# High-fidelity quantum memory using nitrogen-vacancy center ensemble for hybrid quantum computation

## Abstract

We study a hybrid quantum computing system using nitrogen-vacancy center ensemble (NVE) as quantum memory, current-biased Josephson junction (CBJJ) superconducting qubit fabricated in a transmission line resonator (TLR) as quantum computing processor and the microwave photons in TLR as quantum data bus. The storage process is seriously treated by considering all kinds of decoherence mechanisms. Such a hybrid quantum device can also be used to create multi-qubit W states of NVEs through a common CBJJ. The experimental feasibility and challenge are justified using currently available technology.

###### pacs:

03.67.Bg, 76.30.Mi, 42.50.PqWith experimental progress in fabrication of solid-state systems and in manipulation of qubits in atomic, molecular and optical systems, a lot of efforts have been given to explore the possibility of building hybrid quantum devices (1). Combining the advantages of component systems in a compatible experimental setup, hybrid quantum device may allow us to focus on the quantum coherent interface between different kinds of qubits. Examples include storage of quantum information from fragile qubits, e.g., superconducting qubit (2), to long-lived quantum memories such as ultracold Rb atomic ensemble (3) and polar molecular ensemble (4); (5). To achieve appreciable coupling strengths, these systems often resort to large electric-dipole interactions (3); (4), e.g., employing optically excited Rydberg states (3).

However, magnetic-dipole interactions may be more desirable due to sufficiently long coherence times achieved in systems with spin states storing quantum information (6). Besides, electronic spin degrees of freedom may provide excellent quantum memory owing to their weak magnetic interaction with the environment. For example, the nitrogen-vacancy (NV) centers in diamond have significant electronic spin lifetime, narrow-band optical transitions, as well as the possibility of coherent manipulation at room temperature (7); (8). Several recent highlights include two parallel experiments demonstrating strong magnetic couplings ( 10 MHz) of a transmission line resonator (TLR) to an ensemble of NV centers (9), and to substitution (P1) centers (10), where the strong spin-field coupling benefits from the enhancement of the coupling strength by a factor for an ensemble consisting of qubits. It implies the possibility of using electronic spin ensemble to construct a good quantum memory for superconducting qubits.

In this work, we study a hybrid quantum device which uses nitrogen-vacancy center ensemble (NVE) as a spin-based quantum memory and uses current-biased Josephson junction (CBJJ) phase qubit (11); (12); (13); (14); (15) fabricated in a TLR (16) as a quantum computing processor, which implements rapid quantum logic gates. The spin qubits are coupled to quantized modes of TLR through an effectively enhanced magnetic-dipole interaction, which enables high-quality storage of arbitrary quantum states of a CBJJ. Considering decoherence in experimentally available systems, we show the feasibility to achieve a high-fidelity quantum memory with either resonant interaction (RI) or dispersive interaction (DI) by modulating the external parameters of the CBJJ to independently tune both the CBJJ’s transition frequency and CBJJ-TLR coupling strength. We also discuss a potential application of such a hybrid quantum device for preparing multi-qubit W states of NVEs.

As illustrated in Fig. 1(a), the hybrid quantum device we study is the CBJJ-TLR-NVE system governed by . The employed CBJJ is designed using Josephson junctions to make low loss nonlinear oscillator so that one of the key elements of the CBJJ, i.e., sufficiently large anharmonicity, could be obtained for preventing qubit operations from exciting other transitions in these energy levels. So the Hamiltonian of CBJJ can be simply modeled as , where is the Pauli spin operator, and is the lowest energy-level spacing of the CBJJ. With the energy of to be energy zero point, the energy-level spacings can be expressed as and with the plasma oscillation frequency at the bottom of the well (12), the bias current , the critical current , the junction capacitance , and the flux quantum .

The Hamiltonian of the microwave-driven TLR (with length , inductance , capacitance , wiring capacitors , and coupling capacitors ) is where is the annihilation (creation) operator of the full-wave mode, and the frequency of this mode is slightly renormalized by the wiring and coupling capacitor as with and . The voltage and current distribute inside the TLR as

(1) |

where , and the small phase meets the condition . The TLR can be capacitively coupled to the CBJJ by the coupling capacitors and junction capacitance . This resonator acts as the channel to control, couple, and read out the states of the qubit (17). In this circuit quantum electrodynamic architecture, the CBJJ-TLR interaction Hamiltonian can be written as with () the CBJJ’s rising (lowering) operator. is a tunable qubit-resonator coupling strength (14) due to the flexibility of CBJJ-qubits, allowing access to a wide range of tunable experimental parameters for and . A similar system was presented in Ref. (15) for the realization of entanglement between two distant NVEs.

