caltech.edu/ Laser Interferometer Gravitational Wave Observatory

8. Interference of Molecules M. Arndt et al., Nature 401, 680-682, 14.October 1999 Source Grating Detector

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. Bose-Einstein Condensation High Temperature Like classical particles BEC Low Temp. Quantum wavepackets T < T c First Rb BEC, JILA, 1995

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. 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. Absorptive Imaging CCD Illuminate sample with collimated laser Atoms absorb light => Image “shadow” on camera BEC Probe Only Subtracted Image 50  m BEC

14. Optical Lattice Laser shifts energy levels Lower energy of ground state |g> |e> Standing Wave Periodic Potential Atoms trapped in high intensity

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. Atomic Tunnel Array Output Array Output: Measured pulse period of ~1.1 msec is in excellent agreement with calculated  J = mg  z/  (  z=  / 2) .

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. 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. 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. Lattice potential vs. double well Ng  / 

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. 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. 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. 3-D Picture

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. 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. 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. Quantum State Dynamics 1 ms 5 ms 9 ms 13 ms 17 ms 21 ms 25 ms 29 ms

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

34. Quantum State Dynamics 1ms 4ms 6ms 8ms 10ms 12ms 15ms 19ms 23ms

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

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

37. Quantum State Dynamics: Exp’t Vary Low Lattice Level:  ~ (Ng  ) 1/2

38. Quantum State Dynamics: Exp’t 3ms 7ms 13ms 19ms Max Value: 42 E r Hold at: 11 E r

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

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

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

43. Adiabatic and Non-Adiabatic

44. Time-dependent Variational Calculation Wavefunction parameterized in terms of mean and variance of atom number and phase for each lattice site: Model allows for calculation of time evolution of quantum state. Valid for   < 1 rad.  ~ ( Tunneling energy / mean field energy ) Time dependent equations for variational parameters: where Lattice wavefunction:

46. 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 

47. TOP Trap Quadrupole Trap B ~ B ' q x Tightly confining, but spin-flip losses Apply rotating bias field  ~ 10 kHz Time-averaged potential Harmonic: U ~ B ' q 2 B rot Circle of Death (Time Orbiting Potential)

48. Evaporative Cooling Remove hot atoms from trap => Remaining sample gets colder TOP Evaporation Reduce rotating field Circle of Death moves in Forced RF Evaporation Drive transition to un-trapped state