Optical Sensing Principles

This section bridges the gap between classical electromagnetism and the general workflow of sensing.

1. The Sensing Workflow

1. State Preparation: Initializing the probe (e.g., a laser beam).

2. Parameter Encoding: The physical parameter $\theta$ interacts with the probe, modifying its state (e.g., phase shift).

   $$E(t, \theta) \approx E(t) + \theta \frac{\partial E(t, \theta)}{\partial \theta}$$

3. Detection: Measuring the probe.

4. Estimation: Using the data to estimate $\theta$ (the result $X$).

2. Classical Wave Optics

• Maxwell’s Equations: Govern the propagation of electromagnetic waves.

  $$\nabla^2 E - \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = 0$$

• Plane Wave Solution:

  $$\vec{E}(t, \vec{r}) = \vec{E}_0 e^{i(\vec{k}\cdot\vec{r} - \omega t)}$$

• Intensity: $I = |\mathcal{E}_0|^2$ (Proportional to the square of the complex amplitude).

• Polarization: Describes the orientation of the oscillating electric field.

  • $|H\rangle$: Horizontally polarized.
  • $|V\rangle$: Vertically polarized.

3. Interaction with Material (Non-Linear Optics)

When light interacts with a material, it induces a dipole moment.

• Polarization Density ($\vec{P}$):

  $$\vec{P} = \epsilon_0 (\chi^{(1)}\vec{E} + \chi^{(2)}\vec{E}^2 + \chi^{(3)}\vec{E}^3 + \dots)$$
  • $\chi^{(1)}$: Linear susceptibility (Refraction, Absorption).
  • $\chi^{(2)}$: Second-order non-linearity. Responsible for Second Harmonic Generation (SHG), where two photons of frequency $\omega$ combine to form one photon of $2\omega$.

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Photodetection and Noise

Detecting light involves converting photons into electron-hole pairs. This process is inherently statistical.

1. Key Detector Metrics

• Responsivity ($R$): Output electrical current per unit optical power ($A/W$).

  $$R = \frac{e \lambda \eta}{hc}$$
  • $\eta$: Quantum Efficiency (fraction of photons converted to electrons).
  • $\lambda$: Wavelength.

• Sensitivity: The smallest change in the parameter $\theta$ that can be detected.

• SNR (Signal-to-Noise Ratio):

  $$SNR = \frac{\text{Signal Power}}{\text{Noise Power}}$$

2. Noise Mechanisms

• Shot Noise: Arises from the discrete nature of photons (Poissonian statistics) and charge carriers.

  $$i_{shot} = \sqrt{2 e I_{avg} \Delta f}$$
  • $e$: Electron charge ($1.6 \times 10^{-19}$ C)
  • $\Delta f$: Bandwidth

• Thermal (Johnson) Noise: Caused by thermal agitation of electrons in a resistor.

  $$V_{thermal} = \sqrt{4 k_B T R \Delta f}$$
  • $k_B$: Boltzmann constant.
  • $T$: Temperature (Kelvin).

• Flicker Noise ($1/f$): Dominates at low frequencies, often due to material defects.

3. Calculation Example (from notes)

Scenario: A Silicon Avalanche Photodiode (APD) with Gain $M=100$.

• Given:

  • Responsivity $R = 0.6$ A/W (at 850 nm)
  • Incident Power $P = 5 \mu W$
  • Bandwidth $\Delta f = 1$ MHz

• Primary Photocurrent:

  $$I_{primary} = P \times R = 5 \mu W \times 0.6 A/W = 3 \mu A$$

• Amplified Current (Gain $M=100$):

  $$I_{gain} = 3 \mu A \times 100 = 300 \mu A$$

• Shot Noise Calculation (using amplified current):

  $$i_{shot} = \sqrt{2 \cdot (1.6 \times 10^{-19}) \cdot (300 \times 10^{-6}) \cdot (10^6)}$$ $$i_{shot} \approx 9.8 \times 10^{-9} A = 9.8 nA$$

4. Important Detectors

• PIN Photodiode: Standard detector, no internal gain.
• APD (Avalanche Photodiode): Uses high reverse bias to create gain (impact ionization). Good for low light but adds excess noise.
• SPAD (Single Photon Avalanche Diode): Operated above breakdown voltage (Geiger mode) to detect single photons.
• SNSPD (Superconducting Nanowire Single Photon Detector): Extremely sensitive, operates at cryogenic temps (0.5K - 4K), very low noise (dark counts).

