Phase-covariant cloning via adiabatic passage in fiber-nanocavity system

TL;DR

This paper presents a scheme for long-range quantum phase-covariant cloning between two qubits in a fiber-nanocavity system using adiabatic passage, which is robust against decay and spontaneous emission.

Contribution

It introduces a novel adiabatic passage method for quantum cloning that avoids populating cavity photons and NV center excited states, enhancing robustness.

Findings

High fidelity achieved despite realistic imperfections

No cavity or NV center excited states populated during process

Robust against cavity decay and spontaneous emission

Abstract

We propose an effective scheme for realizing a long-range quantum state phase-covariant cloning between two qubits in fiber-nanocavity system via an adiabatic passage. Since no cavity (fiber) photons or excited levels of the nitrogen-vacancy (NV) center are populated during the whole process, the scheme is immune to the decay of cavity (fiber) and spontaneous emission of the NV center. The strictly numerical simulation shows that the fidelity is high even in the presence of realistic imperfections.

Equations

$$H$$

$$H_{NVc}$$

$$H_{NVc}^{{}^{\prime }}$$

$$H_{fc}=\nu \sum _{k=0}^2(b^{+}{a}_k+H.c.),$$

$$H_{total}=H_{NVc}^{{}^{\prime }}+H_{fc},$$

$$|{\varphi }_0⟩=|fgg000⟩|0{⟩}_f,$$

$$|{\varphi }_1⟩=|ggg100⟩|0{⟩}_f,$$

$$|{\varphi }_2⟩=|ggg000⟩|1{⟩}_f,$$

$$|{\varphi }_3⟩=|ggg010⟩|0{⟩}_f,$$

$$|{\varphi }_4⟩=|ggg001⟩|0{⟩}_f,$$

$$|{\varphi }_5⟩=|gfg000⟩|0{⟩}_f,$$

$$|{\varphi }_6⟩=|ggf000⟩|0{⟩}_f.$$

$$|\psi ⟩=\frac{1}{\sqrt{K}}(\frac{1}{{\lambda }_0}|{\varphi }_0⟩+\frac{1}{{\lambda }_1}|{\varphi }_5⟩+\frac{1}{{\lambda }_2}|{\varphi }_6⟩-\frac{1}{\nu }|{\varphi }_2⟩).$$

$$\underset{t\to -\infty }{\lim }\frac{{\Omega }_0(t)}{\Omega (t)}=0,$$

$$\underset{t\to \infty }{\lim }\frac{\Omega (t)}{{\Omega }_0(t)}=1.$$

$$|\psi {⟩}_W=\frac{1}{\sqrt{3}}(|fgg⟩+|gfg⟩+|ggf⟩)|000{⟩}_c|0{⟩}_f.$$

$$|\psi ⟩$$

$$|\psi {⟩}_W$$

$$|{\psi }_{(0)}⟩$$

$$|{\psi }_{(t)}⟩$$

$$\Omega (t)$$

$${\Omega }_0(t)$$

$$F=⟨{\Psi }_{(t)}||{\Psi }_{ideal}⟩,$$

$$\dot{\rho }$$

Full text

phase-covariant cloning,adiabatic passage,nitrogen-vacancy center, nanocavity

I Introduction

It is well known that the fundamental for quantum information science is no-cloning theorem, which states that an unknown quantum state cannot be cloned perfectly W1982 . However, we can try to achieve a state as close as possible to the input state. Recently, different schemes of quantum cloning have been proposed, and various quantum cloning machines were designed for different tasks R1998 ; V1998 . For the universal quantum cloning machine (UQCM), the input can be arbitrary qubits, the fidelity is optimal and does not depend on the input qubit Buzek1996 . It has been reported in the NMR system Cummings2002 and linear optics system Huang2001 ; A2002 . Different form UQCM, phase-covariant quantum cloning machine D2000 restricts the input state of the equatorial qubit $|\psi\rangle=(|0\rangle+e^{i\phi}|1\rangle)/\sqrt{2}$, which is located on the equator of the Bloch sphere. Phase-covariant quantum cloning machine has been reported in the solid state system with a single electron spin at room temperature Pan2011 , and the corresponding fidelity even reached $85.4\%$.

The key of quantum cloning is entanglement. Especially, tripartite entanglement states have been used to realize quantum cloning. The W state is one important tripartite entangled state, which was proposed by D$\ddot{u}$r and has some interesting propertiesDur2000 . In recent experiments, it was shown that three-qubit W states can be realized in optical systems Eibl2004 and ion traps Roos2004 . Since W state retains bipartite entanglement when any one of the three qubits is traced out, it can be used in quantum information processing Dr1998 and to test quantum nonlocality without inequality Zheng2002 ; Cabello2002 . What’s more, the schemes using W states to realize phase-covariant quantum cloning have been studied in various physical systems Zheng2005 ; Shen2012 .

The main obstacle to realizing multi-particle entanglement and quantum information processing is decoherence. In the process of decoherence in cavity QED, atomic spontaneous emission and cavity decay take effect. How to reduce the decoherence is an important problem. Adiabatic techniques are the answers since they feature a certain robustness, and in systems of $\bigwedge$ type one can avoid a transient large population in the excited state. Recently, the techniques of stimulated Raman adiabatic passage K1998 and fractional stimulated Raman adiabatic passage N1999 have been extensively used for realizing the quantum information processing X2006 ; Yang2010 ; Song2010 .

Motivated by these works, we present a new scheme to generate W state of NV centers in fiber-cavity coupling system via the adiabatic passage. Meanwhile, we realize phase-covariant cloning using the W state. In this scheme, we use the ground states of NV centers, and the cavities (fibers) field remains in the vacuum state throughout the operation. So NV center’s spontaneous emission and cavity decay can be efficiently suppressed. Another advantage of the scheme is that the interaction time does not need to be accurately controlled as long as the adiabatic condition is fulfilled, which makes it more applicable to current laboratory techniques. Our scheme may offer promising perspectives for entanglement generation and quantum information processing.

II Physical model

The experimental setup for generating W state of NV centers in the coupled system, as shown in Fig. 1(a). Three NV centers (labeled $NV_{0}$, $NV_{1}$, $NV_{2}$) are separately trapped in three distant optical cavities (labeled $C_{0}$, $C_{1}$, $C_{2}$) via an optical fiber coupler. The fiber coupler connects three fibers, at the end of which connect three cavities. The ground state of NV center is a spin triplet, which labeled as ${}^{3}A$. There is a zero-field splitting (2.88GHz) with a zero-field splitting between the state $\left|0\right\rangle$ ($m_{s}=0$) and $\left|\pm 1\right\rangle$ ($m_{s}=\pm 1$) Togan2010 . When an external magnetic field are added along the NV center symmetry axis, the levels $\left|\pm 1\right\rangle$ can be split by $2\pi\times 200$ MHz, while the $\left|0\right\rangle$ state was not affected. For simplicity, we denote the following states: $\left|0\right\rangle=\left|g\right\rangle$ and $\left|-1\right\rangle=\left|f\right\rangle$ are the ground states. while $\left|1\right\rangle=\left|e\right\rangle$ is the excited state. As shown in Fig. 1(b), the transition $\left|f\right\rangle\leftrightarrow\left|e\right\rangle$ is derived by two classical fields with the Rabi frequencies $\Omega_{k}$ and $\Omega_{k}^{{}^{\prime}}$, and the corresponding detunings $\Delta_{d}$ and $\Delta_{d}^{{}^{\prime}}$. The transition $\left|g\right\rangle\leftrightarrow\left|e\right\rangle$ is coupled to the cavity mode with the coupling strength $g$, and the corresponding detuning is $\Delta_{c}$. In the interaction picture, the Hamiltonian describing NV centers and cavities interaction is ($\hbar=1$)

[TABLE]

where $a$ is the annihilation operator for the cavity mode. Under the large detuning condition, i.e., $|\Delta_{c}|,|\Delta_{d}|,|\Delta_{d}^{{}^{\prime}}|\gg g,\Omega_{k},\Omega_{k}^{{}^{\prime}}$, the upper level $|e\rangle$ can be adiabatically eliminated. We set the parameters $\Omega_{k}=\Omega_{k}^{{}^{\prime}}$ and $\Delta_{d}=-\Delta_{d}^{{}^{\prime}}$ to eliminate the Stark shift induced by the classical lasers. The cavities are initially in vacuum states, so the Stark shift induced by the cavity mode is discarded. Furthermore, choosing the detunings appropriately $\Delta_{c}=\Delta_{d}=\Delta$, the dominant Raman transition is induced by the classical field and the cavity mode, while the other Raman transitions are far off-resonant, and can be neglected. Then, the Hamiltonian describing NV centers and cavities interaction is rewritten as

[TABLE]

where $\lambda_{k}=g\Omega_{k}/\Delta$ is the effective coupling strength of the Raman transition induced by the classical field and the cavity mode.

In this model, the optical cavities are connected by identical single-mode fibers. We assume that all the fibers connected to the coupler have the same transverse mode. In the short fiber limit, $l\bar{\nu}/(2\pi c)\leq 1$ Serafini2006 , where $l$ is the length of the fiber and $\bar{\nu}$ is the decay rate of the cavity fields into the fiber modes, only one resonant mode $b$ of the fiber interacts with the cavity modes. In the case, the coupling between the fiber mode and cavity fields is modeled by the interaction Hamiltonian

[TABLE]

where $\nu$ is coupling strength of the cavity mode to the fiber mode, and $b^{+}$ is the creation operator for the fiber mode. In the interaction picture, the total Hamiltonian describing NV centers, cavities and fibers interaction is given by

[TABLE]

III W state preparation and phase covariant cloning

Now, we show how to generate W state of NV centers using the physical model above. For an initial state of the system $|fgg000\rangle|0\rangle_{f}$, the single excitation subspace $\forall$ can be spanned by the following state vectors: $\{|\phi_{0}\rangle,|\phi_{1}\rangle,|\phi_{2}\rangle,|\phi_{3}\rangle,|\phi_{4}\rangle,|\phi_{5}\rangle,|\phi_{6}\rangle\}$, with

[TABLE]

The Hamiltonian $H_{total}$ has the following dark state with null eigenvalue as well

[TABLE]

where $K=(1/{\lambda_{0}^{2}}+1/{\lambda_{1}^{2}}+1/{\lambda_{2}^{2}}+1/{\nu^{2}})$ is the normalized factor.

According to the adiabatic theory, if the whole system is initially prepared in the dark state, under the adiabatic condition, it will evolve in the dark state subspace. We control the Rabi frequencies of classical fields $\Omega_{1}=\Omega_{2}=\Omega$ to satisfy the effective coupling strengths $\lambda_{1}=\lambda_{2}$. At the beginning, the Rabi frequencies are assumed $\Omega\gg\Omega_{0}$. Then we slowly decrease $\Omega(t)$ and simultaneously increase $\Omega_{0}(t)$ to obtain $\Omega(t)/\Omega_{0}(t)=1$ at the time $T$. Meanwhile, assuming $\lambda_{1}=\lambda_{0}\ll\nu$, we can discard the last term in the Eq. (6) above. From the discussion above, the pulse shapes must satisfy

[TABLE]

and

[TABLE]

As a result, we can obtain the target state

[TABLE]

The Eq. (9) above denotes that the fiber and cavity field are in vacuum states, and the three NV centers are in three-particle W state.

Then, we show that the idea can be generalized to the generation of the multi-NV-centers W state. $N$ ($N>3$) NV centers are separately trapped in each of the distant cavities connected by fibers, as shown in Fig. 2. We assume that the system is initially in the state $|f_{0}g_{1}g_{2}\cdot\cdot\cdot g_{N}\rangle|0\rangle_{c}\rangle|0\rangle_{f}$, where $|0\rangle_{c}\rangle|0\rangle_{f}=\Pi_{k=0}^{N}|0_{k}\rangle_{c}|0_{k}\rangle_{f}$ denotes all the cavity fields and fiber fields are in the vacuum states. Similar to the Eq. (6), we can get the dark state

[TABLE]

where $K=(1/{\lambda_{0}^{2}}+1/{\lambda_{1}^{2}}+\cdot\cdot\cdot+1/{\lambda_{N}^{2}}+1/{\nu^{2}})$ is the normalized factor. Choosing $\Omega_{1}=\Omega_{2}=\cdot\cdot\cdot\Omega_{N}=\Omega$, we get $\lambda_{1}=\lambda_{2}=\cdot\cdot\cdot\lambda_{N}=\lambda$. At the beginning, we assume $\Omega\gg\Omega_{0}$. Then we slowly decrease $\Omega(t)$ and simultaneously increase $\Omega_{0}(t)$ to obtain $\Omega(t)/\Omega_{0}(t)=1$ at the time $T$. Meanwhile, assuming $\lambda_{0}=\lambda\ll\nu$, we can discard the last term in the Eq. (10) above. The final state is

[TABLE]

So $N$ NV centers are prepared in an entangled state with the cavity mode and fiber mode left in the vacuum states.

The quantum cloning scheme can be implemented based on the W state previously prepared. Again, we assume that all the cavities and the optical fiber channel are initially both in the vacuum state, and only one NV center of the first cavity is prepared in the arbitrary orbital state of the Bloch sphere. The system initial state is written as

[TABLE]

Under the above-mentioned conditions, the dark state $|g_{0}g_{1}g_{2}\cdot\cdot\cdot g_{N}\rangle|0\rangle_{c}\rangle|0\rangle_{f}$ undergoes no changes. On the other hand, $|f_{0}g_{1}g_{2}\cdot\cdot\cdot g_{N}\rangle|0\rangle_{c}\rangle|0\rangle_{f}$ evolves to the state of the Eq. (10). So the system evolves to the state

[TABLE]

All the cavity and fiber field are always in vacuum states, so this scheme is robust against the cavity and fiber decay.

IV discussion

It should be noted that preparing W state of the NV centers is the key step to achieving the phase-covariant quantum cloning. For this reason, it is necessary to consider the feasibility of generating the W state of the NV centers. In the above derivations, we reasonably select the experimental parameters to satisfy the adiabatic passage. To make the scheme experimentally feasible, we discuss the Rabi frequencies and other correlative parameters. The time-dependent pulses we introduce are as follows:

[TABLE]

where $\Omega_{m}$, $t_{k}(k=0,1)$ and $t_{p}$, are the peak, the time delay and the width of the laser pulses, respectively. We utilize the fidelity and the fidelity of the prepared states is given by

[TABLE]

where $\left|\Psi_{ideal}\right\rangle$ is the ideal three-NV centers W state in Eq. (10), and $\left|\Psi_{(t)}\right\rangle$ is the final state described by the Schrödinger equation $i(d\left|\Psi_{(t)}\right\rangle/dt)=H\left|\Psi_{(t)}\right\rangle$, where $H$ is the full Hamiltonian governed by Eq. (1).

Fig. 3, we plot the dependences of the fidelity of three-NV centers W state versus the scaled parameters of the laser pulses, i.e, the scaled peak $\Omega_{m}/g$, the scaled time delay $gt_{k}(k=0,1)$ and the scaled width $g{t_{p}}$. As we can see from the subfigures, when the parameters are set in a significant range, the fidelity reaches to unit one. The Fig. 3(a) shows that small $\Omega_{m}$ will reduce the fidelity seriously. Increasing the Rabi frequency $\Omega_{m}$ will get a raise in the effective coupling strength of the Raman transition induced by the classical field and the cavity mode, and the effective Hamiltonian in Eq. (2) takes effect. The excited state can be eliminated, so the decoherence is reduced. We can see from the Fig. 3(b) that the fidelity decreases rapidly when $t_{0}-t_{1}$ is too large or too small. Obviously, when $t_{1}$ is close to $t_{0}$ or deviate much from $t_{0}$, the condition $\Omega(t)/\Omega_{0}t=1$ can not be satisfied at the end of evolution. In Fig. 3(c), we can find when $40<gt_{p}<60$, the fidelity is larger than 0.98. It is because the large deviation can not ensure the Eq. (7) and Eq. (8) to take effect. Hence, a high-fidelity W state can be obtained as long as the experiment parameters are set in the correspond range: $\Omega_{m}=g$, $gt_{0}=150$, $gt_{1}=90$ and $gt_{p}=50$.

