
- Photonics Research
- Vol. 10, Issue 7, 1640 (2022)
Abstract
1. INTRODUCTION
Implementing a controllable and strong enough coupling to a quantum unit at a single-quanta level is a very desirable basic goal in quantum information processing (QIP) [1–8]. This type of strong coupling can first ensure complete and fast control of the qubits directly or indirectly at the single-quanta level [9,10], which also underlies applications of quantum simulation [11,12], manipulation [13,14], and metrology [15]. Second, such interactions can also be applied to explore many interesting and essential physics [16], such as single photon or phonon technology [17–19] and chiral quantum science [20–22].
Working as a point defect in diamond, the nitrogen-vacancy (NV) center integrated in a hybrid quantum system has recently emerged as one of the leading candidates for QIP. It is desirable thanks to its excellent spin properties [23–26], such as solid-state spins with atom-like properties and without an additional trap device [27,28], precise implantation and easy scalability [29,30], and longer coherence times even at ambient conditions [24,25,31], in addition to the convenient preparation, manipulation, and readout of its quantum state [32,33]. Significant theoretical and experimental investigations have been carried out using NV spins in hybrid systems to realize quantum simulation and quantum state manipulating [34–41]. In recent years, more and more attention has been devoted to the application of NV centers in the quantum acoustics area, which also leads to a growing interest in studying and exploiting coherent spin–phonon coupling [2,42,43]. However, it is still a huge challenge to significantly enhance the spin–phonon coupling at a single-quanta level by the means currently available [32].
In this work, we present a combined scheme to enhance the spin–phonon coupling in a hybrid setup, which consists of a single NV spin and three optical cavities dispersively coupled with three mechanical resonators (MRs) [44]. To further enhance the spin–phonon coupling in this spin–cavity–resonator tripartite system, there are two key points in our proposal. First, we can modify the spatial distribution of the electric field
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2. MFC AND MPA
A. MFC
In a traditional cavity quantum electrodynamics (C-QED) system, the realistic dipole coupling to a single quantum emitter (QE) can be written as
We stress that this type of MFC coupling mechanism may be a general proposal, and we can demonstrate it in the traditional optomechanical system and also can introduce it to other optomechanics-like systems; the photon–magnon (lattice) system and the photon–phonon (lattice) system, for example.
B. MPA
Parametric amplification [including optical parametric amplification (OPA) and MPA] has been another hot topic related to the enhancement of coherent coupling since it was first proposed in an optomechanical system. In particular, a milestone achievement on this topic has also been reported and demonstrated in a hybrid ion trap system. This type of progress reinforces our confidence in the feasibility of the OPA or MPA, and we believe that in this work MPA will be a more feasible choice to further enhance the coherent coupling to a single QE. To realize MPA, the core point lies in the modulation of the time-dependent spring constant [32,50]; i.e.,
In short, we believe that the MFC and MPA are both general physical ideas, and that this proposal can first show a joint cooperation of the MFC and MPA via a hybrid design to further enhance the EQ–phonon coupling. To induce this joint enhancement effect of the spin–phonon coupling to a feasible hybrid quantum system, we believe that optomechanics [44,58] or an optomechanics-like system will be a suitable choice; for example, the lattice of a photon–phonon coupling system or a hybrid optics–acoustics system [35,59–67]. Here, let’s consider the traditional optomechanical system and then discuss this joint scheme in detail.
