"squeezed states in bose-einstein condensate"
TRANSCRIPT
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)
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
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".
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
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 = / 2p = / 2 x
x
h
hh
h
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Uncertainty and Light
Light Wave:
Uncertainty: E t / 2Energy- Time Uncertainty
Energy: Amplitude of wave
Time: Phase of wave
h
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Squeezed States
N
N
n
ne
n
n
2/121
!
2
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
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Michelson Interferometer
LaserBeamSplitter
Mirror
L
Light from two arms overlaps, interferes
Can measure changes in path length difference (L)
Can measure phase shifts ()
Detector
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
http://www.ligo.caltech.edu/
Laser Interferometer Gravitational Wave Observatory
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Interference of Molecules
M. Arndt et al., Nature 401, 680-682, 14.October 1999
Source Grating Detector
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Atom Interferometry
N1
N2
General Scheme:
Detectors for Rotation, Acceleration, Gravity Gradients, etc.
Beam splitters/ gratings
Atom Beam
Improve by using Squeezed States?
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Bose-Einstein Condensation
High TemperatureLike classical particles
BEC
Low Temp.Quantum wavepackets
T < Tc
First Rb BEC, JILA, 1995
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Interfering BEC
M.R. Andrews et al., Science 275, 637 (1997)
(Ketterle group, MIT)
Two BEC's created in trap
Let fall, overlap, interfere
Fringes in overlap region
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Path to BEC
Laser Cooling
Cool atoms to ~ 100 K
Trap ~ 108 - 109 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
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Absorptive Imaging
CCD
Illuminate sample with collimated laser
Atoms absorb light => Image “shadow” on camera
BEC Probe Only
Subtracted Image
50m
BEC
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Optical Lattice
Laser shifts energy levels
Lower energy of ground state|g>
|e>
Standing Wave
Periodic Potential
Atoms trapped in high intensity
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Optical Lattice
Uo
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
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Atomic Tunnel Array Output
Array Output:
Measured pulse period of ~1.1 msec is in excellent agreement with calculated J = mgz/ (z=/2).
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Tunnel Array
Tunnel array:
• Under appropriate conditions, atoms tunnel from lattice sites to the continuum.
• Waves interfere to form pulses in region A.
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Tunnel Array
• Wavefunction of atoms at qth lattice site:
• Each site has a probability of tunneling out of lattice, into continuum:
Emission of deBroglie waves.
)](exp[ tin qqq
tqt Jqq )()0()( 2/ mgJ
• Relative macroscopic phase q(t) depends on initial phase at t=0 and on g.
Macroscopic Quantum State
Phase
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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 = (aL+ aR
+ aR+ aL) + g (NL
2 + NR2)
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Ground states
Assume | = cn|n, N-n
Left trap Right trap
n0 20 40 60 80 100
Pn
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
n0 20 40 60 80 100
Pn
0.0
0.1
0.2
0.3
0.4
0.5
Ngnon-interacting)
| = {(aL+ + aR
+)/2}N|vac
Ng
For Ng| = {aL
+} N/2{ aR
+}N/2|vac
Note: Squeezed solutions can not be obtained with Gross-Pitaevskii Eq., which assumes a coherent state and large N.
Squeezed
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Lattice site0 5 10 15 20 25 30
Rel
ativ
e N
umbe
r and
fluc
tuat
ions
0
10
20
30
40
50
60
Lattice potential
Ansatz,| = |i (i indexes lattice site)where,|i ~ exp -{(n-n0)2/} |n
Use variational method to find ground-state:
Example solution:
“Soft” Bose-Hubbard model
30 lattice sitesNg~50 atoms/site (center)
Lattice plus harmonic potential
Vary n0,
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Lattice potential vs. double well
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10 100 1000
Nkd
Num
ber V
aria
nce
Lattice calculation (numerical)
Double well (exact)
Ng/
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Quantum Optics and BEC
Coherent state:
n
ne
n
n
2/121
!
