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Standard Quantum Limit (SQL)

Shot noise and radiation pressure noise

 

noise spectrum

 

Above figure shows the quantum-noise curves of a gravitational-wave detector with different light power in the unit of the strain sensitivity (= displacement sensitivity / baseline). Quantum noise consists of shot noise and quantum radiation pressure noise. Shot noise is inversely proportional to the signal response of the interferometer. As the signal at high frequencies cancels during the circulation in the arm cavity, the shot-noise curves increases by f at high frequencies. Radiation pressure noise is white force noise, thus it increases by f^-2 at low frequencies.

 

Raising the incident light power, shot noise decreases and radiation pressure increases. Consequently the square sum of two cannot exceeds the black f^-1 line, which is called the SQL. Shot noise represents the displacement information of the test mass. As we increase the precision by raising the power, back action of the measurement, appearing as radiation pressure noise, makes the sensitivity worse. This phenomenon shows what is predicted by the Heisenberg's uncertainty principle of the displacement and momentum.

 

Displacement and momentum

Quantum Mechanics tells us that displacement (x) and momentum (p) are operators that do not commute (i.e. they are correlated):

formula 2

This results in the Heisenberg's uncertainty principle:

formula 3

Let us assume the test mass is free (suspension is ignored). The momentum at time t influences the displacement at a later time, namely

formula 4

This means that the displacements at different time do not commute. The minimum value of the fluctuation is then given by

formula 5
Here the measurement time is given as
formula 6
and the power spectrum is given by
formula 7
Therefore, we have
formula 8

This is the SQL. In a Michelson interferometer with Fabry-Perot arm cavities, we use m/4 instead of m as our target motion is the differential-mode motion of 4 mirrors. And we divide this by L^2 (L is the arm length) and take the square root of it when we show the sensitivity in strain.

 

Why can we exceed the limit?

It is because we are actually measuring the force with a gravitational-wave detector. Those x and p shown above are the initial location and the initial momentum, respectively. If we want to measure the initial x and p, Quantum Mechanics does not allow us to increase the accuracy for both x and p; we cannot exceed the SQL. However, the initial information has already disappeared in the gravitational-wave detector. Environmental decoherence has hidden the quantum property, which we actually do not need to measure the gravitational waves.

 

Nevertheless, the SQL exists in the gravitational-wave detector if we aim at the conventional measurement of the phase quadrature component (see QND for the introduction of the vacuum fluctuation and the quadrature). Unlike the operator x, the output field operator commutes at different times. But there are two components in the phase quadrature that do not commute, which results in the limit. Each of them is a quadrature component of the vacuum fluctuation coming in from the dark port of the interferometer. One is the source of shot noise and the other is the source of radiation pressure noise. Since the latter appears as a result of the opto-mechanical coupling, the SQL reads the same value as the one derived above. This, however, does not impose a fundamental limit, and hence there can be various ways to overcome the SQL in the sensitivity curve of the gravitational-wave detector.

 

If we aim at the measurement of the initial x and p, or those quantities recovered by the measurement (conditional x and p), the SQL will be the limit that cannot be overcome. The measured variance of x and p, together with their covariance, will follow the Heisenberg's uncertainty principle. Interestingly, in order to prepare this conditional state, we need an interferometer whose classical noise is lower than the SQL (the beatable SQL). So, the SQL is finally important.

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