Abstract
We explore the possibility of dynamical quantum phase transitions (DQPTs) occurring during the temporal evolution of a quenched transverse field Ising chain coupled to a particle loss type of bath (local in Jordan-Wigner fermion space) using two versions of the Loschmidt overlap (LO), namely, the fidelity induced LO and the interferometric phase induced LO. The bath, on the one hand, dictates the dissipative evolution following a sudden quench and on the other, plays a role in dissipative mixed state preparation in the later part of the study. During a dissipative evolution following a sudden quench, no trace of DQPTs are revealed in both the fidelity and the interferometric phase approaches; however, remarkably the interferometric phase approach reveals the possibility of inter-steady state DQPTs in passage from one steady state to the other when the system is subjected to a quench after having reached the first steady state. We further probe the occurrences of DQPTs when the system evolves unitarily after being prepared in a mixed state of engineered purity by ramping the transverse field in a linear fashion in the presence of the bath. In this case though the fidelity approach fails to indicate any DQPT, the interferometric approach indeed unravels the possibility of occurrence of DQPTs which persists even up to a considerable loss of purity of the engineered initial state as long as a constraint relation involving the dissipative coupling and ramping time (rate) is satisfied. This constraint relation also marks the boundary between two dynamically inequivalent phases; in one the LO vanishes for the critical momentum mode (and hence DQPTs exist) while in the other no such critical mode can exist and hence the LO never vanishes.
Highlights
The study of dynamics of quantum many-body systems driven out of equilibrium is a frontier area of recent research both from the experimental as well as the theoretical viewpoints1–27. (For review articles, we refer to28–33)
Let us recall that a metric space can be defined as an ordered pair (S, d), where S is a set and d is a metric on S; one defines a notion of distance function given by d: S × S →, such that for any x, y, z ∈ S, the conditions listed below hold true: d(x, y) ≥ 0 non‐negativity; (1a) d(x, y) = 1 ⇒ x = y identity of indiscernibles d(x, y) = d(y, x) symmetry; (1c) d(x, z) ≤ d(x, y) + d(y, z) triangle inequality
(We note that in Eq (1b), one may define the identity as d(x, y) = 0 iff x = y.) In the subsequent discussions, dynamical quantum phase transitions (DQPTs) are studied through the zeros of the Loschmidt overlap (LO) constructed out of two different metrics, the density matrix fidelity and the interferometric distance
Summary
The study of dynamics of quantum many-body systems driven out of equilibrium is a frontier area of recent research both from the experimental as well as the theoretical viewpoints1–27. (For review articles, we refer to). In a DQPT, non-analyticities manifest in the subsequent real-time dynamics of a quantum many-body system generated by the time-independent final Hamiltonian following a sudden or a slow ramping of one of the parameters of the Hamiltonian. An analogy can be drawn between DQPTs and equilibrium classical phase transitions by analysing the lines of Fisher zeros in the one-dimensional situation, (see also36,37) in the complex time plane These non-analyticities have been detected experimentally in a string of ions simulating interacting transverse field Ising models. We establish that remarkably, after a second quench starting from the first steady state the system approaches another steady state with possible occurrences of DQPTs in the inter-steady state dynamics These DQPTs are manifested only in the interferometric phase induced LO which preserves the Bloch sphere structure unlike the fidelity induced LO. The occurrence of this inter-steady state DQPT is unique to our study and to the best of our knowledge has not been reported before
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