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{{short description|Quantum effect of uncertainty}}
{{Multiple issues|
{{More footnotes|date=November 2009}}
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'''Quantum noise''' is a type of [[Physics:Noise (spectral phenomenon)|noise]] in a quantum system due to quantum mechanical phenomena such as quantized fields and [[Physics:Uncertainty principle|the uncertainty principle]].<ref name="pathfinder">{{Cite web |title=Quantum Noise – nmiccg |url=https://www.pathfinderdigital.com/quantum-noise |access-date=2025-09-19 |website=PathFinder Digital}}</ref> This principle says that some [[Physics:Observable|observables]] cannot simultaneously be known with arbitrary precision. This [[Physics:Quantum indeterminacy|indeterminate state of matter]] introduces a fluctuation in the value of properties of a quantum system, even at zero temperature.<ref>{{Cite web |last=Ball |first=Philip |date=2018-09-18 |title=Putting quantum noise to work |url=https://physicsworld.com/putting-quantum-noise-to-work/ |access-date=2025-09-19 |website=Physics World |language=en-GB}}</ref> These fluctuations in the absence of thermal noise are known as [[Physics:Zero-point energy|zero-point energy]] fluctuations.

Quantum noise can also come from the discrete nature of the small quantum constituents such as [[Physics:Electron|electron]]s and quantum effects, such as [[Physics:Photocurrent|photocurrent]]s. An example of this form of quantum noise is [[Shot noise|shot noise]] as coined by J. Verdeyen<ref name="Verdeyen">{{Cite book |last=Verdeyen |first=Joseph Thomas |title=Laser Electronics |date=1995 |publisher=Prentice-Hall |isbn=978-0-13-101668-2 |edition=3rd |location=Englewood Cliffs (N.J.)}}</ref> which comes from the discrete arrival of photons or electrons in a detector. Because these quanta arrive randomly in time, even a perfectly steady current or light beam exhibits fluctuations in the detected signal.

In most systems, classical noise dominates over quantum noise, because classical fluctuations are several orders of magnitude larger, and it masks the effects of quantum noise. Quantum noise generally only becomes visible after suppressing the effects of conventional noise sources such as thermal fluctuations, mechanical vibrations, and industrial noise by cooling a system to a millikelvin range and using extremely low-noise electronics. This is why quantum noise is present in [[Superconducting quantum computing|superconducting circuits]] and in the [[Astronomy:LIGO|LIGO]] [[Astronomy:Gravitational wave|gravitational wave]] [[Astronomy:Observatory|observatory]], but not in many conventional settings.

At absolute zero temperature, classical noise vanishes. However, unlike classical noise, quantum noise cannot be completely eliminated as it arises directly from fundamental tenets of [[Physics:Quantum mechanics|quantum mechanics]]. The uncertainty principle requires any [[Engineering:Amplifier|amplifier]] or [[Physics:Detector|detector]] to have some noise, setting a fundamental limit on the accuracy of these instruments.<ref name="A A Clerk PDF">{{harvnb|Clark|2008}}</ref> Despite this fact, experimental physicists still define an "ideal" amplifier or detector as one that optimizes the fundamental quantum noise inequality, known as a "quantum-limited detector".<ref name="A A Clerk PDF" />

Noise is of practical concern for precision engineering and engineered systems approaching the standard quantum limit. Typical engineered consideration of quantum noise is for [[Physics:Quantum nondemolition measurement|quantum nondemolition measurement]] and [[Physics:Quantum point contact|quantum point contact]]. So quantifying noise is useful.<ref name="Verdeyen" /><ref name="Clark A 2010">
{{harvnb|Clerk|Devoret|Girvin|Marquardt|Schoelkopf|2010}}</ref><ref name="Henry C H 1996">
{{Cite journal |last1=Henry |first1=Charles H. |last2=Kazarinov |first2=Rudolf F. |date=July 1, 1996 |title=Quantum noise in photonics |journal=[[Physics:Reviews of Modern Physics|Reviews of Modern Physics]] |volume=68 |issue=3 |pages=801–853 |bibcode=1996RvMP...68..801H |doi=10.1103/RevModPhys.68.801 |issn=0034-6861}}</ref>

