Fast macroscopic-superposition-state generation by coherent driving

We propose a scheme to generate macroscopic superposition states (MSSs) in spin ensembles, where a coherent driving field is applied to accelerate the generation of macroscopic superposition states. The numerical calculation demonstrates that this approach allows us to generate a superposition of two classically distinct states of the spin ensemble with a high fidelity above 0.97 for 300 spins. For a larger spin ensemble, though the fidelity slightly declines, it maintains above 0.84 for an ensemble of 500 spins. The time to generate an MSS is also estimated, which shows that the significantly shortened generation time allows us to achieve such MSSs within a typical coherence time of the system.

Figure 1

(i) Plots of time evolution of the fidelity $F(J;\tilde{\omega },\varphi ;\tau )$, the relative phase ${\gamma }_0^{\prime }$, and the displacement angle ${\delta }_0$ and (ii) the $Q$ functions $Q(\alpha ,\beta )\equiv \frac{2J+1}{4\pi }{\left|\langle {\mathrm{\Phi }}_{\mathrm{CSS}}(J;\alpha ,\beta )|\mathrm{\Psi }(J;\tilde{\omega },\varphi ;\tau )\rangle \right|}^2$ corresponding to the initial state and the first local maximum of the fidelity for $J=50$, $\tilde{\omega }=0.0204\pi$, and $\varphi =0.024\pi$. (i) Time dependences of $F(J;\tilde{\omega },\varphi ;\tau )$, ${\gamma }_0^{\prime }$, and ${\delta }_0$ are indicated by the black solid curve, the red dashed curve, and the green dots, respectively. The yellow and red shaded regions represent the intervals $F(J;\tilde{\omega },\varphi ;\tau )\ge 0.99$ and ${\delta }_0\ge 0.95\pi$, respectively. (ii) The color at the point indicated by the polar and azimuthal angles of $(\alpha ,\beta )$ represents $\frac{4\pi }{2J+1}Q(\alpha ,\beta )$ according to the right gauge. The time evolution of the fidelity and the $Q$ functions for $J=74.5$ and $J=200$ are shown in Figs. 9 and 10, respectively.

Figure 2

Total spin dependences of (i) the set of the rescaled driving frequency and the driving phase, (ii) the fidelity, (iii) the displacement angle, and (iv) the rescaled evolution time. In the plots (i)–(iv), there is discontinuity between $J=150$ and 174.5.

Figure 3

Interference fringes of the quantum fluctuations in ${\widehat{\sigma }}_x^{\otimes N}$. The red solid curves and the blue dashed curves indicate the interference fringes produced by the perfect MSS given in Eq. (8) and pure MSSs $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$, respectively. The black dots with error bars represent the mean values and the standard variances of the quantum fluctuations in ${\widehat{\sigma }}_x^{\otimes N}$ and ${\tau }_{\mathrm{opt}}$ of $\left|\mathrm{\Psi }\right(J;{\tau }_{\mathrm{opt}})\rangle$ represents the optimized evolution time for $\overline{J}$. (i),(ii) The fringes produced by the MSSs with $5\%$ of the Gaussian fluctuations in the number of spins for $N=149$ and 200. (iii),(iv) The fringes produced by the MSSs with $5\%$ of the uniform distribution in the magnitude of the driving field $\mathrm{\Omega }$.

Figure 4

Discrete Fourier transformation of ${\widehat{\sigma }}_x^{\otimes N}-1$. The red solid curves and the blue dashed curves indicate the spectra produced by the perfect MSS given in Eq. (8) and pure MSSs $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$. The black dots represent the mean values of the spectra and ${\tau }_{\mathrm{opt}}$ of $\left|\mathrm{\Psi }\right(J;{\tau }_{\mathrm{opt}})\rangle$ represents the optimized evolution time for $\overline{J}$. The black solid lines indicate the frequencies of the interference fringes for the perfect MSS, i.e., $\omega =\pm 4Jcos\beta$ in Eq. (8). (i),(ii) The spectra produced by the MSSs with $5\%$ of the Gaussian fluctuations in the number of spins for $N=149$ and 200 at each step of rotation.

