Trapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization. Diego Porras

Trapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization Diego Porras

Outline Spin-boson trapped ion quantum simulators o Rabi Lattice/Jahn-Teller models o Gauge symmetries o Complexity of spin-boson Hamiltonians Non-equilibrium physics: equilibration and thermalization 3 (G. - 20 Clos, m D. Porras, U. Warring, T. Schaetz, arxiv:1509.07712) o Eigenstate Thermalization in the spin-boson model o Inverse Participation Ratio and ergodicity o Experimental results: testing predictions by Eigenstate Thermalization o Outlook

Spin-boson trapped ion QS: Rabi-Lattice models Trapped ion quantum simulators Adiabatic elimination of vibrational modes leads to effective spin-spin interactions H Ising = H ph + B x 2 j σ j x + 1 2 j,l J j,l σ j z σ l z D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004) A. Friedenauer, H. Schmitz, J.T. Glueckert, D. Porras, T. Schaetz, Nat. Phys. 4, 757 (2008)

Spin-boson trapped ion QS: Rabi-Lattice models Strong spin-phonon couplings lead to phenomena beyond quantum magnetism. Vibrational modes Active elements in the quantum simulation 3-20 m H Rabi Lattice = n ω n a n a n + Ω x 2 j σ j x + g j,n M j,n σ j z (a n + a n ) D. Porras, P.A.Ivanov, F. Schmidt-Kaler, Phys. Rev. Lett. 108, 235701 (2012) P. Nevado, D. Porras, Phys. Rev. A 93, 013625 (2016)

Spin-boson trapped ion QS: Rabi-Lattice models Spin-phonon entangled phases Interplay between magnetic and structural phases (Jahn- Teller effect) 3-20 m H Rabi Lattice = n ω n a n a n + Ω x 2 j σ j x + g j,n M j,n σ j z (a n + a n ) D. Porras, P.A.Ivanov, F. Schmidt-Kaler, Phys. Rev. Lett. 108, 235701 (2012) P. Nevado, D. Porras, Phys. Rev. A 93, 013625 (2016)

Spin-boson trapped ion QS: Rabi-Lattice models Spin-phonon entangled phases Interplay between magnetic and structural phases (Jahn- Teller effect) Coulomb coupling between ions 3-20 m H Rabi Lattice = n ω n a n a n + Ω x 2 j σ j x + g j,n M j,n σ j z (a n + a n ) D. Porras, P.A.Ivanov, F. Schmidt-Kaler, Phys. Rev. Lett. 108, 235701 (2012) P. Nevado, D. Porras, Phys. Rev. A 93, 013625 (2016)

Spin-boson trapped ion QS: gauge symmetry Phononic modes can also be used to simulate minimal models with gauge symmetries (local discrete or continuous symmetries) Example: replace the magnetic field by a quantum field in a trapped ion Ising chain H I = 1 2 J i,j σ i z σ j z + Ω x 2 σ j x H IR = 1 2 J i,j σ z z j σ j+1 + g a j + a j + σ j x + δ a j + a j j j P. Nevado, D. Porras, Phys. Rev. A 92, 013624 (2015)

Spin-boson trapped ion QS: gauge symmetry H IR = 1 2 J i,j σ z z j σ j+1 + g a j + a j + σ j x + δ a j + a j j j Continuous transition First order phase transition P. Nevado, D. Porras, Phys. Rev. A 92, 013624 (2015)

Spin-boson trapped ion QS: gauge symmetry H IR = 1 2 J i,j σ z z j σ j+1 + g a j + a j + σ j x + δ a j + a j j j P. Nevado, D. Porras, Phys. Rev. A 92, 013624 (2015)

Spin-boson trapped ion QS: single spin in a vibrational bath Single spin coupled to a quantum bath H spin boson = Ω R 2 σ x + n g n σ z (a n + a n + ) + n ω n a n + a n 3-20 m D. Porras, F. Marquardt, J. von Delft, J.I. Cirac, Phys Rev. A 78, 010101 (2008)

Spin-boson trapped ion QS: single spin in a vibrational bath Single spin coupled to a quantum bath H spin boson = Ω R 2 σ x + n g n σ z (a n + a n + ) + n ω n a n + a n Non-equilibrium phase transition as a function of the coupling strength 3-20 m Quantum revivals (phonons bounce at the ends of the ion chain) D. Porras, F. Marquardt, J. von Delft, J.I. Cirac, Phys Rev. A 78, 010101 (2008)

