1. Introduction
Quantum telecommunication, first proposed by Bennett et al [1], is known as a technique for the transfer of an unknown quantum state from a transmitter at one location to a receiver at some distance away. To fulfill quantum teleportation between two separate observers, named Alice (transmitter) and Bob (receiver), a few conditions should be provided, including (i) preparing an unknown state (atom or field) that aimed to be teleported to the Alice lab, (ii) providing a nonlocal quantum channel, (iii) making a Bell-state measurement (BSM) by Alice, (iv) communicating the measurement result via a classical channel to Bob. This quantum process plays a key role in the processes of quantum information and communication [2, 3]. This protocol attracted a lot of attention in the field of quantum information; for instance, using the entangled squeezed states [4, 5], pairs of entangled photons by parametric down-conversion process [6–8], genuinely entangled multipartite states [9], nonmaximally entangled resources [10], and quantum dot single-photon source [11]. In this line, various schemes have been proposed to perform the teleportation process in different contexts, such as running wave fields [12–16], trapped ions [17–19], superconducting circuits [20, 21], trapped wave fields inside high-quality cavities [22–26], continuous-variable quantum states [27], and entangled coherent states [28–30].
In this regard, a variety of schemes in the context of cavity QED have been proposed for teleportation purposes. In a few cases, an experimentally feasible scheme for teleporting an unknown atomic state between two high-quality cavities has been investigated in [22], wherein the cavities contain a superposition of microwave field states initially prepared in a maximally entangled state. Also, to teleport an unknown atomic state, two schemes have been investigated in the microwave and optical regimes [23]. Furthermore, a similar scheme, which is based on the Raman atom cavity field interaction, has been performed for the teleportation of an unknown atomic state [31]. Very recently, one of us has paid attention to this issue in various schemes [32–34].
As stated before, the BSM method is one of the four requirements in quantum teleportation, and so is commonly used [35–41]. However, as is well-known, one of the main challenges in the realization of this protocol is the BSM which is not generally a straightforward task in experiments [1]. This fact has encouraged researchers in the field to implement this protocol without BSM. As some important alternative methods, we may refer to the related literature in the continuation. For instance, quantum teleportation of an unknown qubit state has been investigated approximately and conditionally, in which an additional qubit has been used [42]. A scheme for teleportation of zero- and one-photon entangled states from a two-mode high-quality cavity to another one has been proposed in [43]. A similar scheme for faithful teleporting of an unknown qutrit state using an extra qutrit, as well as zero- and one-photon entangled states between two cavities, has been studied in [44]. Approximate and conditional teleportation of an unknown entangled state of two qubits in cavity QED has been examined in [45]. Also, the teleportation of an unknown two-qubit entangled state has been investigated, wherein a cluster state is utilized as a quantum channel [46]. The teleportation of an unknown state of a qubit, conditionally and approximately via two-photon interaction in a cavity QED has been discussed in [47]. A similar scheme, however, with multi-photon interaction has been studied approximately and conditionally in [48].
In direct relation to the above-discussed literature, the Jaynes–Cummings model (JCM) describes the interaction between a qubit and a single-mode quantized field (cavity QED method) in the presence of rotating wave approximation (RWA) [49]. However, the RWA and thus the standard JCM is only valid whenever one works in the weak atom-field coupling regime [50]. In fact, in the strong atom-field coupling regime [51–53], the contribution of counter rotating terms can no longer be ignored and therefore should be taken into account. A lot of studies have been performed to study the effects of such terms in various aspects and for various purposes. For instance, in quantum open systems, the effects of these terms on the non-Markovianity of the system have been investigated in [54]. The influence of the mentioned terms on various physical quantities such as entanglement dynamics [55–60] and atomic population inversion [59, 61, 62] have been discussed in different quantum interacting systems. In this respect, due to the fact that (i) the atom-field interaction has a key role in a variety of teleportation protocols dealing with the cavity QED method, and (ii) since it may be possible to perform atom-field interaction in the strong regimes, in order to investigate the effects of the mentioned terms in the teleportation of unknown states of a qubit (or a field) in the strong coupling regime, in the present work we use the Rabi model [51, 63, 64] instead of JCM in the necessary processes of performing the protocol. By this, we want to show that, even in the strong regime wherein the JCM is no longer valid, quantum teleportation can be performed, while no BSM has been utilized.
