Spin coherent quantum transport of electrons between defects in diamond

TL;DR

This paper proposes a method for coherent quantum transport of electron spins between NV centers in diamond using spatial stimulated adiabatic passage, enabling potential on-chip quantum communication.

Contribution

It introduces a novel approach for transporting electron spins between distant NV centers in diamond via optical control and adiabatic passage, advancing quantum bus development.

Findings

High-fidelity transport over hundreds of nanometers

Transport timescales of hundreds of nanoseconds

Potential for on-chip quantum communication

Abstract

The nitrogen-vacancy color center in diamond has rapidly emerged as an important solid-state system for quantum information processing. While individual spin registers have been used to implement small-scale diamond quantum computing, the realization of a large-scale device requires development of an on-chip quantum bus for transporting information between distant qubits. Here we propose a method for coherent quantum transport of an electron and its spin state between distant NV centers. Transport is achieved by the implementation of spatial stimulated adiabatic Raman passage through the optical control of the NV center charge states and the confined conduction states of a diamond nanostructure. Our models show that for two NV centers in a diamond nanowire, high fidelity transport can be achieved over distances of order hundreds of nanometres in timescales of order hundreds of nanoseconds. Spatial adiabatic passage is therefore a promising option for realizing an on-chip spin quantum bus.

Equations

$$|1⟩$$

$$|2⟩$$

$$|3⟩$$

$${\psi }_{n,i}(\vec{r})=F_{n,i}(\vec{r})e^{i{\vec{K}}_i\cdot \vec{r}}{u}_i(\vec{r}),$$

$$F_{n,i}=\sqrt{\frac{8V_c}{w^{2}L}}\sin \left(\frac{n_x\pi x}{w}\right)\sin \left(\frac{n_y\pi x}{w}\right)\sin \left(\frac{n_z\pi z}{L}\right),$$

$$E_n=\frac{{\hslash }^2{\pi }^2}{2}\left(\frac{n_x^2}{m_{\parallel }{w}^2}+\frac{n_y^2}{m_{\perp }{w}^2}+\frac{n_z^2}{m_{\perp }{L}^2}\right),$$

$$\Delta E_c=\frac{3{\hslash }^2{\pi }^2}{2m_{\perp }{L}^2}.$$

$$d_{\text{wire}}=F_0({\vec{R}}_A)e^{-S/2}{d}_{\text{bulk}},$$

$$F_0(s)=\sqrt{\frac{8V_{\text{sc}}}{w^{2}L}}\cos \left(\frac{\pi s}{2L}\right),$$

$$\Omega =\frac{e^{-S/2}}{r\hslash }\sqrt{\frac{8P}{n_{D}c{ϵ}_0\pi }}{F}_0(s)d_{\text{bulk}},$$

$${\hat{\mathcal{H}}}_{\text{ep}}={\Xi }_d\vec{\nabla }\cdot {\vec{u}}_f,$$

$${\Gamma }_{\text{SE}}=2\frac{{\omega }^3{d}_{\text{wire}}^2}{3\pi {ϵ}_{0}n\hslash c^3}.$$

$${\omega }_m^2={\pi }^2{c}_l^2\left(\frac{m_x^2+m_y^2}{w^2}+\frac{m_z^2}{L^2}\right),$$

$${\Gamma }_{\text{ep}}=\frac{2\pi }{{\hslash }^2}\sum _{n,m}$$

$$\times n_B({\omega }_m,T)\rho ({\omega }_n-{\omega }_m),$$

$$\rho ({\omega }_n-{\omega }_m)=\frac{\gamma /\pi }{({\omega }_n-{\omega }_m{)}^2+{\gamma }^2},$$

$${\Gamma }_{\text{ep}}=$$

$$\times {\int }_0^{\pi }{\int }_0^{2\pi }{G}_n\left(\frac{{\omega }_n}{c_l},\theta ,\varphi \right)\sin (\theta )\text{d}\theta \text{d}\varphi ,$$

Full text

Introduction

Diamond-based quantum information processing (QIP) is possible due to the remarkable properties of the nitrogen-vacancy (NV) defect center. The spin-states of individual NV centers can be optically addressed for initialization and read-out[1] and possess the longest room-temperature spin-coherence time of any solid-state defect[2]. Cryogenic temperatures further enhance these spin properties by up to four orders of magnitude[3] and enable spin-photon entanglement through coherent optical transitions[4]. This includes adiabatic control of the NV spin using optical pulses[5]. These properties have been used to operate individual spin registers for small-scale quantum computing including demonstrations of error correction[6], simple algorithms[7] and simulations[8]. Spin registers have been realized through clusters of NV centers as well as single NV centers coordinated with clusters of paramagnetic defects such as 13C atoms and substitutional nitrogen ($\text{N}_{\text{S}}$) centers[9, 10, 11, 12, 13, 14, 15].

