Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction

We propose an efficient method to generate a Greenberger-Horne-Zeilinger entangled state of $n$ photons in $n$ microwave cavities (or resonators) via resonant interaction to a single superconducting qutrit. The deployment of a qutrit, instead of a qubit, as the coupler enables us to use resonant interactions exclusively for all qutrit-cavity and qutrit-pulse operations. This unique approach significantly shortens the time of operation, which is advantageous for reducing the adverse effects of qutrit decoherence and cavity decay on the fidelity of the protocol. Furthermore, the protocol involves no measurement on either the state of the qutrit or cavity photons. We also show that the protocol can be generalized to other systems by replacing the superconducting qutrit coupler with different types of physical qutrits, such as an atom in the case of cavity QED, to accomplish the same task.

Figure 1

Illustration of qutrit-cavity resonant interaction. The cavity mode is resonant with the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transition of the qutrit. $g$ is the coupling constant between the cavity mode and the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transition. In (a), the cavity mode is decoupled from the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition of a phase qutrit as long as the large detuning condition $\Delta \gg g^{\prime }$ is satisfied. Here, $\Delta$ is the detuning between the cavity-mode frequency and the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition frequency, and $g^{\prime }$ is the coupling constant between the cavity mode and the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition. In (b), the dipole matrix element between $\left|0\right.〉$ and $\left|1\right.〉$ can be made much weaker than that between $\left|1\right.〉$ and $\left|2\right.〉$ by increasing the barrier height of the double-well potential. Thus the coupling between $\left|0\right.〉$ and $\left|1\right.〉$ via the cavity mode is negligible. Note that the coupling strength $g$ may vary when the qutrit couples with different cavities or resonators. Thus, $g$ is replaced by $g_i$ to denote the coupling strength between the qutrit and cavity $i$ ($i=1,2,\cdots ,n$).

Figure 2

(a) Diagram of a superconducting qutrit $A$ (a circle at the center) and $n$ cavities. Each red circle represents a one-dimensional coplanar waveguide resonator which is capacitively coupled to the coupler qutrit $A$, as shown in (b). (b) The diagram on the left side is equivalent to the diagram on the right side.

Figure 3

Illustration of qutrit-cavity or qutrit-pulse interaction. (a) Cavity $i$ is resonant with the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transition of qutrit $A$ when ${\omega }_{c,i}={\omega }_{21}$ with coupling constant $g_i$ but off-resonant with the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition with coupling constant $g_i^{\prime }$ and detuning $\Delta ={\omega }_{10}-{\omega }_{c,i}.$ (b) Cavity $j$ of frequency ${\tilde{\omega }}_{c,j}$ is off-resonant with the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ ($\left|0\right.〉\leftrightarrow \left|1\right.〉$) transition of qutrit $A$ with coupling constant ${\tilde{g}}_j$ (${\tilde{g}}_j^{\prime }$) and detuning ${\Delta }_j={\omega }_{21}-{\tilde{\omega }}_{c,j}$ (${\Delta }_j^{\prime }={\omega }_{10}-{\tilde{\omega }}_{c,j}$)$.$ (c) The situation when a microwave classical pulse of frequency $\omega ={\omega }_{21}$ is applied to qutrit $A$ but is off-resonant with the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition with detuning ${\Delta }_{\mu w}={\omega }_{10}-\omega .$ The corresponding Rabi frequencies are ${\Omega }_{21}$ and ${\Omega }_{10},$ respectively. (d) A microwave pulse of frequency $\omega ={\omega }_{20}$ is applied to qutrit $A$ with the corresponding Rabi frequency ${\Omega }_{20}.$ Note that for (c), the coupling of the pulse to the $\left|0\right.〉\leftrightarrow \left|2\right.〉$ transition is negligible due to the fact that the pulse is highly detuned from the $\left|0\right.〉\leftrightarrow \left|2\right.〉$ transition frequency. For the same reason, for (d), the coupling of the pulse to the $\left|0\right.〉\leftrightarrow \left|1\right.〉$ transition and the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transitions is negligible as well.

Figure 4

Fidelity vs $b=\Delta /g^{\prime }$. Refer to the text for the parameters used in the numerical calculation. Here, $g_{kl}$ are the coupling strengths between cavities $k$ and $l$ ($k\ne l;$ $k,l=1,2,3,4$), which are taken to be the same for simplicity. In each plot, the red dots, green squares, and blue diamonds correspond to $n=2,3,$ and 4, respectively.

Figure 5

Diagram of $n$ identical cavities and an atom, $A$ (red dot). Atom $A$ is sent through or moved into each cavity for an interaction time $\pi /\left(2g\right).$ Before arriving in cavity $n-1,$ atom $A$ is addressed by a classical pulse [with frequency $\omega =$ ${\omega }_{21},$ initial phase $\pi ,$ and duration $\pi /\left(2{\Omega }_{21}\right)$] after it leaves each cavity [see the pink frame with an arrow]. When the atom $A$ exits the cavity $n-1$, two pulses are applied to it. The first pulse has frequency $\omega =$ ${\omega }_{20},$ initial phase $-\pi /2,$ and duration $\pi /\left(2{\Omega }_{20}\right)$ [see the blue frame with an arrow], while the second pulse has frequency $\omega ={\omega }_{21},$ initial phase $\pi /2,$ and duration $\pi /\left(2{\Omega }_{21}\right)$. Here, $g$ is the coupling constant between the cavity mode and the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transition of atom $A;$ ${\omega }_{20}$ and ${\omega }_{21}$ are the $\left|2\right.〉\leftrightarrow \left|0\right.〉$ transition frequency and the $\left|2\right.〉\leftrightarrow \left|1\right.〉$ transition frequency of atom $A,$ respectively. In addition, ${\Omega }_{21}$ (${\Omega }_{20}$) is the Rabi frequency of the pulse associated with the $\left|1\right.〉\leftrightarrow \left|2\right.〉$ transition ($\left|0\right.〉\leftrightarrow \left|2\right.〉$ transition) of atom $A$.