Scattering of charged particles on two spatially separated time-periodic optical fields

We consider a monoenergetic beam of moving charged particles interacting with two separated oscillating electric fields. Time-periodic linear potential is assumed to model the light-particle interaction using a nonrelativistic, quantum mechanical description based on Gordon-Volkov states. Applying Floquet theory, we calculate transmission probabilities as a function of the laser field parameters. The transmission resonances in this Ramsey-like setup are interpreted as if they originated from a corresponding static double-potential barrier with heights equal to the ponderomotive potential resulting from the oscillating field. Due to the opening of new “Floquet channels,” the resonances are repeated at input energies when the corresponding frequency is shifted by an integer multiple of the exciting frequency. These narrow resonances can be used as precise energy filters. The fine structure of the transmission spectra is determined by the phase difference between the two oscillating light fields, allowing for the optical control of the transmission.

Figure 1

Schematic view of the setup we consider. In region 1, we have an incident monoenergetic free particle wave propagating towards the oscillating time-dependent potential localized in a finite region with length $L$, inducing reflected and transmitted waves in regions 1 and 5, respectively. In the interaction regions (2 and 4), superpositions of Gordon-Volkov states [see Eq. (11)] are present.

Figure 2

Cycle-averaged transmission probability $〈T〉$ as a function of the energy of the incoming particle $E_0$. Parameters in atomic units are $F_0=0.00488, L=200, d=400, {\phi }_0=\pi$, and $\omega =0.057322$, corresponding to a wavelength of 800 nm.

Figure 3

Cycle-averaged transmission probability $〈T〉$ (red solid line) and transmission probability for the corresponding static double barrier (blue dashed line, see the main text for more details) as a function of the energy of the incoming particle $E_0$ in atomic units. Parameters are $F_0=0.1, L=10, d=30, \omega =0.2$, and ${\phi }_0=\pi$. For low energies, the cycle-averaged transmission probability has peaks at the same input energies where transmission resonances occur in the static double-barrier model.

Figure 4

Probability densities of the first three localized states in the double-barrier system. Localized states with increasing eigenenergies are denoted by blue, red, and green lines, respectively. As a reference, the potential barriers are drawn by black dashed lines.

Figure 5

Cycle-averaged transmission probabilities $〈T〉$ as a function of the incoming particle's energy $E_0$ in atomic units. Parameters: $L=10, d=10$, and ${\phi }_0=\pi$. The open circles and triangles correspond to the first and second scattering resonances, respectively.

Figure 6

Cycle-averaged transmission probability $〈T〉$ as a function of incoming particle's energy $E_0$ in atomic units. Parameters: $F_0=0.1, L=10, d=10, \omega =0.3$, and ${\phi }_0=\pi$. All the insets show the individual current contributions $j_n^T$ of the Floquet channels to the transmission probability for different values of $E_0.$

Figure 7

Transmission probabilities as a function of the phase difference ${\phi }_0$ for different lengths $d$ of region 3. System parameters: $E_0=0.025, {\lambda }_{\mathrm{dB}}=28.0993, F_0=0.1, L=10$, and $\omega =0.2$.

Figure 8

Density plot of $j(x,t)$ as a function of time and coordinate $x.$ For parameters $E_0=0.06, F_0=0.1, L=10, d=10$, and $\omega =0.2,$ panels (a) and (b) correspond to the maximal and minimal transmission probabilities, respectively. Numerically, $\langle T\rangle =0.8941$ at ${\phi }_0=3.3772$ in panel (a), and $\langle T\rangle =0.392$ at ${\phi }_0=0.7226$ in panel (b). Horizontal dashed lines indicate the boundaries of the different regions, and vertical ones correspond to the zeros of the classical force exerted on the charged particle. The sign of this force is also shown in the various space-time regions.