Individual spin registers are ultimately limited by the number paramagnetic defects addressable by a single NV center (5-10 strongly coupled qubits[16]). Development of large-scale QIP therefore requires the use of quantum buses to entangle a network of interconnected clusters. At cryogenic temperatures optical quantum buses have been used to entangle spin registers separated by macroscopic distances[4, 17, 18]. Unfortunately this option is not viable for scalable QIP as optical buses cannot be compactly incorporated on chip. One scalable suggestion is intra-cluster spin transport along a chain of paramagnetic defects[19, 20]. Despite the promise of room-temperature operation, engineering the spin chain is difficult with existing fabrication techniques. Recently, Doherty et al[21] have proposed entanglement using semi-classical electron transport between two NV-${}^{14}\text{N}_{\text{S}}$ pairs embedded in a diamond nanowire. This scheme is severely limited as semi-classical transport is prone to erroneous capture from $\text{N}_{\text{S}}^{+}$ defects, surface traps[22] or surface emission[23].

Consequently, here we propose an on-chip spin quantum bus for diamond QIP using spatial stimulated adiabatic Raman passage (STIRAP). Spatial STIRAP involves the transport of massive particles between spatially distinct locations using conventional STIRAP type coupling. It was originally explored in the context of electronic transfer in double quantum dots[24, 25] and for general methods of spatial adiabatic passage which promise robust quantum information transport[26, 27]. The conventional STIRAP technique is a method of population transfer between the two lowest levels of a $\Lambda$ type atomic system[28]. By applying optically coupling fields in the so-called counter intuitive direction, where the unpopulated transition is coupled before the populated transition, the system can be maintained in the optical dark state, which suppresses population in the central (excited) state. This suppression of population in the central state, which will ideally have zero population in the adiabatic limit, leads to the suppression of the effects of spontaneous emission. In the context of NV centers, STIRAP has been proposed[29] and demonstrated[30] for single qubit control, however we are not aware of any proposals for the spatial transport of electrons between NV centers of the form considered here.

In our proposal for spatial STIRAP the two disconnected states of the $\Lambda$ scheme correspond to an electron occupying one of two distant NV centers embedded in a diamond nanostructure. As depicted in Figure 1 (A), we envision mediating adiabatic passage through states of the conduction band minimum, which are discretized by the confining potential of the nanostructure. While all nanostructures lead to discretization, we will consider nanowires as an archetypal structure. We have identified two different designs for realizing a nanowire confining potential, via the potential of a diamond surface or an electrostatic potential applied by electrodes in a bulk structure. Schematics for the transport scheme in these designs are presented in Figure 1. While STIRAP is performed similarly in both cases, the susceptibility to different decoherence mechanisms varies. We also predict that quantum transport will maintain coherence of the electron’s spin. Spin transport is known theoretically to be excellent in diamond due to an indirect band gap, low spin-orbit coupling, low electron-phonon (e-p) scattering and low background impurity spins[31]. Here, under STIRAP we explicitly exploit the low background impurities and low spin-orbit coupling of the conduction band to achieve spin coherent transport.

This proposal for quantum transport is fundamentally different from existing schemes for spatial STIRAP. As coupling is mediated optically rather than through manipulation of tunneling amplitudes[27], the spatially separated states are truly disconnected and there is no occupation of the intermediary space during transport. In this sense, it is colloquially akin to teleportation of a massive particle. Furthermore, to our knowledge this is the first spatial STIRAP proposal which includes coherent spin transport.

