Coherent electric field control of orbital state of a neutral nitrogen-vacancy center

The coherent control of the orbital state is crucial for realizing the extremely-low power manipulation of the color centers in diamonds. Herein, a neutrally-charged nitrogen-vacancy center, NV0, is proposed as an ideal system for orbital control using electric fields. The electric susceptibility in the ground state of NV0 is estimated, and found to be comparable to that in the excited state of NV−. Also, the coherent control of the orbital states of NV0 is demonstrated. The required power for orbital control is three orders of magnitude smaller than that for spin control, highlighting the potential for interfacing a superconducting qubit operated in a dilution refrigerator.

where h is the Plank's constant, λ is the spin-orbit interaction parameter, Lz = σ z , L± = are the strain eigenstates, and S z = (1/2)σ z is the spin operator for the 1/2 spin.Because NV 0 has the same wave function symmetry as the excited state of NV − due to the dynamic Jahn-Teller effect (the effective C 3v symmetry), the effects of the strain and electric fields on |e x,y ⟩ o are considered to be similar to that in the excited state of NV − [1][2][3].Therefore, the discussion on the excited state of NV − in Ref. [2] is applied on NV 0 as outlined below.
In the |e x,y ⟩ o basis, the strain and electric field Hamiltonian can be written as: where ϵ A 1 , ϵ E 1 , and ϵ E 2 represent the strain in the energy dimensions with symmetric indices, d ⊥ denotes the in-plane electric susceptibility, and d ∥ represents the electric susceptibility along the NV axis.In the |±⟩ o basis: (5) To simplify this analysis, only the subspace of spin in the state |↑⟩ s is considered because the strain and electric fields do not directly affect the spin.* E-mail: kurokawa-hodaka-hm@ynu.ac.jp † E-mail: kosaka-hideo-yp@ynu.ac.jp

A. Strain and DC electric fields
First, the state where only the strain is present in the system is considered.By redefining the axis of rotation, the strain Hamiltonian can be expressed as follows: The electric fields are rewritten as shown below because they may not necessarily align in the same direction as the strain: where E ⊥ represents the electric fields with the same direction as ϵ ⊥ , and E ′ ⊥ represents the electric fields perpendicular to E z and E ′ ⊥ .To diagonalize these equations, H 0 + H strain + H electric , the energy eigenvalues are obtained as: The corresponding eigenvectors without normalization can be written as:

B. Strain and AC electric fields
The AC electric fields are applied in such a way that they resonate with the eigenstates of the strain, |± ′ ⟩ o (E = 0).Therefore, the AC electric fields are treated as a perturbation to the eigenstates of the strain.
The effects of the electric fields in the basis of |± ′ ⟩ o with normalized eigenvectors can be expressed as: where L± are the rising (lowering) operators in the |± ′ ⟩ o basis.The equation with the redefined axis after introducing Subsequently, considering the sinusoidal driving fields, we move to the rotating frame with electric /h.The equation under the rotating wave approximation is Therefore, Rabi oscillation can be observed between the orbital states when ∆ = 2 λ 2 + ϵ 2 ⊥ − f d ∼ 0. Also, we briefly mention the derivation of the Autler-Townes splitting in the main text.
When we set ∆ = 0, H After the diagonalization of the Hamiltonian, the resultant energy eigenvalues of the dressed states are ±d ⊥ E ′′ ⊥ /2.The energy splitting between two dressed states is d ⊥ E ′′ ⊥ , corresponding to the Rabi frequency at that driving amplitude.Thus, we can estimate d ⊥ from the energy splitting if we know

II. SUPPLEMENTARY METHODS
The electric fields and applied voltage are estimated based on a simulation of the electric field distribution using the finite element method (FEM) (COMSOL Multiphysics, COM-SOL).The electric fields at the NV center are estimated to be (E X , E Y , E Z ) = (12497.6 V/m, -26122.3V/m, -7973.57V/m) per 1 V for the electrode used for DC voltage application, where E X , E Y , E Z are the electric fields in the laboratory frame, as shown in Fig. S1 (a).For the electrode used for ac voltage application, (E X , E Y , E Z ) = (13763.6V/m, -18844.1 V/m, -1079.8V/m) per 1 V.The electric fields in the NV frame, E x , E y , and E z , are defined as |±⟩ o ⟨∓| o are the orbital operator in the |±⟩ o = ∓1/ √ 2(|e x ⟩ o ± i |e y ⟩ o ) basis, |e x ⟩ o and |e y ⟩ o FIG. S1. (a)Electric field distribution around the electrode simulated using FEM software using an applied voltage of 1 V.The unit of the color bar is V/m.(b) Relationship between the laboratory frame and frame of the NV center.C, N, and V correspond to the carbon atom, nitrogen atom, and vacancy, respectively.The direction of the NV center is estimated from the measurement of the optically-detected magnetic resonance.Only two of the four NV axes can be distinguished through measurement; however, the distinction between the two types of axes only affects the sign of the electric field.Therefore, the NV axis is defined as illustrated in the figure.z NV is along the NV axis, and x NV and y NV are defined following the definition of |e x,y ⟩ in Ref.[2].(c) θ is defined as shown in the figure.
where V rms is the root-mean-square (RMS) voltage, P is the microwave power, and R is the resistance.When 1 µW microwave is applied to the electrode at the end of the 50 Ω transmission line, V rms = 7.1 mV.Assuming that the end of the electrode is an approximate open circuit (Z = ∞), the voltage amplitude at the electrode is 7.1 × √ 2 × 2 = 20 mV.The factor √ 2 is introduced to convert the RMS voltage to the voltage amplitude, wherein a factor of 2 is used because the circuit is open at the end of the electrode.