Topological Frequency Conversion in Strongly Driven Quantum Systems

When a physical system is subjected to a strong external multifrequency drive, its dynamics can be conveniently represented in the multidimensional Floquet lattice. The number of Floquet lattice dimensions equals the number of irrationally-related drive frequencies, and the evolution occurs in response to a built-in effective “electric” field, whose components are proportional to the corresponding drive frequencies. The mapping allows us to engineer and study temporal analogs of many real-space phenomena. Here, we focus on the specific example of a two-level system under a two-frequency drive that induces topologically nontrivial band structure in the 2D Floquet space. The observable consequence of such a construction is the quantized pumping of energy between the sources with frequencies ${\omega }_1$ and ${\omega }_2$. When the system is initialized into a Floquet band with the Chern number $C$, the pumping occurs at a rate $P_{12}=-P_{21}=(C/2\pi )\hslash {\omega }_1{\omega }_2$, an exact counterpart of the transverse current in a conventional topological insulator.

Popular Summary

A major goal of quantum condensed-matter physics is to control the electronic and atomic states of many particles. One of the most exciting emerging means for achieving such control is a periodic drive, such as a laser or time-varying magnetic field. A periodic drive effectively raises the number of dimensions of the system, which can lead to new phenomena. The role of this extra emergent dimension, however, remains little used. Interestingly, there is another example where extra dimensions emerge: Quasicrystals, which are aperiodic materials, can be thought of as projections of higher-dimensional periodic crystals onto lower dimensions. Most simply, a one-dimensional quasicrystal can be constructed by superimposing two periodic but mutually incommensurate potentials. We show a surprising consequence of combining these two methods for increasing the dimensionality of a system.

By subjecting a single spin-1/2 particle to two elliptically polarized periodic waves, we can realize a type of topological insulator known as the chiral Bernevig-Hughes-Zhang model, which usually resides in two spatial dimensions. Just as a quantized Hall conductance is the earmark of spatial topological phenomena, the signature of temporal topological phenomena arising from incommensurate drives is energy pumping. We show that by combining drives into a topological temporal texture, the system pumps energy between the driving fields, drawing energy from one and feeding it into the other. The pumping effect will occur for rational and irrational frequency combinations. The energy-pumping rate is topologically quantized (independent of drive amplitudes) and can be as large as one megawatt for a one-millimeter magnetic particle.

This general principle could be used, for instance, to convert photons between distinct photonic modes in optical cavities when coupled by a small magnetic particle.

Figure 1

We present a model that uses 2D topological insulator band structure as a guide to engineering the quantized energy pumping between different frequency modes. (a) A spin-$1/2$ particle driven by two sources with incommensurate frequencies. In the topological phase, the spin trajectory (green line) fully covers the Bloch sphere (blue). The intermode coupling induced by the spin dynamics pumps energy from one drive (red) to the other (blue) at a nearly quantized rate. The lower panel shows effective magnetic-field trajectories for (b) $m=0$ (gapless phase between two topological ones), (c) $m=1.8$ (just inside the topological phase), and (d) $m=2.2$ (just outside the topological phase). The mass $m$ is depicted as a purple arrow [static $B_z$ field; see Eq. (15) for the definition].

Figure 2

The Floquet harmonics in a two-drive system give rise to a two-dimensional space. Each spot represents the $n_1{\omega }_1+n_2{\omega }_2=\overrightarrow{n}·\overrightarrow{\omega }$ harmonic. If the frequencies are incommensurate, then the space is infinite. If the ratio ${\omega }_1/{\omega }_2$ is rational, however, the infinite lattice contracts to a cylinder (strip marked in red, with periodic boundary conditions across the strip). The periodic drives appear to exert a force in the $\overrightarrow{n}$ space along the $\overrightarrow{\omega }$ direction. The motion that arises because of Berry curvature is normal to the force and is indicated by the $d\overrightarrow{n}/dt$ arrow.

Figure 3

The Floquet zone for two drives. In analogy to a 2D band structure, we draw the Berry curvature (color plot with side legend) vs the two offset phases ${\phi }_1$ and ${\phi }_2$. The periodic drives are akin to a motion along this Floquet zone with $\overrightarrow{\phi }={\overrightarrow{\phi }}_0+\overrightarrow{\omega }t$. In the case of a rational frequency ratio, the system will explore a closed periodic path through the Floquet zone. Drawn here is an example of $3{\omega }_1=2{\omega }_2$. The pumping effect will be roughly the integral of the Berry curvature along the path times ${\omega }_1{\omega }_2$ [Eq. (22)].

Figure 4

The energy flow and fidelity for the model of Eq. (28) in the strong-drive regime, $\eta =2$, ${\omega }_1=0.1$, and ${\omega }_2=\gamma {\omega }_1\approx 0.1618$. (a) The total work done by drives 1 and 2 as a function of time for different $m$’s. Each pair of lines is displaced on the time axis for clarity. The time axis is given in cycles of drive 1, while the work is normalized by ${\omega }_2$. The quantized rate corresponds to slope 1. (b) Fidelity vs time of a state initialized in the instantaneous eigenstate of $\mathcal{H}(t=0)$ and measured against the eigenstate of $\mathcal{H}(t)$. All fidelities start at 1 at $t=0$ and are offset vertically for clarity. (c) The power pumped as a function of the parameter $m$ averaged up to time $t=2000$. The quantized pumping is marked by the gray line at $dE/dt={\omega }_1{\omega }_2/2\pi$. The initial phases for these plots are ${\varphi }_1=\pi /10$, ${\varphi }_2=0$. The strong-drive regime realizes the topological pumping prediction up to $m\sim 1.8$. Both fidelity and pumping deteriorate close to the phase transition because of the closing of the band gap.

