Fast and high-fidelity generation of photonic Greenberger-Horne-Zeilinger states in circuit QED
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Jingwei Chen1,4, Guan-Hao Feng1,4, Hong-Hao Zhang,1,5, L F Wei,1,2,3,51School of Physics, Sun Yat-sen University, Guangzhou 510275, China 2Photonics Laboratory and Institute of Functional Materials, College of Science, Donghua University, Shanghai 201620, China 3Information Quantum Technology Laboratory, School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
First author contact:4J W Chen and G H Feng contributed equally to this work.5Authors to whom any correspondence should be addressed. Received:2019-09-18Revised:2019-10-6Accepted:2019-10-21Online:2019-12-27
Abstract A fast scheme to generate Greenberger-Horne-Zeilinger states between different cavities in circuit QED systems is proposed. To implement this scheme, we design a feasible experimental device with three qubits and three cavities. In this device, all the couplings between qubit and qubit, cavity and qubit are tunable and are independent with frequencies, and thus the shortcut to adiabaticity technique can be directly applied in our scheme. It is demonstrated that the GHZ state can be generated rapidly with high fidelity in our scheme. Keywords：GHZ states;shortcut to adiabaticity;quantum state engineering;circuit QED
PDF (621KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Jingwei Chen, Guan-Hao Feng, Hong-Hao Zhang, L F Wei. Fast and high-fidelity generation of photonic Greenberger-Horne-Zeilinger states in circuit QED. Communications in Theoretical Physics, 2020, 72(1): 015102- doi:10.1088/1572-9494/ab4fd9
1. Introduction
Quantum entanglements have drawn more and more attention in recent years, as they are crucial not only to quantum information [1–6], but also to fundamental studies of physics [7, 8]. Generation of entangled states has been widely studied, for instance in [9, 10]. Various quantum entangled states, including Bell states [11], NOON states [12, 13], Greenberger-Horne-Zeilinger (GHZ) states [4, 8, 14] and W states [15], have been investigated in both theory and experiment, due to the potential wide applications of these entangled states. For example, the GHZ states could be used to test quantum mechanics against local hidden theory without using Bell’s inequality [16, 17]. The GHZ states also play a significant role in applied sciences, such as quantum computing [18], quantum communication [19], and quantum metrology [20]. Therefore, generating GHZ states becomes a considerable problem.
Generally, a GHZ state can be written as [14]$ \begin{eqnarray}| {\rm{GHZ}}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 0{\rangle }^{\bigotimes N}+| 1{\rangle }^{\bigotimes N}),\end{eqnarray}$with $N\geqslant 3$. The simplest GHZ state is a 3-qubit GHZ state, read as$ \begin{eqnarray}| {\rm{GHZ}}\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle ).\end{eqnarray}$In order to generate GHZ states, a lot of schemes have been proposed in various physical systems, such as atoms [17], superconducting qubits [21], and photons [14]. In this article, we propose a fast scheme to generate photonic GHZ states in three individual cavities. In our scheme, three qubits are tunablely coupled to three cavities, respectively, and the central qubit is tunablely coupled to the other two qubits. Therefore, the shortcut to adiabaticity (STA) technique [22–26] could be applied in this scheme. The STA technique is an optimizing version of the rapid adiabatic passage (RAP) technique. The basic idea of the STA technique is to design a suitable auxiliary driving to eliminate diabatic transitions which might exist in the original RAP technique [22]. For a generic quantum system governed by a Hamiltonian H(t) and prepared initially in an energy eigenstate $| {\lambda }_{n}(0)\rangle $, which is defined as $H(t)| {\lambda }_{n}(t)\rangle ={E}_{n}(t)| {\lambda }_{n}(t)\rangle $, we can introduce an auxiliary driving$ \begin{eqnarray}\begin{array}{rcl}{H}_{1}(t) & = & i{\hslash }\displaystyle \sum _{n}(| {\partial }_{t}{\lambda }_{n}\rangle \langle {\lambda }_{n}| -\langle {\lambda }_{n}| {\partial }_{t}{\lambda }_{n}\rangle | {\lambda }_{n}\rangle \langle {\lambda }_{n}| )\\ & = & i{\hslash }\displaystyle \sum _{m\ne n}\displaystyle \sum _{n}\displaystyle \frac{| {\lambda }_{m}\rangle \langle {\lambda }_{m}| {\partial }_{t}H| {\lambda }_{n}\rangle \langle {\lambda }_{n}| }{{E}_{n}-{E}_{m}},\end{array}\end{eqnarray}$to eliminate diabatic transitions. This is the so-called transitionless driving algorithm. In this algorithm, the adiabatic condition of the original RAP is unnecessary. Thus, the transitionless evolution of the system along the instantaneous eigenstate $| {\lambda }_{n}(t)\rangle $ could be driven rapidly with high fidelity [22, 25].
This paper is organized as follows. In section 2 we propose a generic theoretical scheme, in which three coupled qubits are coupled respectively with three individual cavities, to rapidly generate high-fidelity photonic GHZ states. In section 3 we design a specific device of this generic scheme with experimental circuit QED systems. The conclusion is given in section 4.
2. The scheme
In this section we propose a generic scheme to generate photonic GHZ states by using the STA technique. Consider a model with three qutrits (labeled as qutrit 1, 2, and 3, respectively) and three cavities (labeled as cavity a, b, and c, respectively). The energy level structure of each qutrit is V-type, as shown in figure 1(a). The transition between $| g\rangle $ and $| e\rangle $ is coupled to the relevant cavity. Besides, the transition between $| g\rangle $ and $| a\rangle $ is used to couple the two successive qutrits (i.e. qutrit 1 and qutrit 2, qutrit 2 and qutrit 3). The Hamiltonian of this model is given by$ \begin{eqnarray}H={H}_{q}+{H}_{C}+{H}_{{int}},\end{eqnarray}$with$ \begin{eqnarray}{H}_{q}=\displaystyle \sum _{i=1,2,3}\left({\hslash }{\omega }_{i}^{g}| g{\rangle }_{i}\langle g{| }_{i}+{\hslash }{\omega }_{i}^{e}| e{\rangle }_{i}\langle e{| }_{i}+{\hslash }{\omega }_{i}^{a}| a{\rangle }_{i}\langle a{| }_{i}\right),\end{eqnarray}$$ \begin{eqnarray}{H}_{C}={\hslash }{\omega }_{a}{a}^{\dagger }a+{\hslash }{\omega }_{b}{b}^{\dagger }b+{\hslash }{\omega }_{c}{c}^{\dagger }c,\end{eqnarray}$$ \begin{eqnarray}\begin{array}{rcl}{H}_{{int}} & = & {\hslash }{g}_{1}(| e{\rangle }_{1}\langle g{| }_{1}a+| g{\rangle }_{1}\langle e{| }_{1}{a}^{\dagger })+{\hslash }{g}_{2}\left(| e{\rangle }_{2}\langle g{| }_{2}b\right.\\ & & \left.+| g{\rangle }_{2}\langle e{| }_{2}{b}^{\dagger }\right)+{\hslash }{g}_{3}\left(| e{\rangle }_{3}\langle g{| }_{3}c\right.\\ & & \left.+| g{\rangle }_{3}\langle e{| }_{3}{c}^{\dagger }\right)+{\hslash }{g}_{12}| a{\rangle }_{1}\langle a{| }_{1}| a{\rangle }_{2}\langle a{| }_{2}\\ & & +{\hslash }{g}_{23}| a{\rangle }_{2}\langle a{| }_{2}| a{\rangle }_{3}\langle a{| }_{3}.\end{array}\end{eqnarray}$Here, ${\omega }_{i}^{g}$, ${\omega }_{i}^{e}$, and ${\omega }_{i}^{a}$ are the frequencies of state $| g{\rangle }_{i}$, $| e{\rangle }_{i}$, and $| a{\rangle }_{i}$ of qutrit i, respectively. ωa, ωb, and ωc are the frequencies of cavity a, b, and c, respectively. ${a}^{(\dagger )}$, ${b}^{(\dagger )}$, and ${c}^{(\dagger )}$ are the annihilation (creation) operators of cavity a, b, and c, respectively. g1, g2, and g3 represent the coupling strength between qutrit 1 and cavity a, qutrit 2 and cavity b, qutrit 3 and cavity c, respectively. g12 and g23 represent the coupling strength between qutrit 1 and qutrit 2, qutrit 2 and qutrit 3, respectively.
