Quantum and classical phase-space dynamics of a free-electron laser

In a quantum mechanical description of the free-electron laser (FEL), the electrons jump on discrete momentum ladders, while they follow continuous trajectories according to the classical description. In order to observe the transition from quantum to classical dynamics, it is not sufficient that many momentum levels are involved. Only if additionally the initial momentum spread of the electron beam is larger than the quantum mechanical recoil, caused by the emission and absorption of photons, the quantum dynamics in phase space resembles the classical one. Beyond these criteria, quantum signatures of averaged quantities like the FEL gain might be washed out.

Figure 1

Small-signal contribution ${\mathcal{W}}^{\left(1\right)}$, Eq. (B6), to the Wigner function of an electron in the FEL for the fixed phase $\theta =\pi$ and as a function of the relative and normalized momentum $(\wp -\overline{\wp })/\mathrm{\Delta }\wp$. We compare the results for three different values of the quantum parameter $\alpha$, that is $\alpha =16$ (solid yellow line), 100 (red dashed line), and 400 (blue dotted line), to the classical distribution function $f_{\text{cl}}^{\left(1\right)}$, Eq. (B7) (blue line) [40]. In all cases, we have chosen the values $\varepsilon =1$, $\tau =0.01$, $\overline{\wp }=\pi$, and $\mathrm{\Delta }\wp =0.1$ for the normalized field amplitude, the dimensionless time, the mean initial momentum, and the initial momentum spread, respectively. Although $\alpha =16\gg 1$ indicates that we are close to the classical regime the corresponding curve for ${\mathcal{W}}^{\left(1\right)}$ significantly differs from the classical result. Only after increasing $\alpha$ to $\alpha =400$, we observe an agreement between the quantum and the classical theory. As apparent from Eq. (6) the important parameter that governs this transition is not $\alpha$ but rather the ratio $\hslash k/\mathrm{\Delta }p$ of recoil $\hslash k$ and momentum spread $\mathrm{\Delta }p$. Indeed, for $\alpha =16$ this parameter is with $\hslash k/\mathrm{\Delta }p=1.25$ of the order of unity and we cannot neglect quantum corrections while $\alpha =400$ leads to the decreased value $\hslash k/\mathrm{\Delta }p=0.25\ll 1$.

Figure 2

Comparison of Wigner function $\mathcal{W}$ and classical phase-space distribution $f_{\text{cl}}$ both as functions of dimensionless position and momentum, $\theta$ and $\wp$, respectively, for different values of the quantum parameter $\alpha$ and of the initial momentum spread $\mathrm{\Delta }\wp$. In all cases, we have chosen the time $\tau =\pi$, which classically corresponds to half of a rotation near to the center of the potential. Each row corresponds to a different initial momentum width, that is $\mathrm{\Delta }\wp =0.1$, 1.0, and 2.0. The marginals on the left show the initial momentum distribution (dotted) and the evolved ones for $\alpha =1/3$ and 10 as well as the corresponding classical result (solid lines). The second and third column show the evolved Wigner functions for $\alpha =1/3$ and $\alpha =10$, the rightmost column displays the classical phase-space distribution. In the case of small $\alpha$ and $\mathrm{\Delta }\wp$ the discrete structure in momentum is visible in (b), where we have $\hslash k/\mathrm{\Delta }p=8.66$. With an increased $\alpha$ more levels are involved in (c) and the shape of the corresponding classical distribution in (d) starts to appear, even though there are a lot of interference structures. This behavior is consistent with the value of $\hslash k/\mathrm{\Delta }p=1.58$, which is of the order of unity. A broader initial momentum distribution washes out the discrete levels, but still the Wigner functions for $\alpha =1/3$ in (f) with $\hslash k/\mathrm{\Delta }p=0.87$, and in (j) with $\hslash k/\mathrm{\Delta }p=0.43$ are quite different from their classical counterparts in (h) and in (l), respectively. For large $\alpha =10$ and $\mathrm{\Delta }\wp =1$, that is, $\hslash k/\mathrm{\Delta }p=0.16$, there are only a few interference fringes left in (g). An even wider initial distribution with $\mathrm{\Delta }\wp =2$, corresponding to $\hslash k/\mathrm{\Delta }p=0.08$, makes them nearly invisible such that Wigner function in (k) and classical distribution in (l) resemble each other.

Figure 3

Deviation $\mathcal{W}-f_{\text{cl}}$ of the Wigner function from the classical distribution function depending on dimensionless position and momentum, $\theta$ and $\wp$, respectively, at three different times. In all plots, we have chosen the values $\alpha =10$ and $\mathrm{\Delta }\wp =2$, which are the configurations of Figs. 2 and 2. For a small time $\tau =\pi /12$, the interference fringes have a width proportional to the initial distribution, but a low amplitude. In the course of time, the amplitude grows (note the different scaling of the colormap for each plot) due to the increasing magnitude of the higher-order derivatives of the Wigner function. Those, in turn, are a consequence of the narrow width in momentum space that appears first in the proximity to the separatrix due to dispersion. The appearing interference fringes are also narrow, leading to a self-reinforcing effect. However, the amplitude of the deviation remains in the depicted case about one order of magnitude smaller than the maximal values of $\mathcal{W}$ in Fig. 2.