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Estimation Theory & Limits

This section deals with “how well” we can measure something, bounded by statistics.

1. Statistics of Detection

• Poisson Distribution: Describes photon counting (random arrival of photons).

  $$P(n|\lambda) = \frac{e^{-\lambda}\lambda^n}{n!}$$
  • Variance = Mean ($\sigma^2 = \mu$).

2. Fisher Information ($I(\theta)$)

Fisher Information quantifies how much “information” a random variable $X$ carries about an unknown parameter $\theta$.

• It measures the sensitivity of the probability distribution $p(x|\theta)$ to changes in $\theta$.

  $$I(\theta) = E \left[ \left( \frac{\partial \ln p(x|\theta)}{\partial \theta} \right)^2 \right] = -E \left[ \frac{\partial^2 \ln p(x|\theta)}{\partial \theta^2} \right]$$

• Interpretation: High Fisher Information $\rightarrow$ Sharp peak in distribution $\rightarrow$ High sensitivity (Lower uncertainty).

3. Cramer-Rao Bound (CRB)

The fundamental limit on the precision of any measurement. The variance of an unbiased estimator $\hat{\theta}$ is bounded by the inverse of the Fisher Information.

$$\text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)}$$

• This implies that Noise (Variance) and Information are inversely related.

4. Shannon Entropy

A measure of randomness or uncertainty in a variable $X$.

$$H(X) = - \sum p(x) \log_2 p(x)$$

• For a Gaussian distribution with variance $\sigma^2$:

  $$H(x) = \frac{1}{2} \log(2\pi e \sigma^2)$$

• Connection: Fisher information for a Gaussian parameter $\mu$ is $1/\sigma^2$.

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Quantum Fundamentals & Operators

This section expands on the mathematical framework, including operators, measurement, and gates.

1. Operators and Observables

Physical quantities (energy, momentum, spin) are represented by Operators acting on the Hilbert space.

• Hermitian Operator ($\hat{A}$): An operator that is equal to its conjugate transpose ($\hat{A} = \hat{A}^\dagger$).

  • These represent Observables because their eigenvalues are always real.
  • $\hat{A}|\lambda_n\rangle = \lambda_n |\lambda_n\rangle$ (Eigenvalue equation).

• Mean / Expectation Value:

  $$\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle = \sum \lambda_n |a_n|^2$$

• Standard Deviation (Uncertainty):

  $$\Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2}$$

• Commutators: Two operators commute if $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} = 0$. If they do not commute (e.g., Position and Momentum, or Pauli matrices), they cannot be simultaneously measured with arbitrary precision.

2. Pauli Matrices & Gates

The fundamental operators for qubits (Spin-1/2 systems).

• Pauli-X ($\sigma_x$): Bit flip. Exchanges $|0\rangle \leftrightarrow |1\rangle$.

  $$\sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

• Pauli-Y ($\sigma_y$):

  $$\sigma_y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$$

• Pauli-Z ($\sigma_z$): Phase flip.

  $$\sigma_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

• Hadamard Gate ($\hat{H}$): Creates superposition.

  $$\hat{H} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$ $$\hat{H}|0\rangle = |+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad \hat{H}|1\rangle = |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$$

3. Measurement Theory

• Projective Measurement:
  • Defined by Projection Operators $\hat{\Pi}_n = |\lambda_n\rangle\langle\lambda_n|$.
  • Born Rule: Probability of outcome $\lambda_n$ is $P(\lambda_n) = |\langle \lambda_n | \psi \rangle|^2 = \langle \psi | \hat{\Pi}_n | \psi \rangle$.
  • Post-Measurement State: $|\psi'\rangle = \frac{\hat{\Pi}_n |\psi\rangle}{\sqrt{P(\lambda_n)}}$ (Normalized).
• POVM (Positive Operator-Valued Measure):
  • A more general measurement formalism.
  • Set of operators $\{\hat{M}_n\}$ such that $\sum \hat{M}_n = \hat{I}$ and $\hat{M}_n \ge 0$.
  • Probability $P(n) = \text{Tr}(\hat{M}_n \rho)$.

