Autoresonant excitation of space-time quasicrystals in plasma

Photonic crystals [1, 2, 3, 4] built from conventional materials are routinely employed to focus, polarize, and manipulate light pulses. A periodic array of alternating dielectrics and plasmas called plasma photonic crystals (PPCs) has been the subject of much interest [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] owing to its optical properties. Besides crystals, Levine and Steinhardt [20] introduced quasicrystals — materials with properties that are ordered in space but do not possess an exact periodicity. Photonic quasicrystals have been studied extensively [21, 22, 23, 24, 25, 26, 27, 28]. Moreover, crystals can be periodic not just in space but also in time. For example, the optical properties of photonic time crystals with a refractive index varying periodically in time have been investigated in Refs. [29, 30, 31, 32, 33, 34, 35, 36]. The optical properties of the structures possessing periodicity both in space and time have been studied in Refs. [36, 37, 38, 39, 40, 41, 42, 43].

Plasma photonic crystals have a major drawback where the plasmas are contained within solid material: they break down at high field intensity and are, therefore, incapable of controlling the laser pulses that are essential for many high energy density science applications. Purely plasma based structures, on the other hand, can withstand high intensity pulses. Many concepts for plasma-based optical elements have been proposed and built [44]. Plasma channels [45] routinely focus lasers for particle acceleration experiments. Over two decades ago, it was realized that resonant interactions [46] could be used for compression of intense pulses (replacing large compressor gratings). Plasma mirrors [47, 48, 49, 50] are routinely used [51, 52] at the National Ignition Facility (NIF) to improve performance in inertial fusion experiments. Laser-sculpted plasma grating structures [53, 54] as well as polarization control using plasma structures [55, 56, 57, 58] have been investigated. A significant challenge of plasma gratings is that their efficacy depends on the maximum variation in the index of refraction that can be achieved at the required spatial scale. Plasma density is well below critical density, and the density modulation should be as large as possible. This naturally leads to an exploration of nonlinear waves.

In this paper, we propose formation and control of space-time quasicrystalline structures in plasma through excitation of strongly nonlinear large amplitude multiphase ion acoustic waves that modulate plasma density in the desired way. It is known that a linear standing wave of the form $U(x,t)\propto\cos(kx)\cos(\omega t)$, which is periodic in both time and space, is formed by superposing two linear traveling waves of the same frequency $\omega$, but propagating in the opposite direction. Nonlinearity of the media in some cases allows a generalization of linear standing waves to a waveform $U(x,t)=F\left(kx-\omega t,kx+\omega t\right)$, where $F$ is a $2\pi$-periodic nonlinear function of two phase variables and is also periodic in time and space. Recently, it was demonstrated that such nonlinear structures could be formed in plasma in the form of electron plasma [59] and ion acoustic [60] waves. It is also known that even more complex multiphase constructs of the form $U(x,t)=F\left(\theta_{1},\theta_{2},\ldots,\theta_{N}\right),$ where each phase is $\theta_{i}=k_{i}x-\omega_{i}t$, exist in other physical systems described by integrable nonlinear wave equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger (NLS), and sine-Gordon (SG) equations [61]. Generally, multiphase functions $F$ are very nontrivial and can be described by a complex analysis based on the Inverse Scattering Transform (see, for example, Ref. [62] for the KdV case). The multiphase waves are $2\pi$-periodic in each of the phase variables, but if at least two of $k_{i}$ or $\omega_{i}$ are not commensurate, function $F$ is not exactly periodic in time and/or space and, thus, comprise a family of nontrivial space-time quasicrystals. But how can one excite such a multiphase wave in a given physical system? Because of the complexity of the waveform, a direct realization of a multiphase wave requires setting up precise initial conditions, making it an impractical, if not impossible, strategy for experiments even in the case of just two phases $\theta_{1}$, $\theta_{2}$. A possible way to circumvent the experimental difficulty of setting up the precise initial conditions is exploiting phase locking with small amplitude chirped-frequency traveling waves. In the past this approach was used in exciting multiphase solutions for integrable systems: KdV [63], NLS [64], SG [65], and the periodic Toda lattice [66]. The autoresonant excitation was also demonstrated for single phase large amplitude ion acoustic waves [67, 68]. This suggests that perhaps the autoresonance could be used in a more general way to excite multiphase solutions for ion acoustic waves.

