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 interactions Over 8 orders of magnitude! Repulsive or attractive PRL Trapping in optical potential Optical 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 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 K Rb

Experimental setup

Internal structure Rotational Hamiltonian Integer Angular momenta Linear level spacing Spherical Symmetry Hyperfine Hamiltonian Lots of terms, most small Nuclear Quadrupole dominates Nuclear Quadrupole Diagonal in F=N+I Mixing of rotational/nuclear spin states Parameters taken from DFT/experiment

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 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 (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 (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 (2010) Wall and Carr NJP (2009)

Stark Spectra

Experimental Progress Molecules at edge of quantum degeneracy 87Rb-40K, JILA Absolute ground state STIRAP procedure Hyperfine state is important! A single hyperfine state is populated Can be chosen via experimental cleverness

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 (2003) What does it say about HMHH?

Hubbard Parameters Choose 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 B N s = 2

Decoupled: Entanglement and structure factors EB N s = 2

Now couple single to many molecule physics EB N s = 2 DC Ground state structure DC+AC Ground state structure Dynamics

Coupled: Entanglement and Structure Factors EB N s = 2

Route II: Turning on Internal State Structure N s = 4 EB DC Ground state structure DC+AC Ground state structure Dynamics

Entanglement and Structure Factor EB N s = 4

Route II.3 E B N s = 4 DC Ground state structure DC+AC Ground state structure Dynamics

Route II.4 N s = 4 E B

Physical Scales for this Problem