Figure 4

Distance $d_{\text{cl}}$, Eq. (8), between Wigner and classical phase-space distribution. A large distance (close to unity) means that the distributions are very different, while a small value (close to zero) indicates that they are similar. (a) Distance $d_{\text{cl}}$ as a function of the dimensionless time $\tau$ for the values $\alpha =0.3$ (dotted lines) and 10 (solid lines) of the quantum parameter as well as for small (blue) and large (red) initial momentum spread $\mathrm{\Delta }\wp$. Note that these are the same configurations as for the first and third row of Fig. 2. The distance increases with increasing values of $\hslash k/\mathrm{\Delta }p=1/\left(2\sqrt{\alpha }\mathrm{\Delta }\wp \right)$. For large values of this parameter, the distance rises quickly, while it remains small at short times for small values of $\hslash k/\mathrm{\Delta }p$. After a longer time, the distance suddenly increases even for small $\hslash k/\mathrm{\Delta }p$ but saturates afterwards. In (b) and (c), we draw $d_{\text{cl}}$ as a function of the initial momentum width $\mathrm{\Delta }\wp$ and of the quantum parameter $\alpha$ at times $\tau =\pi /2$ and $\tau =\pi$, respectively, corresponding to the grey vertical lines in plot (a). For small values of $\alpha$ and $\mathrm{\Delta }\wp$, that is the lower left corner, $d_{\text{cl}}$ is maximized and close to unity, while it decreases for large values of $\alpha$ and $\mathrm{\Delta }\wp$ towards the upper right corner, that is, the evolution becomes more classical. At the earlier time $\tau =\pi /2$, the decrease is steep, while it is more gradual at the later time $\tau =\pi$. Even if the distance is very small at an earlier time, this does not mean that it will be small also for later times.

Figure 5

Quantum corrections to the FEL gain: we have drawn the small-signal gain for a cold (above), Eq. (11), and a warm electron beam (below), Eq. (12), as functions of $\overline{\wp }\tau =2k\overline{p}t/m$ and $\overline{\wp }/\mathrm{\Delta }\wp =\overline{p}/\mathrm{\Delta }p$, respectively. In the cold-beam case, $\mathrm{\Delta }\wp \tau \ll 1$ we compare the classical gain (black line) to gain curves including the lowest-order quantum corrections $[m=1$ in Eq. (11)] for the values ${\omega }_{\text{r}}t=1$ (blue, dotted line) and ${\omega }_{\text{r}}t=2$ (red, dashed line), respectively, of the recoil parameter ${\omega }_{\text{r}}t$. We observe that for increasing values of ${\omega }_{\text{r}}t$ the positions of minimum and maximum are slightly shifted to the left and to the right, respectively, while their magnitude is decreased. The behavior in the warm beam case with $\mathrm{\Delta }\wp \tau \gg 1$, is qualitatively similar. However, for our—already moderate—choice of $\mathrm{\Delta }\wp \tau =10$ visible quantum effects [again in lowest order, that is $m=1$ in Eq. (12)] occur not until ${\omega }_{\text{r}}t=3$ (blue, dotted line) and are still quite small for ${\omega }_{\text{r}}t=7$ (red, dashed line). Hence, quantum corrections in the warm beam case are small in comparison to a cold beam in accordance to Eq. (13). We conclude that a large momentum spread suppresses quantum effects in the FEL gain.

Figure 6

Comparison of the FEL gain $G$ as a function of the dimensionless time $\tau$ computed with initial mean momentum $\overline{\wp }=1.6$ and for different values of the quantum parameter $\alpha$ as well as for narrow ($\mathrm{\Delta }\wp =0.1$) and broad ($\mathrm{\Delta }\wp =2$) initial momentum distributions. The solid lines correspond to the result obtained from the Wigner function, while the dotted ones are calculated from the classical distribution function with the same initial state. The latter curves feature oscillations around a saturation value with a frequency based on the rotation period of the bounded trajectories and are independent of $\alpha$. For $\alpha =1$, as shown in (a), the results from both theories are similar for short times, especially for the wide momentum spread with $\hslash k/\mathrm{\Delta }p=0.25$, in contrast the narrow momentum spread with $\hslash k/\mathrm{\Delta }p=5.00$. For $\alpha =10$, as shown in (b), the quantum and classical behavior of the gain is qualitatively very similar and agrees very well at least until the first maximum is reached. Note the large value of $\hslash k/\mathrm{\Delta }p=1.58$ for $\mathrm{\Delta }\wp =0.1$ in contrast to $\hslash k/\mathrm{\Delta }p=0.08$ for $\mathrm{\Delta }\wp =2$. The good agreement comes from the fact that even substantial quantum features of the Wigner function, see for instance Fig. 2, average out in the gain, since it is obtained from the change of mean momentum. Further, we observe that the point in time where the curves diverge appears later, when $\alpha$ is increased.