Coupling of magnetic and optomechanical structuring in cold atoms

Self-organized phases in cold atoms as a result of light-mediated interactions can be induced by coupling to internal or external degrees of the atoms. There has been growing interest in the interaction of internal spin degrees of freedom with the optomechanical dynamics of the external center-of-mass motion. We present a model for the coupling between magnetic and optomechanical structuring in a $J=1/2\to J^{\prime }=3/2$ system in a single-mirror feedback scheme, which is representative of a larger class of diffractively coupled systems such as longitudinally pumped cavities and counterpropagating-beam schemes. For negative detunings, a linear stability analysis demonstrates that optical pumping and optomechanical driving cooperate to create magnetic ordering. However, for long-period transmission gratings the magnetic driving will strongly dominate the optomechanical driving, unless one operates very close to the existence range of the magnetic instability. At small lattice periods, in particular at wavelength-scale periods, the optomechanical driving will dominate.

Figure 1

Experimental scheme for single-mirror feedback. A cloud of length $L$ is driven by a pump laser beam. A plane mirror at a distance $d$ retroreflects the transmitted light back into the cloud. The theoretical treatment assumes $d\gg L$. The optical axis and quantization axis are in the $z$ direction. Structuring is depicted here in the $x$ direction but could, by symmetry, be anywhere in the transverse $x-y$ plane. See text for further explanation.

Figure 2

$J=1/2\to J^{\prime }=3/2$ transition. $a=1/3$ denotes the relative strengths of weak and strong transitions. $a^{\prime }=2/9<a$ allows us to account for a lower optical pumping efficiency as some cycles will return to the original state. The main part of the dynamics takes place in the stretched states with maximum modulus of the magnetic quantum number.

Figure 3

Schematic illustration of the interplay of magnetic and optomechanical degrees of freedom. The plots depict atomic states and optical fields and potentials in transverse space $x$ measured in units of the lattice period $\mathrm{\Lambda }$. (a) The intensity of the ${\sigma }_{+}$ [solid red (dark gray) line] and ${\sigma }_{-}$ (solid black line) components of optical field. Dotted red (dark gray) and black lines are the corresponding potentials for spin-up and spin-down atoms, respectively. The green (light gray) lines indicate the resulting spatially antiphased bunching in density for spin-up (dotted) and spin-down (solid) atoms. The atomic density is narrower than the intensity for cold (enough) atoms. (b) Black line: Atomic magnetization (with dominance of spin-up atoms where positive) resulting from optical pumping from the optical spin patterns [solid red and black lines in (a)] at low saturation. Green (light gray) line: more peaked structure of magnetization if atoms also bunch. The profiles of the curves are artist impressions and are not calculated self-consistently. Here, and in the following, the information in parenthesis indicates gray levels in black and white print version.

Figure 4

Effective relaxation rate (solid lines) vs lattice period from Eq. (6) for the temperatures relevant for [17, 19, 20] compared to diffusive modeling (dotted lines) for different diffusion constants. Black (corresponding to the situation in [17]): solid line, $T=290\mu \mathrm{K}$; dotted line, $8.7\times {10}^{-7}{\mathrm{m}}^2$/s. Red (dark gray; corresponding to the situation in [20]): solid line, $T=200\mu \mathrm{K}$; dotted line, $7\times {10}^{-7}{\mathrm{m}}^2$/s. Green (light gray; corresponding to the situation in [19]): solid line, $T=120\mu \mathrm{K}$, dotted line, $3\times {10}^{-7}{\mathrm{m}}^2$/s.

Figure 5

Threshold saturation parameter for one circular component vs temperature for $\mathrm{\Delta }=-8.6, \mathrm{\Lambda }=100\mu \mathrm{m}$. (a) Black lines ($b_0=80$): solid line, threshold for magnetic ordering; dotted line, threshold for optomechanical bunching. The thresholds for combined magnetic and optomechanical driving of magnetic ordering are essentially indistinguishable from the black solid line for magnetic ordering alone at this scale. Red (dark gray) lines ($b_0=70$): solid line, threshold for magnetic ordering; dotted line, threshold for optomechanical bunching (both are above their black counterparts). The cyan (light gray) solid line [just below the red (dark gray) solid line] represents the combined threshold for magnetic and optomechanical driving for negative detuning and $b_0=70$; the green (light gray) solid line [just above the red (dark gray) solid line] represents positive detuning. (b) Same as (a), but for $b_0=69.31$. The cyan line is now dotted to indicate the difference from the green line. The parameter $\sigma$ changes from infinity at $T=0$ to 4.3 at $T=300\mu \mathrm{K}$ via 12.5 at $T=100\mu \mathrm{K}$ and 8.4 $T=150\mu \mathrm{K}$.

Figure 6

Threshold saturation parameter for one circular component vs lattice period for $\mathrm{\Delta }=-8.6, b_0=80, T=150\mu \mathrm{K}$. Solid black line, threshold for magnetic ordering; dashed black line, threshold for optomechanical bunching; cyan (light gray) dotted line, combined threshold for magnetic and optomechanical driving of magnetic ordering for negative detuning; and green (light gray) solid line, combined threshold for positive detuning $\mathrm{\Delta }=+8.6$.

Figure 7

Threshold saturation parameter for one circular component vs the saturation parameter for one molasses beam for $\mathrm{\Delta }=-8.6, b_0=80, T=150\mu \mathrm{K}, \mathrm{\Lambda }=100\mu \mathrm{m}$. Horizontal solid black line, threshold for magnetic ordering without molasses; horizontal dashed black line, threshold for optomechanical bunching; red (dark gray) line, the threshold for magnetic ordering including the repumping of the molasses.