QSim Quantum simulation with ultracold atoms

Transcription

1 APS Tutorial 7 QSim Quantum simulation with ultracold atoms Lecture 1: Lecture 2: Lecture 3: Lecture 4: Introduction to quantum simulation with ultracold atoms Hubbard physics with optical lattices Ultracold bosons in optical lattices: an overview Quantum simulation & quantum information J. H. Thywissen B. DeMarco A.-M. Rey I. Deutsch

2 An introduction to Quantum simulation with ultracold atoms Joseph H. Thywissen University of Toronto 20 March 2011 APS March meeting Dallas, TX Quantum simulator?

3 Problem: what is the minimal surface given fixed edges?

4 Soap films a simulation to find minimal surfaces. Problem: what is the minimal surface given fixed edges? Answer: construct a wire grid and dip it in soap! minimal surface for tetrahedral edges.

5 Answer precedes the explanation Lagrange: calculus of variations 1760: poses minimal surface problem Plateau: soap film simulations, c.1840 Initiates a Golden Age of mathematical study of minimal surfaces. Riemann, Weierstraß, Schwarz, others: fail to find answer to surfaces of least area. Douglas: solves in (Fields Medal 36) J. A. F. Plateau

6 Answer precedes the explanation Lagrange: calculus of variations Plateau s laws 1760: poses minimal surface problem Plateau: soap film simulations, c.1840 Initiates 2. a Constant Golden curvature Age of mathematical study of minimal surfaces. Riemann, Weierstraß, Schwarz, others: fail to find answer to surfaces of least area. 1. Smooth surfaces 3. Soap films always meet in threes, and they do so at an angle of 120 o, forming an edge ( Plateau Border ). 4. These Plateau Borders meet in fours at an angle of arccos(-1/3) to form a Douglas: solves in vertex. (Fields Medal 36) J. A. F. Plateau

7 What is simulation? Provides the answer to a mathematical problem or model Typically done (today) on a classical digital computer. Does not solve the model -- does not tell us why. (unlike calculation?) Empirical rules might be learned; and further simulations (various initial conditions, etc) could address questions. Experiment? Yes, but we know Hamiltonian.

8 APS Tutorial 7 QSim Quantum simulation with ultracold atoms Lecture 1: Introduction to quantum simulation with ultracold atoms Outline J. H. Thywissen Lecture 2: Lecture 3: Lecture 4: Hubbard physics with optical lattices Ultracold bosons in optical lattices: an overview Quantum simulation & quantum information I. What B. is DeMarco Simulation? II. Length scales A.-M. Rey III.Example - strongly interacting I. Deutsch fermions Tutorial 7 slides online:

9 Classical simulation of a quantum system Typically on a computer...a device that cannot be in a superposition or entangled state. {more about this in Lecture 4.} Methods typically used are numerical integration of the Schrödinger Eq. (or mean field extension, such as GP Eq.) Monte Carlo (QMC) simulations However QMC fails* for many-body fermion problems, or excited states of bose systems, due to sign problem. Feynman: Use a quantum system to simulate another quantum system [1981] *or has exponential scaling

10 Quantum simulation (QSim) When classical simulation is inefficient, using a quantum system may be the only option. Not universal quantum computing...eg, couldn t factor a number. Certain models natural fits for atoms Hubbard Model: optical lattices 1D models: extremely elongated traps 2D models: pancake traps Universal interactions: unitarity-limited Fermi gas

11 Neutral atom Hamiltonian Ĥ = dr ˆΨ (r) ] [ 2 2m 2 + U(r) ˆΨ(r)+ 1 2 drdr ˆΨ (r) ˆΨ (r )V (r r ) ˆΨ(r ) ˆΨ(r) V: Inter-atomic potential is deep, complex, and unique to each atom pair) U: Trapping potential not reminiscent of textbooks, where we typically worked in a box (U=0) How could this Hamiltonian be useful to simulate other systems?

12 1st simplification: low-energy limit Dilute atoms scatter pair-wise, because their typical spacing R = n 1/3 is much smaller than the potential range r0 Below 0.1mK, atom pairs do not have enough E to overcome the p-wave centrifugal barrier inter-atomic potential, Cs Two-body collision l =1 m1 m2 R l =0 V l (R) =V (R)+ 2 l(l + 1)/(2µR 2 ) R (nm)

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14 S-wave ( l =0) scattered wave function For elastic scattering, must be ψ k ( r )=e i k r a 1+ika e ikr r + plane wave spherical wave The scattering term has an amplitude f k = [1/a + ik] 1 scattering amplitude from which you find the phase & cross-section k θ σ =4π f k ( n) 2 1/a

15 S-wave ( l =0) scattered wave function For elastic scattering, must be ψ k ( r )=e i k r + a 1+ika e ikr r Only one free parameter! scattering length a plane wave spherical wave The scattering term has an amplitude f k = [1/a + ik] 1 scattering amplitude from which you find the phase & cross-section k θ σ =4π f k ( n) 2 1/a

