Spin coherent quantum transport of electrons between defects in diamond

1 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 the 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 (N_S) centers [9], [10], [11], [12], [13], [14], [15].

Individual spin registers are ultimately limited by the number of paramagnetic defects addressable by a single NV center (5–10 strongly coupled qubits [16]). The 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 challenging for scalable QIP as optical buses are difficult to compactly incorporate on chip while maintaining low losses. One scalable suggestion is intracluster 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 semiclassical electron transport between two NV–¹⁴N_S pairs embedded in a diamond nanowire. This scheme is severely limited as semiclassical transport in a nanowire is prone to erroneous capture from Ns+ 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 Λ-type atomic system [28]. By applying optically coupling fields in the so-called counterintuitive 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 Λ scheme correspond to an electron occupying one of two distant NV centers embedded in a diamond nanostructure. As depicted in Figure 1A, 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. The schematics for the transport scheme in these designs are presented in Figure 1. Although 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.

Figure 1:

Design schematics for quantum transport of an electron between NV centers using STIRAP.

(A) Λ-level scheme for two NV centers in a diamond nanowire. Adiabatic passage is mediated by mutual coupling to the conduction band minimum state, |ψ⟩. Applying a Stokes laser pulse before a pump laser pulse with some overlapping time τ results in adiabatic quantum transport of the electron and its spin state from NV A to NV B. (B) Two NV centers labeled A and B are separated by a distance s in a surface confined nanowire of length L and width w. A potential difference of Φ is applied to electrode plates affixed to either end of the wire. This generates a gradient electric potential (graphed as U), which lifts the degeneracy of states |NV⁻⟩_A|NV⁰⟩_B and |NV⁰⟩_A|NV⁻⟩_B in the STIRAP level scheme. To avoid erroneous excitations to higher conduction band states, the pump and Stokes lasers must individually address each NV center and therefore be spatially separated. (C) Schematic of an electrostatically confined nanowire. The two NV centers are aligned coaxial and perpendicular to the surface of a diamond substrate. A square nanoelectrode of width w is affixed to the surface above the NV centers, whereas a grounded electrode plate is affixed to the opposite side of the substrate. Applying a small positive voltage Φ to the nanoelectrode generates a nanowire confining potential (red equipotentials), which extends an effective depth L into the substrate.

This proposal for quantum transport is fundamentally different from existing schemes for spatial STIRAP. As coupling is mediated optically rather than through the 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 the teleportation of a massive particle. Furthermore, to our knowledge, this is the first spatial STIRAP proposal that includes coherent spin transport.

In Section 1, we present a proposal for implementing STIRAP in a diamond nanowire including a spin initialization and a 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, ΔE_c/ħ≫Ω≫Γ, where ΔE_c defines the energy gap between the first and second conduction states, Ω is the Rabi frequency of the optical transition between defect and conduction states, and Γ is the optical decoherence rate. As a conservative estimate, these constraints can be considered satisfied when ΔE_c/ħ≳10Ω≳10Γ. In Section 2, we calculate ΔE_c by modeling the effect of the nanowire confining potential on the diamond conduction band. Effective mass theory (EMT) is used to derive a level scheme and determine the 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 Ω that can be practically achieved. We then identify optical decoherence mechanisms that contribute to Γ. Finally, in Section 4, we evaluate Ω, ΔE_c, and Γ at cryogenic temperatures for both 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.

2 Quantum transport in diamond nanowires

STIRAP is a method for evolving a three-state system from an initial state |1⟩ to a final state |3⟩ [28], [32]. This is achieved through coupling to a mutual intermediary state, |2⟩, although the direct transition |1⟩→|3⟩ may be forbidden. To do so, a pump laser pulse with Rabi frequency Ω_P(t) is tuned to the |1⟩→|2⟩ transition, whereas a Stokes laser pulse with Rabi frequency Ω_S(t) is tuned to the |2⟩→|3⟩ transition. Applying the Stokes laser pulse before the pump laser pulse with some overlapping time τ results in an adiabatic evolution of the state from |1⟩→|3⟩ without any population of |2⟩. The requirement for adiabaticity is given by Ω_τ≫1, where Ω2=ΩP2+ΩS2 is the effective Rabi frequency.

We propose using STIRAP for quantum transport of an electron between two NV centers. In Figure 1, 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

|1〉= |NV−〉A|NV0〉B|2〉= |NV0〉A|NV0〉B|ψ〉|3〉= |NV0〉A|NV−〉B,

where |NV⁻⟩ and |NV⁰⟩ correspond to the ³A₂ and ²E electronic ground states of the negative and neutral charge states of the NV centers and |ψ⟩ is a nanowire conduction band minimum state. A schematic of the STIRAP level scheme is presented in Figure 2. The adiabatic passage between states |1⟩ and |3⟩ therefore constitutes the electron transport from NV A to NV B with no occupation of the intermediary space. The degeneracy of states |1⟩ and |3⟩ may be lifted by applying a potential difference Φ between each end of the length L wire. For a defect separation distance s, this induces an energy splitting of ΔE_d=eΦs/L.

