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Optical Lattices

Optical Lattices. Prof. Christopher Foot University of Oxford. Gregynog Summer School, 2006. EPSRC QIP IRC. Centre for Quantum Computing & Technology, University of Oxford. Why do experiments with optical lattices? .

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Optical Lattices

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  1. Optical Lattices Prof. Christopher Foot University of Oxford Gregynog Summer School, 2006 EPSRC QIP IRC Centre for Quantum Computing & Technology, University of Oxford.

  2. Why do experiments with optical lattices? • Feasible to build apparatus for quantum simulation, e.g. 100 qubits/spins within the next 3 to 5 years. • Study of quantum phases and other properties of a multi-particle quantum systems we need lower fidelity than for QIP. Broadly speaking 1 - F≤ 1 % = factor of 100 less stringent than minimum for useful QIP. • Architecture of lattices well matched to CMP problems • QIP with cluster states: One-way computation

  3. Outline of optical lattice lecture • What is an optical lattice? • Description of Mott-Insulator experiment • AC Stark effect (light shift) and the dipole force • Estimate of the interaction energy of two neutral atoms at the same site in an optical lattice. • Implementation of spin-dependent interactions in an optical lattice  large scale entanglement • Description of experiments at Oxford • Direct Quantum Simulation of Condensed Matter (useful with around 100 qubits)

  4. Optical lattice formed by counter-propagating laser beams

  5. Optical lattice potential • Light shift (a.c. Stark effect): U I / where I = intensity of light, and = is the frequency detuning from resonance. • Scattering (spontaneous emission ), R I /2 Therefore work at large frequency detuning and (moderately) high intensity.

  6. Details of atomic physics,e.g. light shift or a.c. Stark effectOUP, 2005

  7. Optical lattice potential loaded with atoms from a Bose-Einstein Condensate • A simple argument shows that loading one atom into ground state (lowest vibrational level) of an optical lattice requires a phase-space density (PSD) approximately equal to the critical PSD for the quantum degeneracy, i.e. the temperature and density required for BEC (in the case of atoms that are bosons, e.g. 87Rb).

  8.  wavelength of light dB de Broglie wavelength dB = h / Mv dB ~ /2 n = 8 / 3 =1013 cm-3 v = 2h / MdB = 1 cm s-1 `melting´ of crystal T = 1 μK Conditions for quantum degeneracy in Rb

  9. Cooling of atoms • Laser cooling Approach the recoil limit, ER = ħ2k2 / 2M = h2 / 2M2 at which atom has momentum equal to that of single photon of the cooling light • Evaporative cooling in a magnetic trap to reach quantum degeneracy.

  10. View of the magnetic trap and quartz vacuum cell BEC 105 rubidium atoms. Temperature ~ 50 nK Density of BEC ~ 1014 cm-3

  11. Why such interest in optical lattices? • An optical lattice with one, or more, atoms per site is a strongly correlated system, cf. condensed matter systems. This requires loading from a quantum degenerate gas, e.g. BEC. (Atoms move independently in a sparsely populated lattice.)

  12. Coherence of BEC Atom laser = coherent source of matter waves Cf. laser light Nathan Smith et al, Oxford

  13. Spacing of peaks corresponds to two photon momentum, 2ħk Diffraction of atoms from 1-D optical lattice BEC 1-D standing wave Cf. diffraction from a grating Giuseppe Smirne et al, Oxford 54mm waist

  14. Expt. in Munich by Greiner, Bloch, Hänsch et al. How do we know that coherence disappears because of the MI transition and not some other decoherence mechanism?

  15. Mott-Insulator transition Expt. in Munich by Greiner, Bloch, Hänsch et al.

  16. Number squeezing • Number and phase are conjugate quantum variables    (or think in terms of amplitude and phase in quantum optics) c.f. position and momentum xp  ħ (or equivalently xk  1 ) .

  17. Some bookwork • Calculation of the light shift, or AC Stark shift. Correspondence between time-dependent perturbation theory and the standard result of 2nd order time-independent theory. • Frequency dependence of the light shift, or dipole force • See Atomic Physics, Foot (OUP, 2005), Appendix A and Chapters 7 and 9.

  18. Radiation forces on an atom

  19. bj … bosonic destruction operator for a particle in lattice site j … number of particles in lattice site j defined as The Bose-Hubbard model

  20. BEC phase J >U: The Mott insulator– loading from a BEC Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002 /U n=3 Mott insulator J << U n=2 Quantum freezing superfluid Mott n=1 melting J/U

  21. For simple two-site system (double well) a b J U

  22. * Initially, system in binomial state Adiabatically turn down coupling, J. As the state evolves to: • So called Fock state, or number-squeezed state. • Sub-quantum-limited interferometry. For U<0, get CAT STATES: |N,0> and |0,N>. (degenerate ground states) Dunningham and Burnett, Jaksch

  23. Some more bookwork • Reminder about quantum tunnelling in a double well potential • Number states = eigenstates when U » J • Elementary description of Mott-Insulator quantum phase transition • Mathematical analogy between tunnelling and transitions in a 2-level system

  24. Quantum information processing with neutral atoms This lecture Initial state preparation: One atom in each well Next lecture Controlled interaction: Collision gate leads to state 1 acquiring a pi-phase shift

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