Long-distance quantum information transfer with strong coupling hybrid solid system

In this paper, we demonstrate how information can be transferred among the long-distance memory units in a hybrid solid architecture, which consists the nitrogen-vacancy (NV) ensemble acting as the memory unit, the LC circuit acting as the transmitter (receiver), and the flux qubit acting as the interface. Numerical simulation demonstrates that the high-fidelity quantum information transfer between memory unit and transmitter (receiver) can be implemented, and this process is robust to both the LC circuit decay and NV ensemble spontaneous emission.

Scientific RepoRts | 5:17025 | DOI: 10.1038/srep17025 Motivated by the recent papers [23][24][25][26][27][28] , here, we elaborate a different proposal to realize QIT and LDQC with simple physical set-ups. As shown in Fig. 1, Alice and Bob have a same device, respectively, which consists a NV ensemble, a flux qubit, and a LC circuit. The NV ensemble acts as the information memory unit, the flux qubit acts as the interface, and the LC circuit is a transmitter (receiver) of information. In the large detuning regime, the degrees of freedom of the flux qubit can be eliminated, and we obtain the effective coupling between the NV ensemble and the LC circuit. And the entanglement of the two subsystems is induced by a flux qubit. Initially, the information is stored in the memory unit of Alice. Then, the information is transferred to transmitter by means of evolution of the system. Through the antenna radiation of the LC circuit, the information is transferred in free-space. At distant sites, the information is received by the receiver of Bob, then stored in the memory unit. So, the LDQC between two spatially-distant memory units has been achieved.

Results
System and Model. The model as shown in Fig. 1. The flux qubit can be described as a two-level system 29,30 , the Hamiltonian is (setting ħ = 1) is the energy spacing of the two classical current states, I p is persistent current of the flux qubit, Φ 0 = h/2e is the magnetic-flux quantum, Φ = Φ α /2 + Φ β is the external magnetic flux applied in the qubit; Δ is the energy gap between the two states at the degeneracy point; Pauli x are defined in terms of the classical current where = a |↻⟩ and = b |↺⟩ denote the states with clockwise and counterclockwise currents in the loop. After transformation to the eigenbasis of the flux qubit, the Hamiltonian can be rewritten as , where Ω = D − g e μ B B z is the energy gap between the ground state sublevels 0 and 1 with the magnetic field B z , and g e and μ B are the Lande factor and the Bohr magneton, respectively.
The NV ensemble couples to the flux qubit via the magnetic field created. The Hamiltonian for flux qubit coupled to a NV ensemble can be represented by J(S + + S − )σ z with the coupling strength

Alice
antenna Bob , here J k is the coupling strength between the flux qubit and NV centers. After a trivial change of basis on the flux qubit and we make a rotating wave approximation, the direct interaction Hamiltonian of the flux qubit and the NV ensemble is J(S + σ − + S − σ + ) 23 .
The LC circuit is described by a simple harmonic oscillator Hamiltonian ωa † a with resonance frequency ω = / LC 1 , where a † and a are the plasmon creation and annihilation operators, respectively. In addition, since the interaction between a flux qubit and an LC circuit via the mutual inductance M has been experimentally realized 34 , the physical features have been widely studied both in theory 35 and in experiments 36,37 . The interaction Hamiltonian is g′ (a † + a)σ z with coupling strength At the eigenbasis of the flux qubit, neglecting the small diagonal terms, the interaction Hamiltonian can be written g(aσ + + a † σ − ) with effective coupling constant ε = ′∆/ + ∆ g g 2 2 under the rotating-wave approximation and the condition that Δ > ε is satisfied.
According to the above mentions, in the Schrödinger picture, the total Hamiltonian of a single device can be written as For convenience, the Hamiltonian of the Eq. (1) can be divided into two parts: the free term are satisfied (i.e., in the large detuning regime), the effective Hamiltonian of the Eq. (1) is obtained by Fröhlich-Nakajima transformation 38,39 . The expression of the effective Hamiltonian is is an anti-Hermitian operator, which satisfies . Obviously, the Eq. (4) and (5) represent the entangled states between the LC circuit and the NV ensemble. If the information is stored in NV ensemble, we can read out it by measuring quantum states of the LC circuit.
In the interaction picture, the Hamiltonian (3)  can be emitted by the antennary radiation. At the distant receiving terminal, the information is received by another LC circuit (receiver), then stored in NV ensemble (memory unit), that is, 0 . In other word, we realize a LDQC between Alice and Bob without using data bus (fibers, transmission line resonator, or nanomechanical resonator). Moreover, Alice can act as a base station, and Bob can act as a user. We can realize the quantum communication from one base station to many users. The channel of our scheme is electromagnetic wave, which has been widely used in the field of communications. We now discuss the dominant noise of the channel due to microwave photons loss. When the electromagnetic wave transmits in free-space, the signal power of the receiver can be written as where P t is the transmitted power, G t indicates the gain of antenna-transmitter, G r expresses the gain of antenna-receiver, d is the distance between the transmitter and the receiver, and λ′ is the wavelength. In order to avoid the loss of the channel, we should shorten the distance d between the transmitter and the receiver, or increase the wavelength λ′ .