Figure 2:

Nanowire energy-level diagram for the STIRAP transport protocol.

The electron is transported from the initial defect state |NV⁻⟩_A|NV⁰⟩_B to the target defect state |NV⁰⟩_A|NV⁻⟩_B using laser pulses with Rabi frequencies Ω_P and Ω_S to mediate adiabatic passage with the fourfold degenerate conduction band minima states. Implicit in the labeling of conduction band states is the combined electronic state of the defects, |NV⁰⟩_A|NV⁰⟩_B. Not to scale.

Let us now consider the transportation of the electron spin during STIRAP. This requires a protocol for spin initialization of both |NV⁻⟩_A and |NV⁰⟩_B, which we now present in turn. First, suppose we wish to transport the spin-state |↑⟩. We restrict our attention to photoionization by absorption of a single photon from an optical pulse and do not consider the two-photon ionization process of the NV⁻ [33]. As the ionizing pulse cannot differentiate between the two electrons occupying the e-orbitals of |NV⁻⟩_A, both must be initialized into |↑⟩. This can be achieved through the optical polarization of |NV⁻⟩_A into the m_s=0 triplet state followed by the application of a microwave control pulse. The resulting spin-state can be expressed in the two e-orbital basis as |+1⟩=|↑⟩|↑⟩. Hence, through this protocol, we can guarantee the ionization of a spin-up electron during STIRAP.

Now, consider the initialization of the |NV⁰⟩_B spin-state. After STIRAP, electron transportation can only be validated if the read-out of the |NV⁻⟩_B spin-state agrees with the spin initialized at |NV⁻⟩_A. Otherwise, it is impossible to discern if the charging of |NV⁰⟩_B into |NV⁻⟩_B was instead due to the erroneous electron capture. It is therefore essential for the read-out that |NV⁰⟩_B is initialized into either the |↑⟩ or |↓⟩ spin eigenstates. To see how this could be achieved, consider that the |NV⁻⟩_B spin-state after transport (assuming |NV⁰⟩_B was initialized as |↑⟩) is given by |↑⟩|↑⟩=|+1⟩. Similarly, we obtain the m_s=0 triplet-state when |NV⁰⟩_B is initialized as |↓⟩. 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. The application of a >2.6 eV photoionizing pulse will then eject an electron with |↑⟩ spin, leaving NV B in the neutral charge state with |↑⟩ spin. Equivalently, the |↓⟩ spin state can be prepared for NV B prepared in the m_s=−1 state. Although 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 nonconserving 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 polarization can be swapped back onto the electron spin of the neutral charge state via microwave control. Such a scheme is now feasible given that recent progress has identified the NV⁰ spin Hamiltonian [34].

Transportation also requires that NV⁰ spin relaxation is long with respect to the STIRAP operation time. As of yet, the spin relaxation of the NV⁰ ground state has not been directly observed due to the apparent absence of an electron paramagnetic resonance (EPR) signal [35]. Although this could potentially imply a high rate of spin relaxation, the new work by Barson et al. indicated that this is likely not the case. Rather, the apparent absence of an EPR signal is due to strain broadening in ensembles [34]. Drawing analogy between the similar electronic groundstates of the NV⁰ and Si-V⁻ centers, we expect spin relaxation to be not overly fast. This is because within the Si-V⁻ center, e-p scattering is known to cause spin dephasing but not relaxation [36]. 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 coaxial 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 (CVD) [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 the successful implementation of STIRAP. First, coherent transport demands that 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 ΔE_c must be significantly greater than the Rabi frequency, ΔE_c/ħ≫Ω. Second, the optical decoherence rate Γ sets a bound on Ω, as it induces an unknown phase on the electron wave function as the transition energy fluctuates. Provided that the Rabi frequency is large (Ω≫Γ), 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.

3 Diamond nanowire conduction band

The confining potential of the nanowire leads to the discretization of the conduction band suitable for STIRAP. These confinement effects are well described through EMT [38], as detailed in Supplementary Section A. As shown in Figure 1, the nanowire was modeled as a square prism of monocrystalline diamond with dimensions of w×w×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 K→i between the Γ and X points in each orthogonal axis of the fcc Brillouin zone. The dispersion relationship at each K→i is modeled using diamond’s anisotropic mass tensor and is referred to as a valley [39].

