Quantum States and Phases in Dissipative Many-Body Systems with Cold Atoms

Transcription

1 ASC Lectures, October 14/ , Arnold Sommerfeld Center, Munich Quantum States and Phases in Dissipative Many-Body Systems with Cold Atoms UNIVERSITY OF INNSBRUCK Cold Atoms Engineering Condensed Matter Many-Body States Quantum Optics Dissipation/Driving IQOQI AUSTRIAN ACADEMY OF SCIENCES SFB Coherent Control of Quantum Systems Sebastian Diehl IQOQI Innsbruck

2 Lecture Overview Main theme: Dissipation can be turned into a favorable, controllable tool in cold atom many-body systems. Condensed Matter Many-Body States Cold Atoms Engineering Quantum Optics Dissipation/Driving Part I: Quantum State Engineering in Driven Dissipative Many-Body Systems Proof of principle: Driven Dissipative BEC Application I: Nonequilibrium phase transition from competing unitary and dissipative dynamics Application II: Cooling into antiferromagnetic and d-wave states of fermions - Collaboration: H. P. Büchler, A. Daley, A. Kantian, B. Kraus, A. Micheli, A. Tomadin, W. Yi, P. Zoller today Part II: Dissipative Generation and Analysis of 3-Body Hardcore Models Mechanism Experimental prospects, ground state preparation Application I: phase diagram for attractive 3-hardcore bosons Application II: atomic color superfluid for 3-component fermions - Collaboration: M. Baranov, A. J. Daley, M. Dalmonte, A. Kantian, J. Taylor, P. Zoller tomorrow

3 Outline Part I: Quantum State Engineering in Driven Dissipative Many-Body Systems Cold Atoms Engineering Condensed Matter Many-Body States Quantum Optics Dissipation/Driving Introduction: Open Systems in Quantum Optics Driven Dissipative BEC: - Mechanism for pure DBEC: Many-Body Quantum Optics - Physical Implementation of DBEC: Reservoir Engineering, Bogoliubov bath Application I: Competition of unitary vs. dissipative dynamics - first look: weak interactions - strong interactions: nonequilibrium phase transition Application II: Targeting pure fermion states - An excited many-body state: η-condensate - Antiferromagnetic and d-wave fermion states References: SD, A. Micheli, A. Kantian, B. Kraus, H.P. Büchler, P. Zoller, Nature Physics 4, 878 (2008); B. Kraus, SD, A. Micheli, A. Kantian, H.P. Büchler, P. Zoller, Phys. Rev. A 78, (2008) F. Verstraete, M. Wolf, I. Cirac, Nature Physics 5, 633 (2009)

4 Quantum State Engineering in Many-Body Systems thermodynamic equilibrium - standard scenario of condensed matter & cold atom physics H E g = E g E g ρ e H/k BT T 0 E g E g Hamiltonian (many body) cooling to ground state Hamiltonian Engineering: interesting ground states quantum phases driven / dissipative dynamical equilibrium - quantum optics drive system bath dρ dt = i [H, ρ] + Lρ competing dynamics ρ(t) t ρ ss!? = D D mixed state pure state ( dark state ) master equation steady state Liouvillian Engineering: many body pure states / driven quantum phases mixed states ~ finite temperature useful an interesting fermion states

5 Open Quantum Systems

6 Open Quantum Systems drive system environment / bath H B = dω ωb ωb ω continuum bath of harmonic oscillators quantum jump operators polynomial in system operators linear bath operator coupling to the system Three approximations: (1) Born approximation: (2) Markov approximation: (3) Rotating wave approximation: detuning system frequency reservoir bandwidth e! g

7 Open Quantum Systems Eliminate bath degrees of freedom in second order time-dependent perturbation theory (Born approximation) Tr bath system bath quantum jump operators effective system dynamics from Master Equation (zero temperature bath) Liouvillian operator in Lindblad form Structure: second order perturbation theory mnemonic: norm conservation but: Purity is not conserved go for pure state: -- purity