The term "quantum noise" is often used in the fields of [[Quantum information|quantum information]] and [[Quantum computing|quantum computing]] as an umbrella term for unwanted environmental disturbances that affect quantum systems and cause [[Physics:Quantum decoherence|decoherence]].<ref>{{Cite web |title=What is Quantum Noise |url=https://www.quera.com/glossary/noise |access-date=2025-09-19 |website=www.quera.com}}</ref><ref>{{Cite web |last=Mura |first=Maria Teresa Della |date=2024-01-29 |title=Quantum Noise: Overcoming This Obstacle is Crucial for the Evolution of Quantum Computing |url=https://tech4future.info/en/quantum-noise-quantum-computing/ |access-date=2025-09-21 |website=Tech4Future |language=en}}</ref><ref>{{Cite web |title=Quantum noise |url=https://qsnp.eu/glossary/quantum-noise/ |access-date=2025-09-21 |website=QSNP |language=en-GB}}</ref> An isolated quantum system, such as a [[Qubit|qubit]], has a state that will evolve deterministically. But in an [[Open system (systems theory)|open system]], such as those found in nature, the qubit interacts with uncontrolled [[Degrees of freedom|degrees of freedom]] in its environment, introducing fluctuations which are commonly referred to as quantum noise.<ref>{{Citation |last1=Krantz |first1=Philip |title=A quantum engineer's guide to superconducting qubits |date=2021-07-07 |arxiv=1904.06560 |last2=Kjaergaard |first2=Morten |last3=Yan |first3=Fei |last4=Orlando |first4=Terry P. |last5=Gustavsson |first5=Simon |last6=Oliver |first6=William D. |journal=Applied Physics Reviews |volume=6 |issue=2 |article-number=021318 |doi=10.1063/1.5089550 }}</ref><ref name="Clark A 2010" /> This is distinct from the above definition, which specifically concerns ''intrinsic noise due to the nature of quantum mechanics'', not all environmental sources of noise and decoherence. In practice, however, definitions of quantum noise often include environmental or external disturbances affecting quantum systems.<ref name="pathfinder" />

==Principles==

=== Noise theory ===
A signal's noise is quantified as the [[Fourier transform]] of its [[Autocorrelation|autocorrelation]].
The autocorrelation of a signal is given as
<math display="block">G_{vv}(t-t') = \langle V(t)V(t')\rangle ,</math>
which measures when our signal is positively, negatively or not correlated at different times <math>t</math> and <math>t'</math>.
The time average, <math> \langle V(t) \rangle </math>, is zero and our <math>V(t)</math> is a voltage signal. Its Fourier transform is
<math display="block">V(\omega) = \frac{1}{\sqrt{T}}\int_{0}^{T} V(t)e^{i\omega t}dt </math>
because we measure a voltage over a finite time window. The [[Wiener–Khinchin theorem]] generally states that a noise's power spectrum is given as the autocorrelation of a signal, i.e.,
<math display="block">S_{vv}(\omega) = \int_{-\infty}^{+\infty}e^{i\omega t} G_{vv}dt = \int_{-\infty}^{+\infty}e^{i\omega t} \langle |V(\omega)|^2\rangle dt </math>The above relation is sometimes called the power spectrum or spectral density.
In the above outline, we assumed that
* Our noise is ''stationary'' or the probability does not change over time. Only the ''time difference'' matters.
* Noise is due to a very large number of fluctuating charge so that the central limit theorem applied, i.e., the noise is Gaussian or [[Normal distribution|normally distributed]].
* <math>G_{vv}</math> decays to zero rapidly over some time <math>\tau_c</math>.
* We sample over a sufficiently large time, <math>T</math>, that our integral scales as a random walk <math>\sqrt{T}</math>. So our <math>V(\omega)</math> is independent of measured time for <math>T \gg \tau_c</math>. {{pb}} Said in another way, <math>G_{vv}(t-t') \to 0</math> as <math> |t-t'| \gg \tau_c</math>.

One can show that an ideal "top-hat" signal, which may correspond to a finite measurement of a voltage over some time, will produce noise across its ''entire spectrum'' as a sinc function. Even in the classical case, noise is produced.

