High-gain quantum free-electron laser: Emergence and exponential gain

We derive an effective Dicke model in momentum space to describe collective effects in the quantum regime of a free-electron laser (FEL). The resulting exponential gain from a single passage of electrons allows the operation of a Quantum FEL in the high-gain mode and avoids the experimental challenges of an x-ray FEL oscillator. Moreover, we study the intensity fluctuations of the emitted radiation, which turn out to be super-Poissonian.

Figure 1

Difference between the single-electron model for the low-gain regime (left) and the many-electron model for the high-gain FEL (right): In the low-gain regime we consider only a single electron interacting with the laser field and simply multiply the resulting change of the laser field with the number $N$ of electrons in the bunch. For the simultaneous interaction of all $N$ electrons with the laser field the motion of each electron influences the dynamics of the laser field which in turn acts back on the motion of all electrons. Hence, the electrons are indirectly coupled to each other due to their common interaction with the laser field.

Figure 2

Creation of entangled superposition states in the FEL for $N=2$ electrons: On the left-hand side, the first electron with momentum $p_1$ (above) absorbs a laser photon and emits a wiggler photon which leads to the increased momentum $p_1+q$ with $q\equiv 2\hslash k$ denoting the discrete recoil. In contrast, the second electron (below) does not scatter and thus maintains its initial momentum $p_2$. The second possible event is depicted on the right-hand side: Here the first electron does not interact with the field and its momentum $p_1$ stays unchanged, while the second electron is scattered from a laser and a wiggler photon yielding the final momentum $p_2+q$. Superimposing these two possibilities leads us finally to the entangled state in Eq. (3). For didactic reasons, we have used in this figure a representation, where besides the laser mode, described by ${\widehat{a}}_{\text{L}}$ and ${\widehat{a}}_{\text{L}}^{†}$, also the wiggler field is quantized with the photon annihilation and creation operators, ${\widehat{a}}_{\text{W}}$ and ${\widehat{a}}_{\text{W}}^{†}$, respectively.

Figure 3

Exponential gain of a high-gain Quantum FEL for short times: We have drawn the mean photon number $\langle \widehat{n}\rangle =n_{\text{sp}}$ emerging from spontaneous emission, Eq. (25), as a function of the wiggler length $L$ in multiples of the gain length $L_g$, Eq. (26). We, moreover, study the behavior of $\langle \widehat{n}\rangle$ for four different values of the deviation $\mathrm{\Delta }$, Eq. (12), from resonance. Besides the start-up from vacuum, that is, from $\langle \widehat{n}\left(0\right)\rangle =0$, we observe that the growth of the mean photon number is decreased for increasing values of $\mathrm{\Delta }$.

Figure 4

Gain functions in the classical and in the quantum regime: We have drawn the increments $\text{Im}{\lambda }_{+}$ for the intensity growth in a classical FEL (top) and in a Quantum FEL (bottom), respectively, both as functions of the momentum $p$ divided by the recoil $q$. For the former case we numerically solve the cubic characteristic equation from Ref. [7] for ${\alpha }_N=10$, while we use our analytic expression from Eq. (28) for the latter case with ${\alpha }_N=0.2$. The classical curve reaches its maximum at $p=0$ and is a smooth and broad function that covers many multiples of $q$. In contrast, its quantum counterpart is sharply peaked at the quantum resonance, $p=q/2$, and is very narrow, covering a range of momenta which is smaller than $q$. Hence, the gain bandwidth in a Quantum FEL is much smaller than in a classical FEL.

Figure 5

Variance $\mathrm{\Delta }{n}^2$, Eq. (29), of the photon number normalized to the expectation value $\langle \widehat{n}\rangle$, Eq. (24), as a function of the length $L$ of the wiggler in units of the gain length $L_g$, Eq. (26), for exact resonance $p=q/2$. We compare the situations in which the field is initially in (i) a Fock state (blue line), (ii) a coherent state (red line), and (iii) a thermal state (green line), with each of them being described by the same mean photon number, that is, $\langle \widehat{n}\left(0\right)\rangle =100$. In all cases we observe a super-Poissonian behavior of the photon statistics which moves away from a Poisson statistics with $\mathrm{\Delta }{n}^2=\langle \widehat{n}\rangle$ (dotted line) for increasing values of $L$. We find the largest deviation from a Poissonian for an initially thermal field, while the case of an initial Fock state shows the smallest deviation.

Figure 6

Gain function of a high-gain Quantum FEL including higher-order corrections: We have drawn the increment $\text{Im}{\lambda }_{+}$ for the growth of the mean photon number against the initial momentum $p$ of the electrons divided by the recoil $q$ for two different values of the quantum parameter, that is, ${\alpha }_N=0.1$ (top) and ${\alpha }_N=0.5$ (bottom). In both plots, we have compared the first-order solution (black line), Eq. (23), corresponding to the two-level approximation and the expression Eq. (32), in third order (red line) to the numerical solution of the cubic equation (blue dashed line), Eq. (33), from Ref. [6]. In the deep quantum regime, ${\alpha }_N=0.1$, we observe that all three curves approximately agree with a nearly perfect matching of the third-order solution, Eq. (32), with the result of Ref. [6]. If we increase the quantum parameter to ${\alpha }_N=0.5$, the two-level approximation significantly differs from the other two curves. However, the third-order solution and the result from Ref. [6] still agree very well despite a small shift that originates from even higher orders of ${\alpha }_N$. Moreover, while the two-level approximation leads to a maximized gain at $p=q/2$, the maximum in third order has moved to the left for increasing values of ${\alpha }_N$. This behavior is consistent with the transition of the quantum resonance at $p=q/2$ to the classical one, $p=0$, in Fig. 4.