Figure 1.
New window|Download| PPT slide Figure 1.(a) Energy level schematic of V-type qutrits. The transition between states $| g\rangle $ and $| e\rangle $ is used to generate photons in the relevant cavity, and the transition between states $| g\rangle $ and $| a\rangle $ is used to coupled with the other qutrit. (b) A general process to generate GHZ states with three qutrits.
With this model setup, we can first prepare the GHZ state of three qutrits via the general process shown in figure 1(b). Since the transition between $| g\rangle $ and $| e\rangle $ is used to generate photons in the cavity, we focus on the subspace $\{| g\rangle ,| a\rangle \}$ of every qutrit in this step. Assuming the state of qutrits is initialized in $| g\rangle | g\rangle | g\rangle $, we apply the Hadamard gate to every qutrit, then the state of qutrits becomes$ \begin{eqnarray}| g\rangle | g\rangle | g\rangle \to \displaystyle \frac{1}{2\sqrt{2}}(| g\rangle +| a\rangle )(| g\rangle +| a\rangle )(| g\rangle +| a\rangle ).\end{eqnarray}$Then we apply the conditional phase (c-Phase) gate cU01 to qutrit 1 and qutrit 2, and the state of qutrits evolves to$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2\sqrt{2}}(| g\rangle +| a\rangle )(| g\rangle +| a\rangle )(| g\rangle +| a\rangle )\\ \to \,\displaystyle \frac{1}{2\sqrt{2}}\left(| g\rangle | g\rangle \right.\left.-\ | g\rangle | a\rangle +| a\rangle | g\rangle +| a\rangle | a\rangle \right)\otimes (| g\rangle +| a\rangle ).\end{array}\end{eqnarray}$Applying another c-Phase gate cU10 to qutrit 2 and qutrit 3, the state of qutrits becomes$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2\sqrt{2}}(| g\rangle | g\rangle -| g\rangle | a\rangle +| a\rangle | g\rangle +| a\rangle | a\rangle )\otimes (| g\rangle +| a\rangle )\\ \to \,\displaystyle \frac{1}{2\sqrt{2}}\left(| g\rangle | g\rangle | g\rangle +| g\rangle | a\rangle | g\rangle +| a\rangle | g\rangle | g\rangle -| a\rangle | a\rangle | g\rangle \right.\\ \quad \left.+\ | g\rangle | g\rangle | a\rangle -| g\rangle | a\rangle | a\rangle +| a\rangle | g\rangle | a\rangle +| a\rangle | a\rangle | a\rangle \right)\\ =\ \displaystyle \frac{1}{2\sqrt{2}}\left[(| g\rangle +| a\rangle )| g\rangle (| g\rangle +| a\rangle )\right.\\ \left.\,\,+\,(| g\rangle -| a\rangle )| a\rangle (| g\rangle -| a\rangle )\right]\end{array}\end{eqnarray}$Finally, by applying the Hadamard gates to qutrit 1 and qutrit 3, respectively, we obtain the GHZ state of qutrits [27]$ \begin{eqnarray}\displaystyle \frac{1}{\sqrt{2}}(| g\rangle | g\rangle | g\rangle +| a\rangle | a\rangle | a\rangle ).\end{eqnarray}$
Similarly, if the initial state of qutrits is $| a\rangle | a\rangle | a\rangle $, the state of qutrits will evolve as$ \begin{eqnarray}| a\rangle | a\rangle | a\rangle \to \displaystyle \frac{1}{\sqrt{2}}(| g\rangle | g\rangle | g\rangle -| a\rangle | a\rangle | a\rangle ).\end{eqnarray}$
Next, we show how to use the qutrits to generate the GHZ state of photons in cavities. The state of the system is initialized to $| g\rangle | g\rangle | g\rangle | 0\rangle | 0\rangle | 0\rangle $. First, the state of three qutrits evolves to the GHZ state$ \begin{eqnarray}| g\rangle | g\rangle | g\rangle | 0\rangle | 0\rangle | 0\rangle \to \displaystyle \frac{1}{\sqrt{2}}(| g\rangle | g\rangle | g\rangle +| a\rangle | a\rangle | a\rangle )| 0\rangle | 0\rangle | 0\rangle .