Discussion
For a really physical system, we should take account of decoherence effects. As Alice and Bob have the same dissipation mechanism, here, we only discuss the decoherence effects of Alice. The flux qubit worked in large detuning regime and prepared in the ground state. The decoherence effective of the flux qubit is omitted. Thus, we only consider the decay of LC circuit, the dephasing and relaxation of NV ensemble. Following the standard quantum theory of the damping, the Markovian master equation is where the Lindblad term  ρ ( ) presents the decay of the LC circuit and the decoherence of the NV ensemble, and the detailed expression is  ρ ρ ρ ρ with the decay rate κ of the LC circuit, and the dephasing rate γ ϕ and the relaxation rate γ of the NV ensemble. Fidelity is a direct measure to characterize how accurate the information transfer from NV ensemble to LC circuit, and its expression is 1 is a target state to be stored in the LC circuit. In Fig. 2, we plot the fidelity F as a function of the dimensionless time λt with the different decay rate κ, the dephasing rate γ ϕ and the relaxation rate γ. This figure shows that the high-fidelity QIT could be achieved in the weak decoherence case.
The experiment 24 has reported that the coupling strength between flux qubit and NV ensemble is J = 70 MHz, the Lande factor is g e = 2, the Bohr magneton is μ B = 14 MHz/mT, and the magnetic field is B z = 2.6 mT. Besides, the strong coupling of a LC circuit and a flux qubit has been implemented 37 . In this experiment 37 , the coupling strength between the flux qubit and the LC circuit is g = 119 MHz, the frequency of the LC circuit is ω = 2.723 GHz, and the decay rate of the LC circuit is κ = 0.45 MHz. Through adjusting the frequency ω q of the flux qubit, the large-detuning between the flux qubit and the LC circuit (NV ensemble) can be well satisfied. Also, the resonant condition Ω ′ = ω′ is satisfied at the proper frequency ω q , see in Fig. 3. According to the above value of parameters, we can estimate the effective coupling strength between NV ensemble and LC circuit λ ~ 10 MHz. Thus, the strong coupling between NV ensemble and LC circuit is realized. So, we can estimate the time t p ~ 0.16 μs, which is shorter than the decoherence time of the NV ensemble approaching 1 s 40 and the flux qubits coherence time T 2 ≃ 20 μs 41 .
In summary, we have proposed a hybrid solid architecture, which can realize the strong coupling between a NV ensemble and a LC circuit by a flux qubit. We have also shown the high-fidelity quantum information transfer between the NV ensemble and the LC circuit. In addition, the LDQC can be implemented using this architecture by the antenna radiation. The proposed architecture opens a way for quantum communication from one base station to many users.