16 Pseudo-potential Two interaction potentials V and V are equivalent if they have the same scattering length So: after measuring a for the real system, we can model with a very simple potential. Replace interaction V ( R) potential with delta function! V ( R)=gδ( R) where Actually, to avoid divergences you need g = 4π 2 m a V ( R)=gδ( R) R (R ) regularized

17 Neutral atom Hamiltonian (revisited) Ĥ = dr ˆΨ (r) ] [ 2 2m 2 + U(r) ˆΨ(r)+ 1 2 drdr ˆΨ (r) ˆΨ (r )V (r r ) ˆΨ(r ) ˆΨ(r) Can write V(..) as pseudo potential: V ( R)=gδ( R) R (R ) in limit of dilute ( ) R r 0 and ultracold ( T 100µK).

18 Neutral atom Hamiltonian (revisited) Ĥ = dr ˆΨ (r) ] [ 2 2m 2 + U(r) ˆΨ(r)+ 1 2 drdr ˆΨ (r) ˆΨ (r )V (r r ) ˆΨ(r ) ˆΨ(r) Can write V(..) as pseudo potential: V ( R)=gδ( R) R (R ) What about the trap? in limit of dilute ( ) R r 0 and ultracold ( T 100µK).

19 2nd simplification: Local chemical potential What if a cold gas were a distribution of local creatures? snake line of ants {scare the ant at the front of the line, and the last ant won t rattle its tail...}

20 2nd simplification: Local chemical potential What if a cold gas were a distribution of local creatures? snake line of ants {scare the ant at the front of the line, and the last ant won t rattle its tail...} Recipe: µ µ local = µ U( r)

21 Local chemical potential: µ local = µ U( r) how to use your Stat. Mech. textbook Thomas Fermi density profiles: n = 1 6π 2 [ 2mEF 2 ideal quantum gas functions: ] 3/2 n TF = (2m)3/2 6π 2 3 [E F U( r)] 3/2 for zero-temperature fermions in semiclassical limit. z = e βµ z = e β(µ U( r)) n = λ 3 T f 3/2(z) at finite temperature ( β =1/k B T), where z=fugacity. Similar Thomas Fermi expression for bosons: µ = gn textbook local µ textbook local µ textbook local µ n TF = 1 g [µ U( r)]

22 Validity of local chemical potential A local density approximation (LDA). Not a good approximation when: -tunneling can occur through barriers -long-range order affected (eg, phase coherence) -gradients perturb states (eg, localized states [AM Rey]) -long-range interactions (Coulomb etc) In those cases, QSim model must include trapping potential. However in some important cases works well: -important length scales (eg, Fermi length or lattice constant) much smaller than trap size -Far from edges, compared to healing length ξ: ξ =1/ 8πna such that 2 2mξ 2 = gn

23 Cold neutral gases: length scales inter-atomic potential range, r0: 2 nm scattering length, a -low-field (background) 5 nm -near a Feshbach resonance 100 nm to 1000 nm thermal de Broglie wavelength: 100 nm average inter-particle spacing: 100 nm -same length scale as 1/kF lattice constant: 400 nm ground state width: 100Hz (typ. magnetic trap) 10kHz (single site of optical lattice) cloud size: µm

24 Cold neutral gases: length scales inter-atomic potential range, r0: 2 nm scattering length, a -low-field (background) 5 nm -near a Feshbach resonance 100 nm to 1000 nm thermal de Broglie wavelength: 100 nm average inter-particle spacing: 100 nm -same length scale as 1/kF Quantum degeneracy lattice constant: 400 nm ground state width: 100Hz (typ. magnetic trap) 10kHz (single site of optical lattice) cloud size: µm

25 Cold neutral gases: length scales inter-atomic potential range, r0: 2 nm scattering length, a -low-field (background) 5 nm -near a Feshbach resonance 100 nm to 1000 nm thermal de Broglie wavelength: 100 nm average inter-particle spacing: 100 nm -same length scale as 1/kF Quantum degeneracy lattice constant: 400 nm ground state width: 100Hz (typ. magnetic trap) Simulation space 10kHz (single site of optical lattice) cloud size: µm

26 Cold neutral gases: length scales (in traps) inter-atomic potential range, r0: 2 nm scattering length, a -low-field (background) 5 nm -near a Feshbach resonance 100 nm to 1000 nm thermal de Broglie wavelength: 100 nm average inter-particle spacing: 100 nm -same length scale as 1/kF ground state width: 100Hz (typ. magnetic trap) cloud size: µm Simulation space

27 QSim in local µ picture: µdensity energy U(r) position position

28 QSim in local µ picture: µdensity energy U(r) position position uniform H, simulated with local µ & T.

29 QSim in local µ picture: µdensity energy bosons (for single component): [ ] Ĥ = ˆΨ 2 2m 2 U(r) ˆΨ + g 2 ˆn2 fermions (for 2-component gas): position Ĥ = σ ˆΨ σ [ 2 ] 2m 2 ˆΨ σ + g ˆn ˆn position uniform H, simulated with local µ & T.

