Single shot readout of nuclear spins

As we have seen in section1.2.2, the optical readout of the electron spin also destroys its state due to polarization. The detected signal is the relative fluorescence rate, which only yields qualitative information on the electron spin state. Here, we will discuss optical single shot readout of nuclear spins [82, 41, 42], and the influence of the electronic dynamics during readout onto the nuclear spin.

Readout of nuclear spins is achieved by correlating the electron spin state with the nuclear spin, and optical readout of the electron spin [36]. For strongly coupled nuclear spins, where the hyperfine splitting is larger than the linewidth of the electron spin transition, this correlation is created by a frequency selective electron spinπ pulse. This pulse flips the state of the electron spin conditional on the nuclear spin state, which is a C_nNOT_e gate. The quantum logic readout sequence is shown in fig. 1.4c. It is important to note that for the readout laser pulse, the number of detected photons is either according to the distribution formS = 0 (bright distribution) or formS =±1 (dark distribution), and the wavefunction of the nuclear spin collapses onto the corresponding eigenstate. The average number of detected photons hni per readout pulse is up to hni ≈ 0.1 for m_S = 0 and hni ≈0.07 for m_S =±1, with a standard deviationσ=^qhni according to the Poisson distribution. This means that there is a large overlap of the two distributions, such that state determination after one readout pulse is not possible.

While the electron spin state is destroyed during readout, we will see below that for an appropriate external magnetic field and position of the nucleus relative to the NV the nuclear spin eigenstates can be robust against the optical readout of the electron spin. In this case, the nuclear spin state survives the readout process, and stays in its initial state, while the electron spin is re-initialized into m_S = 0. Therefore, another application of the nuclear readout sequence will yield a number of detected photons with the same statistical distribution (bright or dark) as for the first readout step. By applying repetitive readout of the nuclear spin, the random numbers of detected photons with always the same distribution are summed up. Consequently, the relative standard deviation σ/hni is reduced, such that eventually the dark and bright distributions are well separated and single shot state determination is possible.

Fig. 1.4a shows a fluorescence time trace of the NV during repetitive readout of the¹⁴N nuclear spin within a magnetic field of ≈ 0.62 T. There, low fluorescence corresponds to the m_I = +1 nuclear spin state, and high fluorescence to m_I = 0,−1. As we can

1.2. The Nitrogen-Vacancy defect in diamond

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Figure 1.4.: Nuclear spin single shot readout. a, Fluorescence time trace of the NV during repetitive readout of the¹⁴N nuclear spin and corresponding histogram. These measurements were performed with a solid immersion lens, increasing the detected fluorescence by a factor of≈3 (see section 4.2). The red line is the most likely state evolution obtained by a hidden Markov model [83]. The orange line shows the threshold for single shot readout. b, Histogram of the fluorescence time trace. The red lines are Gaussian fits. c, Measurement sequence for nuclear spin readout. d, Measurement sequence for nuclear spin operations with single shot readout. e, Rabi oscillation of the ¹⁴N nuclear spin measured by single shot readout.

see, the lifetime of the nuclear spin states is much longer than the time needed for state determination, which enables single shot readout. Fig. 1.4b shows a histogram of measurement results for this time trace. The two peaks correspond to the two nuclear spin states m_I = +1 and m_I 6= +1 (i.e. m_I = 0,−1). Placing a threshold between these two peaks allows for state determination of a single measurement point in the time trace, by checking whether the number of photons is below or above this threshold.

Due to the overlap of the two distributions, the readout fidelityF will be limited, here it is F = 0.958. This projective, single shot readout is also used for initialization of the nuclear spin. Thereby, the initialization fidelity can be increased by shifting the threshold to lower photon count numbers. This will remove results which are likely wrong (cf. fig.

1.4b), at the expense of successful initialization events. A typical measurement sequence for Rabi oscillation of the nuclear spin is shown in fig. 1.4d. Two consecutive single shot measurements are correlated by taking the average result of the second measurement, if the first measurement yielded e.g. state m_I= +1. Thereby, the effect of the rf pulse in between these two measurements is obtained. Fig. 1.4d shows the spin flip probability of the ¹⁴N nuclear spin during resonant rf irradiation measured by single shot readout.

The lifetime of nuclear spins during optical readout is limited by interactions with the electron spin. The hyperfine interaction can be split into two parts, which lead to two different flipping mechanisms of the nuclear spin. The first part are A_xx and A_yy terms of the hyperfine tensor in (1.8), which leads to S₊I₊, S₊I−, S−I₊, S−I− terms in the Hamiltonian. These lead to mixing of electron and nuclear spin states, i.e. if we consider the m_S = 0,−1 electron spin states and a nuclear spin 1/2 with states m_I =−,+, the eigenstates can be written asα_i|m_S= 0, m_I=±i+β_i|m_S =−1, m_I=±i, with different prefactorsα_i,β_i for each eigenstate. After each readout laser pulse, the electron spin is polarized into m_S = 0, which is not an eigenstate due to the mixing. Thus, the electron and nuclear spin states will coherently evolve, effectively destroying the nuclear spin state. The flipping rater₁ due to this mechanism scales inversely quadratically with the electron Zeeman splitting γ_eB [41, 42],

r1 ∝ ≈ 2A²_⊥

2A²_⊥+ (D_i−γ_eB)², (1.10) whereA⊥ = (A_xx+A_yy)/2,D_i the electron zero-field splitting for ground state or excited state, andB the magnetic field aligned along the NV axis. The second part are theA_zx and Azy terms of the hyperfine tensor. These terms lead to nuclear spin eigenstates which depend on the electron spin state. Thus, whenever the electron spin flips, the nuclear spin will start to evolve in the new eigenbasis, effectively destroying its state.

The flipping rate r2 due to this mechanism will scale inversely quadratically with the nuclear Zeeman splitting γ_nB [42],

r₂ ∝ ≈ A²_zx+A²_zy

A²_zx+A²_zy+ (A_zz−γ_nB)². (1.11)

1.2. The Nitrogen-Vacancy defect in diamond

Usually, the flipping rate r₂ will be the dominant one, as it scales with the nuclear Zeeman splitting, contrary to r₁ which scales with the much larger electron Zeeman splitting. However, the Azx and Azy terms are highly position dependent. Specifically, these terms are zero for nuclear spin positions on the axis of the NV and for positions on the equatorial plane of the electron spin. Thus, for the ¹⁴N nuclear spin, the dominant flipping mechanism is spin state mixing [41].