==== Classical to quantum noise ====
To study quantum noise, one replaces the corresponding classical measurements with quantum operators, e.g.,
<math display="block"> S_{xx}(\omega) = \int_{-\infty}^{+\infty}e^{i\omega t} \langle \hat{x}(t) \hat{x}(0) \rangle dt ,</math>
where <math> \langle \cdot \rangle </math> are the quantum statistical average using the [[Density matrix|density matrix]] in the Heisenberg picture.

=== Heisenberg microscope ===
{{main|Physics:Heisenberg's microscope}}

Quantum noise can be illustrated by considering a Heisenberg microscope where an atom's position is measured from the scattering of photons. The uncertainty principle is given as,

<math display="block">\Delta x_{imp} \Delta p_{BA} \gtrsim \hbar.</math>

Where the <math>\Delta x_{imp}</math> is the uncertainty in an atom's position, and the <math>\Delta p_{BA}</math> is the uncertainty of the momentum or sometimes called the [[Physics:Back action (quantum)|backaction]] (momentum transferred to the atom) when near the [[Physics:Quantum limit|quantum limit]]. The precision of the position measurement can be increased at the expense of knowing the atom's momentum. When the position is precisely known enough backaction begins to affect the measurement in two ways. First, it will impart momentum back onto the measuring devices in extreme cases. Secondly, we have decreasing future knowledge of the atom's future position. Precise and sensitive instrumentation will approach the uncertainty principle at sufficiently control environments.

=== Heisenberg uncertainty and noise ===
The Heisenberg uncertainty implies the existence of noise.<ref name="Gardiner Zoller">{{harnvb|Gardiner|Zoller|2004}}</ref> An operator with a hermitian conjugate follows the relationship, <math>A A^{\dagger} \ge 0 </math>. Define <math>A</math> as <math>A = \delta x +\lambda e^{i\theta}\delta y</math> where <math>\lambda </math> is real. The <math>x</math> and <math>y</math> are the quantum operators. We can show the following,

<math display="block"> \langle \delta x^2 \rangle \langle \delta y^2 \rangle \ge \frac{1}{4} |\langle [\delta x, \delta y]\rangle|^2 + |\langle [\delta x, \delta y]_+ \rangle|^2</math>
where the <math> \langle \cdot \rangle</math> are the averages over the wavefunction and other statistical properties. The left terms are the uncertainty in <math>x</math> and <math>y</math>, the second term on the right is to covariance or <math>\langle \delta x \delta y + \delta y \delta x \rangle</math> which arises from coupling to an external source or quantum effects. The first term on the right corresponds to the [[Commutator]] relation and would cancel out ''if the x and y commuted''. That is the origin of our quantum noise.

It is demonstrative to let <math>x</math> and <math>y</math> correspond to position and momentum that meets the well known commutator relation, <math>[x,p]=i\hbar</math>. Then our new expression is,

<math display="block">\Delta x \Delta y \ge \sqrt{\frac{1}{4} \hbar^2 + \sigma_{xy}^2 } </math>

Where the <math> \sigma_{xy}</math> is the correlation. If the second term on the right vanishes, then we recover the Heisenberg uncertainty principle.

===Harmonic motion and weakly coupled heat bath===

Consider the motion of a simple harmonic oscillator with mass, <math>M</math>, and frequency, <math>\Omega</math>, coupled to some heat bath which keeps the system in equilibrium. The equations of motion are given as,

<math display="block"> x(t) = x(0)\cos(\Omega t) + p(0)\frac{1}{M\Omega}\sin(\Omega t) </math>

The quantum autocorrelation is then,

<math display="block">\begin{align}
G_{xx} &= \langle \hat{x}(t) \hat{x}(0) \rangle \\
& = \langle \hat{x}(0) \hat{x}(0) \rangle \cos(\Omega t) + \langle \hat{p}(0)\hat{x}(0)\rangle \sin(\Omega t)
\end{align} </math>

Classically, there is no correlation between position and momentum. The uncertainty principle requires the second term to be nonzero. It goes to <math>i\hbar/2</math>.
We can take the equipartition theorem or the fact that in equilibrium the energy is equally shared among a molecule/atoms degrees of freedom in thermal equilibrium, i.e.,

