Shortcut to adiabaticity in spinor condensates

We devise a method to shortcut the adiabatic evolution of a spin-1 Bose gas with an external magnetic field as the control parameter. An initial many-body state with almost all bosons populating the Zeeman sublevel $m=0$ is evolved to a final state very close to a macroscopic spin-singlet condensate, a fragmented state with three macroscopically occupied Zeeman states. The shortcut protocol, obtained by an approximate mapping to a harmonic oscillator Hamiltonian, is compared to linear and exponential variations of the control parameter. We find a dramatic speedup of the dynamics when using the shortcut protocol.

Figure 1

Sketch of the proposed experimental protocol. A spin-1 BEC is prepared at large positive values of the quadratic Zeeman energy (QZE) $q$ with all atoms in the $m=0$ state. In this regime, the initial state is very close to the ground state (arbitrarily close as $q\to \infty$). In the adiabatic method, the QZE is slowly reduced in such a way that the state of the system always remains close to the instantaneous ground state. Stopping the QZE ramp at $q=0$, the system ends up in a total spin-singlet ground state with strong spin correlations. Stopping the ramp at a large, negative value of $q$, we prepare instead a twin Fock state with half the atoms in the Zeeman states $m=\pm 1$. In this paper we target the production of the singlet state and examine this procedure and alternative ramps which are not adiabatic but result in a state close to the ground state in a much shorter time.

Figure 2

Upper panel: Fluctuation of the number of particles in the $m=0$ manifold, $\mathrm{\Delta }{N}_0/N$, computed for three numbers of atoms. Middle panel: Energy of the ground state of the exact system compared with the energy of the ground state obtained in Eq. (10) for different values of the parameter $q$. When $q{N}^2{U}_s^{-1}<2$, the system becomes $U_s$ dominated, and thus, the energy is constant [see Eq. (10)]. It is noteworthy that in both the fragmented ($U_s$-dominated) and the BEC ($q$-dominated) regimes, the approximate value of the energy agrees well with the exact one. Lower panel: Wave function squared of the ground state compared to the value obtained with the continuous approximation described in the text. The middle and lower panels were obtained for a system of $N=100$ spins. In all cases, $U_s/h=104.13$ Hz. For the lower panel, we have used $q=U_s$.

Figure 3

Comparison between the linear, exponential, and shortcut ramps. In (a) and (b) $U_s/h=104.13$ Hz, $q_0=U_s$, and $q_f/q_0={10}^{-3}$. (a) We consider a slow evolution, with $t_f=1$ s; (b) a shorter evolution is considered, $t_f=0.01$ s.

Figure 4

Mean squared spin $\langle \widehat{\mathit{S}}{}^2\rangle /N$ as a function of the time computed for systems with $N=100$ and $N=1000$ spins evolved following either the shortcut protocol, a linear ramp, or an exponential ramp. All cases describe the evolution of the exact solution of the Schrödinger equation in situations where the evolution is clearly not adiabatic. Only the shortcut protocol brings the system to a state with lower total spin $S$. In (a)–(d) the initial state corresponds to $N^2{q}_0/U_s=N^2$ and, thus, is deep in the BEC phase (see Fig. 1 in [9]). (a, c) A state at $N^2{q}_f/U_s=0.01N^2$; (b, d) a final state closer to the fragmented phase $N^2{q}_f/U_s=0.001N^2$. (a, b) Computed with $N=100$; (c, d) computed with $N=1000$. Horizontal black and light-gray solid lines show the level of fluctuations in the ground state for $q=q_f$ and the reference value $\langle \widehat{\mathit{S}}{}^2\rangle /N=1/N$, respectively. In all cases, $U_s/h=104.13$ Hz and $t_f=0.01$ s.

Figure 5

Time evolution of the wave function. The radii of circles indicate the value of $|c_S{|}^2$ for each $S$ at each time $t/t_f$. Blue circles correspond to a system evolved using the shortcut protocol and red circles correspond to a system evolved with an exponential ramp. (a, c) Evolution of a system from $q_0/h=104.13$ Hz to $q_f={10}^{-2}{q}_0$ in $t_f=0.01$ s and (b, d) evolution of a system from $q_0/h=104.13$ Hz to $q_f={10}^{-3}{q}_0$ in $t_f=0.01$ s; this figure can be directly compared to Fig. 4. Since only the even solution is considered, the wave function $c_S$ at odd $S$ is identically 0.

Figure 6

Value of $\langle \widehat{\mathit{S}}{}^2\rangle /N$ after the time evolution, i.e., $t=t_f$, for the three protocols considered: (a) shortcut, (b) linear, and (c) exponential. In all cases only the results that give $\langle \widehat{\mathit{S}}{}^2\rangle /N<0.12$ are shown. (d) The contour lines corresponding to $\langle \widehat{\mathit{S}}{}^2\rangle /N=0.12$. In all cases we have $N=500$ particles and an initial value of $q_0/h=0.1U_s/h=10.413$ Hz.

Figure 7

$\mathrm{\Delta }{N}_0/N$ for the same conditions as in Fig. 6. Results obtained from the three protocols: (a) the shortcut, (b) the linear ramp, and (c) the exponential ramp. (d) The contour lines of the three cases in (a)–(c) for $\mathrm{\Delta }{N}_0/N=0.15$. All these plots have been realized using a system of $N=500$ particles and $q_0/h=0.1U_s/h=10.413$ Hz.

Figure 8

$\langle \widehat{\mathit{S}}{}^2\rangle /N$ as a function of time computed for systems evolved following the shortcut protocol, a linear ramp, or an exponential ramp. We compare a system of $N=1000$ spins (exact solution) and a system of $N=40000$ spins (approximate solution). The system is evolved from a state with $q_0/h=10U_s/h=71$ Hz to $q_f={10}^{-3}{q}_0$ in $t_f=0.1$ s. The shortcut protocol also works for this set of parameters, although it is, in principle, valid only for $q\left(t\right)\ll U_s$, in the sense that it provides a clear gain over simpler exponential or linear ramps. For $N=1000$ and $N=40000$ the corresponding values of $N^2{q}_f/U_s$ are ${10}^4$ and $16\times {10}^6$. The solid black line is the ground-state value for $q=q_f$.