Squeezed states of light representation in the phasor plane
For amplitude-squeezed light, the variance in the amplitude quadrature, \mathbf Q_\lambda, is reduced. This is particularly useful for absorption measurements. In conventional methods employing quasi-classical states, improving measurement accuracy often requires increasing beam intensity, which can damage sensitive samples. Squeezed light offers an alternative by directly reducing the quantum noise in the amplitude measurement. For amplitude measurements using balanced homodyne detection, the relative uncertainty with squeezed light is given by:
\frac{\Delta Q_\lambda}{\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf{Q}_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle} = \frac{e^{-R_\lambda}}{2\alpha_\lambda^\prime}
This is a factor e^{-R_\lambda} smaller than what can be achieved with a quasi-classical state of the same amplitude \alpha_\lambda^\prime. A squeezed state with real amplitude \alpha_{\lambda R} and squeezing parameter R_\lambda > 0 has approximately the same average photon number, and therefore the same power, as a quasi-classical state with amplitude \alpha_\lambda^\prime. This is shown by calculating the average photon number for a squeezed state:
\langle \boldsymbol \alpha_\lambda, R_\lambda | \mathbf N_{\lambda} | \boldsymbol \alpha_\lambda, R_\lambda \rangle = \left(\alpha_\lambda^\prime\right)^2 + \sinh^2 \left(R_\lambda\right)
For large \alpha_\lambda^\prime, the \sinh^2 \left(R_\lambda\right) term becomes negligible, and the power is comparable to the quasi-classical state power |\alpha_\lambda^\prime|^2.
Conversely, for phase-squeezed light, the variance in the phase quadrature, \mathbf P_\lambda, is reduced (corresponding to negative R_\lambda). This leads to enhanced precision in phase measurements, which is critical for applications like interferometry. The uncertainty in phase measurement using phase-squeezed states is:
\Delta \varphi = \frac{\Delta P_\lambda}{\sqrt{2\hbar}} \frac{1}{\alpha_\lambda^\prime} = \frac{e^{R_\lambda}}{2\alpha_{\lambda R}}
Phase-squeezed states allow for phase determination with an uncertainty below the standard quantum limit, improving the sensitivity of interferometers for detecting small phase variations.
Squeezed states of light offer a pathway to surpass the limitations of classical light in precision measurements. By reducing quantum noise in specific quadratures without increasing beam power, they enable more sensitive measurements for delicate samples and in power-constrained scenarios, opening new avenues for advanced quantum technologies.
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