Dynamic control for nanostructures through slowly ramping parameters

We propose a nanostructure control method which uses slowly ramping parameters. We demonstrate the dynamics of this method in both a nonlinear classical system and a quantum system. When a quantum mechanical two-level atom (quantum dot) is irradiated by an electric field with a slowly increasing frequency, there exists a sudden transition from ground (excited) to excited (ground) state. This occurs when the ramping rate is smaller than the square of the Rabi frequency. The transition arises when its “instant frequency”—the time derivative of the driving field phase—matches the resonance frequency, satisfying the Fermi golden rule. We also find that the parameter ramping is an efficient control manner for classical nanomechanical shuttles. For ramping of driving amplitudes, the shuttle's mechanical oscillation is amplified and even survives when the ramping is stopped outside the original oscillation region. This strange oscillation is due to the entrance into a multistable dynamic region in phase space. For ramping of driving frequencies, an onset of oscillation arises when the instant frequency enters the oscillation region. Thus, regardless of being classical or quantum, the instant frequency is physically relevant. We discuss in which conditions the dynamic control is efficient.

Figure 1

(a) Schematic figure for the nanomechanical shuttle and its circuit diagram. (b) Phase diagram which shows dynamical regions (black). Arnold's tongues are labeled with an integer number $p=1,2,3,...$.

Figure 2

Shuttle position $x$ as a function of time-dependent driving amplitude $V_a\left(t\right)$ (a) when the parameter $V_a$ is swept from A to B for different ramping rates ${\epsilon }_V=0.05$ mV/ns (upper panel) and ${\epsilon }_V=0.005$ mV/ns (lower panel). (b) The same plot when the parameters are swept from B to A. (c) The $x-V_a$ curve when $V_a$ is swept from C to D (upper panel) and from D to C (lower panel) at ramping rate ${\epsilon }_V=0.005$ mV/ns. The dotted lines denote the boundaries of the Arnold's tongues.

Figure 3

(a) (Left) Shuttle position when the voltage amplitude increases consistently (${\epsilon }_V=0.005$ mV/ns) and stops at point B of Fig. 1. (Right) An enlarged view of the phase diagram. (b) The basin of attraction at point $A_{\mathrm{out}}$ which occupies all the space and the limit cycle at point $A_{\mathrm{in}}$. (c) Limit cycle of the shuttle at point $B_{\mathrm{in}}$ and the basin of attraction at the point $B_{\mathrm{out}}$. (d) The amplitude of oscillation as a function of the driving voltage amplitude for the Arnold tongue of $p=3$. Note that there are two stable limit cycle motions in this regime. Thus, when we increase the driving voltage from C to D, the shuttle has large enough oscillation amplitude to perform the large amplitude limit cycle (red). When the driving voltage decreases from D to C, the shuttle takes small amplitude oscillation and eventually the oscillation amplitude decays out (black). (e) Two stable limit cycle trajectories are shown for $V_a=255$ mV.

Figure 4

(a) Shuttle position $x$ as a function of the time-dependent frequency $\omega \left(t\right)={\omega }_{\mathrm{i}}+{\epsilon }_{\omega }t$ where ${\epsilon }_{\omega }=2\pi \times 50$ MHz/600 $\mu \mathrm{s}$. Note that the jump arises at the midpoint between ${\omega }_{\mathrm{i}}$ and ${\omega }_{-}$. (b) Comparison between the calculated jump frequency ${\omega }_{\mathrm{jump}}$ and $({\omega }_{\mathrm{i}}+{\omega }_{-})/2$ for $\gamma =0.005{\omega }_0, V_a=190\mathrm{mV}$ (filled circles), $\gamma =0.025{\omega }_0, V_a=150\mathrm{mV}$ (filled squares), and $\gamma =0.05{\omega }_0, V_a=120\mathrm{mV}$ (filled diamonds).

Figure 5

(a) Schematic figure for a two-level atom with a slowly varying frequency field. (b) The probability for the system to be in the excited state $P_2\left(t\right)={\left|C_2\left(t\right)\right|}^2$ where $C_1(t=0)=1$ and $C_2(t=0)=0$, initially. (c) $P_2\left(t\right)$ when the atom is at the ground state at $t=0$. Here, the energy spacing is larger than the Rabi frequency, ${\omega }_0=100\mathrm{\Omega }$, and we set various initial frequencies ${\omega }_i=0$ (black), $30\mathrm{\Omega }$ (red), and $70\mathrm{\Omega }$ (green). Note that, regardless of the initial frequency, the two-level system shows a clear transition to the excited level when the instant frequency $d\varphi /dt$ becomes the resonance frequency ${\omega }_0$. (Insets) Same data with a larger time window.

Figure 6

Time trace of $P_2$ for various ${\omega }_0=100\mathrm{\Omega }$ (a), $10\mathrm{\Omega }$ (b), and $\mathrm{\Omega }$ (c). Note that the clear transition at the resonance frequency $\dot{\varphi }={\omega }_0$ vanishes when the resonance frequency is as small as the Rabi frequency. (d) The time series of $P_2$ for various ramping rates ${\epsilon }_{\omega }$. Note that increasing the ramping rate does not destroy the sudden jump to the second level, but the saturated value of the probability decreases.

Figure 7

Saturated probability for the two-level atom to be at the excited state from full calculation (black line) and the first-order perturbation calculation (red line) as a function of ${\mathrm{\Omega }}^2/{\epsilon }_{\omega }$. We set ${\omega }_0=100\mathrm{\Omega }$ and ${\omega }_i=0$ but the results do not depend on these parameters.