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 FEL is more feasible than self-amplified spontaneous emission.

By identifying three constants of motion we solve in the present article the long-time dynamics of a Quantum FEL in the high-gain regime, within (i) an analytical approximation and (ii) a numerical simulation. Moreover, we discuss fundamental requirements [12] to realize such a device in an experiment.
Employing momentum-jump operators we showed in Ref. [13] that the dynamics of a high-gain Quantum FEL is effectively governed by the Dicke Hamiltonian [14]. We briefly review this model in Sec. II. In order to solve the resulting equations of motion we applied in Ref. [13] a parametric approximation [15] and obtained an exponential growth of the laser intensity along the wiggler length. However, this approximation breaks down when the number of emitted photons becomes large after a certain interaction time. Therefore, we derive in Sec. III solutions beyond the short-time limit.
As a first result of these studies, we find that in the quantum regime each electron emits at most only a single photon [3], in contrast to the multi-photon processes dominating the classical regime [16]. Moreover, we derive expressions for the saturation length and consider deviations from resonance.
Secondly, we discuss the variance of the photon number. For a start-up from vacuum, the field possesses almost chaotic statistics. In case of a seeded Quantum FEL, the behavior of the variance depends strongly on the initial field state. A Fock state or a coherent state with a high photon number lead to comparably narrow photon distributions in the course of time. In contrast, the statistics evolving from a thermal state remains broad, but becomes much narrower than a thermal distribution when the intensity assumes its maximal value.
Our results allow us in Sec. IV to identify the challenges for a Quantum FEL experiment, and to explain the necessity for an optical undulator [17,18]. Indeed, it was argued in Ref. [12] that the combined influence of space charge and spontaneous emission into all modes prevents an effective Quantum FEL operation for more than several gain lengths which drastically reduces the possible laser intensity. We can circumvent this loss of intensity if we consider a seeded FEL instead of selfamplified spontaneous emission (SASE). This effect is a direct consequence of the decreased saturation length in a seeded Quantum FEL so that one drops below the problematic limit put forward in Ref. [12].
Finally, we summarize our main results and conclude in Sec. V. To keep this article self-contained we add the Appendices A and B which are devoted to the detailed calculations associated with our analytical approximation and the numerical approach, respectively.

II. QUANTUM FEL: BASIC BUILDING BLOCKS
In Ref. [13] we have formulated the FEL dynamics [19,20] in terms of collective jump operators for the electron momentum. In the following, we review this description, where each electron populates levels on a discrete momentum ladder induced by the quantum mechanical recoil ≡ 2ℏ the electron experiences when it scatters from the fields. Here ℏ denotes the reduced Planck constant while is the wave number of the laser and wiggler field in the co-moving Bambini-Renieri frame [21] where both coincide.

A. Collective Hamiltonian
By performing a rotating wave-like approximation [13] we found that the motion of each electron in the quantum regime is restricted to only two momentum levels, that is the excited state with momentum , close to the resonant momentum = /2, and the ground state with − − /2. In the language of collective operators we therefore defined where we sum over projection operators for all electrons with | ( ) denoting the momentum eigenstate of the -th electron. Indeed,ˆ+,ˆ−, andˆsatisfy the commutation relations for angular momentum [22,23], that is [ˆ+,ˆ−] = 2ˆand [ˆ,ˆ±] = ±ˆ±. Hence, we identify the jump operators with the ladder operatorsˆ+ andˆ− of angular momentum, whiled escribes its -component [24]. The restriction to two momentum levels leads us to the dimensionless Dicke Hamiltonian [14] eff ≡ where the dynamics of the laser mode is described by the photon annihilation, creation, and number operator,ˆL,ˆ † L , andˆ≡ˆ † LˆL , respectively. These field operators fulfill the commutation relation [ˆL,ˆ † L ] = 1. Moreover, we have recalled the dimensionless parameter ≡ / r as the ratio of the coupling constant for electrons and fields [10,25] and the recoil frequency [10] r ≡ 2 /(2 ℏ) with denoting the mass of an electron. In addition, ≡ ( − /2)/( /2) describes the scaled detuning of the initial momentum of the electrons from resonance at = /2. We emphasize that also the time variable ≡ r in this description is in a dimensionless form.
The approximation leading to the effective Hamiltonian of Eq. (3) is only valid, provided the quantum parameter (compare to Table I) satisfies the inequality which defines the quantum regime of the FEL. Moreover, we require that the detuning from resonance is small, that is < 1. For a realistic electron beam, where each electron has a different initial momentum we additionally require Δ < for the initial momentum spread Δ of the electrons.

