Quantum noise in Mach-Zehnder interferometers
In precision measurements, the Mach-Zehnder interferometer is a tool used for detecting small changes in optical path lengths. By splitting a beam of light and recombining it after it traverses two different paths, variations in arm lengths can be inferred from the interference pattern. A common technique to enhance sensitivity involves measuring the difference between the two output signals of the interferometer.
Classically, we might expect that increasing the intensity of the input light source would indefinitely improve measurement precision. However, this intuition clashes with reality. There is a fundamental noise limit, often referred to as shot noise, that arises even with intense light sources. This noise limits the signal-to-noise ratio (SNR), hindering our ability to detect very small phase shifts \varepsilon.
In a semi-classical description, the signal in a Mach-Zehnder interferometer, when operating at optimal sensitivity, is proportional to the phase shift \varepsilon and the input photon number N_2:
\langle \mathbf N_{\lambda_6} - \mathbf N_{\lambda_5} \rangle \approx N_2 \varepsilon
The noise, in this view, is attributed to the statistical fluctuations in photon detection, leading to a noise level proportional to the square root of the input photon number \sqrt{N_2}. The resulting SNR is then:
\text{SNR} = \varepsilon \sqrt{N_2}
While this equation suggests that increasing N_2 improves SNR, it does not explain the fundamental origin of this noise or how to overcome this limit. A full quantum mechanical treatment is needed to understand the true nature of noise in interferometers.
By considering the quantum operators for photon annihilation and creation, \mathbf a_\lambda and \mathbf a_\lambda^\dag respectively, we can analyze the interferometer’s output in terms of quantum fields. Even when one input port of the interferometer is seemingly unused, it is still interacting with the vacuum field.
Based on the quantum description of light, it is possible to calculate that the fluctuations in the output signal are not simply due to detector noise, but originate from the interference of the input laser field with vacuum fluctuations entering the interferometer through the empty input port, so the noise is a consequence of quantum vacuum fluctuations.
If vacuum fluctuations are the source of noise, then manipulating the quantum state of light entering the “vacuum” port offers a pathway to reduce noise and enhance sensitivity. Squeezed light, a non-classical state of light with reduced fluctuations in one quadrature at the expense of increased fluctuations in the other, becomes a potential tool for surpassing the classical shot noise limit and pushing the boundaries of precision measurements.
For more insights into this topic, you can find the details here.