In Section 1 we present a proposal for implementing STIRAP in a diamond nanowire including a spin initialization and measurement protocol. These spin protocols play the essential role of identifying the transported electrons (thereby distinguishing them from background sources) and will be the means for which spin entanglement is mediated. We identify a feasibility condition for transport, $\Delta E_{c}/\hbar\gg\Omega\gg\Gamma$, where $\Delta E_{c}$ defines the energy gap between the first and second conduction states, $\Omega$ is the Rabi frequency of the optical transition between defect and conduction states, and $\Gamma$ is the optical decoherence rate. As a conservative estimate, these constraints can be considered satisfied when $\Delta E_{c}/\hbar\gtrsim 10\Omega\gtrsim 10\Gamma$. In Section 2 we calculate $\Delta E_{c}$ by modeling the effect of the nanowire confining potential on the diamond conduction band. Effective mass theory is used to derive a level scheme and determine energies of discretized conduction states. In section 3 we present the first ab-initio calculations for the photoionization mechanism of the NV center. This is used to evaluate values of $\Omega$ that can be practically achieved. We then identify optical decoherence mechanisms that contribute to $\Gamma$. Finally, in Section 4 we evaluate $\Omega$, $\Delta E_{c}$ and $\Gamma$ at cryogenic temperatures for both of the nanowire designs introduced previously. The dimensions of the wires are optimized to maximize the Rabi frequency relative to decoherence and to satisfy the STIRAP feasibility constraints.

1 Quantum transport in diamond nanowires

STIRAP is a method for evolving a three-state system from an initial state $|1\rangle$ to a final state $|3\rangle$[28, 32]. This is achieved through coupling to a mutual intermediary state, $|2\rangle$, even though the direct transition $|1\rangle\rightarrow|3\rangle$ may be forbidden. To do so, a pump laser pulse with Rabi frequency $\Omega_{P}(t)$ is tuned to the $|1\rangle\rightarrow|2\rangle$ transition while a Stokes laser pulse with Rabi frequency $\Omega_{S}(t)$ is tuned to the $|2\rangle\rightarrow|3\rangle$ transition. Applying the Stokes laser pulse before the pump laser pulse with some overlapping time $\tau$ results in an adiabatic evolution of the state from $|1\rangle\rightarrow|3\rangle$ without any population of $|2\rangle$. The requirement for adiabaticity is given by $\Omega\tau\gg 1$, where $\Omega^{2}=\Omega_{P}^{2}+\Omega_{S}^{2}$ is the effective Rabi frequency.

We propose using STIRAP for quantum transport of an electron between two NV centers. As presented in Figure 1, the NVs are embedded within the nanowire confining potential and labeled as $A$ and $B$. This allows us to define a three-level system given by

[TABLE]

where $|\text{NV}^{-}\rangle$ and $|\text{NV}^{0}\rangle$ correspond to the ${}^{3}\text{A}_{2}$ and ${}^{2}\text{E}$ electronic ground states of the negative and neutral charge states of the NV centers and $|\psi\rangle$ is a nanowire conduction band minimum state. A schematic of the STIRAP level scheme is presented in Figure 2. Adiabatic passage between states $|1\rangle$ and $|3\rangle$ therefore constitutes electron transport from NV $A$ to NV $B$ with no occupation of the intermediary space. The degeneracy of states $|1\rangle$ and $|3\rangle$ may be lifted by applying a potential difference $\Phi$ between each end of the length $L$ wire. For a defect separation distance $s$, this induces an energy splitting of $\Delta E_{d}=e\Phi s/L$.

Let us now consider transportation of the electron spin during STIRAP. This requires a protocol for spin initialization of both $|\text{NV}^{-}\rangle_{A}$ and $|\text{NV}^{0}\rangle_{B}$ which we now present in turn. Firstly, suppose we wish to transport the spin-state $|\uparrow\rangle$. We restrict our attention to photoionization by adsorption of a single photon from an optical pulse and do not consider the two-photon ionization process of the $\text{NV}^{-}$[33]. As the ionizing pulse cannot differentiate between the two electrons occupying the $e$-orbitals of $|\text{NV}^{-}\rangle_{A}$, both must be initialized into $|\uparrow\rangle$. This can be achieved through optical polarization of $|\text{NV}^{-}\rangle_{A}$ into the $m_{s}=0$ triplet state followed by application of a microwave control pulse. The resulting spin-state can be expressed in the two $e$-orbital basis as $|+1\rangle=|\uparrow\rangle|\uparrow\rangle$. Hence, through this protocol we can guarantee ionization of a spin-up electron during STIRAP.

