Self-Stopping of Light

Here, we show that light can bring itself to a complete standstill (self-stop) via self-interaction mediated by the resonant nonlinearity in a fully homogeneous medium. An intense few-cycle pulse, entering the medium, may reshape to form a strongly coupled light-matter bundle, in which the energy is transferred from light to the medium and back periodically on the single-cycle scale. Such oscillating structure can decelerate, alter its propagation direction, and even completely stop, depending on the state of its internal degrees of freedom. This phenomenon is expected to occur in the few-cycle strong-field regime when the Rabi oscillation frequency becomes comparable with the frequency of the incoming light.

Figure 1

Self-stopping light. (a),(b) Evolution of the field $E(z,t)$ (a) of a short pulse entering a resonant medium, and its inversion $n(z,t)$ (b). After a reach transient dynamics the light self-stops at around $z\approx 8{\lambda }_{12}$. The points $t_1$ and $t_2$ indicate quantitative changes in the dynamical regime. (c) Dynamics of the energy in the field $\mathcal{E}(t)$. (d) The transmitted light fraction $W_{\mathrm{out}}/W_{\mathrm{in}}$ (black line) and the position of the stop point (orange line) as a function of the particle density $N_0$. (e) Evolution of the field spectrum $I(z,\omega )$ with $z$. (f) Spectra $I(z,\omega )$ at certain $z$ positions: solid black line: $z=4{\lambda }_{12}$ (before the entrance); solid blue line: $z=6.2{\lambda }_{12}$; solid orange line $z=8{\lambda }_{12}$. For comparison, also the spectrum of a conventional SIT soliton (black dashed curve), as well as the linear losses $\alpha$ (red solid line) normalized to the maximum ${\alpha }_0$ and the linear group velocity $v_{\mathrm{gr}}(\omega )$ (red dotted curve) in units of $c$ for the small-signal situation are presented.

Figure 2

Formation of LMO near the entrance to the medium. (a)–(f) Evolution of the field (a), inversion (b), and atomic polarization (c) on the scale of multiple (a)–(c) and half (d)–(f) of period of LMO oscillation. Horizontal line in (a)–(c),(i) marks $t=18\mathrm{fs}$ as the position where the collapse takes place and LMO is born. In (d)–(f), dashed lines indicate zeros of $E(z,t)$. The consequent half-oscillations of the field are displaced by $\mathrm{\Delta }z$ in space. For comparison, in (g),(h) the evolution of a more conventional SIT soliton structure (two copropagating $2\pi$ solitons) is shown. (i) The dynamics of the energy in field $\mathcal{E}(t)$ for LMO (blue line) and the SIT soliton (orange line). Inset in (i) shows the field (red solid line) inversion (red dashed line) and atomic polarization (red dotted line) in dependence on time for one of the SIT solitons at $z=5.8\mu \mathrm{m}$. In (e), red lines show the time dependence of the field, polarization and inversion for $z=6.18\mu \mathrm{m}$ [to be compared with the SIT soliton case in (i)].

Figure 3

Details of LMO self-stopping dynamics. (a)–(c) The dynamics of the field (a), inversion (b), and atomic polarization (c) in the range where it self-stops. (d)–(f) and insets to (a)–(c) show the dynamics of $E$, $n$, and $P$ on the single-cycle scale near exemplary time positions marked in (a) by dotted horizontal lines [$t\approx 35\mathrm{fs}$ (d)–(f) and $t\approx 46\mathrm{fs}$ [insets to (a)–(c)]. Solid lines in (a)–(f) show the evolution of the center of mass (CM) of the field.

Figure 4

Self-stopping of LMO as a relaxation to the energy minimum. (a) The dependence of the diffractive energy $H_{\mathrm{diff}}$ of LMO in dependence on the displacement $\mathrm{\Delta }z$. Dots show the numerically calculated energies for every particular oscillation tracked along the position of CM in Fig. 3, the solid orange line shows the quadratic fit to the data points. The insets show the undeformed LMO ($\mathrm{\Delta }z=0$, having the minimal energy) as well as deformed LMO ($\mathrm{\Delta }z\ne 0$, out of the energy minimum) advancing by $\mathrm{\Delta }z$ every half-period $T$ and thereby moving, at the same time slowly relaxing to the minimum and thus to the undeformed structure. (b) Blue dots: velocity $v$ of LMO calculated from the CM curve in Fig. 3; orange circles: the velocity suggested from the displacement as $v=\mathrm{\Delta }z/T$. Black dots indicate the fit to the relaxation dynamics obtained from the oscillator model following from the potential in (a) as described in text.