The Hamiltonian of a NVE reads , where is the collective spin operator for the NVE with , and . The operator can create symmetric Dicke excitation states , among which the NVE could be defined in the lowest two states and . All the spins in NVE interact symmetrically with a single mode of electromagnetic field because the mode wavelength is larger than the spatial dimension of the NVE if the spin ensemble is placed near the TLR’s field antinode. So the NVE can be coupled to the TLR by magnetic-dipole coupling with the corresponding Hamiltonian , which is a Jaynes-Cummings-type interaction with , and being the single NV vacuum Rabi frequency.

In the following, we investigate how to realize the NVE-based quantum memory using RI or DI. The Hamiltonian for this hybrid system is given in units of by,

(2) | |||||

In the RI case, as shown in Fig. 2(a), we should tune the frequency of the CBJJ to be resonant with the TLR and NVE, namely, , and for simplicity we assume . In the frame rotating with the TLR frequency , Eq. (2) becomes , by which an arbitrary state of CBJJ is first converted into a superposition state of the microwave photon in the TLR, and then transferred to the NVE as with and the normalized complex numbers. Once the single microwave photon is absorbed by the NVE, our goal is achieved, where the total operation time is with natural numbers.

In the DI case, as shown in Fig. 2(b), the frequency of TLR should be changed to be detuned from the zero-field splitting of the NV centers by . Additionally, using fast control of bias current and junction capacitance of the CBJJ, we may tune the frequency from being resonant with TLR to a large-detuning case . As a result, both CBJJ and NVE can dispersively interact with the resonator. In this limit, using the Fröhlich’s transformation (18), (Eq. (2)) is rewritten as

(3) | |||||

where is the effective CBJJ-NVE coupling rate. If the TLR is initially prepared in the vacuum state then the Hamiltonian becomes

(4) | |||||

Assuming and , we have the effective Hamiltonian in the interaction picture as , which implies that the photon-assisted transitions cannot happen in practice, but the CBJJ-NVE interaction is effectively induced. Different from the RI case, the DI case is usually mediated by the exchange of virtual photons rather than real photons, which could effectively avoid the TLR-induced loss. Therefore the DI method does not require exact time control for the coupling of the photon qubits to the CBJJ and to the NVE. In this way, we can also achieve our goal of NVE-based quantum memory, i.e., , where the degree of freedom of the TLR mode has been eliminated, and the total operation time is .

Taking account of the decoherence effects, we simulate the dynamics of the transfer process by integrating the full phenomenological quantum master equation

(5) |

where , and is the TLR decay rate. and are the spontaneous emission rate and quantum tunneling rate of the level of the CBJJ, respectively, and is the pure dephasing rate of the CBJJ. We characterize the transfer process for some given initial states of the CBJJ by the conditional fidelity of the quantum state according to the expression , where is the target state to be stored in the NVE. In Fig. 2(c), we plot the fidelity as a function of the TLR decay rate and CBJJ decay rate in the RI case. One can find that high fidelity could be achieved in the weak decoherence case, and the influence from CBJJ seems more serious than that from TLR. In the DI case, the TLR-induced loss could be effectively suppressed due to the large-detuning. As shown in Fig. 2(d), we can also obtain high fidelity if the CBJJ decay rate is within the region .

In the RI case, we assume and introduce the parameter . In real experiments, is inevitably non-zero, which leads to errors. But this imperfection is nearly negligible (See Fig. 3(a)) if is within the region . Moreover, the state transmission process accelerates for , but decelerates in the case of . In the DI case, the fidelity is plotted in Fig. 3(b) against the operating time for different initial states of the CBJJ, where the nearly perfect state transfer with the fidelity close to 0.97 is available if the parameters are chosen as above, and the fidelity maximum is nearly unchanged with different initial states of the CBJJ. Noted that the decoherence effects resulted from the imperfect spin polarization of the NVE could also reduce the fidelity. Nevertheless, experimental progress has demonstrated the possibility to suppress the spin depolarization by using suffciently long optical pulse (19).