Complexity of a spin-boson quantum simulator Bosonic modes lead to a drastic increase of the complexity of a quantum simulator Example: single spin-boson model H sb = Ω R 2 σ x + n g n σ z (a n + a n + ) + n ω n a n + a n dim H = 2 1 + n c N modes n c = cut-off in the phonon number N modes = 5 ions n c = 20 (assuming a n + a n 2 ) dim H = 8 10 5 (equivalent 23 spins) Out of reach with state-of-the-art full exact diagonalization

Complexity of a spin-boson quantum simulator Complex (or numerically intractable) does not mean necessarily interesting However, spin-boson simulators can be used to study open questions in nonequilibrium physics Does a finite closed system system thermalize? ρ 0 = ρ BATH ρ t How does thermalization scale with systemsize, coupling, energy...? What are the time-scales for thermalization? G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Complexity of a spin-boson quantum simulator Complex (or numerically intractable) does not mean necessarily interesting However, spin-boson simulators can be used to study open questions in nonequilibrium physics Most of what we know about thermalization is from numerics ρ 0 = ρ BATH ρ t Can we use a trapped ion quantum simulator to check conjectures/theories on thermalization with larger systems? G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Trapped ion spin-boson model: controllable non-linearity The trapped ion spin-boson model can be simulated directly in polaron form: H s b = n ω n a n + a n + ω z 2 σ z + Ω R 2 σ+ e ik Lz + σ e ik Lz e i n η n (a n +a n + ) η LD 1 Ω R 2 (σ+ 1 i n η n a n + a n + + H. c. ) linear coupling (weak coupling regime) (single phonon emission/absorption) (ω n = relevant eigenfrequencies in the vibrational bath)

Trapped ion spin-boson model: controllable non-linearity The trapped ion spin-boson model can be simulated directly in polaron form: H s b = n ω n a + n a n + ω z 2 σ z + Ω R 2 σ+ e i n η n (a n +a +) n + σ e i n η n (a n +a + n ) η LD > 1 Ω R 2 (σ+ (1 + η n O 1 a n, a n + + η n η m O 2 a n, a n + + ) + H. c. ) (many-phonon emission/absorption) Larger phase-space for thermalization (many-body vibrational bath)

Thermalization of closed quantum systems Classical statistical physics arise a result of ergodicity: {x n t, p n (t)} a system visits all the microscopic states with the same energy (microcanonical ensemble) with the same probability Can an isolated quantum system be described by a microcanonical average? Ψ t H t i = e ħ Ψ 0 Specifically, are time-averages of operators equivalent to microcanonical averages? μ T A = 1 T 0 T Ψ(t) A Ψ t dt μ micro (A) = de n p E n ψ n A Ψ n μ T A =? μ micro (A)

Thermalization of closed quantum systems A second aspect of thermalization is equilibration: we expect time-fluctuations around the microcanonical average to be small and decrease with system size δ T A = 1 T 0 T Ψ t A Ψ t μ T A 2 dt 0?? A typical experiment with a trapped ion simulator will measure time-averages and fluctuations of a local observable, e.g., the z-spin component: σ z t δ(σ z ) μ(σ z ) t

Eigenstate Thermalization Eigenstate Thermalization Hypothesis for non-integrable systems Thermalization occurs at the level of single eigenstates J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Formally expressed as a condition on matrix element of energy eigenstates: A n,n = ψ n A ψ n A(E n ) A(E n ) is a smooth function of the energy Eigenstate Thermalization implies that each Hamiltonian eigenstate is effectively a microcanonical ensemble A n,n = A E n does not depend on the details of eigenstate n takes the same value as a microcanonical ensemble with energy E n

Eigenstate Thermalization Eigenstate Thermalization Hypothesis for non-integrable systems Thermalization occurs at the level of single eigenstates J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Formally expressed as a condition on matrix element of energy eigenstates: A n,n = ψ n A ψ n A(E n ) A(E n ) is a smooth function of the energy Eigenstate Thermalization implies that each Hamiltonian eigenstate is effectively a microcanonical ensemble A n,n = A E n A n+1,n+1 = A E n+1 A n+2,n+2 = A E n+2