This paper is arranged in this order; in section 2 , the teleportation of an unknown state of a qubit to a second qubit in the strong coupling regime is introduced. Then, we teleport an unknown field state in a similar manner between two high-quality cavities (see figure 1). Finally, we present a summary of concluding remarks in section 3 .
Figure 1. The scheme related to teleportation of unknown states of a qubit and a single-mode field in a strong coupling regime without Bell-state measurement. |
2. Quantum teleportation model
Various unconditional schemes for teleportation protocol have been introduced and investigated [65, 66]. However, the introduced schemes that have not still been realized experimentally are not infrequent. This is partly due to the fact that most of the schemes need a BSM (this measurement is a hard task practically) [33, 34, 67]. Therefore, this challenge motivated many researchers to propose teleportation schemes without BSM [25, 42, 44–48]. In one of these studies, Zheng investigated the teleportation of an unknown qubit and field states based on the cavity QED method using the JCM (under the RWA), approximately and conditionally [42]. In the present paper, we are going to extend this scheme using the Rabi model (in which the counter rotating terms are taken into account) which is governed in the strong coupling regime. Such regimes are now easy to access [52, 53], and so we have enough motivation to pay attention to our teleportation protocol. The Rabi Hamiltonian of a qubit interacting with a single-mode quantized field reads as follows [51]:5 ), the relations are simplified to5 ), the relations can be written as
$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{\rm{R}}} & = & \nu {\hat{a}}^{\dagger }\hat{a}+\displaystyle \frac{\omega }{2}{\hat{\sigma }}_{z}\\ & & +g\left({\hat{a}}^{\dagger }+\hat{a}\right)\left({\hat{\sigma }}^{+}+{\hat{\sigma }}^{-}\right),\end{array}\end{eqnarray}$
where ν (ω) and g are the frequency of the field (atomic transition) and atom-field coupling constant, respectively. Furthermore, $\hat{a}$ $({\hat{a}}^{\dagger })$ is the bosonic annihilation (creation) operator of the field and ${\hat{\sigma }}^{+}$ (${\hat{\sigma }}^{-}$) is the raising (lowering) operator associated with the qubit. Applying a special unitary transformation operator on the above Hamiltonian, the Bloch–Siegert Hamiltonian of a qubit in cavity QED for strong coupling regime is achieved as below [51] $\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{\mathrm{BS}} & = & \nu {\hat{a}}^{\dagger }\hat{a}+\displaystyle \frac{{\omega }_{q}}{2}{\hat{\sigma }}_{z}\\ & & +g\left({\hat{\sigma }}^{-}{\hat{a}}^{\dagger }+{\hat{\sigma }}^{+}\hat{a}\right)+\mu {\hat{\sigma }}_{z}{\hat{a}}^{\dagger }\hat{a},\end{array}\end{eqnarray}$
