Collective atomic recoil laser as a synchronization transition

We consider here a model previously introduced to describe the collective behavior of an ensemble of cold atoms interacting with a coherent electromagnetic field. The atomic motion along the self-generated spatially periodic force field can be interpreted as the rotation of a phase oscillator. This suggests a relationship with synchronization transitions occurring in globally coupled rotators. In fact, we show that whenever the field dynamics can be adiabatically eliminated, the model reduces to a self-consistent equation for the probability distribution of the atomic “phases.” In this limit, there exists a formal equivalence with the Kuramoto model, though with important differences in the self-consistency conditions. Depending on the field-cavity detuning, we show that the onset of synchronized behavior may occur through either a first- or second-order phase transition. Furthermore, we find a secondary threshold, above which a periodic self-pulsing regime sets in, that is immediately followed by the unlocking of the forward-field frequency. At yet higher, but still experimentally meaningful, input intensities, irregular, chaotic oscillations may eventually appear. Finally, we derive a simpler model, involving only five scalar variables, which is able to reproduce the entire phenomenology exhibited by the original model.

Figure 1

(Color online) Left panel: Bifurcation diagram of the ACM model for $\mu =40$, $C=5$, $\Delta =0$. Solid, dashed, and dotted-dashed lines correspond to $b$, $f$, and $R$ as a function of $\sigma$. Right panel: the parameter $\xi$ of the force field.

Figure 2

(Color online) Velocity of the density grating ($-\omega$, solid line) and average velocity of the atoms ($v_a$, dashed line) vs the scaled temperature $\sigma$ and the same parameter values as in Fig. 1.

Figure 3

(Color online) Phase diagram separating locked or drifting from partially locked regimes for $C=5$.

Figure 4

Bifurcation diagram for the backward (a) and forward (b) field amplitudes as a function of $\mu$. The other parameters are $C=20$, $\sigma =1$, and $\Delta =0$. The meaning of the various regions is discussed in the text.

Figure 5

Phase space portraits of the typical behavior observed in region II [(a),(d), $\mu =2.3$], region III [(b),(e), $\mu =6$], and region IV [(c),(f) $\mu =9$]. Parameters are those of Fig. 4.

Figure 6

(Color online) Locus of the secondary Hopf bifurcation as a function $\mu$ and $\sigma$, for different values of $C$. As $C$ decreases, the threshold for self-oscillation goes toward infinite values of $\mu$.

Figure 7

Bifurcation diagram for the backward field amplitude $b$ as a function $\mu$. the fixed parameters are $C=20$ and $\Delta =0$. Panels (a)–(d) correspond to $\sigma =1$,2,3, and 4, respectively. As $\sigma$ increases, a region of bistability indicated by the vertical dashed lines becomes more and more visible thus indicating a subcritical secondary Hopf bifurcation.

Figure 8

A projection of $F$ in the complex plane, for the critical value ${\mu }_c=2.752$, when the forward field unlocks from the input field. Parameters are those of Fig. 4 The cycle goes through the point $F=0$.

Figure 9

(Color online) A projection of two limit cycles just below and above the critical point, ${\mu }_c=2.752$ where the forward field unlocks from the input field. The other parameter values are those of Fig. 5. The arrows indicate the direction of the motion: below (above) the transition, in the (un)locked regime, the upper (lower) semicircle is followed.

Figure 10

(Color online) The behavior of the instantaneous slope of the potential, just below ($\mu =2.75$ dashed line) and above ($\mu =2.76$ the transition point). The different amplitude of the two curves is due to a different distance from the critical point. Parameters are those of Fig. 4.

Figure 11

(Color online) Frequency of the forward (dashed line) and backward (solid line) fields, referred to that of the input field, as a function of $\mu$. Parameters are the same as in Fig. 4. An enlargement of the $\overline{\dot{\beta }}$ behavior is plotted in the inset.

Figure 12

Bifurcation diagram of the steady states of Eqs. (38) as a function of $\mu$, for $C=20$ and $\sigma =1$. Panels (a)–(d) correspond to $\Delta =0$, 2, 3, and $-2$, respectively. The dashed lines denote the border of the bistability regions whenever there is one.

Figure 13

(Color online) Spinodal decomposition of the steady states. Dashed lines refer to the original full model (for $\sigma =1$ and $C=20$); solid lines refer to the simplified model (38). The two shaded regions correspond to mono- and bistable regimes in the full model. The circle denotes the tricritical point ${\Delta }_c$.

Figure 14

(Color online) Locus of the secondary instability as a function $\mu$ and $\sigma$, for different values of $C$. Predictions of the simplified model, Eqs. (38), with dashed lines, are compared with those of the exact model, Eqs. (10, 11, 12, 13, 14), with solid lines. Parameters are those of Fig. 6.