Abstract
We consider the exact time-evolution of a broad class of fermionic open quantum systems with both strong interactions and strong coupling to wide-band reservoirs. We present a nontrivial fermionic duality relation between the evolution of states (Schrödinger) and of observables (Heisenberg). We show how this highly nonintuitive relation can be understood and exploited in analytical calculations within all canonical approaches to quantum dynamics, covering Kraus measurement operators, the Choi-Jamiołkowski state, time-convolution and convolutionless quantum master equations and generalized Lindblad jump operators. We discuss the insights this offers into the divisibility and causal structure of the dynamics and the application to nonperturbative Markov approximations and their initial-slip corrections. Our results underscore that predictions for fermionic models are already fixed by fundamental principles to a much greater extent than previously thought.
Highlights
We extend the scope to the much broader class of models of the form Htot = H + HR + HT where only the following assumptions are made: (I) The multiple fermionic reservoirs described by HR are noninteracting with structureless, infinitely wide bands, each one being separately in equilibrium at the initial time. (II) The coupling to the fermions in the system is bilinear in the field operators, HT = rl dω trl dl†crω + h.c., and independent of the energy ω of the fermionic modes in the reservoirs. (III) The system Hamiltonian H obeys parity superselection, [H, (−1)N ] = 0, and as a result so does the total system
We have shown that this works for essentially all canonical approaches used in quantum transport, open-system dynamics and quantum information theory without introducing any assumptions except for wide-band coupling to the reservoirs
In the operational approaches we derived additional fermionic sum rules for measurement and jump operators [Eqs. (40), (59)] and their scalars coefficients [Eqs. (41), (42), (60)], and nontrivial cross-relations for these entire sets of operators [Eqs. (39), (61)]. Combining the latter approaches with fermionic duality naturally led us to consider a new type of divisibility of the dynamics: We noted that the Schrödinger and Heisenberg time-local generators through their quantum-jump coefficients encode both ordinary causal and anti-causal divisibility, respectively
Summary
The dynamics of open quantum systems is a problem of interest in a range of research fields. In the weak coupling limit close inspection reveals [10] that one can use fermionic duality to set up a relation between two dual physical systems which both have nonnegative decay constants but otherwise different physical parameters. We will show that the anti-Hermitian coupling Hamiltonian causes the reduced dynamics Π (t) to violate complete positivity, giving a clear operational meaning to the vague notion of an “unphysical” system This is important since it will allow us to identify which contributions to the evolution of the dual system are unavoidably unphysical, a question that cannot be answered directly using the original derivation of the duality in Ref. This provides the simplest yet nontrivial illustration of the general results derived in the subsequent sections.
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