Semiclassical theory of synchronization-assisted cooling

We analyze the dynamics leading to radiative cooling of an atomic ensemble confined inside an optical cavity when the atomic dipolar transitions are incoherently pumped and can synchronize. Our study is performed in the semiclassical regime and assumes that cavity decay is the largest rate in the system dynamics. We identify three regimes characterizing the cooling. At first hot atoms are individually cooled by the cavity friction forces. After this stage, the atoms' center-of-mass motion is further cooled by the coupling to the internal degrees of freedom while the dipoles synchronize. In the latest stage dipole-dipole correlations are stationary and the center-of-mass motion is determined by the interplay between friction and dispersive forces due to the coupling with the collective dipole. We analyze this asymptotic regime by means of a mean-field model and show that the width of the momentum distribution can be of the order of the photon recoil. Furthermore, the internal excitations oscillate spatially with the cavity standing wave forming an antiferromagnetic-like order.

Figure 1

(a) Atoms are transversally driven by an incoherent pump at rate $w$ and strongly couple with the mode of a standing-wave cavity. (b) The relevant internal states are the two metastable states $|g\rangle$ and $|e\rangle$, which can be the ground and excited states of the intercombination line of an alkali-earth metal atom or two sublevels of a hyperfine multiplet. The dipolar transitions strongly couple to the cavity mode with position-dependent strength $gcos\left(kx\right)$, with $g$ the vacuum Rabi frequency and $k$ the cavity wave number. The cavity decay rate is the largest parameter of the dynamics, i.e., $\kappa \gg g,w$.

Figure 2

(a) Example of the time evolution of the one-particle momentum width $\mathrm{\Delta }p$ and schematic overview of the regimes characterizing the dynamics of synchronization-assisted cooling. The width $\mathrm{\Delta }p$ determines the characteristic time scale $T_e$ of the atoms' external motion, which is inversely proportional to the mean Doppler shift $k\mathrm{\Delta }p/m$. The time scales of reference are determined by cavity decay, $T_C\sim {\kappa }^{-1}$, and by the spins pump rate, $T_i\sim 1/w$. Initially, $T_e\ll T_i$ and the atoms' center-of-mass motion is cooled by the cavity forces. When the atoms are sufficiently cold that $T_e$ becomes comparable to $T_i$, the cooling dynamics is determined by the nonadiabatic coupling of the external motion with the spin dynamics. The final stages are characterized by the regime $T_e\gg T_i$ and exhibit temperatures that are orders of magnitude smaller than $\kappa$. (b) The correlations between internal and external degrees of freedom give rise to a position-dependent expectation value of population inversion (solid line) that oscillates with the cavity intensity ${cos}^2\left(kx\right)$ and is maximum at the nodes.

Figure 3

Dynamics of the width $\langle p^2\rangle$ of the single-atom momentum distribution [in units of ${\left(\hslash k\right)}^2]$ as a function of time (in units of $1/{\omega }_R$). The solid curves are determined by integrating Eqs. (10), (11) (using the cumulant expansion, see Appendix pp2) and (17) assuming that initially all atoms are in the excited state and are uniformly spatially distributed, while their momentum distribution is thermal with width (a) $\langle p^2\left(0\right)\rangle =500{\left(\hslash k\right)}^2$; (b) $\langle p^2\left(0\right)\rangle =50{\left(\hslash k\right)}^2$; (c) $\langle p^2\left(0\right)\rangle =5{\left(\hslash k\right)}^2$. The dashed and dotted lines are the corresponding simulations obtained by integrating the equations after setting in Eq. (17) $F=F_j^{\left(1\right),\text{sc}}$(dotted line) and $F=F_j^{\left(0\right),\text{sc}}$ (dashed line). The parameters are $N=100$, $\kappa =780{\omega }_R$, $N{\mathrm{\Gamma }}_C=40{\omega }_R$, $\mathrm{\Delta }=\kappa /2$, $w=N{\mathrm{\Gamma }}_C/4$.

Figure 4

Correlation $\langle {\widehat{X}}^{†}\widehat{X}\rangle$ as a function of time (in units of $1/{\omega }_R$). This quantity signifies the occurrence of synchronization. Subplots (a)–(c) respectively correspond to the dynamics of subplots (a)–(c) of Fig. 3.

Figure 5

(a) Momentum distribution $f_{\text{st}}\left(p\right)$ as a function of $p$ (in units of $\hslash k$) and resulting from the dynamics of Fig. 3 (a) at time $t\simeq 2000{\omega }_R^{-1}$. The solid and dashed lines illustrate the momentum distributions obtained by considering the full force and only the adiabatic component, respectively, in Eq. (17). Subplot (b) shows the momentum distribution (in logarithmic scale) over the full initial range of values, demonstrating the existence of long tails. These are responsible for the discrepancy observed in the asymptotic limit of the corresponding curves in Fig. 3.