Here, we demonstrate, using a warm fluid plasma model, that a two-phase, strongly nonlinear ion acoustic wave can be generated and controlled. The wave is created by starting from zero and autoresonantly driving the system with two small amplitude chirped-frequency ponderomotive traveling waves. We show that slow passage of the driving waves through resonances in the plasma results in the continuing autoresonant (phase locked with both drives) excitation of the wave. The system sustains this double autoresonance as the driving frequencies vary in time by increasing the amplitude of the excited waveform, creating an extremely large amplitude space-time quasicrystal in plasma. This result is surprising and suggests some degree of integrability in the problem. We conjecture that this integrability is related to the fact that in the limit of small amplitude, ion acoustic waves are reduced to the KdV-type equation [69]. Since there exist many continuous physical systems approximated by the KdV, NLS, and SG equations, we expect that autoresonant space-time quasicrystals can be formed similarly in all these systems.

The paper is organized as follows. In Sec. II, we formulate the problem within a warm fluid plasma model and present a fully nonlinear numerical solution. In Sec. III, by using the Lagrangian formulation of the fluid equations and applying Whitham’s averaged variational principle [70], we derive an analytical weakly nonlinear theory and demonstrate that it agrees well with the fully nonlinear simulations. In Sec. IV, we use our weakly nonlinear theory to study how to choose the parameters required to excite a two-phase autoresonant solution as well as discuss its threshold nature. Finally, in Sec. V, we summarize and discuss our results.

For convenience, let us introduce two new potentials $\psi$ and $\sigma$ via $u=\psi_{x}$, $n=1+\sigma_{x}$. The system of the nonlinear equations (1)–(3) in terms of the new variables is then

Equations (7)–(9) can be derived from the Lagrangian variational principle $\delta\left(\int Ldxdt\right)=0$ with the corresponding Lagrangian density $L$ given by

The Lagrangian form of the problem together with the slow adiabatic synchronization (autoresonance) procedure we employ to excite multiphase waves suggest that it should be possible to use Whitham’s averaged variational principle [70] to derive weakly nonlinear analytical results for our problem.

We proceed by writing the following ansatz describing the two-phase [$\theta_{1}=k_{1}x-\int\omega_{1}\left(t\right)dt$, $\theta_{2}=k_{2}x-\int\omega_{2}\left(t\right)dt$] solutions for the potentials $\sigma$, $\psi$, $\varphi$:

Here, we view the amplitudes $\tilde{A}_{10}$, $\tilde{A}_{01}$, $\tilde{B}_{10}$, $\tilde{B}_{01}$, $C_{10}$, $C_{01}$ as the first-order coefficients, while the amplitudes $\tilde{A}_{20}$, $\tilde{A}_{02}$, $\tilde{B}_{20}$, $\tilde{B}_{02}$, $C_{20}$, $C_{02}$, $\tilde{A}_{11}$, $\tilde{A}_{1,-1}$, $\tilde{B}_{11}$, $\tilde{B}_{1,-1}$, $C_{11}$, $C_{1,-1}$, and $C_{00}$ as the second-order coefficients. It is also convenient to write the driving potential $\varphi_{d}$ in the form with explicit phase mismatches $\Phi_{i}=\theta_{i}-\theta_{d,i}$ ($i=1,2$) between phases of the solutions $\theta_{1}$, $\theta_{2}$ and the driving phases $\theta_{d,1}$, $\theta_{d,2}$:

Furthermore, here and in the following it is assumed that the phases $\theta_{1}$, $\theta_{2}$ are varying rapidly, while all the coefficients in our ansatz and $\Phi_{1}$, $\Phi_{2}$ are slow functions of time.

In the linear case without phase mismatches we have the following solutions:

The next crucial step is to find the averaged Lagrangian density over the rapidly varying phases $\theta_{1}$, $\theta_{2}$:

After long but straightforward calculations one can obtain the average of Eq. (10) over the rapidly varying phases. This derivation of the averaged Lagrangian can be found in Appendix A.

Equations (38)–(41) have the same form as a weakly nonlinear system studied in Ref. [73]. Such a system is described by the following Hamiltonian for the effective action ($I_{1}$, $I_{2}$) and angle ($\Phi_{1}$, $\Phi_{2}$) variables:

As shown in Ref. [73], the possibility of double autoresonance is determined by the signs of $D=ac-b^{2}$ and $\alpha_{1}\alpha_{2}$: for the double autoresonance to occur, they must have the same sign. From Eqs. (38)–(41), we see that in the case of the double autoresonance the asymptotic large $t$ solutions for the actions are given by

Thus, in addition, for the double autoresonance to actually happen the asymptotic actions $\bar{I}_{1}$, $\bar{I}_{2}$ defined above must be positive.