B. Constants of motion
Despite of several attempts [22,[26][27][28][29][30][31] no closed analytic solution for the dynamics dictated by the Dicke Hamiltonian has been found. Hence, we resort to approximations and numerical methods. Hereby, we closely follow the lines of the existing literature [22,28,29,32] on this problem. In addition, we study the effect of a nonzero detuning from resonance.
The key ingredient of our approach is the identification of three constants of motion [28]. From the Hamiltonian in Eq. (3) we straightforwardly derive that the total angular momentum as well as the total number of excitationŝ ≡ˆ+ˆ (6) are constants of motion which commute with each other, that is ˆ,ˆ = 0.
Moreover, the Hamiltonian ≡ˆe ff (7) is independent of time and thus itself constitutes a third constant.

C. Our Approaches
We use these quantities to investigate the long-time dynamics of a high-gain Quantum FEL in an analytical as well as a numerical approach. The former one is carried out in detail in App. A and relies [28] on the decoupling of the Heisenberg equations of motion of the photon-number operatorˆwith the help ofˆ,ˆ, andˆ. To solve the resulting differential equation of non-commuting operators we approximate the operators by c-numbers [28,29]. By this procedure we find an analytical expression for the mean photon number ≡ ˆ in terms of a Jacobi elliptic function [22,29,33].
In contrast, the approach of App. B towards a numerical solution [32] is based on the evolution of time-dependent state vectors as solutions of the Schrödinger equation. For that, we expand the state of the laser field into Fock states | with photon number while the collective state of the electrons is described by the basis | , , where corresponds to the total angular momentum and to its -component. Sinceˆand are constant, the set , , of three independent quantum numbers is reduced to a single one. As a consequence, the Schrödinger equation leads to a three-term recurrence relation for the expansion coefficients which we straightforwardly solve by the numerical diagonalization of a tri-diagnoal matrix of dimension ( + 1) × ( + 1).

III. LONG-TIME DYNAMICS
In the following we present our results of the analytical approximation and the numerical simulation for the long-time dynamics of a high-gain Quantum FEL. Here we first focus on the time evolution of the mean number of photons and concentrate on the dependence of the saturation intensity and length on the number of electrons. We conclude by discussing the variance of the photon distribution of the Quantum FEL.

