Spin squeezing in internal bosonic Josephson junctions via enhanced shortcuts to adiabaticity

We investigate a time-efficient and robust preparation of spin-squeezed states—a class of states of interest for quantum enhanced metrology—in internal bosonic Josephson junctions with a time-dependent nonlinear coupling strength between atoms in two different hyperfine states. We treat this state-preparation problem, which had previously been addressed using shortcuts to adiabaticity (STA), using the recently proposed analytical modification of this class of quantum control protocols that has become known as the enhanced STA (eSTA) method. We characterize the state-preparation process by evaluating the time dependence of the coherent spin-squeezing and number-squeezing parameters and the target-state fidelity. We show that the state-preparation times obtained using the eSTA method compare favorably to those found in previously proposed approaches. We also demonstrate that the increased robustness of the eSTA approach—compared to its STA counterpart—leads to additional advantages for potential experimental realizations of strongly spin-squeezed states in internal bosonic Josephson junctions.

Figure 1

A pictorial illustration of an internal BJJ: an external electromagnetic field coherently transfers atoms between the two relevant hyperfine states (modes), denoted by $|{\psi }_1⟩$ and $|{\psi }_2⟩$, via Rabi rotations; $\hslash \mathrm{\Omega }$ is the corresponding constant Rabi-coupling strength. The interspecies interaction strength $U_{12}(t)$, which is assumed to be time dependent, can be varied using an external magnetic field in the presence of a Feshbach resonance.

Figure 2

The time evolution of (a) the control function $\mathrm{\Lambda }(t)$, (b) the number squeezing ${\xi }_N^2(t)$, (c) the coherent spin squeezing ${\xi }_S^2(t)$ in decibels, and (d) the fidelity $F(t)$. Enhanced STA applied to the discrete Hamiltonian (control function ${\mathrm{\Lambda }}_2$, $\nu =5$; red, solid lines), the STA scheme (control function ${\mathrm{\Lambda }}_0$; black, dashed lines), and the reference adiabatic scheme (control function ${\mathrm{\Lambda }}_{ad}$; green, dashed-dotted lines). $N=10$, $t_f=0.1t_R$, $\mathrm{\Lambda }(0)=0$, and $\mathrm{\Lambda }(t_f)=50$.

Figure 3

The fidelities $F$ versus the final time $t_f$ for eSTA applied to Hamiltonian $H_2$ (${\mathrm{\Lambda }}_2$, $\nu =5$; red, solid lines—${\tilde{\mathrm{\Lambda }}}_2$, $\nu =1$; blue, dotted lines), eSTA applied to the approximated Hamiltonian $H_1$ (${\mathrm{\Lambda }}_1$; orange, small dashed line), the STA scheme (${\mathrm{\Lambda }}_0$; black, dashed lines) and, as a reference, the adiabatic scheme (${\mathrm{\Lambda }}_{ad}$; green, dashed-dotted lines) with different particle numbers: (a) $N=10$ and (b) $N=100$. $\mathrm{\Lambda }(0)=0$ and $\mathrm{\Lambda }(t_f)=50$.

Figure 4

(a) The sensitivity $S_m$ for a systematic amplitude error. (b) The sensitivity $S_t$ for a systematic time shift. (c) The imperfection $\eta$; the inset shows the sensitivity $S_p$ for phase noise. eSTA applied to the Hamiltonian $H_2$ (${\mathrm{\Lambda }}_2$, $\nu =5$; red, solid lines—${\tilde{\mathrm{\Lambda }}}_2$, $\nu =1$; blue, dotted lines), eSTA applied to the approximated Hamiltonian $H_1$ (${\mathrm{\Lambda }}_1$; orange, small dashed line), and the STA scheme (${\mathrm{\Lambda }}_0$; black, dashed lines). Different particle numbers: (i) $N=10$ and (ii) $N=100$. The best result is obtained for ${\mathrm{\Lambda }}_2$, $\nu =5$ corrections and eSTA applied to the original Hamiltonian (solid red line), as it ensures the lowest sensitivity to systematic errors and the highest fidelity.