Quantum dynamics generated by the two-axis countertwisting Hamiltonian

We study the quantum dynamics generated by the two-axis countertwisting Hamiltonian from an initial spin coherent state in a spin-$1/2$ ensemble. A characteristic feature of the two-axis countertwisting Hamiltonian is the existence of four neutrally stable and two saddle unstable fixed points. The presence of the latter is responsible for a high level of squeezing. The squeezing is accompanied by the appearance of several quantum states of interest in quantum metrology with Heisenberg-limited sensitivity, and we show fidelity functions for some of them. We present exact results for the quantum Fisher information and the squeezing parameter. Although the overall time evolution of both changes strongly with the number of particles, we find that they have regular dynamics for short times. We explain scaling with the system size by using a Gaussian approach.

Figure 1

Mean-field trajectories of the two-axis countertwisting Hamiltonian $\widehat{\mathcal{H}}=-\hslash \chi ({\widehat{S}}_z{\widehat{S}}_y+{\widehat{S}}_y{\widehat{S}}_z)$. Two unstable saddle fixed points are located at $z=0$ and $\phi =0,\pi$. Four neutrally stable center fixed points correspond to $(\phi ,z)=(\pm \pi /2,\pm 1/\sqrt{2})$. Colored spots visualize the spin coherent states around the classical unstable (red) and stable (green) fixed points.

Figure 2

Scaling of (a) the squeezing parameter and (b) the quantum Fisher information with the total number of particles $N$.

Figure 3

Top: Time evolution of the fidelity (26) to the EWSS, Twin-Fock, Berry-Wiseman, and optimized over $\alpha$ Yurke states. The maxima of respective fidelity functions are: ${\mathcal{F}}_{\mathrm{BW},\max }=0.9999$ for the BW state; ${\mathcal{F}}_{\mathrm{EWSS},\max }=0.9965$ for the EWSS state; ${\mathcal{F}}_{\mathrm{Y},\max }=0.9936$ for the Yurke state with $\alpha =0.678$; and ${\mathcal{F}}_{\mathrm{TF},\max }=0.8732$ for the twin-Fock state. Bottom: Probability distributions (27) with Fock states at the maximum of the respective fidelity function for (a) Berry-Wiseman, (b) EWSS, (c) Yurke, and (d) Twin-Fock states. Red crosses correspond to the ideal case given by Eqs. (22, 23, 24, 25). The total number of particles is $N=50$.

Figure 4

Top: Arrows indicate the locations of quantum states that are plotted in bottom panels on the time evolution of the quantum Fisher information. Bottom: The Husimi (left column) and Wigner (right column) functions at particular moments of time: (a) the initial coherent spin state; (b) the maximum of the BW fidelity function; (c) the maximum of the EWSS fidelity function; (d) the best spin squeezing; (e) the maximum of the Yurke fidelity function; (f) the maximum of the quantum Fisher information; (g) the maximum of the twin-Fock fidelity function; and (h) the second local minimum of the quantum Fisher information. The total number of particles is $N=50$.

Figure 5

(a) The spin squeezing parameter and (b) the quantum Fisher information for different numbers of particles $N$ around a stable fixed point. Here: solid red line is for $N=100$, dotted black line for $N=50$ and dot-dashed blue line for $N=20$.