Non-abelian Geometric Quantum Energy Pump

TL;DR

The paper introduces a non-abelian geometric quantum energy pump realized via a transitionless geometric quantum drive, confining dynamics to a degenerate subspace and employing a Kato gauge potential $\mathcal{A}_t$ to counterdiabatically guide evolution on a smooth control manifold $\boldsymbol{\varphi}$. The pumped energy into drive $\mu$ is given by $E_\mu(\boldsymbol{\varphi}_t)=\int_0^t ds\, \dot{\varphi}_s^\nu \dot{\varphi}_s^\mu \langle \psi(s) | \mathcal{F}_{\nu\mu}^n(\boldsymbol{\varphi}_s) | \psi(s) \rangle$, where $\mathcal{F}_{\mu\nu}^n=i\Pi_{\boldsymbol{\varphi}}[\partial_\mu \Pi_{\boldsymbol{\varphi}},\partial_\nu \Pi_{\boldsymbol{\varphi}}]\Pi_{\boldsymbol{\varphi}}$ is the non-abelian Berry curvature in the $n$th subspace. Averaging over initial phases yields a phase-coherent pumping power controlled by the Euler class $\chi_{\nu\mu}=\int d\varphi^\nu d\varphi^\mu\, \mathrm{Eu}_{\nu\mu}/(2\pi)$ with $\mathrm{Eu}_{\nu\mu}=-iF^{12}_{\nu\mu}$, and the initial-state amplitudes and relative phase $\delta\phi$ set the amplitude via $\sin(\delta\phi)$. A concrete tripod artificial-atom realization is analyzed, with the KGP in the dark-state subspace given by $\mathcal{A}_t = \frac{i}{\Omega^{2}}\sum_{jk}(\dot{\Omega}_{j}\Omega_{k}-\dot{\Omega}_{k}\Omega_{j})|g_j\rangle\langle g_k|$, and prospects for quantum transduction, charging, and metrological sensing of phase coherence are discussed. The work demonstrates tunable, topologically governed energy transfer that can operate beyond adiabatic limits and be implemented across platforms such as trapped atoms, superconducting circuits, and semiconductor quantum dots.

Abstract

We introduce a non-abelian geometric quantum energy pump realized by a transitionless geometric quantum drive--a time-dependent Hamiltonian supplemented by a counterdiabatic term generated by a prescribed trajectory on a smooth control manifold--that coherently transports states within a degenerate subspace. When the coordinates of the trajectory are independently addressable by external drives, the net energy transferred between drives is set by the non-abelian Berry-curvature tensor. The trajectory-averaged pumping power is separately controlled by the initial state and by the Hamiltonian topology through the Euler class. We outline an implementation with artificial atoms, which are realizable on various platforms including trapped atoms/ions, superconducting circuits, and semiconductor quantum dots. The resulting energy pump can serve as a quantum transducer or charger, and as a metrological tool for measuring phase coherences in quantum states.

Figures (2)

• Figure 1: (a) Transitionless geometric quantum drive: a trajectory (green dashed line) $\boldsymbol{\varphi}_t$ starting from $\boldsymbol{\varphi}_0$ on a smooth manifold $M$, and a time-dependent Hamiltonian $H_0(\boldsymbol{\varphi}_t) + \mathcal{A}_t(\boldsymbol{\varphi}_t)$ with $\mathcal{A}_t$ the Kato gauge potential of $H_0$. $\mathcal{C}_t$ is a closed loop in the counterclockwise direction starting at $\boldsymbol{\varphi}_0$ as indicated by the orange dashed line. $\boldsymbol{\delta \varphi}^{\mu}$ is a vector, which is taken to be infinitesimally short, along the $\mu$th component of the local coordinate $\boldsymbol{\varphi}$. (b) An artificial atom is controlled by $H_0(\boldsymbol{\varphi}_t) + \mathcal{A}_t(\boldsymbol{\varphi}_t)$, where we assumed $\boldsymbol{\varphi}_t = (\varphi_t^1, \varphi_t^2)$, in which both components can be independently controlled by different protocols, for $H_0$ and $\mathcal{A}_t$, as indicated by blue and red sprial arrows. (c) Tripod system is a concrete realization of the setting in (b). $H_0$ consists of three degenerate levels ${\ket{g_{1,2,3}}}$ which are coupled (blue arrows) to the excited state $\ket{e}$ at detuning $\Delta$ with, $\boldsymbol{\varphi}_t$-dependent coupling strength $\Omega_{1,2,3}$. At large detuning, $\mathcal{A}_t$ consists couplings between states in the degenerate subspace, as indicated by the red dashed lines.
• Figure 2: (a) Pumped energy $E_2(\boldsymbol{\varphi}_t)$ into drive 2 of the two-tone drive example. Five trajectories at different random initial phases are shown in colors. The black dashed line is phase-averaged result based on Eq. (\ref{['eq:central']}). The parameters used are $m = 0.5, c = 1/\sqrt{2}, \delta\phi = \pi/2, p/q = 3/2$. (b) Pumped energy $\overline{E}_2(t)$ into drive 2 averaged over 400 random initial phases sampled uniformly from $0$ to $2\pi$. The ratio is fixed at $p/q=3/2$. The different sets of other parameters that control the energy pumping power are indicated in different color. The black dashed line is the analytically result. (c,d) Standard deviation $\sigma_{E_2}$ of $E_2$ at different $p/q$ ratios for the 400 trajectories from different initializations, plotted with different time scale. We fix $c=1/\sqrt{2}$, $\delta\phi = \pi/2$, $m = 0.5$. The common parameters in all figures are $\omega = 0.4$, $\Delta = 1$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof