Nuclear-Spin Dependent Photon Scattering

2

(S_xS_z+S_zS_x)

!

, (6.29)

where the transition energyEgis shifted byωlto the detuningδ =Eg−ωl. In the Hamiltonian Hc[Eq. (6.23)], the transformation causes a shift of the cavity frequencyωctoδc =ωc−ωl, which is the detuning of the laser frequency from the cavity mode,

Hc⁰=δca^†a+д

τ₊a+τ−a^†

. (6.30)

6.4.2 Nuclear-Spin Dependent Photon Scattering

Excluding the NV center, the external laser field and the optical cavity are independent objects that do not interact with each other. However, the situation significantly changes if an absorber that interacts with both fields is located inside the cavity. One can imagine, e.g., a scenario in which

5The constant termEg/2 in Eq. (6.17) will be omitted from now on.

6.4 Controlled Quantum Gate between Nuclear Spins

a photon from the laser field⁶is absorbed by the NV center, which is thereby excited from the ground to the excited state described by the HamiltonianHl⁰[Eq. (6.28)]. Due to the coupling to the cavity viaHc⁰[Eq. (6.30)], the NV center itself can now emit a photon into the cavity mode. This two-step process can also be seen as scattering of a laser photon into the cavity mode, however mediated by an intermediate excitation of the NV center. As we demonstrate below, the rate of such scattering process depends on the nitrogen nuclear-spin state.

We assume the laser frequency to be sufficiently detuned from any optical transition such that the NV center is only excited virtually. Any real population of the excited state offers the possibility of spontaneous decay, making the whole process incoherent. For the case of only virtual excitation, one can use the technique of quasi-degenerate perturbation theory in the framework of Schrieffer-Wolff (SW) transformations [221,222] to effectively decouple the low-and high-energy subspaces of the HamiltonianH1⁰in Eq. (6.27). Here, the high-energy subspace is the orbital excited-state manifold. The effects of the coupling between the two subspaces is incorporated in the structure of the transformed states. The HamiltonianH1⁰can be separated into a block-diagonal partH1⁽⁰⁾that only acts within the low- and high-energy subspace, respectively, H₁⁽⁰⁾=He⁰+Hn+Hhf+δca^†a, (6.31) and an off-diagonal partV¹that connects the two subspaces of the total Hilbert space,

V1=Ωτ₊+Ω^∗τ−+д

τ₊a+τ−a^†

, (6.32)

i.e.H1⁰=H1⁽⁰⁾+V1. For the SW transformation, we construct an anti-Hermitian operatorS1that is defined by the condition [S¹,H1⁽⁰⁾]=V¹, and apply the unitary transformation of the Hamiltonian H₁⁰according to

HH1=e^−S1H1⁰e^S1 ≈H1⁽⁰⁾+ 1 2

[V1,S1], (6.33)

keeping only the lowest-order contribution of the off-diagonal interaction partV1(see Appendix C). The approximation made here is justified if the off-diagonal elements are small compared to

the energy gap between the low- and high-energy subspace. The transformed HamiltonianHH¹is block-diagonal and only acts within the low- and high-energy subspace, respectively.

For the time being, we neglect the contributions proportional to the transversal spin-spin couplings∆¹ and∆²in Eq. (6.29). The physics behind the scattering process can be more il-lustratively depicted in this case and do not change qualitatively. Furthermore, one can write down the SW transformation explicitly. To eventually obtain quantitative results, the spin-spin contributions will be included again. The anti-Hermitian operatorS¹, that is determined by the

6Although we describe the laser as a classical radiation field, we use the termphotonin the context of NV center optical excitation.

89

Chapter 6: Long-Range Two-Qubit Gate

Figure 6.9–NV center energy level diagram with involved frequencies.(a) Hyperfine levels of themS = −1 manifolds in the ground (|gsi) and excited (|esi) state (blue shading) for a¹⁴N nuclear spin. The energy difference between themI =0 states isE^g−∆. The laser of frequency ω^lis detuned from themI =0 optical transition byδ^l. The difference between the cavity and the laser frequency isδ^c. Also indicated are the computational basis states|0i=|mS=−1,mI =0iand

|1i=|mS =−1,mI = +1iin the ground state. (b) Equivalent level diagram for a¹⁵N nuclear spin.