A. Time evolution of mean photon number
In App. A we derive the approximate expression for the mean photon number ≡ ˆ of a Quantum FEL as a function of the length of the wiggler, where cn denotes a Jacobi elliptic function [33]. The quantities + and − are roots of the right-hand side of the differential equation (A9) for . The explicit expressions for these roots are given in Eq. (A10) and depend on the parameters of this differential equation, that 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 start-up from vacuum the photon number saturates at about 10 gain lengths. There, the approximate solution takes on the value = , that is each electron has emitted exactly one photon, while the numerically computed maximum is at about 0.8 . 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 0 and 0 + with a significantly shorter periodicity than for the start-up from vacuum. is the initial photon number 0 , the electron number , and the detuning / from resonance in units of the quantum parameter.
Moreover, we have recalled from Refs. [3,13] the gain length of a Quantum FEL. In addition, the Jacobi elliptic function cn is characterized by the modulus = ( + , − ) from Eq. (A15), whereas ≡ ( ) describes a complete elliptic integral of first kind [33]. We emphasize that the initial state of the laser field enters in the approximation, Eq. (8), only through the mean photon number 0 . In Fig. 1 we compare the analytical approximation for the mean photon number = ˆ to the numerical solution, for exact resonance = /2 and with = 10 4 electrons [34].
Here we consider the start-up from vacuum (top) as well as a seeded FEL evolving from a Fock state with 0 = 10 3 photons (bottom).
In both cases we observe an exponential start-up in accordance with Ref. [13]. However, this growth saturates for increasing values of the wiggler length leading to a local maximum of the photon number. The mean photon number then decreases until it reaches its initial value, before it again increases in an oscillatory-like behavior.
For start-up from vacuum we deduce from the analytical approach that each electron emits at most one photon, that is max = , in contrast to the smaller value max 0.8 obtained by the numerical simulation. The second maximum of the numerical solution, however, is even stronger suppressed compared to the analytical approximation.
We interpret these deviations as the result of entangled Dicke states for the electron momenta [13,14] which the numerical solution takes into account. The oscillations of the analytical solution between 0 and indicate that in this model all electrons are in the ground state when the maximum photon number is reached. In the exact treatment, however, the electrons entangle with each other due to their common interaction with the laser field. This entanglement prevents a product state, where each electron is in the ground state, decreasing the maximum photon number.
Despite these differences between the numerical and the approximate results, the latter one is helpful to estimate the qualitative behavior of the dynamics. This feature becomes even more important in the discussion of experimentally relevant length scales in Sec. IV, when we increase the electron number to more realistic values where a numerical diagonalization becomes impracticable. In this limit, we have to resort to the predictions given by the expression in Eq. (8).
In contrast, for a seeded Quantum FEL the numerical and analytical solution for the mean photon number agree very well, at least for a Fock state. In this situation we observe oscillations of the mean photon number between 0 and 0 + . We note that the periodicity of these oscillations is much shorter than for the start-up from vacuum.
So far, we have only considered the case of the initial state of the laser field being given by a Fock state. We now study the influence of the photon statistics of the initial state on the time evolution of the mean photon number and refer to App. B for details. Therefore, we show in Fig. 2 the numerically evaluated expectation value = ˆ as a function of , with the initial state of the field given by (i) a Fock state, (ii) a coherent state, and (iii) a thermal state. To make a meaningful comparison we have chosen in the three cases the same initial mean photon number 0 = 500 while assuming = 5000 electrons. A coherent state is usually employed to model the output of a coherent light source such as a laser. In contrast, a thermal state describes a random, incoherent source, for example a light bulb [35,36].
We observe that a coherent state leads approximately to the same behavior as a Fock state, that is an oscillation of the photon number between 0 and + 0 . However, the curve corresponding to a thermal input state is different: Here the Similar to a Fock state, a coherent state with a high photon number possesses a sharply peaked photon distribution . We obtain the mean photon number through expressed by Eq. (B15). Here = ( ) denotes the mean photon number for the initial Fock state .
Hence, mainly contributions with photon numbers close to the mean 0 are relevant. Since the dynamics of these contributions occurs on similar time scales, the averaging process yields results identical to the ones originating from a delta-peaked photon distribution, that is, for a Fock state.
In contrast, the photon distribution of a thermal state is very broad and, moreover, is not symmetric around 0 [36]. Hence, we have to average over many differently weighted contributions which corresponds to many different time scales of the resulting dynamics.
Additionally, the probabilities for a thermal state increase for photon numbers close to zero. There, the dynamics of is drastically different from high initial photon numbers as apparent in Fig. 1. This mixing of many different time scales, combined with the influence of the qualitatively different behavior for small photon numbers explains the difference in Fig. 2 between a thermal input state and a coherent or a Fock state.

B. Saturation intensity and length
In this section we investigate the magnitude max and the position max of the first maximum of the mean photon number shown in Fig. 1, that is the saturation intensity and the saturation length, respectively, of a Quantum FEL. Our analysis distinguishes the two cases of exact resonance and near resonance.

