quantum noise

{{main|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,

$$\backslash Delta\; x\_\{imp\}\; \backslash Delta\; p\_\{BA\}\; \backslash gtrsim\; \backslash hbar.$$

Where the $\backslash Delta\; x\_\{imp\}$ is the uncertainty in an atom's position, and the $\backslash Delta\; p\_\{BA\}$ is the uncertainty of the momentum or sometimes called the backaction (momentum transferred to the atom) when near the 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.

Noise is of practical concern for precision engineering and engineered systems approaching the standard quantum limit. Typical engineered consideration of quantum noise is for quantum nondemolition measurement and quantum point contact. So quantifying noise is useful.

{{cite journal

|author1=Clerk, A. A. |author2=Devoret, M. H. |author3=Girvin, S. M. |author4=Marquardt, Florian |author5=Schoelkopf, R. J.

|title=Introduction to quantum noise, measurement, and amplification

|doi=10.1103/RevModPhys.82.1155

|journal=Rev. Mod. Phys.

|volume=82

|page=1155--1208

|year=2010|issue=2|arxiv=0810.4729

|bibcode=2010RvMP...82.1155C

|s2cid=119200464

}}

{{cite journal

|author1=Henry, Charles H. |author2=Kazarinov, Rudolf F.

|title=Quantum noise in photonics|doi=10.1103/RevModPhys.68.801

|journal=Rev. Mod. Phys.

|volume=68

|page=01–853

|year=1996

|issue=3

|bibcode=1996RvMP...68..801H

}}

A signal's noise is quantified as the Fourier transform of its autocorrelation.

The autocorrelation of a signal is given as

$$G\_\{vv\}(t-t\text{'})\; =\; \backslash langle\; V(t)V(t\text{'})\backslash rangle\; ,$$

which measures when our signal is positively, negatively or not correlated at different times $t$ and $t\text{'}$.

The time average, $\backslash langle\; V(t)\; \backslash rangle$, is zero and our $V(t)$ is a voltage signal. Its Fourier transform is

$$V(\backslash omega)\; =\; \backslash frac\{1\}\{\backslash sqrt\{T\}\}\backslash int\_\{0\}^\{T\}\; V(t)e^\{i\backslash omega\; t\}dt$$

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.,

$$S\_\{vv\}(\backslash omega)\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}e^\{i\backslash omega\; t\}\; G\_\{vv\}dt\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}e^\{i\backslash omega\; t\}\; \backslash langle\; |V(\backslash omega)|^2\backslash rangle\; dt$$

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 normally distributed.
• $G\_\{vv\}$ decays to zero rapidly over some time $\backslash tau\_c$.
• We sample over a sufficiently large time, $T$, that our integral scales as a random walk $\backslash sqrt\{T\}$. So our $V(\backslash omega)$ is independent of measured time for $T\; \backslash gg\; \backslash tau\_c$. {{pb}} Said in another way, $G\_\{vv\}(t-t\text{'})\; \backslash to\; 0$ as $|t-t\text{'}|\; \backslash gg\; \backslash tau\_c$.

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.

To study quantum noise, one replaces the corresponding classical measurements with quantum operators, e.g.,

$$S\_\{xx\}(\backslash omega)\; =\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}e^\{i\backslash omega\; t\}\; \backslash langle\; \backslash hat\{x\}(t)\; \backslash hat\{x\}(0)\; \backslash rangle\; dt\; ,$$

where $\backslash langle\; \backslash cdot\; \backslash rangle$ are the quantum statistical average using the density matrix in the Heisenberg picture.

The Heisenberg uncertainty implies the existence of noise.{{Cite book |author=Crispin W. Gardiner and Paul Zoller |title=Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics |edition=3rd |publisher=Springer |year=2004 |isbn=978-3540223016}} An operator with a hermitian conjugate follows the relationship, $A\; A^\{\backslash dagger\}\; \backslash ge\; 0$. Define $A$ as $A\; =\; \backslash delta\; x\; +\backslash lambda\; e^\{i\backslash theta\}\backslash delta\; y$ where $\backslash lambda$ is real. The $x$ and $y$ are the quantum operators. We can show the following,

$$\backslash langle\; \backslash delta\; x^2\; \backslash rangle\; \backslash langle\; \backslash delta\; y^2\; \backslash rangle\; \backslash ge\; \backslash frac\{1\}\{4\}\; |\backslash langle\; [\backslash delta\; x,\; \backslash delta\; y]\backslash rangle|^2\; +\; |\backslash langle\; [\backslash delta\; x,\; \backslash delta\; y]\_+\; \backslash rangle|^2$$

where the $\backslash langle\; \backslash cdot\; \backslash rangle$ are the averages over the wavefunction and other statistical properties. The left terms are the uncertainty in $x$ and $y$, the second term on the right is to covariance or $\backslash langle\; \backslash delta\; x\; \backslash delta\; y\; +\; \backslash delta\; y\; \backslash delta\; x\; \backslash rangle$ 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 $x$ and $y$ correspond to position and momentum that meets the well known commutator relation, $[x,p]=i\backslash hbar$. Then our new expression is,

$$\backslash Delta\; x\; \backslash Delta\; y\; \backslash ge\; \backslash sqrt\{\backslash frac\{1\}\{4\}\; \backslash hbar^2\; +\; \backslash sigma\_\{xy\}^2\; \}$$

Where the $\backslash sigma\_\{xy\}$ is the correlation. If the second term on the right vanishes, then we recover the Heisenberg uncertainty principle.