Computational basis chosen as|0i=|mS=−1,mI =−1/2iand|1i=|mS =−1,mI = +1/2i.

condition [S1,H₁⁽⁰⁾]=V1, is calculated to be S1=

Ω

∆S_z²−∆hfS_zI_z−δ−1

τ₊+д

∆S_z²−∆hfS_zI_z+δc−δ−1

τ₊a

−h.c. (6.34) At this point, we can fully decouple the orbital ground and excited state by calculatingHH¹ in Eq. (6.33). We are only interested in the low-energy dynamics of the system and therefore restrict our considerations to the respective subspace of the total Hilbert space. The dimension of the Hilbert space is further reduced by taking only themS =−1 subspace into account and neglecting the spin states withm_S = 0 andm_S = +1. This restriction is justified since a magnetic field is chosen that separates the spin states with different quantum numberm_S well in energy. In doing so, we can replace the spin operatorSzby its eigenvaluemS =−1 in the following. The effective low-energy part ofHH1acting on the orbital ground state withm_S =−1 is given by

HH1^(gs) =−

A^gs_q +γⁿB

Iz+QI_z²+δ^ca^†a +1

2 дΩ

(∆^hfIz−δ^l)⁻¹+(∆^hfIz+δ^c−δ^l)⁻¹

a^†+h.c.

, (6.35)

where we define the laser detuningδl=δ−∆from them_I =0 orbital transition. An overview about the involved energies is shown in Fig. 6.9. In Eq. (6.35) we omit all constant terms and neglect small energy shifts proportional toд²(Lamb shift) and|Ω|²(Stark shift).

On the basis of previous experimental work [49,50,151], in which the nitrogen nuclear spin

6.4 Controlled Quantum Gate between Nuclear Spins

has been utilized as qubit, we choose the computational basis as

|0i=|m_S =−1,m_I =0i, (6.36)

In this form, we see that the cavity excitation and disexcitation crucially depends on the laser detuningδland the nuclear-spin state of the nitrogen atom. The scattering into and out of the cavity can especially be completely suppressed for one of the two nuclear-spin states, e.g., if the laser frequency is chosen such thatδ^l =δ^c/2. In this case, a laser photon can only be scattered if the nitrogen nuclear spin is in statem_I = +1, i.e. in the qubit state|1i. The effective Hamiltonian describing this situation, also referred to asm_I = +1-scattering, is

HH₁^(gs)=

Q−A^gs_q +γnB

|1i h1|+δca^†a+д⁰|1i h1|a^†+(д⁰)^∗|1i h1|a. (6.39) For the process that a photon is scattered into the cavity or vice versa, we thus obtain an effective coupling strength

The second possibility is to suppress the scattering mechanism if the NV nuclear spin is in the qubit state|0i. This can be achieved by adjusting the laser frequency to a detuning ofδ^l=∆^hf +δ^c/2.

We find the same effective scattering rateд⁰[Eq. (6.40)] for this case. However, we concentrate on the first case in the following, where the detuning allows scattering only if the qubit is state|1i. The transversal spin-spin interaction terms proportional to∆¹and∆²have so far not been included in the derivation of the effective low-energy HamiltonianHH₁^(gs). Quantitative predictions regarding the developed scattering mechanism can only be made if these terms are taken into account. We can perform the SW transformation in a similar fashion as described in above, determining the anti-Hermitian matrixS1from the full HamiltonianH1⁰in Eq. (6.27). In doing so, we have to assume sufficiently large detunings |δ^l| > |Ω| and |δ^l−δ^c| > д such that the off-diagonal terms inV¹are small compared to the energy gap between the low- and high-energy subspace. We therefore assume an initially empty cavity that is maximally populated by one photon, and only if the NV center is in the ground state. In the excited state, we need to include the m_S =0 andm_S = +1 spin states, giving a 10-dimensional Hilbert space. We consider the scenario

91

Chapter 6: Long-Range Two-Qubit Gate

Figure 6.10–Laser detunings enabling scattering only from the state|1i.(a) Five solutions ofδl, for which the matrix element for scattering of a laser photon from themI =0 state is zero, as a function ofδ^c. (b) Effective coupling strength ˜д=дΩf(δ^c)for the solutionsδ^lshown in (a).

Solution 3 (solid line) is used for further calculations.

when scattering is only possible if the nuclear spin qubit is in state|1i. Compared to the unique solution for the laser detuningδlpreviously, we find five solutions such that the scattering matrix element for themI =0 state is zero, shown in Fig.6.10. For all solutions, the effective ground state Hamiltonian in the case ofm_I = +1 scattering has the same form as given in Eq. (6.39),

HH1^(gs) =

Q−A^gs_q +γⁿB

|1i h1|+δ^ca^†a+д˜|1i h1|a^†+(д)˜ ^∗|1i h1|a, (6.41) however with a different effective coupling strength ˜д=дΩf(δc), where∆^hf/(∆²_hf−(δc/2)²)is replaced by a different detuning-dependent partf(δc). In the following derivation, we choose the solution gives rise to the largest value of ˜д(solution 3 in Fig.6.10), and eventually minimizes the two-qubit gate time in the following.