Displacement measurement Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a
general position measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body M. The position x of M is probed periodically with time interval \vartheta. We assume mass M large enough to neglect the displacement inflicted by the pulses regular (classical)
radiation pressure in the course of measurement process. Then each j-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position x(t_j) at the moment of reflection: {{NumBlk|:| \hat{\phi}_j^{\mathrm{refl}} = \hat{\phi}_j - 2 k_p\hat{x}(t_j) \,, |}} where k_p=\omega_p/c, \omega_p is the light frequency, j=\dots,-1,0,1,\dots is the pulse number and \hat{\phi}_j is the initial (random) phase of the j-th pulse. We assume that the mean value of all these phases is equal to zero, \langle\hat{\phi}_j\rangle=0, and their
root mean square (RMS) uncertainty (\langle\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2} is equal to \Delta\phi. The reflected pulses are detected by a phase-sensitive device (the
phase detector). The implementation of an optical phase detector can be done using
e.g. homodyne or
heterodyne detection schemes (see Section 2.3 in There is additionally a quantum limit for
phase noise, reachable only by a
laser at high noise frequencies. In
spectroscopy, the shortest wavelength in an X-ray spectrum is called the quantum limit. ==Misleading relation to the classical limit==