8 Open Quantum Systems Stochastic Interpretation: Quantum Jumps e quantum jump operators decay! g damped Rabi oscillations time evolution of upper state population of driven dissipative two-level system (single run) Averaging over quantum trajectories generates all correlation functions Engineer the jump operators

9 Driven Dissipative BEC

10 Dark States in Quantum Optics Goal: pure BEC as steady state solution, independent of initial density matrix: ρ(t) BEC BEC for t Such situation is well-known quantum optics (three level system): optical pumping (Kastler, Aspect, Cohen-Tannoudji; Kasevich, Chu;...) ρ(t) t g + g + Driven dissipative dynamics purifies the state g + is a dark state decoupled from light c α g + = 0 Dark state is Eigenstate of jump operators with zero Eigenvalue Time evolution stops when system is in DS: pure steady state

11 An Analogy Λ-system: three electronic levels (VSCPT by Aspect, Cohen-Tannoudji; Kasevich, Chu) 1 atom on 2 sites dark state bright state 1 2 J ~ dissipative Josephson junction (a 1 + a 2 ) vac (a 1 a 2 ) vac symmetric anti-symmetric in-phase out-of-phase pumping into symmetric state phase locking : like a BEC

12 Driven Dissipative lattice BEC nearest neighbours Consider jump operator: ci j = (a i + a j )(ai a j ) (1) BEC state is a dark state: ci j BEC! = 0 i "N 1! BEC! = a! vac! N!! (ai a j ) a! = a! (ai a j ) + δi! δ j!! (2) BEC state is the only dark state: (ai + a j ) has no eigenvalues (ai a j ) has unique zero eigenvalue (ai a j ) i (1 eiqeλ ) aq q!!

13 Driven Dissipative lattice BEC (3) Uniqueness: BEC> is the only stationary state (sufficient condition) If there exists a stationary state which is not a dark state, then there must exist a subspace of the full Hilbert space which is left invariant under the set {c α } (4) Compatibility of unitary and dissipative dynamics D be an eigenstate of H, H D = E D ρ(t) t D D Long range order in many-body system from quasi-local dissipative operations Uniqueness: Final state independent of initial density matrix Criteria are general: jump operators for AKLT states (spin model), eta-states (fermions), d-wave states (fermions, next lecture)

14 A. Griessner, A. Daley et al. PRL 2006; NJP 2007 (noninteracting atom) Physical Realization: Reservoir Engineering driven two-level atom + spontaneous emission Quantum optics ideas/techniques laser atom photon e!! optical photon reservoir: vacuum modes of the radiation field (T=0) g much lower energy scales... ω 2π Hz

15 A. Griessner, A. Daley et al. PRL 2006; NJP 2007 (noninteracting atom) Physical Realization: Reservoir Engineering driven two-level atom + spontaneous emission Quantum optics ideas/techniques laser atom photon reservoir: vacuum modes of the radiation field (T=0) e!! optical photon g? (many body) cold atom systems much lower energy scales... ω 2π Hz

16 A. Griessner, A. Daley et al. PRL 2006; NJP 2007 (noninteracting atom) Physical Realization: Reservoir Engineering driven two-level atom + spontaneous emission trapped atom in a BEC reservoir laser atom e photon!! optical photon 1 BEC 0 phonon g laser assisted atom + BEC collision reservoir: vacuum modes of the radiation field (T=0) ω 2π Hz ω bd 2π khz

17 A. Griessner, A. Daley et al. PRL 2006; NJP 2007 (noninteracting atom) Physical Realization: Reservoir Engineering driven two-level atom + spontaneous emission trapped atom in a BEC reservoir laser atom e photon!! optical photon 1 BEC 0 phonon g laser assisted atom + BEC collision reservoir: vacuum modes of the radiation field (T=0) ω 2π Hz reservoir: Bogoliubov excitations of the BEC (at temperature T) ω bd 2π khz

