Single-particle model of a strongly driven, dense, nanoscale quantum ensemble

We study the effects of interatomic interactions on the quantum dynamics of a dense, nanoscale, atomic ensemble driven by a strong electromagnetic field. We use a self-consistent, mean-field technique based on the pseudospectral time-domain method and a full, three-directional basis to solve the coupled Maxwell-Liouville equations. We find that interatomic interactions generate a decoherence in the state of an ensemble on a much faster time scale than the excited-state lifetime of individual atoms. We present a single-particle model of the driven, dense ensemble by incorporating interactions into a dephasing rate. This single-particle model reproduces the essential physics of the full simulation and is an efficient way of rapidly estimating the collective dynamics of a dense ensemble.

Figure 1

(a) When the polarization of an electromagnetic field sets the quantization axis of an atom, the effective quantum system is a two-level system with the direction of the transition dipole oriented along that polarization direction. (b) When the polarization of an incident control field is different from the quantization axis of an atom, the effective quantum system is a four-level system with a dipole transition oriented along each field component. $\mathrm{\Omega }$'s represent the field-atom interaction frequency, $\mathrm{\Delta }$'s are the detuning between the frequency of the driving field and the transition frequency of the 2LS, and $\gamma$'s are the spontaneous emission rates from the excited states to the ground state.

Figure 2

Fourier transform of the electric field over a 200 fs time window at $\vec{r}=(0,13\text{nm},0)$, a point outside a 10 nm radius spherical ensemble of atoms centered at the origin. Each atom in the ensemble has an energy level structure as shown in Fig. 1, with energy spacing between the ground and excited states of 1 eV, and spontaneous emission rates of 2.95 MHz. The number density of atoms in the ensemble is $4\times {10}^{27}$ atoms per cubic meter. The incident plane wave electromagnetic wave of frequency 241 THz and electric field amplitude 1.5 GV/m is polarized in the $\widehat{y}$ direction and propagates along the $\widehat{z}$ direction. (a) Frequency components that appear in the time window 0–200 fs after the start of excitation include a distinct blueshifted peak. (b) Frequency components that appear in the time window 100–300 fs after the start of excitation. Notice that the blueshifted frequency components have died out.

Figure 3

Snapshots of the spatial distribution of free-current density components ${\vec{J}}_y\left(\vec{r}\right)$ [row (a)] and ${\vec{J}}_x\left(\vec{r}\right)$($\text{A}/{\text{m}}^2$) [row (b)] in the $x\text{−}y$ plane (with $\widehat{y}$ being horizontal) bisecting a 10 nm nanosphere of atoms at times (i) 10 fs, (ii) 100 fs, and (iii) 250 fs after start of excitation. Parameters of the ensemble and the incident field are the same as in Fig. 2. The snapshots show that, at early times, there are ordered patterns in the spatial distribution of the free current density and, as time goes on, disorder sets in due to interatomic interactions, finally ending in a disordered “phase.”

Figure 4

(a) Spatially averaged populations in the $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$-directional excited states and (b) ensemble-averaged purity for a 10 nm radius nanosphere of atoms with atomic number density $N_A=4.0\times {10}^{27}{\text{m}}^{-3}$. Other parameters of the ensemble and the incident field are the same as in Fig. 2.

Figure 5

Spatially averaged populations (${\overline{\rho }}_{xx}$, ${\overline{\rho }}_{yy}$, ${\overline{\rho }}_{zz}$) in the $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$-directional excited states for a 10 nm radius spherical ensemble of atoms with varying number densities. Parameters of the incident field are the same as in Fig. 2.

Figure 6

Effective ensemble disorder-onset rates (${\gamma }_{\mathrm{ens}}$) for a 10 nm radius spherical ensemble of atoms with varying number density ($N_a$), for two different amplitudes of the driving plane-wave electromagnetic wave. The spontaneous emission rate of a single atom in the ensemble is 2.95 MHz. The data are fit to a logistic function as in Eq. (10). The saturation value of the disorder-onset rate $L$ increases as the amplitude of the driving field increases. (See Table 2.)

Figure 7

Modified decoherence structure in the single-particle model of a driven atomic ensemble. When significant interatomic interactions are present in an ensemble, it becomes possible for the spontaneously emitted radiation from the excited state of one atom to excite-state population from the ground state of a nearby atom. This emission followed by absorption is modeled by a dephasing process between states that have electric-dipole forbidden transitions (${\delta }_{ij}$'s in red). These dephasing rates do not affect the total state population; they only reduce the overall coherence of the single-particle state that models the ensemble.

Figure 8

Comparison between single-particle model calculation and the mean-field PSTD calculation of spatially averaged excited-state populations for a 10 nm radius spherical ensemble of atoms. The amplitude of the driving electromagnetic wave is $E=1.5\times {10}^9$ V/m. The number density of atoms in the ensemble are (a) $4.0\times {10}^{27}$ and (b) $2.5\times {10}^{27}$ atoms per cubic meter.