Construction of a qudit using Schrödinger cat states and generation of hybrid entanglement between a discrete-variable qudit and a continuous-variable qudit

We show that a continuous-variable (CV) qudit can be constructed using quasiorthogonal cat states of a bosonic mode, when the phase encoded in each cat state is chosen appropriately. With the constructed CV qudit and the discrete-variable (DV) qudit encoded with Fock states, we propose an approach to generate the hybrid maximally entangled state of a CV qudit and a DV qudit by using two microwave cavities coupled to a superconducting flux qutrit. This proposal relies on the initial preparation of a superposition of Fock states of one cavity and the initial preparation of a cat state of the other cavity. After the initial state of each cavity is prepared, this proposal requires only two basic operations, i.e., the first operation employs the dispersive coupling of both cavities with the qutrit while the second operation uses the dispersive coupling of only one cavity with the qutrit. The entangled state production is deterministic and the operation time decreases as the dimensional size of each qudit increases. In addition, during the entire operation, the coupler qutrit remains in the ground state and thus decoherence from the qutrit is significantly reduced. As an example, we further discuss the experimental feasibility for generating the hybrid maximally entangled state of a DV qutrit and a CV qutrit based on circuit QED. This proposal is universal and can be extended to accomplish the same task, by using two microwave or optical cavities coupled to a natural or artificial three-level atom.

Figure 1

(a) Schematic diagram of two microwave cavities coupled to a flux qutrit (FQ). Each square represents a one-dimensional or three-dimensional cavity. The circle in the middle represents the flux qutrit, which is capacitively or inductively coupled to each cavity. (b) The three levels of the flux qutrit form a $\mathrm{\Lambda }$-type level structure, where the transition between the two lowest levels can be made weak by increasing the barrier between the two potential wells.

Figure 2

(a) Cavity 1 is dispersively coupled to the $|g\rangle \leftrightarrow |f\rangle$ transition with coupling constant $g_1$ and detuning ${\mathrm{\Delta }}_1={\omega }_{c_1}-{\omega }_{fg}>0$, while cavity 2 is dispersively coupled to the $|e\rangle \leftrightarrow |f\rangle$ transition with coupling constant $g_2$ and detuning ${\mathrm{\Delta }}_2={\omega }_{c_2}-{\omega }_{ef}>0$. (b) Cavity 1 is dispersively coupled to the $|g\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\mu }_1$ and detuning ${\delta }_1={\omega }_{fg}-{\tilde{\omega }}_{c_1}>0$, while the frequency of cavity 2 is adjusted such that cavity 2 is decoupled from the qutrit. Here, ${\tilde{\omega }}_{c_1}$ is the adjusted frequency of cavity 1.

Figure 3

Diagram of two 1D microwave cavities capacitively coupled to a superconducting flux qutrit (FQ). Each cavity here is a one-dimensional transmission line resonator. The flux qutrit consists of three Josephson junctions and a superconducting loop.

Figure 4

(a) Illustration of the unwanted coupling between cavity 1 and the $|e\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\tilde{g}}_1$ and detuning ${\tilde{\mathrm{\Delta }}}_1={\omega }_{c_1}-{\omega }_{fe}>0$ as well as the unwanted coupling between cavity 1 and the $|g\rangle \leftrightarrow |e\rangle$ transition with coupling constant $g_1^{\prime }$ and detuning ${\mathrm{\Delta }}_1^{\prime }={\omega }_{c_1}-{\omega }_{eg}>0$. Illustration of the unwanted coupling between cavity 2 and the $|g\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\tilde{g}}_2$ and detuning ${\tilde{\mathrm{\Delta }}}_2={\omega }_{fg}-{\omega }_{c_2}>0$ as well as the unwanted coupling between cavity 2 and the $|g\rangle \leftrightarrow |e\rangle$ transition with coupling constant $g_2^{\prime }$ and detuning ${\mathrm{\Delta }}_2^{\prime }={\omega }_{c_2}-{\omega }_{eg}>0$. (b) Illustration of the unwanted coupling between cavity 1 and the $|e\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\tilde{\mu }}_1$ and detuning ${\tilde{\delta }}_1={\tilde{\omega }}_{c_1}-{\omega }_{fe}>0$ as well as the unwanted coupling between cavity 1 and the $|g\rangle \leftrightarrow |e\rangle$ transition with coupling constant ${\mu }_1^{\prime }$ and detuning ${\delta }_1^{\prime }={\tilde{\omega }}_{c_1}-{\omega }_{eg}>0$. Illustration of the unwanted coupling between cavity 2 and the $|e\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\mu }_2$ and detuning ${\delta }_2={\omega }_{fe}-{\tilde{\omega }}_{c_2}>0$, the unwanted coupling between cavity 2 and the $|g\rangle \leftrightarrow |f\rangle$ transition with coupling constant ${\tilde{\mu }}_2$ and ${\tilde{\delta }}_2={\omega }_{fg}-{\tilde{\omega }}_{c_2}>0$, as well as the unwanted coupling between cavity 2 and the $|g\rangle \leftrightarrow |e\rangle$ transition with coupling constant ${\mu }_2^{\prime }$ and detuning ${\delta }_2^{\prime }={\omega }_{eg}-{\tilde{\omega }}_{c_2}>0$.

Figure 5

Fidelity versus $T$ for $g_{cr}/g_{\max }=0,0.001$, and 0.01. In the numerical simulation, we set ${\kappa }^{-1}=10\mu \mathrm{s}$. Here, $\kappa$ is the cavity decay rate, and $g_{cr}$ is the crosstalk strength between the two cavities.

Figure 6

Fidelity versus ${\kappa }^{-1}$ for $T=1.5$, 2.5, and $5.0\mu \mathrm{s}$. In the numerical simulation, we set $g_{\mathrm{cr}}=0.01g_{max}$.

Figure 7

Fidelity versus $x$ for ${\kappa }^{-1}=10$, 30, and $50\mu \mathrm{s}$. In the numerical simulation, we set $T=2.5\mu \mathrm{s}$ and $g_{\mathrm{cr}}=0.01g_{max}$.

Figure 8

Fidelity versus $\delta \tau /\tau$ for ${\kappa }^{-1}=10$, 30, and $50\mu \mathrm{s}$. In the numerical simulation, we set $T=2.5\mu \mathrm{s}$ and $g_{cr}=0.01g_{max}$.