Resonances in a periodically driven bosonic system

Periodically driven systems are a common topic in modern physics. In optical lattices specifically, driving is at the origin of many interesting phenomena. However, energy is not conserved in driven systems, and under periodic driving, heating of a system is a real concern. In an effort to better understand this phenomenon, the heating of single-band systems has been studied, with a focus on disorder- and interaction-induced effects, such as many-body localization. Nevertheless, driven systems occur in a much wider context than this, leaving room for further research. Here, we fill this gap by studying a noninteracting model, characterized by discrete, periodically spaced energy levels that are unbounded from above. We couple these energy levels resonantly through a periodic drive, and discuss the heating dynamics of this system as a function of the driving protocol. In this way, we show that a combination of stimulated emission and absorption causes the presence of resonant stable states. This will serve to elucidate the conditions under which resonant driving causes heating in quantum systems.

Figure 1

(a) The value $|U_{i,2000}|$ of the propagator (in units of ${10}^{-6}$) for $\mu =0,\nu =20,\theta =1/20,M=1$. This is the absolute value of the $2000\text{th}$ column of the propagator. The highest weight is at $i=2000$ since the propagator becomes diagonal at large frequencies. The overlap with the low-energy states is higher than with the high-energy states, which offsets the infinite tail on the right side. (b) Same as in (a) but for a restricted range of values so that the resonance structure is clearly visible. The peaks have a spacing of 40, which is precisely twice the driving frequency.

Figure 2

The value of $\mathrm{\Delta }{E}_i$ for the first 100 values of $i$. It can be seen that $\mathrm{\Delta }{E}_i$ is positive for $i<4$, and is negative for larger $i$. Although $\mathrm{\Delta }{E}_i$ stays negative for large $i$, it asymptotically tends towards zero.

Figure 3

(a) The components $|v_i|$ of the $1024\text{th}$ eigenvector $v$ of $U$ for $\mu =0,\nu =20,\theta =1/20,M=1$. The corresponding eigenvalue has unit norm. The vertical lines serve as a guide to the eye, and show that the function has extremely sharp peaks, with most weights indistinguishable from zero. (b) A zoom of (a) to show the small scale structure. A smaller range of components is depicted, and the matrix elements are multiplied by 10, so the scales are comparable. The decay of the peaks is now visible, but still rapid.

Figure 4

The real part of $\psi \left(x\right)-f_{1024}\left(x\right)$ is shown for the first quarter of the interval. The function $f_{1024}\left(x\right)$ is substracted to display the various beat frequencies present in $\psi \left(x\right)$. The beat has a period of $1/20$, which is the driving frequency, as expected.

Figure 5

The components $|v_i|$ of an eigenvector $v$ different from that in Fig. 3, but for the same parameters. The corresponding eigenvalue has norm smaller than unity. The peak weight is one of the eigenstates that gains energy over a cycle.