\end{eqnarray}$Then we excite the ground state $| g\rangle $ to $| e\rangle $ of all the three qutrits, and transfer the populations to the relevant cavities,$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{\sqrt{2}}\left(| g\rangle | g\rangle | g\rangle +| a\rangle | a\rangle | a\rangle \right)| 0\rangle | 0\rangle | 0\rangle \\ \to \ \displaystyle \frac{1}{\sqrt{2}}\left(| e\rangle | e\rangle | e\rangle +| a\rangle | a\rangle | a\rangle \right)| 0\rangle | 0\rangle | 0\rangle \\ \to \ \displaystyle \frac{1}{\sqrt{2}}\left(| g\rangle | g\rangle | g\rangle | 1\rangle | 1\rangle | 1\rangle +| a\rangle | a\rangle | a\rangle | 0\rangle | 0\rangle | 0\rangle \right).\end{array}\end{eqnarray}$Repeating the processes of generating the GHZ state of qutrits, we have$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{\sqrt{2}}\left(| g\rangle | g\rangle | g\rangle | 1\rangle | 1\rangle | 1\rangle +| a\rangle | a\rangle | a\rangle | 0\rangle | 0\rangle | 0\rangle \right)\\ \to \ \displaystyle \frac{1}{2}\left[\left(| g\rangle | g\rangle | g\rangle +| a\rangle | a\rangle | a\rangle \right)| 1\rangle | 1\rangle | 1\rangle \right.\\ \quad \left.+\ \left(| g\rangle | g\rangle | g\rangle -| a\rangle | a\rangle | a\rangle \right)| 0\rangle | 0\rangle | 0\rangle \right]\\ =\ \displaystyle \frac{1}{2}\left[| g\rangle | g\rangle | g\rangle \left(| 1\rangle | 1\rangle | 1\rangle +| 0\rangle | 0\rangle | 0\rangle \right)\right.\\ \quad \left.+\ | a\rangle | a\rangle | a\rangle \left(| 1\rangle | 1\rangle | 1\rangle -| 0\rangle | 0\rangle | 0\rangle \right)\right].\end{array}\end{eqnarray}$Finally, we measure the state of one of the qutrits, and the state of cavities will collapse to the GHZ state $(| 1\rangle | 1\rangle | 1\rangle +| 0\rangle | 0\rangle | 0\rangle )/\sqrt{2}$ (or $(| 1\rangle | 1\rangle | 1\rangle -| 0\rangle | 0\rangle | 0\rangle )/\sqrt{2}$) when the result is $| g\rangle $ (or $| a\rangle $).
All the steps of our scheme can be implemented by the STA technique [26]. For example, if we apply a time-dependent driving$ \begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{1}(t) & = & \displaystyle \frac{{\hslash }}{2}\left[{{\rm{\Omega }}}_{1}(t)\left(| a{\rangle }_{1}\langle g{| }_{1}+| g{\rangle }_{1}\langle a{| }_{1}\right)\right.\\ & & \left.+2{{\rm{\Delta }}}_{1}(t)| a{\rangle }_{1}\langle a{| }_{1}\right],\end{array}\end{eqnarray}$to qutrit 1, where the two pulses ω1(t) and Δ1(t) are chosen properly to tune the mix angle $\theta (t)=\arctan [{{\rm{\Omega }}}_{1}(t)/{{\rm{\Delta }}}_{1}(t)]/2$ from zero to π/4, then we can drive the state of the system from the state $| g,g,g,0,0,0\rangle $ to the $(| g,g,g,0,0,0\rangle +| a,g,g,0,0,0\rangle )/\sqrt{2}$. In this process, all possible diabatic transitions can be eliminated if we simultaneously apply an extra driving$ \begin{eqnarray}{\hat{H}}_{1}^{{\prime} }=\displaystyle \frac{{\hslash }}{2}{{\rm{\Omega }}}_{1}^{{\prime} }(t)(i| a{\rangle }_{1}\langle g{| }_{1}-i| g{\rangle }_{1}\langle a{| }_{1}),\end{eqnarray}$where ${{\rm{\Omega }}}_{1}^{{\prime} }(t)$$=\,\left[{{\rm{\Omega }}}_{1}(t){\dot{{\rm{\Delta }}}}_{1}(t)-{\dot{{\rm{\Omega }}}}_{1}(t){{\rm{\Delta }}}_{1}(t)\right]$/$\left[{{\rm{\Omega }}}_{1}^{2}(t)+{{\rm{\Delta }}}_{1}^{2}(t)\right]$. However, when we want to realize the swap gate between the qutrit and the relevant cavity, the coupling strength between the qutrit and the cavity should be tunable and be independent with the frequencies. Given this problem, we should design a suitable physical device to demonstrate our scheme.
