"squeezed states in bose-einstein condensate"

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".


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


Uncertainty and Light

Light Wave:

Uncertainty: E t / 2Energy- Time Uncertainty

Energy: Amplitude of wave

Time: Phase of wave

h


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


Michelson Interferometer

LaserBeamSplitter

Mirror

L

Light from two arms overlaps, interferes

Can measure changes in path length difference (L)

Can measure phase shifts ()

Detector


http://www.ligo.caltech.edu/

Laser Interferometer Gravitational Wave Observatory


Interference of Molecules

M. Arndt et al., Nature 401, 680-682, 14.October 1999

Source Grating Detector


Atom Interferometry

N1

N2

General Scheme:

Detectors for Rotation, Acceleration, Gravity Gradients, etc.

Beam splitters/ gratings

Atom Beam

Improve by using Squeezed States?


Bose-Einstein Condensation

High TemperatureLike classical particles

BEC

Low Temp.Quantum wavepackets

T < Tc

First Rb BEC, JILA, 1995


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


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


Absorptive Imaging

CCD

Illuminate sample with collimated laser

Atoms absorb light => Image “shadow” on camera

BEC Probe Only

Subtracted Image

50m

BEC


Optical Lattice

Laser shifts energy levels

Lower energy of ground state|g>

|e>

Standing Wave

Periodic Potential

Atoms trapped in high intensity


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


Atomic Tunnel Array Output

Array Output:

Measured pulse period of ~1.1 msec is in excellent agreement with calculated J = mgz/ (z=/2).


Tunnel Array

Tunnel array:

• Under appropriate conditions, atoms tunnel from lattice sites to the continuum.

• Waves interfere to form pulses in region A.


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


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)


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


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,


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/


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


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


Squeezed State Formation

(a)

(b)

(c)

(d)

(e)

(f)

8 Er

18 Er

44 Er

ramp = 200 msec

Lattice strengthHarmonic trap off

Density image


3-D Picture


Squeezed State Formation

Peak Contrast vs. Well Depth

• Fit gaussians to cross sections; Peak width determines contrast

• Vary condensate density by changing TOP gradient


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


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):


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


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


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


Ultimate Goal: Squeezed State Atom Interferometry

Have Shown:

Quantum Phase Transition



1ms 4ms 6ms 8ms 10ms 12ms 15ms 19ms 23ms


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


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


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


Quantum State Dynamics: Exp’t

3ms

7ms

13ms

19ms

Max Value: 42 Er Hold at: 11 Er


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


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


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


Adiabatic and Non-Adiabatic


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 )


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


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


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)


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


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


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

"squeezed states in bose-einstein condensate"

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