18 Physical Realization Schematic b Rabi frequency 1 a1 (1) In practice 2 a2 Coherent excitation with opposite sign of Rabi frequency Ω b (a1 a2 ) + h.c. antisymmetric ci j = (a i + a j )(ai a j ) level structure: optical superlattice coherent excitation: Raman laser laser

19 Physical Realization Schematic In practice reservoir laser b ωbd 1 a1 2 a2 (2) Dissipative decay back: coupling of upper level to reservoir κ(a 1 + a 2 )b (rk + rk ) symmetric ci j = (a i + a j )(ai a j ) TBEC! ωbd: effective zero temperature reservoir k BEC = reservoir of Bogoliubov excitations coupling to system: interspecies interaction short coherence length in bath provides quasi-local dissipative processes, but not mandatory for our setup to work

20 Physical Realization Comments: 1 2 Long range phase coherence from quasi-local dissipative operations - Coherent drive: locks phases - Dissipation: randomizes - Conspiracy: purification (3) adiabatic elimination of auxiliary level, trace out the bath Effective single band jump operators c12 = (a 1 + a 2 )(a1 a2 ) Many sites: Array of dissipative junctions The coherence of the driving laser is mapped on the matter system Setting is therefore robust

21 Competition of unitary vs. dissipative dynamics

22 Effects of finite interactions H = J <i,j> a i a j + U i a 2 i a2 i dissipative dynamics favors pure BEC state interacting Hamiltonian dynamics not compatible Competition treating interactions in weak coupling dρ dt = i [H, ρ] + Lρ 3D: true (depleted) condensate, fixed phase: Bogoliubov theory 1,2D: phase fluctuations destroy long range order: Luttinger theory Strong coupling, 3D mixed state Gutzwiller Ansatz

23 Weak Coupling: Linearized jump operators momentum momentum space jump operators are nonlocal nonlinear objects c q,λ = 1 M d/2 (1 + e ike λ)(1 e i(k+q)e λ)a k a k+q k k+q k In a linearized theory the reduce to (any dimension) c q,λ = f q,λ a q f q,λ = 2 n(1 e iqe λ) q Interpretation: bosonic mode operators: depopulation of momentum q in favor of condensate zero mode explicit: f q=0,λ = 0 lead to momentum dependent decay rate accumulation k = 0 κ q = κ f q,λ 2 q 2 λ

24 Many-Body Master Equation Interpretation: How close are we to the GS of the Hamiltonian? Diagonalize H consider equation for single mode Bogoliubov / hydrodynamic excitation E q κ q t ρ = i E 2 [d d,ρ] π a linear sound mode q π a +2κ(u 2 dρd + v 2 d ρd σuv(d ρd + dρd) + anticommutator term cooling heating v 2 q, u 2 q = v 2 q + 1 N, N + 1 squeezing generalized Bogoliubov coefficients cf. thermal reservoir Intrinsic heating/cooling, though reservoir is at T = 0

25 Characterization of Steady State: Density Operator linearized ME exactly solvable: Gaussian density operator for each mode expressible as mixed state with ρ k = exp ( β k b k b k with squeezed operators b (Bogoliubov transformation) ) coth 2 (β k /2) = κ2 k + (ε k +Un) 2 κ 2 k + E2 k E q κ q d qd q at low momenta, resemblance to thermal state: π a linear sound mode q π a β k E k T eff, T eff = Un 2 role of temperature played by interaction

26 Correlations in various dimension: 3D Steady state: condensate depletion: n D = n n 0 = 1 2 Z dq v 0 (Un) 2 κ 2 q + E 2 q small depletion justifies Bogoliubov theory squeezing and mixing effects tied to interaction strength (unlike th. equilibrium) Approach to the steady state: n 0,eq n 0 (t) Un 8J 1 2κn t 1 power-law: Many-body effect due to mode continuum sensitive probe to interactions: cf. for noninteracting system universal at late times n 0,eq n 0 (t) t 3/2

