2 The distribution driven by a chirped frequency wave

Consider a one-dimensional problem of a tenuous electron distribution $f(v,x,t)$ driven by a travelling longitudinal field $E(t)\sin (kx-\int \unicode[STIX]{x1D714}_{d}\,\text{d}t)$ having slowly varying amplitude and frequency. Inclusion of the self-field of the distribution requires addition of the Poisson equation, the problem studied previously in application to Bernstein–Greene–Kruscal modes (Khain & Friedland Reference Khain and Friedland2010) and we will discuss the expected effect of this inclusion on the resonant capture probability in § 5. If one neglects the self-field, the single particle trajectory in this problem is governed by

Therefore, defining $I=kv$ , $\unicode[STIX]{x1D6E9}=kx$ , $\unicode[STIX]{x1D6E9}_{d}=\int \unicode[STIX]{x1D714}_{d}\,\text{d}t$ , $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6E9}-\unicode[STIX]{x1D6E9}_{d}$ and $\unicode[STIX]{x1D700}=keE/m$ , we have

which is the same as (1.2) and (1.3) with $a=1$ , so our resonant Hamiltonian is

where $H_{0}(I)=I^{2}/2$ and $\unicode[STIX]{x1D6FA}=I$ . Remarkably, the same Hamiltonian describes the classical evolution of the angular momentum of a polar molecule (in the rigid rotor approximation) driven by an electric field rotating in the plane of molecular rotation (Marcus, Friedland & Zigler Reference Marcus, Friedland and Zigler2005). Therefore, our equations are isomorphic to those encountered in the problem of capture of an ensemble of molecules into a continuing rotational resonance, the configuration known as the optical centrifuge (Karczmarek et al. Reference Karczmarek, Wright, Corkum and Ivanov1999).

We start our analysis by assuming a linear driving frequency chirp

where $\unicode[STIX]{x1D714}_{0}$ is the initial driving frequency. We introduce dimensionless time $\unicode[STIX]{x1D70F}=\sqrt{\unicode[STIX]{x1D6FC}}t$ and normalized $J=I/I_{c}$ , where $I_{c}$ is some characteristic value of $I$ . In these variables, the evolution equations (2.3) and (2.4) become

where $J_{0}=\unicode[STIX]{x1D714}_{0}/I_{c}$ and the two dimensionless parameters are

These parameters can be associated with three characteristic time scales in the problem, i.e. the chirping time scale $T_{ch}=\unicode[STIX]{x1D6FC}^{-1/2}$ (this scale is only relevant in the chirp-driven problem), the time scale associated with the amplitude of the driving perturbation $T_{d}=I_{c}/\unicode[STIX]{x1D700}$ and the scale of the nonlinearity $T_{nl}=I_{c}^{-1}$ . In terms of these time scales, the two parameters in (2.7) and (2.8) are $P_{1}=T_{ch}/T_{d}$ and $P_{2}=T_{ch}/T_{nl}$ , referred to as the driving and nonlinearity parameters in the following. Note that the normalized driving frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{d}/I_{c}$ can be written as $\unicode[STIX]{x1D714}=J_{0}+\unicode[STIX]{x1D70F}/P_{2}$ .

Figure 1. Passage through resonance with the cold beam distribution. (a) Initial distribution, (b) $P_{1}=0.1,P_{2}=30$ and (c) time varying $P_{1}$ case. The initial distribution is strongly perturbed, while 15.7 % of the particles are trapped in autoresonance with the drive for the time varying $P_{1}$ .

Next, we present the results of numerical simulations of our driven system for two initial particle distributions. Figure 1 illustrates the evolution of an initially monoenergetic particle beam $J=1$ at $\unicode[STIX]{x1D70F}=0$ with uniformly distributed values of $\unicode[STIX]{x1D6F7}$ , as shown in figure 1(a). Figure 1(b) shows the strongly perturbed distribution of these particles at $\unicode[STIX]{x1D70F}_{f}=60$ after passage through resonance (at $\unicode[STIX]{x1D70F}=15$ ) in the case of $P_{1}=0.1$ and $P_{2}=30.$ We choose the initial driving frequency $\unicode[STIX]{x1D714}(0)=J_{0}=0.5$ , so the final driving frequency is $\unicode[STIX]{x1D714}_{f}=2.5$ . Figure 1(c) shows the final distribution for the same parameters and initial conditions as in figure 1(a), but $P_{1}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D6FD}=0.0067$ (note that at the time $\unicode[STIX]{x1D70F}=15$ of crossing of the resonance, in this case $P_{1}=0.1$ ). The final distribution in this case is also strongly perturbed but, in contrast to figure 1(b), 15.7 % of the initial distribution in figure 1(c) forms a separated region in phase space, which is trapped in resonance and enters the autoresonant stage as its averaged $\overline{J}$ is accelerated, following the driving frequency continuously $\overline{J}\approx 0.5+0.0067\unicode[STIX]{x1D70F}/P_{2}$ .

