Phase Space Crystals: A New Way to Create a Quasienergy Band Structure

A novel way to create a band structure of the quasienergy spectrum for driven systems is proposed based on the discrete symmetry in phase space. The system, e.g., an ion or ultracold atom trapped in a potential, shows no spatial periodicity, but it is driven by a time-dependent field coupling highly nonlinearly to one of its degrees of freedom (e.g., $\sim q^n$). The band structure in quasienergy arises as a consequence of the $n$-fold discrete periodicity in phase space induced by this driving field. We propose an explicit model to realize such a phase space crystal and analyze its band structure in the frame of a tight-binding approximation. The phase space crystal opens new ways to engineer energy band structures, with the added advantage that its properties can be changed in situ by tuning the driving field’s parameters.

Figure 1

Quasienergy $g$ in phase space. (a) $g\propto H_R$ versus $\mathrm{Re}[a]$ and $\mathrm{Im}[a]$ for power $n=10$ and driving strength $\mu =0.4{\mu }_c$. For nonzero driving, the quasienergy is invariant under discrete phase space rotations $e^{i\theta }\to e^{i(\theta +\tau )}$ where $\tau =2\pi /n$. (b) A cut through the bottom of the quasienergy $g$ in (a). There are $n$ stable states (yellow closed curves) and $n$ saddle points (unstable states, between two stable states) arranged periodically in angular direction. A local coordinate system ($x$, $p$) is defined near the bottom of a stable state. (c) Quasienergy $g$ versus radius $r$ (left) and angle $\theta$ (right). Stable states are confined by the radial potential barrier $U_r$ and the angular potential barrier $U_{\theta }$. For the latter, we plot two wells between $\theta =-\tau$ and $\theta =\tau$. The localized states confined in each well (green lines) are coupled by quantum tunneling.

Figure 2

Quasienergy band structure. (a) Quasienergy spectrum changing from quasicontinuous in the absence of driving (left) to a band structure induced by finite driving (right). The gaps start to open from the bottom of the spectrum. (b) Quasienergy band structure in the reduced Brillouin zone. Each red dot represents one quasienergy level. There are $n$ levels in each band. (c) Width of the lowest band $d_1$, and the asymmetry factor $\delta$ versus driving. Numerical (triangles) and approximate (lines) results are compared. The parameters are $\lambda =1/205$, $n=10$ for all the figures, and for (b) we choose $\mu =0.3{\mu }_c$.

Figure 3

Emission spectrum: The top, middle, and bottom figures are the emission spectrum for the first band, the second band, and the interband, respectively. The parameters are $\lambda =1/205$, temperature $\mathrm{T}=0.5\hslash {\omega }_0/k_B$, driving $\mu =0.4{\mu }_c$, and damping $\kappa ={10}^{-4}\lambda$.