Classical and quantum simulation of dissipative quantum many-body systems

Transcription

1 Classical and quantum simulation of dissipative quantum many-body systems David Mesterházy Albert Einstein Center for Fundamental Physics, University of Bern Phys. Rev. A 93 (206) 02602, New J. Phys. 8 (206) 0730, arxiv:703.xxxxx Tr ρ t ρ(t = 0) ρ(t) March 2, 207 SIGN Workshop, Seattle

2 Dissipation as a resource! Dissipative quantum computing [Verstraete, Wolf & Cirac] [Kraus et al.] Dissipative quantum Church-Turing theorem [Kliesch, Barthels, Gogolin, Kastoryano & Eisert] Every time evolution generated by a k-local Liouvillian can be simulated by a unitary quantum circuit with resources scaling polynomially in the system size N and simulation time. Quantum memory via engineered dissipation [Pastawski, Clemente & Cirac] Dissipative state preparation [Verstraete, Wolf & Cirac] Nonequilibrium phase transitions driven by dissipation [Diehl et al.]...

3 Dissipation as a resource! Dissipative quantum computing [Verstraete, Wolf & Cirac] [Kraus et al.] Dissipative quantum Church-Turing theorem [Kliesch, Barthels, Gogolin, Kastoryano & Eisert] Every time evolution generated by a k-local Liouvillian can be simulated by a unitary quantum circuit with resources scaling polynomially in the system size N and simulation time. Quantum memory via engineered dissipation [Pastawski, Clemente & Cirac] Dissipative state preparation [Verstraete, Wolf & Cirac] Nonequilibrium phase transitions driven by dissipation [Diehl et al.]... What can dissipation do for you?

4 Dissipation as a resource! Dissipative quantum computing [Verstraete, Wolf & Cirac] [Kraus et al.] Dissipative quantum Church-Turing theorem [Kliesch, Barthels, Gogolin, Kastoryano & Eisert] Every time evolution generated by a k-local Liouvillian can be simulated by a unitary quantum circuit with resources scaling polynomially in the system size N and simulation time. Quantum memory via engineered dissipation [Pastawski, Clemente & Cirac] Dissipative state preparation [Verstraete, Wolf & Cirac] Nonequilibrium phase transitions driven by dissipation [Diehl et al.]... } This talk

5 Markovian quantum master equation Time evolution of open quantum system described in terms of a density matrix ρ, whose dynamics is assumed to be governed by a Markovian quantum master equation (alternatively GKS-L, or simply Lindblad equation) d dt ρ = Lρ = i[h, ρ] + L λ ρ λ λ =,..., d 2 N, where d N denotes the dimension of the N-particle Hilbert space. This represents an effective description of the system where the degrees of freedom associated to the environment have been integrated out (no memory effects). Written in Lindblad form, the action of L λ on ρ is expressed in terms of jump operators L λ ( L λ ρ = γ λ L λ ρl λ { L 2 λ L λ, ρ }) Instead of a generic coupling to an external environment (e.g., dephasing noise), we inquire about the dynamics induced by engineered dissipative couplings.

6 Experimental platforms Ion trap experiments [Barreiro et al.] [Schindler et al.] Rydberg atoms [Weimer et al.] Ultracold atoms...

7 Experimental platforms Ion trap experiments [Barreiro et al.] [Schindler et al.] Rydberg atoms [Weimer et al.] Ultracold atoms... Example: State symmetrization (dissipative BEC condensation) via coherent driving of atoms in optical lattice

8 Computational methods Exact diagonalization Time-dependent variational Monte Carlo [Transchel, Milsted & Osborne et al.] Cluster Monte Carlo techniques [Banerjee, Hebenstreit, Huffman, Chandrasekharan, Wiese et al.] t-dmrg [White & Feiguin] Classical statistical, or truncated Wigner approximations (TWA) [Polkovnikov et al.] MPS for nonequilibrium steady states [Cui, Cirac, Bagñuls] Mean-field type approximations... Exact Monte Carlo methods are typically not applicable due to a severe sign problem, although some problems may still be tractable (H = 0). Other methods typically biased towards low entanglement or small system sizes. Benchmark on large system sizes?

