Magnetic ordering of nitrogen-vacancy centers in diamond via resonator-mediated coupling
- Bo-Bo Wei^{1, 2},
- Christian Burk^{3},
- Jörg Wrachtrup^{3} and
- Ren-Bao Liu^{1, 2, 4, 5}Email author
Received: 6 March 2015
Accepted: 9 July 2015
Published: 21 July 2015
Abstract
Nitrogen-vacancy centers in diamond, being a promising candidate for quantum information processing, may also be an ideal platform for simulating many-body physics. However, it is difficult to realize interactions between nitrogen-vacancy centers strong enough to form a macroscopically ordered phase under realistic temperatures. Here we propose a scheme to realize long-range ferromagnetic Ising interactions between distant nitrogen-vacancy centers by using a mechanical resonator as a medium. Since the critical temperature in the long-range Ising model is proportional to the number of spins, a ferromagnetic order can be formed at a temperature of tens of millikelvin for a sample with ∼10^{4} nitrogen-vacancy centers. This method may provide a new platform for studying many-body physics using qubit systems.
Keywords
1 Introduction
The negatively charged nitrogen-vacancy (NV) centers in high-purity diamond have been considered as a promising candidate for solid state quantum information processing due to their long coherence time [1–3] and high feasibility in initialization, control, and readout of their spin states [4]. Simulation of many-body physics using NV centers, in analogue to cold atom physics [5, 6], has been proposed [7–10]. To realize phase transitions in NV center qubit systems under realistic temperatures, however, sufficiently strong interactions between NV centers located sufficiently close (<30 nm) is till highly challenging [7, 8, 11]. A new opportunity is to use resonators as mediators [7, 12–15], which has potential of coupling NV centers at distance.
In this paper, we propose to realize long-range coupling between many separated NV centers via a mechanical resonator. A remarkable feature of the long-range interacting system is: the critical temperature for ferromagnetic phase transitions is proportional to the number of spins, so a ferromagnetic order could be formed at a temperature of tens of millikelvin (mK) for a sample with ∼10^{4} NV centers, even though the mediated coupling between two NV centers is less than 200 kHz (∼10 μK).
2 Microscopic model
We make use of the two Zeeman states of the electron spin \(|1\rangle\) and \(|0\rangle\) and define a pseudo-spin by \(\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle0|\) and the corresponding spin flip operators \(\sigma^{+} =|1\rangle\langle0|\) and \(\sigma^{-}=|0\rangle\langle1|\). In each NV center there is a ^{14}N nuclear spin which interacts with the on-site NV center spin. The Hamiltonian of the NV center near the degenerate point (\(\delta\approx0\)) is \(H_{\mathrm{NV}}=\sum_{j}[\Delta _{N}(I_{j}^{z})^{2}-\gamma_{N}I_{j}^{z} B_{\mathrm{NV}}+A I_{j}\cdot\sigma_{j}+\delta\sigma _{j}^{z}]\), where \(\Delta_{N}=5.1\mbox{ MHz}\) is the ^{14}N nuclear spin quadrupole splitting [18], \(\gamma_{N}\) is the gyromagnetic ratio of the nitrogen nuclear spin and \(A\approx2\mbox{ MHz}\) is the hyperfine coupling between the electronic spin and the ^{14}N nuclear spin [18].
The motion of the mechanical resonator is described by the Hamiltonian \(H_{r}=\omega_{r}b^{\dagger}b\), with \(\omega_{r}\) as the frequency of the fundamental vibration mode of the resonator and b as the corresponding annihilation operator. For example a silicon nitride string resonator has dimensions \((325\times0.35\times0.1 )\mbox{ $\upmu$m}\) with \(\omega_{r}=2\pi\times1.0\mbox{ MHz}\) and \(Q =1.3\times10^{6}\) at room temperature [19].
The magnetic field felt by the electronic spin of the NV centers can be approximated by a magnetic dipole [20], \(\vec {B}(d_{j}-x(t))=\vec{B}(d_{j})-G_{m}\vec{x}(t)+o(x^{2})\), where \(d_{j}\) is the distance between the equilibrium position of the resonator and jth NV center, \(G_{m}\) is the magnetic field gradient at the position of the NV centers and x is the amplitude of mechanical resonator oscillation (\(x\sim10^{-12}\mbox{ m}\) at temperature of mK). This magnetic field will induce a Zeeman shift to the NV spins with Hamiltonian \(H_{z}=\sum_{j}\eta_{j}\sigma_{j}^{x}(b+b^{\dagger})\) and \(\eta_{j}=\gamma _{e}|G_{m}|a_{0}\), where \(\gamma_{e}\) is the gyromagnetic ratio for electron and \(a_{0}=\sqrt{\hbar/2m\omega_{r}}\) is the amplitude of zero point fluctuations for a resonator of mass m.
Due to the distance distribution of the NV centers to the magnet, the coupling of the NV centers to the mechanical resonator have a distribution. A magnetic tip with size of 100 nm produces a magnetic gradient \(G_{m}\sim7.8\times10^{6}\mbox{ T/m}\) at a distance 25 nm away from the tip [21]. In such case \(\eta/2\pi\) could reach 200 kHz. For a sample with typical distance of NV centers ∼20 nm, the neighbor NV centers have a direct dipolar interaction with strength ∼5 kHz (∼0.25 μK), much less than the critical temperature to be discussed later. For the sake of simplicity, in the following we assume the coupling between the mechanical resonator and NV centers to be uniform and neglect the direct dipolar coupling between NV centers. Without these assumptions, however, the results in this paper would only be quantitatively affected. The coupling constant \(\eta/2\pi\sim200\mbox{ kHz}\) considerably exceeds both the electronic spin decoherence rate (∼kHz) of the NV centers and the intrinsic damping rate of the mechanical resonator, \(\gamma=\omega_{r}/Q\) [19]. We neglect the interaction between the mechanical resonator and the ^{14}N nuclear spins since it is three orders of magnitude smaller than that between the NV center and the mechanical resonator.
3 Effective long-range Ising model
4 Ferromagnetic ordering
5 Conclusion
In summary, we propose a scheme to realize ferromagnetic ordering of distant nitrogen-vacancy centers by using a mechanical resonator to mediate long-range Ising-type interaction. The critical temperature for the ferromagnetic phase transition in the long-range Ising model is proportional to the number of spins, so the ferromagnetic order could be formed at the temperature of tens of millikelvin for a sample with ten thousand nitrogen-vacancy centers. In addition, it may also be possible to use a superconducting resonator [24, 25] as a medium to realize the long-range ferromagnetic coupling between the NV centers. Since the interactions between the NV centers mediated by a superconducting resonator is usually small compared to that by a mechanical resonator, high density NV samples are required to observe the magnetic ordering in the NV centers.
Declarations
Acknowledgements
We are grateful to P Bertet for useful discussions. This work was supported by National Basic Research Program of China Grant 2014CB921402, Hong Kong RGC/GRF 401413, Hong Kong RGC/CRF CUHK4/CRF/12G, and CUHK VC’s One-Off Discretionary Fund.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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