Figure 5

Dependence of the gap $\tilde{\mathrm{\Delta }}(J;\tau )$ between $|{\varepsilon }_1(J;\tau )\rangle$ and $|{\varepsilon }_2(J;\tau )\rangle$ on the phase of the driving field ${\tilde{\omega }}_{\mathrm{opt}}\tau +{\varphi }_{\mathrm{opt}}$. (i) The gap $\tilde{\mathrm{\Delta }}(J;\tau )$ with respect to ${\tilde{\omega }}_{\mathrm{opt}}\tau +{\varphi }_{\mathrm{opt}}$ for $J=50\text{–}250$. The dots, the triangles, and the thin diamonds mark $\tau =0$, $\tau =0.2{\tau }_{\mathrm{opt}}$, $\tau =0.4{\tau }_{\mathrm{opt}}$, $\tau =0.6{\tau }_{\mathrm{opt}}$, $\tau =0.8{\tau }_{\mathrm{opt}}$, $\tau ={\tau }_{\mathrm{opt}}$ on the curves representing $\tilde{\mathrm{\Delta }}(J;\tau )$. (ii) The gap functions $\tilde{\mathrm{\Delta }}(J;\tau )$ for $J=50\text{–}250$ coincide with each other by shifts in the phase of the driving field and enlargements (or shrinks) of the magnitude of the gap energy.

Figure 6

Time evolution of the $Q$ functions of $|{\varepsilon }_1(J;\tau )\rangle$ and $|{\varepsilon }_2(J;\tau )\rangle$ for $J=74.5$ and $J=200$. The color hue represents $\frac{4\pi }{2J+1}Q(\alpha ,\beta )$ whose gauge is the same as Figs. 1. The dotted white curves indicate the contours of ${\tilde{h}}_{\mathrm{opt}}(J;\tau )$ in the mean-field limit, which is given by ${\tilde{E}}_{\mathrm{opt}}(J;\alpha ,\beta ;\tau )=J[\frac{1}{2}{cos}^2\beta +rcos\alpha sin\beta cos({\tilde{\omega }}_{\mathrm{opt}}\tau +{\varphi }_{\mathrm{opt}})]$, and the values on the contours represent $\frac{2}{J}{\tilde{E}}_{\mathrm{opt}}(J;\alpha ,\beta ;\tau )$. The solid white curves represent the energy contours ${\tilde{E}}_{\mathrm{opt}}(J;\alpha ,\beta ;\tau )=rJcos({\tilde{\omega }}_{\mathrm{opt}}\tau +{\varphi }_{\mathrm{opt}})$.

Figure 7

$J$ dependence of the probability distribution of the initial state $|{\mathrm{\Phi }}_{\mathrm{CSS}}(J;0,\frac{\pi }{2})\rangle$ on the highest-energy eigenstate $|{\varepsilon }_1(J;0)\rangle$ (black solid curve with dots) and the second-highest-energy eigenstate $|{\varepsilon }_2(J;0)\rangle$ (red dashed curve with triangles) of the Hamiltonian $\tilde{h}(J;0)$ and the other eigenstates (blue dotted curve with thin diamonds). The probability on $|{\varepsilon }_1(J;0)\rangle$ monotonically and slowly decreases with respect to $J$ and converges toward $\sim 0.5$, while the probability on $|{\varepsilon }_2(J;0)\rangle$ stays at zero. The probability distributing on the other eigenstates monotonically and slowly increases and converges toward $\sim 0.5$.

Figure 8

Plots of the relative phases ${\gamma }_M^{\prime }$'s of $|{\varepsilon }_1(J;\tau )\rangle$ and $|{\varepsilon }_2(J;\tau )\rangle$ as functions of $\tau$ and $\frac{M}{J}$ for (i) $J=74.5$ and (ii) $J=200$. The red dots and the blue triangles represent ${\gamma }_M^{\prime }$'s for $|{\varepsilon }_1(J;\tau )\rangle$ and $|{\varepsilon }_2(J;\tau )\rangle$, respectively. The right-hand sides of blue shaded planes parallel to the $\frac{M}{J}-{\gamma }_M^{\prime }$ planes are the time regions where $\tilde{\mathrm{\Delta }}(J;\tau )<O\left({10}^{-6}\right)$, i.e., the region where the gaps can be regarded to be closed. We note that we plot ${\gamma }_M^{\prime }$'s for every five points for $J=74.5$ and for every 20 points for $J=200$ with respect to $M/J$ for the sake of visibility of the points. For both $J=74.5$ and 200, ${\gamma }_M^{\prime }=0$ for $|{\varepsilon }_1(J;\tau )\rangle$ and ${\gamma }_M^{\prime }=\pi$ for $|{\varepsilon }_2(J;\tau )\rangle$ when the gaps are open. When $J=200$ and the gap closes, ${\gamma }_M^{\prime }$ can be considered to be indefinite, since the $Q$ functions of $|{\varepsilon }_1(J=200;\tau ={\tau }_{\mathrm{opt}})\rangle$ and $|{\varepsilon }_2(J=200;{\tau }_{\mathrm{opt}})\rangle$ in Fig. 6 imply that they are close to coherent spin states, which are separable, and the probabilities either on $|J;\pm M\rangle$ become $\sim 0$.