Figure 5

Energy pumping between two frequency sources for varying $\eta$ and $m$. The parameters of the system for this plot are ${\omega }_1=0.1$ and ${\omega }_2=\gamma {\omega }_1$. The pumping rate was averaged over the time range $t<10000$. The theoretical (ideal) value for the chosen parameters is $|P_1-P_2|={\omega }_1{\omega }_2/\pi \approx 0.00515036$.

Figure 6

The energy flow in the strong-drive regime, $\eta =2$, for rational frequency ratio, ${\omega }_2=(3/2){\omega }_1$, with Floquet state initialization. (a) The total work done by drives 1 and 2 as a function of time for different $m$’s. Each pair of lines is displaced on the time axis for clarity. (b) The power pumped as a function of the parameter $m$ averaged up to time $t=2000$. The pumping in the commensurate case could be stronger than its incommensurate counterpart. Furthermore, pumping may persist in the nontopological regime since the system does not average the Berry curvature of the entire Floquet zone. In the case of the offset angles chosen, even when $m=2.2$, the system explores regions with net positive Berry curvature. The gray line is at $dE/dt={\omega }_1{\omega }_2/2\pi =0.002387$, signifying the pumping due to Berry curvature of the Chern number $C=1$ band. The initial phases for these plots are ${\varphi }_1=\pi /10,{\varphi }_2=0$.

Figure 7

Pumping profile for different offset phases for ${\omega }_2=(3/2){\omega }_1$. (a) Accumulated work for Floquet initial states. (b) Average power for $t<2000$. We take $m=1$, $\eta =1$, and ${\varphi }_1=n\pi /5$ with $n=0$, 1, 2, 3, 4. The Floquet zone path of this system for one offset phase combination is shown in Fig. 3. Floquet eigenstate initial conditions were used. For these parameters, there were no significant differences with instantaneous eigenstate initialization. Here, too, ${\omega }_1{\omega }_2/2\pi =0.002387$.

Figure 8

A comparison of the energy pumping rates vs initial phase of drive ${\omega }_1$, ${\varphi }_1$ (while holding ${\varphi }_2=0$), for rational drive frequency ratios with and without temporal disorder. The results shown for the disordered cases are averaged over several realizations, and the disorder applied is a frequency random walk with temporal correlations as described in Sec. 6 along with the additional parameters of the simulation. (a) The topological regime, with $m=1.4$. The disorder brings the average energy pumping rate back to the universal and initial-phase-independent value of ${\omega }_1·{\omega }_2/2\pi$. (b) The nontopological regime, with $m=2.6$. Although the rational frequency ratio allows for significant pumping well within the trivial driving phase, the effect lacks the topological protection, and disorder eliminates the pumping. The results in both plots are averaged over 20 realizations. The error bars represent the standard deviation of the average.

Figure 9

Convergence of Floquet eigenfunctions for ${\omega }_1/{\omega }_2=p_i/q_i\to r$, where $r$ is irrational [the golden mean, $\gamma =1.6180\dots$ (blue line) and $\sqrt{2}$ (orange line)]. Note that $|\mathrm{\Delta }\mathrm{\Psi }|$ is calculated relative to the best rational approximation used ($2584/1597$ for $\gamma$ and $3363/2378$ for $\sqrt{2}$); it is approximately linear in $|p/q-r|$. Parameters of the Hamiltonian are described in the text.

Figure 10

Comparison of evolution starting from instantaneous (solid lines) and Floquet (dotted lines) eigenstates. Rational frequencies in panels (a)–(c) are approximations of the golden mean $\gamma$. The Hamiltonian scale is $\eta =1$ and ${\omega }_1=0.1$, with ${\omega }_2$ stated on the plots. Panel (c) illustrates how the energy transfer for the rational drive depends on the relative phase, ${\varphi }_2-{\varphi }_1=0$, 1.5, 3, 4.5, starting from the respective instantaneous eigenstates. The variation is due to sampling of different closed lines in the phase Brillouin zone. Panel (d) illustrates how initializing into a Floquet eigenstate (computed for ${\omega }_2=144/89{\omega }_1$) enhances pumping at long times for the irrational drive.

Figure 11

Same as Fig. 10, but for $\eta =0.5$.

Figure 12

Same as Fig. 10, but for $\eta =0.2$.

Figure 13

Work (a) and fidelity (b) in the intermediate drive regime for $\eta =1$. We initialized the system with the instantaneous eigenstate of $\mathcal{H}(0)$, where the initial phases for these plots are ${\varphi }_1=\pi /10$, ${\varphi }_2=0$. (c) Power pumped averaged for $t<2000$ as a function of gap parameter $m$.

Figure 14

Work (a) and fidelity (b) in the intermediate drive regime for $\eta =0.5$. We initialized the system with the instantaneous eigenstate of $\mathcal{H}(0)$, where the initial phases for these plots are ${\varphi }_1=\pi /10$, ${\varphi }_2=0$. (c) Power pumped averaged for $t<2000$ as a function of gap parameter $m$.

Figure 15

Work and fidelity in the weak-drive regime. (a) Energy transfers for $\eta =0.1$. (b) Fidelities of the initial state, an instantaneous eigenstate of $\mathcal{H}(0)$, and the instantaneous eigenstates of $\mathcal{H}(t)$. The initial phases for these plots are ${\varphi }_1=\pi /10,{\varphi }_2=0$. (c) Power pump averaged for $t<2000$ as a function of gap parameter $m$.