<math display="block">\frac{1}{2}M\Omega^2 \langle x^2\rangle = \frac{1}{2}k_\text{B} T</math>

In the classical autocorrelation, we have

<math display="block">G_{xx} = \frac{k_\text{B}T}{M\Omega^2}\cos(\Omega t)
\to
S_{xx}(\omega) = \pi \frac{k_\text{B} T}{M\Omega^2}[\delta(\omega - \Omega) +\delta(\omega +\Omega)]</math>

while in the quantum autocorrelation we have

<math display="block">G_{xx} = \left( \frac{\hbar}{2M\Omega}\right) \left\{n_{BE}(\hbar\Omega) e^{i\Omega t} + [ n_{BE}(\hbar \Omega) +1 ]e^{-i\Omega t} \right \}
\to
S_{xx}(\omega) = 2\pi \left( \frac{\hbar}{2M\Omega}\right) [n_{BE}(\hbar \Omega)\delta(\omega - \Omega) +[n_{BE}(\hbar \Omega)+1]\delta(\omega +\Omega)]</math>

Where the fraction terms in parentheses is the zero-point energy uncertainty. The <math> n_{BE}</math> is the Bose-Einstein population distribution. Notice that the quantum <math>S_{xx}</math> is asymmetric in the due to the imaginary autocorrelation. As we increase to higher temperature that corresponds to taking the limit of <math>k_BT \gg \hbar\Omega </math>. One can show that the quantum approaches the classical <math> S_{xx}</math>. This allows <math display="inline"> n_{BE} \approx n_{BE}+1 \approx \frac{k_\text{B}T}{\hbar \Omega}</math>

==== Physical interpretation of spectral density ====
Typically, the positive frequency of the spectral density corresponds to the flow of energy ''into the oscillator'' (for example, the photons' quantized field), while the negative frequency corresponds to the emitted of energy ''from the oscillator''. Physically, an asymmetric spectral density would correspond to either the net flow of energy from or to our oscillator model.

=== Linear gain and quantum uncertainty ===
Most optical communications use [[Physics:Amplitude modulation|amplitude modulation]] where the quantum noise is predominantly the [[Shot noise|shot noise]]. A [[Physics:Laser|laser]]'s quantum noise, when not considering shot noise, is the uncertainty of its [[Physics:Electric field|electric field]]'s amplitude and phase. That uncertainty becomes observable when a [[Physics:Quantum amplifier|quantum amplifier]] preserves phase. The phase noise becomes important when the energy of the [[Physics:Frequency modulation|frequency modulation]] or [[Physics:Phase modulation|phase modulation]] is comparable to the energy of the signal (frequency modulation is more robust than amplitude modulation due to the additive noise intrinsic to amplitude modulation).

==== Linear amplification ====
An ideal noiseless gain cannot exit.<ref name="Emmanuel D 1994">{{Cite book |last=Desurvire |first=Emmanuel |title=Erbium-Doped Fiber Amplifiers: Principles and Applications |date=1994 |publisher=[[Company:Wiley (publisher)|Wiley]] |isbn=978-0-471-58977-8 |location=New York}}</ref> Consider the amplification of stream of photons, an ideal linear noiseless gain, and the Energy-Time uncertainty relation.

<math display="block">\Delta E \Delta t \gtrsim \hbar/2 </math>

The photons, ignoring the uncertainty in frequency, will have an uncertainty in its overall phase and number, and assume a known frequency, i.e., <math>\Delta \phi = 2\pi \nu \Delta t </math> and <math>\Delta E = h\nu\Delta n </math>. We can substitute these relations into our energy-time uncertainty equation to find the number-phase uncertainty relation or the uncertainty in the phase and photon numbers.
<math display="block">\Delta n \Delta \phi > 1/2 </math>

Let an ideal linear noiseless gain, <math>G</math>, act on the photon stream. We also assume a unity [[Engineering:Quantum efficiency|quantum efficiency]], or every photon is converted to a photocurrent. The output will be following with no noise added.