Figure 9

Rescaled-time dependences of fidelity $F(J;\tilde{\omega },\varphi ;\tau )$, relative phase ${\gamma }_0^{\prime }$, and displacement angle ${\delta }_0({\alpha }_0,{\beta }_0)$ for (i) $J=74.5$ and (ii) $J=200$. The vertical scales on the left-hand sides and the right-hand sides are for $F(J;\tilde{\omega },\varphi ;\tau )$ and the two angles, ${\gamma }_0^{\prime }$ and ${\delta }_0({\alpha }_0,{\beta }_0)$, respectively. The black solid curves, the green dots, and the red dashed curves represent $F(J;\tilde{\omega },\varphi ;\tau )$, ${\gamma }_0^{\prime }$, and ${\delta }_0({\alpha }_0,{\beta }_0)$, respectively. The yellow shaded regions and the red shaded region with a left-right arrow express the intervals satisfying $F(J;\tilde{\omega },\varphi ;\tau )\ge 0.99$ and the interval where an almost perfect cat state $\frac{1}{\sqrt{2}}\left(\right|J,J\rangle +|J,-J\rangle )$ is generated, i.e., the region with $F(J;\tilde{\omega },\varphi ;\tau )\ge 0.99$ and ${\delta }_0({\alpha }_0,{\beta }_0)\ge 0.95$. The $Q$ functions at which the black and thin dotted lines, (a)–(e), are illustrated in Figs. 10 and $\tau ={\tau }_{\max }$, at which the first local maximum of the fidelity $F_{\max }(J;\tilde{\omega },\varphi )$ is achieved, is indicated by the black and thin dashed line. The driving-field parameters are set to be $\tilde{\omega }=0.0204\pi$ and $\varphi =0.024\pi$ for $J=50$, $\tilde{\omega }=0.0174\pi$, and $\varphi =0.012\pi$ for $J=74.5$, and $\tilde{\omega }=0.0151\pi$ and $\varphi =-0.0128\pi$ for $J=200$.

Figure 10

Time evolution of the $Q$ functions $Q(\alpha ,\beta )\equiv \frac{2J+1}{4\pi }{\left|\langle {\mathrm{\Phi }}_{\mathrm{CSS}}(J;\alpha ,\beta )|\mathrm{\Psi }(J;\tilde{\omega },\varphi ;\tau )\rangle \right|}^2$ for $J=50$, $J=74.5$, and $J=200$. The driving-field parameters are set to be the same as Figs. 9. The color at the point indicated by the polar and azimuthal angles of $(\alpha ,\beta )$ represents $\frac{4\pi }{2J+1}Q(\alpha ,\beta )$ according to the gauge shown in Fig. 1. For $J=50$, the driving field parameters are set to be $\tilde{\omega }=0.0204\pi$ and $\varphi =0.024\pi$ and the snapshots are taken at the rescaled elapsed times of (a) $\tau =0$, (b) 4, (c) 8, (d) 12, and (d) 16. For $J=74.5$ and $J=200$, the driving field parameters are given by $\tilde{\omega }=0.0174\pi$ and $\varphi =-0.012\pi$ and $\tilde{\omega }=0.0151\pi$ and $\varphi =-0.0128\pi$, respectively, and the snapshots are taken at (a) $\tau =0$, (b) 5, (c) 10, (d) 15, and (d) 20.

