Generation of long-lived $W$ states via reservoir engineering in dissipatively coupled systems

Very recently, dissipative coupling was discovered, which develops and broadens methods for controlling and utilizing light-matter interactions. Here, we propose a scheme to generate the tripartite $W$ state in a dissipatively coupled system, where one qubit and two resonators simultaneously interact with a common reservoir. With appropriate parameters, we find the $W$ state is a dark state of the system. By driving the qubit, the dissipatively coupled system will evolve from the ground state to the tripartite $W$ state. Because the initial state is the ground state of the system and no measurement is required, our scheme is easy to implement in experiments. Moreover, the $W$ state decouples from the common reservoir and thus has a very long lifetime. This scheme is applicable to a wide class of dissipatively coupled systems and we specifically illustrate how to prepare the $W$ state in a hybrid qubit-photon-magnon system by using this scheme.

Figure 1

(a) A schematic showing a qubit and two resonators (with annihilation operators $\sigma , a$, and $b$, respectively) simultaneously interacting with a common reservoir. To generate the $W$ state, a drive pulse with Rabi frequency ${\mathrm{\Omega }}_d$ pumps the qubit. (b) Mediated by the common reservoir in (a), the three subsystems are dissipatively coupled together, which can be described using a Lindblad superoperator $\mathcal{L}\left[o\right]$ in the master equation, cf. Eqs. (2) to (4). (c) Diagram of a hybrid qubit-photon-magnon system. A superconducting transmon qubit, a superconducting transmission-line resonator, and a ferrimagnetic YIG sphere are dissipatively coupled via an open waveguide, where the transmon qubit is driven by a microwave pulse with Rabi frequency ${\mathrm{\Omega }}_d$.

Figure 2

Time evolution of the fidelity ${\mathcal{F}}_X=\mathrm{Tr}\left(\rho \right|X\rangle \langle X\left|\right)$ for the initial states $|X\rangle =|D_1\rangle$ (black solid curve), $|X\rangle =|D_2\rangle$ (red dashed curve), and $|X\rangle =|B\rangle$ (blue dotted curve), calculated using Eq. (2). Here the parameters are set as ${\omega }_{\sigma }/\tau ={\omega }_a/\tau ={\omega }_b/\tau =500$ and ${\eta }_{\sigma }={\eta }_a={\eta }_b=1$.

Figure 3

(a) Time evolution of the fidelity $\mathcal{F}$ of the prototype $W$ state $|W_3^{\left(1\right)}\rangle =\left(\right|e00\rangle +|g10\rangle +|g01\rangle )/\sqrt{3}$, with ${\mathrm{\Omega }}_d/\tau =0.01$ and $t_0=+\infty$, where the red dot denotes the maximum value ${\mathcal{F}}_{\max }=0.985$ of the fidelity $\mathcal{F}$ at time $\tau t_{\max }=273$. (b) The maximal fidelity ${\mathcal{F}}_{\max }$ of $|W_3^{\left(1\right)}\rangle$ versus the Rabi frequency ${\mathrm{\Omega }}_d/\tau$, where the corresponding time $\tau t_{\max }$ (for reaching ${\mathcal{F}}_{\max }$) versus ${\mathrm{\Omega }}_d/\tau$ is shown in the inset. (c) Time evolution of the fidelity $\mathcal{F}$ of $|W_3^{\left(1\right)}\rangle$ for drive pulses with different shapes: ${\mathrm{\Omega }}_d/\tau =0.01, \tau t_0=273$ (black solid curve); ${\mathrm{\Omega }}_d/\tau =0.05, \tau t_0=54.1$ (red dashed curve); and ${\mathrm{\Omega }}_d/\tau =0.1, \tau t_0=26.9$ (green dotted curve). (d) Time evolution of the fidelity $\mathcal{F}$ for different $W$ states: the Agrawal $W$ state $|W_3^{\left(2\right)}\rangle =\left(\sqrt{2}\right|e00\rangle +|g10\rangle +|g01\rangle )/2$ with ${\mathrm{\Omega }}_d/\tau =0.001, \tau t_0=2211.1$ (black solid curve); the common $W$ state $|W_3^{\left(3\right)}\rangle =\left(2\right|e00\rangle -|g10\rangle -|g01\rangle )/\sqrt{6}$ with ${\mathrm{\Omega }}_d/\tau =0.001, \tau t_0=1934.7$ (red dashed curve); and the Bell state $|W_3^{\left(4\right)}\rangle =\left(\right|e00\rangle +|g01\rangle )/\sqrt{2}$ with ${\mathrm{\Omega }}_d/\tau =0.001, \tau t_0/=2211.1$ (green dotted curve). These results in (a)–(d) are obtained using the master equation in Eq. (2) with the system Hamiltonian $H_s^{\left(d\right)}$ given in Eq. (12) and the initial state $|g00\rangle$. For the four $W$ states, $|W_3^{\left(1\right)}\rangle , |W_3^{\left(2\right)}\rangle , |W_3^{\left(3\right)}\rangle$, and $|W_3^{\left(4\right)}\rangle$, the corresponding values of $({\eta }_{\sigma },{\eta }_a,{\eta }_b)$ can be found in Table 1. Other parameters are ${\omega }_{\sigma }/\tau ={\omega }_a/\tau ={\omega }_b/\tau ={\omega }_d/\tau =500$.