<math display="block">n_0 \pm \Delta n_0 \to Gn_0 \pm G\Delta n_0 </math>

The phase will be modified too,

<math display="block">\phi_0 \pm \Delta\phi_0 \to \phi_0 +\theta + \Delta\phi_0 ,</math>
where the <math>\theta</math> is the overall accumulated phase as the photons traveled through the gain medium.
Substituting our output gain and phase uncertainties, gives us
<math display="block">\Delta n_0 \Delta \phi_0 > 1/2G .</math>

Our gain is <math>G>1</math>, which is a contradiction to our uncertainty principles. So a ''linear noiseless amplifier cannot increase its signal without noise''.
A deeper analysis done by H. Heffner
showed the minimum noise power output required to meet the Heisenberg uncertainty principle is given as<ref name="Heffner H">{{cite journal|last=Heffner|first=Hubert|title=The Fundamental Noise Limit of Linear Amplifiers| doi=10.1109/JRPROC.1962.288130|journal=Proceedings of the IRE| volume=50 | page=1604-1608 | year=1962 | issue=7|bibcode=1962PIRE...50.1604H | s2cid=51674821}}</ref>
<math display="block">P_n = h \nu B (G-1)</math>
where <math>B </math> is half of the [[Full width at half maximum|full width at half max]], the <math>\nu</math> frequency of the photons, and <math>h</math> is the [[Physics:Planck constant|Planck constant]]. The term <math>h\nu B_0/2</math> with <math>B_0 = 2 B</math> is sometimes called quantum noise <ref name = "Emmanuel D 1994"/>

== Types of Quantum Noise ==

=== Shot noise ===
{{Main|Shot noise}}
In precision optics with highly stabilized lasers and efficient detectors, quantum noise refers to the fluctuations of signal.

The random error of interferometric measurements of position, due to the discrete character of photons measurement, is another quantum noise. The uncertainty of position of a probe in probe microscopy may also attributable to quantum noise; but not the dominant mechanism governing resolution.

In an electric circuit, the random fluctuations of a signal due to the discrete character of electrons can be called quantum noise.<ref name="Gardiner Zoller" />

An experiment by S. Saraf, et .al.
<ref name = "S Saraf" >{{cite journal
|last1=Saraf |first1=Shally
|last2=Urbanek |first2=Karel
|last3=Byer |first3=Robert L.
|last4=King |first4=Peter J.
|title=Quantum noise measurements in a continuous-wave laser-diode-pumped Nd:YAG saturated amplifier.
|doi=10.1364/ol.30.001195
|journal=[[Physics:Optics Letters|Optics Letters]]
|volume=30
|year=2005
|issue=10
|pages=1195–1197
|pmid=15943307
|bibcode=2005OptL...30.1195S
|url=https://authors.library.caltech.edu/6950/
|archive-date=2022-05-25
|access-date=2021-12-23
|archive-url=https://web.archive.org/web/20220525192057/https://authors.library.caltech.edu/6950/
|url-status=dead
}}</ref>
demonstrated shot noise limited measurements as a demonstration of quantum noise measurements. Generally speaking, they amplified a Nd:YAG free space laser with minimal noise addition as it transitioned from linear to nonlinear amplification. The experiment required Fabry-Perot for filtering laser mode noises and selecting frequencies, two separate but identical probe and saturating beams to ensure uncorrelated beams, a zigzag slab gain medium, and a balanced detector for measuring quantum noise or shot-noise limited noise.

==== Shot Noise Power ====
The theory behind noise analysis of photon statistics (sometimes called the ''forward Kolmogorov equation'') starts from the Masters equation from Shimoda ''et al.''<ref name="Shimoda K 1957">{{cite journal|last1=Shimoda |first1=Koichi |last2=Takahasi |first2=Hidetosi |last3=H. Townes |first3=Charles |title=Fluctuations in Amplification of Quanta with Application to Maser Amplifiers |doi=10.1143/JPSJ.12.686 |journal=Journal of the Physical Society of Japan |volume=12 |page=686-700 |year=1957 |number=5 |bibcode=1957JPSJ...12..686S}}</ref>

