Conditional displacement interaction in transversal direction from the quantum Rabi model

Abstract

We investigate the realization of conditional displacement interaction in transversal direction from the quantum Rabi model by adjusting parameters of external magnetic fields. The special interaction is derived in the system of qubit(s) coupled to a resonator. We explore the implementation of quantum gates and the generation of superposed coherent states based on the transversal conditional displacement interaction, and consolidate the investigations numerically. We also show the special interaction can be realized by using the quantum Rabi model with qubit–qubit coupling.

Appendices

A The derivation of effective Hamiltonian in Eq. (9)

In this appendix, we give the detailed derivation of Eq. (9) from Eq. (2). For the sake of simplification, we set unified qubit frequency (i.e., $\omega _{q}^{m}=\omega _{q})$ and unified coupling strength (i.e., $g_{m}=g$). In order to obtain the effective Hamiltonian, we introduce the periodical modulation of the transition frequency in the following form:

$$\begin{aligned} \omega _{q}^{m}(t) = \omega _{q} + \varepsilon _{m}\sin \left( \omega _{m}t-\phi _{m}\right) . \end{aligned}$$

(27)

Then, the two-qubit QRM with the frequency modulation field can be recast as follows:

$$\begin{aligned} H(t)= & {} \sum _{m=1}^{2}\frac{\omega _{q}^{m}(t)}{2}\sigma _{z}^{m}+ \omega _{r}a^{\dag }a+g\left( a^{\dag }+a\right) \sum _{m=1}^{2}\sigma _{x}^{m}. \end{aligned}$$

(28)

Moving to the rotating frame defined by time-dependent transformation $\mathcal {U}(t) = \mathcal {U}_{1}(t)\mathcal {U}_{2}(t)$ with

$$\begin{aligned}&\mathcal {U}_{1}(t) = \exp \left[ -i\left( \sum _{m}\frac{\omega _{q}}{2}\sigma _{z}^{m} + \omega _{r}a^{\dag }a\right) t\right] , \end{aligned}$$

(29a)

$$\begin{aligned}&\mathcal {U}_{2}(t) = \exp \left[ i\sum _{m}\frac{\alpha _{m}}{2}\cos (\omega _{m}t-\phi _{m})\sigma _{z}^{m}\right] , \end{aligned}$$

(29b)

where $\alpha _{m}=\varepsilon _{m}/\omega _{m}$. For simplification, we set $\phi _{m}=\phi$ and $\omega _{m}=\omega$. In the rotating framework, the transformed Hamiltonian reads

$$\begin{aligned} \tilde{H}(t) = \mathcal {U}^{\dag }(t)H\mathcal {U}(t)-i\mathcal {U}^{\dag }(t)\partial _{t}\mathcal {U}(t) \, . \end{aligned}$$

(30)

The first term in Eq. (30) reads

$$\begin{aligned} \mathcal {U}^{\dag }(t)H\mathcal {U}(t)&= \mathcal {U}^{\dag }(t)\left( g(a^{\dag }+a)\sum _{m=1}^{2}\sigma _{x}^{m}\right) \mathcal {U}(t)\nonumber \\&=g\sum _{m=1}^2\left( a^{\dag }\sigma _{-}^{m}e^{i(\omega _{r}-\omega _{q}+\alpha _{m}\cos (\omega t-\phi ))t} +a^{\dag }\sigma _{+}^{m}e^{i(\omega _{r}+\omega _{q}-\alpha _{m}\cos (\omega t-\phi ))t} + h.c.\right) \nonumber \\&= g\sum _{m=1}^2\left( a^{\dag }\sigma _{-}^{m}e^{i\delta _{-}t}e^{i\alpha _{m}\cos (\omega t-\phi )} +a^{\dag }\sigma _{+}^{m}e^{i\delta _{+}t}e^{-i\alpha _{m}\cos (\omega t-\phi )} + h.c.\right) , \end{aligned}$$

(31)

where $\delta _{\pm } = \omega _{r}\pm \omega _{q}$ and $\sigma _{\pm }^{m}\equiv (\sigma _{x}^{m}\pm i \sigma _{y}^{m})/2$. In order to obtain the second term in Eq. (30), we should calculate the derivative of the operator exponential. Let $\exp (A(t))$ be the operator exponential. If $[\partial _{t}A(t),A(t)]=0$, the derivative of $\exp (A(t))$ reads

$$\begin{aligned} \partial _{t}\exp (A(t)) = \left( \partial _{t}A(t)\right) \exp (A(t))=\exp (A(t))\left( \partial _{t}A(t)\right) \, . \end{aligned}$$

(32)

By means of Eq. (32), the derivative of the $\mathcal {U}(t)$ reads

$$\begin{aligned} \partial _{t}\mathcal {U}(t) = \mathcal {U}(t)\left[ -i\left( \sum _{m}\frac{\omega _{q}}{2}\sigma _{z}^{m} + \omega _{r}a^{\dag }a\right) -i\sum _{m}\frac{\varepsilon _{m}}{2}\sin (\omega _{m}t-\phi _{m})\sigma _{z}^{m}\right] \, .\nonumber \\ \end{aligned}$$