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

$$x(t)\; =\; x(0)\backslash cos(\backslash Omega\; t)\; +\; p(0)\backslash frac\{1\}\{M\backslash Omega\}\backslash sin(\backslash Omega\; t)$$

The quantum autocorrelation is then,

$$\backslash begin\{align\}$$

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}

Classically, there is no correlation between position and momentum. The uncertainty principle requires the second term to be nonzero. It goes to $i\backslash hbar/2$.

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.,

$$\backslash frac\{1\}\{2\}M\backslash Omega^2\; \backslash langle\; x^2\backslash rangle\; =\; \backslash frac\{1\}\{2\}k\_\backslash text\{B\}\; T$$

In the classical autocorrelation, we have

$$G\_\{xx\}\; =\; \backslash frac\{k\_\backslash text\{B\}T\}\{M\backslash Omega^2\}\backslash cos(\backslash Omega\; t)$$

\to

S_{xx}(\omega) = \pi \frac{k_\text{B} T}{M\Omega^2}[\delta(\omega - \Omega) +\delta(\omega +\Omega)]

while in the quantum autocorrelation we have

$$G\_\{xx\}\; =\; \backslash left(\; \backslash frac\{\backslash hbar\}\{2M\backslash Omega\}\backslash right)\; \backslash left\backslash \{n\_\{BE\}(\backslash hbar\backslash Omega)\; e^\{i\backslash Omega\; t\}\; +\; [\; n\_\{BE\}(\backslash hbar\; \backslash Omega)\; +1\; ]e^\{-i\backslash Omega\; t\}\; \backslash right\; \backslash \}$$

\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)]

Where the fraction terms in parentheses is the zero-point energy uncertainty. The $n\_\{BE\}$ is the Bose-Einstein population distribution. Notice that the quantum $S\_\{xx\}$ is asymmetric in the due to the imaginary autocorrelation. As we increase to higher temperature that corresponds to taking the limit of $k\_BT\; \backslash gg\; \backslash hbar\backslash Omega$. One can show that the quantum approaches the classical $S\_\{xx\}$. This allows $n\_\{BE\}\; \backslash approx\; n\_\{BE\}+1\; \backslash approx\; \backslash frac\{k\_\backslash text\{B\}T\}\{\backslash hbar\; \backslash Omega\}$

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.

Most optical communications use amplitude modulation where the quantum noise is predominantly the shot noise. A laser's quantum noise, when not considering shot noise, is the uncertainty of its electric field's amplitude and phase. That uncertainty becomes observable when a quantum amplifier preserves phase. The phase noise becomes important when the energy of the frequency modulation or 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).

An ideal noiseless gain cannot exit. {{Cite book |author=Desurvire, Emmanuel |title=Erbium-Doped Fiber Amplifiers. Principles and Applications |edition=1st |publisher=Wiley-Interscience |year=1994 |isbn=978-0471589778}} Consider the amplification of stream of photons, an ideal linear noiseless gain, and the Energy-Time uncertainty relation.

$$\backslash Delta\; E\; \backslash Delta\; t\; \backslash gtrsim\; \backslash hbar/2$$

The photons, ignoring the uncertainty in frequency, will have an uncertainty in its overall phase and number, and assume a known frequency, i.e., $\backslash Delta\; \backslash phi\; =\; 2\backslash pi\; \backslash nu\; \backslash Delta\; t$ and $\backslash Delta\; E\; =\; h\backslash nu\backslash Delta\; n$. 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.

$$\backslash Delta\; n\; \backslash Delta\; \backslash phi\; >\; 1/2$$

Let an ideal linear noiseless gain, $G$, act on the photon stream. We also assume a unity quantum efficiency, or every photon is converted to a photocurrent. The output will be following with no noise added.

$$n\_0\; \backslash pm\; \backslash Delta\; n\_0\; \backslash to\; Gn\_0\; \backslash pm\; G\backslash Delta\; n\_0$$

The phase will be modified too,

$$\backslash phi\_0\; \backslash pm\; \backslash Delta\backslash phi\_0\; \backslash to\; \backslash phi\_0\; +\backslash theta\; +\; \backslash Delta\backslash phi\_0\; ,$$

where the $\backslash theta$ is the overall accumulated phase as the photons traveled through the gain medium.

Substituting our output gain and phase uncertainties, gives us

$$\backslash Delta\; n\_0\; \backslash Delta\; \backslash phi\_0\; >\; 1/2G\; .$$

Our gain is $G>1$, 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{{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| s2cid=51674821}}

$$P\_n\; =\; h\; \backslash nu\; B\; (G-1)$$

where $B$ is half of the full width at half max, the $\backslash nu$ frequency of the photons, and $h$ is the Planck constant. The term $h\backslash nu\; B\_0/2$ with $B\_0\; =\; 2\; B$ is sometimes called quantum 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.C. W. Gardiner and Peter Zoller, Quantum Noise, Springer-Verlag (1991, 2000, 2004)

An experiment by S. Saraf, et .al.

