Fragility of queezed states of light
Squeezed states are a recent development used for advanced quantum measurements, offering reduced quantum noise in one quadrature at the expense of increased noise in the other. This noise reduction is what makes squeezed states so appealing for applications like gravitational wave detection, where minute signals need to be extracted from background noise. However, the effectiveness of squeezing is highly sensitive to optical losses. This fragility is a significant hurdle in deploying squeezed states outside specialized laboratory settings.
Optical losses, whether from absorption in optical components, scattering, or imperfect detector efficiency, inevitably degrade squeezing.
To understand how losses impact squeezed states, I consider a model using a beam splitter. In this model, a beam splitter represents the loss mechanism, diverting a portion of the light, which corresponds to the loss, away from the primary beam path.
Imagine a squeezed state entering input (1) of a beam splitter, while the other input (2) is in a vacuum state. The output of interest is (4), representing the detected signal after loss. Mathematically, the annihilation operator at output (4), \mathbf a_{\lambda_4}, can be expressed in terms of the annihilation operators at input ports (1) and (2), \mathbf a_{\lambda_1} and \mathbf a_{\lambda_2}, and the beam splitter transmission (t) and reflection (r) coefficients:
\mathbf a_{\lambda_4} = t\mathbf a_{\lambda_1} - r \mathbf a_{\lambda_2}
Considering a squeezed state |\boldsymbol \alpha_\lambda, R_\lambda \rangle_1 at input (1) and vacuum | \mathbf 0 \rangle_2 at input (2), I can analyze the impact of losses on the squeezed quadrature \mathbf Q_{\lambda 4} = \sqrt{\frac{\hbar}{2}}\left(\mathbf a_{\lambda_4} + \mathbf a_{\lambda_4}^\dag \right).
After some algebra, the average value of the output quadrature becomes:
\langle \mathbf Q_{\lambda 4} \rangle = \sqrt{2\hbar} t e^{-R_{\lambda_1}} \alpha_{{\lambda R}_1}
This shows that the average quadrature amplitude is reduced by the transmission factor t due to losses. The variance of the output quadrature, which dictates the noise level, is given by:
\Delta Q_{\lambda 4} = t^2 e^{-2R_{\lambda_1}} + r^2
The term r^2 arises from the vacuum fluctuations entering through the unused input of the beam splitter. This means that losses effectively introduce vacuum noise into the squeezed state, thus reducing the degree of squeezing achieved. Even starting with a highly squeezed state, losses will inevitably add vacuum noise, pushing the state closer to the standard quantum limit and diminishing the noise reduction benefits.
This analysis shows the challenge in utilizing squeezed states in real-world applications. Maintaining extremely low-loss optical paths and highly efficient detectors is a requirement to preserve the advantages of squeezing. The inherent fragility of squeezed states to losses is a key reason why their practical application remains largely confined to specialized and carefully controlled environments.
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