(33)

Then, the second term in Eq. (30) reads

$$\begin{aligned} -i\mathcal {U}^{\dag }(t)\partial _{t}\mathcal {U}(t)&= -i\mathcal {U}^{\dag }(t)\mathcal {U}(t)\left[ -i\left( \sum _{m}\frac{\omega _{q}}{2}\sigma _{z}^{m} + \omega _{r}a^{\dag }a\right) -i\sum _{m}\frac{\varepsilon _{m}}{2}\sin (\omega _{m}t-\phi _{m})\sigma _{z}^{m}\right] \nonumber \\&= -\left( \sum _{m}\frac{\omega _{q}}{2}\sigma _{z}^{m} + \omega _{r}a^{\dag }a\right) -\sum _{m}\frac{\varepsilon _{m}}{2}\sin (\omega _{m}t-\phi _{m})\sigma _{z}^{m}\nonumber \\&= -\sum _{m=1}^{2}\frac{\omega _{q}^{m}(t)}{2}\sigma _{z}^{m} - \omega _{r}a^{\dag }a. \end{aligned}$$

(34)

Substituting Eq. (31) and Eqs. (34)–(30), we obtain

$$\begin{aligned} \tilde{H}(t) = g\sum _{m=1}^2\left( a^{\dag }\sigma _{-}^{m}e^{i\delta _{-}t}e^{i\alpha _{m}\cos (\omega t-\phi )} +a^{\dag }\sigma _{+}^{m}e^{i\delta _{+}t}e^{-i\alpha _{m}\cos (\omega t-\phi )} + h.c.\right) . \end{aligned}$$

(35)

The exponential term $\exp (\pm \alpha _{m}\cos (\omega t-\phi ))$ can be expanded by means of the following Jacobi–Anger identity [49]:

$$\begin{aligned} \exp \left( \pm i\alpha _{m}\cos (\omega t-\phi )\right) = \sum _{l=-\infty }^{+\infty } (\pm i)^{l}J_{l}(\alpha _{m})\exp (\pm il(\omega t-\phi )). \end{aligned}$$

(36)

Here, $J_{l}(\alpha _m)$ is the Bessel function of first kind. Substituting Eqs. (36)–(35), we obtain

$$\begin{aligned} \tilde{H}(t)&= g\sum _{m=1}^2\left[ a^{\dag }\sigma _{-}^{m}\sum _{l=-\infty }^{+\infty } i^{l}J_{l}(\alpha _{m})e^{i\delta _{-}t}e^{il(\omega t-\phi )}\right. \nonumber \\&\quad \left. + a^{\dag }\sigma _{+}^{m}\sum _{l'=-\infty }^{+\infty } (-i)^{l'}J_{l'}(\alpha _{m})e^{i\delta _{+}t}e^{-il'(\omega t-\phi )} + h.c.\right] . \end{aligned}$$

(37)

Here, $J_{l}(\alpha _m)$ is the Bessel function of first kind. The oscillation frequency in Eq. (30) is $\delta _{-}+l\omega$ and $\delta _{+}-l'\omega$. If we set $\omega =2\omega _{q}$ and $\eta >1$, the lowest oscillation frequency is $|\delta _{-}|=|\eta -1|\omega _{r}$. Hence, many higher-order terms in Eq. (37) can be neglected according to the RWA. Taking the first term in Eq. (37) as an example, we have

$$\begin{aligned} \tilde{H}^{(1)}(t)=g\sum _{m=1}^2\left[ a^{\dag }\sigma _{-}^{m}\sum _{l=-\infty }^{+\infty } i^{l}J_{l}(\alpha _{m})e^{i\delta _{-}t}e^{il(\omega t-\phi )} + h.c.\right] . \end{aligned}$$

(38)

When $l=-1$, the phases are revised as $e^{\pm i(\delta _{-}-\omega )t\pm i\phi }$. By properly choosing parameters, we have $|\delta _{-}-\omega |\gg g|J_{-1}(\alpha _{m})|$, and this tells us that the term can be omitted. When $l=0$, the phases reduce to $e^{\pm i\delta _{-}t}=e^{\pm i(\omega _{r}-\omega _{q})t}$. With proper choice of parameters, it is possible to make $|\delta _{-}|$ not much greater than $g|J_{0}(\alpha _{m})|$ and as a result it cannot be neglected according to the RWA. When $l=1$, the phases are combined as $e^{\pm i(\delta _{-}+\omega ) t\mp i\phi }=e^{\pm i\delta _{+}t\mp i\phi }$. It is possible to make $|\delta _{+}|\gg g|J_{1}(\alpha _{m})|$, and hence it gives a higher-order oscillating term which can be omitted too. Similarly, it can be found that all the terms in $\tilde{H}^{(1)}(t)$ with $l\ne 0$ are higher-order oscillating terms. While for the second term in Eq. (37), all the terms with $l^{\prime }\ne 1$ are also higher-order oscillating terms based on appropriate parameters. Finally, letting $\phi =\pi /2$ and ignoring all the higher-order terms, we can obtain the following effective Hamiltonian:

$$\begin{aligned} \tilde{H}_\mathrm{eff}= & {} \sum _{m=1}^2 g_\mathrm{eff}\left( a^{\dag }e^{i\delta _{-}t}+a e^{-i\delta _{-}t}\right) \sigma _{x}^{m}, \end{aligned}$$

(39)

where $g_\mathrm{eff}=0.5479g$ and this is because we choose $\alpha _{m}=1.4347$ and so $J_{0}(\alpha _{m})=J_{1}(\alpha _{m})=0.5479$. This is just the desired conditional displacement interaction in the transversal direction for two qubits.

