Topological power pumping in quantum circuits

In this article, we develop a description of topological pumps as slow classical dynamical variables coupled by a quantum system. We discuss the cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps based frequency converters. This last case corresponds to a quantum topological coupling between classical modes described by action-angle variables on which we focus. We propose a realization of such a topological coupler based on a superconducting qutrit suitably driven by three modulated drives. A detailed experimental protocol allowing to measure the quantized topological power transfer between the different modes of a superconducting circuit is discussed.

Figure 1

We consider a general gapped and fast system (in blue) coupled to slow variables of the environment (in red). The energy separation of the fast quantum system is assumed to be much larger than that of the slow degrees of freedom, allowing a description of the latter by classical pairs of conjugate variables. While of different nature, we denote generically these variables by $q_{\alpha },p_{\alpha }$. The dynamics of each pair follows from a classical Hamiltonian ${\mathcal{H}}_{\alpha }$. The quantum system couples instantaneously to a single variable $q_{\alpha }$ of each degree of freedom of the classical environment.

Figure 2

(a) Schematic quantum Hall circuit where two LC branches, described by classical conjugate variables $\mathrm{\Phi }$ and $Q$, are connected through a quantum Hall sample. (b) Schematic Thouless pump driven by a phase ${\varphi }_1={\omega }_{1}t$ conjugate to a variable $n_1$, and coupled to an LC branch. Topological pumping gives rise to a current in the $LC$ circuit. (c) Topological mode coupler, or frequency converter, in which two classical modes described by ${\varphi }_1,n_1$ and ${\varphi }_2,n_2$ variables are coupled topologically through a quantum system.

Figure 3

(a) Schematic of the experimental setup. A fluxonium circuit is embedded in a host cavity. The transitions between the first three levels of the fluxonium are driven with a detuning ${\delta }_i$, an amplitude ${\mathrm{\Omega }}_i$, and a modulation frequency ${\omega }_i$ (blue, red, and green). The power of the outgoing signals are recorded with a power spectrum analyzer that provides the instantaneous photon flux of each frequency mode. (b) Spectrum of the driving tones. Each fluxonium transition is driven with two sidebands used to implement a topologically protected power transfer. (c) For decoherence rates smaller than the modulation frequencies ${\omega }_i$, the power of each sideband can be resolved. The quantized power transferred is expressed as a function of the difference of the spectral power $\mathrm{\Delta }{S}_i$ in the sidebands of each reflected driving tone according to Eq. (28).

Figure 4

(a) Schematic representations of a Haldane model on the Lieb lattice and its square Brillouin zone with the four inversion-invariant momenta. The sublattices 0, 1, and 2 respectively have on-site potential energies 0, ${\delta }_1$, and ${\delta }_3$. The regular and bold black lines represent the nearest-neighbor tight-binding amplitudes ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. The dashed black lines depict the next-nearest-neighbor coupling ${\mathrm{\Omega }}_3$. (b) Two allowed configurations of the band parities $p_{\nu }$ at the inversion-invariant momenta. A trivial insulator can be adiabatically connected to an atomic limit, which is necessarily characterized here by positive parity products ${\pi }_{\nu }=+1$. A band insulator exhibiting negative parity products ${\pi }_{\nu }=-1$ is therefore topological. Enumerating all the configurations of the parity eigenvalues allowed by the model parameters leads to the topological phase diagram in panel (c) (see Appendix pp4). (c) Phase diagram of the qutrit model. Each colored region corresponds to a set of band Chern numbers. On the boundaries the gap closes for at least one combination of phases ${\varphi }_1$ and ${\varphi }_2$.

Figure 5

Pumping by a qutrit initialized in its ground state. (a) Filling of the different modes $n_i$ as a function of time, for three different set of pump parameters corresponding to points A, B, and C in Fig. 6. The driving amplitudes are chosen all equals ${\mathrm{\Omega }}_{1,2,3}=\mathrm{\Omega }$. The filling rate are found to depend on the parameters of the pump. (b) The two different combinations of fillings display the topological power $\hslash {\omega }_1{\omega }_2{\mathcal{C}}^{\left(0\right)}/2\pi$ (grey-dashed lines), invariant on the region of stability of the phase diagram, where ${\mathcal{C}}^{\left(0\right)}=4$ is the Chern number of the ground state for all parameter sets A, B, C.