where we have assumed Ω = ω + ν, μ = g2/Ω and ωq = ω + μ. In order to obtain the matrix representation of the above Hamiltonian (considering the basis states ∣e, n⟩ and ∣g, n + 1⟩ for qubit-field system with n, n + 1 as the number of photons, whenever the qubit is respectively in the excited and ground states) one arrives at $\begin{eqnarray}{\hat{H}}_{\mathrm{BS}}=\left(\begin{array}{cc}-\displaystyle \frac{{\omega }_{q}}{2}-(n+1)\mu +(n+1)\nu & g\sqrt{n+1}\\ g\sqrt{n+1} & \displaystyle \frac{{\omega }_{q}}{2}+n(\mu +\nu )\end{array}\right).\end{eqnarray}$
Consequently, the time evolution operator $\hat{U}=\exp (-{\rm{i}}\tau {\hat{H}}_{\mathrm{BS}})$ associated with the considered system can be straightforwardly obtained as, $\begin{eqnarray}\hat{U}=\left(\begin{array}{cc}{y}_{1}(\tau ,n) & {y}_{3}(\tau ,n)\\ {y}_{2}(\tau ,n) & {y}_{4}(\tau ,n)\end{array}\right),\end{eqnarray}$
where we have defined, $\begin{eqnarray}\begin{array}{rcl}{y}_{1}(\tau ,n) & = & \displaystyle \frac{\lambda +\xi }{2\lambda }\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{+}}{2}\right)+\displaystyle \frac{\lambda -\xi }{2\lambda }\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{-}}{2}\right),\\ {y}_{2}(\tau ,n) & = & \displaystyle \frac{g\sqrt{n+1}}{\lambda }\left[\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{-}}{2}\right)-\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{+}}{2}\right)\right],\\ {y}_{3}(\tau ,n) & = & {y}_{2}(\tau ,n),\\ {y}_{4}(\tau ,n) & = & \displaystyle \frac{\lambda -\xi }{2\lambda }\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{+}}{2}\right)\\ & & +\displaystyle \frac{\lambda +\xi }{2\lambda }\exp \left(\displaystyle \frac{{\rm{i}}\tau {\beta }_{-}}{2}\right),\end{array}\end{eqnarray}$
with the definitions ξ = ωq + μ(2n + 1) − ν, $\lambda \,=\sqrt{4{g}^{2}(n+1)+{\xi }^{2}}$ and β± = μ − (2n + 1)ν ± λ. As a result, each of the desired bases state evolves in the following form under the action of the above obtained time evolution operator, $\begin{eqnarray}\begin{array}{rcl}| e,n\rangle & \longrightarrow & {y}_{3}(\tau ,n)| g,n+1\rangle \\ & & +{y}_{4}(\tau ,n)| e,n\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}| g,n+1\rangle & \longrightarrow & {y}_{1}(\tau ,n)| g,n+1\rangle \\ & & +{y}_{2}(\tau ,n)| e,n\rangle ,\end{array}\end{eqnarray}$
where the relations ∣y3(τ, n)∣2 + ∣y4(τ, n)∣2 = 1 = ∣y1(τ, n)∣2 + ∣y2(τ, n)∣2, i.e. the normalization condition of the timed evolved bases, are clearly satisfied. Assuming n = 0 in ( $\begin{eqnarray}\begin{array}{rcl}{y}_{1}(\tau ,0) & = & \displaystyle \frac{{\rm{\Gamma }}}{{\lambda }_{1}}\left[{\lambda }_{1}\cos \left(\displaystyle \frac{{\lambda }_{1}\tau }{2}\right)\right.\\ & & \left.+{\rm{i}}({\omega }_{q}+\mu -\nu )\sin \left(\displaystyle \frac{{\lambda }_{1}\tau }{2}\right)\right],\\ {y}_{2}(\tau ,0) & = & \displaystyle \frac{-2{\rm{i}}g{\rm{\Gamma }}}{{\lambda }_{1}}\sin \left(\displaystyle \frac{{\lambda }_{1}\tau }{2}\right),\\ {y}_{3}(\tau ,0) & = & {y}_{2}(\tau ,0),\\ {y}_{4}(\tau ,0) & = & \displaystyle \frac{{\rm{\Gamma }}}{{\lambda }_{1}}\left[{\lambda }_{1}\cos \left(\displaystyle \frac{{\lambda }_{1}\tau }{2}\right)\right.\\ & & \left.-{\rm{i}}({\omega }_{q}+\mu -\nu )\sin \left(\displaystyle \frac{{\lambda }_{1}\tau }{2}\right)\right],\end{array}\end{eqnarray}$
where ${\rm{\Gamma }}=\cos \left(\tfrac{(\mu -\nu )\tau }{2}\right)+\mathrm{isin}\left(\tfrac{(\mu -\nu )\tau }{2}\right)$, ${\lambda }_{1}\,=\sqrt{4{g}^{2}+{\left({\omega }_{q}+\mu -\nu \right)}^{2}}$. Similarly, if we set n = 1 in ( $\begin{eqnarray}\begin{array}{rcl}{y}_{1}(\tau ,1) & = & \displaystyle \frac{{\rm{\Lambda }}}{{\lambda }_{2}}\left[{\lambda }_{2}\cos \left(\displaystyle \frac{{\lambda }_{2}\tau }{2}\right)\right.\\ & & \left.+{\rm{i}}({\omega }_{q}+3\mu -\nu )\sin \left(\displaystyle \frac{{\lambda }_{2}\tau }{2}\right)\right],\\ {y}_{2}(\tau ,1) & = & \displaystyle \frac{-2\sqrt{2}{\rm{i}}g{\rm{\Lambda }}}{{\lambda }_{2}}\sin \left(\displaystyle \frac{{\lambda }_{2}\tau }{2}\right),\\ {y}_{3}(\tau ,1) & = & {y}_{2}(\tau ,1),\\ {y}_{4}(\tau ,1) & = & \displaystyle \frac{{\rm{\Lambda }}}{{\lambda }_{2}}\left[{\lambda }_{2}\cos \left(\displaystyle \frac{{\lambda }_{2}\tau }{2}\right)\right.\\ & & \left.-{\rm{i}}({\omega }_{q}+3\mu -\nu )\sin \left(\displaystyle \frac{{\lambda }_{2}\tau }{2}\right)\right],\end{array}\end{eqnarray}$
where ${\rm{\Lambda }}=\cos \left(\tfrac{(\mu -3\nu )\tau }{2}\right)+\mathrm{isin}\left(\tfrac{(\mu -3\nu )\tau }{2}\right)$, ${\lambda }_{2}=\sqrt{8{g}^{2}+{\left({\omega }_{q}+3\mu -\nu \right)}^{2}}$.Now, all requirements have been provided in order to perform our goal of teleporting of an unknown state. To achieve the purpose, we assume that the qubit A possesses the following unknown state6 )) reads as8 ), the above relation in (11 ) can be rewritten as follows10 ) and (13 ), the state of the whole system can be expressed as6 ) and (7 ), the state of the whole system evolves to8 ) and (9 ). Now, assuming that ω = ν, ωq + μ − ν = 2g and by performing a measurement on the field state in (15 ), if one arrives at ∣0⟩, the state of the reduced subsystem composed of two qubits A and B
$\begin{eqnarray}| \phi {\rangle }_{{\rm{A}}}={c}_{g}| g{\rangle }_{{\rm{A}}}+{c}_{e}| e{\rangle }_{{\rm{A}}},\end{eqnarray}$
where the unknown coefficients cg and ce are fulfilled the normalization condition (∣cg∣2 + ∣ce∣2 = 1). We want to teleport this state to the qubit B. For this aim, we first send the qubit B, which is initially prepared in the excited state ∣e⟩B, into an initially empty resonant cavity. So, it interacts with the vacuum field ∣0⟩ of this cavity. After the interaction time τB, when qubit B leaves the cavity, the state of the qubit-cavity system (using equation ( $\begin{eqnarray}| \varphi ({\tau }_{{\rm{B}}})\rangle ={y}_{3}({\tau }_{{\rm{B}}},0)| g{\rangle }_{{\rm{B}}}| 1\rangle +{y}_{4}({\tau }_{{\rm{B}}},0)| e{\rangle }_{{\rm{B}}}| 0\rangle .\end{eqnarray}$
