We consider the process of pumping charge through an open quantum system, motivated by the example of a quantum dot with strong repulsive or attractive electron-electron interaction. Using the geometric formulation of adiabatic nonunitary evolution put forward by Sarandy and Lidar, we derive an encompassing approach to ideal charge measurements of time-dependently driven transport, that stays near the familiar approach to closed systems. Following Schaller, Kießlich and Brandes we explicitly account for a meter that registers the transported charge outside the system. The gauge freedom underlying geometric pumping effects in all moments of the transported charge emerges naturally as the calibration of this meter. Remarkably, we find that geometric and physical considerations cannot be applied independently as done in closed systems: physical recalibrations do not form a group due to constraints of positivity (Bochner's theorem). This complication goes unnoticed when considering only the average charge but it is relevant for understanding the origin of geometric effects in the higher moments of the charge-transport statistics. As an application we derive two prominent existing approaches to pumping, based on full-counting-statistics (FCS) and adiabatic-response (AR), respectively, from our approach in a transparent way. This allows us to reconcile all their apparent incompatibilities, in particular the puzzle how for the average charge the nonadiabatic stationary-state AR result can exactly agree with the adiabatic nonstationary FCS result. We relate this and other difficulties to a single characteristic of geometric approaches to open systems: the system-environment boundary can always be chosen to either include or exclude the ideal charge meter. This leads to a physically motivated relation between the mixed-state Berry phase and the entirely different geometric phase of Landsberg.
We consider the process of pumping charge through an open quantum system, motivated by the example of a quantum dot with strong repulsive or attractive electron-electron interaction. Using the geometric formulation of adiabatic nonunitary evolution put forward by Sarandy and Lidar, we derive an encompassing approach to ideal charge measurements of time-dependently driven transport, that stays near the familiar approach to closed systems. Following Schaller, Kießlich and Brandes we explicitly account for a meter that registers the transported charge outside the system. The gauge freedom underlying geometric pumping effects in all moments of the transported charge emerges naturally as the calibration of this meter. Remarkably, we find that geometric and physical considerations cannot be applied independently as done in closed systems: physical recalibrations do not form a group due to constraints of positivity (Bochner's theorem). This complication goes unnoticed when considering only the average charge but it is relevant for understanding the origin of geometric effects in the higher moments of the charge-transport statistics. As an application we derive two prominent existing approaches to pumping, based on full-counting-statistics (FCS) and adiabatic-response (AR), respectively, from our approach in a transparent way. This allows us to reconcile all their apparent incompatibilities, in particular the puzzle how for the average charge the nonadiabatic stationary-state AR result can exactly agree with the adiabatic nonstationary FCS result. We relate this and other difficulties to a single characteristic of geometric approaches to open systems: the system-environment boundary can always be chosen to either include or exclude the ideal charge meter. This leads to a physically motivated relation between the mixed-state Berry phase and the entirely different geometric phase of Landsberg.
We set up a general density-operator approach to geometric steady-state pumping through slowly driven open quantum systems. This approach applies to strongly interacting systems that are weakly coupled to multiple reservoirs at high temperature, illustrated by an Anderson quantum dot. Pumping gives rise to a nonadiabatic geometric phase that can be described by a framework originally developed for classical dissipative systems by Landsberg. This geometric phase is accumulated by the transported observable (charge, spin, energy) and not by the quantum state. It thus differs radically from the adiabatic Berry-Simon phase, even when generalizing it to mixed states, following Sarandy and Lidar. As a key feature, our geometric formulation of pumping stays close to a direct physical intuition (i) by tying gauge transformations to calibration of the meter registering the transported observable and (ii) by deriving a geometric connection from a driving-frequency expansion of the current. Furthermore, our approach provides a systematic and efficient way to compute the geometric pumping of various observables, including charge, spin, energy, and heat. These insights seem to be generalizable beyond the present paper's working assumptions (e.g., Born-Markov limit) to more general open-system evolutions involving memory and strong-coupling effects due to low-temperature reservoirs as well. Our geometric curvature formula reveals a general experimental scheme for performing geometric transport spectroscopy that enhances standard nonlinear spectroscopies based on measurements for static parameters. We indicate measurement strategies for separating the useful geometric pumping contribution to transport from nongeometric effects. A large part of the paper is devoted to an explicit comparison with the Sinitsyn-Nemenmann full-counting-statistics (FCS) approach to geometric pumping, restricting attention to the first moments of the pumped observable. Covering all key aspects, gauge freedom, pumping connection, curvature, and gap condition, we argue that our approach is physically more transparent and, importantly, simpler for practical calculations. In particular, this comparison allows us to clarify how in the FCS approach an ``adiabatic'' approximation leads to a manifestly nonadiabatic result involving a finite retardation time of the response to parameter driving.
