Non-Equilibrium Steady States in Quantum Transport

• Non-Equilibrium Steady States (NESS) are defined as stationary statistical states exhibiting persistent currents despite continuous driving forces.
• The adiabatic partition-free method constructs NESS by slowly introducing a bias into a globally equilibrated quantum system, preventing transient excitations.
• The framework rigorously analyzes spectral decompositions and compares partition-free with partitioned approaches to clarify steady state behavior in quantum transport.

A non-equilibrium steady state (NESS) is a stationary statistical state of a driven system that, while possessing non-vanishing, persistent currents (such as electrical or particle fluxes), maintains time-independent expectation values for observables. In the context of quantum transport in mesoscopic systems, a rigorous construction of a NESS is achieved by starting from global equilibrium in a fully coupled (partition-free) system and adiabatically introducing a bias using time-dependent Hamiltonians and adiabatic wave operators. This approach elucidates both the mathematical and physical subtleties of NESS, establishes precise conditions for the existence of a time-independent steady state, and clarifies the relationship between “partitioned” and partition-free approaches, addressing foundational questions in open quantum system theory.

1. Adiabatic Partition-Free Framework: Setup and Motivation

In the partition-free approach, the system consists of a finite “sample” region coupled to a finite number of leads, with the entire system initially in global equilibrium at $t \to -\infty$, described by the Fermi-Dirac density matrix

$$P_\mathrm{eq}(H) = \frac{1}{1 + \exp\left[(H - \mu)/kT \right]}$$

where $H$ is the full system Hamiltonian.

An electrical bias is introduced not via an instantaneous quench but by slowly switching on a potential difference between the leads. The time-dependent Hamiltonian is constructed as

$K(x(nt)) = H + x(nt) V$

where $x(s)$ is a monotonic switching function with $x(-\infty)=0$ and $x(0)=1$, and $n > 0$ is a small adiabatic parameter. This adiabatic protocol prevents excitation of spurious transients and models experimental scenarios where system-reservoir couplings are always present and only the bias is varied in time.

2. Time Evolution and Propagators

The state evolves according to the quantum Liouville (von Neumann) equation: $i \frac{d}{dt}P_n(t) = [K(x(nt)), P_n(t)]$ with $s$-strong limit initial condition

$\lim_{t \to -\infty} P_n(t) = P_\mathrm{eq}(H)$

Time evolution is controlled via an intertwining family of unitary propagators $W_n(t)$: $$i \frac{d}{dt} W_n(t) = K(x(nt)) W_n(t), \quad W_n(0) = I$$ The density matrix solution is then

$$P_n(t) = W_n(t) P_n(0) W_n(t)^*, \quad P_n(0) = W_n(0)^* P_\mathrm{eq}(H) W_n(0)$$

This formalism permits detailed tracking of the system's trajectory as the bias is switched on, and provides the foundation for subsequent adiabatic analysis.

3. Adiabatic Wave Operators and Strong Limits

A central technical result is the use of adiabatic wave operators to “transfer” the initial equilibrium state to a steady state of the biased system: $$W_+^{(n)} = n\text{-}\lim_{t \to -\infty} e^{itK(1)} W_n(t)$$ where $K(1) = H + V$ is the “final” Hamiltonian after the full bias has been applied.

The existence and properties of these wave operators allow the authors to interchange limits and rigorously define the NESS in the strong operator topology as: $P_\mathrm{ad} = s\text{-}\lim_{n \to 0} P_n(t)$ for any $t$, with the limit independent of $t$.

The steady-state density matrix is decomposed as

$$P_\mathrm{ad} = \sum_{j=1}^N E_\mathrm{eq}(H) E_j(0) + E_\mathrm{eq}(H) E_\mathrm{ac}(K(1))$$

where $E_j(0)$ are (appropriately continued) eigenprojections forming the discrete spectrum of $H$ and $E_\mathrm{ac}(K(1))$ projects onto the absolutely continuous subspace of $K(1)$. The strong convergence of the adiabatic wave operators overcomes spectral subtleties due to the presence of both discrete and continuous parts in $K(1)$, and ensures correct separation of contributions to the final state.

