Multi-Directional Periodic Driving of a Two-Level System beyond Floquet Formalism

TL;DR

The study tackles the exact dynamics of a periodically driven two-level system beyond Floquet truncation. It develops an analytical framework using the $\star$-resolvent formalism and path-sum to derive a compact two-time kernel that encapsulates arbitrary periodic driving, which is then expanded in a non-harmonic Fourier basis with coefficients given by products of generalized Bessel functions. The authors obtain explicit, exact unitary evolution $U(t,s)$ and the transition probability $p(t,s)=|U_{12}(t,s)|^2$, and they analyze the kernel structure, including resonance decompositions into RWA, CR, and OR contributions, as well as the Floquet and effective Hamiltonians $H_F$ and $H_{\mathrm{eff}}$. They also discuss perturbative regimes, showing how the RWA limit emerges and how GBFs determine the generalized Rabi frequencies, with applications to quantum sensing and control. Overall, the method provides a non-perturbative, analytical route to understand and optimize driven two-level systems under complex waveforms, with potential impact on waveform design and parameter estimation in quantum technologies.

Abstract

In this manuscript, we introduce an exact expression for the response of a semi-classical two-level quantum system subject to arbitrary periodic driving. Determining the transition probabilities of a two-level system driven by an arbitrary periodic waveform necessitates numerical calculations through methods such as Floquet theory, requiring the truncation of an infinite matrix. However, such truncation can lead to a loss of significant interference information, hindering quantum sensors or introducing artifacts in quantum control. To alleviate this issue, we use the $\star$-resolvent formalism with the path-sum theorem to determine the exact series solution to Schrödinger's equation, therefore providing the exact transition probability. The resulting series solution is generated from a compact kernel expression containing all of the information of the periodic drive and then expanded in a non-harmonic Fourier series basis given by the divided difference of complex exponentials with coefficients corresponding to products of generalized Bessel functions. The present method provides an analytical formulation for quantum sensors and control applications.

Figures (4)

• Figure 1: Two-Time Kernel: Plots of the real part of the kernel in the two-time domain given by $[0,T] \cup [0,T]$ for period $T=2\pi/\omega$ where the abscissa is the first time variable and the ordinate is the second time variable. The waveforms used are (a) no periodic driving, (b) periodic driving corresponding to the usual Bloch-Siegert Hamiltonian with a tunneling strength of $\Delta = 0.5\omega$, (c) periodic driving corresponding to the usual single harmonically driven two-level system in the longitudinal direction with a tunneling strength of $\Delta = 0.5\omega$ and amplitude of $A = 15\omega$, and (d) periodic driving corresponding to a longitudinal modulation with the first and second harmonics with amplitudes $A_1 = 10\omega$ and $A_2 = 20\omega$, respectively, a constant energy bias $\epsilon_0 = 10\omega$, and transverse modulations given by the first and second harmonics with amplitudes $\Delta_i = 0.25\omega$. As expected, the kernel with no periodic driving suggests that time invariance is not broken, as expected with the information contained in the kernel reducible to a single dimension. However, as soon as periodic driving is introduced, time-invariance is broken and the information contained in the kernel is fully two-dimensional displaying a feature of non-autonomous systems.
• Figure 2: Exact Transition Probability: Plots of the real part of the two-time kernel, two-time transition probability and the transition probability when $s=0$ for two differing waveforms in the interval $[0,T] \cup [0,T]$ for period $T=2\pi/\omega$. In (a)-(c), the typical Bloch-Siegert Hamiltonian was used with energy bias $\epsilon_0 = \omega$ and amplitudes $\Delta_1 = 3\omega$. In (d)-(f), the waveform used is a longitudinal modulation given by the first and third harmonics with amplitudes $A_1 = 13\omega$ and $A_2 = 18\omega$ with $\epsilon_0 = \omega$ and $\Delta = 1.5\omega$. Lastly, (g)-(i) display the results for a longitudinal driving given by the first harmonic with amplitude $A_1 = 13\omega$ and transverse driving given by the first, second, and third harmonics with amplitudes $\Delta_i = 1.5\omega$ and constant energy bias and tunneling given by $\epsilon_0 = \omega$ and $\Delta = 1.5\omega$, respectively. With more complex modulating waveforms, the two-time kernel and the two-time probability display rich interference patterns with specific features reflected in the single time variable probability (that is when $s=0$).
• Figure 3: Generalized Rabi Frequencies: Plots of the generalized Rabi frequencies $|\mathcal{J}_l^\Delta (A_1, A_2)|$ for $-40\omega<A_1,A_2<40\omega$ for various waveforms. The waveforms used are (a) first and second harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx \omega$ , (b) first and second harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx 10\omega$, (c) first and second harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx 10 \omega$ with transverse modulation given by the first and twentieth harmonics, (d) the third and eleventh harmonic in the diagonal with a DC bias strength of $\epsilon_0 \approx \omega$, (e) the third and eleventh harmonic in the diagonal with a DC bias strength of $\epsilon_0 \approx 10\omega$, and (f) the third and eleventh harmonic in the diagonal with a DC bias strength of $\epsilon_0 \approx 10 \omega$ with transverse modulation given by the first and twentieth harmonics. Note here for (a) the Rabi frequencies pick up the GBF structure in the plots with a radially divergent structure as seen in satanin2014amplitudekuklinski2019identities reproduced using our method here. Plot (b) maintains this divergent structure with an ever increasing region of exponential decay associated with higher order GBFs. Plots (d) and (e) seem to remove this divergent structure and appear to be similar in structure, however, if one were to expand the amplitudes to larger magnitudes, the similarity in the structure would vanish and would result in dynamics clearly separated similar to (a) and (b). Lastly, when the longitudinal modulation is turned on, the Rabi frequencies change drastically with overlapping constructive/destructive interference patterns.
• Figure 4: Perturbative Transition Probability from Truncated Kernel: Analytical (left column) and numerical (middle column) plots of the time-averaged upper level occupation probabilities $p(A_1, A_2)$ for $-40\omega<A_1,A_2<40\omega$ with the corresponding time-dependent probabilities displayed in the right column. The waveforms used for (a)-(c) are the first and second harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx \omega$ and a tunneling strength of $\Delta = 0.25 \omega$, (d)-(f) are the third and eleventh harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx \omega$ and a tunneling strength of $\Delta = 0.25 \omega$ with transverse modulation given by the first and twentieth harmonics with amplitudes $\Delta_i = 0.25 \omega$ and (g)-(i) are the first and second harmonics in the diagonal with a DC bias strength of $\epsilon_0 \approx \omega$ and a tunneling strength of $\Delta = 0.25 \omega$ with transverse modulation given by the first, third, fourth, seventh, ninth, tenth, twelfth, thirteenth, fifteenth, seventeenth, eighteenth, and twentieth harmonics with amplitudes $\Delta_i = 0.5 \omega$. We call attention to the accuracy of resonance positions with our analytical treatment using GBFs compared to numerical computations. Furthermore, the magnitudes of the occupation probabilities match well. However, with increasing transverse modulation strength and increasing the number of harmonics, the resonance widths vary greatly from the analytical description. Furthermore, the long time dynamics deviates from the approximation regime, as expected.

Theorems & Definitions (2)

• Theorem 1: Hermite-Genocchi, baxterexponential
• Theorem 2: baxterexponential