3. The device
In this section, let us design a superconducting circuit device to implement our scheme for generating GHZ states in three cavities. According to the previous section, this device contains three qutrits and three cavities, where all the coupling strengths between qutrit and qutrit, qutrit and cavity are tunable and are independent with frequencies. Our device is schematically shown in figure 2. In order to generate GHZ states, we have to modify the device designed in [26], which consists of two qutrits and two cavities and was used to generate NOON states. Now we show how to realize the tunable coupling in this device. First, in order to generate tunable qutrit-cavity coupling strengths, we make use of the structure of so-called tunable-coupling qutrits (TCQs) [26, 28–30]. A TCQ can be treated as a transmon with three islands, described by the Hamiltonian$ \begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{T} & = & \displaystyle \sum _{i=\pm }\left[4{E}_{{Ci}}{\left({\hat{n}}_{i}-{n}_{{gi}}\right)}^{2}-{E}_{{Ji}}\cos ({\hat{\varphi }}_{i})\right]\\ & & +4{E}_{I}{\hat{n}}_{+}{\hat{n}}_{-},\end{array}\end{eqnarray}$where ${\hat{n}}_{+(-)}$ denotes the number of Cooper pairs in the upper (lower) island, ${n}_{g+(-)}$ and ${\hat{\varphi }}_{+(-)}$ are the relevant offset gate charges and the phase differences, respectively. This system has a fixed charge energy EC±=e2/2C±and a fixed interaction energy EI=e2/2CI , but a tunable Josephson-energy ${E}_{J\pm }={E}_{J\pm }^{M}\cos (\pi {{\rm{\Phi }}}_{\pm }/{{\rm{\Phi }}}_{0})$ (with ${E}_{J\pm }^{M}$ the total Josephson energy) since we can control the flux Φ±. We expand $\cos ({\hat{\varphi }}_{i})$ to fourth order, then the Hamiltonian approximates to$ \begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{T} & \approx & \displaystyle \sum _{i=\pm }\left[4{E}_{{Ci}}{\left({\hat{n}}_{i}-{n}_{{gi}}\right)}^{2}-{E}_{{Ji}}\left(1-\displaystyle \frac{1}{2}{\hat{\varphi }}_{i}^{2}+\displaystyle \frac{1}{24}{\hat{\varphi }}_{i}^{4}\right)\right]\\ & & +4{E}_{I}{\hat{n}}_{+}{\hat{n}}_{-},\end{array}\end{eqnarray}$This Hamiltonian can be viewed as two harmonic oscillators with the addition of anharmonic terms and an interaction term. Therefore, we can model the TCQ with annihilation operators ${c}_{\pm }=\tfrac{1}{\sqrt{2}}$$\left[{\left({E}_{J\pm }/8{E}_{C\pm }\right)}^{1/4}{\hat{\varphi }}_{\pm }+i{\left(8{E}_{C\pm }/{E}_{J\pm }\right)}^{1/4}{\hat{n}}_{\pm }\right]$ under the condition ${E}_{J\pm }/{E}_{C\pm }\gg 1$, and thus the Hamiltonian becomes$ \begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{T} & \approx & \displaystyle \sum _{i=\pm }{\hslash }\left[{\omega }_{i}+{\delta }_{i}({c}_{i}^{\dagger }{c}_{i}-1)/2\right]{c}_{i}^{\dagger }{c}_{i}\\ & & +{\hslash }J({c}_{+}{c}_{-}^{\dagger }+{c}_{+}^{\dagger }{c}_{-})\end{array}\end{eqnarray}$with ${\omega }_{\pm }=\sqrt{8{E}_{J\pm }{E}_{C\pm }}/{\hslash }-{E}_{C\pm }/{\hslash }$, ${\delta }_{\pm }=-{E}_{C\pm }/{\hslash }$ and $J={E}_{I}{\left({E}_{J+}{E}_{J-}/{E}_{C+}{E}_{C-}\right)}^{1/4}/\sqrt{2}{\hslash }$.