9 Closed hierarchies of correlation functions Throughout this talk, we consider quantum spin-/2 many-body systems, with spin degrees of freedom s a x = 2 σa x that are localized on N = L d sites of a regular d-dimensional lattice. The equation of motion for n-point correlation functions O z z 2 z n = tr { ρo z z 2 z n } reads d dt O z z 2 z n = tr { ρl } O z z 2 z n = i tr { ρ [ H, O z z 2 z n ]} + 2 λ γ λ tr { ρ ( L [ ] [ λ Oz z 2 z n, L λ + L λ, O ] )} z z 2 z n Lλ where L = i[h, ] + λ L λ is the adjoint map corresponding to the Lindbladian L. In general, this set of equations does not close, i.e., the commutators on the right hand side typically induce operators of higher order, which leads to an infinite hierarchy of correlation functions which cannot be solved without truncating the coupled set of equations. Possible to derive conditions under which the hierarchy of correlation functions closes.

10 Engineered (non-hermitian) jump operators Assume H = 0! Can we still find interesting processes within this class? Yes! Example: Consider non-hermitian jump operators L xy = 2 ( s + x + s + ) ( y s x s ) y acting uniformly on adjacent sites x, y of the regular lattice, i.e., γ xy γ. L xy maps any two-particle spin-singlet state to the spin triplet, while conserving the total spin projection S 3 = x s 3 x along the quantization axis, and annihilates the spin triplet state, i.e., L 2 xy = 0. Simple illustration (N = 4 qubits): ρ(t = 0) ρ(t) time

11 Engineered (non-hermitian) jump operators Assume H = 0! Can we find interesting processes within this class? Yes! Example: Consider non-hermitian jump operators L xy = 2 ( s + x + s + ) ( y s x s ) y acting uniformly on adjacent sites x, y of the regular lattice, i.e., γ xy γ. L xy maps any two-particle spin-singlet state to the spin triplet, while conserving the total spin projection S 3 = x s 3 x along the quantization axis, and annihilates the spin triplet state, i.e., L 2 xy = 0. Simple illustration (N = 4 qubits): Tr ρ t

12 Dissipative BEC condensation The final state ρ(t ) of this purely dissipative process (H = 0) is determined by fixed points ρ of the corresponding dynamic map, i.e., L xy ρ = 0 x,y corresponding to an ensemble of totally symmetric superposition states. Dynamical semigroup is only relaxing (i.e., depends on the initial conditions) and therefore ρ(t ) will depend on initial magnetization. By virtue of the quantum spin-/2 to hardcore boson mapping, L xy can be seen to describe a symmetric delocalization of hardcore bosons over adjacent sites, with a BEC of hardcore bosons as the resulting final state (known by construction).

13 Quasi-exact solution in arbitrary dimensions We provide a semi-analytic solution for the full dynamics starting from a completely incoherent initial ensemble, based on the time-dependence of n-point correlation functions O z z 2 z n = O z z 2 z n tr { ρo z z 2 z n } in particular, products of spin operators O z z 2 z n = i n s zi. Note, while the dissipation rate γ can be factored out (by introducing a rescaled time variable τ = tγ), the finite system size N = L d, provides a time scale that may lead to interesting scaling behavior. This time scale is determined by the eigenvalues of the Lindbladian. More interesting dynamics emerge with competing processes, either by incorporating a nonvanishing Hamiltonian, or different dissipative Lindbladian superoperators.

14 Time-evolution equations for one- and two-point functions In the absence of a Hamiltonian (H = 0) and for bilocal operators L xy acting on nearest-neighbor sites on the regular lattice, we obtain a diffusion equation for the local magnetization s a x = s a x, i.e., τ s a x = 4 xs a x and a linear system for the off-diagonal contributions of two-point correlation functions C xy s + x s y + s x s + y and D xy = s 3 xs 3 y which read τ C xy = τ D xy = 4 ( x + y )C xy 2 δ ( ) x,y Cxy + 4D xy 4 ( x + y )D xy 8 δ ( ) x,y 4Dxy Note that the diagonal contributions are constant in time C xx = 4D xx =.

