Ultracold Atoms and Quantum Simulators

Transcription

1 Ultracold Atoms and Quantum Simulators Laurent Sanchez-Palencia Centre de Physique Théorique Ecole Polytechnique, CNRS, Univ. Paris-Saclay F-28 Palaiseau, France Marc Cheneau Laboratoire Charles Fabry Institut d Optique, CNRS, Univ. Paris-Saclay F-27 Palaiseau, France Laurent Sanchez-Palencia (lsp@cpht.polytechnique.fr) Marc Cheneau (marc.cheneau@institutoptique.fr)

2 Laurent Sanchez-Palencia Research Director at CNRS Prof. CC at Ecole Polytechnique Resp. Quantum Matter group Center for Theoretical Physics (CPHT) Research Ultracold atoms and quantum simulators, quantum technologies ( Understand the physics of strongly-correlated quantum matter and how to simulate it using controlled quantum systems Theory, numerics (PIMC, t-mps, ED, ) and analytics (field theories, RG, ) These lectures 7 3-hour lectures (by LSP and MC) ; Tuesday mornings Exercises each weak n; to be returned at n+; solutions online at n+ All documents (slides, problems, solutions) at Written exam (~3 hours, in general no document)

3 The Saga of Ultracold Atoms at a Glance 8 towards ultra-low temperatures laser cooling (~mk) weakly-interacting quantum gases Bose-Einstein condensation (~nk) strongly correlated 2 quantum matter optical lattices and control of interactions 2 towards quantum simulation dynamical control Historical perspective From classical gases to strongly-correlated quantum systems Several periods (~ years each) Milestone achievements and new perspectives 3

4 The Saga of Ultracold Atoms at a Glance 8 towards ultra-low temperatures laser cooling (~mk) weakly-interacting quantum gases Bose-Einstein condensation (~nk) strongly correlated 2 quantum matter optical lattices and control of interactions 2 towards quantum simulation dynamical control Towards ultra-low temperatures Doppler and Sisyphus cooling schemes (friction force and momentum kicks; temperature limited to the mk range; classical gas) Sub-recoil cooling (eg VSCPT, Raman cooling, side band cooling, ; quantum gas) Moreover, trapping (counteracts Brownian motion; magneto-optical and dipole traps) 4

5 The Saga of Ultracold Atoms at a Glance 8 towards ultra-low temperatures laser cooling (~mk) weakly-interacting quantum gases Bose-Einstein condensation (~nk) strongly correlated 2 quantum matter optical lattices and control of interactions 2 towards quantum simulation dynamical control Weakly-interacting quantum gases Evaporative cooling (quantum gases, weak interactions) Bose-Einstein condensation and degenerate Fermi gases (eg coherence and supefluidity) Moreover, first simulations (Brownian motion, dissipative optical lattices, )

6 The Saga of Ultracold Atoms at a Glance 8 towards ultra-low temperatures laser cooling (~mk) weakly-interacting quantum gases Bose-Einstein condensation (~nk) strongly correlated 2 quantum matter optical lattices and control of interactions 2 towards quantum simulation dynamical control Strongly-correlated quantum matter Control of interactions Optical lattices (non-dissipative ; realization of tight-binding models for solids) Fano-Feshbach resonances Low-dimensional systems 6

7 The Saga of Ultracold Atoms at a Glance 8 towards ultra-low temperatures laser cooling (~mk) weakly-interacting quantum gases Bose-Einstein condensation (~nk) strongly correlated 2 quantum matter optical lattices and control of interactions 2 towards quantum simulation dynamical control Towards quantum simulation Simulating the dynamics of quantum matter in true experiments Thermodynamic equilibrium Out-of-equilibrium physics 7

8 Simulating Interesting Phenomena in Physics Interesting physical systems are usually complex Many bodies, possibly not all identical Complicated microscopic interactions Structure, frustration, quantum entanglement, Cannot be solved exactly at the microscopic level Basic models play a central role Simpler and often reproduce interesting physics May be easier to solve Universal behaviour H=J R, R ' x x z S R S R ' h S R R Simple Hamiltonians do not guarantee simple solutions Numerical simulations Many-body approaches exist (QMC, DMRG, DFT, DMFT, ) Mean-field approaches (non linearities, ) Cannot solve any problem, in particular in the quantum world (exponentially-large Hilbert space, entanglement, sign problem for fermions, )