Figure 2.
New window|Download| PPT slide Figure 2.(a) Schematic and (b) circuit model of our designed circuit QED device to generate photonic GHZ states. Qutrits are transmons consisting of three islands. Every qutrit is coupled with a coplanar waveguide resonator. Two tunable Josephson junctions are used to couple qutrit 1 and qutrit 2, qutrit 2 and qutrit 3, respectively. The single qutrit state can be controlled by additional circuits (not shown in figure).
Next, we apply a unitary transformation $D(\alpha )\,=\exp \left[\alpha (t)\left({c}_{+}{c}_{-}^{\dagger }-{c}_{+}^{\dagger }{c}_{-}\right)\right]$ to the above Hamiltonian. Here,$ \begin{eqnarray}\begin{array}{rcl}\alpha (t) & = & \displaystyle \frac{1}{2}\arctan \left(\displaystyle \frac{2J}{\eta }\right)+\theta ,\\ \eta & = & {\omega }_{+}-{\omega }_{-}-\displaystyle \frac{1}{2}({\delta }_{+}-{\delta }_{-}),\\ \theta & = & \left\{\begin{array}{ll}\pi , & \eta \gt 0;\\ \pi /2, & \eta \lt 0.\end{array}\right.\end{array}\end{eqnarray}$Using this transformation, the Hamiltonian equation (20) can be diagonalized as [28]$ \begin{eqnarray}{\hat{H}}_{T}\approx \displaystyle \sum _{i=\pm }{\hslash }\left[\displaystyle \frac{{\tilde{\delta }}_{i}}{2}\left({\tilde{c}}_{i}^{\dagger }{\tilde{c}}_{i}-1\right)+{\tilde{\omega }}_{i}\right]{\tilde{c}}_{i}^{\dagger }{\tilde{c}}_{i}+{\hslash }{\tilde{\delta }}_{c}{\tilde{c}}_{+}^{\dagger }{\tilde{c}}_{+}{\tilde{c}}_{-}^{\dagger }{\tilde{c}}_{-},\end{eqnarray}$with two modified annihilation operators$ \begin{eqnarray}{\tilde{c}}_{+}={c}_{+}\cos (\alpha )-{c}_{-}\sin (\alpha ),\end{eqnarray}$$ \begin{eqnarray}{\tilde{c}}_{-}={c}_{-}\cos (\alpha )+{c}_{+}\sin (\alpha ),\end{eqnarray}$
Obviously, this Hamiltonian describes a V-type qutrit. We define the levels of this system as follows: the ground state $| g\rangle =| 0{\rangle }_{+}| 0{\rangle }_{-}$, an excited state $| e\rangle ={\tilde{c}}_{+}^{\dagger }| g\rangle $, and an additional state $| a\rangle ={\tilde{c}}_{-}^{\dagger }| g\rangle $. We can obtain the formal coupling strengths between qutrit and cavity as ${\tilde{g}}_{\pm }={g}_{\pm }\cos (\alpha )\mp {g}_{\mp }\sin (\alpha )$, which are tunable and are independent with frequencies of qutrits. Since the parameter J is negative, the coupling strength ${\tilde{g}}_{+}$ can be tuned from zero to the maximum, and thus the relevant transition can be used to excite photons in the cavity [28]. Although we can introduce an additional cavity to generate tunable couplings among qutrits, these effective couplings are typically weak. Therefore, we introduce a tunable Josephson junction to generate a sufficiently strong inter-qutrit coupling S(θ)φ1−φ2− [31–33]. Furthermore, the present coupling strength S can also be tuned by modulating θ and the phase difference across the coupling junction. Specifically,$ \begin{eqnarray}\begin{array}{rcl}S(\theta ){\varphi }_{1-}{\varphi }_{2-} & = & {g}_{12}(\theta )({c}_{-}^{\dagger }+{c}_{-})({d}_{-}^{\dagger }+{d}_{-})\\ & = & {\tilde{g}}_{12}(\theta )({\tilde{c}}_{-}^{\dagger }+{\tilde{c}}_{-})({\tilde{d}}_{-}^{\dagger }+{\tilde{d}}_{-})\\ & \approx & {\tilde{g}}_{12}(\theta )({\tilde{c}}_{-}^{\dagger }{\tilde{d}}_{-}+{\tilde{c}}_{-}{\tilde{d}}_{-}^{\dagger }),\end{array}\end{eqnarray}$where c and d are the annihilation operators of qutrits 1 and 2, respectively. In the last step we have used the rotation wave approximation. Consequently, the tunable qutrit-qutrit coupling can be obtained. In addition, by using the adiabatic interaction between the $| 11\rangle $ and $| 02\rangle $ state, the high-fidelity c-Phase gate can also be obtained based on the STA technique [33, 34].
