Lindbladians with multiple steady states: theory and applications

Abstract

Lindbladians with multiple steady states: theory and applications Victor V. Albert 2017Markovian master equations, often called Liouvillians or Lindbladians, are used to describe decay and decoherence of a quantum system induced by that system's environment. While a natural environment is detrimental to fragile quantum properties, an engineered environment can drive the system toward exotic phases of matter or toward subspaces protected from noise. These cases often require the Lindbladian to have more than one steady state, and such Lindbladians are dissipative analogues of Hamiltonians with multiple ground states. This thesis studies Lindbladian extensions of topics commonplace in degenerate Hamiltonian systems, providing examples and historical context along the way.An important property of Lindbladians is their behavior in the limit of infinite time, and the first part of this work focuses on deriving a formula for the asymptotic projection -the map corresponding to infinite-time Lindbladian evolution. This formula is applied to determine the dependence of a system's steady state on its initial state, to determine the extent to which decay affects a system's linear or adiabatic response, and to determine geometrical structures (holonomy, curvature, and metric) associated with adiabatically deformed steady-state subspaces. Using the asymptotic projection to partition the physical system into a subspace free from nonunitary effects and that subspace's complement (and making a few other minor assumptions), a Dyson series is derived to all orders in an arbitrary perturbation. The terms in the Dyson series up to second order in the perturbation are shown to reproduce quantum Zeno dynamics and the effective operator formalism.

Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

“…From the proof of Proposition 2 of [124] and Theorem 4 (Eqs. (2.39-2.40)) of [142] we know that for a left eigenmode with purely imaginary eigenvalue A † L = −iωA † , we have, Theorem. Let L (s) generate a CPTP map and depend analytically on s with Taylor series…”

“…The proof of the necessity direction in Appx. C makes extensive use of the results and proofs from [124,141,142] regarding the asymptotic subspace of the Liovillian, As(H) and the projector, P , into the corresponding nondecaying part of the Hilbert space, H. Importantly, when there exists a full rank stationary state this projector must be the identity operator, P = I. Compared with Thm 1. there is now no ambiguity about the unitarity of A.…”

“…Proof. We will make use of [141], [124] and [142]. The asymptotic subspace of the Liouvillian As(H) is defined as a subspace of the space of linear operators B(H) such that all initial states ρ(0) in the long-time limit end up in ρ(t → ∞) ∈ As(H).…”

“…The asymptotic subspace of the Liouvillian As(H) is defined as a subspace of the space of linear operators B(H) such that all initial states ρ(0) in the long-time limit end up in ρ(t → ∞) ∈ As(H). The projector P (P 2 = P = P † ) to the corresponding non-decaying part of the Hilbert space H is uniquely defined [124,142] as, for all ρ(t → ∞) ∈ As(H),…”

Exact solutions of interacting dissipative systems via weak symmetries

“…To diagonalize L, we use the formalism of thirdquantization [30][31][32]; relevant details can be found in the SM [33]. One first introduces four new superoperators âL |ρ ≡ |âρ , âR |ρ ≡ |ρâ , â † L |ρ ≡ |â † ρ , and â † R |ρ ≡ |ρâ † which we will refer to as annihilation and creation superoperators.…”