Using the resonance condition ω = ν and assuming that ωq + μ − ν = 2g, one may readily conclude that μ = g and finally we are in the condition g = 2ω. Therefore, keeping in mind the illustration that, if g/ω ≳ 1, atom-field coupling is in the deep-strong regime [51, 52], we are certain that we are working in the strong regime. According to the mentioned conditions and using equation ( $\begin{eqnarray}\begin{array}{rcl}| \varphi ({\tau }_{{\rm{B}}}){\rangle }^{{\prime} } & = & \displaystyle \frac{1}{\sqrt{2}}\left[\sin \left(\displaystyle \frac{g{\tau }_{{\rm{B}}}}{4}\right)-\mathrm{icos}\left(\displaystyle \frac{g{\tau }_{{\rm{B}}}}{4}\right)\right]\\ & & \times \,\sin \left(\sqrt{2}g{\tau }_{{\rm{B}}}\right)| g{\rangle }_{{\rm{B}}}| 1\rangle \\ & & +\displaystyle \frac{1}{\sqrt{2}}\left[\cos \left(\displaystyle \frac{g{\tau }_{{\rm{B}}}}{4}\right)+\mathrm{isin}\left(\displaystyle \frac{g{\tau }_{{\rm{B}}}}{4}\right)\right]\\ & \times & \left[\sqrt{2}\cos \left(\sqrt{2}g{\tau }_{{\rm{B}}}\right)-\mathrm{isin}\left(\sqrt{2}g{\tau }_{{\rm{B}}}\right)\right]| e{\rangle }_{{\rm{B}}}| 0\rangle .\end{array}\end{eqnarray}$
Now, we can adjust the velocity of qubit B, such that the condition $g{\tau }_{{\rm{B}}}=\tfrac{\pi }{2\sqrt{2}}$ is satisfied. Accordingly, we finally arrive at $\begin{eqnarray}| \varphi {\rangle }_{{\rm{B}}}^{{\prime} }=M\left(| g{\rangle }_{{\rm{B}}}| 1\rangle +| e{\rangle }_{{\rm{B}}}| 0\rangle \right),\end{eqnarray}$
where $M=\tfrac{1}{\sqrt{2}}\left[\sin \left(\tfrac{\pi }{8\sqrt{2}}\right)-\mathrm{icos}\left(\tfrac{\pi }{8\sqrt{2}}\right)\right]$ plays the role of normalization coefficient. Consequently, using equations ( $\begin{eqnarray}\begin{array}{rcl}| \psi \rangle & = & M\left({c}_{g}| g{\rangle }_{{\rm{A}}}+{c}_{e}| e{\rangle }_{{\rm{A}}}\right)\\ & & \otimes \left(| g{\rangle }_{{\rm{B}}}| 1\rangle +| e{\rangle }_{{\rm{B}}}| 0\rangle \right).\end{array}\end{eqnarray}$
Next, the qubit A interacts with the single-mode field of the cavity. After the interaction time τA, using equations ( $\begin{eqnarray}\begin{array}{rcl}| \psi ({\tau }_{{\rm{A}}})\rangle & = & {{Mc}}_{g}| g{\rangle }_{{\rm{B}}}\left[{y}_{1}({\tau }_{{\rm{A}}},0)| g{\rangle }_{{\rm{A}}}| 1\rangle \right.\\ & & \left.+{y}_{2}({\tau }_{{\rm{A}}},0)| e{\rangle }_{{\rm{A}}}| 0\rangle \right]\\ & & +{{Mc}}_{g}| e{\rangle }_{{\rm{B}}}| g{\rangle }_{{\rm{A}}}| 0\rangle \\ & & +{{Mc}}_{e}| g{\rangle }_{{\rm{B}}}\left[{y}_{3}({\tau }_{{\rm{A}}},1)| g{\rangle }_{{\rm{A}}}| 2\rangle \right.\\ & & \left.+{y}_{4}({\tau }_{{\rm{A}}},1)| e{\rangle }_{{\rm{A}}}| 1\rangle \right]\\ & & +{{Mc}}_{e}| e{\rangle }_{{\rm{B}}}\left[{y}_{3}({\tau }_{{\rm{A}}},0)| g{\rangle }_{{\rm{A}}}| 1\rangle \right.\\ & & \left.+{y}_{4}({\tau }_{{\rm{A}}},0)| e{\rangle }_{{\rm{A}}}| 0\rangle \right].\end{array}\end{eqnarray}$
