Quantum Sensors & Metrology Beyond the Standard Quantum Limit

Simulation of quantum-enhanced metrology using squeezed states and Ramsey interferometry. Models phase estimation precision scaling, atomic clock stability, and quantum noise spectra for sub-SQL sensing applications.

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🎯 Key Physics Demonstrated

Concept	Description	Equation
Ramsey Interferometry	Population oscillations for phase sensing	$P_{
SQL vs HL Scaling	Standard vs Heisenberg precision limits	SQL: $\sim 1/\sqrt{N}$ → HL: $\sim 1/N$
Squeezed States	Sub-SQL uncertainty in one quadrature	$W(x,p) = \frac{1}{\pi}\exp(-e^{-2r}x^2 - e^{2r}p^2)$
Noise Spectrum	Squeezed light below shot noise	$S(\Omega) < 0.5$ (SQL)
Atomic Clocks	Allan deviation stability analysis	$\sigma_y(\tau) \propto 1/(\nu_0\sqrt{N\tau})$
Bayesian Estimation	Adaptive phase estimation protocol	$P(\phi

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📊 Generated Figures

Figure 1: Ramsey Interferometry

• (a) Ramsey fringes for different detunings
• (b) Phase sensitivity (fringe slope)
• (c) Pulse sequence diagram
• (d) Shot noise simulation

Figure 2: Quantum Metrology Scaling Laws

• (a) SQL vs HL with squeezed state interpolation
• (b) Wineland squeezing parameter
• (c) State-dependent phase estimation
• (d) Quantum Fisher Information

Figure 3: Squeezed State Wigner Function

• (a) 2D Wigner function contour plot
• (b) Cross-sections for different r
• (c) Uncertainty ellipses in phase space
• (d) Quadrature variances vs squeezing

Figure 4: Squeezed Light Noise Spectrum

• (a) Noise spectrum below SQL
• (b) Noise reduction vs squeezing parameter
• (c) Multiple squeezing levels
• (d) Balanced homodyne detection schematic

Figure 5: Atomic Clock Stability

• (a) Allan deviation: Cs, ion, optical lattice
• (b) Scaling with atom number
• (c) Dick effect & laser noise coupling
• (d) Stability budget comparison

Figure 6: Quantum Sensor Architecture

Full pipeline from state preparation through Bayesian estimation

Figure 7: Bayesian Phase Estimation

• (a) Posterior convergence
• (b) Error scaling vs measurements
• (c) Adaptive phase selection
• (d) Cumulative Fisher information

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