Measurement and Readout¶

Quantum measurements in the NV center system involve projective measurements, expectation values, and realistic photon detection.

Projective Measurements¶

Mathematical Framework¶

A projective measurement is described by a set of projection operators \(\{\hat{P}_k\}\) satisfying:

\[\begin{split}\hat{P}_k^\dagger &= \hat{P}_k \quad \text{(Hermitian)} \\ \hat{P}_k^2 &= \hat{P}_k \quad \text{(Idempotent)} \\ \sum_k \hat{P}_k &= \mathbb{1} \quad \text{(Complete)}\end{split}\]

The probability of outcome \(k\) is:

\[p_k = \text{Tr}(\hat{P}_k \hat{\rho})\]

Post-measurement state (for outcome \(k\)):

\[\hat{\rho}_k = \frac{\hat{P}_k \hat{\rho} \hat{P}_k}{p_k}\]

Spin State Projectors¶

For electron spin measurements in the ground state manifold:

\[\hat{P}_{m_s} = |g\rangle\langle g| \otimes |m_s\rangle\langle m_s| \otimes \mathbb{1}_3\]

Explicitly, the \(m_s = 0\) projector in the 18-dimensional space:

\[\hat{P}_{m_s=0}^{(g)} = \sum_{m_I = -1}^{+1} |g, 0, m_I\rangle \langle g, 0, m_I|\]

This corresponds to a 18×18 diagonal matrix with 1s at indices 3, 4, 5 (ground state \(m_s = 0\)).

Electronic State Projectors¶

Ground state projector:

\[\hat{P}_g = |g\rangle\langle g| \otimes \mathbb{1}_9 = \sum_{n=0}^{8} |n\rangle\langle n|\]

Excited state projector:

\[\hat{P}_e = |e\rangle\langle e| \otimes \mathbb{1}_9 = \sum_{n=9}^{17} |n\rangle\langle n|\]

Nuclear Spin Projectors¶

For N14 nuclear spin measurements:

\[\hat{P}_{m_I} = \mathbb{1}_2 \otimes \mathbb{1}_3 \otimes |m_I\rangle\langle m_I|\]

Expectation Values¶

For any observable \(\hat{O}\), the expectation value is:

\[\langle \hat{O} \rangle = \text{Tr}(\hat{O} \hat{\rho})\]

Common Observables¶

Electron spin components:

\[\begin{split}\langle \hat{S}_z \rangle &= \text{Tr}(\hat{S}_z \hat{\rho}) \\ \langle \hat{S}_x \rangle &= \text{Tr}(\hat{S}_x \hat{\rho}) \\ \langle \hat{S}_y \rangle &= \text{Tr}(\hat{S}_y \hat{\rho})\end{split}\]

Spin populations:

\[P_{m_s} = \text{Tr}(\hat{P}_{m_s} \hat{\rho})\]

Coherences:

\[C_{01} = |\rho_{0,1}| = |\langle m_s=0|\hat{\rho}|m_s=1\rangle|\]

Bloch Vector¶

The electron spin state can be visualized on a generalized Bloch sphere:

\[\vec{S} = (\langle \hat{S}_x \rangle, \langle \hat{S}_y \rangle, \langle \hat{S}_z \rangle)\]

For a spin-1 system, \(|\vec{S}| \leq 1\) with equality only for pure states in the spin subspace.

Purity¶

The purity of a quantum state:

\[\gamma = \text{Tr}(\hat{\rho}^2)\]

\(\gamma = 1\): Pure state
\(\gamma = 1/d\): Maximally mixed state (for dimension \(d\))
\(\gamma \in [1/18, 1]\) for NV center

Optical Readout¶

Fluorescence-Based Readout¶

The NV center is read out optically by detecting fluorescence:

Illuminate with 532 nm laser
Collect red fluorescence (637-800 nm)
\(m_s = 0\) is brighter than \(m_s = \pm 1\)