2
nFock 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
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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 phaseSharp interference
Fock State:Large phase varianceInterference washes out
Atoms held in lattice
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Squeezed State Formation
(a)
(b)
(c)
(d)
(e)
(f)
8 Er
18 Er
44 Er
ramp = 200 msec
Lattice strengthHarmonic trap off
Density image
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3-D Picture
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Squeezed State Formation
Peak Contrast vs. Well Depth
• Fit gaussians to cross sections; Peak width determines contrast
• Vary condensate density by changing TOP gradient
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Simple Theory Comparison
Convert B’q, Uo to Ng
Compare to model to extract phase variance
2 = S o2 ~ S (1/N)
Fit (Ng/)C
Theory: C = ½
Fit: C = 0.54(9)
0
10
20
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Fock Coherent
200 ms 150 ms
44 Er
11 Er
13 Er 41 Er 44 -> 11 Er
Adiabatically ramp up to make squeezed state
Ramp down to return to coherent state
Non-Adiabatic (2ms ramp up, 10ms dephasing):
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1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Seconds
0.2
0.4
0.6
0.8
1
1.2
1.4
snaidaR
Relative Phase Spread For Adjacent Wells
Quantum state dynamics
time
Latti
ce d
epth
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
Var
ianc
e
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Quantum State Dynamics
Hold Time (ms)0 5 10 15 20 25 30 35
Con
tras
t
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
8 Er Fit
1 ms 5 ms 9 ms 13 ms 17 ms 21 ms 25 ms 29 ms
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
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
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Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Quantum State Dynamics
1ms 4ms 6ms 8ms 10ms 12ms 15ms 19ms 23ms
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Dephasing Mechanisms
1) Ensemble phase dispersion (inhomogeneous broadening)
2) Coherent-state (self) phase dispersion
3) Squeezing
Trap 2 Trap N
···
(Phasor diagrams)
Trap 1
n
Mean number (thus phase) varies trap-to-trap.
Trap iTrap i
time evolution
Mean-field interaction + initial number variance phase dispersion at each trap
Trap i
control parameter
Trap i
External control parameter used to control quantum many-body state at each trap
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Inhomogeneous Phase Broadening
2ms Hold t
Ramp up in 2ms, hold for variable time
Wells evolve independently
~23 Er
Dephasing Time ~ (Bq )-2 => Harmonic trap
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Quantum State Dynamics: Exp’tC
ontr
ast
Hold Time (ms)
0 10 20 30 40 50
Uf = 8 Er= 2*50 s-1
Uf = 16 Er= 2*36 s-1
Uf = 19 Er= 2*32 s-1
Vary Low Lattice Level: ~ (Ng)1/2
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Quantum State Dynamics: Exp’t
3ms
7ms
13ms
19ms
Max Value: 42 Er Hold at: 11 Er
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
BEC Apparatus
87 Rb F = 2 m = 2 state
Single Vapor Cell MOT
~ 104 atoms in condensate
TOP and RF evaporation
1-D Optical Lattice
850 nm ( = 70 nm)
Focus to 60 m
Absorptive Imaging
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Double-well system
Left trap Right trap
H = (aL+ aR
+ aR+ aL) - g aL
+ aL aR
+ aR
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
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Hold and Release
200 msec ramp
200 msec ramp + 100 msec hold
200 msec ramp + 500 msec hold
ramp hold
Lattice strengthHarmonic trap off
~6 Er depthDensity image High atomic density
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Adiabatic and Non-Adiabatic
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Time-dependent Variational Calculation
Wavefunction parameterized in terms of mean and variance of atom number and phase for each lattice site:
Time dependent equations for variational parameters:
where
Model allows for calculation of time evolution of quantum state. Valid for < 1 rad.
Lattice wavefunction:
~ ( Tunneling energy / mean field energy )
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Bloch Oscillations
• Momentum change dp/dt = -mg
• Wavepackets Bragg diffract from lattice when p = -k. After diffraction p = +k.
• Momentum oscillations with period T=(2k/m)/g
• Frequency is J
ENS, 1996also UT Austin, 1996
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
Squeezed States
N
N
n
ne
n
n
2/121
!
2
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
Kasevich GroupYale Universityhttp://amo.physics.yale.edu
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'
q2
Brot
Circle of Death
(Time Orbiting Potential)
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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
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Quantum Phase Transition
Formation of fragmented state is sudden.
Can be identified by the energy difference between the ground and first excited states.
N = 20, 60 and 100 atoms
Nkd
Paradigm “quantum phase transition”
Related work: D. Jaksch, et al., PRL, 1998. (Mott insulator transition in optical lattices)
E 2-E1 degree of fragmentation
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Mean Field Interactions
• Atom-atom interactions modify the energy of the system
• Mean field: energy/particle changes by an average of
42an/m.
where,
a = scattering lengthn = atomic density = ||2m = atomic mass
2
22
2 42 m
aVmt
i ext
• Gross-Pitaevskii Equation
Mean field energy