To satisfy the condition of adiabatic passage $\nu\gg\Omega g/\Delta$, we should choose the coupling strength $\nu$ and the detuning $\Delta$ to be large enough. The Fig. 4 and Fig. 5 show that when $\nu/g>1$ and $\Delta/g>1$, the fidelity of W state can reach 0.99. Considering the experimental feasibility, we choose $\nu=10g$ and $\Delta=10g$. Moreover, one observes from Fig. 6 that within pulse duration, the fidelity increases with the time going on and approximates unit 1 when $gt>150$. For the reason that the target state approaches a steady state at least, the interaction time need not to be controlled strictly, the same as the parameters. We choose $T/g=200$ as the evolution time with $\nu=10g$, $\Delta=10g$, $\Omega_{m}=g$, $gt_{0}=150$, $gt_{1}=90$ and $gt_{p}=50$. The numerical result shows a good agreement with the expected result.

Considering a realistic experiment,we must pay attention the effect of the spontaneous emission of the NV centers, fiber losses and cavity losses. The master equation of the whole system can be expressed by

[TABLE]

where $\kappa_{c}$ and $\kappa_{f}$ denote the effective decay rate of the cavity and the optical fiber. For simplicity, we assume $\gamma_{k}(k=0,1,2)=\gamma$, where $\gamma$ represents the branching ration of the spontaneous decay from level $\left|e\right\rangle$ to $\left|g\right\rangle$ and $\left|f\right\rangle$ in NV center. By solving the master equation numerically, we obtain the relation of the fidelity of quantum phase-covariant cloning using three NV centers W states versus the scaled ratio $\gamma/g$ and $\kappa/g$ with $\Delta=10g$, $\nu=10g$, $\Omega_{m}=g$, $gt_{p}=50$, $gt_{0}=150$, $gt_{1}=90$, $T=200/g$ in Fig. 7. We see that the fidelity is always larger than 0.955 even $\gamma/g$ and $\kappa/g$ is up to 0.01.

We now want to study the performance of our protocol for quantum optical devices, such as microsphere cavity-fiber coupling system Park2006 . We can adjust the energy levels of different N-V centers using an external magnetic field to get identical N-V centersTogan2010 . When we put the NV centers near the equator of a microsphere cavity, where the NV center interacts with the cavity via the evanescent fields, we can get the coupling constants, which range from hundreds of MHz to several GHz Barclay2009 . In their work, they got a $Q$ factor of the microsphere cavity exceeded $2\times 10^{6}$, which result in a photon leakage rate $\kappa=\omega/Q\sim 2\pi\times 120$ MHz Maze2008 . The spontaneous decay rate of the NV center has been reported is $\gamma\sim 2\pi\times 15$ MHz Santori2006 . Estimating a fiber-nanocavity system with the relevant cavity QED parameters $[g,\gamma,\kappa,\Omega,\Delta,\nu]/2\pi=[1,0.01,0.01,1,10,10]$ GHz, we can get the corresponding fidelity of the W state that can reach $95.5\%$. The operation time of the phase-covariant scheme is about $50ns$ with the parameters above. What’s more, the decoherence time of individual NV centers is longer than $600\mu s$ at room temperature Mizuochi2009 . During the decoherence time of NV center, we can complete the process of phase-covariant cloning. In principle, the performance of this scheme can be improved even further, by using shortcuts to adiabatic passage Chen2010 , which has been used in entangled state preparation and transition Ye-HongLPL2014 .

V conclusion

In summary, based on adiabatic passage, we have proposed a scheme for generating the $N$ NV-centers entangled state in a hybrid system consisting of NV centers, optical fibers, and micro-cavities. What’s more, a scheme of phase-covariant cloning has been realized. By numerical calculation, we have demonstrated that the present scheme is immune to the excited levels and cavity (fiber) photons.

VI acknowledge

This work is supported by the National Fundamental Research Program People’s Republic of China under Grant No. 11405031, and the Research Program of Fujian Education Department under Grant NO. JA14044, and the Research Program of Fuzhou University.