<math display="block">\frac{dP_n}{dx} = a[nP_{n-1}-(n+1)P_n] + b[(n+1)P_{n+1}-nP_n]</math>

where <math>a</math> corresponds to the emission cross section and upper population number product <math>\sigma_e N_2</math>, and the <math>b</math> is the absorption cross section <math>\sigma_a N_1</math>. The above relation is describing the probability of finding <math>n </math> photons in radiation mode <math>|n \rangle</math>. The dynamic only considers neighboring modes <math>| n+1 \rangle </math> and <math> | n-1\rangle </math> as the photons travel through a medium of excited and ground state atoms from position <math>x</math> to <math>x+dx</math>. This gives us a total of ''4 photon transitions'' associated to one photon energy level. Two photon number adding to the field and leaving an atom, <math> |n-1 \rangle \to | n \rangle </math> and <math> |n \rangle \to |n+1 \rangle </math> and two photons leaving a field to the atom <math>|n+1 \rangle \to |n \rangle </math> and <math>|n \rangle \to |n-1 \rangle </math>. Its noise power is given as,

<math display="block">P_d^2 = P_\text{shot}^2 [1+2f_{sp}\eta(G-1)]</math>

Where,
*<math>P_d</math> is the power at the detector,
*<math>P_\text{shot}</math> is the power limited shot noise,
*<math>G</math> the unsaturated gain and is also true for saturated gain,
*<math>\eta</math> is the efficiency factor. That is the product of transmission window efficiency to our photodetector, and quantum efficiency.
*<math>f_{sp}</math> is the spontaneous emission factor that typically corresponds relative strength of spontaneous emission to stimulated emission. A value of unity would mean all doped ions are in the excited state.<ref>{{Cite book |last=Pal |first=Bishnu P. |title=Guided Wave Optical Components and Devices: Basics, Technology, and Applications |date=2006 |publisher=[[Company:Elsevier|Elsevier]] |isbn=978-0-12-088481-0 |location=Amsterdam}}</ref>
Sarif, ''et al.'' demonstrated quantum noise or shot noise limited measurements over a wide range of power gain that agreed with theory.

=== Quantum Back Action ===
{{Main|Physics:Back action (quantum)}}
'''Back action''' is the phenomenon in which the act of measuring a property of a particle directly influences the state of the particle.

In quantum mechanics, operators which do not commute are considered incompatible observables, and carry an associated uncertainty principle:<blockquote><math>\sigma_x\sigma_p\geq\hbar/2</math></blockquote>When measuring these observables, this principle sets a minimum uncertainty in their values.

Each observable operator has a set of eigenstates. The initial state of a system, described by the [[Wave function|wavefunction]], is a linear combination of the full set of its eigenstates. Once measured, the system's [[Physics:Wave function collapse|wavefunction collapses]] to an eigenstate of that observable. It will then evolve in time again. Because the act of measurement altered the state of the observable, it affects the future behavior and any future measurement of the system. This introduces error, and is the concept behind back action.

Back action is a practical source of noise in experiments.<ref>{{Citation |last1=Ghosh |first1=Sohitri |title=Backaction-evading impulse measurement with mechanical quantum sensors |date=2020-08-29 |arxiv=1910.11892 |last2=Carney |first2=Daniel |last3=Shawhan |first3=Peter |last4=Taylor |first4=Jacob M. |journal=Physical Review A |volume=102 |issue=2 |article-number=023525 |doi=10.1103/PhysRevA.102.023525 |bibcode=2020PhRvA.102b3525G }}</ref><ref name="Cripe etal">{{Cite journal |last1=Cripe |first1=Jonathan |last2=Aggarwal |first2=Nancy |last3=Lanza |first3=Robert |last4=Libson |first4=Adam |last5=Singh |first5=Robinjeet |last6=Heu |first6=Paula |last7=Follman |first7=David |last8=Cole |first8=Garrett D. |last9=Mavalvala |first9=Nergis |last10=Corbitt |first10=Thomas |date=25 March 2019 |title=Measurement of quantum back action in the audio band at room temperature |url=https://www.nature.com/articles/s41586-019-1051-4 |journal=Nature |language=en |volume=568 |issue=7752 |pages=364–367 |bibcode=2019Natur.568..364C |doi=10.1038/s41586-019-1051-4 |issn=1476-4687 |pmid=30911169}}</ref> Whenever a probe or measurement device interacts with a system, through photons, electrons, or other carriers, ithe measurement process imparts a random disturbance. In precision instruments, this disturbance appears as an additional noise source that limits sensitivity, known as measurement back-action noise.

