Gauge freedom in observables and Landsberg's nonadiabatic geometric phase: Pumping spectroscopy of interacting open quantum systems

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.

Figure 1

(a) Global gauge transformation ${\widehat{X}}^r\to {\widehat{X}}_g^r={\widehat{X}}^r+g\mathbb{1}$ leaving both currents $I_{X^r}$ and transported observable $\mathrm{\Delta }{X}^r$ invariant. The gray scale bar is translated by fixed amount $g$ relative to the ungauged one ($g=0$, dashed outline). The needle corresponds to the quantum state that from the observable ${\widehat{X}}_g^r$ the measurement expectation value $\langle {\widehat{X}}_g^r\rangle =\langle {\widehat{X}}^r\rangle +g$ indicated. (b) Local gauge transformation ${\widehat{X}}^r\to {\widehat{X}}_g^r={\widehat{X}}^r+g\left[\mathbf{R}\right]\mathbb{1}$, changing the currents to $I_{X_g^r}$ but leaving the transported observable $\mathrm{\Delta }{X}^r$ invariant. Since the meter scale is gauged in a continuous way as function of the parameters $\mathbf{R}$, it returns to its original position every period: $g\left[\mathbf{R}\right(\mathcal{T}\left)\right]=g\left[\mathbf{R}\right(0\left)\right]$.

Figure 2

Fiber bundle space in which the pumping problem (68) and (69) is solved: the plane corresponds to the base space of driving parameters $\mathbf{R}$ containing the driving curve $\mathcal{C}$. The “vertical” space at each point $\mathbf{R}$ is formed by all adiabatic-response expectation values (66), ${\langle {\widehat{X}}_g^r\rangle }^{\text{a}}$, each coordinated by a real value of $g\left[\mathbf{R}\right]$ at that point $\mathbf{R}$.

Figure 3

In a fiber-bundle space, the notion of what is “horizontal” is completely undefined, in contrast to “vertical,” which is naturally the direction along gauge coordinate $g$, the fiber space. “Horizontal” cannot be defined as orthogonal to “vertical” since there is no physically motivated metric in this space. However, the function $A_g\left[\mathbf{R}\right]$ arising in the solution of the physical pumping problem of Eqs. (68) and (69) can be used to define what “horizontal” means at each point, thus defining a geometric “connection.” The nonintegrability of the connection $A_g$ leads to the discontinuity in the horizontal lift of the curve $\mathcal{C}$ into the bundle space. This is the differential-geometric significance of Landsberg's connection for pumping.

Figure 4

Physical meaning of differential geometric notions of Fig. 3 in the pumping problem. The linear space defined by the connection [Eq. (75)] locally determines a plane of “horizontal” vectors in Fig. 3. Pumping corresponds to “parallel transporting” such a vector thereby producing a “horizontal lift” of the closed parameter drive curve $\mathcal{C}$. Physically, this signifies that one calibrates the meter for the observable, drawn in same way as in Fig. 1, such that one maintains zero pumping current $I_{X_g^r}^{\text{a}}$ in this gauge. If there is pumping (the connection is nonintegrable), the lifted curve “breaks” as shown in Fig. 3. Physically, the “vertical” discontinuity, the holonomy, is minus the pumped observable, i.e., the cumulative calibration (purple) required to maintain zero current during one driving period. Along the horizontal lift, the scale (gray) is made to follow the pointer.

Figure 5

Relations between adiabatic-state evolution (ASE, Appendix pp7), adiabatic response (AR, Sec. 3), and full-counting-statistics (FCS, Sec. 5) approach discussed in the text.