B The derivation of evolution operator in Eq. (15)

In this appendix, we present a detailed derivation of Eq. (15) from Eq. (9). If we set $J_{x}=\sigma _{x}^{1}+\sigma _{x}^{2}$, Eq. (9) is recast as follows:

$$\begin{aligned} \tilde{H}_\mathrm{eff}(t)= & {} g_\mathrm{eff}\left( a^{\dag }e^{i\delta _{-}t}+a e^{-i\delta _{-}t}\right) J_{x}. \end{aligned}$$

(40)

The evolution operator of effective Hamiltonian in Eq. (40) can be obtained by means of the Magnus expansion. The evolution operator U(t) satisfies the following differential equation:

$$\begin{aligned} i\partial _{t}U(t)= \tilde{H}_\mathrm{eff}(t)U(t). \end{aligned}$$

(41)

The solution to Eq. (41) can be written as the following Magnus series form:

$$\begin{aligned} U(t)=\mathrm{exp}(\varOmega (t))=\mathrm{exp}(\varOmega _{1}(t)+\varOmega _{2}(t)+\varOmega _{3}(t)+\cdots ), \end{aligned}$$

(42)

where the first three terms of $\varOmega (t)$ are given as follows:

$$\begin{aligned}&\varOmega _{1}(t)=-i\int _{0}^{t}\tilde{H}_{1}\mathrm{d}t_{1}, \end{aligned}$$

(43a)

$$\begin{aligned}&\varOmega _{2}(t)=\frac{(-i)^{2}}{2}\int _{0}^{t}\mathrm{d}t_{1}\int _{0}^{t_{1}}\mathrm{d}t_{2}\left[ \tilde{H}_{1},\tilde{H}_{2}\right] ,\end{aligned}$$

(43b)

$$\begin{aligned}&\varOmega _{3}(t)= \frac{(-i)^{3}}{6}\int _{0}^{t}\mathrm{d}t_{1}\int _{0}^{t_{1}}\mathrm{d}t_{2}\int _{0}^{t_{2}}\mathrm{d}t_{3}\left( \left[ \tilde{H}_{1},\left[ \tilde{H}_{2},\tilde{H}_{3}\right] \right] +\left[ \tilde{H}_{3},\left[ \tilde{H}_{2},\tilde{H}_{1}\right] \right] \right) . \end{aligned}$$

(43c)

Here, we have made the notation simplification that $\tilde{H}_{k}=\tilde{H}(t_{k})$. Substituting Eqs. (40)–(43), we obtain

$$\begin{aligned}&\varOmega _{1}(t)= \frac{g_\mathrm{eff}}{\delta _{-}}\left( a(e^{-i\delta _{-}t}-1)-a^{\dag }(e^{i\delta _{-}t}-1)\right) J_{x},\end{aligned}$$

(44a)

$$\begin{aligned}&\varOmega _{2}(t)= i\frac{g_\mathrm{eff}^{2}}{\delta _{-}^{2}}\left( \delta _{-}t-\sin (\delta _{-}t)\right) J_{x}^{2},\end{aligned}$$

(44b)

$$\begin{aligned}&\varOmega _{k}(t)=0,\quad k\ge 3. \end{aligned}$$

(44c)

Here, we used the commutation relation $[\tilde{H}_{i},\tilde{H}_{j}]=2ig_\mathrm{eff}^{2}\sin \left[ \delta _{-}(t_{j}-t_{i})\right] J_{x}^{2}$. We also can check the relation $[\varOmega _{1}(t),\varOmega _{2}(t)]=0$. Then, the evolution operator U(t) in Eq. (42) can be recast as

$$\begin{aligned} U(t)=\mathrm{exp}(\varOmega _{1}(t))\mathrm{exp}(\varOmega _{2}(t))=D[\beta (t)J_{x}]\mathrm{exp}\left( i\varPhi (t)J_{x}^{2}\right) , \end{aligned}$$

(45)

where $J_{x}=\sigma _{x}^{1}+\sigma _{x}^{2}$, $\beta (t)=(g_\mathrm{eff}/\delta _{-})(1-e^{i\delta _{-}t})$ and $\varPhi (t)=(g_\mathrm{eff}/\delta _{-})^{2}(\delta _{-}t-\sin (\delta _{-}t))$, and $D(\beta )=e^{\beta a^{\dag } - \beta ^{*}a}$.