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{{Short description|Quantum Collection topic on Quantum Relaxation and thermalization}}
{{Quantum book backlink|Statistical mechanics and kinetic theory}}
{{Quantum book backlink|Statistical mechanics and kinetic theory}}
'''Quantum relaxation and thermalization''' describe how a system prepared out of equilibrium evolves toward a stationary state and, under suitable conditions, toward a state consistent with [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]].<ref name="DAlessio2016">{{cite journal
{{Quantum article nav|previous=Physics:Quantum BBGKY hierarchy|previous label=BBGKY hierarchy|next=Physics:Quantum Thermodynamics|next label=Thermodynamics}}
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'''Relaxation and thermalization''' quantum relaxation and thermalization describe how a system prepared out of equilibrium evolves toward a stationary state and, under suitable conditions, toward a state consistent with quantum statistical mechanics. Relaxation refers to the approach of observables toward long-time stationary values, while thermalization means that those values are described by thermal ensembles such as the canonical or microcanonical ensemble. In general, the natural tendency of many-body systems is toward thermal equilibrium, increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems. Relaxation and thermalization in quantum systems describe how nonequilibrium states evolve toward stationary behavior through dephasing, interactions, and coupling to an environment. For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,
</div>
 
<div style="width:300px;">
[[File:Quantum_relaxation_and_thermalization1.jpg|thumb|280px|Quantum Relaxation and thermalization.]]
</div>
 
</div>
 
==Basic idea==
For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,
 
<math>
i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle,
</math>
 
so the microscopic dynamics are reversible.<ref name="Polkovnikov2011">{{cite journal
|last1=Polkovnikov
|first1=Anatoli
|last2=Sengupta
|first2=Krishnendu
|last3=Silva
|first3=Alessandro
|last4=Vengalattore
|first4=Mukund
|title=Colloquium: Nonequilibrium dynamics of closed interacting quantum systems
|journal=Reviews of Modern Physics
|year=2011
|volume=83
|issue=3
|pages=863–883
|doi=10.1103/RevModPhys.83.863
|url=https://link.aps.org/doi/10.1103/RevModPhys.83.863
}}</ref> Yet experimentally relevant observables may still relax because of dephasing between many energy eigenstates, redistribution of correlations, and the practical inaccessibility of detailed phase information in local measurements.<ref name="GogolinEisert2016">{{cite journal
|last1=Gogolin
|first1=Christian
|last2=Eisert
|first2=Jens
|title=Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
|journal=Reports on Progress in Physics
|year=2016
|volume=79
|issue=5
|pages=056001
|doi=10.1088/0034-4885/79/5/056001
|url=https://arxiv.org/abs/1503.07538
}}</ref><ref name="Polkovnikov2011"/>
 
A thermal equilibrium state is commonly represented by the density operator
 
<math>
\rho_{\mathrm{th}} = \frac{e^{-\beta H}}{Z},
\qquad
Z = \mathrm{Tr}\left(e^{-\beta H}\right),
</math>
 
where <math>\beta = 1/(k_B T)</math> and <math>Z</math> is the partition function.<ref name="DAlessio2016">{{cite journal
  |last1=D'Alessio
  |last1=D'Alessio
  |first1=Luca
  |first1=Luca
Line 17: Line 82:
  |doi=10.1080/00018732.2016.1198134
  |doi=10.1080/00018732.2016.1198134
  |url=https://arxiv.org/abs/1509.06411
  |url=https://arxiv.org/abs/1509.06411
}}</ref><ref name="GogolinEisert2016">{{cite journal
}}</ref>
|last1=Gogolin
 