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Density Matrix & Multiple Systems

1. The Density Matrix ($\rho$)

Used to describe both pure and mixed states.

• Pure State: $\rho = |\psi\rangle\langle\psi|$. Purity $\text{Tr}(\rho^2) = 1$.

• Mixed State: $\rho = \sum p_i |\psi_i\rangle\langle\psi_i|$. Purity $\text{Tr}(\rho^2) < 1$.

• Properties:

  1. Hermitian ($\rho = \rho^\dagger$).
  2. Positive Semi-definite (Eigenvalues $\ge 0$).
  3. Trace $\text{Tr}(\rho) = 1$ (Sum of probabilities).

• Von Neumann Entropy: A measure of uncertainty or entanglement.

  $$S(\rho) = -\text{Tr}(\rho \log \rho)$$
  • For a pure state, $S(\rho) = 0$.
  • For a maximally mixed state of dimension $d$, $S(\rho) = \log d$.

2. Multiple Quantum Systems

• Tensor Product ($\otimes$): The joint state space of system A and B is $H_A \otimes H_B$.

  • If $|\psi\rangle_A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $|\phi\rangle_B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$, then:

    $$|\psi\rangle_A \otimes |\phi\rangle_B = \begin{bmatrix} a_1 b_1 \\ a_1 b_2 \\ a_2 b_1 \\ a_2 b_2 \end{bmatrix}$$

• Partial Trace: Used to obtain the state of a subsystem (Reduced Density Matrix) by averaging out the other system.

  $$\rho_A = \text{Tr}_B(\rho_{AB}) = \sum_i \langle i|_B \rho_{AB} |i\rangle_B$$

3. Entanglement

A state is entangled if it cannot be written as a product of individual states ($|\psi\rangle \neq |\phi\rangle_A \otimes |\chi\rangle_B$).

• Bell States (Maximally Entangled):

  $$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$$
  • Strong non-classical correlations.

• GHZ State (3 qubits): $|\text{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}$.

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Measures of Correlation

This section quantifies how coupled two systems are, distinguishing between classical and quantum correlations.

1. Mutual Information ($I(A:B)$)

Quantifies the total correlation (classical + quantum) between subsystem A and B.

$$I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB})$$

• Example: Bell State $|\Phi^+\rangle$

  • The joint state is pure, so $S(\rho_{AB}) = 0$.
  • The reduced states $\rho_A$ and $\rho_B$ are maximally mixed ($I/2$), so $S(\rho_A) = 1$ and $S(\rho_B) = 1$ (in bit base 2).
  • Result: $I(A:B) = 1 + 1 - 0 = 2$. (This is double the correlation max possible for classical bits).

• Example: Fully Mixed Correlated State

  • Classical mixture: $\rho = \frac{1}{2}(|00\rangle\langle00| + |11\rangle\langle11|)$.
  • $S(\rho_{AB}) = 1$ (1 bit of randomness in the choice of 00 vs 11).
  • $S(\rho_A) = 1, S(\rho_B) = 1$.
  • Result: $I(A:B) = 1 + 1 - 1 = 1$.

2. Quantum Discord

Separates correlations into “Classical” and “Quantum” parts.

• Classical Correlation $C(A,B)$: The maximum information we can gain about B by measuring A.

  $$C(A,B) = \max_{\{\Pi_i^A\}} [ S(\rho_B) - \sum p_i S(\rho_{B|i}) ]$$

• Quantum Discord $D(A,B)$: The remaining non-classical correlation.

  $$D(A,B) = I(A,B) - C(A,B)$$
  • For pure entangled states, all correlation is quantum (Discord = Mutual Information).
  • For purely classical states, Discord is zero.