Now consider initialization of the $|\text{NV}^{0}\rangle_{B}$ spin-state. Post STIRAP, electron transportation can only be validated if read-out of the $|\text{NV}^{-}\rangle_{B}$ spin-state agrees with the spin initialized at $|\text{NV}^{-}\rangle_{A}$. Otherwise, it is impossible to discern if charging of $|\text{NV}^{0}\rangle_{B}$ into $|\text{NV}^{-}\rangle_{B}$ was instead due to erroneous electron capture. It is therefore essential for read-out that $|\text{NV}^{0}\rangle_{B}$ be initialized into either the $|\uparrow\rangle$ or $|\downarrow\rangle$ spin eigenstates. To see how this could be achieved, consider that the $|\text{NV}^{-}\rangle_{B}$ spin-state after transport (assuming $|\text{NV}^{0}\rangle_{B}$ was initialized as $|\uparrow\rangle$) is given by $|\uparrow\rangle|\uparrow\rangle=|+1\rangle$. Similarly, we obtain the $m_{s}=0$ triplet-state when $|\text{NV}^{0}\rangle_{B}$ is initialized as $|\downarrow\rangle$. Transport can then be confirmed by performing read-out of the spin-state at NV $B$.

The spin initialization of NV $B$ can be achieved as follows. Consider NV $B$ initially prepared into the negative charge state with spin projection $m_{s}=+1$. Application of a $>2.6$ eV photoionizing pulse will then eject an electron with $|\uparrow\rangle$ spin, leaving NV $B$ in the neutral charge state with $|\uparrow\rangle$ spin. Equivalently, the $|\downarrow\rangle$ spin-state can be prepared for NV $B$ prepared in the $m_{s}=-1$ state. While not proven, it is assumed that photoionization is a spin-conserving process. Even if this is not the case, one way to ameliorate the risk of non-conserving spin transitions is to polarize the N nuclear spin when NV $B$ is in the negative charge state. After the ionizing pulse, this nuclear spin polarisation can be swapped back onto the electron spin of the neutral charge state via microwave control. Such a scheme is now feasible given recent progress has identified the $\text{NV}^{0}$ spin Hamiltonian[34].

Transportation also requires that $\text{NV}^{0}$ spin-relaxation is long with respect to the STIRAP operation time. As of yet the $\text{NV}^{0}$ ground state has not been directly observed through electron paramagnetic resonance. Whilst this would presumably imply a high rate of spin relaxation[35], new work by Barson et al indicates that this is likely not the case. Rather, the apparent absence of EPR is due to strain broadening in ensembles[34]. Furthermore, due to the electronic similarities between the $\text{NV}^{0}$ and $\text{Si-V}^{-}$ center[36], e-p scattering is expected to cause spin dephasing but not relaxation of $\text{NV}^{0}$. Consequently, at cryogenic temperatures NV $B$ should maintain its spin projection throughout transport.

During transport, spin decoherence mechanisms can be neglected. This remains true as long as the spin-quantization axis of the conduction states match those of the NV center and there is no rapid spin dephasing of these conduction states. This is investigated in supplementary section B, where we present symmetry arguments within the closure approximation. We prove that if the NV is co-axial to the wire then the spin-quantization axis is conserved throughout STIRAP. Conveniently, NV centers can be preferentially aligned with a (111) wire axis during fabrication with chemical vapor deposition[37]. For other nanowire directions, the spin-quantization axis of the conduction state can be aligned with the NV center via the application of a magnetic field. The magnitude of this field must be sufficiently large relative to the component of the spin-orbit interaction orthogonal to (111).

There are two further requirements for successful implementation of STIRAP. Firstly, coherent transport demands there be no off-resonant coupling to states other than the conduction band minimum. Hence, the energy splitting between the first and second conduction states $\Delta E_{c}$ must be significantly greater than the Rabi frequency, $\Delta E_{c}/\hbar\gg\Omega$. Secondly, the optical decoherence rate $\Gamma$ sets a bound on $\Omega$ since it induces an unknown phase on the electron wave function as the transition energy fluctuates. Provided the Rabi frequency is large ($\Omega\gg\Gamma$), this dephasing will be sufficiently small to achieve high fidelity transport. The following two sections address each of these feasibility requirements in turn. We first investigate the effects of wire confinement on the conduction band before evaluating the Rabi frequency and identifying sources of optical decoherence.