27 Correlations in various dimension: 1/2D Steady State: quasi-condensates in low temperature phase a xa 0 expi(φ x φ 0 ) { e T eff 8Jn x, d = 1 (x/x 0 ) T eff/4t KT, d = 2 T KT = πjn T eff T eff = Un/2 x 0 = 2κn(T eff J) 1/2 Kosterlitz-Thouless temperature of 2D quasi-condensate Dissipative coupling: only sets cutoff scale steady state well understood as thermal Luttinger liquid similar results for temporal correlations (from ME via quantum regression theorem) weak effect of dissipation on phase fluctuations: E q q,κ q q 2

28 2D: Real Time Evolution Buildup of spatial correlations from disordered state Ψ t (x,0) { e x /ξ t = 0 (x/x 0 ) T eff 4T KT e x2 4ξ πκnt t broadening of Gaussian governed by time-dependent length scale x t = 2(πξ 2 κnt) 1/

29 Strong Coupling: Nonequilibrium Phase Transition with A. Tomadin Analogy to Mott insulator / Superfluid quantum phase transition : enhancement of superfluidity: Hopping J driven dissipation suppression of superfluidity: interaction U interaction U Expect phase transition as function of Differences: Competition of two unitary evolutions vs. competition of unitary and dissipative evolution phase transition (temperature T) quantum phase transition (g)

30 Reminder: Mott Insulator-Superfluid Phase Transition Hopping J favors delocalization in real space: Condensate (local in momentum space!) Fixed condensate phase: Breaking of phase rotation symmetry b i e iϕ Interaction U favors localization in real space for integer particle numbers: Mott state with quantized particle no. no expectation value: phase symmetry intact (unbroken) Competition gives rise to a quantum phase transition as a function of U/J

31 Reminder: Gutzwiller Ansatz Interpolation scheme encompassing the full range J/U. - Main ingredient: product wave function ansatz ψ = i ψ i, ψ i = n f (i) n n i, i ψ ψ i = 1 i complex amplitudes wave function normalization - Limiting cases (homogeous, drop site index, amplitudes chosen real): Mott state with particle number f n = δ n,m f n = m: coherent state: N/n!e N/2 Mott state Coh. state: Poisson Statistics Validity: approximation neglects all spatial correlations - becomes exact in infinite dimensions - reasonable in d=2,3 (T=0)

32 Mixed State Gutzwiller Approach Product ansatz for the density operator (instead of wave function) ρ(t) = ρ i (t), i ρ i (t) = n i m ρ (i) nm(t) nm Project on on-site density operator: ρ k = Tr k ρ Interpretation: off-diagonal: SF diagonal: atom statistics Nonlinear Mean Field Master Equation for reduced density operator (drop index) ρ = i [ ] ZJ( b b + b b) Ub 2 b 2,ρ +Zκ r,r Γ r,r { 2B r ρb r B r B r ρ ρb r B r } B r = {ˆn, b, b, 1} with correlation matrix Γ r,r = ˆn 2 b ˆn bˆn ˆn ˆnb ˆn b 2 b ˆnb b 2 ˆn + 1 b ˆn b b 1 Properties of ME: trace conserving mean particle number conserving Nonlinearity emerging in approximation to linear qm equation: similar GP equation

33 entropy superfluid order parameter Driven Dissipative Phase Transition Dynamic generation of the phase transition from initial coherent state ρ n,n = n n ( n + 1) n+1 1 n = 1, J = 0, zkt = 0, 10 1,..., n = 1, J = 0, zkt = 0, 10 1,..., superfluid phase transition thermal state 0.8 superfluid phase transition thermal state S/S n a 2 / n time 0.2 time U/zK interaction U U/zK interaction U U 0: pure coherent state solution Phase transition: Non-analyticity develops for t above critical point: thermal state: fixed temperature given by mean particle density N; no other scale appears No signatures of Mott physics due to strong mixing effect of U: unlike Bose-Hubbard case of two unitary tendencies at T=0: no superfluid: Mott state purity at T=0:

34 Exact calculations for N=6 sites A. Tomadin entropy SN /log LN n = 1, J = 0 and p.b.c. N=6 N=5 N=4 N=3 N=2 off-diagonal (superfluid) a 1 a 2 Φ[ a 1 a 2 ]/2π n = 1, J = 0 and p.b.c. N=2 N=3 N=6 N=5 N= U/zK interaction U U/zK interaction U

35 Nonequilibrium Phase Diagram U/K transition: Classification? - interaction driven (like quantum PT) - terminates in thermal state (like classical finite temperature PT) Add negative J (via phase imprinting): further competition through dynamical instability - no stable equilibrium state (no dynamical fixed point) - dynamical limit cycle? dynamically unstable: negative curvature of dispersion π/2a unstable thermal stable thermal dynamically unstable unstable condensate stable condensate π/a dynamically stable: positive curvature of dispersion Initialization: Coherent state, U=J=0 follow time evolution of the system

36 Dissipative Driving of Fermions - Excited states: η Condensate - Cooling into Antiferromagnetic and d-wave States

37 Cooling to Excited States: η-condensate η-state: exact excited (i.e. metastable) eigenstate of the two-species Fermi Hubbard Hamiltonian in d dimensions [Yang 89] local doublon H = J i, j,σ η i = f i f i f iσ f jσ +U i f i f i f i f i η-condensate checkerboard superposition η = 1 M d/2 φ i η i i η-particle φ i = ±1 N-η-condensate: H(η ) N 0 = NU(η ) N 0 exact eigenstate, off-diagonal long range order

38 Cooling to Excited States: η-condensate Small scale simulations (open BC) demonstrate η condensation for jumps Interpretation: Quantum Jump picture c (1) i j = (η i η j )(η i + η j ) c (2) i j = n i f i f j + n j f j f i H generates spin-up and down configurations on each pair of sites (for any initial density matrix) c (2) i j c (1) i j associates into local doublons creates checkerboard superposition: η condensate May be conceptually interesting However, these jump operators are two-body: difficult to engineer

39 Motivation: Cooling Fermion Systems High temperature superconductivity - discovered in 1986 (Müller, Bednorz): cuprates show superconductivity at unconventionally high temperature - riddle: attraction from repulsion microscopically, strong Coulomb onsite repulsion still, observe pairing of fermions with d-wave symmetry Experimental phase diagram for cuprates Minimal model: 2d Fermi-Hubbard model - realistic for cuprate high-temperature superconductors? - hard to solve: strongly interacting fermion theory no controlled analytical approach available numerically (classical computer) intractable Quantum simulation of the Fermi-Hubbard model in optical lattices?

40 Quantum Simulation of Fermion Hubbard model Clean realization of fermion Hubbard model possible Detection of Fermi surface in 40K (M. Köhl et al. PRL 94, (2005)) Fermionic Mott Insulators (R. Jördens et al. Nature 455, 204 (2008); U. Schneider et al., Science 322, 1520 (2008)) Cooling problematic: small d-wave gap sets tough requirements BCS superconductors Unitary continuum Fermi gas SF transition Critical temperature for d-wave SF Current lattice experiments Existing proposal: Adiabatic quantum simulation (S. Trebst et al. PRL 96, (2006)) Start from a pure initial state of noninteracting model Adiabatically transform to unknown ground state of interacting model Concrete scheme: find path protected by large gaps: prepare RVB ground state on isolated 2x2 plaquettes Still need to be x cooler couple these plaquettes to arrive at many-body ground state

41 Dissipative Quantum State Engineering Approach Roadmap: (1) Precool the system (lowest Bloch band) (2) Dissipatively prepare pure (zero entropy) state close to the expected ground state: - energetically close - symmetry-wise close - spin-wise close (3) Adapted adiabatic passage to the Hubbard ground state - switch dissipation off - switch Hamiltonian on Precooling Dissipative Cooling Adapted adiabatic passage