Figure 2. Passage through resonance with the water bag distribution. (a) Initial distribution, (b) $P_{1}=0.1,P_{2}=30$ and (c) time varying $P_{1}$ case. Only the boundaries of the initial distribution are perturbed after passage through resonance, while 11.8 % of the distribution is trapped in autoresonance with the drive for time varying $P_{1}$ .

Our second initial distribution, shown in figure 2(a), is uniform in $J$ between $J_{10}=1$ and $J_{20}=2$ and in $\unicode[STIX]{x1D6F7}$ , corresponding to the usual water bag model (Bertrand & Feix Reference Bertrand and Feix1968; Berk, Nielsen & Roberts Reference Berk, Nielsen and Roberts1970). We again use $P_{2}=30$ , chirp the driving frequency as before by starting at $\unicode[STIX]{x1D714}(0)=0.5$ and allow it to cross the Cherenkov resonances with all the particles in the distribution at different times. The final distribution, after passage of $\unicode[STIX]{x1D714}$ through the distribution, in this case is shown for constant $P_{1}=0.1$ in figure 2(b) and increasing $P_{1}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D6FD}=0.0067$ in figure 2(c). One can see in the figure that in both cases, after passage through the resonances, the distributions remained uniform except in the boundary regions and that the whole distribution is shifted on average to lower energies (the phenomenon discussed in Friedland, Khain & Shagalov (Reference Friedland, Khain and Shagalov2006) and Schmit & Fisch (Reference Schmit and Fisch2012)). In addition, for the increasing $P_{1}$ case, some phase space area $\unicode[STIX]{x0394}S$ of the initial distribution is captured into resonance forming a ring of the same uniform initial density. This area is phase locked to the drive, continues its accelerating motion in $J$ as the driving frequency increases and defines the resonant capture probability $P=\unicode[STIX]{x0394}S/[2\unicode[STIX]{x03C0}(J_{20}-J_{10})]$ (in our example $P=11.8\,\%$ ). This capture probability is discussed next.

3 Resonant capture probability

Our theory below is based on analysing the dynamics governed by (2.7) and (2.8), which can be combined into a single second-order equation

where $D=P_{1}P_{2}$ . For constant $D$ , this equation describes a ‘pendulum’ subject to a constant ‘torque’ and is integrable. The pendulum moves in a tilted cosine potential

illustrated in figure 3(a). Figure 3(b) shows the phase space portrait of the dynamics. One can see that the phase space is divided into trapped and untrapped trajectories separated by the separatrix. The area of the separatrix in the $D=\text{const}$ case remains constant and all particles trapped inside the separatrix initially remain trapped at later times despite the variation of the driving frequency, while all the untrapped trajectories remain untrapped. Therefore, if the evolution of the separatrix starts in the empty region of phase space, its area will remain empty during the passage through the water bag distribution. The resonant capture probability in this case is zero, as seen in figure 2(b). Importantly, as the empty separatrix enters the distribution, it mainly disrupts the evolution of particles close to the separatrix, because other particles experience non-resonant forcing which effectively averages to zero. Therefore, in the water bag simulations in the constant $D$ case, we observe a significant disruption of the boundaries of the initial distribution, but as the separatrix enters the bulk of the distribution, the boundaries of the separatrix define a nearly empty hole surrounded by a uniform distribution having initial phase space density, because of the incompressibility of the phase space (Goldstein Reference Goldstein1980). The situation changes drastically when the parameter $D$ varies (increases in our case) in time. We assume adiabatic variation of $D$ (see the discussion of this condition below), which conserves the actions (areas in phase space) associated with all trapped trajectories, but those in the vicinity of the separatrix. As the separatrix area grows, the phase space hole entering the water bag distribution from outside will remain mostly empty and preserves its area, while because of the incompressibility of the phase space fluid, the extension of the hole inside the growing separatrix will be filled by the particles having the initial phase space density. These trapped particles comprise the ring around the hole as seen in figure 2(c). Using these arguments and assuming that at all times the separatrix width $\unicode[STIX]{x0394}J$ in $J$ is small compared to the width of the initial distribution (this is our weak drive assumption, to be discussed in the next section), one can estimate the capture probability of newly trapped particles in the problem as