Eigenstate Thermalization J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Formally expressed as a condition on matrix element of energy eigenstates: A n,n = ψ n A ψ n A(E n ) A(E n ) is a smooth function of the energy leads to the equivalence between time-averages and microcanonical averages: T 1 lim Ψ(t) A Ψ t dt = T T 0 n ψ n Ψ 0 2 A n,n = n ψ n Ψ 0 2 A(E n ) Numerical evidence (e.g. M. Rigol et al., Nature 452, 854-858 (2008) )

Eigenstate Thermalization J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Time fluctuations are related to non-diagonal matrix elements δ T A 2 = n,m (n m) Ψ 0 ψ n 2 Ψ 0 ψ m 2 A n,m 2 The Eigenstate Thermalization Hypothesis also predicts some properties of nondiagonal matrix elements: A n,m = 1 D E density of states (normalization) f A E, ω R n,m random variable 2 R n,m = 1 E = (E n+e m ) 2 ω = E n E m

Eigenstate Thermalization J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Time fluctuations are related to non-diagonal matrix elements δ T A 2 = n,m (n m) Ψ 0 ψ n 2 Ψ 0 ψ m 2 A n,m 2 Eigenstate Thermalization Hypothesis for non-diagonal matrix elements: A n,m = 1 D E f A E, ω R n,m A n,m 0 (E n E m < Γ A ) A n,m = 0 (E n + E m > Γ A ) Smooth function with width Γ A (spectral window of coupled states)

Eigenstate Thermalization J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). M. Srednicki, Phys. Rev. E 50, 888 (1994). Time fluctuations are related to non-diagonal matrix elements δ T A 2 = n,m (n m) Ψ 0 ψ n 2 Ψ 0 ψ m 2 A n,m 2 Eigenstate Thermalization Hypothesis for non-diagonal matrix elements: A n,m = 1 D E f A E, ω R n,m E m Γ A Smooth function with width Γ A (spectral window of coupled states) E n

Quantum chaotic wavefunctions The Eigenstate Thermalization Hypothesis can be qualitatively understood by using chaotic wavefunctions: H s b = n ω n a n + a n + ω z 2 σ z + Ω R 2 σ+ e i n η n (a n +a +) n + σ e i n η n (a n +a + n ) uncoupled eigenstates H 0 φ α = (or ) S n 1 n N spin state phonon number in each mode H I ψ n = α c n α φ α Spin-boson coupling (H I ) leads to many-body eigenstates (spin-phonon entanglement)

Quantum chaotic wavefunctions F Borgonovi, FM Izrailev, LF Santos, VG Zelevinsky, Physics Reports, 2016 The Eigenstate Thermalization Hypothesis can be qualitatively understood by using chaotic wavefunctions: H s b = n ω n a n + a n + ω z 2 σ z + Ω R 2 σ+ e i n η n (a n +a +) n + σ e i n η n (a n +a + n ) H 0 H I If H I turns the system non-integrable, we expect each many-body eigenfunctions to be a linear combination of free eigenstates within a given energy interval Many-body eigenstates (interactions) E n E α uncoupled basis ψ n energy shell (width determined by H I )

Quantum chaotic wavefunctions The Eigenstate Thermalization Hypothesis can be qualitatively understood by using chaotic wavefunctions: H s b = n ω n a n + a n + ω z 2 σ z + Ω R 2 σ+ e i n η n (a n +a +) n + σ e i n η n (a n +a + n ) H 0 H I If H I turns the system non-integrable, we expect many-body eigenfunctions to Many-body eigenstates (interactions) ψ n E n E α uncoupled basis c n α φ α α = 1 C eiθ 1 φ 1 + e iθ 2 φ 2 + ) Approximately random phases

Quantum chaotic wavefunctions In the energy shell model, we approximate many-body wave functions by random, independent variables: ψ n = c n α φ α α c n α = 0 c n α c n (α ) = δ n,n δ α,α D E n Γ α /π E n E α 2 + Γ α 2 Strength function Many-body eigenstates (interactions) E n E α uncoupled basis ψ n Γ α Ω R 2

Quantum chaotic wavefunctions Quantum chaotic wavefunction model confirmed by numerics (N = 4 ions ): c α n 2 α = 20 α = 100 Self-averaging α = 500 α = 1000 E n α = 20 α = 100 α = 500 α = 1000 F α E = n c α n 2 δ ε (E E n ) E

Quantum chaotic wavefunctions The energy shell model can be used to justify the Eigenstate Thermalization Hypothesis A n,n = ψ n A ψ n = c n α c n α φ α A φ α α,α Assume selfaveraging = c n α c n α φ α A φ α α,α = α 1 D E α Γ α /π E n E α 2 + Γα 2 φ α A φ α The energy shell determines the ETH microcanonical ensemble J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).

