Squeezed states of light: quadrature components
For a long time, the precision of optical measurements was thought to be fundamentally limited by the shot noise of stable light beams. This limit is known as the standard quantum limit (SQL). It was believed to be an insurmountable barrier for technologies relying on light measurements.
The SQL arises from the inherent quantum fluctuations in what we considered to be perfectly stable, quasi-classical light beams. These beams, while stable in intensity, still exhibit fluctuations at the quantum level. However, a theoretical breakthrough in the early 1980s suggested a way to go beyond this limitation: squeezed states of light.
The core idea behind squeezed states is to redistribute quantum fluctuations. Instead of having equal fluctuations in all aspects of the light field, we can “squeeze” these fluctuations in a way that reduces them in a specific observable at the expense of increasing them in another, conjugate observable. While directly observing the electric field oscillations at optical frequencies is currently impossible with existing technology, we can measure time-independent observables called quadrature components using balanced homodyne detection.
Squeezed states are characterized by reduced dispersion in one of these quadratures. It allows for more accurate measurements of either the amplitude or phase of an optical wave. For example, consider the quadrature \mathbf Q_{\lambda}, which can be expressed in terms of creation and annihilation operators as:
\mathbf Q_{\lambda} = \sqrt{\frac{\hbar}{2}}\left( \mathbf a_\lambda + \mathbf a_\lambda ^\dag \right)
For squeezed states, the dispersion of this quadrature, \Delta Q_\lambda, can be shown to be:
\Delta Q_\lambda = \sqrt{\frac{\hbar}{2}}e^{-R_\lambda}
where R_\lambda is a squeezing parameter. If R_\lambda > 0, the dispersion \Delta Q_\lambda is reduced compared to the standard quantum limit (\sqrt{\frac{\hbar}{2}}). However, there is a trade-off. The dispersion of the other quadrature, \mathbf P_{\lambda}:
\mathbf P_{\lambda} = -i\sqrt{\frac{\hbar}{2}}\left( \mathbf a_\lambda - \mathbf a_\lambda ^\dag \right)
becomes:
\Delta P_\lambda = \sqrt{\frac{\hbar}{2}}e^{R_\lambda}
The product of these dispersions still respects the Heisenberg uncertainty principle: \Delta Q_\lambda \Delta P_\lambda = \frac{\hbar}{2}. This means we cannot simultaneously reduce the fluctuations in both quadratures. However, for measurements where we are interested in only one quadrature, squeezed states offer a significant advantage.
By using squeezed states and measuring the appropriate quadrature, we can achieve measurement accuracies that surpass the traditional standard quantum limit. This has implications for various fields, including gravitational wave detection, where squeezed light is already being used to enhance the sensitivity of detectors, and quantum metrology.
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