Experimental setups involving optical measurement are limited by both shot noise and backaction noise. In an optomechanical system such as a laser interferometer, measurement back-action noise arises because of fluctuations in the radiation pressure of the light.<ref>{{Cite journal |last=Caves |first=Carlton M. |date=1980-07-14 |title=Quantum-Mechanical Radiation-Pressure Fluctuations in an Interferometer |url=https://link.aps.org/doi/10.1103/PhysRevLett.45.75 |journal=Physical Review Letters |volume=45 |issue=2 |pages=75–79 |doi=10.1103/PhysRevLett.45.75 |bibcode=1980PhRvL..45...75C }}</ref> By increasing the optical power, the shot noise is decreased, but this comes at the cost of increasing backaction, in the form of quantum radiation pressure noise, and the backaction of the randomly-arriving photons’ radiation pressure will become the dominant force on the system.<ref name="Cripe etal" /><ref>{{Cite book |last=Peterson |first=R.W. |url=https://jila.colorado.edu/sites/default/files/2019-03/rwpthesis_jila.pdf |title=Quantum measurement backaction and upconverting microwave signals with mechanical resonators |year=2017}}</ref>

=== Vacuum Fluctuations / Zero-Point Noise ===
{{Main|Physics:Zero-point energy}}

The existence of zero-point energy fluctuations is well-established in the theory of the quantised electromagnetic field.<ref name="Towsend J S 2012">{{Cite book |last=Townsend |first=John S. |url=https://books.google.com/books?id=44lpEQAAQBAJ |title=A Modern Approach to Quantum Mechanics |date=2012 |publisher=University Science Books |isbn=978-1-891389-78-8 |edition=2nd |location=Mill Valley, Calif}}</ref>
Generally speaking, at the lowest energy excitation of a quantized field that permeates all space (i.e. the field mode being in the vacuum state), the root-mean-square fluctuation of field strength is non-zero. This accounts for vacuum fluctuations that permeate all space.

This vacuum fluctuation or quantum noise will effect classical systems. This manifest as quantum decoherence in an entangled system, normally attributed to thermal differences in the conditions surrounding each entangled particle.{{Clarify|date=December 2021}} Because entanglement is studied intensely in simple pairs of entangled photons, for example, decoherence observed in experiments could well be synonymous with "quantum noise" as to the source of the decoherence. Vacuum fluctuation is a possible causes for a quanta of energy to spontaneously appear in a given field or spacetime, then thermal differences must be associated with this event. Hence, it would cause decoherence in an entangled system in proximity of the event.
== Quantum amplifiers ==
A laser is described by the [[Physics:Coherent state|coherent state]] of light, or the superposition of harmonic oscillators eigenstates. Erwin Schrödinger first derived the coherent state for the Schrödinger equation to meet the [[Physics:Correspondence principle|correspondence principle]] in 1926.<ref name="Towsend J S 2012" />

The laser is a quantum mechanical phenomena (see [[Physics:Maxwell–Bloch equations|Maxwell–Bloch equations]], [[Physics:Rotating wave approximation|rotating wave approximation]], and semi-classical model of a two level atom). The [[Physics:Einstein coefficients|Einstein coefficients]] and the laser rate equations are adequate if one is interested in the population levels and one does not need to account for population quantum coherences (the off diagonal terms in a density matrix). Photons of the order of 10<sup>8</sup> corresponds to a moderate energy. The relative error of measurement of the intensity due to the quantum noise is on the order of 10<sup>−5</sup>. This is considered to be of good precision for most of applications.