Finite size scaling of thermalization The energy-shell model also predicts qualitatively the form of non-diagonal matrix elements. Consider in particular the case A = σ z. This is diagonal in the uncoupled spin-boson basis: H s b = n ω n a n + a n + ω z 2 σ z φ α σ z φ α = δ α,α φ α σ z φ α φ α = (or ) S n 1 n N The energy shell model leads to: (σ z ) n,m = ψ n σ z ψ m = α c n α c m α φ α σ z φ α (σ z ) n,m 0 Energy width Γ α, determines spectral width of (σ z ) n,m if E n E m < 2Γ α

Finite size scaling of thermalization We can use the Eigenstate Thermalizaiton Hypothesis together with the energy shell model to predict the scaling of time-fluctuations Assume that the system is initially in an eigenstate of the uncoupled spin-boson model Ψ 0 = φ α δ T σ z 2 = c n α 2 c m α 2 2 O n,m n,m (n m) de n de m Γ α π E n E α 2 + Γ α 2 Γ α π 1 E m E 2 2 α + Γ α D E f σ z E, ω 2 1 D E α Γ α

Finite size scaling of thermalization We can use the Eigenstate Thermalization Hypothesis together with the energy shell model to predict the scaling of time-fluctuations Assume that the system is initially in an eigenstate of the uncoupled spin-boson model Ψ 0 = φ α δ T σ z 2 1 = 1 D E α Γ α D eff number of states ψ n participating in the initial state φ α density of states δ T σ z 1 D eff energy width efficient evaluation: Inverse Participation Ratio D eff = n ψ n Ψ 0 4 1 Related to the upper bound in time-fluctuations found in: N. Linden, S. Popescu, A.J. Short, A. Winter, Phys. Rev. E 79, 0611103 (2009).

Let us summarize the results from our theoretical excursion: Assume that the spin-boson system is initially in a product state, e.g. Ψ 0 = s 0 1 0 N The Eigenstate Thermalization Hypothesis predicts: 1) The system thermalizes efficiently for high values of D eff 2) Time-fluctuations decrease like δ T σ z 1 D eff G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization H s b = Ω R 2 σ + e i n η n (a n +a n +) + σ e i n η n (a n +a n + ) + n ω n a n + a n + ω z 2 σ z Experimental results up to N = 5 ions The system thermalizes better as size is increased G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization In the experiment, vibrational modes are initially at some finite temperature ( a n + a n 1) To calculate the effective dimension we need to average over all states in the initial mixed state ρ 0 = ρ B = D eff = α w α n α Ψ n φ α 4 w α φ α φ α 1 Ergodicity phase diagram as a function of the spin-boson parameters. G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization Theoretical error bars Estimate the effect of low phonon number cut-offs in the numerical calculations Limit of state-of-the art full numerical diagonalization Full time-dependent calculation G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization Thermalization observed in regions of large D eff G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization Experimental results up to N = 4 ions agree well with the scaling predicted by the Eigenstate Thermalization Hypothesis δ σ z 1/ D eff N = 5: Experimental times are too short to obtain the infinite-time average Time-scales for thermalization seem to increase with system size G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization δ σ z 1/ D eff Precise numerical simulation of experimental results out of reach with N = 4, 5 ions G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Experimental observation of thermalization: Outlook Open Questions: What are the time-scales for Eigenstate Thermalization. How do they depend on the effective dimension? Non-equilibrium physics and Eigenstate Thermalization in other spin-boson systems (e.g. circuit-qed) [B. Peropadre, D. Zueco, D. Porras, J.J. García-Ripoll, Phys. Rev. Lett. 111, 243602 (2013)] Applications: o Prepare finite temperature states in quantum simulators. o Dissipative preparation of spin states by sympathetic cooling (laser cooling + thermalization), spin-pumping without cycling transitions. o Dissipative preparation of entangled states. G. Clos, D. Porras, U. Warring, T. Schaetz, Time-resolved observation of thermalization in an isolated quantum system, arxiv:1509.07712

Collaborators: Tobias Schaetz (U. Freiburg) Govinda Clos (U. Freiburg) Ulrich Warring (U. Freiburg) Pedro Nevado-Serrano (U. of Sussex) Samuel Fernandez-Lorenzo (U. of Sussex)