4. Comparison with Partitioned Approaches (Jakšić–Pillet–Ruelle—JPR)

The JPR approach describes NESS by initially decoupling the leads and preparing each in separate equilibria (with, e.g., different chemical potentials), then coupling the entire system at $t=0$ (a quench protocol). The two approaches—partitioned JPR and adiabatic partition-free—yield NESS that are structurally different but closely related.

The adiabatic method starts from a unique, already coupled equilibrium and then drives the system out of equilibrium via slow bias, which is argued to be closer to experimental setups (as noted by Caroli et al., 1971). The present analysis demonstrates that, although the stationary states produced by the two methods are not strictly equivalent, the adiabatic construction furnishes a rigorous and physically appealing NESS. This settles, at least in part, the question of (non)equivalence raised by Caroli and provides operator-theoretic clarity on when and how the steady-state limit is sensitive to the path taken in parameter space.

5. Spectral Decomposition and Sector Analysis

The spectral structure of the final Hamiltonian $K(1)$ dictates the precise form of the NESS. Projections onto the pure point and absolutely continuous subspaces: $H = E_\mathrm{pp}(K)H \oplus E_\mathrm{ac}(K)H$ allow for independent analysis of discrete (bound) states and continuous (scattering) channels. Adiabatic wave operators in both sectors are handled with separate, technically detailed arguments (Propositions 2.7–2.13). In particular, norm convergence estimates (e.g., equation (1.21) in the paper) are essential to isolate discrete versus continuum contributions and interchange limiting procedures, guaranteeing both existence and uniqueness of the steady state.

6. Implications and Physical Significance

This rigorous adiabatic construction shows that a NESS can be realized by an infinitely slow quench of the bias in a coupled quantum system, and that the final steady state does not depend on the switching protocol details (as long as adiabatic conditions are maintained). The methodology precisely identifies the contributions from bound and scattering states and, importantly, justifies the interchangeability of strong and norm limits under suitable conditions on the spectrum.

The analysis provides clarity on the physical realism of the partition-free approach for mesoscopic conductors and simultaneously exposes the limitations of earlier partitioned ensemble methods in scenarios where system–lead coupling is never truly “off” in experiment. The structure of the adiabatic NESS thus obtained is essential in modern quantum transport for understanding the long-time behavior and current-carrying properties of open quantum devices, including the proper treatment of transients and multi-band spectral effects.

7. Mathematical Formulations

A selection of key mathematical relations underpinning the construction is provided below:

Mathematical Object	Definition/Expression	Key Role
Time-dependent Hamiltonian	$K(x(nt)) = H + x(nt)V$	Adiabatic switching protocol
Liouville equation	$i \frac{d}{dt}P_n(t) = [K(x(nt)), P_n(t)]$	Density matrix evolution
Unitary propagator	$i dW_n(t)/dt = K(x(nt)) W_n(t)$, $W_n(0) = I$	Time evolution operator
Adiabatic wave operator	$$W^{(n)}_+ = n\text{-}\lim_{t \to -\infty} e^{itK(1)} W_n(t)$$	Transfer of equilibrium to NESS
Strong limit for NESS	$P_\mathrm{ad} = s\text{-}\lim_{n \to 0} P_n(t)$	Steady-state density matrix
Spectral decomposition	$$P_\mathrm{ad} = \sum_j E_\mathrm{eq}(H)E_j(0) + E_\mathrm{eq}(H)E_\mathrm{ac}(K(1))$$	Discrete/continuous sectors in NESS

Careful spectral decomposition and control of adiabatic limits are essential to the entire construction.

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This mathematical and operator-theoretic framework for adiabatic NESS not only fully delineates the physical and mathematical mechanisms underlying the approach to a non-equilibrium steady state in partition-free quantum transport setups but also lays the groundwork for addressing more general questions of non-equilibrium quantum statistical mechanics and open-system dynamics. The analysis clarifies when different operational definitions of steady states coincide or differ, and precisely characterizes the circumstances under which non-equilibrium steady transport is compatible with underlying time-dependent perturbations.