42 The State to Be Prepared Pauli matrix mean field (product) state two-component spinor d-wave SC What does the state have in common with the expected Hubbard ground state (1) Quantum numbers... - pairing in the singlet channel - phase coherence: delocalization of singlet pairs transformation under spatial rotations: d-wave The state shares the symmetries of Hubbard GS No phase transition will be crossed in preparation process - in the talk, we mainly consider 1-dimensional analog for simplicity:

43 The State to Be Prepared Pauli matrix mean field (product) state two-component spinor What does the state have in common with the expected Hubbard ground state doping not too close to AF (2) Energetically close? Not known, but: - off-site pairing avoids excessive double occupancy cf onsite pairing: - the pairs are quasi-local, i.e. have a short coherence length in accord with observation in cuprates superfluidity decreases due to strong correlations (A. Paramekanti, N. Trivedi, M. Randeria, PRB 70, (2004)) State can be expected to be convenient starting point not too close to half filling

44 Relation to the BCS Wavefunction usually, fixed phase (coherent state) wave function Fixed particle number wavefunction BCS limit BCS amplitude BCS gap chemical potential BEC/molecular limit distinct limits: - localized in momentum space BCS limit dispersion Relation to our state: - delocalized in position space - delocalized in momentum space - localized in position space BEC / molecular limit State shares the symmetries, but can be energetically very different

45 Setting Goal: Construct jump operators with unique mean field dark states: dark state mean field (product) state solve: (sufficient for normal ordered jump operators) Requirements for implementation: non-hermitian particle number conserving quasi-local: j close central site i single-particle operation this is what the eta operators suffered from!

46 Antiferromagnetic Jump Operators Construct jump operators for antiferromagnetism as a preparation Antiferromagnetic Neel state is a product of AF unit cell operators doubly degenerate bipartite lattice with sublattices A,B Set of jump operators (one dimension): Pauli matrices flip! Action of jump operators : Pauli blocking : spin transport flip! 0

47 d-wave Jump Operators Rewrite the d-wave state in terms of AF unit cell operators: shift invariance homogeneous product but delocalized pairs Second equality: interpret the state as a symmetrically delocalized AF Set of jump operators: Action of jump operators : Pauli blocking : spin transport both: phase coherence via delocalization flip & delocalize Combine fermionic Pauli blocking with delocalization as for bosons Pauli blocking is the reason for single particle nature of operators

48 Uniqueness Recall: Unique dark state <-> state reached independent of initial condition Evidence for uniqueness from small scale numerical simulations Entropy evolution Entropy Antiferromagnetism Fidelity perfect mixture fidelity Time Time Entropy time 6 sites, 6 particles Fidelity d-wave Time Fidelity Fidelity 3x3 sites, 4 particles, different random initial states time

49 Uniqueness Understanding can be gained from symmetry considerations Uniqueness of dark state equivalent to uniqueness of ground state (GS) of H is semi-positive an exact GS is the above d-wave (E=0) unique iff no symmetry T such that Symmetries: - Translations - global phase rotations U(1) - global spin rotations SU(2) for, [ ] -> effective Hamiltonian d-wave is an eigenstate to these - additional discrete symmetry on bipartite lattice for spoils uniqueness bipartite (periodic BC) A B B A not bipartite (PBC) Avoid symmetries All three operators needed for uniqueness A B SU(2) symmetry; the jump operators are SU(2) vectors

50 Comments on the effective Hamiltonian Amusing parallel: Above Hamiltonian is a parent Hamiltonian for the d-wave state H is semi-positive an exact unique GS is the above d-wave state(e=0) GS is GS for each separately: projectors on GS completely analogous to e.g. AKLT model there, ground state is valence bond solid with exponentially decaying correlations different: state has long range order due to strong delocalization study excitations mean field decoupling single fermion gap diagonal contributions from normal ordering order parameter-like structure: macroscopically populated -> replace by c-number mean field (-> loose particle number cons.) single fermion excitations are gapped: important for adabatic passage

