4 Linear Quantum Measurement

In Section 3, we discussed the quantum nature of light and fluctuations of the light field observables like phase and amplitude that stem thereof and yield what is usually called the quantum noise of optical measurement. In GW detection applications, where a sensitivity of the phase measurement is essential, as discussed in Section 2.1.3, the natural question arises: is there a limit to the measurement precision imposed by quantum mechanics? A seemingly simple answer would be that such a limit is set by the quantum fluctuations of the outgoing light phase quadrature, which are, in turn, governed by the quantum state the outgoing light finds itself in. The difficult part is that on its way through the interferometer, the light wave inflicts an additional back-action noise that adds up to the phase fluctuations of the incident wave and contaminates the output of the interferometer. The origin of this back action is in amplitude fluctuations of the incident light, giving rise to a random radiation pressure force that acts on the interferometer mirrors along with the signal GW force, thus effectively mimicking it. And it is the fundamental principle of quantum mechanics, the Heisenberg uncertainty principle, that sets a limit on the product of the phase and amplitude uncertainties (since these are complementary observables), thus leading up to the lower bound of the achievable precision of phase measurement. This limit appears to be a general feature for a very broad class of measurement known as linear measurement and is referred to as the SQL [16, 22Jump To The Next Citation Point].

In this section, we try to give a brief introduction to quantum measurement theory, starting from rather basic examples with discrete measurement and then passing to a general theory of continuous linear measurement. We introduce the concept of the SQL and derive it for special cases of probe bodies. We also discuss briefly possible ways to overcome this limit by contriving smarter ways of weak force measurement then direct coordinate monitoring.

 4.1 Quantum measurement of a classical force
  4.1.1 Discrete position measurement
  4.1.2 From discrete to continuous measurement
 4.2 General linear measurement
 4.3 Standard Quantum Limit
  4.3.1 Free mass SQL
  4.3.2 Harmonic oscillator SQL
  4.3.3 Sensitivity in different normalizations. Free mass and harmonic oscillator
 4.4 Beating the SQL by means of noise cancellation
 4.5 Quantum speed meter
  4.5.1 The idea of the quantum speed meter
  4.5.2 QND measurement of a free mass velocity

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