Resonant case
For the time being, we consider exact resonance, that is = /2, and assume that the laser field is initially described by a Fock state. From Eqs. (8) and (A10) we derive the simple relation max 0 + for the maximum photon number which we have already deduced from Fig. 1. There, we have also found that for 0 = 0 the numerical result is lower than the estimated one, at least for our choice of = 10 4 electrons.
In the upper part of Fig. 3 we now examine if this discrepancy between analytics and numerics continues to exist for different values of by plotting the numerically evaluated result for max against . In the limit of a single electron exactly one photon is emitted in accordance with the single-electron approach of Ref. [10]. Here Rabi oscillations between excited and ground state occur and no entangled Dicke states, which we have identified as the reason for a decreased ratio max / , emerge.
However, already for low electron numbers the value max 0.78 is reached which then remains constant for the whole range of the considered values for . We therefore deduce that (i) this behavior continues for increasing values of where numerics starts to become impracticable, and that (ii) the analytical result max = provides us at least with the correct order of magnitude. Already in Ref. [27] it was predicted that the maximum photon number max resulting from the Dicke Hamiltonian with 1 atoms varies between max 0 + for a large initial photon number 0 1 and max 4/5 for 0 = 0 in accordance with our numerical solution.
In Fig. 1 we have also observed a different max for the start-up from vacuum compared to the seeded case. To find an analytical expression for max we now take a closer look at the elliptic function cn in Eq. (8). The first maximum of this function occurs, when its arguments vanish. With the help of this condition and the asymptotic behavior [33] for the saturation length with = /2 and 1 [37]. For 0 = 0 we thus read off a logarithmic growth of max with the electron number , that is max ∝ ln . In the case of a seeded Quantum FEL with 0 1 the important parameter that determines the magnitude of max is the ratio of electron The inset shows the behavior of max for very small values of . For = 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 max apart from a small shift between the two curves, if we consider start-up from vacuum, 0 = 0. In the case of a Quantum FEL seeded with a Fock state and with a fixed ratio of 0 / = 0.1 analytics (green line) and numerics (black, dashed line) also agree well. Here both curves predict the constant value max 5 with denoting the gain length from Eq. (9). number and initial photon number 0 . This scaling explains the difference in Fig. 1 of max for a seeded FEL, and one starting from vacuum because of / 0 . In the lower part of Fig. 3 we depict max as a function of . Here we compare the analytical expression in Eq. (11) to the numerical simulation. For the start-up from vacuum we observe that both curves show the same behavior, apart from a small shift, already apparent in Fig. 1. Hence, we infer that the analytical approximation gives a reasonable estimate also for max . For a seeded FEL with the fixed ratio 0 / = 0.1, the numerical and the analytical curve also match very well with both predicting a constant value of max , in accordance with Eq. (11).

Off-resonant case
We now study the dependence of the maximum photon number on the detuning of the momentum of the electrons from We observe that max decreases and max increases for a growing deviation from resonance = /2. For both, max and 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 max is divided by while its numerical counterpart is divided by 0.78 .
resonance. For that, we restrict ourselves to the start-up from vacuum 0 = 0. According to the analytical solution from Eq. (8), the maximum photon number max is given by the elementary relation valid for 1, where we have used the expression for + from Eq. (A10).
Thus, max is different from zero only for −2 < < 2 , which gives rise to a gain bandwidth of /(2 ) in momentum space, in accordance with the exponential-gain regime [13].
Moreover, from Eq. (A13) we derive for 1 the asymptotic expression for the saturation length.
In Fig. 4 we show both max and max as a function of which reveals a perfect agreement between our analytical results and numerics. We note that max decreases for an increasing deviation from resonance = /2, while the value of max increases.