Figure 6

Topological transitions for bands 0 and 1 in the case of same drive amplitudes ${\mathrm{\Omega }}_{1,2,3}=\mathrm{\Omega }$. For each working point of the phase diagram, we fit the time evolution of the energy $E_1=\hslash {\omega }_1(n_1+n_3)$ by a line on a time $\mathrm{\Delta }t=8\mu \mathrm{s}$ to construct the reduced power rate $2\pi {\dot{E}}_1/\hslash {\omega }_1{\omega }_2$ (red dots) when the qutrit is initialized in band 0, panels [(b),(e),(h)], or in band 1, panels [(c),(f),(i)]. The average populations of the qutrit ${\overline{p}}_{\mu }=\frac{1}{\mathrm{\Delta }t}{\int }_0^{\mathrm{\Delta }t}{\left|\langle {\psi }_{\mu }\left(t\right)|\mathrm{\Psi }\left(t\right)\rangle \right|}^{2}dt$ are displayed to evaluate adiabaticity. [(a)–(c)] Transition line DE, with a transition between bands 1 and 2 at ${\delta }_1/\mathrm{\Omega }=-1$, and between bands 0 and 1 at ${\delta }_1/\mathrm{\Omega }=1$. [(d)–(f)] Transition line HI, with a transition between bands 0 and 1 at ${\delta }_1/\mathrm{\Omega }=-1$, and between bands 1 and 2 at ${\delta }_1/\mathrm{\Omega }=1$. [(g)–(i)] Transition line FG, with a transition between the three bands at ${\delta }_1/\mathrm{\Omega }=\pm \sqrt{2}/2$.

Figure 7

Nonadiabatic effect at longer time. The time evolution for the qutrit prepared in band 0 of the topological energy combination $\hslash {\omega }_1(n_1+n_3)$ and qutrit populations $p_{\mu }\left(t\right)$ in the instantaneous eigenstates are displayed for three working point A, B, and C in the phase diagram of Fig. 6. (a) Point A, case of resonance ${\delta }_1={\delta }_3=0$ where the evolution of the qutrit remains adiabatic during 200 periods so the pumping rate is stable. (b) Point B, limit case beyond which pumping is no longer quantized on Fig. 6, where the evolution of the qutrit is no longer adiabatic after approximately $8\mu \mathrm{s}$ so the pumping rate is no longer quantized. (c) Point C, other limit case beyond which pumping is no longer quantized on Fig. 6, where in this phase the Chern number of band 1 is opposite to band 0 so we see pumping in the other direction when band 1 is mainly populated. Downsampling of data has been applied for clarity of presentation.

Figure 8

Parity product of the bands. The central map represents the minimum value of the lower gap (${\mathrm{\Delta }}_{01}$) or upper gap (${\mathrm{\Delta }}_{12}$) over the Brillouin zone, as a function of the detunings ${\delta }_1$ and ${\delta }_3$. It is obtained from numerical diagonalization of the Bloch Hamiltonian (24). Energy is in units of ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2={\mathrm{\Omega }}_3=\mathrm{\Omega }$. The white areas correspond to values of the detuning for which a band gap closes. They are well described by the dashed (dotted) lines obtained analytically from the closing conditions of gap ${\mathrm{\Delta }}_{01}$ (${\mathrm{\Delta }}_{12}$) at the inversion-invariant momenta $\mathit{\Gamma }$, $\mathit{X}$, $\mathit{Y}$, and $\mathit{M}$ in the Brillouin zone. These degeneracy lines mark the transitions between topologically nonequivalent band insulators, where the parity product ${\pi }_{\nu }$ of certain bands changes signs. The inversion eigenvalues labeling the bands at $\mathit{\Gamma }$, $\mathit{X}$, $\mathit{Y}$, and $\mathit{M}$ are shown in the green boxed insets, as a parity triplet $\mathit{p}=(p_0,p_1,p_2)$ associated with the band energies $E_0<E_1<E_2$. It is only shown for insulating regions that exhibit negative band parity products. In such regions, the band structures do not support any band representation and cannot be adiabatically connected to an atomic limit. The fluxonium qutrit then simulates nontrivial Chern insulators.

Figure 9

Different terms in the variation of the energy $\hslash {\omega }_1(n_1+n_3)$, in the case of resonance ${\delta }_1={\delta }_3=0$, point A in Fig. 6. In blue: Time-integration of the term of variation of the energy of the qutrit $-{\omega }_1\frac{\partial E_{\nu }}{\partial {\varphi }_1}$. In orange: Time-integration of the term of fluctuation of the geometrical coupling $\hslash {\omega }_1{\omega }_2(F_{{\varphi }_1{\varphi }_2}^{\left(\nu \right)}-\frac{{\mathcal{C}}^{\left(\nu \right)}}{2\pi })$. In green: Topological energy transfer at constant rate $\hslash \frac{{\omega }_1{\omega }_2}{2\pi }{\mathcal{C}}^{\left(\nu \right)}$. The fluctuation of the energy of the qutrit is the predominant source of temporal fluctuation of the energy.