Quantum limit
The usage of the term standard quantum limit or SQL is, however, broader than just interferometry.
In principle, any linear measurement of a quantum mechanical observable of a system under study that does not commute with itself at different times leads to such limits.
A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter.
The latter is the system we couple to the Object in order to infer the value of
At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty,
The equality is reached if the system is in a minimum uncertainty state.
, the larger will be perturbation the Meter exerts on the measured observable
Thus, there is a minimal value, or the limit to the precision one can get in such a measurement, provided that
In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body
large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.
-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position
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 [2] and references therein), or other such read-out techniques.
measurement error introduced by the detector is much smaller than the initial uncertainty of the phases
In this case, the initial uncertainty will be the only source of the position measurement error: For convenience, we renormalise Eq.
(1) as the equivalent test-mass displacement: where are the independent random values with the RMS uncertainties given by Eq.
Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to where
are the test-mass momentum values just before and just after the light pulse reflection, and
-th pulse, that plays the role of back action observable
The major part of this perturbation is contributed by classical radiation pressure: with
Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator.
The random part, which cannot be compensated, is proportional to the deviation of the pulse energy: and its RMS uncertainly is equal to with
), one can estimate an additional displacement caused by the back action of the
If we further assume that all light pulses are similar and have the same phase uncertainty, thence
The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation: Therefore, it follows from Eqs.
, the light pulse should have in order not to perturb the mirror too much, should be equal to
in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
In the context of interferometry or other optical measurements, the standard quantum limit usually refers to the minimum level of quantum noise which is obtainable without squeezed states.
In spectroscopy, the shortest wavelength in an X-ray spectrum is called the quantum limit.
Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)