Now we show how to demonstrate the deterministic generation of the GHZ state with the above superconducting device via numerical simulations. First, in figures 3(a) and (b) we show how to realize the population transfer $| g,g,g,0,0,0\rangle \to (| g,g,g,0,0,0\rangle $$+\,| a,g,g,0,0,0\rangle )/\sqrt{2}$ via the STA technique. Here, by using a time-dependent Hamiltonian$ \begin{eqnarray}{H}_{1}=\displaystyle \frac{{\hslash }}{2}\left[{{\rm{\Omega }}}_{1}(t)\left(| a{\rangle }_{1}\langle g{| }_{1}+| g{\rangle }_{1}\langle a{| }_{1}\right)+2{{\rm{\Delta }}}_{1}(t)| a{\rangle }_{1}\langle a{| }_{1}\right],\end{eqnarray}$with two pulses ${{\rm{\Omega }}}_{1}(t)={G}_{0}\exp \left(-{\left(t-\tau \right)}^{2}/{T}_{0}^{2}\right)$, ${{\rm{\Delta }}}_{1}(t)\,={{\rm{\Delta }}}_{0}\exp \left[-{\left(t+\tau \right)}^{2}/{\left({{mT}}_{0}\right)}^{2}\right]$, we can tune the mix angle $\theta (t)=\arctan \left[{\omega }_{1}(t)/{{\rm{\Delta }}}_{1}(t)\right]/2$ of qutrit 1 from 0 to π/4. Simultaneously, we apply an additional counter-diabatic driving$ \begin{eqnarray}{\hat{H}}_{1}^{{\prime} }=\displaystyle \frac{{\hslash }}{2}{{\rm{\Omega }}}_{1}^{{\prime} }(t)(i| a{\rangle }_{1}\langle g{| }_{1}-i| g{\rangle }_{1}\langle a{| }_{1}),\end{eqnarray}$with ${{\rm{\Omega }}}_{1}^{\prime} (t)$$=\,\left[{{\rm{\Omega }}}_{1}(t){\dot{{\rm{\Delta }}}}_{1}(t)-{\dot{{\rm{\Omega }}}}_{1}(t){{\rm{\Delta }}}_{1}(t)\right]$/$\left[{{\rm{\Omega }}}_{1}^{2}(t)+{{\rm{\Delta }}}_{1}^{2}(t)\right]$ to eliminate the unwanted diabatic terms Here, we choose G0=2π×18 MHz, Δ0=2π×75 MHz, τ=1.7ns, T0= 2ns, and m=1.25. The drivings and numerical simulation results are shown in figures 3(a) and (b), respectively. Indeed, all the Hadamard gates for the three qutrits are identical to the above process, and thus we do not need to demonstrate these processes one by one. The next process we should demonstrate is how to transfer the population from qutrit to cavity. Here, a time-dependent Hamiltonian$ \begin{eqnarray}{H}_{2}=\displaystyle \frac{{\hslash }}{2}\left[{g}_{1}\left(| e{\rangle }_{1}\langle g{| }_{1}a+| g{\rangle }_{1}\langle e{| }_{1}{a}^{\dagger }\right)+2{{\rm{\Delta }}}_{1a}| e{\rangle }_{1}\langle e{| }_{1}\right]\end{eqnarray}$with the so-called Allen-Eberly (AE) drivings$ \begin{eqnarray}{g}_{1}(t)={{\rm{\Omega }}}_{0}{\rm{{\rm{sech}} }}(\pi t/2{t}_{0}),\end{eqnarray}$$ \begin{eqnarray}{{\rm{\Delta }}}_{1a}(t)=(2{\beta }^{2}{t}_{0}/\pi )\tanh (\pi t/2{t}_{0}),\end{eqnarray}$is applied to qutrit 1 and cavity a, with ${{\rm{\Delta }}}_{1a}={\omega }_{1}^{e}-{\omega }_{1}^{g}-{\omega }_{a}$. Again, a counter-diabatic driving$ \begin{eqnarray}{\hat{H}}_{2}^{{\prime} }=\displaystyle \frac{{\hslash }}{2}{{\rm{\Omega }}}_{2}^{{\prime} }(t)(i| e{\rangle }_{1}\langle g{| }_{1}a-i| g{\rangle }_{1}\langle e{| }_{1}{a}^{\dagger }),\end{eqnarray}$with ${{\rm{\Omega }}}_{2}^{\prime} (t)$$=\,[{g}_{1}(t){\dot{{\rm{\Delta }}}}_{1a}(t)-{\dot{g}}_{1}(t){{\rm{\Delta }}}_{1a}(t)]$/$\left[{g}_{1}^{2}(t)+{{\rm{\Delta }}}_{1a}^{2}(t)\right]$ is applied to eliminate the diabatic terms. Here, we choose ω0=2π×50 MHz, β=2π×75 MHz, and t0=1ns. The driving and numerical simulation results are shown in figure 3(c) and (d), respectively.