Figure 6

Kurtosis $\mathcal{K}=\langle p^4\rangle /{\langle p^2\rangle }^2$ as a function of time for the dynamics of Fig. 3. The inset shows the kurtosis for the parameters as in Fig. 3: Both curves relax to approximately the same value $\mathcal{K}\approx 2$.

Figure 7

Dynamics of $\langle p^2\left(t\right)\rangle$ (in units of ${\hslash }^2{k}^2$) as a function of time (in units of ${\omega }_R^{-1}$) for (a) $N=100$ and (b) $N=1000$ atoms. The solid lines are obtained by numerically integrating Eqs. (10) and (11) (using the cumulant expansion), and (17), whereas the dashed lines are the predictions of the mean field model in Eqs. (24), (25), and (27). The other parameters are the same as in Fig. 3. Note that $N{\mathrm{\Gamma }}_C=40{\omega }_R$. Accordingly, we rescale the value of ${\mathrm{\Gamma }}_C$ when increasing $N$.

Figure 8

Dynamics of $\langle {\widehat{X}}^{†}\widehat{X}\rangle$ as a function of time (in units of ${\omega }_R^{-1}$ for (a) $N=100$ and (b) $N=1000$, corresponding to the subplots of Fig. 7.

Figure 9

Absolute value of the Laplace transform $S\left(i\omega \right)$, Eq. (28), in arbitrary units and as a function of $\omega$ (in units of ${\omega }_R$) for the curves in Fig. 8. The Laplace transform is evaluated over the same time interval as in Fig. 8.

Figure 10

Spatial dependence of (a) the dipole moment $s\left(x\right)$ [Eq. (29)] and (b) the population inversion $z\left(x\right)$ [Eq. (30)] for $w=N{\mathrm{\Gamma }}_C/4$ and ${\left|X\right|}^2\approx 0.055$. The dashed and dotted lines correspond to the excited state $e\left(x\right)=[1+z(x\left)\right]/2$ and ground state $g\left(x\right)=[1-z(x\left)\right]/2$ population, respectively. The $x$ axis is in units of $1/k$. We verified that this behavior is also predicted by the semiclassical model.

Figure 11

Phase space histogram of the asymptotic dynamics of $N=1000$ particles, the $x$ axis is in unit of $1/k$ and the trajectories are reported modulus the wavelength; the $p$ axis is in units of $\hslash k$. The parameters are the same as in Fig. 7, the time is of the order of $t\approx 50{\omega }_R^{-1}$. Subplot (a) reports 100 trajectories calculated using stochastic differential equations [30] simulating the dynamics of Eqs. (10), (11), and (17). Subplot (b) reports the corresponding mean-field simulations of Eqs. (24), (25), and (27). The black dashed line indicates the trajectory at energy $E_0$, Eq. (40).

Figure 12

(a) Momentum distribution $f_{\text{st}}\left(p\right)$ as a function of $p$ (in units of $\hslash k$) and (b) position distribution $g_{\text{st}}\left(x\right)$ as a function of $x$ (in units of $1/k$ and modulus the wavelength) corresponding to the distribution in Fig. 11. The brown solid line shows the prediction of the semiclassical simulation [corresponding to Fig. 11]; the green dashed line shows the prediction of the mean-field model [Fig. 11].

Figure 13

(a) Stationary momentum width ${\langle p^2\rangle }_{\infty }$ [in units of ${\overline{p}}^2={\left(\hslash k\right)}^{2}N{\mathrm{\Gamma }}_C/\left(2{\omega }_R\right)]$ as a function of $w$ (in units of $N{\mathrm{\Gamma }}_C$) and for $\mathrm{\Delta }=\kappa /2$. Subplot (b) shows the value of the pumping strength $w_{\min }$ (in units of $N{\mathrm{\Gamma }}_C$) that minimizes the temperature for each value of the detuning $\mathrm{\Delta }$ (in units of $\kappa /2$). Subplot (c) reports the corresponding value of the minimum width ${\langle p^2\rangle }_{\min }$ (in units of ${\overline{p}}^2$) as a function of $\mathrm{\Delta }$ (in units of $\kappa /2$).

Figure 14

(a) Dynamics of $cos\left(\arg \right(X\left)\right)$ as a function of time (in units of ${\omega }_R^{-1}$).The time evolution of the order parameter $X$ has been determined using the mean-field model for the same parameters as in Fig. 8. Subplot (b) shows the Laplace transform $F\left(i\omega \right)={\int }_0^{\infty }dt{e}^{i\omega t}cos\left\{\arg \left[X\left(t\right)\right]\right\}$ as a function of the frequency (in units of ${\omega }_R$). Two well-defined sidebands are visible at frequency $\omega \approx \pm 5{\omega }_R=\pm w/2$.