A quantum amplifier is an amplifier which operates close to the quantum limit. Quantum noise becomes important when a small signal is amplified. A small signal's quantum uncertainties in its quadrature are also amplified; this sets a lower limit to the amplifier. A quantum amplifier's noise is its output amplitude and phase. Generally, a laser is amplified across a spread of wavelengths around a central wavelength, some mode distribution, and polarization spread. But one can consider a single mode amplification and generalize to many different modes. A phase-invariant amplifier preserves the phase of the input gain without drastic changes to the output phase mode.
<ref name="inva">{{cite journal
| title= Quantum limit of noise of a phase-invariant amplifier
| first1=D. |last1=Kouznetsov
| first2=D. |last2=Rohrlich
| first3=R. |last3=Ortega
| year=1995
| journal=[[Physical Review A]]
| volume=52
| issue=2
| pages=1665–1669
| doi= 10.1103/PhysRevA.52.1665
| pmid=9912406 |arxiv = cond-mat/9407011 |bibcode = 1995PhRvA..52.1665K | s2cid=19495906 }}</ref>

Quantum amplification can be represented with a [[Unitary operator|unitary operator]], <math>A_\text{out} = U^{\dagger} A_\text{in} U</math> , as stated in D. Kouznetsov 1995 paper.

== Applications ==

=== Experimental suppression using reflective boundaries ===
A study published in ''[[Physics:Physical Review Research|Physical Review Research]]'' (2025) by scientists at [[Organization:Swansea University|Swansea University]] demonstrated a novel method of suppressing quantum noise using reflective boundaries. By placing a nanoparticle at the focal center of a hemispherical mirror, researchers found that the particle became indistinguishable from its mirror image under specific conditions. This configuration prevented extraction of positional information from scattered light, which in turn eliminated the associated quantum [[Physics:Back action (quantum)|backaction]], the disturbance caused by measurement using [[Physics:Photons|photons]].<ref>{{cite journal |last1=Gajewski |first1=Rafał |last2=Bateman |first2=James |title=Backaction suppression in levitated optomechanics using reflective boundaries |journal=Physical Review Research |date=11 April 2025 |volume=7 |issue=2 |article-number=023041 |doi=10.1103/PhysRevResearch.7.023041 |doi-access=free |arxiv=2405.04366 |bibcode=2025PhRvR...7b3041G }}</ref>

This counterintuitive effect occurred precisely when light scattering was maximized, suggesting a fundamental link between information availability and quantum noise. The study opened avenues for highly sensitive quantum sensors, macroscopic quantum state experiments, and applications in space-based quantum physics missions such as MAQRO (Macroscopic Quantum Resonators).<ref>{{cite web |url=https://scitechdaily.com/quantum-noise-vanished-inside-the-mirror-experiment-rewriting-physics/ |title=Quantum Noise? Vanished – Inside the Mirror Experiment Rewriting Physics |publisher=SciTechDaily |date=22 May 2025 }}</ref>

==See also==
*[[Quantum error correction]]
*[[Physics:Quantum optics|Quantum optics]]
*[[Physics:Quantum limit|Quantum limit]]
*[[Shot noise]]
*[[Physics:Quantum harmonic oscillator|Quantum harmonic oscillator]]

==References==
{{reflist}}

==Sources==
* {{Cite web |last=Clark |first=Aashish A. |year=2008 |title=Quantum Noise and quantum measurement |url=https://clerkgroup.uchicago.edu/PDFfiles/LesHouchesNotesAC.pdf |publisher=[[Organization:Oxford University Press|Oxford University Press]] |access-date=13 December 2021}}
* {{Cite journal |last1=Clerk |first1=A. A. |last2=Devoret |first2=M. H. |last3=Girvin |first3=S. M. |last4=Marquardt |first4=Florian |last5=Schoelkopf |first5=R. J. |title=Introduction to quantum noise, measurement, and amplification |journal=[[Physics:Reviews of Modern Physics|Reviews of Modern Physics]] |date=2010 |volume=82 |issue=2 |pages=1155–1208 |doi=10.1103/RevModPhys.82.1155 |arxiv=0810.4729 |bibcode=2010RvMP...82.1155C}}
* {{Cite book |last1=Gardiner |first1=Crispin W. |last2=Zoller |first2=Peter |author-link2=Peter Zoller |url=https://books.google.com/books?id=a_xsT8oGhdgC |title=Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics |date=2004 |publisher=Springer |isbn=978-3-540-22301-6 |edition=3rd |location=Berlin ; Heidelberg}}

{{Quantum mechanics topics}}

[[Category:Quantum optics]]
[[Category:Laser science]]

{{Sourceattribution|Quantum noise}}

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Physics:Quantum noise - Revision history

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