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 ${\displaystyle M}$. The position ${\displaystyle x}$ of ${\displaystyle M}$ is probed periodically with time interval ${\displaystyle \vartheta }$. We assume mass ${\displaystyle M}$ large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.

Simplified scheme of optical measurement of mechanical object position

Then each ${\displaystyle j}$-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position ${\displaystyle x(t_{j})}$ at the moment of reflection:

${\displaystyle {\hat {\phi }}_{j}^{\mathrm {refl} }={\hat {\phi }}_{j}-2k_{p}{\hat {x}}(t_{j})\,,}$

(1)

where ${\displaystyle k_{p}=\omega _{p}/c}$, ${\displaystyle \omega _{p}}$ is the light frequency, ${\displaystyle j=\dots ,-1,0,1,\dots }$ is the pulse number and ${\displaystyle {\hat {\phi }}_{j}}$ is the initial (random) phase of the ${\displaystyle j}$-th pulse. We assume that the mean value of all these phases is equal to zero, ${\displaystyle \langle {\hat {\phi }}_{j}\rangle =0}$, and their root mean square (RMS) uncertainty ${\displaystyle (\langle {\hat {\phi ^{2}}}\rangle -\langle {\hat {\phi }}\rangle ^{2})^{1/2}}$ is equal to ${\displaystyle \Delta \phi }$.

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 scheme (see Section 2.3 in ^[2] and references therein), or other such read-out techniques.

In this example, light pulse phase ${\displaystyle {\hat {\phi }}_{j}}$ serves as the readout observable ${\displaystyle {\mathcal {O}}}$ of the Meter. Then we suppose that the phase ${\displaystyle {\hat {\phi }}_{j}^{\mathrm {refl} }}$ measurement error introduced by the detector is much smaller than the initial uncertainty of the phases ${\displaystyle \Delta \phi }$. In this case, the initial uncertainty will be the only source of the position measurement error:

${\displaystyle \Delta x_{\mathrm {meas} }={\frac {\Delta \phi }{2k_{p}}}\,.}$

(2)

For convenience, we renormalise Eq. (1) as the equivalent test-mass displacement:

${\displaystyle {\tilde {x}}_{j}\equiv -{\frac {{\hat {\phi }}_{j}^{\mathrm {refl} }}{2k_{p}}}={\hat {x}}(t_{j})+{\hat {x}}_{\mathrm {fl} }(t_{j})\,,}$

(3)

where

${\displaystyle {\hat {x}}_{\mathrm {fl} }(t_{j})=-{\frac {{\hat {\phi }}_{j}}{2k_{p}}}}$

are the independent random values with the RMS uncertainties given by Eq. (2).

Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to

${\displaystyle {\hat {p}}_{j}^{\mathrm {after} }-{\hat {p}}_{j}^{\mathrm {before} }={\hat {p}}_{j}^{\mathrm {b.a.} }={\frac {2}{c}}{\hat {\mathcal {W}}}_{j}\,,}$

(4)

where ${\displaystyle {\hat {p}}_{j}^{\mathrm {before} }}$ and ${\displaystyle {\hat {p}}_{j}^{\mathrm {after} }}$ are the test-mass momentum values just before and just after the light pulse reflection, and ${\displaystyle {\mathcal {W}}_{j}}$ is the energy of the ${\displaystyle j}$-th pulse, that plays the role of back action observable ${\displaystyle {\hat {\mathcal {F}}}}$ of the Meter. The major part of this perturbation is contributed by classical radiation pressure:

${\displaystyle \langle {\hat {p}}_{j}^{\mathrm {b.a.} }\rangle ={\frac {2}{c}}{\mathcal {W}}\,,}$

with ${\displaystyle {\mathcal {W}}}$ 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:

${\displaystyle {\hat {p}}^{\mathrm {b.a.} }(t_{j})={\hat {p}}_{j}^{\mathrm {b.a.} }-\langle {\hat {p}}_{j}^{\mathrm {b.a.} }\rangle ={\frac {2}{c}}{\bigl (}{\hat {\mathcal {W}}}_{j}-{\mathcal {W}}{\bigr )}\,,}$

and its RMS uncertainly is equal to

${\displaystyle \Delta p_{\mathrm {b.a.} }={\frac {2\Delta {\mathcal {W}}}{c}}\,,}$

(5)

with ${\displaystyle \Delta {\mathcal {W}}}$ the RMS uncertainty of the pulse energy.

