Tunable Molecular Many-Body Physics and the Hyperfine Molecular Hubbard Hamiltonian

1. 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

2. 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

3. The Bose-Hubbard Model Field operator Hopping • Excellent approximation for deep lattices! • Accounts for SF-MI transition • Simplest nontrivial bosonic lattice model Interaction

4. 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

5. Rb K 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

6. Experimental setup

7. 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

8. 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

9. 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 B

10. 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)

11. 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…

12. 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)

13. Applications 1: Internal state dependence • No AC field->Extended Bose-Hubbard model • Studies of quantum phase equilibria • Dynamics of interactions between phases

14. Applications 2: Quantum dephasing • Exponential envelope on Rabi oscillations • Purely many-body in nature • Emergent timescale

15. 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

16. 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)

17. Stark Spectra

18. 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 http://jila.colorado.edu/yelabs/research/cold.html http://physics.aps.org/viewpoint-for/10.1103/PhysRevLett.101.133005

19. 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”

20. 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?

21. Hubbard Parameters • Choose appropriate Wannier basis, compute overlaps Hopping Internal energy Transitions Interaction

22. Route I: Single and many molecule physics decoupled Ns = 2 E B DC Ground state structure DC+AC Ground state structure Dynamics

23. Decoupled: Entanglement and structure factors Ns = 2 E B

24. Now couple single to many molecule physics Ns = 2 E B DC Ground state structure DC+AC Ground state structure Dynamics

25. Coupled: Entanglement and Structure Factors Ns = 2 E B

26. Route II: Turning on Internal State Structure Ns = 4 E B DC Ground state structure DC+AC Ground state structure Dynamics

27. Entanglement and Structure Factor Ns = 4 E B

28. Route II.3 Ns = 4 E B DC Ground state structure DC+AC Ground state structure Dynamics

29. Route II.4 Ns = 4 E B

30. Physical Scales for this Problem