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arXiv:1806.10422 · Physical Review A

Effective quantum Zeno dynamics in dissipative quantum systems

Vladislav Popkov, Simon Essink, Carlo Presilla A, Gunter Markus Schutz

Published Nov 7, 2018• 39 citations
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Abstract

We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom ${\mathcal{H}}_{0}$ of the system, and targeting a unique dark state in ${\mathcal{H}}_{0}$. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace ${\mathcal{H}}_{0}$, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.

DOI
10.1103/physreva.98.052110

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We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom H0{\mathcal{H}}_{0}H0​ of the system, and targeting a unique dark state in H0{\mathcal{H}}_{0}H0​. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace H0{\mathcal{H}}_{0}H0​, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.
We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZXXZXXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.
We consider an open quantum system with dissipation, described by a Lindblad Master equation (LME). For dissipation locally acting and sufficiently strong, a separation of the relaxation timescales occurs, which, in terms of the eigenvalues of the Liouvillian, implies a grouping of the latter in distinct vertical stripes in the complex plane at positions determined by the eigenvalues of the dissipator. We derive effective LME equations describing the modes within each stripe separately, and solve them perturbatively, obtaining for the full set of eigenvalues and eigenstates of the Liouvillian explicit expressions correct at order 1/Γ included, where Γ is the strength of the dissipation. As an example, we apply our general results to quantum XYZ spin chains coupled, at one boundary, to a dissipative bath of polarization.
We study composite open quantum systems with a finite-dimensional state space HA⊗HB{\mathcal H}_A\otimes {\mathcal H}_BHA​⊗HB​ governed by a Lindblad equation ρ′(t)=Lγρ(t)ρ'(t) = {\mathcal L}_γρ(t)ρ′(t)=Lγ​ρ(t) where Lγρ=−i[H,ρ]+γDρ{\mathcal L}_γρ= -i[H,ρ] + γ{\mathcal D} ρLγ​ρ=−i[H,ρ]+γDρ, and D{\mathcal D}D is a dissipator DA⊗I{\mathcal D}_A\otimes IDA​⊗I acting non-trivially only on part AAA of the system, which can be thought of as the boundary, and γγγ is a parameter. It is known that the dynamics simplifies for large γγγ: after a time of order γ−1γ^{-1}γ−1, ρ(t)ρ(t)ρ(t) is well approximated for times small compared to γ2γ^2γ2 by πA⊗R(t)π_A\otimes R(t)πA​⊗R(t) where πAπ_AπA​ is a steady state of DA{\mathcal D}_ADA​, and R(t)R(t)R(t) is a solution of ddtR(t)=LP,γR(t)\frac{\rm d}{{\rm d}t}R(t) = {\mathcal L}_{P,γ}R(t)dtd​R(t)=LP,γ​R(t) where LP,γR:=−i[HP,R]+γ−1DPR{\mathcal L}_{P,γ} R := -i[H_P,R] + γ^{-1} {\mathcal D}_P RLP,γ​R:=−i[HP​,R]+γ−1DP​R with HPH_PHP​ being a Hamiltonian on HB{\mathcal H}_BHB​ and DP{\mathcal D}_PDP​ being a Lindblad generator over HB{\mathcal H}_BHB​. We prove this assuming only that DA{\mathcal D}_ADA​ is ergodic and gapped. In order to better control the long time behavior, and study the steady states ρˉγ\barρ_γρˉ​γ​, we introduce a third Lindblad generator DP♯{\mathcal D}_P^\sharpDP♯​ that does not involve γγγ, but still closely related to Lγ{\mathcal L}_γLγ​. We show that if DP♯{\mathcal D}_P^\sharpDP♯​ is ergodic and gapped, then so is Lγ{\mathcal L}_γLγ​ for all large γγγ, and if ρˉγ\barρ_γρˉ​γ​ denotes the unique steady state for Lγ{\mathcal L}_γLγ​, then lim⁡γ→∞ρˉγ=πA⊗Rˉ\lim_{γ\to\infty}\barρ_γ= π_A\otimes \bar Rlimγ→∞​ρˉ​γ​=πA​⊗Rˉ where Rˉ\bar RRˉ is the unique steady state for DP♯{\mathcal D}_P^\sharpDP♯​. We show that there is a convergent expansion ρˉγ=πA⊗Rˉ+γ−1∑k=0∞γ−knˉk\barρ_γ= π_A\otimes\bar R +γ^{-1} \sum_{k=0}^\infty γ^{-k} \bar n_kρˉ​γ​=πA​⊗Rˉ+γ−1∑k=0∞​γ−knˉk​ where, defining nˉ−1:=πA⊗Rˉ\bar n_{-1} := π_A\otimes\bar Rnˉ−1​:=πA​⊗Rˉ, Dnˉk=−i[H,nˉk−1]{\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]Dnˉk​=−i[H,nˉk−1​] for all k≥0k\geq 0k≥0.
We propose a solution to the problem of realizing a predefined and arbitrary pure quantum state, based on the simultaneous presence of coherent and dissipative dynamics, noncommuting on the target state and in the limit of strong dissipation. More precisely, we obtain a necessary and sufficient criterion whereby the nonequilibrium steady state (NESS) of an open quantum system described by a Lindblad master equation approaches a target pure state in the Zeno regime, i.e., for infinitely large dissipative coupling. We also provide an explicit formula for the characteristic dissipative strength beyond which the purity of the NESS becomes effective, thus paving the way to an experimental implementation of our criterion. For an illustration, we deal with targeting a Bell state, an arbitrary pure state of NNN qubits, and a spin-helix state of NNN qubits.
Dissipative preparation of a pure steady state usually involves a commutative action of a coherent and a dissipative dynamics on the target state. Namely, the target pure state is an eigenstate of both the coherent and dissipative parts of the dynamics. We show that working in the Zeno regime, i.e., for infinitely large dissipative coupling, one can generate a pure state by a noncommutative action, in the above sense, of the coherent and dissipative dynamics. A corresponding Zeno regime pureness criterion is derived. We illustrate the approach, looking at both its theoretical and applicative aspects, in the example case of an open XXZXXZXXZ spin-1/21/21/2 chain, driven out of equilibrium by boundary reservoirs targeting different spin orientations. Using our criterion, we find two families of pure nonequilibrium steady states, in the Zeno regime, and calculate the dissipative strengths effectively needed to generate steady states which are almost indistinguishable from the target pure states.
We study the critical behavior of the nonequilibrium dynamics and of the steady states emerging from the competition between coherent and dissipative dynamics close to quantum phase transitions. The latter is induced by the coupling of the system with a Markovian bath, such that the evolution of the system's density matrix can be effectively described by a Lindblad master equation. We devise general scaling behaviors for the out-of-equilibrium evolution and the stationary states emerging in the large-time limit for generic initial conditions in terms of the parameters of the Hamiltonian providing the coherent driving and those associated with the dissipative interactions with the environment. Our framework is supported by numerical results for the dynamics of a one-dimensional lattice fermion gas undergoing a quantum Ising transition in the presence of dissipative mechanisms which include local pumping and decay of particles.