We survey the relevant experimental parameters. The two methods above require different conditions in implementation. For example, in the RI case, the TLR with the inductance nH and the capacitance pF leads to a full wave frequency GHz. The CBJJ’s parameters should be tuned to pF, fF, A, and , which make and yield the CBJJ-TLR coupling rate MHz. In the DI case, we may choose GHz and the detuning from the CBJJ and NVE by MHz. As a result, in order to obtain a considerable CBJJ-NVE coupling rate MHz in the case of MHz, we have to tune the CBJJ’s parameters to pF, A, and . The dissipation parameters , and are on the order of hundreds of kHz (11); (12); (20). Therefore, both and are two orders higher than the dissipation rates, which makes reliable quantum memory feasible. Finally, let us go back to the two-level approximation we have made for the CBJJ-qubit. The leakage probability of quantum state population from the subspace to can be estimated as with the level separation . The small value of ensures that the two-level approximation in our scheme is reasonable.

One of the favorable applications of our scheme is to entangle many NVEs by coupling to a common CBJJ, which actually acts as a tunable coupler to mediate the virtual excitation of photons. As shown in Fig. 4(a), the effective Hamiltonian of this system in the DI regime reads , where , and is the number of NVEs. For simplicity, we assume that each NVE is equally coupled to the CBJJ, i.e., . So the state evolution under the Hamiltonian is given by . By choosing = , with the readout result of the CBJJ state to be , we obtain the state of NVEs (21) if the system is initially prepared in the state by optically pumping the electron spins in NVEs. The readout of the phase qubit can be accomplished using single-shot measurements (22). Figs. 4(b) and 4(c) show the fidelity of the state under the consideration of decoherence by the quantum master equation .

In summary, we have put forward a realization of a spin-based quantum memory for CBJJ with currently available techniques. Modulating the external parameters of the CBJJ, we have achieved the quantum information transfer between CBJJ and NVE using RI method or DI method under realistic situation. We argue that our proposal has immediate practical applications in quantum memory and hybrid quantum device with currently available techniques.

This work is supported by NNSF of China under Grants No. 10974225, No. 11004226 and No. 11074070.

### References

- M. Wallquist et al., Phys. Scr. T137, 014001 (2009); A. Imamoğlu, Phys. Rev. Lett. 102, 083602 (2009).
- Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001); A. Wallraff et al., Nature (London) 431, 162 (2004).
- D. Petrosyan et al., Phys. Rev. A 79, 040304(R) (2009).
- K. Tordrup and K. Mølmer, Phys. Rev. A 77, 020301 (2008).
- P. Rabl et al., Phys. Rev. Lett. 97, 033003 (2006); P. Rabl and P. Zoller, Phys. Rev. A 76, 042308 (2007).
- D. Marcos et al., Phys. Rev. Lett. 105, 210501 (2010).
- F. Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004); L. Childress et al., Science 314, 281 (2006); T. Togan et al., Nature (London) 466, 730 (2010); P. Neumman et al., Nat. Phys. 6, 249 (2010).
- F. Shi et al., Phys. Rev. Lett. 105, 040504 (2010); W. L. Yang et al., Appl. Phys. Lett. 96, 241113 (2010); W. L. Yang et al., New. J. Phys. 12, 113039 (2010).
- Y. Kubo et al., Phys. Rev. Lett. 105, 140502 (2010).
- D. I. Schuster et al., Phys. Rev. Lett. 105, 140501 (2010).
- Y. Yu et al., Science 296, 889 (2002).
- J. M. Martinis et al., Phys. Rev. Lett. 89, 117901 (2002).
- A. Blais et al., Phys. Rev. Lett. 90, 127901 (2003); A. M. Zagoskin et al., Phys. Rev. Lett. 97, 077001 (2006).
- Y. Hu et al., Phys. Rev. A 75, 012314 (2007).
- W. L. Yang et al., Phys. Rev. A 83, 022302 (2011).
- A. Blais et al., Phys. Rev. A 69, 062320 (2004).
- J. M. Gambetta, A. A. Houck, and A. Blais, Phys. Rev. Lett. 106, 030502 (2011).
- H. Fröhlich, Phys. Rev. 79, 845 (1950); H. R. Zhang et al., Phys. Rev. A 80, 062308 (2009); Q. Ai et al., Phys. Rev. A 78, 022327 (2008).
- L. Jiang, et al., Science 326, 267 (2009); P. Neumann, et al., Science 329, 542 (2010).
- L. Du et al., New. J. Phys. 12, 063015 (2010); L. Frunzio et al., IEEE Trans. Appl. Supercond. 15, 860 (2005).
- Z. J. Deng, M. Feng, and K. L. Gao, Phys. Rev. A 73, 014302 (2006).
- M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006).