High-gain quantum free-electron laser: Long-time dynamics and requirements

We solve the long-time dynamics of a high-gain free-electron laser in the quantum regime. In this regime each electron emits at most one photon on average, independently of the initial field. In contrast, the variance of the photon statistics shows a qualitatively different behavior for different initial states of the field. We find that the realization of a seeded quantum free-electron laser is more feasible than self-amplified spontaneous emission.

Figure 1

Mean photon number $n=\langle \widehat{n}\rangle$ of a high-gain Quantum FEL as a function of the wiggler length $L$ scaled in units of the gain length $L_g$. We compare the analytical approximation (red line) from Eq. (8) to the numerical solution (blue, dashed line) of Eq. (B4) for the startup from vacuum (top), that is, $n_0=0$, as well as for a seeded FEL (bottom) with $n_0={10}^3$ photons in an initial Fock state. For both plots we have assumed resonance, $p=q/2$, and the value $N={10}^4$ for the number of electrons. In all cases the curves show an exponential growth for short times which saturates and leads to a first maximum followed by further oscillations. For the startup from vacuum the photon number saturates at about ten gain lengths. There, the approximate solution takes on the value $n=N$, that is, each electron has emitted exactly one photon, while the numerically computed maximum is at about $n\cong 0.8N$. In addition, the locations of these maxima for analytics and numerics are slightly shifted with respect to each other. For an FEL seeded by a large Fock state, numerics and analytics agree very well. In this case, we obtain complete oscillations of the mean photon number between the values $n_0$ and $n_0+N$ with a significantly shorter periodicity than for the startup from vacuum.

Figure 2

Mean photon number $n=\langle \widehat{n}\rangle$ of a seeded high-gain Quantum FEL as a function of the wiggler length $L$ in units of the gain length $L_g$. We depict the numerical solution for $n$ with respect to three different initial states for the laser field, that is, a Fock state (red line), a coherent state (blue, dashed line), and a thermal state (black, dotted line). The latter two states are described by the photon distributions in Eqs. (B6) and (B7). In all three examples we have chosen the values $n_0=500$ and $N=5000$ for the initial mean photon number and the number of electrons, respectively. While a Fock state and a coherent state lead to the same oscillatory behavior of the photon number between $n_0$ and $N+n_0$, the situation for a thermal state is different. Here the first maximum of $n$ attains the decreased value of approximately $0.8N$ instead of $1.1N$. Moreover, we do not observe full oscillations, but the photon number seems to approach a constant value of approximately $0.5N$.

Figure 3

The first maximum $n_{\text{max}}$ (top) of the photon number and the corresponding saturation length $L_{\text{max}}$ (bottom), both as functions of the number $N$ of electrons and for resonance $p=q/2$. In the top panel, which covers only the startup from vacuum with $n_0=0$, we observe that $n_{\text{max}}$ very quickly reaches the constant value of $n_{\text{max}}=0.78N$ in contrast to the analytical approximation which predicts $n_{\text{max}}=N$. The inset shows the behavior of $n_{\text{max}}$ for very small values of $N$. For $N=1$ we find that the electron emits one photon due to the Rabi oscillation in the single-electron approach of Ref. [10]. At the bottom, the comparison of analytics, Eq. (11) (red line) to numerics (blue, dashed line) reveals the same logarithmic behavior of $L_{\text{max}}$ apart from a small shift between the two curves, if we consider startup from vacuum, $n_0=0$. In the case of a Quantum FEL seeded with a Fock state and with a fixed ratio of $n_0/N=0.1$ analytics (green line) and numerics (black, dashed line) also agree well. Here both curves predict the constant value $L_{\text{max}}\cong 5L_g$ with $L_g$ denoting the gain length from Eq. (9).

Figure 4

Saturation length $L_{\text{max}}$ (upper curves) and the corresponding maximum photon number $n_{\text{max}}$ (lower curves), as functions of the momenta $p$ in the neighborhood of resonance for $N={10}^3$ electrons and for $n_0=0$. We observe that $n_{\text{max}}$ decreases and $L_{\text{max}}$ increases for a growing deviation from resonance $p=q/2$. For both, $L_{\text{max}}$ and $n_{\text{max}}$, we find perfect agreement between analytics, that is, the red curve and the green curve, respectively, and numerics, that is, the blue dotted and the black dotted curve, respectively. We emphasize that the analytical and the numerical curves are normalized to their respective values at resonance which, however, differ from each other. For example, the analytical result for $n_{\text{max}}$ is divided by $N$ while its numerical counterpart is divided by $0.78N$.

Figure 5

Variance $\mathrm{\Delta }{n}^2$ of the photon-number distribution in a high-gain Quantum FEL as a function of the wiggler length $L$ scaled in units of the gain length $L_g$. For the underlying numerical simulation we have assumed resonance $p=q/2$, $N={10}^4$ electrons, and startup from vacuum, that is, $n_0=0$. We obtain a qualitatively similar behavior as for the mean photon number (compare to gray, dashed line and right axis), that is, exponential growth, local maximum, and decrease in an oscillatory-like fashion. However, the structure of $\mathrm{\Delta }{n}^2$ is more complicated. For example, close to $L_{\text{max}}$ a dip occurs, where $\mathrm{\Delta }{n}^2\cong 0.05N^2$ while $n_{\text{max}}\cong 0.8N$. Hence, the value of $\mathrm{\Delta }{n}^2$ is smaller, but roughly of the order of magnitude of $n_{\text{max}}^2$ corresponding to an almost chaotic behavior of the laser field.

Figure 6

Fano factor ${\sigma }^2\equiv \mathrm{\Delta }{n}^2/\langle \widehat{n}\rangle$ of the photon-number distribution in a seeded high-gain Quantum FEL as a function of the wiggler length $L$ scaled in units of the gain length $L_g$ for two different initial states, that is, a Fock state (red line) and a coherent state (blue, dashed line). For the underlying numerical simulations we have assumed resonance $p=q/2$, $N={10}^3$ electrons, and an initial mean photon number of $n_0={10}^2$. In both cases we observe super- as well as sub-Poissonian behavior which are separated by the horizontal dotted line at ${\sigma }^2=1$. We note that the first minimum of this normalized variance occurs in the vicinity of the first maximum of the mean photon number (compare to the gray solid and dotted lines as well as to the right axis) at $L\cong 5L_g$. Moreover, we find that the fluctuations for the initially coherent case drastically increase at around ten gain lengths, while only slowly increasing for the case of a Fock state.

Figure 7

Variance $\mathrm{\Delta }{n}^2$ of the photon number in a seeded high-gain Quantum FEL as a function of the wiggler length $L$ scaled in units of the gain length $L_g$. We have normalized the variance with respect to ${\langle \widehat{n}\rangle }^2$ to investigate the evolution of the fluctuations in an initial thermal state. The variance $\mathrm{\Delta }{n}^2$ is of the same order of magnitude as ${\langle \widehat{n}\rangle }^2$. However, this ratio attains a significant minimum when the first maximum of the mean photon number is reached (compare to gray, dashed line and right axis).