Strong correlations and quantum vortices for ultracold atoms in rotating lattices

Murray HollandJILA

(NIST and Dept. of Physics , Univ. of Colorado-Boulder)

Acknowledgments

•Stefano Giogini•Marilu Chiofalo •Rajiv Bhat•Lincoln Carr•John Cooper•Rajiv Bhat•Brandon Peden•Ron Pepino•Brian Seaman•Jochen Wachter

Special thanks: Erich Mueller

Outline

1. Released momentum distribution of a Fermi gas in the BCS-BEC crossover

2. Strongly interacting atoms in a rotating optical lattice

[Two recent papers: look on cond-mat or PRL]

Dashed: before field ramp

Solid: after field ramp

Calculated momentum distribution at unitarity: homogeneous gas

2

2

F

Calculated release energy: homogeneous gas

F5

3

Column integrated momentum distribution in a trap

BCS

BEC

Solid: Theory

Crosses: Experiment

[Regal et al. PRL 95, 250404 (2005).]

6.0,0,7.0,711

akF

Calculated release energy in a trap

Blue: Leggett ground state + expansion

Red circles: Regal/Jin experiment

[PRL 95, 250404 (2005)]

Green: two-body physics

F8

3

Rotating a BEC

• Observation of vortex formation (Cornell, JILA)

• BEC rotated using Stirring (ENS,JILA)Two-component condensates ( JILA)

• Main features- Quantized vortices with depleted cores- Formation of vortex lattice

• Both have gotten close (>0.99) to the Quantum Hall regime • Can optical lattices help enter the strongly correlated QH regime?

The Quantum Hall effect in cold gases

int2222

2

1

2

1Urmm

m Ap

xy yxi

Urmm

H

int22

2)1(

2

1

2

• For a 2D system at =

int22)1(

2

1U

mH yx

rΩA

• Very similar to one-body Hamiltonian for 2D electrons in a magnetic field

CoulombyxQH Vm

H 22)1( ''2

1

Ground State is the variational Laughlin state of general form

2/||exp)()( 2k

ji k

qji zzzz

)()( jiji rrrrV

N=5 atoms

Contact interactionsq=2

)1(86420 NNLLaughlin

Quantum Phase Transitions

• Quantum Phases (Greiner et.al., 2001)

a. Superfluid- Hopping dominates- Particles delocalized- Coherent- Number density on each site uncertain

b. Mott Insulator-Onsite energy dominates-Particles localized-No phase correlation between sites

-Integer number of particles on each site

t/U

BECs in Optical Lattices

Rotating Bose-Einstein Condensates

BECin a 2D rotating lattice

BEC in a rotating 2D lattice experimentally realized at JILA

• Scheme proposed by-J. Reijnders et al, PRL 93, 060401 (2004) -H. Pu et al, PRL 94, 190401 (2005)

• First experimental realization at JILA• Lattice spacing ~ 10• Filling factor (particles/site)~103-104

• BEC in Rotating lattice schematic (Cornell, 2005)

Outline

Theory• Hamiltonian• Methods

Cross-checks• Center depletion •MI-SF phase diagram

Results• Interaction effects• Symmetries• QPTs

Summary and future work

System described using modified Bose-Hubbard Hamiltonian

• Particle field operator expanded using site specific annihilation/ creation operators and a Wannier basis

ˆ (x) ˆ a iw0(x x i)i

• Two approximations

-Tight binding approximation (Only nearest neighbor terms considered)-Only lowest Bloch band occupied

ˆ H t (aia j aia j

)i, j

ni U

2ni(ni 1)

i

i

i K ij (aia j aia j

)i, j

I. Hopping IV. RotationII. Chemical potential

III. Interaction energy

ˆ H d2r ˆ 2

2M2

g

2ˆ ˆ V lat L

ˆ

• Hamiltonian in a rotating frame of reference

System described using modified Bose-Hubbard Hamiltonian

I.

ji

jiji aaaat,

)( Hopping: KE associated with particles hopping from one site to the next

i

inII. Chemical potential part of onsite energy

U

2ni(ni 1)

i

III. Interaction part of onsite energy: Proportional to number of pairs

ji

jijiij aaaaKi,

)(IV. Rotation: Preferred hopping in one direction. Kij is a dimensionless geometric factor

)( ijjiij yxyxCK

• indicates angular velocity of rotating frame. xi and yi are coordinates from center of rotation.

Methodology: Diagonalize and examine ground state

Write down many-body HamiltonianKE, PE, interaction, rotation

Second quantize HamiltonianModified Bose-Hubbard Hamiltonian

Specify basis setProduct basis using n Fock states per site

Exactly diagonalize

Examine groundstateUse energy eigenvalues and a toolbox of operators (current, number

density and discrete rotational symmetry) to examine the ground state

Center depletion seen in odd lattice

• Axis of rotation goes through central site in a 3X3 lattice• Center number density gets depleted once rotation sets in

Results

Interaction restricts current flow

Discrete rotational symmetry states

Quantum phase transitions between states of different symmetry

Results

Interaction restricts current flow

Discrete rotational symmetry states

Quantum phase transitions between states of different symmetry

Interaction restricts current flow

• Current decreases with increasing interaction as particles find it more difficult to cross each other

• Same currents for 1 and 3 particles indicate Fermionization

•Two-state approximation correctly describes strongly repulsive particles in a lattice

Number of particles in lattice

J12

1 2

34

Results

Interaction restricts current flow

Discrete rotational symmetry states

Quantum phase transitions between states of different symmetry

Discrete rotational symmetry

• Schematic for 4X4 lattice

• 4X4 square lattice: Each site can have at most one particle (strongly repulsive bosons)

• The ground state must be consistent with the symmetry of the lattice(4-fold here). A Discrete Rotational Operator R exists such that

4/24 mier

r

R

R

• R commutes with the lattice potential and with H. Energy eigenstates are simultaneous eigenstates.

• Symmetry states are labeled by index m

Discrete changes in energy as system adopts higher rotational energy symmetries 1 Particle in

4X4 lattice

• Discrete jumps in energy derivative due to level crossings

• Discontinuous changes in ground state symmetry

n=1

=0.1: Rotation yet to enter the system

Current

Number density

Number density and current

=0.2: Rotation sets in the center

Current

Number density

CurrentNumber density

=0.4: Number density moves out

=0.8: Direction of current in rotating frame changes

Current

Number density

Results

Interaction restricts current flow

Discrete rotational symmetry states

Quantum phase transitions between states of different symmetry

Avoided level crossings for transitions between same symmetry states

•Symmetry of the system dependent on filling• Particles prefer to be spread out

• 4 particles in 16 sites

Level crossings for transitions between different symmetry states

• Filling not commensurate with 4-fold symmetry of system• Energy level crossings for many particles as function of a Hamiltonian parameter is a non-trivial result characterizing Quantum Phase Transitions

• 5 particles in 16 sites

Summary

• Our current work on rotation is at the junction of interesting areas of research - rotating BECs and BECs in optical lattices.

• Future work: Rotating lattices play a two fold role by enhancing interactions and restricting number of particles per vortex and may help to open up strongly correlated regimes (FQHE) for exploration.

•Momentum distribution measurements have provided detailed quantitative information revealing limitations and strengths of the Leggett ground state

• Shape independent, time-dependent theory with no adjustable parameters