Orientation-to-alignment conversion and spin squeezing

The relationship between orientation-to-alignment conversion (a form of atomic polarization evolution induced by an electric field) and the phenomenon of spin squeezing is demonstrated. A “stretched” state of an atom or molecule with maximum angular-momentum projection along the quantization axis possesses orientation and is a quantum-mechanical minimum-uncertainty state, where the product of the equal uncertainties of the angular-momentum projections on two orthogonal directions transverse to the quantization axis is the minimum allowed by the uncertainty relation. Application of an electric field for a short time induces orientation-to-alignment conversion and produces a spin-squeezed state, in which the quantum state essentially remains a minimum-uncertainty state, but the uncertainties of the angular-momentum projections on the orthogonal directions are unequal. This property can be visualized using the angular-momentum probability surfaces, where the radius of the surface is given by the probability of measuring the maximum angular-momentum projection in that direction. Brief remarks are also given concerning collective-spin squeezing and quantum nondemolition measurements.

Figure 1

Angular-momentum probability surfaces (see text) showing an $F=2$ atomic state initially stretched along $\widehat{\mathbf{x}}$ and evolving in the presence of an electric field along $\widehat{\mathbf{z}}$. A complete cycle of orientation-to-alignment conversion is shown.

Figure 2

The initial part of the evolution shown in Fig. 1 as viewed from the $x$ direction, showing the process of squeezing. The solid line is a polar plot of $\Delta F_{y,\nu }$ as a function of azimuth, with the angle $\nu =\pi /2-\delta$ of the minimum-uncertainty axis indicated by the dashed line.

Figure 3

The squeezing parameter ${\xi }_R$ for $F=2$ as a function of time. The solid curve shows ${\xi }_R$ for the most strongly squeezed axis ($\nu =\frac{\pi }{2}-\delta$), while the dashed curve gives ${\xi }_R$ for the orthogonal “antisqueezing” axis. A value ${\xi }_R<1$ (below the gray line) indicates a squeezed state. The dotted curve shows the product of the squeezing parameters for the two orthogonal axes, with a value of 1 indicating a minimum-uncertainty state.

Figure 4

The angle $\nu =\frac{\pi }{2}-\delta$ of the minimum uncertainty axis as a function of time for a state with $F=2$. The $\pi /2$ shift halfway through the cycle corresponds to the mirror symmetry seen between the first and second halves of the cycle in Fig. 1.

Figure 5

Logarithmic plot of the minimum value of ${\xi }_R$ with respect to $t$ as a function of $F$ when $\nu =\frac{\pi }{2}-\delta$. The dependence quickly approaches a power law (${\xi }_R\propto F^{-1/3}$). Note that the case of $F=1$ has an apparently anomalously small value of ${\xi }_R$. This is a special case in which the optimum squeezing parameter is achieved as $\chi t$ approaches $\pi /2$, when the spin projection $\langle F_x\rangle$ goes to zero.

Figure 6

Comparison of AMPS for the (a) “single-axis twisting” and (b) “two-axis countertwisting” Hamiltonians at times equal to $1/8$ of the respective quantum-beat periods for an $F=2$ state initially stretched along $\widehat{\mathbf{x}}$. Twisting about the $z$ axis introduces “swirliness” in (a); simultaneous twisting about the $y$ axis (b) cancels this effect.