Abstract
We study a periodically driven qubit coupled to a quantized cavity mode. Despite its apparent simplicity, this system supports a rich variety of exotic phenomena, such as topological frequency conversion as recently discovered in [Martin et al, PRX 7, 041008 (2017)]. Here we report on a qualitatively different phenomenon that occurs in this platform, where the cavity mode's oscillations lock their frequency to a rational fraction $r/q$ of the driving frequency $\Omega$. This phenomenon, which we term quantum frequency locking, is characterized by the emergence of $q$-tuplets of stationary (Floquet) states whose quasienergies are separated by $\Omega/q$, up to exponentially small corrections. The Wigner functions of these states are nearly identical, and exhibit highly-regular and symmetric structure in phase space. Similarly to Floquet time crystals, these states underlie discrete time-translation symmetry breaking in the model. We develop a semiclassical approach for analyzing and predicting quantum frequency locking in the model, and use it to identify the conditions under which it occurs.
Highlights
In recent years, periodic driving has been explored as a way to create desirable properties in otherwise ordinary systems [1,2,3,4,5,6]
III, we identified the conditions for quantum frequency locking: namely, it arises for parameters in the adiabatic or Floquet regimes where the effective Hamiltonian Heff (x, p) [Eq (8)] has extrema at nonzero amplitude in phase space
For the driven qubit-cavity system we consider in this work, we show below that δε ∼ e−d/ξ, where d denotes the separation in phase space between the extrema of the effective Hamiltonian Heff (x, p), while ξ ∼ 1 denotes the scale of quantum fluctuations
Summary
Periodic driving has been explored as a way to create desirable properties in otherwise ordinary systems [1,2,3,4,5,6]. When the qubit is driven close to resonance with a rational multiple of the cavity’s resonance frequency, the cavity mode oscillates with frequency locked to r /q This phenomenon has a wide range of interesting implications. A robust effect, which does not require fine tuning [56] It persists both for weak and strong qubit-cavity coupling, and for finite ranges of the driving frequency [see Fig. 1(c)]. Our proposal provides a complementary realization that can capitalize on recent advances in control of few-level quantum systems in Rydberg atoms, quantum dots, and superconducting qubits interacting with microwave cavities [60,61,62] In this way our results provide a direct path for realizing quantum frequency locking on readily available experimental platforms. Ad are dimensionless numbers denoting the effective Zeeman field strength and driving amplitude, respectively This model was shown to support topological frequency conversion in Refs. We expect that our following discussion generalizes to cavities with multiple modes [17]
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