Wikipedia / Quantum limit
Wikipedia / Quantum limit
A quantum limit in physics is a limit on measurement accuracy at quantum scales. Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring quantum states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes.
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. In short, it is the Heisenberg uncertainty principle that is the cause. A more detailed explanation would be that any measurement in quantum mechanics, of this system, e.g. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp.38–42 of and modifying the results of subsequent measurements. This is known as back action (quantum) of the Meter on the system under measurement.
. This one is known as measurement imprecision or measurement noise. Because of the Heisenberg uncertainty principle, this imprecision cannot be arbitrary and is linked to the back-action perturbation by the uncertainty relation:
stands for expectation value in whatever quantum state the system is. The equality is reached if the system is in a minimum uncertainty state. The consequence for our case is that the more precise is our measurement, i.e. Therefore, the readout of the meter will, in general, consist of three terms:
are uncorrelated.
The terms "quantum limit" and "standard quantum limit" are sometimes used interchangeably. Usually, "quantum limit" is a general term which refers to any restriction on measurement due to quantum effects, while the "standard quantum limit" in any given context refers to a quantum limit which is ubiquitous in that context.
Examples
Measuring gravitational radiation
An interferometer measuring distances between mirrors is used to detect gravitational waves For a mirror of mass measured over will be limited in accuracy to .
Displacement measurement
Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.
at the moment of reflection:
, and their root mean square.
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 and references therein), or other such read-out techniques.
. In this case, the initial uncertainty will be the only source of the position measurement error:
For convenience, we renormalise Eq. () 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
of the Meter. The major part of this perturbation is contributed by classical radiation pressure:
the mean energy of the pulses. 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
the RMS uncertainty of the pulse energy.
later:
Its uncertainty will be simply
pulses, i.e. its displacement, we will have to deal with three additional uncertainties that limit precision of our estimate:
where we assumed all contributions to our measurement uncertainty.
Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? 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. ( and due to back action also satisfy the uncertainty relation:
. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:
) and the limit of displacement measurement error will reduce to:
which is known as the Standard Quantum Limit for the measurement of free mass displacement.
This example represents a simple particular case of a linear measurement. This class of measurement schemes can be fully described by two linear equations of the form~() and ( 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).
Usage in quantum optics
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.
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
Note that due to an overloading of the word "limit", the classical limit is not the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the Armstrong limit). In "classical limit", "limit" is used in the sense of a limiting process. (Note that there is no simple rigorous mathematical limit which fully recovers classical mechanics from quantum mechanics, the Ehrenfest theorem notwithstanding. Nevertheless, in the phase space formulation of quantum mechanics, such limits are more systematic and practical.)