Figure 1

Basic concept for generating Dicke states using rapid adiabatic passage. (a) (left) Relevant states. A motional Fock state $|\downarrow \downarrow \rangle |1\rangle$ is coupled to $|\downarrow \uparrow \rangle |0\rangle$ and $|\uparrow \downarrow \rangle |0\rangle$ with a red-sideband pulse. (right) After applying a basis transformation (Morris–Shore transformation) to the upper two states, the system is decomposed into a two-state system and an uncoupled (dark) state. (b) Dressed state potentials for the two-state system plotted against the detuning of the laser from the red-sideband resonance. In rapid adiabatic passage, the ions move from the left along the potential (indicated by the solid curve) to $|D_2^{(1)}\rangle |0\rangle =(1/\sqrt{2})(|\downarrow \uparrow \rangle +|\uparrow \downarrow \rangle )|0\rangle$. Here, the Rabi frequencies are assumed to be constant for simplicity, while they are usually varied in actual experiments.Reuse & Permissions

Figure 2

(a) Result of projection (population) measurement after Dicke-state generation. A histogram of fluorescence counts obtained by shining a 397-nm laser beam for 7 ms after the generation of a Dicke state is shown. The left, middle, and right peaks ($~$0, 70, and 140 counts, respectively) correspond to $P_{\uparrow \uparrow },P_{\downarrow \uparrow }+P_{\uparrow \downarrow },$ and $P_{\downarrow \downarrow }$, respectively, where $P_{s_1{s}_2}$($s_1,s_2=\downarrow ,\uparrow$) represents the population in the respective basis state. From this, ${\rho }_{\downarrow \uparrow ,\downarrow \uparrow }+{\rho }_{\uparrow \downarrow ,\uparrow \downarrow }=0.74\pm 0.06$ is obtained. (b) Result of parity measurement after Dicke-state generation. The measured parity after Dicke-state generation followed by the application of a $\pi /2$ pulse is plotted against the phase of the $\pi /2$ pulse phase (open circles). The solid curve is a sinusoidal fit with period $\pi$ to the measured values. From this, $2\mathrm{Re}({\rho }_{\downarrow \uparrow ,\uparrow \downarrow })=0.58\pm 0.02$ is obtained. Filled circles represent parities after Dicke-state generation followed by two $\pi /2$ pulses, and the phase of the second pulse is variable. The sinusoidal oscillation with period $\pi$ indicates the presence of $(1/\sqrt{2})(|\downarrow \downarrow \rangle +|\uparrow \uparrow \rangle )$ after the first $\pi /2$ pulse.Reuse & Permissions

Figure 3

Effects of ac Stark shifts due to optical pulses for rapid adiabatic passage. (a) Energy levels used for the calculation (horizontal solid lines). $|D\rangle$ represents $|D_2^{(1)}\rangle =(1/\sqrt{2})(|\downarrow \uparrow \rangle +|\uparrow \downarrow \rangle )$. The bold arrow indicates the transition used for rapid adiabatic passage and the other arrows represent couplings considered here for calculating the ac Stark shifts. (b) Variation of the envelope (solid curve) and detuning (dashed line) of an optical pulse for rapid adiabatic passage against time. The envelope is assumed to be a Gaussian function, and the detuning is assumed to be linear. (c) Variation of adiabatic energies of relevant levels against time (solid curves). They approach each other near 0.11 ms, as well as near 0.3 ms. The dashed curves are shown for comparison; they represent adiabatic energies for an ideal case in which Hamiltonian matrix elements for carrier transitions ($|\downarrow \downarrow \rangle |1\rangle$–$|D\rangle |1\rangle$,  $|D\rangle |0\rangle$–$|\downarrow \downarrow \rangle |0\rangle$, and $|D\rangle |0\rangle$–$|\uparrow \uparrow \rangle |0\rangle$) are set to be zero. These potentials only approach each other near 0.3 ms. (d) $|{\alpha }_{\mathit{ji}}(t)/{\omega }_{\mathit{ji}}(t)|{}^2$ indicating violation of adiabaticity is plotted (see text).Reuse & Permissions

Figure 4

Verification of robustness in generation of Dicke states using rapid adiabatic passage. Gaussian pulses producing time-dependent Rabi frequency $\Omega (t)={\Omega }_{\mathrm{peak}}\exp [-t^2/2{\sigma }^2]$ are applied as RAP pulses and values of obtained fidelity (circles) are estimated, with (a) the pulse width $2\sigma$ and (b) the peak Rabi frequency ${\Omega }_{\mathrm{peak}}/2\pi$ varied. Here, the Rabi frequency is for the carrier transition. ${\Omega }_{\mathrm{peak}}/2\pi$ is fixed to 145 kHz in (a) and $2\sigma$ is fixed to 244 $\mu$s in (b). The total duration of each RAP pulse is 2.36 times the pulse width $2\sigma$, and during the pulse duration the excitation laser is swept from $-100$ kHz below to 100 kHz above the resonance of the red-sideband transition. The error bars represent 68% confidence intervals. The horizontal dashed lines at a fidelity of 0.5 represent the threshold for inseparable states (see text). Crosses and asterisks indicate values of ${\rho }_{\downarrow \uparrow ,\downarrow \uparrow }+{\rho }_{\uparrow \downarrow ,\uparrow \downarrow }$ and $2\text{Re}({\rho }_{\downarrow \uparrow ,\uparrow \downarrow })$, respectively, which are used for discussions about infidelity in Sec. 7.Reuse & Permissions