2 Diamond nanowire conduction band

The confining potential of the nanowire leads to discretization of the conduction band suitable for STIRAP. These confinement effects are well described through effective mass theory (EMT)[38], as detailed in supplementary section A. As shown in Figure 1, the nanowire was modeled as a square prism of mono-crystalline diamond with dimensions $w\times w\times L$ ($w<L$). The two NV centers are separated by a distance $s$ and located on the longitudinal. Consider that bulk diamond possesses six equivalent conduction band minima, located at the $k$-points $\vec{K}_{i}$ between the $\Gamma$ and $X$ points in each orthogonal axis of the fcc Brillouin zone. The dispersion relationship at each $\vec{K}_{i}$ is modeled using diamond’s anisotropic mass tensor and is referred to as a valley[39].

In the effective mass framework, the $n^{\text{th}}$ conduction band state for the $i^{\text{th}}$ valley of the nanowire is given by

[TABLE]

where $F_{n,i}$ is an envelope function determined by the confining potential, and $u_{i}$ is the bulk-diamond Bloch function. For simplicity we approximate the confining potential as an infinite square well. While more realistic confining potentials such as a finite well would result in perturbations to the electronic structure, the key features are well described within our approximate model. This leads to the following envelope function

[TABLE]

where $n=(n_{x},n_{y},n_{z})$ and $V_{c}$ is the volume of the diamond unit cell.

The true eigenstates of the nanowire conduction band are linear combinations of the valley wavefunctions 1 which respect its $\text{D}_{4\text{h}}$ symmetry. As presented in supplementary section A, symmetry analysis reveals four degenerate conduction band minima states formed from the valleys with $\vec{K}_{i}$ perpendicular to the wire axis (with symmetries $\text{A}_{1g}$, $\text{B}_{1g}$ and $\text{E}_{u}$). Similarly, the valley states with $\vec{K}_{i}$ parallel to the wire axis form degenerate eigenstates with $\text{A}_{1g}$ and $\text{A}_{2u}$ symmetry. For $L\gg w$, the energy of the two-fold degenerate states exceeds that of the four-fold degenerate states and so only the latter are relevant for STIRAP. For the simple case of a wire aligned along a (100) axis, the eigenenergies of the four-fold conduction band minima states are given by

[TABLE]

where $m_{\perp}$ and $m_{\parallel}$ are the transverse and longitudinal effective masses of an electron in bulk diamond. For $L\gg w$, this corresponds to

[TABLE]

In Section 4 we use Equation 4 to assess the feasibility condition $\Delta E_{c}/\hbar\gg\Omega$ for the two different nanowire designs.

EMT is limited as it cannot account for coupling between valley states via the wire confining potential. Known as the valley-orbit interaction, this coupling leads to a fine splitting of the conduction band minima states[40]. While the resulting eigenstates may be determined by symmetry, the magnitude of the splitting is notoriously difficult to quantify but assumed to be small[41, 42, 43, 44]. A proper treatment would require detailed ab-initio calculations, which are left as an opportunity for future work. The final electronic consideration is splitting induced by the applied electric field. As explored in supplementary section A, this Stark effect does not influence states of the conduction band minimum. While it does produce a small splitting of the $\text{A}_{1g}$ and $\text{A}_{2u}$ states, for $L\gg w$ this poses no impediment to the performance of STIRAP.

3 Optical processes in diamond nanowires

Assessing the feasibility of STIRAP requires comparing attainable photoionization Rabi frequencies to optical decoherence mechanisms. In this section we first present calculations of the photoionization Rabi frequency including Franck-Condon effects. We then identify and characterize three optical decoherence mechanisms; erroneous capture by $\text{N}_{\text{S}}^{+}$ defects, e-p scattering and spontaneous emission.

3.1 Photoionization Rabi frequency

The conduction dipole moment was first obtained for the photoionization of bulk NV centers using ab-initio calculations. Density functional theory was performed using the VASP plane-wave code with a 512-atom diamond supercell[45, 46, 47, 48] and Heyd-Scuseria-Ernzerhof functionals[49]. The energy of the ${}^{3}\text{A}_{2}$ ground state of the $\text{NV}^{-}$ center and the ${}^{2}\text{E}$ ground state of the $\text{NV}^{0}$ center (plus an ionized electron) were calculated as a function of generalized atomic coordinates. As detailed in supplementary section C, the calculations produced an ionization energy of $\omega=2.6$ eV (close to experimental values[33]) and a Huang-Rhys factor of $S=1.39$. Direct calculation of the optical matrix elements yields a transition dipole moment of $d_{\text{bulk}}=0.085\ e\cdot\text{\AA}$, normalised to the volume of the 512-atom supercell.

