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CLARENDON LABORATORY PHYSICS DEPARTMENT UNIVERSITY OF OXFORD and CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE Quantum Simulation Dieter Jaksch Outline Lecture 1: Introduction Lecture 2: Optical lattices Analogue simulation: Bose-Hubbard model and artificial gauge fields. Digital simulation: using cold collisions or Rydberg atoms. Lecture 4: Tensor Network Theory (TNT) Bose-Einstein condensation, adiabatic loading of an optical lattice. Hamiltonian Lecture 3: Quantum simulation with ultracold atoms What defines a quantum simulator? Quantum simulator criteria. Strongly correlated quantum systems. Tensors and contractions, matrix product states, entanglement properties Lecture 5: TNT applications TNT algorithms, variational optimization and time evolution Remarks References to stock market models: Melvin Lax, Wei Cai, and Min Xu, Random Processes in Physics and Finance, Oxford University Press, 2006 Neil F. Johnson, Paul Jefferies, and Pak Ming Hui, Financial Market Complexity, Publ. Oxford University Press, 2003 Brief introduction: T.H. Johnson, Non-equilibrium strongly-correlated dynamics, DPhil thesis, Email: [email protected] Remarks Lattice systems/crystals: strong correlations can be achieved at any density by quenching the kinetic energy Continuum systems/gases: for finite range interactions the system will become weakly interacting if mean separation between particles much larger than range of the interaction Thanks to Prof J. Walraven for pointing this out Hubbard models 𝑎𝑖† 𝑎𝑗 𝐻 = −𝑡 𝑖,𝑗 𝑈 + ℎ. 𝑐. + 2 𝑎𝑖† 𝑎𝑖† 𝑎𝑖 𝑎𝑖 𝑖 Landay theory fails for too strong interactions: perturbation series, renormalization fails to converge Universal behaviour is often encountered near phase transitions, this motivates the study of toy models Atoms Rydberg CMP CMP cooled Bio s – ms 𝜇s – ns ps – fs ps – fs ps - fs Hz - kHz MHz THz THz THz nK nK 300K mK 300K 1 − 10 104 − 106 1 − 10 104 − 106 1 – 10 Coherence 𝛾, 𝜅 Hz kHz THz GHz THz Driving frequency Ω kHz MHz THz THz THz Experiment Time Energy 𝐸/ℎ Temperature 𝑇 Ratio 𝐸/𝑘𝐵 𝑇 200Hz ⟺ 10nK ⟺ 1peV Quantum simulation TWO PARTICLES IN TWO BOXES 2nd quantization formalism for bosons Single particle Hilbert space ℋ with complete set of orthogonal functions 𝜙𝑖 . We denote a quantum state of identical bosons with 𝑛𝑖 particles in the single particle orbital 𝜙𝑖 by | … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … 〉 We define annihilation operators 𝑎𝑖 which destroy a particle in the 𝑖-th orbital. It act according to 𝑎𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … = 𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 − 1, 𝑛𝑖+1 … Similarly operators 𝑎𝑖† create a particle in the 𝑖-th mode and act according to 𝑎𝑖† … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … = 𝑛𝑖 + 1 … 𝑛𝑖−1 , 𝑛𝑖 + 1, 𝑛𝑖+1 … Their commutation relations are those of harmonic oscillator ladder operators 𝑎𝑖 , 𝑎𝑖† = 𝛿𝑖,𝑗 The combination 𝑛𝑖 = 𝑎𝑖† 𝑎𝑖 counts the number of particles in orbital 𝑖. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 2nd quantization formalism for bosons It acts according to 𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … = 𝑛𝑖 … 𝑛𝑖−1 , 𝑛𝑖 , 𝑛𝑖+1 … A general pure many body state of the system is given by a coherent superposition of configuration states Ψ = 𝑐 𝑛𝑖 | 𝑛𝑖 〉 𝑛𝑖 Mixed states are build as incoherent mixtures of pure states as usual 𝜌= 𝑑𝑖𝑗 Ψ𝑖 〈Ψj | 𝑖,𝑗 It is also useful to introduce the vacuum state 𝑣𝑎𝑐 = 0,0,0,0, ⋯ = |0〉 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Particle energies (I) Potential energy E1 E0 Kinetic energy A particle gains energy by hopping between different states a a+1 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Particle energies (II) Interaction energy n particles in the same state Each particle interacts with n-1 particles in the same state E1 E0 Interactions between particles in different states E1 E0 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Each particle interacts with all particles in the other state Two atoms in two sites Hamiltonian Basis states L R |20〉 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. L R |11〉 L R |02〉 Hamiltonian and ground state Matrix form Ground state Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Density matrix The density matrix corresponding to the ground state wave function is 𝜌 = Ψ 〈Ψ| For the case 𝐽 = 0 and 𝑈 > 0 this density matrix is diagonal in the single particle orbitals 𝜙𝐿 and 𝜙𝑅 given by 𝜌 = 1,1 𝐿𝑅 1,1 = 0 0 0 1 0 0 0 0 0 For the case 𝑈 = 0 and 𝐽 > 0 the density matrix is diagonal in a different basis given by symmetric and antisymmetric linear combinations ∝ 𝜙𝐿 ± 𝜙𝑅 1 0 0 𝜌 = 2,0 ± 2,0 = 0 0 0 0 0 0 For general values of 𝐽 and 𝑈 no such single particle matrix exists. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Single particle density matrix This is defined by 𝜌1 = 𝑎𝐿† 𝑎𝐿 𝑎𝑅† 𝑎𝐿 𝑎𝐿† 𝑎𝑅 𝑎𝑅† 𝑎𝑅 For the ground state 𝜌1212 1.0 0.8 0.6 0.4 0.2 𝑈 J 𝐽 U 0 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 5 10 15 20 Particle number fluctuations How well defined is the particle number in each site? 2 Δ𝑛 n2 0.5 0.4 0.3 0.2 0.1 𝑈 J 𝐽 U 0 5 10 This is given by Δ𝑛2 = 𝑛𝐿2 − 𝑛𝐿 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 15 2 20 Compare: Mean field approximation Assume the particles are not correlated Ψ = 𝑐𝐿 𝐿 1 + 𝑐𝑅 𝑅 1 ⊗ 𝑐𝐿 𝐿 2 + 𝑐𝑅 𝑅 2 Ψ = 𝑐𝐿2 𝐿𝐿 + 𝑐𝐿 𝑐𝑅 ( 𝐿𝑅 + 𝑅𝐿 ) + 𝑐𝑅2 |𝑅𝑅〉 With this ansatz the Schrödinger equation (obtained from the principle of least action) becomes non-linear in the remaining coefficients 𝐻= 𝑐𝐿 2 𝑈 −𝐽 −𝐽 𝑐𝑅 2𝑈 Ψ𝑀𝐹 𝑐𝐿 = 𝑐 𝑅 The resulting approximate ground state is not correlated and describes one particle moving in a mean background field created by the other particle Dimension of vectors increases linearly with size. |𝐿〉 |𝑅〉 Reduction of degrees of freedom is achieved by ignoring correlations and introducing non-linearity. Often valid for sufficiently weak interactions Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Particle number fluctuations Δ𝑛𝐿 1 Mean field approach 0.5 Exact solution 0 5 10 𝑈/𝐽 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Limit of large interactions U Mean field theory gives (independently of 𝑈) 1 Ψ = 𝐿 2 1 + 𝑅 1 ⊗ 𝐿 2 + 𝑅 2 Compared to the exact solution Ψ ∝ 𝐿 1 ⊗ 𝑅 2 + 𝑅 1 ⊗ 𝐿 2 Mean field theory predicts wrongly on-site particle number fluctuations Particle-particle correlations Coherence Superfluidity We need to account for correlations when interactions are strong. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Quantum simulation BOSE-EINSTEIN CONDENSATION Bose-Einstein condensation (BEC) Prediction 1924 by Bose and Einstein Helium Strong interactions shield BEC Alkali gases Weak interactions Small temperature Mean field description 99% of particles in same orbital Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Single particle orbitals and Hamiltonian Single particle Hamiltonian 𝒑2 𝐻1 = +𝑉 𝒙 2𝑚 Single particle orbitals (ordered by ascending energy) 𝐻1 𝜙𝑖 𝒙 = 𝜖𝑖 𝜙𝑖 𝒙 Many body Hamiltonian using operators defined through 𝜙𝑖 𝜖𝑖 𝑎𝑖† 𝑎𝑖 𝐻𝑖𝑑 = 𝑖 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Ideal Bose gas Define field operator 𝜓 𝑥 = 𝜙𝑖 𝑥 𝑎𝑖 𝑖 Hamiltonian 𝐻𝑖𝑑 2 𝒑 = ∫ d𝒙 𝜓 † 𝒙 + 𝑉 𝒙 𝜓(𝒙) 2𝑚 Field operator commutation relation 𝜓 𝒙 , 𝜓 † 𝒙′ Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. = 𝛿(𝒙 − 𝒙′ ) Eigenstates A general eigenstate And the ground state in this notation is With 𝑁 the total number of particles in the system Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Symmetry breaking Particle number conserving ground state Ψ0 = 𝑁, 0,0, ⋯ = † 𝑁 𝑎0 𝑁! |𝑣𝑎𝑐〉 Symmetry breaking coherent state by 𝑎0 → 𝑁 = 𝛼 𝛼 = 𝛼2 − 2 𝛼 𝑎0† 𝑒 𝑒 𝑣𝑎𝑐 = 𝛼𝑛 𝛼2 𝑒− 2 𝑛 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 𝑛! |𝑛, 0,0, ⋯ 〉 Symmetry breaking There is no physical reason for choosing 𝛼 real or to believe that each realization gives the same phase 1 𝜌= 2𝜋 2𝜋 𝑑𝜑 |𝛼 𝑒 𝑖𝜑 〉〈𝛼𝑒 𝑖𝜑 | 0 This density matrix is identical to the mixture 𝜌= 𝑝 𝑛 𝑛, 0,0,0, ⋯ 〈𝑛, 0,0,0, ⋯ | 𝑛 Here 𝑝 𝑛 is a Poisson distribution with mean 𝛼 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 2 BEC? Need to consider single particle density matrix 〈𝑎0† 𝑎0 〉 𝜌1 = Tr2,…,N 𝜌 = 〈𝑎† 𝑎 〉 1 0 ⋮ 〈𝑎0† 𝑎1 〉 … 〈𝑎1† 𝑎1 〉 ⋮ … ⋱ Does 𝜌1 have an eigenvalue of order N in a thermal state with finite temperature BEC Depends on the density of single particle orbitals Interactions deplete the BEC Photons are usually not considered a BEC Laser: not a thermal state Black body radiation: photons emitted and absorbed Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Isotropic harmonic trap - recap Energy Degeneracy 𝑗ℏ𝜔 𝑗+1 𝑗+2 2 ⋮ ⋮ 2ℏ𝜔 6 ℏ𝜔 3 0 1 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Thermal properties – harmonic trap Total number of particles Thermal population with 𝛽 = 1/𝑘𝑇 Ground state population 𝑍 𝑛0 = 1−𝑍 Limits on the values of the fugacity 0 ≤ 𝑍 = exp 𝛽𝜇 ≤ 1 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Thermal properties – harmonic trap Take out 𝑛0 and use continuum approximation for rest ∞ 𝑁 = 𝑛0 + 0 𝑗 2 𝑛𝑗 𝑑𝑗 2 Integrals gives the Bose function (PolyLog in Mathematica) 𝑔3 𝑍 𝑁 = 𝑛0 + 𝛽ℏ𝜔 𝜁 3/2 ≈ 2.61 3 𝑔1/2 𝑔3/2 𝑔3 1 = 𝜁 3 𝜁 3 ≈ 1.202 𝑔3 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Thermal properties – harmonic trap Critical temperature Fraction of particles in the ground state Appropriate statistical ensemble? Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Two particle interactions Molecular potential ‘Approximation’ by a contact potential Define Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Physical interpretation contact interaction Molecular interaction potential 𝑎𝑠 > 0 𝑎𝑠 𝑎𝑠 Valid for 𝑛𝑎𝑠3 ≪1 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 𝑎𝑠 < 0 The Gross-Pitaevskii equation Hamiltonian Assume ground state to be still of the form And minimize the expected energy to find orbital Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The Gross-Pitaevskii equation Write Minimize To find Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Thomas-Fermi approximation Neglect the kinetic energy 𝜑0 𝒙 = 𝜇−𝑉 𝒙 𝑁𝑔 0 for 𝜇 > 𝑉(𝒙) for 𝜇 < 𝑉(𝒙) 1 Valid if the so-called healing length 𝜉 = 8𝜋𝑎𝑠 𝑁 −2 𝑅3 is much smaller than the size of the condensate 𝑅, i.e. 𝑅 ≫ 𝜉. For harmonic trap an inverted parabola. The condensate size is given by 𝑉 𝑅 = 𝜇. 𝑉(𝒙) 𝑛 𝒙 = 𝜑0 𝒙 2 Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 𝑅 Time dependent GPE Carry out minimization over action Note: No chemical potential Mean field nonlinear Schroedinger equation Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Quantum simulation OPTICAL LATTICES Recap: Trapped bosonic atoms, Hamiltonians Atoms in a given internal state |𝑎〉 are described by a bosonic field operator 𝜓 𝒙 which fulfils canonical commutation relations 𝜓 𝒙′ , 𝜓 † (𝒙) = 𝛿 𝒙 − 𝒙′ . The single particle Hamiltonian contains a kinetic and the trapping energies 𝐻0 = ∫ d𝒙 𝜓† 𝒑2 𝒙 + 𝑉 𝒙 𝜓(𝒙) 2𝑚 The particles interact via a contact type potential whose effect (at low temperatures) can be described by a Dirac Delta function 𝑉 𝒙, 𝒙′ 4𝜋𝑎𝑠 ℏ2 = 𝛿(𝒙 − 𝒙′ ) 𝑚 This leads to the interaction Hamiltonian of the form 𝐻int = 1 ∫ d𝒙d𝒙′ 𝜓 † 𝒙 𝜓 † 𝒙′ 𝑉 𝒙, 𝒙′ 𝜓 𝒙 𝜓 𝒙′ 2 2𝜋𝑎𝑠 ℏ2 = ∫ d𝒙 𝜓 † 𝒙 𝜓 † 𝒙 𝜓 𝒙 𝜓 𝒙 𝑚 Here 𝑎𝑠 is the s-wave scattering length. The total Hamiltonian is 𝐻 = 𝐻0 + 𝐻int Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The optical lattice potential Periodic Potential created by a standing wave Near the potential minimum Period of half a wave length 𝜆 𝑎= 2 See Lecture Course by Prof Helene Perrin Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Bloch functions Bloch wave functions Where 𝑢𝑞𝑛 are periodic eigenfunctions of With energies of See Lecture course by Prof Pierre Clade Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Single particle problem in 1D Mathieu equation for the mode functions (~ dimensionless parameters) Bloch bands with normalizable Bloch wave functions in the stable regions Stable regions a) V0 = 5 ER b) V0 = 10 ER c) V0 = 25 ER Lowest band: E(0)q ∝ cos(q) Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Wannier functions These are mode functions pertaining to a certain Bloch band and localized at a lattice site Note: This definition is not unique because of the arbitrary phase in the Bloch wave functions. The degree of localization depends strongly on their choice. See e.g. R. Walters et al., Phys. Rev. A 87, 043613 (2013) and reference therein. At small temperatures only the lowest Bloch band n=(0,0,0) will be occupied Wannier functions See Lecture course by Prof Pierre Clade Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Wannier functions leaking into other sites Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.