C. Variance of photon number
A quantum mechanical observable is not only characterized by its mean value, but also by its higher moments. The numerical solution of Eq. (B4) enables us to calculate these For the underlying numerical simulation we have assumed resonance = /2, = 10 4 electrons, and start-up from vacuum, that is 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 Δ 2 is more complicated. For example, close to max a dip occurs, where Δ 2 0.05 2 while max 0.8 . Hence, the value of Δ 2 is smaller, but roughly of the order of magnitude of 2 max corresponding to an almost chaotic behavior of the laser field. moments for the observables of a high-gain Quantum FEL. We now study the second moment, that is the variance of the photon number which is a measure of the intensity fluctuations of the emitted radiation.
In Fig. 5 we depict the variance of the photon number as a function of the wiggler length for resonance = /2, and for the start-up from vacuum with = 10 4 electrons. Similar to the mean value in Fig. 1, the variance shows an oscillatory behavior [27]. However, compared to the mean value (dotted line) the curve corresponding to the variance displays a richer structure with a dip close to max .
Here we find Δ 2 0.05 2 while max 0.8 . Hence, the variance is smaller, but of the same order of magnitude, as the square of the mean value and we deduce a nearly chaotic behavior of the radiation field [15]. The situation, however, changes, when we consider a seeded FEL illustrated in Fig. 6. Here we compare the Fano Factor [35] ≡ Δ 2 / ˆ depending on the wiggler length for two different initial states of the radiation field, that is a Fock state [27] and a coherent state characterized by the same mean number of photons.
While the variance for a Fock state vanishes, a coherent state possesses a Poissonian photon statistics with Δ 2 = ˆ . As time evolves we obtain in both cases super-but also sub-Poissonian photon statistics, in contrast to the broad distribution originating from the start-up from vacuum displayed in Fig. 5. Both curves show a sub-Poissonian statistics close to the first maximum of the mean photon number indicating a non-classical state of light. Moreover, we obtain a drastic increase of the fluctuations for larger values of , when the field initially was 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 = /2, = 10 3 electrons, and an initial mean photon number of 0 = 10 2 . In both cases we observe super-as well as sub-Poissonian behavior which are separated by the horizontal dotted line at 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 5 . Moreover, we find that the fluctuations for the initially coherent case drastically increase at around 10 gain lengths, while only slowly increasing for the case of a Fock state. in a coherent state. In contrast, for an initial Fock state a growth of the fluctuations is hardly visible.
In Fig. 7 we study the second moment of the field when it starts from a thermal state where Δ 2 = ˆ 2 . Although the variance of the photon number remains of the same order of magnitude as the square of the mean value, the corresponding ratio attains a prominent minimum close to the saturation length where the mean intensity assumes a maximum. Here the corresponding statistics with Δ 2 < ˆ 2 deviates significantly from the thermal distribution of the initial state. Hence, the intensity noise of the emitted light is at least partially decreased.

IV. EXPERIMENTAL REQUIREMENTS
In this section we address the challenges associated with realizing a Quantum FEL. Already from our elementary onedimensional description we can establish the most relevant experimental conditions. Constraints which go beyond the limits of our model are only mentioned here, but are studied in more detail in Ref. [12].
In Table I we have summarized the important parameters of our model of a Quantum FEL in the Bambini-Renieri frame as well as in the laboratory frame [39]. In addition, we have expressed these quantities in terms of the 'universal scaling' of Ref. [3], where¯≡ / with the Pierce parameter [38], is the analogue of our quantum parameter [40].

