Rabi-Bloch oscillations in spatially distributed systems: Temporal dynamics and frequency spectra

We consider one-dimensional chains of two-level quantum systems coupled via tunneling. The chain is driven by the superposition of dc and ac fields in the strong coupling regime. Based on the fundamental principles of electrodynamics and quantum theory, we have developed a generalized model of quantum dynamics for such interactions, free of rotating-wave approximation. The system of equations of motion was studied numerically. We analyzed the dynamics and spectra of the inversion density, dipole current density, and tunneling current density. In the case of resonant interaction with the ac component, the particle dynamics exhibits itself in the oscillatory regime, which may be interpreted as a combination of Rabi and Bloch oscillations with their strong mutual influence. Such scenario for an obliquely incident ac field dramatically differs from the individual picture of both types of oscillations due to the interactions. This effect is counterintuitive because of the existence of markedly different frequency ranges for such two types of oscillations. These dynamics manifest themselves in multiline spectra in different combinations of Rabi and Bloch frequencies. The effect is promising as a framework of a new type of spectroscopy in nanoelectronics and electrical control of nanodevices.

Figure 1

(a) General illustration of the periodic two-level atomic chain used as a model, indicating RBO. It is excited with obliquely incident ac field in the strong coupling regime and driven with dc voltage applied at the ends. The dipole moments (red arrows) are identical and oriented arbitrarily with respect to the axis. (b) The electrons are localized inside the quantum wells which are separated by potential barriers of identical form and value. The neighboring atoms are coupled via interatomic tunneling with different values of penetration $t_{a,b}$ at the ground and excited states (green arrows). Ground and excited energy levels of single two-level atom separated by the transition energy $\hslash {\omega }_0$.

Figure 2

Space-time distribution of inversion density. Here, (a–f) notation corresponds to Table 1. Here, the quantum transition frequency is taken as the frequency unit, ${\mathrm{\Omega }}_B=3.9\times {10}^{-3}$ (corresponds to $E_{\mathrm{dc}}=1.95\mathrm{kV}/\mathrm{cm}$), ${\mathrm{\Omega }}_R=2.5\times {10}^{-2}$, $t_a=3.5\times {10}^{-2}$, interatomic distance $a=20\mathrm{nm}$. The initial state of the chain is an excited single Gaussian wave packet. Gaussian initial position and width are $p^{\prime }=80$, $g=20$, respectively, $ka=-0.624$. Number of the atoms, $N=128$.

Figure 3

Space-time distribution of inversion density for short times. Here, (c,e) notation corresponds to Table 1. Spatial beatings correspond to the Floquet harmonics in the electron diffraction by the lattice induced by the ac field at the case of oblique incidence. As a result, the initial single Gaussian decays into the set of subpackets, which synchronously move and coherently oscillate in the large-time regime (Fig. 2). All parameters are identical to Fig. 2.

Figure 4

Frequency spectra of the inversion density. Here, (a–f) notation corresponds to Table 1. The quantum transition frequency is taken as the unit. All parameters are identical to Fig. 2. For better discrimination, the results presented in this figure are multiplied by the factor ${10}^{-2}$.

Figure 5

Space-time distribution of the density of dipole current. Here, (b–f) notation corresponds to Table 1. All parameters are identical to Fig. 2. Inserts show scaled-up space-time distribution in the areas marked via the black rectangles at the main panels. For better discrimination, the results presented in this figure are multiplied by the factor ${10}^4$.

Figure 6

Frequency spectra of dipole current. Here, (b–f) notation corresponds to Table 1. The quantum transition frequency is taken as the unit. All other parameters are identical to Fig. 2.

Figure 7

Space-time distribution of the tunnel current density. Here, (a,c–f) notation corresponds to Table 1. All parameters are identical to Fig. 2. For better discrimination, the results presented in this figure are multiplied by the factor ${10}^4$.

Figure 8

Frequency spectra of tunnel current. Here, (b–f) notation corresponds to Table 1. The quantum transition frequency is taken as the unit. All other parameters are identical to Fig. 2.

Figure 9

Plot of inversion against a dimensionless time at the atom with number 40. Here, (c–f) notation corresponds to Table 1. Direct solution: blue line; RWA solution: red line. The initial state of the chain is an excited single Gaussian wave packet. All units and parameters are identical to Fig. 2. One can see that RWA correctly reproduces the qualitative dynamics of inversion, but allows rather high inaccuracies in detail description.

Figure 10

Plot of inversion against a dimensionless time at the atom with number 60. Here, (c–f) notation corresponds to Table 1. Direct solution: blue line; RWA solution: red line. The initial state of the chain is an excited single Gaussian wave packet. All units and parameters are identical to Fig. 2.

Figure 11

Plot of inversion against a dimensionless time at the atom with number 80. Here, (c–f) notation corresponds to Table 1. Direct solution: blue line; RWA solution: red line. The initial state of the chain is an excited single Gaussian wave packet. All units and parameters are identical to Fig. 2.

Figure 12

Space-time distribution of inversion density for the initial state in the form of the Gaussian analog of the inversion trapped state given by Eq. (34). Here, the quantum transition frequency is taken as the frequency unit. Gaussian initial position and width are $p^{\prime }=80$, $g=20$, respectively, Here, interatomic distance $a=20\mathrm{nm}$, $t_a=3.5\times {10}^{-2}\mathrm{eV}$, $t_b=3.5\times {10}^{-3}\mathrm{eV}$, and number of atoms is 128. (i) BO case: ${\mathrm{\Omega }}_B=3.9\times {10}^{-3}$, ${\mathrm{\Omega }}_R=0$. No BOs have been observed, because of their small amplitude due to the weak tunneling penetration over the barrier for the ground state; (ii) RO case: ${\mathrm{\Omega }}_B=0$, ${\mathrm{\Omega }}_R=2.5\times {10}^{-2}$, standing wave ($k=0$). No ROs have been observed, because of the special interference mechanism of inversion trapping. Progressive motion of the wave packet is dictated by the nonzero value of its initial momentum. (iii) RBO case: ${\mathrm{\Omega }}_B=3.9\times {10}^{-3}$, ${\mathrm{\Omega }}_R=2.5\times {10}^{-2}$, standing wave ($k=0$). Both RO and BO are observed and progressive motion is suppressed due to the combined action of ac and dc fields; (iv) RO case: ${\mathrm{\Omega }}_R=2.5\times {10}^{-2}$, ${\mathrm{\Omega }}_B=0$, traveling wave ($ka=-0.624$). No ROs have been observed, similar to case (ii). Progressive motion of the wave packet is accompanied by the spatial beatings dictated by diffraction on the lattice induced by the traveling wave; (v) RBO case: ${\mathrm{\Omega }}_B=3.9\times {10}^{-3}$, ${\mathrm{\Omega }}_R=2.5\times {10}^{-2}$, traveling wave ($ka=-0.624$). Both RO and BO are observed similar to case (iii), and accompanied by the spatial beatings similar to case (iv); Case (v): the simultaneous action of both dc and ac fields with $k\ne 0$.