Classical and quantum particle dynamics in univariate background fields

We investigate deviations from the plane wave model in the interaction of charged particles with strong electromagnetic fields. A general result is that integrability of the dynamics is lost when going from lightlike to timelike or spacelike field dependence. For a special scenario in the classical regime we show how the radiation spectrum in the spacelike (undulator) case becomes well-approximated by the plane wave model in the high-energy limit, despite the two systems being Lorentz inequivalent. In the quantum problem, there is no analogue of the WKB-exact Volkov solution. Nevertheless, WKB and uniform-WKB approaches give good approximations in all cases considered. Other approaches that reduce the underlying differential equations from second to first order are found to miss the correct physics for situations corresponding to barrier transmission and wide-angle scattering.

Figure 1

Comparison of the harmonic structure in the undulator and plane wave spectra, as a function of emitted photon frequency ${\omega }^{\prime }$, for two values of incoming electron $\gamma$. The plane wave frequency is optical, ${\omega }_l=2\times {10}^{-6}\mathrm{m}$, and the undulator frequency, i.e. the inverse of the undulator period, is fixed such that $k.p$ is the same for both fields, see (14). As $\gamma$ increases the spectra start to overlap.

Figure 2

Upper panel: normalized potential function $V$ for $\kappa =-1$ in the Sauter field, and energy eigenvalue $\mathcal{E}=1/32$. Lower panel: U.WKB wavefunction for $\hslash =1/100$, shown together with the potential and energy eigenvalue. The wavefunction is exponentially damped under the barrier, i.e. when $\mathcal{E}<V$, and oscillatory when $\mathcal{E}>V$, above the potential. The numerical solution of (33) is indistinguishable from the U.WKB approximation on the scale shown.

Figure 3

Comparison of the longitudinal quasimomentum component in (42). The real and imaginary parts of the exact quantum solution are shown with green/solid and orange/dashed lines, respectively. Blue/solid: WKB (equal to exact classical). Yellow/dotted: small $Q$ expansion. Here $Q$ is large, i.e. $\hslash$ is small. The vertical grey line shows the position of the barrier top, $A=2Q$. For $A>2Q$, above the barrier, the exact quantum expression is well estimated by the semiclassical WKB result, and the small $Q$ expansion is not as accurate. For $A<2Q$, under the barrier, the band and gap structure is clearly visible. Neither approximation sees the band/gap structure, either over or under the barrier, where the gaps are respectively broad and narrow [54].

Figure 4

The quasimomentum as in Fig. 3, but for $Q$ small, i.e. $\hslash$ large. The small $Q$ (strong coupling) expansion is accurate both above and below the barrier. Even above the barrier, $A\gtrsim 2Q$, the small $Q$ expansion provides a better estimate than the WKB result, as $\hslash$ is not small.

Figure 5

Comparison of the U.WKB, perturbative (top panel) and RO (bottom panel) wavefunctions, for the example of Fig. 2, which corresponds to the expansion parameter of the perturbative and RO approaches being $\epsilon =\pm 1/50$. The RO wavefunction more closely tracks the U.WKB wavefunction as we approach the turning point, but the difference is small. Under the barrier, neither the perturbative nor RO wavefunctions fall off as expected. This is consistent with those approximations begin essentially high-energy.

Figure 6

Comparison of the perturbative (blue/dotted), RO (yellow/dashed) and U.WKB wavefunctions (green/solid) for the Sauter pulse, with parameters $a_0=1$, $\omega =2\times {10}^{-6}\mathrm{m}$, $|p_{\perp }|=a_{0}m$, $|p_z|=a_{0}m/2$. This corresponds to $\hslash ={10}^{-6}$, $\mathcal{E}=1/32$ and $\epsilon =2\times {10}^{-6}$. Top left: The wavefunctions agree asymptotically, far to the left of the turning point where $\mathcal{E}>V$. Top right: As we approach the turning point where $\mathcal{E}=V$, at $\varphi \simeq -1.38$, the U.WKB and RO wavefunctions agree, but the perturbative wavefunction begins to deviate from them. Bottom left: Closer to the turning point, the RO wavefunction also starts to deviate from the U.WKB wavefunction. Bottom right: to the right of the turning point we have $\mathcal{E}<V$ and the U.WKB wavefunction is exponentially suppressed, whereas the perturbative and RO wavefunctions continue to oscillate.

Figure 7

The effective potential (47) compared with $V(\varphi )$. Top panel: above the barrier, the perturbative and RO effective potentials track $V(\varphi )$, but with a series of divergences where $G(\varphi )$ vanishes. There are no such divergences for the U.WKB potential, which is indistinguishable from $V(\varphi )$ on the scale shown. As the wavefunctions penetrate the barrier, the trends change. Neglecting the divergences for clarity, the bottom panel shows that the perturbative and RO wavefunctions see a negative potential, which explains why they oscillate faster under the barrier than above. (The RO potential has been scaled for presentation purposes.)

Figure 8

Comparison of the numerical solution of the Schrödinger equation in the sech-type potential (48) with analytic approximations. $\mathcal{E}=0.995$ (horizontal dashed line) just below the peak of the potential (black dashed line), $\kappa =-1$, and initial conditions $G(-5)=1$, $G^{\prime }(-5)=0$. Upper panel: perturbative approximation. Lower panel: RO approximation. The parameters correspond to $\epsilon \simeq 4\times {10}^{-2}$. Even when the incident energy is only just below the peak of the potential, the perturbative and RO approximations fail to capture the physics of the Schrödinger equation.