Download Controlling atom motion through the dipole

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
Trapping in optical potential
Optical potential couples to
dynamical polarizability of object
Simple 2-state picture:
AC Stark effect
Potential proportional to intensity
Extremely tunable interactions
Over 8 orders of magnitude!
Repulsive or attractive
PRL 102 090402
The Bose-Hubbard Model
Field operator
Hopping
Interaction
• Excellent approximation for deep lattices!
• Accounts for SF-MI transition
• Simplest nontrivial bosonic lattice model
Diatomic Molecules
3 energy scales
Electronic 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
Rb
K
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Experimental setup
Internal structure
Rotational Hamiltonian
Integer Angular momenta
Linear level spacing
Spherical Symmetry
Nuclear Quadrupole
Diagonal in F=N+I
Mixing of rotational/nuclear spin states
Parameters taken from DFT/experiment
Hyperfine Hamiltonian
Lots of terms, most small
Nuclear Quadrupole dominates
External Fields
Stark 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
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 anisotropic
Reducible rank-2 tensor
Write in terms of irreducible rank-0 and rank-2 components
Tunneling depends on rotational mode
Different “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 oscillations
Purely many-body in nature
Emergent timescale
Applications 3: Tunable complexity
Many interacting degrees of freedom
Can 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 degeneracy
87Rb-40K, JILA
Absolute ground state
STIRAP procedure
http://jila.colorado.edu/yelabs/research/cold.html
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
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 scaling
Find 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 Parameters
Choose appropriate Wannier basis, compute overlaps
Hopping
Internal energy
Transitions
Interaction
Ns = 2
Route I: Single and many
molecule physics decoupled
DC Ground state
structure
Dynamics
DC+AC Ground
state structure
E
B
Decoupled: Entanglement andE
structure factors
Ns = 2
B
Ns = 2
Now couple single to many
molecule physics
DC Ground state
structure
Dynamics
DC+AC Ground
state structure
E
B
Ns = 2
Coupled: Entanglement and
Structure Factors
E
B
Ns = 4
Route II: Turning on Internal
State Structure
DC Ground state
structure
Dynamics
DC+AC Ground
state structure
E
Ns = 4
Entanglement and Structure
Factor
E
Ns = 4
Route II.3
DC Ground state
structure
Dynamics
DC+AC Ground
state structure
E
Ns = 4
Route II.4
E
Physical Scales for this
Problem