Figure 3. The effective potential equation (3.2) (a) for $P_{1}P_{2}=3$ and the phase space portrait of the associated dynamics (b). The boundary of the red filled area in the bottom panel is the separatrix. The equal energy lines in the bottom panel are separated by energy steps of $\unicode[STIX]{x03C0}$ .

where $S_{in}$ and $S_{out}$ are the instantaneous areas of the separatrix at the entrance and exit from the initial distribution. Here the area of the separatrix in ( $J,\unicode[STIX]{x1D6F7}$ ) phase space is

where $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6F7}^{s}$ is the velocity of the ‘pendulum’ on the separatrix. The latter is found by observing that the energy at the separatrix is

where $\unicode[STIX]{x1D6F7}^{\ast }$ is the value of $\unicode[STIX]{x1D6F7}$ ( $\text{mod}(2\unicode[STIX]{x03C0})$ ) at the local maximum of the quasipotential (3.2) and that $\sin \unicode[STIX]{x1D6F7}^{\ast }=-1/D$ , $\cos \unicode[STIX]{x1D6F7}^{\ast }=-\sqrt{1-D^{-2}}$ . Then,

or by using (3.5) and introducing $\unicode[STIX]{x1D717}=\unicode[STIX]{x1D6F7}^{s}-$ $\unicode[STIX]{x1D6F7}^{\ast }$ , we have

where $A=(D^{2}-1)^{1/2}$ . Therefore, the area of the separatrix (3.4) can be evaluated from

Figure 4. The rescaled area of the separatrix $P_{2}S$ versus parameter $D=P_{1}P_{2}$ (blue solid line) and its asymptotic value $16D^{1/2}$ (red dashed line).

We show $P_{2}S$ versus $D$ in figure 4. Note that no separatrix (trapped trajectories) exists if $D<1$ , $S$ is an increasing function of $P_{1}$ and asymptotically, as $D$ becomes large, $P_{2}S\rightarrow 16D^{1/2}$ . In the example in figure 2, (3.3) yields $Q=12.6\,\%$ , in good agreement with simulations. The developments above can be also used for calculating the trapping probability for resonant passage through an arbitrary initial distribution $F_{0}(J)$ (normalized as $\int F_{0}(J)\,\text{d}J=1/2\unicode[STIX]{x03C0}$ ). Indeed, define the resonant value $J_{r}$ , which follows the driving frequency continuously

If the driving frequency increases during an infinitesimal time step $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F},$ then $J_{r}$ increases by $\unicode[STIX]{x1D6FF}J_{r}=\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}/P_{2}$ and the separatrix area $S$ changes by $\unicode[STIX]{x1D6FF}S$ , the fraction of the trapped particles will change by

In all examples in this work, we assume that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}P_{1}>0$ , so $\unicode[STIX]{x1D6FF}S>0$ and, therefore, (3.10) describes the fraction of newly trapped particles. Nevertheless, if the instantaneous $\text{d}S/\text{d}J_{r}$ is negative, i.e. $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}P_{1}<0$ , equation (3.10) describes the fraction of particles leaving the separatrix, i.e. detrap. By integration of (3.10), after a finite time, the trapping probability is

Figure 5. The chirp-driven phase space distribution at different times: (a) the initial shifted Gaussian distribution at $\unicode[STIX]{x1D70F}=0$ , (b) a resonant phase space hole in the distribution at $\unicode[STIX]{x1D70F}=90$ and (c) a resonant bump in the tail of the distribution at $\unicode[STIX]{x1D70F}=180$ . The colour coding represents probability density in phase space.