The bulk photoionization dipole moment can be used to determine the corresponding moment in a nanowire. Noting that this moment is identical for both the pump and Stokes pulse, we calculate in supplementary section C that

[TABLE]

where $\vec{R}_{A}$ is the position of NV $A$. For defects centered at opposite ends of the wire, the normalization function in Equation (5) can be simplified as

[TABLE]

where $V_{\text{sc}}=2.837\text{ nm}^{3}$ is the volume of the 512-atom supercell. Assuming a laser power $P$ with radial spot-size $r$, the effective photoionization Rabi frequency within a diamond nanowire is

[TABLE]

where $n_{D}$ is the index of refraction in diamond. The achievable spot size and power are related to each other depending on the technical aspects of the optics employed. A conservative estimate says that a diffraction limited spot size $r=200$ nm can be attained with a lasing power of $P=100$ mW. These parameters will be used in Section 4 to evaluate the photionization Rabi frequency.

3.2 Decoherence mechanisms

One source of optical decoherence is erroneous capture of conduction band electrons by $\text{N}_{\text{S}}^{+}$ defects. Typically $\text{N}_{\text{S}}$ defects are formed during CVD growth due to N-based gases present in the plasma[50]. These defects act as electron donors, readily ionizing to $\text{N}_{\text{S}}^{+}$ to fill electron traps from surface defects[22]. The $\text{N}_{\text{S}}^{+}$ capture rate can be estimated as $\Gamma_{\text{cap}}=\rho_{\text{N}_{\text{S}}^{+}}\rho_{\text{e}}L^{2}w\sigma\sqrt{k_{B}T/m^{*}}$, where $\sigma\approx 3-7\ \text{nm}^{2}$ is the $\text{N}_{\text{S}}^{+}$ capture cross section, $k_{B}$ is Boltzmann’s constant, $T$ is temperature, $m^{*}$ is the isotropic mean of the effective mass components, $\rho_{\text{N}_{\text{S}}^{+}}$ is the density of $\text{N}_{\text{S}}^{+}$ defects and $\rho_{\text{e}}$ is the probability density of the conduction band states[21]. We may conservatively estimate that $\rho_{\text{e}}L^{2}w<1$ and thus the capture rate is independent of volume. Assuming $T=4$ K and $\rho_{\text{N}_{\text{S}}^{+}}\approx 1$ ppb we find that $\Gamma_{\text{cap}}\lesssim 2.5$ MHz. This places a constraint on the photoionization Rabi frequency for STIRAP in any nanostructure. In this analysis we have assumed passivation of all surface traps by $\text{N}_{\text{S}}$ donors. If this is not the case surface defects must be considered as another source of electron capture.

A second source of decoherence can be attributed to e-p scattering of the conduction band states. We consider only first order processes involving adsorption and emission of acoustic phonons with higher energy conduction states. Higher order interactions can be considered negligible at $4$ K. The e-p scattering rate, $\Gamma_{\text{ep}}$, may be determined through Fermi’s golden rule with the Hamiltonian

[TABLE]

where $\Xi_{d}=8.7$ eV is the diamond deformation potential[51] and $\vec{u}_{f}$ is the nanowire phonon field. Explicit rates for $\Gamma_{\text{ep}}$ are specific to the nanowire design and are presented in the following section.

A fundamental source of optical decoherence is that due to spontaneous emission. The spontaneous emission rate for transitions $|2\rangle\rightarrow|1\rangle$ and $|2\rangle\rightarrow|3\rangle$ may be estimated using conventional expressions as[52]

[TABLE]

Note that this decoherence rate is dependent on wire geometry through the normalization function $F$ as per Equation (5). We now evaluate the Rabi frequency and total decoherence rate within both nanowire designs – surface confinement and electrostatic confinement – and optimize wire dimensions to satisfy the STIRAP feasibility conditions.