15 Example: Nonequilibrium BEC from an incoherent ensemble The system is initially prepared in the infinite-temperature ensemble ρ(τ = 0) and afterwards quenched to zero temperature, where the system is driven by the continuous application of quantum jump operators L xy. At time τ = 0: C xy (τ = 0) = D xy (τ = 0) = 0, while off-diagonal correlations are generated for τ > 0. x y ρ(τ ) = 2 N N n=0 C p (τ ) = 2 δ p,0 + 2N ( ) N N/2, N/2 + n N/2, N/2 + n n d = 3, N = 36 3

16 Finite-size scaling of the dissipative gap From the spectrum of the linear operator M that generates the time-evolution for two-point functions C xy and D xy, i.e., λ Spec M, we may define the dissipative gap = max λ Re λ > 0 which characterizes the asymptotic approach towards the final state. D N2

17 Finite-size scaling of the dissipative gap From the spectrum of the linear operator M that generates the time-evolution for two-point functions C xy and D xy, i.e., λ Spec M, we may define the dissipative gap = max λ Re λ > 0 which characterizes the asymptotic approach towards the final state. D N2 2D N log N

18 Finite-size scaling of the dissipative gap From the spectrum of the linear operator M that generates the time-evolution for two-point functions C xy and D xy, i.e., λ Spec M, we may define the dissipative gap = max λ Re λ > 0 which characterizes the asymptotic approach towards the final state. 3D N D N2 2D N log N

19 Finite-size scaling of the dissipative gap From the spectrum of the linear operator M that generates the time-evolution for two-point functions C xy and D xy, i.e., λ Spec M, we may define the dissipative gap = max λ Re λ > 0 which characterizes the asymptotic approach towards the final state. 3D N D N2 2D N log N N d 2, d < 2, N log N, d = 2, N, d > 2.

20 Finite-size scaling of the dissipative gap This behavior appears to be universal the scaling exponents are independent of the underlying lattice structure (square vs. triangular) and of the chosen boundary conditions (free vs. periodic).

21 Quantifying multi-particle entanglement via correlation functions The target state is a macroscopically entangled BEC. We need to assess the efficacy of the proposed state-preparation protocol. What possibilities are there to quantify entanglement in many-particle systems? Different measures of entanglement typically require full information about the (reduced) density matrix a daunting task since the number of matrix elements grows exponentially with the size of the considered (sub)system. Can we find criteria for multi-particle entanglement that can be measured only by a small set of macroscopic operators?

22 Quantifying multi-particle entanglement via correlation functions The target state is a macroscopically entangled BEC. We need to assess the efficacy of the proposed state-preparation protocol. What possibilities are there to quantify entanglement in many-particle systems? Different measures of entanglement typically require full information about the (reduced) density matrix a daunting task since the number of matrix elements grows exponentially with the size of the considered (sub)system. Can we find criteria for multi-particle entanglement that can be measured only by a small set of macroscopic operators? Yes! Possible to relate a multi-particle entanglement measure (entanglement depth) to two-point (spin-spin)-correlations functions.

23 Lindblad dynamics in the S 3 = 0 sector At initial time we prepare the system in a fully mixed state of the following form: ρ(τ = 0) = ( ) N s + x N/2 s + x 2 s + x N/2 Ω Ω s x s x 2 s x N/2 x <x 2 <...<x N/2 N Ω Starting from this initial state, we run the Lindblad process which has the Dicke state N/2, 0 as its final state, i.e., ρ(τ ) = N/2, 0 N/2, 0 Note that the Lindblad process conserves S 3 and therefore: D p=0 = 0 ( S 3 ) 2 = 0... essential to derive stringent bounds on entanglement depth.

24 Multi-particle entanglement dynamics in S 3 = 0 sector To verify genuine M-particle entanglement at any given time in the time-evolved state, we need to check whether the following inequality is satisfied: C p=0 > (M + )/(2N). We can use this to define the cumulative entanglement distribution: E = max ( 2N Cp=0 2, ) C p=0 (τ ) = (N + 2)/(2N) E(τ = ) = N d = 2, N = 26 2 C p=0 (τ = 0) = /N E(τ = 0) =

25 Lindblad dynamics in the S 3 = N/2 + n sector Similar to the previous case, we consider an initial fully mixed state in the following form: ρ(τ = 0) = Asymptotically, the system is driven to the state ( ) N s + x n s + x 2 s + x n Ω Ω s x s x 2 s x n x <x 2 <...<x n N ρ(τ ) = N/2, N/2 + n N/2, N/2 + n In contrast to the previous example D p=0 0 and ( S 3 ) 2 0, which does not yield strong enough bounds on the entanglement. Other quantifiers needed...

