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Quantum nondemolition (QND) measurement is a special type of measurement of a quantum system in which the uncertainty of the measured observable does not increase from its measured value during the subsequent normal evolution of the system. This necessarily requires that the measurement process preserves the physical integrity of the measured system, and moreover places requirements on the relationship between the measured observable and the self-Hamiltonian of the system. In a sense, QND measurements are the "most classical" and least disturbing type of measurement in quantum mechanics.

Most devices capable of detecting a single particle and measuring its position strongly modify the particle's state in the measurement process, e.g. photons are destroyed when striking a screen. Less dramatically, the measurement may simply perturb the particle in an unpredictable way; a second measurement, no matter how quickly after the first, is then not guaranteed to find the particle in the same location. Even for ideal, "first-kind" projective measurements in which the particle is in the measured eigenstate immediately after the measurement, the subsequent free evolution of the particle will cause uncertainty in position to quickly grow.

In contrast, a momentum (rather than position) measurement of a free particle can be QND because the momentum distribution is preserved by the particle's self-Hamiltonian p²/2m. Because the Hamiltonian of the free particle commutes with the momentum operator, a momentum eigenstate is also an energy eigenstate, so once momentum is measured its uncertainty does not increase due to free evolution.

Note that the term "nondemolition" does not imply that the wave function fails to collapse.

QND measurements are extremely difficult to carry out experimentally. Much of the investigation into QND measurements was motivated by the desire to avoid the standard quantum limit in the experimental detection of gravitational waves. The general theory of QND measurements was laid out by Braginsky, Vorontsov, and Thorne^[1] following much theoretical work by Braginsky, Caves, Drever, Hollenhorts, Khalili, Sandberg, Thorne, Unruh, Vorontsov, and Zimmermann.

Technical definition

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

| ψ ⟩ \approx \sum_{i} | A_{i} ⟩_{𝒮} | R_{i} ⟩_{ℛ}

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

Allow time-dependence to denote the Heisenberg picture observables:

A (t) = e^{i t H_{𝒮}} A e^{- i t H_{𝒮}} .

A sequence of measurements of $A$ are said to be QND measurements if and only if^[1]

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

for any $t_{n}$ and $t_{m}$ when measurements are made. If this property holds for any choice of $t_{n}$ and $t_{m}$ , then $A$ is said to be a continuous QND variable. If this only holds for certain discrete times, then $A$ 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.

Discussion

An observable $A$ which is conserved under free evolution,

\frac{d}{d t} A (t) = \frac{i}{ℏ} [H_{𝒮}, A] = 0,

is automatically a QND variable. A sequence of ideal projective measurements of $A$ 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.

Criticism

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]

