Circuit QED: generation of two-transmon-qutrit entangled states via resonant interaction

Abstract

We present a way to create entangled states of two superconducting transmon qutrits based on circuit QED. Here, a qutrit refers to a three-level quantum system. Since only resonant interaction is employed, the entanglement creation can be completed within a short time. The degree of entanglement for the prepared entangled state can be controlled by varying the weight factors of the initial state of one qutrit, which allows the prepared entangled state to change from a partially entangled state to a maximally entangled state. Because a single cavity is used, only resonant interaction is employed, and none of identical qutrit–cavity coupling constant, measurement, and auxiliary qutrit is needed, this proposal is easy to implement in experiments. The proposal is quite general and can be applied to prepare a two-qutrit partially or maximally entangled state with two natural or artificial atoms of a ladder-type level structure, coupled to an optical or microwave cavity.

Appendix

We here give a derivation on the state transformations induced by the application of the pulses, for the operations of steps 1, 5, and 9 described above.

For step 1: A microwave pulse of {$\omega _{10},-\pi /2,\pi /\left( 4 {{\varOmega }}_{10}\right)$} and then a microwave pulse of {$\omega _{21},-\pi /2,\pi /\left( 2{{\varOmega }}_{21}\right)$} were applied to qutrit 2 (Fig. 2a). The first pulse results in $\left| 0\right\rangle _{2}\rightarrow \left( \left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}$ according to Eq. (2), while the second pulse leads to $\left| 1\right\rangle _{2}\rightarrow \left| 2\right\rangle _{2}$ according to Eq. (3). After the two pulses, we thus have the state transformations:

$$\begin{aligned} \left| 0\right\rangle _{2}\overset{p1}{\rightarrow }\left( \left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}\overset{ p2}{\rightarrow }\left( \left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}, \end{aligned}$$

(17)

where the first transformation is obtained after applying the first pulse, while the second transformation is achieved after applying the second pulse. Note that the state $\left| 0\right\rangle _{2}$ remains unchanged during the second pulse because the pulse is highly detuned (or decoupled) from the $\left| 1\right\rangle _{2}\rightarrow \left| 2\right\rangle _{2}$ transition of qutrit 2.

For step 5: The microwave pulses of {$\omega _{10},-\pi /2,\pi /\left( 2 {{\varOmega }}_{10}\right)$}, {$\omega _{21},\pi /2,\pi /\left( 4 {{\varOmega }}_{21}\right)$}, {$\omega _{10},\pi /2,3\pi /\left( 4 {{\varOmega }}_{10}\right)$}, and then {$\omega _{21},\pi /2,\pi /\left( 2{{\varOmega }}_{21}\right)$} were applied to qutrit 2 in turn (Fig. 2d). According to Eqs. (2) and (3), the first pulse results in $\left| 0\right\rangle _{2}\rightarrow \left| 1\right\rangle _{2},$ the second pulse results in $\left( \left| 1\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\rightarrow \left| 1\right\rangle _{2}$ and $\left( -\left| 1\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\rightarrow \left| 2\right\rangle _{2},$ the third pulse results in $\left| 1\right\rangle _{2}\rightarrow \left( \left| 0\right\rangle _{2}-\left| 1\right\rangle _{2}\right) / \sqrt{2}$ but nothing to the state $\left| 2\right\rangle _{2}$, and the last pulse leads to $\left| 1\right\rangle _{2}\rightarrow -\left| 2\right\rangle _{2}$ and $\left| 2\right\rangle _{2}\rightarrow \left| 1\right\rangle _{2}.$ Based on these results, we can obtain the following state transformations:

$$\begin{aligned}&\left( \left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\overset{p1}{\rightarrow }\left( \left| 1\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\nonumber \\&\quad \sqrt{2}\overset{ p2}{\rightarrow }\left| 1\right\rangle _{2}\overset{p3}{\rightarrow } \left( \left| 0\right\rangle _{2}-\left| 1\right\rangle _{2}\right) / \sqrt{2}\overset{p4}{\rightarrow }\left( \left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}, \nonumber \\&\quad \left( -\left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\overset{p1}{\rightarrow }\left( -\left| 1\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\overset{ p2}{\rightarrow }\left| 2\right\rangle _{2}\overset{p3}{\rightarrow } \left| 2\right\rangle _{2}\overset{p4}{\rightarrow }\left| 1\right\rangle _{2},\quad \end{aligned}$$

(18)

where the first, second, third, and last transformations are obtained after applying the first, second, third, and the last pulses, respectively.

For step 9: A microwave pulse of {$\omega _{21},\pi /2,\pi /\left( 2{{\varOmega }}_{21}\right)$} and then a microwave pulse of {$\omega _{10},\pi /2,\pi /\left( 4{{\varOmega }}_{10}\right)$} were applied to qutrit 2 (Fig. 2a). The first pulse results in $\left| 2\right\rangle _{2}\rightarrow$ $\left| 1\right\rangle _{2}$ and $\left| 1\right\rangle _{2}\rightarrow$ $-\left| 2\right\rangle _{2}$, while the second pulse leads to $\left( \left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}\rightarrow$ $\left| 0\right\rangle _{2}$ and $\left( -\left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}\rightarrow \left| 1\right\rangle _{2}$ but nothing to the state $\left| 2\right\rangle _{2}.$ Based on these results, one can have the following state transformations:

$$\begin{aligned}&\left( \left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\overset{p1}{\rightarrow }\left( \left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}\overset{ p2}{\rightarrow }\left| 0\right\rangle _{2}, \nonumber \\&\quad \left( -\left| 0\right\rangle _{2}+\left| 2\right\rangle _{2}\right) /\sqrt{2}\overset{p1}{\rightarrow }\left( -\left| 0\right\rangle _{2}+\left| 1\right\rangle _{2}\right) /\sqrt{2}\overset{ p2}{\rightarrow }\left| 1\right\rangle _{2}, \nonumber \\&\quad \left| 1\right\rangle _{2}\overset{p1}{\rightarrow }-\left| 2\right\rangle _{2}\overset{p2}{\rightarrow }-\left| 2\right\rangle _{2}, \end{aligned}$$

(19)

where the first transformation is obtained after the first pulse, while the second transformation is achieved after the second pulse.

Note that the p1, p2, p3,  and p4 above represent the first, second, third, and fourth pulses, respectively.