Tunable Molecular Many-Body Physics and the Hyperfine Molecular

Hubbard Hamiltonian

Michael L. Wall

Department of Physics

Colorado School of Mines

in collaboration with

Lincoln D. Carr

Motivation: Ultracold atoms in optical lattices

Extremely tunable interactionsOver 8 orders of magnitude!

Repulsive or attractive

PRL 102 090402

Trapping in optical potentialOptical potential couples to

dynamical polarizability of object

Simple 2-state picture: AC Stark effect

Potential proportional to intensity

The Bose-Hubbard Model

• Excellent approximation for deep lattices!• Accounts for SF-MI transition• Simplest nontrivial bosonic lattice model

Field operator

Hopping

Interaction

Diatomic Molecules

3 energy scalesElectronic potential

Vibrational excitations

Rotational excitations

Rough scaling based on powers of m_e/M_N

At ultracold temps neglect all except for rotational terms

Focus on Heteronuclear Alkali Dimers

No spin or orbital angular momentum:

Rotational energy scale determined by B~GHz

Heteronuclear->permanent dipole moment d~1D

Dynamical polarizability is anisotropic

K

Rb

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are needed to see this picture.

Experimental setup

Internal structure

Rotational Hamiltonian

Integer Angular momenta

Linear level spacing

Spherical Symmetry

Hyperfine HamiltonianLots of terms, most small

Nuclear Quadrupole dominates

Nuclear QuadrupoleDiagonal in F=N+I

Mixing of rotational/nuclear spin states

Parameters taken from DFT/experiment

External FieldsStark effect

Breaks rotational symmetry

Couples N->N+1

Dipole moments induced along field direction

1D~0.5 GHz/(kV/cm)

Zeeman effect

Rotational coupling-small

Nuclear spin coupling-large

New handle on system

Dipolar control

Separation of dipolar and hyperfine degrees of freedom Selection rule for nuclear spin projection along E-field

Dipole strongly couples to E field, insensitive to B field

Reverse for Nuclear spin-rotate using B field

Dipole character “smeared” across many states

E B

E

B

What does the dipole get us?

Resonant dipole-dipole interaction

Anisotropic and long range

Dominates rethermalization via inelastic collisions

Ultracold chemistry->bad news for us!

Stabilize using DC field and reduced geometry

Coupling to AC microwave fields

Dynamics!

Easy access to internal states

PRA 76 043604 (2007)

Optical lattice effects

Dynamical polarizability is anisotropicReducible rank-2 tensor

Write in terms of irreducible rank-0 and rank-2 components

Tunneling depends on rotational modeDifferent “effective mass”

Put this all together…

The Hyperfine Molecular Hubbard Hamiltonian

Energy offsets from single particle spectra

Tunneling dependent on rotational mode

Nearest neighbor Dipole-Dipole interactions

Transitions between states from AC driving

Wall and Carr PRA 82 013611 (2010)

Applications 1: Internal state dependence

No AC field->Extended Bose-Hubbard model

Studies of quantum phase equilibria

Dynamics of interactions between phases

Applications 2: Quantum dephasing

Exponential envelope on Rabi oscillationsPurely many-body in nature

Emergent timescale

Applications 3: Tunable complexity

Many interacting degrees of freedomCan dynamically alter the number and timescale

Interplay of spatial and internal dof->Emergence

“Quantum complexity simulator”

Quantitative discussion in the works

Conclusions/Further research

• Cold atoms are great “quantum simulators”• Molecules have interesting new structure that

can be controlled• Emergent behavior, complexity simulator• Future work will quantify complexity, study

different molecular species, include loss terms related to chemistry, study dissipative quantum phase transitions, etc.

• Wall and Carr PRA 82 013611 (2010)

• Wall and Carr NJP 11 055027 (2009)

Stark Spectra

Experimental Progress

Molecules at edge of quantum degeneracy87Rb-40K, JILA

Absolute ground state

STIRAP procedure

Hyperfine state is important!A single hyperfine state is populated

Can be chosen via experimental cleverness

http://physics.aps.org/viewpoint-for/10.1103/PhysRevLett.101.133005

http://jila.colorado.edu/yelabs/research/cold.html

How do we simulate such a Hamiltonian?

We want to solve the Schroedinger eqn.

Question: How big is Hilbert space?Answer 1: Big

Exponential scaling->exact diagonalization difficult

Answer 2: Too big

Finite range Hamiltonians can’t move states “very far”

All eigenstates of such Hamiltonians live on a tiny submanifold of full Hilbert space

In 1D, restate as: critical entanglement bounded by

Perform variational optimization in class of states with restricted entanglement->”Entanglement compression”

Time-Evolving Block Decimation

Variational method in the class of Matrix Product States

Polynomial scalingFind ground states of nearest-neighbor Hamiltonians

Simulate time evolution (still difficult)

Google “Open source tebd”

Original paper G. Vidal PRL 91 147902 (2003)

What does it say about HMHH?

Hubbard ParametersChoose appropriate Wannier basis, compute overlaps

Hopping

Internal energy

Transitions

Interaction

Route I: Single and many molecule physics decoupled

DC Ground state structure DC+AC Ground

state structure

Dynamics

E BNs = 2

Decoupled: Entanglement and structure factors

E BNs = 2

Now couple single to many molecule physics

E BNs = 2

DC Ground state structure DC+AC Ground

state structure

Dynamics

Coupled: Entanglement and Structure Factors

E BNs = 2

Route II: Turning on Internal State Structure

Ns = 4E B

DC Ground state structure DC+AC Ground

state structure

Dynamics

Entanglement and Structure Factor

E BNs = 4

Route II.3 E

BNs = 4

DC Ground state structure DC+AC Ground

state structure

Dynamics

Route II.4Ns = 4

E

B

Physical Scales for this Problem