In the effective mass framework, the nth conduction band state for the ith valley of the nanowire is given by

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. Although 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

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 to its D_4h symmetry. As presented in Supplementary Section A, symmetry analysis reveals four degenerate conduction band minima states formed from the valleys with K→i perpendicular to the wire axis (with symmetries A_1g, B_1g, and E_u). Similarly, the valley states with K→i parallel to the wire axis form degenerate eigenstates with A_1g and A_2u symmetry. For L≫w, the energy of the twofold degenerate states exceeds that of the fourfold degenerate states, so only the latter are relevant for STIRAP. For the simple case of a wire aligned along a (100) axis, the eigenenergies of the fourfold conduction band minima states are given by

where m_⊥ and m_|| are the transverse and longitudinal effective masses of an electron in bulk diamond. For L≫w, this corresponds to

In Section 4, we use Equation (4) to assess the feasibility condition ΔE_c/ħ≫Ω 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]. Although 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 the states of the conduction band minimum. Although it does produce a small splitting of the A_1g and A_2u states, for L≫w, this poses no impediment to the performance of STIRAP.

4 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 Ns+ defects, e-p scattering and spontaneous emission.

5 Optimization of nanowire design

5.1 Surface confinement

As displayed in Figure 1A, electronic confinement may be realized through the fabrication of freestanding 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 δ-doping [58]. A pair of NV centers suitable for STIRAP can then be identified by confocal microscopy. Ideally, the NVs will be coaligned along a (111) axis and situated at depths greater than ~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 Γ_ep by approximating u→f through the elasticity theory. Acoustic nanowire phonon modes may be classified as dilational, flexural, torsional, and shear [60]. However, only dilational modes possess nonzero divergence and contribute to the e-p scattering as per Hamiltonian (8). The dilational modes u→d,m have angular frequencies quantized by m=(m_x, m_y, m_z), which are given by

(10)ωm2=π2cl2(mx2+my2w2+mz2L2),

where c_l is the longitudinal speed of sound in diamond. The scattering rate can then be calculated as

(11)Γep=2πℏ2∑n, mΞd2|Mn, m|2ℏ2ρCw2Lωm                   ×nB(ωm,T)ρ(ωn−ωm),

where M_n,m is the overlap integral between the electron and phonon wavefunctions, ρ_C is the density of diamond, and n_B is the Bose-Einstein distribution. The density of states, ρ, is assumed to be a Lorentzian, given by

(12)ρ(ωn−ωm)=γ/π(ωn−ωm)2+γ2,

where γ≈ω_m/Q for Q is the quality factor. For diamond microcantilevers, Q>10⁶ has been observed [61].

We now optimize the dimensions of a surface-confined nanowire with respect to the STIRAP feasibility condition ΔE_c/ħ≫Ω≫Γ. In Figure 3A, we present the photoionization Rabi frequency derived in Equation (7) as a function of wire dimension. In Figure 3B, we compare the Rabi frequencies to the total decoherence rate Γ=Γ_cap + Γ_SE + Γ_ep for varying nanowire dimensions. We assume that Q=10⁶, T=4 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, although similar results likely hold for other wire directions.

Figure 3:

Dimensional dependence of optical properties within (100) aligned diamond nanowires.

(A) Density plot of the photoionization frequency Ω as a function of wire dimension. (B) Density plot of the ratio of the photoionization frequency to the optical decoherence rate (Ω/Γ) as a function of dimension for a surface confined nanowire. (C) Density plot of the ratio of the photoionization frequency to the optical decoherence rate (Ω/Γ) as a function of nanowire dimension for an electrostatically confined nanowire. Note that we have omitted calculations of Ω and Ω/Γ for wires with w>L.

Figure 3B 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 3A [cf. Equation (7)]. The sporadic variations in Ω/Γ can be attributed to the resonance between electronic and photonic 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 μm and w<0.2 μm satisfy the STIRAP feasibility condition with at least Ω>10Γ, reaching a maximum of Ω≈20Γ for short wires 0.3 μm long. Note that for all wire dimensions considered in Figure 3B, ΔE_c/ħ>10³Ω; therefore, off-resonant coupling does not pose an impediment to STIRAP.

6 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 by implementing STIRAP for the optical control of the NV center charge states and the confined nanowire conduction states. In contrast to existing spatial STIRAP protocols that 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, ΔE_c/ħ≫Ω≫Γ, requiring the photoionization 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 photoionization spectroscopy to assess the spectral linewidth of discretized states in the nanowire conduction band. Ultimately, this will determine whether deterministic photoionization to the conduction band minimum is feasible. A second key experiment is to directly the measure the spin-relaxation time of the NV⁰ center to confirm that it is sufficiently long to enact the STIRAP protocol. 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 the precise calculation of valley-orbit and spin-orbit interactions. This would require the development of new computational tools for simulating mesoscopic diamond structures.