Figure 3.
New window|Download| PPT slide Figure 3.Implementation of transition from: (a) and (b), $| g,g,g,0,0,0\rangle $ to $(| g,g,g,0,0,0\rangle +| a,g,g,0,0,0\rangle )/\sqrt{2};$ (c) and (d), $| e,g,g,0,0,0\rangle $ to $| g,g,g,1,0,0\rangle $.
Finally, we investigate the fidelity of the generation with the parameter variations. Without loss of the generality, we focus on the implementation of transition from $| e,g,g,0,0,0\rangle $ to $| g,g,g,1,0,0\rangle $. The fidelity of this step can be defined as the population of the state $| g,g,g,1,0,0\rangle $. We modify the AE drivings with three dimensionless parameters w0, w1, and w2, and the drivings become$ \begin{eqnarray}{g}_{1}^{{\prime} }(t)={w}_{1}{{\rm{\Omega }}}_{0}{\rm{{\rm{sech}} }}(\pi t/2{t}_{0}),\end{eqnarray}$$ \begin{eqnarray}{{\rm{\Delta }}}_{1a}^{{\prime} }{(t)=(2({w}_{2}\beta )}^{2}{t}_{0}/\pi )\tanh (\pi t/2{w}_{0}{t}_{0}).\end{eqnarray}$These parameters vary from 0.9 to 1.1. The results of numerical calculations are shown in figure 4. It is shown that the STA technique can achieve high fidelity in any case. Since all the steps of our scheme are implemented via the STA technique, we can achieve sufficient high fidelity of generation of the GHZ state.
Figure 4.
New window|Download| PPT slide Figure 4.Fidelity of the implementation of transition from $| e,g,g,0,0,0\rangle $ to $| g,g,g,1,0,0\rangle $.
4. Conclusion
In summary, we theoretically demonstrate that photonic GHZ states between three cavities can be deterministically generated by using the STA technique. Since the operation speed of this generating scheme is significantly faster than that of the adiabatic passage technique, and also has more robustness than that of the usual Rabi oscillation technique to the fluctuations of the pulse parameters, the GHZ state generation could be sufficiently high fidelity and feasible.
In order to further demonstrate our general proposal, we design a circuit QED device which contains three qutrits and three cavities. By combining the structure of TCQ and gmon, every qutrit is tunably coupled with a relevant cavity, and qutrit 2 is tunably coupled with qutrit 1 and qutrit 3 via Josephson junctions. Therefore, our general scheme can be feasibly implemented in this experimental device. The high fidelity of this scheme is shown by numerical simulation. We hope that our scheme for generating a high-fidelity photonic GHZ state can be used in various applications.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11974290 and 11875327, the Natural Science Foundation of Guangdong Province under Grant No. 2016A030313313, the Fundamental Research Funds for the Central Universities, and the Sun Yat-Sen University Science Foundation.
,1,5, L F Wei
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