A. Need for optical undulator
From Table I we deduce the condition for the quantum regime in the laboratory frame, where e denotes the electron density while 0 , W , and 0 are the wiggler parameter, the wavelength of the wiggler in the laboratory frame, and the ratio of the kinetic energy to the rest energy of an electron, respectively. This relation implies that either increasing the electron energy 0 , or lowering the wiggler wavelength W leads to a decreasing value of the quantum parameter.
At the same time, we have to ensure that the gain length, given in Table I does not become unfeasibly large. We observe that high 0 as well as small W lead to a large gain length. However, while the scaling of with the energy 0 is quadratic, the dependence on the undulator wavelength scales only with −1/2 W . Hence, in order to satisfy Eq. (15) and at the same time minimizing the gain length , Eq. (16), we propose to operate a Quantum FEL with a small undulator wavelength and with a moderately high electron energy [41].
The requirement of a small undulator wavelength quite naturally forces us to employ an optical undulator [17], where the periodic array of magnets is replaced by counterpropagating laser fields [17,18,42,43]. For an efficient interaction between electrons and wiggler we also need according to Eq. (16) a relatively large value of the wiggler parameter 0 , that is a high intensity of the counterpropagating 'pump' laser.
However, a laser with the required intensity would not operate in a continuous way, but rather in a pulsed mode which decreases the interaction length. To overcome this problem the authors in Ref. [18] proposed a 'traveling-wave Thompson scattering' (TWTS) scheme. Instead of the usual head-on geometry for electrons and optical undulator, TWTS uses a side-scattering geometry, where a laser pulse with a tilted front interacts under an optimal angle with the electron beam. This procedure can considerably enhance the interaction length.

B. Electron beam
Apart from the condition in Eq. (15) on the quantum parameter which has led to constraints for the wiggler, we have the inequality Δ < for the momentum spread of the electrons which reflects itself in the required quality of the electron beam. According to Table I this condition translates into the inequality (17) for the relative energy spread of the electron beam in the laboratory frame with C denoting the Compton wavelength of the electron.
Nevertheless, we note that also here a short wiggler wavelength is advantageous since it raises the upper bound of the inequality. Equation (17) represents an ambitious requirement on the quality of the electron beam [10,44].
Indeed, for an efficient operation of a Quantum FEL the momentum spread Δ has to be smaller than the gain bandwidth 2 , that is due to momentum selectivity [13]. Since 1, Eq. (18) even lowers the maximally allowed energy spread.
Further experimental challenges due to the interaction geometry and intensity fluctuations of the pump laser as well as the required properties of the electron beam taking into account three-dimensional effects are discussed in more detail in Ref. [12].

C. Space charge vs spontaneous emission
So far, we have considered the dynamics of a Quantum FEL governed by a unitary time evolution neglecting processes which destroy the strong correlation between momentum jumps of the electrons and the emission or absorption of photons. Decoherence mechanisms of this kind can be for example spontaneous emission into all modes of the radiation field [45], or space-charge effects [46,47] due to the Coulomb interaction between the electrons. As these processes may eventually prevent the operation of a Quantum FEL, strong limits were imposed in Ref. [12] on the parameter regime for the electron beam and the wiggler field. We now show with the help of our results of Sec. III that these constraints can be weakened. I. Parameters of a Quantum FEL in the Bambini-Renieri (BR) frame (left), in the universal scaling of Ref. [3] (center), and in the laboratory frame (right). We consider the five adjustable parameters wavelength W of the undulator, its dimensionless field amplitude 0 , electron density e , dimensionless energy 0 of the electrons, and associated spread Δ 0 . Here, C and e are the Compton wavelength of the electron and the classical electron radius, respectively. In addition, we have introduced the longitudinal dimensionless energy of the electrons in the wiggler which is connected by the relation = 0 (1 + 2 0 ) −1/2 to the free energy and the wiggler parameter. BR frame universal scaling laboratory frame The discussion in Ref. [12] relied on estimating the typical length scales of the decoherence processes by classical theories. Hence, we consider the rates p and sp , that is the plasma wave number [38,47] of the electron beam and the inverse decay length [45,48,49], respectively, which both are listed in Table I.
To ensure a coherent time evolution over the total length of the wiggler, both processes have to occur on longer length scales than the interaction. Hence, in accordance with Ref. [12] we require the inequalities 1/(2 sp ) > and 1/ p > , that is sp < 1/2 and p < 1. In addition, we demand in the quantum regime that multi-photon processes are suppressed, that is 1. However, the parameters p , sp , and are not independent of each other due to their mutual dependence on 0 , e , W , and 0 . Indeed, we can relate [12] the wiggler length (19) to these three parameters and the gain length. Here we have used the definitions from Table I and introduced [50] the fine-structure constant f ≡ 2 e / C .
We emphasize that the values of the parameters on the righthand side of Eq. (19) can be chosen independently from each other. However, once this choice is made, the interaction length on the left-hand side is fixed.
In order to get an estimate for / we set p and sp to their upper bounds, that is p = 1 and sp = 0.5, respectively. For the example of = 0.25 we obtain from Eq. (19) the value g 5.9 (estimated limit) (20) for the maximally allowed interaction length. However, from Eq. (11) we derive the saturation length max g 23.5 (start-up from vacuum) (21) for start-up from vacuum with 0 = 0 and for = 10 9 electrons. This choice for is a typical number [38,51] for electron bunches in an FEL. Since max the coherent time evolution breaks down long before saturation is reached, and the maximally possible intensity is extremely decreased.
In contrast, for a seeded FEL with 0 = 0.1 we deduce from Eq. (11) the saturation length max g 5.1 (seeded FEL) (22) which is of the same order of magnitude as the allowed interaction length from Eq. (19) and thus the maximum intensity can be reached. As a consequence, we believe in accordance with Ref. [12] that the focus of the research on Quantum FELs should shift from SASE to seeded FELs. We emphasize, however, that this discussion is based on arguments borrowed from classical theories and that a rigorous quantum theory covering the full interplay between multi-photon processes, spontaneous emission, and space charge is necessary to prove our statements.