51 Arbitrary phase coherent pairing states Any pairing product state can be characterized by 3 quantum numbers pairing momentum pairing distance Examples: s-wave BCS eta-state d-wave like state Jump operators constructed for all k, mu, and n >0 (displayed just for completeness...) arbitrary n > 0 pairing states can be targeted d-wave not distinguished, but off-site pairing special symmetries of the state inherited by the parent Hamiltonian

52 Implementation of d-wave jump operators Decisive property: single-particle nature of the jump operators Implement Fourier transformed operators: Basic physical ingredients: Dissipation: Emission in cavity Use Earth Alkaline atoms in state dependent superlattice Lattice system: xy plane Engineering requirements: Spin imprinting: Light Polarization Momentum transfer: Laser angle (incoherent beams) cos q dependence: Quantum Interference

53 Implementation of d-wave jump operators Level scheme: Earth Alkaline atoms spont. emission: cavity mode momentum transfer spin imprinting Bloch bands Quantum Interference: physical spin atom confinement via optical lattice cos q: onsite processes interfere destructively

54 Adapted Adiabatic Passage Assume we have prepared zero entropy d-wave Want to connect to Hubbard ground state Adiabatic passage (purely Hamiltonian dynamics):? Adapted adiabatic passage ramping slowly: remain in ground state Adapted adiabatic passage: Two ingredients gap protection from auxiliary Hamiltonian parent Hamiltonian has d-wave eigenstate and is gapped: add detuning to the effective Hamiltonian Fidelity fidelity to Hubbard GS ramp parameters U J Γ Δ Time (1/U f ) probabilistic ground state preparation dissipative and Hubbard dynamics compete focus on time before first jump: state prepared with probability Time (1/U f ) Probability Time (1/U f ) preparation probability

55 Summary Part I By merging techniques from quantum optics and many-body systems: Driven dissipation can be used as controllable tool in cold atom systems. Pure states with long range correlations from quasilocal dissipation Many-body dark state, independent of initial density matrix Laser coherence mapped on matter system System steady state has zero entropy Nonequilibrium phase transition driven via competition of unitary and dissipative dynamics driven by interactions (like quantum phase transition) terminates into thermal state (like classical phase transition) Strong potential applications for fermionic quantum simulation cool into zero entropy d-wave state as intial state for Fermi-Hubbard model single particle operations due to Pauli blocking realistic setting using earth alkaline atoms in a cavity Condensed Matter Many-Body States Cold Atoms Engineering Quantum Optics Dissipation/Driving

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57 Optical Lattices AC-Stark shift - Consider an atom in its electronic ground state exposed to laser light at fixed position x. - The light be far detuned from excited state resonances: ground state experiences a secondoder AC-Stark shift δe g = α(ω)i with α(ω) - dynamic polarizability of the atom for laser frequency ω, I E 2 - light intensity. - Example: two-level atom { g, e }. Rabi frequency For standing wave laser configuration E, ACδE g = Ω2 4 detuning from resonance = ω ω eg Ω { } < 0, δe g < 0 > 0, δe g > 0 For standing wave laser configuration E( x, t) = ee sin kx e iωt + h.c., AC- Stark shift is a function of position: It generates an optical potential V opt ( x) δe g ( x) = Ω2 ( x) 4 V 0 sin 2 kx (k = 2π/λ) V 0

58 Effective Lattice Hamiltonian Start from our model Hamiltonian, add optical potential:! $ % " # " 2 H= µ + V (x) + Vopt (x) ax + gn x ax 2m x Periodicity of the optical potential suggests expansion of field operators into localized lattice periodic Wannier functions (complete set of orthogonal functions) ax =! i,n band index wn (x xi )bi,n minimum position For low enough energies (temperature), we can restrict to lowest band: Then we obtain the single band Bose-Hubbard model!!!! 1!i n i + 2 U H = J bi bj µ n i + n i (n i 1)! T, U, J! 4V0 ER, ER = k 2 /(2m) n = 0 J = U =g!!!i,j" i i i dxw0 (x)(!2 /2m" Vopt (x))w0 (x λ/2) dx w0 (x) 4 n i = b i bi U J