Counterdiabatic driving in spin squeezing and Dicke-state preparation

A method is presented to transfer a system of two-level atoms from a spin coherent state to a maximally spin squeezed Dicke state, relevant for quantum metrology and quantum information processing. The initial state is the ground state of an initial linear Hamiltonian that is gradually turned into a final quadratic Hamiltonian whose ground state is the selected Dicke state. We use compensating operators to suppress diabatic transitions to unwanted states that would occur if the change were not slow. We discuss the possibilities of constructing the compensating operators by sequential application of quadratic Hamiltonians available in experiments.

Figure 1

Bloch sphere graphs of (a) the full compensating Hamiltonian of Eq. (1) and (b) the Hamiltonian of Eq. (2) compensating for transitions out of the ground state. The left figures correspond to ${\chi }_{\max }t=0.5$ and the right to ${\chi }_{\max }t=1.5$ of the time evolution described by Eqs. (13, 14, 15). In (a) and (b) the lighter (darker) color corresponds to the higher (lower) expectation values of the Hamiltonian in the spin coherent states. The lines with arrows represent a torque exerted by the Hamiltonian on the collective spin as $d\langle \vec{J}\rangle /dt=i\langle [H,\vec{J}]\rangle$ with the mean value calculated in the spin coherent state. (c) shows the $Q$ function of the instantaneous ground state of the Hamiltonian (13) with $N=30$ at the same instants of time as the Hamiltonian plots.

Figure 2

Bloch sphere graphs show the expectation value in the spin coherent states of the compensating Hamiltonians (a) $L_1$, (b) $L_2$, (c) $L_3$, and (d) $L_4$. The lighter (darker) color corresponds to the higher (lower) value. The lines with arrows represent the torque exerted by the Hamiltonian on the collective spin, similar to Fig. 1.

Figure 3

Time dependence of the coefficients ${\alpha }_k$ of Eq. (3) with operators $L_k$ given by Eqs. (16, 17, 18, 19), with $N=30$.

Figure 4

Time dependence of the coefficients ${\alpha }_1$ of Eq. (3) provided that only operator $L_1$ is available, with $N=30$.

Figure 5

Time dependence of the fidelity [expressed as ${\text{log}}_{10}(1-F)]$ for the state without any compensation, with compensation with operator $L_1$, etc., up to compensation with all four operators $L_1--L_4$, with $N=30$. The line marked with $H_B$ is achieved with the fully compensating Hamiltonian of Eq. (1) computed numerically. This indicates the numerical precision (exact calculation would yield $1-F=0$).

Figure 6

Time dependence of squeezing for the state without any compensation, with compensation with operator $L_1$, etc., up to compensation with all four operators $L_1$–$L_4$, with $N=30$. The line marked with $H_B$ is the result of fidelity achieved with the fully compensating Hamiltonian of Eq. (1). The inset shows the probability distribution of $J_z$ of the initial spin coherent state (narrow red bars) and of the final state achieved by compensation with four operators $L_1$–$L_4$ (wide blue bar).

Figure 7

Time dependence of the coefficients ${\alpha }_k$ of Eq. (3) for preparation of the Dicke state $|J_z=n\rangle$ with $N=30$ and $n=5$ for (a) the Hamiltonian (13) with $H_c=-J_x$ and (b) $H_c=\vec{c}·\vec{J}$ with $\vec{c}$ given by Eq. (24).

Figure 8

Final fidelity [expressed as ${\text{log}}_{10}(1-F)]$ for preparation of the Dicke states $|J_z=n\rangle$ dependent on $n$ with $N=30$ for (a) the Hamiltonian (13) with $H_c=-J_x$ and (b) the Hamiltonian $H_c=\vec{c}·\vec{J}$ with $\vec{c}$ given by Eq. (24). The inset shows the statistics of $J_z$ of the initial state (narrow red bars) and of the final state (wide blue bar) with $n=5$.

Figure 9

Time dependence of the coefficients ${\alpha }_k$ of Eq. (3) obtained from averaged values of Eqs. (26) and (27) for equally distributed atomic numbers between $N=50$ and 70 to be used with operators of Eqs. (16, 17, 18, 19) and to aim for the state $|J_z=0\rangle$.

Figure 10

Distribution of $J_z$ after evolution with the Hamiltonian (13) and compensating terms (wide blue bars) compared with the initial distribution of the spin coherent state (narrow red bars). The protocol is robust to atom number fluctuations and results are shown for four different atom numbers, driven by the coefficients ${\alpha }_k$ shown in Fig. 9 determined for a uniform distribution of atomic numbers from $N=50$ to 70.