Staircase in magnetization and entanglement entropy of spinor condensates

Staircases in response functions are associated with physically observable quantities that respond discretely to continuous tuning of a control parameter. A well-known example is the quantization of the Hall conductivity in two-dimensional electron gases at high magnetic fields. Here, we show that such a staircase response also appears in the magnetization of spin-1 atomic ensembles evolving under several spin-squeezing Hamiltonians. We discuss three examples, two mesoscopic and one macroscopic, where the system's magnetization vector responds discretely to continuous tuning of the applied magnetic field or the atom density, thus producing a magnetization staircase. The examples that we consider are directly related to Hamiltonians that have been implemented experimentally in the context of spin and spin-nematic squeezing. Thus, our results can be readily put to experimental test in spin-1 ferromagnetic ${}^{87}\mathrm{Rb}$ and antiferromagnetic ${}^{23}\mathrm{Na}$ condensates.

Figure 1

Staircase in the one-axis twisting Hamiltonian: (a) shows the ground-state magnetization as a function of the strength of the applied field $p$ for constant interaction $\chi$ in the one-axis twisting Hamiltonian $H=\chi S_z^2-p{S}_z$, with and without the perturbation $\epsilon S_x$. The blue curve (the smooth curve) corresponds to the Hamiltonian with perturbation and the black curve (the staircase curve with sharp edges) corresponds to the Hamiltonian without the perturbation. (b) shows the corresponding ground-state and the first excited-state energies around the level crossings between $m=-1$ and $m=0$, as well as $m=0$ and $m=+1$. In the absence of the perturbation, there are true level crossings, but when the perturbation is added, gaps open and thereby smooth the staircase. The term $\epsilon S_x$ is also responsible for changing the system's magnetization, which is otherwise conserved. (c) shows the entanglement entropy of the local ground state as a function of the control parameter $p/\chi$. The black curve (the curve without cusps) shows the entanglement without the perturbation a written in Eq. (1), while the blue curve (the curve with cusps) shows the entanglement with the perturbation for $\frac{\epsilon }{\chi }=0.02$.

Figure 2

Convexity of the energy: The minimum energy eigenvalue $E_m$ of the ferromagnetic Hamiltonian $H=c{S}^2+q{Q}_{zz}-p{S}_z$ is a convex function of the magnetization $m$. For the purpose of this illustration, we have used $N=10$. The minima of these curves correspond to the ground-state magnetization. Because $p$ is the coefficient of a linear term in $m$, changing it has the effect of shifting the minimum. The four values of $p/\left|c\right|$ have their minima are different values of $m$, leading to a staircase response of the ground-state magnetization as $p/\left|c\right|$ is changed.

Figure 3

Staircase in the magnetization direction: (a) shows the ground-state magnetization vector of an antiferromagnetic condensate with Hamiltonian $H=c{S}^2+p{S}_x+\alpha Q_{xz}$, for three different values of $c$ with $N=100$. The last term in the Hamiltonian induces the tilting of the magnetization vector by specific angles, depending on where the system is on the staircase. (b) shows the tilt angle for $N=20$ as a function of $c/p$, a staircase, but in contrast with the previous examples, this time it is not only in the magnitude of magnetization, but also in the direction. The blue curve shows the smoothened staircase after adding an $\epsilon Q_{xx}$ perturbation, with $\epsilon =0.02p$. The inset shows the ground-state entanglement entropy as a function of the control parameter. In both (a) and (b), $\alpha =0.1p$. The blue curve (the smooth curve) corresponds to the Hamiltonian with the $Q_{xx}$ perturbation and the black curve (the curve with the sharp edges) corresponds to the Hamiltonian without the $Q_{xx}$ perturbation.