For more details on the above equation, see equations ( $\begin{eqnarray}\begin{array}{rcl}|\psi ({\tau }_{{\rm{A}}}){\rangle }^{{\rm{{\prime} }}} & = & N\left\{\displaystyle \frac{M}{\sqrt{2}}\left[\sin \left(\displaystyle \frac{g{\tau }_{{\rm{A}}}}{4}\right)-{\rm{icos}}\left(\displaystyle \frac{g{\tau }_{{\rm{A}}}}{4}\right)\right]\right.\\ & & \times \,\sin \left(\sqrt{2}g{\tau }_{{\rm{A}}}\right){c}_{g}|g{\rangle }_{{\rm{B}}}|e{\rangle }_{{\rm{A}}}+{{Mc}}_{g}|e{\rangle }_{{\rm{B}}}|g{\rangle }_{{\rm{A}}}\\ & & +\displaystyle \frac{M}{\sqrt{2}}\left[\cos \left(\displaystyle \frac{g{\tau }_{{\rm{A}}}}{4}\right)+{\rm{isin}}\left(\displaystyle \frac{g{\tau }_{{\rm{A}}}}{4}\right)\right]\\ & & \times \left.\left[\sqrt{2}\cos \left(\sqrt{2}g{\tau }_{{\rm{A}}}\right)-{\rm{isin}}\left(\sqrt{2}g{\tau }_{{\rm{A}}}\right)\right]{c}_{e}|e{\rangle }_{{\rm{B}}}|e{\rangle }_{{\rm{A}}}\Space{0ex}{3.2ex}{0ex}\right\},\end{array}\end{eqnarray}$
where N is a normalization coefficient. If $g{\tau }_{{\rm{A}}}=\tfrac{\pi }{2\sqrt{2}}$, the above relation can be rewritten as follows $\begin{eqnarray}\begin{array}{rcl}| \psi {\rangle }^{{\prime} } & = & N\left\{{M}^{2}\left[{c}_{g}| g{\rangle }_{{\rm{B}}}+{c}_{e}| e{\rangle }_{{\rm{B}}}\right]| e{\rangle }_{{\rm{A}}}\right.\\ & & \left.+{{Mc}}_{g}| e{\rangle }_{{\rm{B}}}| g{\rangle }_{{\rm{A}}}\right\}.\end{array}\end{eqnarray}$
Now, we perform an atomic measurement on the qubit A. If this qubit is detected in the excited state ∣e⟩A, the qubit B is exactly in the initial state of the qubit A, and we successfully achieved the purpose, that is $\begin{eqnarray}| \phi {\rangle }_{{\rm{B}}}={c}_{g}| g{\rangle }_{{\rm{B}}}+{c}_{e}| e{\rangle }_{{\rm{B}}}.\end{eqnarray}$
It can be readily found that the success probability of teleportation and the fidelity2(2The evaluation of fidelity is based on the method available in [42–44].) are 0.25 and 1, respectively. This value of success probability (post-selection probability) is typically acceptable in schemes for quantum teleportation [32, 42, 44].As the next protocol, in order to teleport an unknown state of a quantized field between two distant optical cavities, each containing a single-mode field, we want to teleport the state of cavity C, which at first has been prepared in the following unknown state6 ) and (8 ), the state of qubitcavity system can be written as follows19 ) and (20 ), can be expressed as follows6 ) and (7 ), the evolution of the state of the whole system results in8 ) and (9 ) for more details in (22 ). Supposing that ω = ν, ωq + μ − ν = 2g, and via an implementation a measurement on the field state of the cavity C, if it projects into the vacuum state (∣0⟩C), the state of the reduced subsystem consisting of the field corresponding to cavity D and the inside qubit results in
$\begin{eqnarray}| {\rm{\Psi }}{\rangle }_{{\rm{C}}}=\alpha | 0{\rangle }_{{\rm{C}}}+\beta | 1{\rangle }_{{\rm{C}}},\end{eqnarray}$