V. CONCLUSIONS
In the present article we have studied the mean intensity as well as the intensity fluctuations in a high-gain Quantum FEL. The reduction to two momentum levels limits the maximum intensity to a single emitted photon per electron. We have found that the necessary wiggler length for this maximum is significantly decreased if we consider a seeded FEL instead of SASE. Hence, the experimental realization of the former seems more feasible, especially with regard to the problematic requirements pointed out in Ref. [12]. Our results have also shown why the short wavelength of an optical undulator is necessary to fulfill the most important constraints for the operation of a high-gain Quantum FEL.
Moreover, we have observed that the time-evolved intensity fluctuations are mainly determined by the initial field stateranging from super-to sub-Poissonian statistics in case of a Fock or a coherent state with a high photon number, to a very broad photon distribution for vacuum.
To refine our model to more realistic scenarios we have to take space charge and spontaneous emission into all modes into account. In addition, relativistic effects such as slippage [52] of the radiation pulse over the electron bunch have to be included. However, these topics go beyond the scope of our article and are subject to further studies.

Dynamics of number operator
Our analytical approach is based on the decoupling of the Heisenberg equations of motion for the photon-number operator with the help of three constants of motion, in analogy to Ref. [28]. are constants of motion. We note that the three constantsˆ, , andˆ, as well as the photon-number operatorˆ≡ˆ † LˆL mutually commute with each other.
The second derivative ofˆwith respect to time reads and after insertingˆe ff from Eq. (A2) and calculating the nested commutator we arrive at (A7) When we express the right-hand side purely in terms of the operatorsˆ,ˆ,ˆ, andˆwe finally obtain the second-order differential equation [28] d 2d The dynamics ofˆis indeed decoupled from the electron operators. Unfortunately, we cannot solve Eq. (A8) by integration since it is a nonlinear equation of operators instead of numbers. Hence,ˆand dˆ/d do not necessarily commute.

Approximating operators as c-numbers
In order to find an estimate for the mean photon number ≡ ˆ we approximate the constant operators in Eq. (A8) by their expectation values at = 0, that is = ( +1), = 0 + , and = − 0 with = = /2. Here we have assumed that the field starts with the photon number = 0 , and initially all electrons are in the excited state close to = /2. Strictly speaking, this approximation is only valid as long as products of operators result in products of expectation values [28] when we form the total expectation value of Eq. (A8).
Then, we multiply the resulting c-number equation by d /d and integrate over time . This procedure yields the equation of motion By setting d /d = 0 in Eq. (A9) we observe that the maximum photon number is given by + . The other two roots of Eq. (A9), that is = 0 and = − − , correspond to the minimum and initial value of , and to an unphysical negative photon number, respectively.
We proceed by integrating Eq. (A9) and arrive at the expression which describes an elliptic integral.