3. POTENTIAL SCHEME FOR THE JOINT MFC AND MPA
We have designed this hybrid setup, as illustrated in Fig. 1(a), with three identical optical cavities with frequency
Figure 1.Schematic of our hybrid system. (a) Three identical optical cavities with frequency
In addition, as illustrated in Fig. 1(a), a single NV center is placed inside the central cavity. In the optical frequency domain, the optical mode
Therefore, according to the Appendices A–C, we can get the total Hamiltonian to describe this hybrid system
In Eq. (1), under the rotating frame with frequency
According to Appendix D, we can simplify the total Hamiltonian for this system, and obtain an effective Hamiltonian with the tripartite interactions (spin–photon–phonon):
In view of this Hamiltonian in the squeezed frame, its effective triple coupling strength is strengthened with the rate
4. ENHANCING THE SPIN–PHONON COUPLING
We consider that the cavity is pumped with a large coherent field with an average photon number
Next, in the interaction picture (IP), we transfer Eq. (3) into an equivalent expression with the relations
Whether it is a J–C model or an anti J–C model, we can get an enhanced coupling strength with the effective coupling strength
To enhance the coherent coupling
In Fig. 2, we plot the spin–phonon coupling enhancement
Figure 2.(a) Spin–phonon coupling enhancement
Furthermore, in Fig. 3, we also plot the dynamical population of the phonon number operator
Figure 3.Dynamical population of the phonon number
5. ENHANCEMENT OF PHOTON–SPIN–PHONON INTERACTION
On the other hand, if
For the second case, such as the red sideband, when the resonance condition satisfies
In this tripartite interaction quantum system, the effective coupling strength is
In Fig. 4, we make the simulations on this tripartite interaction system according to Eq. (7) and Eq. (8), and then plot the dynamical population of the phonon number operator
Figure 4.(a), (b) Dynamical population of the spin operator
6. APPLICATION OF THIS PROPOSAL
A. Entangling Collective NV Spins Dynamically
In this section, the first potential application for this proposal is that we can entangle separated NV spins. We assume a certain number of NV spins are set separately in this central optical cavity. According to Eq. (3), we can obtain the effective Hamiltonian as
Here, we assume
Here, the coupling is identical
Note that because this operator has no contribution from the mechanical modes, in this instance the system gets insensitive to the states of the mechanical modes. Starting from the initial state of the mechanical mode and NV spins
Figure 5.Dynamical fidelity of the target entangled GHZ state for four NV spins, in which, the initial state is
B. Local Cooling One Supermode of Triple Resonators with an NV Ensemble
On the other hand, we stress that another potential application on this scheme is to cool down the mechanical supermode to its ground state efficiently with the NV center ensemble (NVE). Here, we assume a number of NV centers are set inside the central optical cavity, which form an NVE. Taking the Eqs. (4) and (10) into consideration, we can obtain the effective Hamiltonian in IP for this hybrid system:
Similarly, we can also ignore this weak system disorder adverse effect according to the advanced processing techniques. Then, we can rewrite this effective Hamiltonian as
Figure 6.Dynamical population of this mechanical supermode
Figure 7.Dynamical evolution of the mechanical population of the left local mode
7. EXPERIMENTAL PARAMETERS
To examine the feasibility of our scheme in a realistic experiment, we now discuss the relevant experimental parameters. We consider a high-quality optical cavity with frequency
8. CONCLUSION
In summary, we propose a protocol to further enhance the spin–phonon coupling at the single-quanta level with two methods jointly working together: the MFC and MPA. Importantly, in our scheme, we can enhance the coherent spin–phonon interaction not only by the optical field intensity with rate
Acknowledgment
Acknowledgment. Yuan Zhou thanks Peng-Bo Li for valuable discussions. Part of the simulations are coded in Python using the QuTiP library [93,94].
APPENDIX A: THE HAMILTONIAN OF MECHANICAL MODES WITH SECOND-ORDER NONLINEAR INTERACTION
The Hamiltonian for the
Here, we assume that these are three identical MRs, with an intrinsic frequency
Furthermore, in this scheme, the central mechanical mode
In this scheme, the coefficient
APPENDIX B: THE HAMILTONIAN OF SPIN–CAVITY AND CAVITY–CAVITY INTERACTIONS
As illustrated in Fig.
APPENDIX C: THE INTERACTIONS BETWEEN THE MECHANICAL MODES AND CAVITY MODES
In this scheme, the three cavity modes
APPENDIX D: THE EFFECTIVE HAMILTONIAN DERIVATION FOR THIS WHOLE SYSTEM
Considering the Hamiltonian in Eq. (
Here, the relevant coefficients are defined, respectively, as
The mechanical part of this Hamiltonian can also be diagonalized by the canonical transformation
Applying a unitary transformation with the definition
Then, we can also diagonalize the cavity-mode part in the Hamiltonian
When the spectral separation between the supermodes is much larger than the mechanical frequency (
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