4 Optimization of nanowire design

4.1 Surface Confinement

As displayed in Figure 1 (A), electronic confinement may be realized through fabrication of free-standing nanowires[53, 54] or nanopillars[55] using reactive ion etching. To avoid injection of spurious electrons, it is necessary that the electrodes affixed to the wire ends are insulating and separated from the diamond surface with a high resistivity contact. NV centers may be introduced into wires through either N ion-implantation[56, 57] or N $\delta$-doping[58]. A pair of NV centers suitable for STIRAP can then be identified through confocal microscopy. Ideally, the NVs will be co-aligned along a (111) axis and situated at depths greater than $\sim 50$ nm (this is necessary for obtaining optimal bulk-like properties[59]).

We now derive an expression for the decoherence rate due to e-p scattering. In supplementary section D, we present calculations of $\Gamma_{\text{ep}}$ by approximating $\vec{u}_{f}$ through elasticity theory. Acoustic nanowire phonon modes may be classified as dilational, flexural, torsional and shear[60]. However, only dilational modes possess non-zero divergence and contribute to the e-p scattering as per Hamiltonian (8). The dilational modes $\vec{u}_{d,m}$ have angular frequencies quantized by $m=(m_{x},m_{y},m_{z})$ which are given by

[TABLE]

where $c_{l}$ is the longitudinal speed of sound in diamond. The scattering rate can then be calculated as

[TABLE]

where $M_{n,m}$ is the overlap integral between the electron and phonon wavefunctions, $\rho_{C}$ is the density of diamond and $n_{B}$ is the Bose-Einstein distribution. The density of states, $\rho$, is assumed to be a Lorentzian, given by

[TABLE]

where $\gamma\approx\omega_{m}/Q$ for $Q$ the quality factor. For diamond microcantilevers, $Q>10^{6}$ has been observed[61].

We now optimize the dimensions of a surface confined nanowire with respect to the STIRAP feasibility condition $\Delta E_{c}/\hbar\gg\Omega\gg\Gamma$. In Figure 3 (A) we present the photionization Rabi frequency derived in Equation (7) as a function of wire dimension. In Figure 3 (B) we compare the Rabi frequencies to the total decoherence rate $\Gamma=\Gamma_{\text{cap}}+\Gamma_{\text{SE}}+\Gamma_{\text{ep}}$ for varying nanowire dimension. We assume that $Q=10^{6}$, $T=4\ \text{K}$ and that each NV center is positioned 100 nm from its respective wire end. For simplicity we have only considered wires aligned along a (100) axis, though similar results likely hold for other wire directions.

Figure 3 (B) indicates that in general decoherence effects are minimized for smaller wire dimensions. This can be attributed to the normalization of the Rabi frequency, which scales inversely proportional with wire volume as evident in Figure 3 (A) (c.f., Equation 7). The sporadic variations in $\Omega/\Gamma$ can be attributed to resonance between electronic and phononic levels. Consulting equation (11), certain wire dimensions produce an increased density of resonant states which amplifies the rate of e-p scattering. Off resonance, the dominant decoherence mechanism is electron capture. A range of wire dimensions with $L<0.6$ $\mu\text{m}$ and $w<0.2$ $\mu\text{m}$ satisfy the STIRAP feasibility condition with at least $\Omega>10\Gamma$, reaching a maximum of $\Omega\approx 20\Gamma$ for short wires 0.3 $\mu\text{m}$ long. Note that for all wire dimensions considered in Figure 3 (B), $\Delta E_{c}/\hbar>10^{3}\Omega$, and therefore off-resonant coupling does not pose an impediment to STIRAP.

4.2 Electrostatic confinement

An alternative means for realizing a nanowire confining potential is through electrodes in a bulk structure. Consider the diamond substrate presented in Figure 1 (B) where two co-aligned NV centers can be identified through confocal microscopy. To generate the confining potential, a square nano-electrode with width $w$ is affixed to the surface directly above the NV centers. On the opposing surface of the diamond substrate an electrode plate is fixed and grounded. Applying a positive potential to the square electrode will then generate a confining potential. This localizes the electron density near the surface into a wire-like geometry. The effective length of this wire may be tuned through the magnitude of the potential. Conveniently, the potential also generates an electric field gradient which can be used to energetically distinguish each NV center during STIRAP. These electrostatic properties are demonstrated through simulations using COMSOL Multiphysics® software presented in supplementary section E.