where we have ∣α∣2 + ∣β∣2 = 1. We want to teleport the above state to the optical cavity D. To do this task, a qubit is first prepared in the excited state ∣e⟩ and the cavity D contains a single-mode field in the vacuum ∣0⟩D. The qubit is sent into the cavity D and so the interaction starts. Provided the conditions, ω = ν, ωq + μ − ν = 2g and $g{\tau }_{{\rm{D}}}=\tfrac{\pi }{2\sqrt{2}}$ hold, after which the qubit leaves the cavity (in the interaction time τD), using equations ( $\begin{eqnarray}| {\rm{\Psi }}{\rangle }_{q-{\rm{D}}}=M\left(| g\rangle | 1{\rangle }_{{\rm{D}}}+| e\rangle | 0{\rangle }_{{\rm{D}}}\right).\end{eqnarray}$
Therefore, the state of the whole system, using equations ( $\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}{\rangle }_{{\rm{C}}-q-{\rm{D}}} & = & M\left((\alpha | 0{\rangle }_{{\rm{C}}}+\beta | 1{\rangle }_{{\rm{C}}}\right)\\ & & \otimes \left((\left|| g\rangle | 1{\rangle }_{{\rm{D}}}+| e\rangle \right|0{\rangle }_{{\rm{D}}}\right).\end{array}\end{eqnarray}$
Now, after the time τC in the interaction process of the qubit with the cavity C containing a single-mode field, with the help of equations ( $\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}({\tau }_{{\rm{C}}}){\rangle }_{{\rm{C}}-q-{\rm{D}}} & = & M\alpha | 1{\rangle }_{{\rm{D}}}| g\rangle | 0{\rangle }_{{\rm{C}}}\\ & + & M\alpha | 0{\rangle }_{{\rm{D}}}\left[{y}_{3}({\tau }_{{\rm{C}}},0)| g\rangle | 1{\rangle }_{{\rm{C}}}\right.\\ & & \left.+{y}_{4}({\tau }_{{\rm{C}}},0)| e\rangle | 0{\rangle }_{{\rm{C}}}\right]\\ & & +M\beta | 1{\rangle }_{{\rm{D}}}\left[{y}_{1}({\tau }_{{\rm{C}}},0)| g\rangle | 1{\rangle }_{{\rm{C}}}\right.\\ & & \left.+{y}_{2}({\tau }_{{\rm{C}}},0)| e\rangle | 0{\rangle }_{{\rm{C}}}\right]\\ & & +M\beta | 0{\rangle }_{{\rm{D}}}\left[{y}_{3}({\tau }_{{\rm{C}}},1)| g\rangle | 2{\rangle }_{{\rm{C}}}\right.\\ & & \left.+{y}_{4}({\tau }_{{\rm{C}}},1)| e\rangle | 1{\rangle }_{{\rm{C}}}\right].\end{array}\end{eqnarray}$
See equations ( $\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}({\tau }_{{\rm{C}}}){\rangle }_{q-{\rm{D}}}^{{\prime} } & = & {N}^{{\prime} }\left\{\Space{0ex}{3.7ex}{0ex}M\alpha | 1{\rangle }_{{\rm{D}}}| g\rangle \right.\\ & & +\displaystyle \frac{M}{\sqrt{2}}\left[\cos \left(\displaystyle \frac{g{\tau }_{{\rm{C}}}}{4}\right)+\mathrm{isin}\left(\displaystyle \frac{g{\tau }_{{\rm{C}}}}{4}\right)\right]\\ & \times & \left[\sqrt{2}\cos \left(\sqrt{2}g{\tau }_{{\rm{C}}}\right)-\mathrm{isin}\left(\sqrt{2}g{\tau }_{{\rm{C}}}\right)\right]\\ & & \times \,\alpha | 0{\rangle }_{{\rm{D}}}| e\rangle +\displaystyle \frac{M}{\sqrt{2}}\left[\sin \left(\displaystyle \frac{g{\tau }_{{\rm{C}}}}{4}\right)-\mathrm{icos}\left(\displaystyle \frac{g{\tau }_{{\rm{C}}}}{4}\right)\right]\\ & & \left.\times \,\sin \left(\sqrt{2}g{\tau }_{{\rm{C}}}\right)\beta | 1{\rangle }_{{\rm{D}}}| e\rangle \Space{0ex}{3.7ex}{0ex}\right\},\end{array}\end{eqnarray}$
where ${N}^{{\prime} }$ is a normalization coefficient. If $g{\tau }_{{\rm{C}}}=\tfrac{\pi }{2\sqrt{2}}$, the above relation may be simplified as below $\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}{\rangle }_{q-{\rm{D}}}^{{\prime} } & = & {N}^{{\prime} }\left\{{M}^{2}\left[\alpha | 0{\rangle }_{{\rm{D}}}+\beta | 1{\rangle }_{{\rm{D}}}\right]| e\rangle \right.\\ & & \left.+M\alpha | 1{\rangle }_{{\rm{D}}}| g\rangle \right\}.\end{array}\end{eqnarray}$