Abstract The understanding of out-of-equilibrium fluctuation relations in small open quantum systems has been a focal point of research in recent years. In particular, for systems with adiabatic time-dependent driving, it was shown that the fluctuation relations known from stationary systems do no longer apply due the geometric nature of the pumping current response. However, the precise physical interpretation of the corrected pumping fluctuation relations as well as the role of many-body interactions remained unexplored. Here, we study quantum systems with many-body interactions subject to slow time-dependent driving, and show that fluctuation relations of the charge current can in general not be formulated without taking into account the total energy current put into the system through the pumping process. Moreover, we show that this correction due to the input energy is nonzero only when Coulomb-interactions are present. Thus, fluctuation response relations offer an until now unrevealed opportunity to probe many-body correlations in quantum systems. We demonstrate our general findings at the concrete example of a single-level quantum dot model, and propose a scheme to measure the interaction-induced discrepancies from the stationary case.
For an open quantum system, we investigate the pumped current induced by a slow modulation of control parameters on the basis of the quantum master equation and full counting statistics. We find that the average and the cumulant generating function of the pumped quantity are characterized by the geometrical Berry-phase-like quantities in the parameter space, which is associated with the generator of the master equation. From our formulation, we can discuss the geometrical pumping under the control of the chemical potentials and temperatures of reservoirs. We demonstrate the formulation by spinless electrons in coupled quantum dots. We show that the geometrical pumping is prohibited for the case of noninteracting electrons if we modulate only temperatures and chemical potentials of reservoirs, while the geometrical pumping occurs in the presence of an interaction between electrons.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
We develop a theory of adiabatic response for open systems gov-erned by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for resid-ual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissi-pative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to de-phasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect. Two frameworks that provide insight and understanding of transport coef-ficients are Kubo’s theory of linear response [1, 2] and the theory of adiabatic response [3, 4, 5]. Both have a quantum version and a classical version, agree when there is overlap, and endow the non-dissipative transport coefficients with the geometric meaning of the adiabatic (Berry’s) curvature [6]. A notable success of this approach to “geometrization of quantum re-sponse ” has been its application to the integer quantum Hall effect where 1 ar
We study charge and spin transport through an interacting quantum wire, caused by backscattering of electrons off an effective impurity potential with a periodic time dependence. The adiabatic regime of this pump for charge and spin is shown to depend on the presence of interactions in the wire. For the noninteracting case, the transported quantities in a period are calculated and found to be adiabatic (independent of the pumping frequency) for all frequencies $\ensuremath{\Omega}\ensuremath{\ll}{v}_{F}/\mathcal{l},$ where l is the range of the scattering potential and vF is the Fermi velocity; this result is along the lines of adiabatic pumping in mesoscopic systems. On the other hand, we show that for a wire with repulsive electron-electron interactions, the adiabatic regime is confined to $\ensuremath{\Elzxh}\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\Gamma}},$ where ${\ensuremath{\omega}}_{\ensuremath{\Gamma}}$ is an energy scale set by the backscattering potential. Using symmetry and scaling properties of the quantum wire Hamiltonian, we relate the charge Q and spin S transported through the wire in a period $2\ensuremath{\pi}/\ensuremath{\Omega}$ to an integral involving quasistatic backscattering conductances. We also show that the pumped charge Q (or the spin S) is quantized in the adiabatic limit if the conductance of the system is zero at the stable fixed point of the renormalization-group (RG) transformations. The quanta transported are given by the winding number of a complex coupling $\ensuremath{\Gamma}(\ensuremath{\Omega}t)$ as it traverses a closed path C enclosing the origin $\ensuremath{\Gamma}=0.$ No adiabatic charge or spin is transported when the RG fixed point corresponds to a perfectly conducting wire, or if the path C does not enclose the point $\ensuremath{\Gamma}=0.$ By contrast, for a RG marginal conductance---which is the case for noninteracting electrons---the charge transported in a cycle is not quantized, rather it is proportional to the area enclosed by the path or circuit C, in agreement with previous studies of noninteracting mesoscopic systems. Finite size, temperature, and frequency effects on the transported quantities follow from the relation between adiabatic transport coefficients (backscattering conductances) and the quasistatic conductance of the system.