In contrast to surface confined wires, electron scattering within electrostatically confined wires is due to interactions with bulk phonons. Calculation of $\Gamma_{\text{ep}}$ using Fermi’s golden rule and the Hamiltonian (8) is presented in supplementary section D. We find that

[TABLE]

where $G_{n}$ is a complicated expression for the overlap integral.

A further benefit of the electrostatic confinement design is that decoherence due to $\text{N}_{\text{S}}^{+}$ capture can be mitigated through inclusion of a sacrificial donor layer. $\text{N}_{\text{S}}$ defects distant from the confining potential can be $\delta$-doped into the substrate during CVD growth[58]. The sacrificial defects donate electrons to the surface traps until charge conservation occurs, reducing the density of $\text{N}_{\text{S}}^{+}$ available for erroneous capture within the confining potential[22]. We conservatively estimate that the presence of a sacrificial layer can reduce $\Gamma_{\text{cap}}$ by 95%.

In Figure 3 (C) we present the dimensional dependence of $\Omega/\Gamma$ for an electrostatically confined nanowire aligned along the (100) axis. We have that $G_{n}\propto w^{-4}$ in Equation 13 and therefore e-p scattering dominates the Rabi frequency for wire widths less than approximately $0.4\ \mu\text{m}$. For larger wire dimensions, both e-p scattering and spontaneous emission are negligible and the dominant source of decoherence is due to $\text{N}^{+}_{\text{S}}$ capture. The STIRAP feasibility conditions are satisfied when $0.5\ \mu\text{m}<w<0.9\ \mu\text{m}$ and $0.75\ \mu\text{m}<L<1.25\ \mu\text{m}$ which yields $\Omega>10\Gamma$. The ratio $\Omega/\Gamma$ is optimized at $\approx 25$ for wire dimensions of $L=0.75\ \mu\text{m}$ and $w=0.6\ \mu\text{m}$. Furthermore, for all wire dimensions $\Delta E_{c}/\hbar\gg 10^{3}\Omega$ and hence cross-talk with higher energy states in the conduction band is negligible.

Conclusion

In this paper we propose spatial STIRAP to realize an on-chip spin quantum bus for scalable diamond QIP devices. Our scheme considers coherent quantum transport of an electron and its spin-state between two NV centers embedded in a diamond nanowire. This is achieved through implementing STIRAP for optical control of the NV center charge states and the confined nanowire conduction states. In contrast to existing spatial STIRAP protocols which manipulate tunneling amplitudes between distant sites, our proposal realizes quantum transport without any occupation of the intermediary space. Furthermore, to our knowledge this is the first implementation of spatial STIRAP including spin-state transport.

A proposal for our scheme was first presented along with a protocol for initializing the NV spin-states. We identified a feasibility condition for implementing STIRAP, $\Delta E_{c}/\hbar\gg\Omega\gg\Gamma$, requiring the photionization Rabi frequency to exceed optical decoherence rates and be limited by coupling with higher-energy conduction states. For the latter, EMT was employed to model the effects of the nanowire confining potential on the diamond conduction band. We then presented the first ab-initio calculations of the NV photoionization Rabi frequency. This allowed the STIRAP feasibility condition to be evaluated in two nanowire designs, which achieved electronic confinement through either a surface structure or electrodes. By optimizing the Rabi frequency relative to the decoherence rate for varying wire dimension, we conclude that STIRAP is feasible in both designs in timescales on the order of hundreds of nanoseconds.

The first experimental stages should perform photionization spectroscopy to assess the spectral line width of discretized states in the nanowire conduction band. Ultimately, this will determine whether deterministic photoionization to the conduction band minimum is feasible. This study has also introduced multiple avenues for future theoretical research. For example, a detailed ab-initio model of the nanowire electronic structure would allow for precise calculation of valley-orbit and spin-orbit interactions. This would require development of new computational tools for simulating mesoscopic diamond structures.

Acknowledgements

We acknowledge funding from the Australian Research Council (DE170100169, DP170103098, FT160100357, CE170100026). C.A.M acknowledges support from the National Science Foundation through grants NSF-1619896, NSF-1547830, and from Research Corporation for Science Advancement through a FRED Award. AA acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 820394 (project Asteriqs). This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government.