Now, by performing a next measurement on the qubit, if this qubit is detected in the state ∣e⟩, the cavity D is exactly in the initial state of the cavity C, i.e. $\begin{eqnarray}| {\rm{\Psi }}{\rangle }_{{\rm{D}}}=\alpha | 0{\rangle }_{{\rm{D}}}+\beta | 1{\rangle }_{{\rm{D}}}.\end{eqnarray}$
Similar to the previously explained procedure for the atomic teleportation, the success probability of the field teleportation is 0.25 with the maximum amount of fidelity, 1.Summing up, as is shown, in both cases, either atom or field state teleportation, the success probability and the fidelity of teleportation are obtained as 0.25 and 1, respectively. Comparing with the results in [42] wherein the success probability (fidelity) read as 0.25 (F ≳ 0.987) it is found that our system possesses a bit better values. This improvement of fidelity value in our work compared to [42], may originate from the presence of counter rotating terms, i.e. in the different used Hamiltonians of the two desired protocols, as well as the conditions governing the proposed schemes, such as choosing the value of interaction time, the resonance conditions and the relationship between the frequencies in each of these schemes. Despite this fact, noticing that we have done the procedure without BSM is another advantage of the model, and considering the counter rotating terms is the distinguishable feature of our present work.
3. Summary and conclusions
In this paper, we have developed the proposed scheme in [42] wherein the teleportation of an unknown qubit state as well as a field state have been performed in the weak coupling regime with RWA, using the cavity QED method. This means that, in our proposed scheme, the Rabi model has been utilized instead of JCM, for a strong coupling regime for the system consisting of a qubit interacting with a single-mode quantized field. In this regard, using the Bloch–Siegert Hamiltonian, which has been obtained via applying a particular unitary transformation on the Rabi Hamiltonian, we calculated the time evolution of the system by achieving the corresponding time evolution operator. In the continuation, a scheme has been proposed for the teleportation of an unknown state of a qubit using an additional qubit, however, without the BSM and RWA, so that an unknown state of a qubit that interacts with a single-mode quantized field within a cavity is teleported to a second qubit in a distant cavity field. At last, we similarly teleported an unknown state of the single-mode quantized field between two cavities (each of them contains a single-mode quantized field). It is shown that this teleportation protocol was successfully performed and leads to a probability of success of 0.25, with the maximum possible amount of fidelity.