Assuming the mirror is free (which is a fair approximation if time interval between pulses is much shorter than the period of suspended mirror oscillations, ${\displaystyle \vartheta \ll T}$), one can estimate an additional displacement caused by the back action of the ${\displaystyle j}$-th pulse that will contribute to the uncertainty of the subsequent measurement by the ${\displaystyle j+1}$ pulse time ${\displaystyle \vartheta }$ later:

${\displaystyle {\hat {x}}_{\mathrm {b.a.} }(t_{j})={\frac {{\hat {p}}^{\mathrm {b.a.} }(t_{j})\vartheta }{M}}\,.}$

Its uncertainty will be simply

${\displaystyle \Delta x_{\mathrm {b.a.} }(t_{j})={\frac {\Delta {p}_{\mathrm {b.a.} }(t_{j})\vartheta }{M}}\,.}$

If we now want to estimate how much has the mirror moved between the ${\displaystyle j}$ and ${\displaystyle j+1}$ pulses, i.e. its displacement ${\displaystyle \delta {\tilde {x}}_{j+1,j}={\tilde {x}}(t_{j+1})-{\tilde {x}}(t_{j})}$, we will have to deal with three additional uncertainties that limit precision of our estimate:

${\displaystyle \Delta {\tilde {x}}_{j+1,j}={\Bigl [}(\Delta x_{\rm {meas}}(t_{j+1}))^{2}+(\Delta x_{\rm {meas}}(t_{j}))^{2}+(\Delta x_{\rm {b.a.}}(t_{j}))^{2}{\Bigr ]}^{1/2}\,,}$

where we assumed all contributions to our measurement uncertainty statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence ${\displaystyle \Delta x_{\rm {meas}}(t_{j+1})=\Delta x_{\rm {meas}}(t_{j})\equiv \Delta x_{\rm {meas}}=\Delta \phi /(2k_{p})}$.

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:

${\displaystyle \Delta {\mathcal {W}}\Delta \phi \geq {\frac {\hbar \omega _{p}}{2}}\,.}$

Therefore, it follows from Eqs. (2 and 5) that the position measurement error ${\displaystyle \Delta x_{\mathrm {meas} }}$ and the momentum perturbation ${\displaystyle \Delta p_{\mathrm {b.a.} }}$ due to back action also satisfy the uncertainty relation:

${\displaystyle \Delta x_{\mathrm {meas} }\Delta p_{\mathrm {b.a.} }\geq {\frac {\hbar }{2}}\,.}$

Taking this relation into account, the minimal uncertainty, ${\displaystyle \Delta x_{\mathrm {meas} }}$, the light pulse should have in order not to perturb the mirror too much, should be equal to ${\displaystyle \Delta x_{\mathrm {b.a.} }}$ yielding for both ${\displaystyle \Delta x_{\mathrm {min} }={\sqrt {\frac {\hbar \vartheta }{2M}}}}$. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:

${\displaystyle \Delta {\tilde {x}}_{j+1,j}\geqslant {\Bigl [}2(\Delta x_{\rm {meas}})^{2}+{\Bigl (}{\frac {\hbar \vartheta }{2M\Delta x_{\rm {meas}}}}{\Bigr )}^{2}{\Bigr ]}^{1/2}\geqslant {\sqrt {\frac {3\hbar \vartheta }{2M}}}\,.}$

This is the Standard Quantum Limit for such a 2-pulse procedure. In principle, if we limit our measurement to two pulses only and do not care about perturbing mirror position afterwards, the second pulse measurement uncertainty, ${\displaystyle \Delta x_{\rm {meas}}(t_{j+1})}$, can, in theory, be reduced to 0 (it will yield, of course, ${\displaystyle \Delta p_{\rm {b.a.}}(t_{j+1})\to \infty }$) and the limit of displacement measurement error will reduce to:

${\displaystyle \Delta {\tilde {x}}_{SQL}={\sqrt {\frac {\hbar \vartheta }{M}}}\,,}$

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~(3) and (4), provided that both the measurement uncertainty and the object back-action perturbation (${\displaystyle {\hat {x}}_{\mathrm {fl} }(t_{j})}$ and ${\displaystyle {\hat {p}}^{\mathrm {b.a.} }(t_{j})}$ 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.^[4]

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.^[5]