9 Simulation : From Numerics to Quantum Systems Why don t we let Nature work for us? Design a quantum system exactly governed by a pre-defined Hamiltonian Ĥmodel Let the system evolve under Ĥmodel towards its ground state (cooling) or a thermal state (coupling to a bath), or study its time-dependent dynamics Measure relevant quantities so as to consider Ĥmodel is solved R.P. Feynman, Int. J. Theor. Phys. 2, 467 (82) ; S. Lloyd, Science 273, 73 (6) Requirements and challenges Build up : Create a quantum system (bosons, fermions, spins, ) that can be manipulated by external fields Quantum engineering : Design the desired Hamiltonian with at least one control parameter (eg benchmarking) Initialization : Prepare the system in a well-known initial state (pure or mixed) Detection : Sufficiently accurate and various measurements

10 Towards Quantum Simulation New promising platforms Quantum optics [Aspuru-Guzik & Walther, Nat. Phys. 8, 28 (22)] Superconducting circuits [Houck et al., Nat. Phys. 8, 22 (22)] Magnetic insulators [Ward et al., J. Phys : Condens. Matter 2, 44 (23)] Ultracold atoms and ions [Bloch et al., Nat. Phys. 8, 267 (22); Blatt & Roos, Nat. Phys. 8, 277 (22)] Ultracold atoms A major playground Almost any parameter can be controlled experimentally

11 Towards Quantum Simulation with Ultracold Atoms BEC-BCS crossover in strongly-correlated Fermi gases Quantum gases in arbitrary dimensions Bose- and Fermi-Hubbard models in optical lattices Lattice spin Hamiltonians Artificial gauge fields Disordered quantum systems Out-of-equilibrium dynamics

12 Content of the Course Lecture (LSP) Overview of the course. Reminder of statistical physics (ideal, classical and quantum gases). Lecture 2 (MC) Bose-Einstein condensation of the ideal gas. Lecture 3 (LSP) Weakly-interacting Bose-Einstein condensates. Introduction to the second quantization formalism. Lecture 4 (LSP) Microscopic theory of the Bose gas and Bogoliubov approach Lecture (MC) Optical lattices: From one-body to many-body physics Lecture 6 (LSP) Degenerate Fermi gases: Ideal and interacting gases Lecture 7 (MC) Introduction to the physics of synthetic gauge fields 2

13 Non-exhaustive Literature Quantum mechanics and statistical physics [] J.-L. Basdevant, J. Dalibard, and M. Joffre, Mécanique Quantique (Presse de l Ecole Polytechnique; available also in English at Springer, 26). [2] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique, parts, 2 & 3. [3] L. D. Landau and E. M. Lifshitz, Statistical Physics, parts & 2 (Elsevier, Oxford, 8). [4] B. Diu, D. Lederer, and B. Roulet, Physique Statistique (Hermann, Paris, 6). Ultracold atoms [] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 28). [2] L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon press, Oxford, 24). 3

14 Statistical Physics : A Reminder 4

15 Microscopic Description Classical gases One-body state of particle j : l j =( r j, p j) p l3 N-body state of the gas : Λ ={ l,..., l N } l Dynamics governed by the Newton equations d r j p j = dt m and l2 l4 d p j j ( { r j, p j }, t ) =F dt (or, equivalently, by the Hamilton equations) r Quantum gases One-body state of particle : l = α c α (l) α j N-body state of the gas : Λ { l,..., l N } Dynamics governed by the Schrödinger equation iℏ 2 Λ = c(λ) + c2(λ) 2 c2(λ) d Λ ^ = H (t ) Λ dt measurement (and quantum jumps induced by measurements) c(λ) In both case, the dynamics is essentially deterministic (up to quantum measurements)