@@ Line 1: / Line 1: @@
 {{Short description|Type of measurement of a quantum system}}
-'''Quantum nondemolition''' ('''QND''') '''measurement''' is a special type of [[Physics:Measurement in quantum mechanics|measurement]] of a [[Physics:Quantum mechanics|quantum]] system in which the uncertainty of the measured [[Physics:Observable|observable]] does not increase from its measured value during the subsequent normal evolution of the system.  This necessarily requires that the measurement process preserves the physical integrity of the measured system, and moreover places requirements on the relationship between the measured observable and the self-Hamiltonian of the system. In a sense, QND measurements are the "most classical" and least disturbing type of measurement in quantum mechanics.
-Most devices capable of detecting a single particle and measuring its position strongly modify the particle's state in the measurement process, e.g. photons are destroyed when striking a screen.  Less dramatically, the measurement may simply perturb the particle in an unpredictable way; a second measurement, no matter how quickly after the first, is then not guaranteed to find the particle in the same location. Even for ideal, [[Physics:Measurement in quantum mechanics#History of the measurement concept|"first-kind"]] [[Projection-valued measure|projective measurements]] in which the particle is in the measured eigenstate immediately after the measurement, the subsequent free evolution of the particle will cause uncertainty in position to quickly grow.
+{{Quantum book backlink|Measurement and information}}
-In contrast, a ''momentum'' (rather than position) measurement of a [[Physics:Free particle|free particle]] can be QND because the momentum distribution is preserved by the particle's self-Hamiltonian ''p''<sup>2</sup>/2''m''.  Because the Hamiltonian of the free particle commutes with the [[Physics:Momentum operator|momentum operator]], a momentum eigenstate is also an energy eigenstate, so once momentum is measured its uncertainty does not increase due to free evolution.
+<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
-Note that the term "nondemolition" does not imply that the [[Wave function|wave function]] fails to [[Physics:Wave function collapse|collapse]].
+<div style="width:280px;">
+__TOC__
+</div>
+<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
+'''Quantum nondemolition''' ('''QND''') '''measurement''' is a special type of measurement of a [[Physics:Quantum mechanics|quantum]] system in which the uncertainty of the measured observable does not increase from its measured value during the subsequent normal evolution of the system.  This necessarily requires that the measurement process preserves the physical integrity of the measured system, and moreover places requirements on the relationship between the measured observable and the self-Hamiltonian of the system. In a sense, QND measurements are the "most classical" and least disturbing type of measurement in quantum mechanics.
+Most devices capable of detecting a single particle and measuring its position strongly modify the particle's state in the measurement process, e.g. photons are destroyed when striking a screen.  Less dramatically, the measurement may simply perturb the particle in an unpredictable way; a second measurement, no matter how quickly after the first, is then not guaranteed to find the particle in the same location. Even for ideal, "first-kind" projective measurements in which the particle is in the measured eigenstate immediately after the measurement, the subsequent free evolution of the particle will cause uncertainty in position to quickly grow.
+In contrast, a ''momentum'' (rather than position) measurement of a free particle can be QND because the momentum distribution is preserved by the particle's self-Hamiltonian ''p''<sup>2</sup>/2''m''.  Because the Hamiltonian of the free particle commutes with the momentum operator, a momentum eigenstate is also an energy eigenstate, so once momentum is measured its uncertainty does not increase due to free evolution.
+Note that the term "nondemolition" does not imply that the wave function fails to collapse.
 QND measurements are extremely difficult to carry out experimentally.  Much of the investigation into QND measurements was motivated by the desire to avoid the standard quantum limit in the experimental detection of gravitational waves.  The general theory of QND measurements was laid out by [[Biography:Vladimir Braginsky|Braginsky]], Vorontsov, and Thorne<ref name="Braginsky1980">
@@ Line 20: / Line 30: @@
 | s2cid = 19278286
   }}</ref> following much theoretical work by Braginsky, Caves, Drever, Hollenhorts, Khalili, Sandberg, Thorne, Unruh, Vorontsov, and Zimmermann.
+</div>
+<div style="width:300px;">
+[[File:Quantum_nondemolition_measurement_concept_map.svg|thumb|280px|nondemolition measurement in the Quantum Collection.]]
+</div>
+</div>
 ==Technical definition==
@@ Line 27: / Line 44: @@
 where <math>\vert A_i \rangle_\mathcal{S}</math> are the eigenvectors of <math>A</math> corresponding to the possible outcomes of the measurement, and <math>\vert R_i \rangle_\mathcal{R}</math> are the corresponding states of the apparatus which record them.
-Allow time-dependence to denote the [[Physics:Heisenberg picture|Heisenberg picture]] observables:
+Allow time-dependence to denote the Heisenberg picture observables:
 :<math>A(t) = e^{i t H_\mathcal{S}} A e^{-i t H_\mathcal{S}}.</math>
-A sequence of measurements of <math>A</math> are said to be QND measurements [[If and only if|if and only if]]<ref name="Braginsky1980" />
+A sequence of measurements of <math>A</math> are said to be QND measurements if and only if<ref name="Braginsky1980" />
 :<math>[A(t_n),A(t_m)] = 0</math>
 for any <math>t_n</math> and <math>t_m</math> when measurements are made.  If this property holds for ''any'' choice of <math>t_n</math> and <math>t_m</math>, then <math>A</math> is said to be a ''continuous QND variable''.  If this only holds for certain discrete times, then <math>A</math> is said to be a ''stroboscopic QND variable''.
@@ Line 44: / Line 61: @@
 is automatically a QND variable.  A sequence of ideal projective measurements of <math>A</math> 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.<ref name="Qi2016">{{cite journal |title=Dispersive response of atoms trapped near the surface of an optical nanofiber with applications to quantum nondemolition measurement and spin squeezing |doi=10.1103/PhysRevA.93.023817 |journal=Physical Review A |year=2016 |first1=Xiaodong |last1=Qi |first2=Ben Q. |last2=Baragiola |first3=Poul S. |last3=Jessen |first4=Ivan H. |last4=Deutsch |volume=93 |issue=2 |article-number=023817 |arxiv=1509.02625 |bibcode=2016PhRvA..93b3817Q |s2cid=17366761 }}</ref> People usually use [[Physics:Optical depth|optical depth]] or [[Biology:Cooperativity|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,<ref name="Qi2018">{{cite journal |title=Enhanced cooperativity for quantum-nondemolition-measurement–induced spin squeezing of atoms coupled to a nanophotonic waveguide |doi=10.1103/PhysRevA.97.033829 |journal=Physical Review A |year=2018 |first1=Xiaodong |last1=Qi |first2=Yuan-Yu |last2=Jau |first3=Ivan H. |last3=Deutsch |volume=97 |issue=3 |article-number=033829 |arxiv=1712.02916 |bibcode=2016PhRvA..93c3829K |s2cid=4941311 }}</ref> and hence an enhanced precise quantum measurement with little disruption to the quantum system.
+To implement QND measurements on atomic systems, the measurement strength (rate) is competing with atomic decay caused by measurement backaction.<ref name="Qi2016">{{cite journal |title=Dispersive response of atoms trapped near the surface of an optical nanofiber with applications to quantum nondemolition measurement and spin squeezing |doi=10.1103/PhysRevA.93.023817 |journal=Physical Review A |year=2016 |first1=Xiaodong |last1=Qi |first2=Ben Q. |last2=Baragiola |first3=Poul S. |last3=Jessen |first4=Ivan H. |last4=Deutsch |volume=93 |issue=2 |article-number=023817 |arxiv=1509.02625 |bibcode=2016PhRvA..93b3817Q |s2cid=17366761 }}</ref> 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,<ref name="Qi2018">{{cite journal |title=Enhanced cooperativity for quantum-nondemolition-measurement–induced spin squeezing of atoms coupled to a nanophotonic waveguide |doi=10.1103/PhysRevA.97.033829 |journal=Physical Review A |year=2018 |first1=Xiaodong |last1=Qi |first2=Yuan-Yu |last2=Jau |first3=Ivan H. |last3=Deutsch |volume=97 |issue=3 |article-number=033829 |arxiv=1712.02916 |bibcode=2016PhRvA..93c3829K |s2cid=4941311 }}</ref> and hence an enhanced precise quantum measurement with little disruption to the quantum system.
 ==Criticism==
@@ Line 67: / Line 84: @@
 {{Reflist}}
-==See also==
+== See also ==
-* [[Physics:Interaction-free measurement|Interaction-free measurement]]
+{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
 ==External links==

Physics:Quantum nondemolition measurement: Difference between revisions

Latest revision as of 23:35, 23 May 2026

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