The fluorescence rate depends on the spin state:

\[\begin{split}R_{m_s=0} &= \eta \cdot \gamma_{\text{rad}} \cdot P_e^{(m_s=0)} \\ R_{m_s=\pm 1} &= \eta \cdot \gamma_{\text{rad}} \cdot P_e^{(m_s=\pm 1)} \cdot (1 - p_{\text{ISC}})\end{split}\]

where:

\(\eta\): Collection efficiency
\(\gamma_{\text{rad}}\): Radiative decay rate
\(P_e\): Excited state population
\(p_{\text{ISC}}\): Intersystem crossing probability

Contrast¶

The readout contrast is defined as:

\[C = \frac{R_{m_s=0} - R_{m_s=\pm 1}}{R_{m_s=0}}\]

Typical values: \(C \approx 20-30\%\)

Signal-to-Noise Ratio¶

For \(N\) photons detected, the SNR for distinguishing \(m_s = 0\) from \(m_s = \pm 1\):

\[\text{SNR} = \frac{C \cdot N}{\sqrt{N}} = C \sqrt{N}\]

Single-shot readout requires \(\text{SNR} > 1\), i.e., \(N > 1/C^2 \approx 10-25\) photons.

Photon Detection¶

Photon Statistics¶

For a coherent optical field, detected photons follow Poisson statistics:

\[P(n) = \frac{\bar{n}^n e^{-\bar{n}}}{n!}\]

where \(\bar{n}\) is the mean photon number.

Detector Models¶

The simulator includes realistic detector effects:

Detection efficiency \(\eta\):

APD: 5-20%
SNSPD: 70-95%

Dark counts: Background counts even without signal

\[N_{\text{dark}} = R_{\text{dark}} \cdot T_{\text{int}}\]

Dead time \(\tau_d\): Detector is blind after each detection

\[R_{\text{measured}} = \frac{R_{\text{true}}}{1 + R_{\text{true}} \cdot \tau_d}\]

Timing jitter \(\sigma_t\): Uncertainty in photon arrival time

Afterpulsing: Spurious counts following a detection

Second-Order Correlation¶

The \(g^{(2)}(\tau)\) function characterizes photon statistics:

\[g^{(2)}(\tau) = \frac{\langle I(t) I(t+\tau) \rangle}{\langle I(t) \rangle^2}\]

For single NV centers:

\(g^{(2)}(0) < 0.5\): Single-photon emission (antibunching)
\(g^{(2)}(\tau \to \infty) = 1\): Uncorrelated at long times

Fano Factor¶

The Fano factor characterizes count fluctuations:

\[F = \frac{\text{Var}(n)}{\langle n \rangle}\]

\(F = 1\): Poissonian (coherent light)
\(F < 1\): Sub-Poissonian (single photon source)
\(F > 1\): Super-Poissonian (bunched light)

ODMR Spectroscopy¶

Continuous Wave ODMR¶

In CW-ODMR, the fluorescence is monitored while sweeping the microwave frequency:

\[S(f_{\text{MW}}) = S_0 \left(1 - C \cdot L(f_{\text{MW}} - f_0)\right)\]

where \(L(f)\) is a Lorentzian lineshape:

\[L(f) = \frac{\Gamma^2/4}{f^2 + \Gamma^2/4}\]

with linewidth \(\Gamma \approx 1/(\pi T_2^*)\).

The resonance frequencies are:

\[f_\pm = D \pm \gamma_e B_z / (2\pi)\]

Pulsed ODMR¶

In pulsed ODMR:

Initialize with laser pulse → \(m_s = 0\)
Apply MW \(\pi\)-pulse
Readout with laser pulse

The signal shows the population transfer efficiency as a function of MW frequency.

Ramsey Interferometry¶

The Ramsey sequence measures free precession:

\[P_{m_s=0}(\tau) = \frac{1}{2}\left(1 + e^{-\tau/T_2^*} \cos(\Delta \omega \cdot \tau)\right)\]

where \(\Delta\omega = \omega_{\text{MW}} - \omega_0\) is the detuning.

From the oscillation frequency, the local field can be determined with sensitivity:

\[\delta B \approx \frac{1}{\gamma_e \sqrt{T_2^* \cdot T_{\text{meas}}}}\]