26 Competing thermal bath Allow for local spin flips via additional jump processes that are accounted for by local operators L ± x = s ± x (introduce in addition an external magnetic field, i.e., H = h x s 3 x 0). Assuming a thermal occupation of the bath, the spin flip rates are related γ ± x are related via the Boltzmann factor exp ( 2h/T), i.e., γ + /κ = n T, γ /κ = n T +, n T ( e 2h/T ) T denotes the bath temperature, κ is the dissipative coupling, and we assume spatial homogeneity (γ ± x = γ ± ). The equations of motion for correlation functions receive additional contributions from the spin flip processes L ± x : d dτ S3 x = d dτ C xy = d dτ D xy = 4 xs 3 x κ γ (2n T + )S 3 x κ 2γ, ( ) x + y Cxy 4 2 δ ( ) κ x,y Cxy + 4D xy γ (2n T + )C xy, ( ) x + y Dxy δ ( ) 2κ x,y Dxy D xx γ (2n T + )D xy κ ( S 3 x + S 3 ) y 2γ while C xx = 4D xx =.

27 Competing thermal bath In the thermodynamic limit N, single spin flips eventually destroy any long-range order and therefore dominate for any finite value of γ/κ. Immediate implications for dynamic stability of dissipative state preparation protocols.

28 Competing (dissipative) order Different proposals exits for purely dissipative (quantum) phase transitions. Typically Lindblad process constructed in analogy with some Hamiltonian, e.g., XXZ Hamiltonian ( H/J = s + 2 x s y + s x s + ) y + ( λ)s 3 x s 3 y x,y Two competing classes of jump operators: Conserves m = N Conserves m s = N x s 3 x, i.e., x( ) x s 3 x, i.e., L () xy = Ψ xy () L (2) xy = Ψ + Ψ xy (2) L (3) xy = Ψ xy (3) L (4) xy = λ xy (4) L () xy = λ xy () L (6) xy = λ xy (6) L (7) xy = λ xy (7)

29 Simple illustration (N = 4 qubits): λ λ 0 ρ(t = 0) ρ(t)

30 Competing (dissipative) order In contrast to the limiting cases λ = 0,, we observe a unique fixed point ρ in the intermediate domain (dynamic semigroup is uniquely relaxing). Crossover at intermediate values of 0 < λ λ < between long-range ordered phase (λ = 0) and antiferromagnetic order (λ = ). Simple illustration (N = 4 qubits): Tr[ρ* log(ρ*)] Cp=0 0. Tr ρ * λ - λ 0.2 D p=0 Dp= λ - λ

31 Competing (dissipative) order No phase transition at any finite, nonvanishing value of λ: m(t ) = 0, m s (t ) = 0. d = 3, N = 60 3 λ λ At odds with mean-field and variational methods!

32 Competing (dissipative) order No phase transition at any finite, nonvanishing value of λ: m(t ) = 0, m s (t ) = 0. However, dynamics shows three distinct regimes distinguished by the characteristic time-scales associated to the asymptotic decay towards the steady-state, i.e., dissipative gap: Region I Region II N d 2, d < 2, N log N, d = 2, N, d > 2. λ λ N 2 d

33 Summary We have solved explicitly for the real-time dynamics of the state-preparation protocol leading to a macroscopically entangled BEC. The finite-size scaling of the dissipative gap is independent of the underlying lattice structure and boundary conditions (well understood analytically). In view of experimental realizations our findings indicate that thermal fluctuations are not too prohibitive for generating long-range order via engineered dissipation, at least for not too large systems. But the situation is hopeless for N. Closed hierarchies allow us to study the competition of dissipative dynamics that generate distinct types of order and benchmark numerical approaches to real-time dynamics. Open questions: Different spin representations, possibly leading to a wider class of possible stationary ensembles? Steady-state phase transitions driven purely by dissipation? Interesting playing ground for Monte Carlo methods...