Fast-forward problem in quantum mechanics

We show the way to speed up the time evolution of a wave function (WF), i.e., to fast-forward the WF in microscopic and macroscopic quantum mechanics, by controlling the driving potential with resultant regulation of the additional phase of the WF, so that a target state is obtained in a shorter time. We first present a general framework of the fast-forwarding of a WF in quantum mechanics and provide an example of the fast-forward of a WF in two-dimensional (2D) free space. Then the framework of the fast-forward is extended to macroscopic quantum mechanics described by the nonlinear Schrödinger equation. We show the fast-forward of (i) transport of Bose-Einstein condensates trapped by a moving 2D harmonic potential and (ii) propagation of a soliton both in free space and through a potential barrier (: macroscopic quantum tunneling).

Figure 1

Time dependence of a magnification factor $\alpha \left(t\right)$. In time domains (I), (II), and (III), $\alpha$ increases from 1, keeps a constant value larger than 1, and decreases back to 1, respectively.

Figure 2

(Color online) Schematic diagram of the time-evolution of the amplitude of wave packets moving in the $x$ direction. Horizontal and vertical axes, respectively, represent time and position. (a) Standard force-free evolution; and (b) corresponding fast-forwarded evolution characterized by $\alpha \left(t\right)$ in Fig. 1. The lines in the figures represent the path of the center of mass. Time domains (I), (II), and (III) in (b) correspond to (I), (II), and (III) in Fig. 1, respectively.

Figure 3

(Color online) Time dependence of $\alpha$ in the case of Eq. (23) with $\overline{\alpha }=1.5$. The broken line represents the value of the time-averaged $\alpha$.

Figure 4

(Color online) Sequence (a)–(f) of the driving potential $V_{\mathit{FF}}$ for $-5⩽x⩽35$ and $-16⩽y⩽16$ during the fast-forward of the Gaussian wave packet in 2D free space. The wave packets in (a) and (f) show the amplitude of the fast-forwarded state at the initial $(t=0)$ and final $(t=T_{\mathit{FF}})$ time of the fast-forward, respectively. Space and time coordinates are scaled by $L={10}^{-2}\times \mathit{\text{linear}}$ dimension of a device and $\tau ={10}^{-2}\times \mathit{\text{phase}}$ coherent time, respectively, and we put $\frac{\hslash }{m_0}=1(\times L^2{\tau }^{-1})$. These prescriptions are also employed in Fig. 5.

Figure 5

(Color online) $x$ dependence of the real part of WFs at $y=0$. (a) The standard state ${\Psi }_0$ at $t=T=12.5$; and (b) the fast-forwarded state ${\Psi }_{\mathit{FF}}$ at $t=T_{\mathit{FF}}=12.5\times 2∕3$.

Figure 6

Time-dependence of the center of mass of the WF (solid line) and position of the moving harmonic potential (dotted line). (a) and (b) correspond to the parameters $v_{\max }=1.25$, $T_1=12.5$ and $v_{\max }=2.5$, $T_1=6.25$, respectively. Space and time coordinates are scaled by $L={10}^{-2}\times \mathit{\text{the}}$ system size and $\tau ={10}^{-2}\times \mathit{\text{the}}$ dissipation time, respectively, and we put $\frac{\hslash }{m_0}=1(\times L^2{\tau }^{-1})$, $\omega =1.0(\times {\tau }^{-1})$, and $\frac{c_0}{\hslash }=0.5(\times L^2{\tau }^{-1})$. These prescriptions are also employed in Figs. 7, 8, 9.

Figure 7

Driving potential (thick lines) for the acceleration of the transport of a WP and ${\mid {\Psi }_{\mathit{FF}}\mid }^2$ (thin lines). (a) to (d) correspond to $t=\frac{T_{\mathit{FF}}}{3}\times n$ for $n=0,1,2,3$, respectively. Here $\overline{\alpha }=2$ and $T_{\mathit{FF}}=6.25$.

Figure 8

The time-dependence of the center of mass of ${\Psi }_0$ (broken line) and ${\Psi }_{\mathit{FF}}$ (solid line).

Figure 9

Time-dependence of the fidelity in Eq. (30). The inset represents $x$ dependence of the amplitude of ${\Psi }_{\mathit{FF}}$ and the target state at the final time. They are overlapping precisely.

Figure 10

Spatiotemporal dependence of the driving potential $V_{\mathit{FF}}$.

Figure 11

Spatiotemporal dependence of the amplitude of ${\Psi }_0$ and ${\Psi }_{\mathit{FF}}$, which are identical at the initial time. The dotted and solid lines correspond to ${\mid {\Psi }_0\mid }^2$ and ${\mid {\Psi }_{\mathit{FF}}\mid }^2$, respectively. The spatial distributions are plotted at every time intervals $\Delta t=1∕60(\times \tau )$.

Figure 12

Time-dependence of the fidelity in Eq. (36). The inset represents the amplitude of ${\Psi }_{\mathit{FF}}^s$ and the target state at the final time of the fast-forward. They are overlapping almost exactly.

Figure 13

Parameter dependence of the fidelity $F$. (a) $T$, $\overline{\alpha }$ dependence and (b) $v$, $q$ dependence.

Figure 14

(a)–(d) Snapshots of ${\mid {\Psi }_{\mathit{FF}}\mid }^2$ corresponding to the soliton scattered by the Gaussian potential. (e)–(h) Corresponding potential $V_{\mathit{FF}}$.

Figure 15

(a)–(d) Snapshots of ${\mid {\Psi }_{\mathit{FF}}\mid }^2$ corresponding to the soliton scattered by a rectangular potential. (e)–(h) Corresponding potential $V_{\mathit{FF}}$.