Figure 11

Fidelity $F_{\max }(J;\tilde{\omega },\varphi )$ of the first local maximum and its corresponding displacement angle ${\delta }_{\max }(J;\tilde{\omega },\varphi )$ as functions of the driving-field frequency and phase, $\tilde{\omega }$ and $\varphi$, for (i) $J=50$, (ii) $J=74.5$, (iii) $J=100$, (iv) $J=124.5$, (v) $J=150$, (vi) $J=174.5$, (vii) $J=200$, (viii) $J=224.5$, and (ix) $J=250$. The $z$ axes represent $F_{\max }(J;\tilde{\omega },\varphi )$ and the color on the $F_{\max }(J;\tilde{\omega },\varphi )$ surface and the plane at the bottom of the plot indicate the magnitude of the displacement angle whose gauge is shown on the right-hand side of (ix). There are two parameter regions with high fidelity with ${\delta }_{\max }(J;\tilde{\omega },\varphi )\gtrsim 0.4\pi$ for $J=100-150$, while ${\delta }_{\max }(J;\tilde{\omega },\varphi )$ decreases to be ${\delta }_{\max }(J;\tilde{\omega },\varphi )\sim 0$ in the region with smaller $\tilde{\omega }$ for $J=174.5-250$.

Figure 12

Plots of $J$ dependence of the angles ${\alpha }_{\mathrm{opt}}$, ${\beta }_{\mathrm{opt}}$, and ${\gamma }_{\mathrm{opt}}^{\prime }$, which are represented by the red dots, the blue triangles, and the green thin diamonds, respectively. The relative phase ${\gamma }_{\mathrm{opt}}^{\prime }=0$ for all $J$.

Figure 13

Fringes of $\sqrt{\langle {\left(\mathrm{\Delta }{\widehat{\sigma }}_x^{\otimes N}\right)}^2\rangle }$ generated by $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ with the spin-number fluctuations of $2\%$ and $10\%$ for $\overline{J}=74.5$ and $\overline{J}=200$. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS $|{\mathrm{\Phi }}_{\mathrm{MSS}}(J;{\alpha }_{\mathrm{opt}},{\beta }_{\mathrm{opt}},{\gamma }_{\mathrm{opt}})\rangle$ and $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ without spin-number fluctuation. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments, where the spin-number fluctuation is given by Eq. (C9) and $\left|\mathrm{\Psi }\right(J;{\tau }_{\mathrm{opt}})\rangle$ is prepared after the optimized evolution time ${\tau }_{\mathrm{opt}}$ for $\overline{J}$.

Figure 14

Fringes of $\sqrt{\langle {\left(\mathrm{\Delta }{\widehat{\sigma }}_x^{\otimes N}\right)}^2\rangle }$ generated by $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ with the fluctuation in the driving-field magnitude of $2\%$ and $10\%$ for $\overline{J}=74.5$ and $\overline{J}=200$. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS $|{\mathrm{\Phi }}_{\mathrm{MSS}}(J;{\alpha }_{\mathrm{opt}},{\beta }_{\mathrm{opt}},{\gamma }_{\mathrm{opt}})\rangle$ and $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ without the fluctuation in $\mathrm{\Omega }$, i.e., $\mathrm{\Omega }=\overline{\mathrm{\Omega }}$. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments with the $\mathrm{\Omega }$ distributed randomly between $(1\pm \sigma )\overline{\mathrm{\Omega }}$.

Figure 15

Fringes of $\sqrt{\langle {\left(\mathrm{\Delta }{\widehat{\sigma }}_x^{\otimes N}\right)}^2\rangle }$ generated by $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ with the fluctuation in the nonlinear interaction energy $\lambda$ of $2\%$, $5\%$, and $10\%$ for $\overline{J}=74.5$ and $\overline{J}=200$. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS $|{\mathrm{\Phi }}_{\mathrm{MSS}}(J;{\alpha }_{\mathrm{opt}},{\beta }_{\mathrm{opt}},{\gamma }_{\mathrm{opt}})\rangle$ and $|{\mathrm{\Psi }}_{\mathrm{opt}}\left(J\right)\rangle$ without the fluctuation in $\lambda$, i.e., $\lambda =\overline{\lambda }$. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments with the $\lambda$ distributed randomly between $(1\pm \sigma )\overline{\lambda }$.