where $\unicode[STIX]{x1D70F}_{1}$ and $\unicode[STIX]{x1D70F}_{2}$ are the initial and final times, respectively, and $Q_{0}$ is the fraction of the particles inside the separatrix initially. For $\text{d}S/\text{d}J_{r}>0$ , this result can be interpreted probabilistically (Neishtadt Reference Neishtadt1975), viewing $(1/2\unicode[STIX]{x03C0})(\text{d}S(J_{r})/\text{d}J_{r})$ as the probability of resonant capture of uniformly distributed initial conditions in a thin strip in phase space of size $2\unicode[STIX]{x03C0}\,\text{d}J_{r}$ as the resonant action $J_{r}$ passes this strip. Note that (3.11) reduces to (3.3) in the water bag model, where $F_{0}(J)=1/(2\unicode[STIX]{x03C0}(J_{20}-J_{10}))$ for $J_{10}<J<J_{20}$ and zero elsewhere. As a more complex example, we consider a shifted Gaussian initial electron distribution, $F_{0}(J)=[4\unicode[STIX]{x03C0}\sqrt{2\unicode[STIX]{x03C0}}]^{-1}\exp [-(J-2)^{2}/8]$ . We again use parameters $P_{1}=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D6FD}=0.0067,$ $P_{2}=30$ and show in figure 5 the distribution $F(J,\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F})$ at the initial, intermediate and final times of the simulation, as the driving frequency $\unicode[STIX]{x1D714}$ varies from 0 to 6. The formation of the resonant region inside the separatrix is seen in the figure, with colours indicating the origin in $J$ of different trapped particles. Additional information from this simulation is presented in figure 6, showing the distribution of the electrons in $J$ , i.e. $f(J,\unicode[STIX]{x1D70F})=\int F(J,\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D6F7}$ at times corresponding to those in figure 5. One can see the deviation of $f(J,\unicode[STIX]{x1D70F})$ from the initial distribution (dashed line) and the appearance of a resonant bump in the distribution which includes resonantly trapped particles. Finally, figure 7 compares the theoretical prediction (3.11) for the trapping probability $Q$ in this example for different initial driving frequencies $\unicode[STIX]{x1D714}(0)=J_{0}$ with simulations, showing a very good agreement.

Figure 6. The distribution $f(J,\unicode[STIX]{x1D70F})$ of the electrons (blue solid lines) at the times shown in figure 5. The initial distribution is shown by red dashed lines.

Figure 7. The resonant capture probability for initially Gaussian electron distribution versus initial resonant $J_{0}$ . The theoretical prediction is shown by the blue solid line, the yellow diamonds represent the results of the numerical simulations.

4 Validity conditions

We have made the adiabaticity and weak drive assumptions in developing the theory above and discuss these assumptions next. Both will be analysed in $D=P_{1}P_{2}\gg 1$ case (the width of the separatrix in $\unicode[STIX]{x1D6F7}$ in this case is nearly $2\unicode[STIX]{x03C0}$ ). In estimating the validity conditions of the theory we linearize equations (2.7), (2.8) in $\unicode[STIX]{x1D6F7}$

where $\unicode[STIX]{x1D6FF}J=J-J_{r}$ . The characteristic frequency of oscillations of $\unicode[STIX]{x1D6FF}J$ and $\unicode[STIX]{x1D6F7}$ in this system is $\unicode[STIX]{x1D708}=D^{1/2}$ . Then the adiabaticity condition for deeper trapped orbits is $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}/\unicode[STIX]{x1D708}^{2}\ll 1$ , yielding (for $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}P_{1}=\unicode[STIX]{x1D6FD}$ )

This condition preserves the area of the empty phase space hole inside the separatrix passing a water bag distribution, while the remaining area of the separatrix in this model is filled uniformly by trapped particles, as described above. Finally, we estimate the weak drive condition. The amplitude of oscillations of $\unicode[STIX]{x1D6FF}J$ is $\unicode[STIX]{x0394}J=P_{1}\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}/\unicode[STIX]{x1D708}$ . By using $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}\sim O(1)$ and assuming $|J-J_{0}|\sim O(1)$ for typical values of $J$ in the distribution, the small $\unicode[STIX]{x0394}J$ (weak drive) condition $\unicode[STIX]{x0394}J\ll |J-J_{0}|$ can be written as

The inequalities (4.3) and (4.4) in the trapping regime comprise the validity conditions of our theory and are satisfied in all our numerical examples. We illustrate these conditions in $P_{1,2}$ parameter space in figure 8. The condition $D=1$ (the separatrix exists for $D>1$ only) is also shown in the figure by dashed line. The region of the applicability of the theory is coloured in blue.

Figure 8. The region of validity of the theory in $P_{1,2}$ parameter space is coloured in blue. The black lines represent the weak drive and adiabaticity (for $\unicode[STIX]{x1D6FD}=2$ ) limits and the limit of the existence of the separatrix is shown by the dashed line.