There are two possible situations, depending on whether $k_{1}$ and $k_{2}$ have the same sign. (1) If $k_{1}k_{2}>0$, we have $b<0$, $a<0$, $c<0$. In this case $D$ can have both positive and negative values. Figure 6 shows the lines for $D=0$ in the plane formed by the absolute values of the phase velocities of the driving waves $\left(\omega_{1}/\left|k_{1}\right|,\omega_{2}/\left|k_{2}\right|\right)$ for different values of the ion thermal velocity $u_{th}$. We can see that the ion thermal velocity $u_{th}$, though not an extremely sensitive parameter, nevertheless determines the regions of positive and negative values of $D$. Thus the cold ion model should be used carefully and the thermal ion velocity should in general be taken into account. In the region where $D>0$, for the double autoresonance to occur $\alpha_{1}\alpha_{2}$ must be positive. In addition, for positive $\bar{I}_{1}$, $\bar{I}_{2}$ we must have $\left|b\right|/\left|a\right|<\alpha_{2}/\alpha_{1}<\left|c\right|/\left|b\right|$. In contrast, if $D<0$, we must have $\alpha_{1}\alpha_{2}<0$. However, in this case it is impossible for both $\bar{I}_{1}$ and $\bar{I}_{2}$ to be positive at the same time. Thus, if $k_{1}k_{2}>0$, the double autoresonance is possible only in the region where $D>0$; in this region we also must have $\alpha_{1}\alpha_{2}>0$ and $\left|b\right|/\left|a\right|<\alpha_{2}/\alpha_{1}<\left|c\right|/\left|b\right|$. (2) The second possibility is $k_{1}k_{2}<0$, then we have $b>0$, $a<0$, $c<0$. In this case $D$ is always positive, and, consequently, for double autoresonance we must have $\alpha_{1}\alpha_{2}>0$. In addition, to have positive $\bar{I}_{1}$, $\bar{I}_{2}$ only the case when both $\alpha_{1},\alpha_{2}$ are positive works. Thus, in the case where $k_{1}k_{2}<0$, the double autoresonance is possible when $\alpha_{1},\alpha_{2}>0$. We do not deal with the degenerate case of $k_{1}=k_{2}$ in this paper.

We have demonstrated by means of nonlinear numerical simulations that it is possible to create quasicrystalline spatiotemporal structures in plasma by exciting large amplitude two-phase ion acoustic waves nonlinearly phased locked into the corresponding small amplitude traveling wave drives with chirped frequencies. We have used the Lagrangian formulation and Whitham’s averaged variational method to derive analytical results describing the weakly nonlinear evolution of the system. We have applied the weakly nonlinear analytical theory to determine the parameters necessary for the successful excitation and control of multiphase waves. The nonlinearly excited quasicrystalline structures remain even after we turn off the drive. Thus, the space-time quasicrystalline structure in plasma can be excited and then used independently for the purpose of plasma photonics experiments. While our warm fluid model does not have dissipation or noise due to collisions or Landau damping, for example, and, generally speaking, its long-time stability must be addressed by kinetic or particle-in-cell (PIC) simulations, it is apparent that, depending on the time scales of the problem, this structure can be considered as at least a dissipative space-time crystal. We also note that we do not address in this paper whether the driven space-time crystal is a “true” time crystal in the sense of spontaneous symmetry breaking as proposed in Refs. [76, 77]; for more discussion regarding this see Refs. [78, 35, 79].

It is expected that, using our technique, similar structures can be driven in other systems, for example dust acoustic waves in complex plasmas [80]. We expect that multiphase solutions when the number of drives exceeds two are also possible; however, it will be harder to analyze such a system analytically. Finally, we point out that beyond any practical application as a plasma photonic (or accelerating) structure, the possibility of exciting multiphase solutions for in general non-integrable warm ion acoustic waves system is in itself an important fundamental result in the field of nonlinear dynamics. These techniques developed for the excitation of large amplitude ion acoustic waves may be applied to other partial differential equation systems that can support multiphase nonlinear waves.

This work was supported by NSF-BSF Grant No. 1803874 and US-Israel Binational Science Foundation Grant No. 2020233.