30 Feshbach resonances How can we tune the scattering length a? We can tune a molecular bound state into resonance with the free atoms, and affect net phase acquired during the collision. Result is indistinguishable from tuning the single-channel square well: it s only the phase that matters.

31 Feshbach resonances How can we tune the scattering length a? We can tune a molecular bound state into resonance with the free atoms, and affect net phase acquired during the collision. Result is indistinguishable from tuning the single-channel square well: it s only the phase that matters.

32 Feshbach resonances How can we tune the scattering length a? We can tune a molecular bound state into resonance with the free atoms, and affect net phase acquired during the collision. Result is indistinguishable from tuning the single-channel square well: it s only the phase that matters.

33 Feshbach resonances single-channel model Tune the square well potential & calculate a: potential V R b We find: 1. Resonances at bv =(n +1/2)π when each new bound state appears. 2. Mostly a>0. Near a resonance when a<0 (eg, Li.) a R b V

34 Feshbach resonances Near resonance the scattering length can be described as ( a(b) =a bg 1 B B 0 ) Example: 6 Li s-wave cross section is σ 0 = 4π k 2 sin2 η 0 For a>0, a bound state exists with binding energy E b = 2 2µa 2

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37 Length scales (in res.) inter-atomic potential range, r0: 2 nm thermal de Broglie wavelength: 100 nm average inter-particle spacing: 100 nm -same length scale as 1/kF scattering length, a -at Feshbach resonance: divergent ground state width: 100Hz (typ. magnetic trap) cloud size: µm Simulation space

38 Length scales (in T=0) inter-atomic potential range, r0: 2 nm average inter-particle spacing: 100 nm -same length scale as 1/kF scattering length, a -at Feshbach resonance: divergent ground state width: 100Hz (typ. magnetic trap) cloud size: µm Simulation space

39 Length scales (in T=0) inter-atomic potential range, r0: 2 nm average inter-particle spacing: 100 nm -same length scale as 1/kF scattering length, a -at Feshbach resonance: divergent ground state width: 100Hz (typ. magnetic trap) cloud size: µm Simulation space Only one length scale left in the problem! Universal

40 Unitarity limit: a >> R If the scattering length far exceed any physical length scale of the problem, it cannot be important. Inter-particle spacing d only length scale left: must determine all interaction energies! In fact, EF is the energy scale associated with d -for both fermions *and* bosons! [Ho 2004] -so restate this condition as a k 1 F where for both bosons and fermions k F (6πn) 1/3

41 Cross section at unitarity Near a Feshbach resonance, a diverges. The scattering cross section departs from its low-ka form: σ = 4πa2 1+k 2 a 2 4π k 2 This is just a manifestation of the optical theorem, which says that complete reflection corresponds to a finite scattering length. In terms of the de Broglie wavelength, σ res = λ 2 db/π You may be more familiar with the resonant atom-photon cross section (which has different constants because it is a vector instead of scalar field): σ res = 3 2π λ2 L

42 Quantum simulation at unitarity For a many-body system, resonant interactions also saturate but are less easy to quantify. Certainly it is the case that a divergent a can no longer be a relevant physical quantity to the problem. For fermions, the only remaining length scale is k 1 F. This means that interaction energies must scale with the Fermi E. In particular, for resonant attractive interactions, where β 0.58 µ Local = (1 + β)ɛ F has been measured in various experiments. Using the LDA to integrate over the profile, we find µ U = 1+βE F 0.65E F for a

43 Perspective: What can cold atoms teach us? APS March meeting: 10,000 CM physicists. 100-yr-old field (SC observed in 1911 by Kammerling-Onnes) Traditional CM approach: see phenomenon (eg, superconductivity) search for theory (eg, BCS model) Ultracold atoms: quantum many-body physics (eg, BEC) know Hamiltonian

44 Quantum simulation with neutral atoms Conclusion: Emulation of simple models relies on a separation of length (or energy) scales. Contact interaction when dilute and ultracold R k 1 F r 0 T 100µK Universal (no dependence on interactions) when unitaritylimited a k 1 F Simulate uniform physics when LDA valid µ local = µ U( r) Single-band model for high lattice depths {next 2 lectures}

45 Quantum simulation at the University of Toronto Postdoc Bose- Fermi mixture experiment position available! Site-resolved optical lattice experiment

46 Thank you!

47 Addendum How to cool atoms?

48 Laser system

49 sympathetic cooling on a chip Aubin et al, Nature Phys. (2006)