{{cite journal

|author=Saraf, Shally and Urbanek, Karel and Byer, Robert L. and King, Peter J.

|title=Quantum noise measurements in a continuous-wave laser-diode-pumped Nd:YAG saturated amplifier.

|doi=10.1364/ol.30.001195

|journal= Optics Letters

| volume=30

| year=2005|issue=10

|pages=1195–1197

|pmid=15943307

|bibcode=2005OptL...30.1195S

|url=https://authors.library.caltech.edu/6950/

}}

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.

The theory behind noise analysis of photon statistics (sometimes called the forward Kolmogorov equation) starts from the Masters equation from Shimoda et al.{{cite journal|author=Shimoda, Koichi and Takahasi, Hidetosi and H. Townes, 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}}

$$\backslash frac\{dP\_n\}\{dx\}\; =\; a[nP\_\{n-1\}-(n+1)P\_n]\; +\; b[(n+1)P\_\{n+1\}-nP\_n]$$

where $a$ corresponds to the emission cross section and upper population number product $\backslash sigma\_e\; N\_2$, and the $b$ is the absorption cross section $\backslash sigma\_a\; N\_1$. The above relation is describing the probability of finding $n$ photons in radiation mode $|n\; \backslash rangle$. The dynamic only considers neighboring modes $|\; n+1\; \backslash rangle$ and $|\; n-1\backslash rangle$ as the photons travel through a medium of excited and ground state atoms from position $x$ to $x+dx$. 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, $|n-1\; \backslash rangle\; \backslash to\; |\; n\; \backslash rangle$ and $|n\; \backslash rangle\; \backslash to\; |n+1\; \backslash rangle$ and two photons leaving a field to the atom $|n+1\; \backslash rangle\; \backslash to\; |n\; \backslash rangle$ and $|n\; \backslash rangle\; \backslash to\; |n-1\; \backslash rangle$. Its noise power is given as,

$$P\_d^2\; =\; P\_\backslash text\{shot\}^2\; [1+2f\_\{sp\}\backslash eta(G-1)]$$

Where,

• $P\_d$ is the power at the detector,
• $P\_\backslash text\{shot\}$ is the power limited shot noise,
• $G$ the unsaturated gain and is also true for saturated gain,
• $\backslash eta$ is the efficiency factor. That is the product of transmission window efficiency to our photodetector, and quantum efficiency.
• $f\_\{sp\}$ 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. {{Cite book | editor=Bishnu P. Pal | title=Guided Wave Optical Components and Devices: Basics, Technology, and Applications| edition=1st | publisher=Academic| year=2006 | isbn=978-0-12-088481-0}}

Sarif, et al. demonstrated quantum noise or shot noise limited measurements over a wide range of power gain that agreed with theory.

{{Main|Zero-point energy}}

The existence of zero-point energy fluctuations is well-established in the theory of the quantised electromagnetic field.{{Cite book | author=John S Townsend. | title=A Modern Approach to Quantum Mechanics| edition=2nd | publisher=University Science Books| year=2012 | isbn=978-1891389788 }}

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.{{dubious|date=April 2015}}

A laser is described by the 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 correspondence principle in 1926.

The laser is a quantum mechanical phenomena (see Maxwell–Bloch equations, rotating wave approximation, and semi-classical model of a two level atom). The 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 10⁸ corresponds to a moderate energy. The relative error of measurement of the intensity due to the quantum noise is on the order of 10⁻⁵. 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.

{{cite journal

| title= Quantum limit of noise of a phase-invariant amplifier

| author=D. Kouznetsov |author2=D. Rohrlich |author3=R.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 }}

Quantum amplification can be represented with a unitary operator, $A\_\backslash text\{out\}\; =\; U^\{\backslash dagger\}\; A\_\backslash text\{in\}\; U$ , as stated in D. Kouznetsov 1995 paper.

A study published in Physical Review Research (2025) by scientists at 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 backaction, the disturbance caused by measurement using photons.{{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 |doi=10.1103/PhysRevResearch.7.023041 |doi-access=free |arxiv=2405.04366 }}

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).{{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 }}

• Quantum error correction
• Quantum optics
• Quantum limit
• Shot noise
• Quantum harmonic oscillator

{{reflist}}

• Clerk, Aashish A. Quantum Noise and quantum measurement. Oxford University Press.
• Clerk, Aashish A., et al. Introduction to Quantum Noise, measurement, and amplification,Reviews of Modern Physics 82, 1155-1208.
• Gardiner, C. W. and Zoller, P. Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer, 2004, 978-3540223016

• C. W. Gardiner and Peter Zoller, [https://books.google.com/books?id=a_xsT8oGhdgC Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics], Springer-Verlag (1991, 2000, 2004).

{{Quantum mechanics topics}}

Category:Quantum optics

Category:Laser science