Figure 4

(a) The maximal fidelity ${\mathcal{F}}_{\max }$ of the prototype $W$ state $|W_3^{\left(1\right)}\rangle =\left(\right|e00\rangle +|g10\rangle +|g01\rangle )/\sqrt{3}$ versus the Rabi frequency ${\mathrm{\Omega }}_d/\tau$ with ${\gamma }_b^{-1}=5\mathrm{\mu s}$, where the corresponding time $t_{\max }$ (for reaching ${\mathcal{F}}_{\max }$) versus ${\mathrm{\Omega }}_d/\tau$ is shown in (b). (c) Time evolution of the fidelity $\mathcal{F}$ of $|W_3^{\left(1\right)}\rangle$ for different relaxation times of the magnon mode: ${\gamma }_b^{-1}=1\mathrm{\mu s}$ (black solid curve); ${\gamma }_b^{-1}=5\mathrm{\mu s}$ (red dashed curve); and ${\gamma }_b^{-1}=20\mathrm{\mu s}$ (green dotted curve). The shapes of the corresponding drive pulses are ${\mathrm{\Omega }}_d/\tau =0.0452, t_0=0.478\mathrm{\mu s}$; ${\mathrm{\Omega }}_d/\tau =0.0221, t_0=0.976\mathrm{\mu s}$; and ${\mathrm{\Omega }}_d/\tau =0.0137, t_0=1.572\mathrm{\mu s}$, respectively. (d) Time evolution of the fidelity $\mathcal{F}$ for different $W$ states in the case of ${\gamma }_b^{-1}=5\mathrm{\mu s}$: the Agrawal $W$ state $|W_3^{\left(2\right)}\rangle =\left(\sqrt{2}\right|e00\rangle +|g10\rangle +|g01\rangle )/2$ with ${\mathrm{\Omega }}_d/\tau =0.0221, t_0=0.795\mathrm{\mu s}$ (black solid curve); the common $W$ state $|W_3^{\left(3\right)}\rangle =\left(2\right|e00\rangle -|g10\rangle -|g01\rangle )/\sqrt{6}$ with ${\mathrm{\Omega }}_d/\tau =0.0281, t_0=0.547\mathrm{\mu s}$ (red dashed curve); and the Bell state $|W_3^{\left(4\right)}\rangle =\left(\right|e00\rangle +|g01\rangle )/\sqrt{2}$ with ${\mathrm{\Omega }}_d/\tau =0.0221, t_0=0.795\mathrm{\mu s}$ (green dotted curve). These results in (a)–(d) are obtained using the master equation in Eq. (14) with the system Hamiltonian $H_s^{\left(d\right)}$ given in Eq. (12) and the initial state $|g00\rangle$. Other parameters are ${\gamma }_{\sigma }^{-1}={\gamma }_a^{-1}=60\mathrm{\mu s}, {\gamma }_{\phi }^{-1}=25\mathrm{\mu s}, \tau /2\pi =20\mathrm{MHz}$, and ${\omega }_{\sigma }/2\pi ={\omega }_a/2\pi ={\omega }_b/2\pi ={\omega }_d/2\pi =5\mathrm{GHz}$.

Figure 5

(a) The optimum fidelity ${\mathcal{F}}_{\max }^{\left(\mathrm{opt}\right)}$ of the prototype $W$ state $|W_3^{\left(1\right)}\rangle =\left(\right|e00\rangle +|g10\rangle +|g01\rangle )/\sqrt{3}$ versus the cooperative decay rate $\tau /2\pi$ for different relaxation times of the magnon mode, ${\gamma }_b^{-1}=1\mathrm{\mu s}$ (black square), ${\gamma }_b^{-1}=5\mathrm{\mu s}$ (red circle), and ${\gamma }_b^{-1}=20\mathrm{\mu s}$ (green triangle). (b) The optimum fidelity ${\mathcal{F}}_{\max }^{\left(\mathrm{opt}\right)}$ of the prototype $W$ state $|W_3^{\left(1\right)}\rangle$ versus the deviation $\delta$ from different subsystems with $\tau /2\pi =20\mathrm{MHz}$ and ${\gamma }_b^{-1}=5\mathrm{\mu s}$. Here, $o=2(1+\delta )\sigma -a-b$ (black square), $o=2\sigma -(1+\delta )a-b$ (red circle), and $o=2\sigma -a-(1+\delta )b$ (green triangle). In (a,b), each data point is calculated using the master equation in Eq. (14), where ${\mathrm{\Omega }}_d={\mathrm{\Omega }}_d^{\left(\mathrm{opt}\right)}$ and $t_0=t_{\max }^{\left(\mathrm{opt}\right)}$. Other parameters are the same as in Fig. 4.