Quantum nondemolition measurement

Let ${\displaystyle A}$ be an observable for some system ${\displaystyle {\mathcal {S}}}$ with self-Hamiltonian ${\displaystyle H_{\mathcal {S}}}$. The system ${\displaystyle {\mathcal {S}}}$ is measured by an apparatus ${\displaystyle {\mathcal {R}}}$ which is coupled to ${\displaystyle {\mathcal {S}}}$ through interactions Hamiltonian ${\displaystyle H_{\mathcal {RS}}}$ for only brief moments. Otherwise, ${\displaystyle {\mathcal {S}}}$ evolves freely according to ${\displaystyle H_{\mathcal {S}}}$. A precise measurement of ${\displaystyle A}$ is one which brings the global state of ${\displaystyle {\mathcal {S}}}$ and ${\displaystyle {\mathcal {R}}}$ into the approximate form

${\displaystyle \vert \psi \rangle \approx \sum _{i}\vert A_{i}\rangle _{\mathcal {S}}\vert R_{i}\rangle _{\mathcal {R}}}$

where ${\displaystyle \vert A_{i}\rangle _{\mathcal {S}}}$ are the eigenvectors of ${\displaystyle A}$ corresponding to the possible outcomes of the measurement, and ${\displaystyle \vert R_{i}\rangle _{\mathcal {R}}}$ are the corresponding states of the apparatus which record them.

Allow time-dependence to denote the Heisenberg picture observables:

${\displaystyle A(t)=e^{-itH_{\mathcal {S}}}Ae^{+itH_{\mathcal {S}}}.}$

A sequence of measurements of ${\displaystyle A}$ are said to be QND measurements if and only if[1]

${\displaystyle [A(t_{n}),A(t_{m})]=0}$

for any ${\displaystyle t_{n}}$ and ${\displaystyle t_{m}}$ when measurements are made. If this property holds for any choice of ${\displaystyle t_{n}}$ and ${\displaystyle t_{m}}$, then ${\displaystyle A}$ is said to be a continuous QND variable. If this only holds for certain discrete times, then ${\displaystyle A}$ is said to be a stroboscopic QND variable. For example, in the case of a free particle, the energy and momentum are conserved and indeed continuous QND observables, but the position is not. On the other hand, for the harmonic oscillator the position and momentum satisfy periodic in time commutation relations which imply that x and p are not continuous QND observables. However, if one makes the measurements at times separated by an integral numbers of half-periods (τ = kπ/ω), then the commutators vanish. This means that x and p are stroboscopic QND observables.

An observable ${\displaystyle A}$ which is conserved under free evolution,

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}A(t)=-{\frac {i}{\hbar }}[H_{\mathcal {S}},A]=0,}$

is automatically a QND variable. A sequence of ideal projective measurements of ${\displaystyle A}$ will automatically be QND measurements.

To implement QND measurements on atomic systems, the measurement strength (rate) is competing with atomic decay caused by measurement backaction.[2] People usually use optical depth or cooperativity to characterize the relative ratio between measurement strength and the optical decay. By using nanophotonic waveguides as a quantum interface, it is actually possible to enhance atom-light coupling with a relatively weak field,[3] and hence an enhanced precise quantum measurement with little disruption to the quantum system.

It has been argued that the usage of the term QND does not add anything to the usual notion of a strong quantum measurement and can moreover be confusing because of the two different meanings of the word demolition in a quantum system (losing the quantum state vs. losing the particle). [4]

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