Solution of elliptic integral
Next we invert the elliptic integral from Eq. (A11) in order to express the mean photon number as a function of the dimensionless time . Our result is then presented in terms of Jacobi elliptic functions [33].
These special functions are defined as inverse functions of the elliptic integral of first kind which is characterized by the amplitude and the modulus with 0 < < 1.
There exists a set of elliptic functions, for example sn ≡ sin and cn ≡ cos which are called sine amplitude and cosine amplitude, respectively. They vary between −1 and 1 and are 4 -periodic, with ( ) ≡ ( /2, ) denoting a complete elliptic integral of first kind [33].
We note that shows the asymptotic behavior [33] for approaching unity, that is → 1. We now return to the solution of the integral in Eq. (A11) and observe for 0 and / < 2 the relative ordering + ≥ ≥ 0 > − − for the roots of the denominator. Following Ref. [33] we proceed by performing the substitution where ≡ √︂ denotes the modulus corresponding to the sn function.
With the help of the substitution in Eqs. (A14) and (A15) we perform the integration in Eq. (A11). Solving the resulting expression for leads us finally to after using several fundamental identities [33] for Jacobi elliptic functions. Moreover, we have expressed our result in terms of the length ≡ and the gain length ≡ /(2 √ ) by recalling the relation = /(2 ) from Ref. [13].
Hence, we expand the total state vector [32] |Ψ( ) ≡ in terms of the single quantum number with the expansion coefficients = ( ). We note that varies between = 0 and = 0 + due to the constraint − /2 ≤ ≤ /2.
With the help of Eq. (B2) and the Schrödinger equation governed by the Hamiltonian from Eq. (3) we derive the equation of motion for the probability amplitudes where the coefficients are independent of . The solution of this linear system of differential equations relies on the diagonalization of a tridiagonal matrix with the dimension ( + 1) × ( + 1) which can be straightforwardly achieved by numerical methods. For this purpose, we choose the initial state | 0 , /2, /2 , that is all electrons initially are in the excited state while the laser field starts from = 0 , which leads to the initial condition (0) = δ , 0 for Eq. (B4).

Arbitrary initial state
So far, we have only considered a Fock state as initial field state. We easily generalize our approach to an arbitrary initial state characterized by the photon statistics .
In the present article, we are mainly interested in the photon statistics (B9) The coefficients , denote the matrix elements of the initial density operator for the laser field in photon number representation. The diagonal elements of this matrix define the photon statistics, that is ≡ , . In order to calculate the expectation value of any function = (ˆ) of the number operatorˆwe trace over the electrons and the laser field, that is corresponding to the case with initial photon number .This result also emerges by solving Eq. (B4), and calculating the expectation value of (ˆ) with respect to the state in Eq. (B3). Hence, we finally obtain the elementary expression for the expectation value of = (ˆ). This result arises due to our choice of the initial state | /2, /2 of the electrons and due to the fact that there is only one independent quantum number, . 38, 233 (1988), analogous results for the mean photon number were shown in Ref. [9], at least for the start-up from vacuum. In addition, we study in our article a seeded Quantum FEL and consider higher moments of the photon statistics as well as a nonzero detuning from resonance. [40] The universal scaling is motivated by a classical theory rather than a quantum mechanical one. In contrast, our approach adopts a different perspective by introducing the quantum parameter as the ratio of the two important frequency scales of the FEL interaction. Moreover, occurs in a natural way as the expansion parameter of the asymptotic expansion in Ref. [13]. However, from an experimental point of view it is often advantageous to employ the scaling of Ref. [3].