16 Macroscopic Description The microscopic dynamics is untractable Too many particles, N N A 6 23 One cannot perform calculations, nor store information Microscopic dynamics much faster than the macroscopic dynamics (scale separation) Useful information limited to a small number of variables (P, T, N, M, ) These variables are related by heuristic equations (equation of state), eg P Ω=N k B T Thermodynamic approach Heuristic but remarkably efficient and universal First law de=δ W + δ Q Second law ds= Third law S class (T =)= δq +δ S crea with δ Screa T but S quant (T =)=k B ln(g) Thermodynamics is essentially irreversible (non deterministic)

17 Microscopic versus Macroscopic Descriptions There are many more microstates than macrostates Hereafter, we use the macroscopic variables E (and N) A simple example : 3 non-interacting particles ; ladder one-body spectrum macrostate of energy E=e 6e corresponding microstates (realizations) 4e e 3e 2e e Λ Λ2 Λ3 Λ4 Λ In general, the number of microstates grows exponentially with N (energy grows algebraic, eg E N Σ j E j; Hilbert space grows exponentially, ie H N j H j )

18 Boltzmann s Statistical Physics Establish the link between the microscopic and macroscopic descriptions Focus on the slow dynamics (of the macrostate) and get rid of the rapid dynamics (of the microstates) The microstate is random and the system performs fast erratic jumps between the microstates. It is thus relevant to attribute them a probability distribution PΛ. The macrostates we observe are those with a dominant number of microstate realizations N.B. : Probabilities of macro- and microstates : P(E)= PΛ Λ, E Λ= E Basic principles Boltzmann's principle : The best statistical description of a complex system is the one that attributes the same probability to all the accessible microstates if no particular constraint favors some microstates with respect to other ones. Ergodicity principle : Time and statistical averages are equal, O t= O stat. O t= lim t meas t meas t meas dt O(t ) ^ O stat = P Λ Λ O Λ Λ

19 The Gibbs Canonical Ensembles Microcanonical ensemble Isolated systems (no exchange of energy ; no exchange of matter) B S U : universe B : bath S : system U Probability of a many-body microstate : P Λ = W (E,Δ E; N, Δ N ) Statistical entropy : S=k B ln [ W ( E, Δ E ; N, Δ N ) ]

20 The Gibbs Canonical Ensembles Canonical ensemble Closed systems (exchange of energy ; no exchange of matter) B U : universe B : bath S : system U S Probability of a many-body microstate : P Λ = exp ( β E Λ ) ZC Canonical partition function : Z C = exp ( β E Λ ) Λ

21 The Gibbs Canonical Ensembles Grand canonical ensemble Open systems (exchange of energy ; exchange of matter) B U S Probability of a many-body microstate : P Λ = U : universe B : bath S : system exp ( β E Λ +α N Λ ) Z GC Canonical partition function : Z GC = exp ( β E Λ +α N Λ ) Λ

22 Discernable versus Indiscernable Counting

23 Counting N-body Quantum States A simple and instructive example 2 non-interacting particles 3 possible one-body states Probability that the two particles are in the same state? bosons fermions /2 /2 2 3

24 Counting N-body Quantum States A simple and instructive example 2 non-interacting particles 3 possible one-body states Probability that the two particles are in the same state? bosons discernible fermions /3 /2 2/3 /2 2 4

25 Counting N-body Quantum States A simple and instructive example 2 non-interacting particles 3 possible one-body states Probability that the two particles are in the same state? discernible fermions bosons /3 /2 2/3 /2 2

26 Counting N-body Quantum States A simple and instructive example 2 non-interacting particles 3 possible one-body states Probability that the two particles are in the same state? discernible bosons fermions /3 /2 2/3 /2 2 6

27 Counting N-body Quantum States A simple and instructive example 2 non-interacting particles 3 possible one-body states Probability that the two particles are in the same state? discernible bosons fermions /3 /2 2/3 /2 Bose amplification Pauli exclusion 2 7

28 Classical versus Quantum de Broglie Thermal Wave Length Quantum degenerate regime : d T n l lt a lt classical gas quantum gas 2 8