Exact quantum transport in non-Markovian open Gaussian systems

Abstract

We build an exact framework to evaluate heat, energy, and particle transport between Gaussian reservoirs mediated by a quadratic quantum system. By combining full counting statistics with newly developed non-Markovian master equation approaches, we introduce an effective master equation whose solution can be used to generate arbitrary moments of the heat statistics for any number of reservoirs. This theory applies equally to fermionic and bosonic systems, holds at arbitrarily strong coupling, and resolves out-of-equilibrium transient dynamics determined by the system's initial state. In the steady-state, weak-coupling limit, we recover results analogous to those of the well-known Landauer-Büttiker formalism. We conclude our discussion by demonstrating an application of the method to a prototypical fermionic system. Our results uncover a regime of transient negative heat conductance contingent upon the initial system preparation, providing a clear signature of non-trivial out-of-equilibrium dynamics.

Figures (5)

• Figure 1: Graphical representation of a quantum system described by a quadratic Hamiltonian ${\cal H}_{S}$ interacting with arbitrary modes of multiple reservoirs ${\cal H}_{\alpha}$ via potentials ${\cal V}_{\alpha}$. Each reservoir is initialized at thermal and chemical equilibrium at inverse temperature and chemical potential $\beta_\alpha,\, \mu_\alpha$. The unitary evolution of the system-environment compound is generated by the total Hamiltonian ${\cal H}_{}$.
• Figure 2: Graphical representation of the Keldysh contour on the complex time plane.
• Figure 3: A summary of the results presented in Secs. \ref{['sec:II']} to \ref{['sec:IV']}. Star-marked boxes contain equations introduced in the present work. The equations valid in the weak-coupling regime (left) are set apart from the exact results (right), which are valid for arbitrary system-environment coupling strength. The exact dynamics of the system is captured by the GME introduced in Ref. d2025exact, characterized by the dressed GF ${\cal G}$, that can be obtained by “dressing" the contour GF [Eq. \ref{['eq:bare-GF']}] via the Dyson equation \ref{['eq:bare-dyson']}. In the weak-coupling limit, we recover the standard “memoryless" Redfield equation breuer2007bookdeVega2017review. The tilting procedure, which maps ${\cal C} \to \tilde{{\cal C}}$ and ${\cal G} \to \tilde{{\cal G}}$, allows us to study the fluctuating thermodynamics of the system-environment compound via a similar machinery Espositoreview. Indeed, the MGF \ref{['eq:MGF']}, which time evolution is described by $\tilde{{\cal G}}$ via the tGME \ref{['eq:tGME']}, contains all the information on the heat statistics of the compound. Finally, by evaluating the first-order moments of such a distribution, we obtain exact transport equations governed by heat kernels ${\mathfrak g}_\alpha$, which are in turn obtained from the “memoryless" kernels ${\mathfrak c}_\alpha$ by dressing them according to the Dyson form \ref{['eq:goth-g-dyson-contour']}.
• Figure 4: Schematic representation of the model analyzed in Sec. \ref{['sec:VI']}, showing the current flow within the system-environment compound.
• Figure 5: (a)-(d) Exact (red dots) and weak-coupling (black squares) current flow $I_Q^{(L\to R)}$ [Eq. \ref{['eq:LtoR-current']}] for different values of the coupling constant $\eta$ and different system initialization (depicted in the inset). The system parameters are $\varepsilon_1 = 0.5$, $\varepsilon_2=1.0$, $\Delta = 0.7$. The reservoirs are initialized at inverse temperatures $\beta_L = 1$ and $\beta_R = 10$. Other numerical parameters are specified in App. \ref{['app:G']}. When the system is initialized in $|1\rangle_1 \otimes |0\rangle_2$ [(b) and (d)] we observe transient anomalous heat flow from the colder reservoir to the hotter one. In the strong-coupling regime [(c) and (d)] the w.c. approximation fails to capture the full current dynamics. (f)-(g) Pictorial representation of the phenomenology behind the emergence of negative heat conductance at small $t$: when site $1$ is full and site $2$ is empty [(g)], Pauli exclusion prevents heat exchange between the hotter reservoir and the system, and the latter is forced to absorb heat from the colder reservoir, thus resulting in net heat flow against the thermal bias.

Theorems & Definitions (18)

• proof : Proof of Eq. \ref{['eq:MGF-exp']}
• proof : Proof of Eq. \ref{['eq:heat-from-MGF']}
• Lemma A.1
• proof : Proof of Lemma \ref{['lemma:1']}
• Definition A.1: effective conjugate
• Corollary A.1
• proof : Proof of Corollary \ref{['cor:1']}
• proof : Proof of Eqs. \ref{['eq:C-tord']} to \ref{['eq:C-<']}
• Theorem B.1: Wick
• Theorem B.2: Fundamental theorem of calculus on the Keldysh contour