|first1=Christian
==Equilibration versus thermalization==
|last2=Eisert
Equilibration and thermalization are related but distinct concepts.<ref name="GogolinEisert2016"/> A system '''equilibrates''' when expectation values of observables become nearly time-independent for most late times, even if the exact state continues to evolve unitarily. It '''thermalizes''' when those stationary values coincide with predictions from an appropriate thermal ensemble determined by conserved quantities such as energy.<ref name="RigolNature2008">{{cite journal
|first2=Jens
|title=Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
|journal=Reports on Progress in Physics
|year=2016
|volume=79
|issue=5
|pages=056001
|doi=10.1088/0034-4885/79/5/056001
|url=https://arxiv.org/abs/1503.07538
}}</ref> Relaxation refers to the approach of observables toward long-time stationary values, while thermalization means that those values are described by thermal ensembles such as the canonical or microcanonical ensemble.<ref name="RigolNature2008">{{cite journal
  |last1=Rigol
  |last1=Rigol
  |first1=Marcos
  |first1=Marcos
Line 45: Line 100:
  |url=https://www.nature.com/articles/nature06838
  |url=https://www.nature.com/articles/nature06838
}}</ref>
}}</ref>
In general, the natural tendency of many-body systems is toward [[Physics:thermal equilibrium|thermal equilibrium]], increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems.<ref name="Polkovnikov2011">{{cite journal
|last1=Polkovnikov
|first1=Anatoli
|last2=Sengupta
|first2=Krishnendu
|last3=Silva
|first3=Alessandro
|last4=Vengalattore
|first4=Mukund
|title=Colloquium: Nonequilibrium dynamics of closed interacting quantum systems
|journal=Reviews of Modern Physics
|year=2011
|volume=83
|issue=3
|pages=863–883
|doi=10.1103/RevModPhys.83.863
|url=https://link.aps.org/doi/10.1103/RevModPhys.83.863
}}</ref><ref name="NandkishoreHuse2015">{{cite journal
|last1=Nandkishore
|first1=Rahul
|last2=Huse
|first2=David A.
|title=Many-Body Localization and Thermalization in Quantum Statistical Mechanics
|journal=Annual Review of Condensed Matter Physics
|year=2015
|volume=6
|pages=15–38
|doi=10.1146/annurev-conmatphys-031214-014726
|url=https://arxiv.org/abs/1404.0686
}}</ref>
<div style="float:right; border:1px solid #ccc; padding:4px; background:#ffffe6; margin:0 0 1em 1em; width:450px;">
[[File:Quantum_relaxation_and_thermalization1.jpg]]
<div style="font-size:90%; line-height:1.4; padding-top:4px;">
Relaxation and thermalization in quantum systems describe how nonequilibrium states evolve toward stationary behavior through dephasing, interactions, and coupling to an environment.
</div>
</div>
==Basic idea==
For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,
<math>
i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle,
</math>
so the microscopic dynamics are reversible.<ref name="Polkovnikov2011"/> Yet experimentally relevant observables may still relax because of dephasing between many energy eigenstates, redistribution of correlations, and the practical inaccessibility of detailed phase information in local measurements.<ref name="GogolinEisert2016"/><ref name="Polkovnikov2011"/>
A thermal equilibrium state is commonly represented by the density operator
<math>
\rho_{\mathrm{th}} = \frac{e^{-\beta H}}{Z},
\qquad
Z = \mathrm{Tr}\left(e^{-\beta H}\right),
</math>
where <math>\beta = 1/(k_B T)</math> and <math>Z</math> is the partition function.<ref name="DAlessio2016"/>
==Equilibration versus thermalization==
Equilibration and thermalization are related but distinct concepts.<ref name="GogolinEisert2016"/> A system '''equilibrates''' when expectation values of observables become nearly time-independent for most late times, even if the exact state continues to evolve unitarily. It '''thermalizes''' when those stationary values coincide with predictions from an appropriate thermal ensemble determined by conserved quantities such as energy.<ref name="RigolNature2008"/>


Thus a system can relax without becoming thermal in the ordinary Gibbs sense. This distinction is central in modern nonequilibrium quantum theory.<ref name="DAlessio2016"/><ref name="GogolinEisert2016"/>
Thus a system can relax without becoming thermal in the ordinary Gibbs sense. This distinction is central in modern nonequilibrium quantum theory.<ref name="DAlessio2016"/><ref name="GogolinEisert2016"/>
Line 176: Line 172:


==Magnetic resonance as a relaxation example==
==Magnetic resonance as a relaxation example==
A concrete and experimentally important example of quantum relaxation occurs in [[Physics:Magnetic resonance imaging|magnetic resonance imaging]] (MRI) and [[Physics:Nuclear magnetic resonance spectroscopy|nuclear magnetic resonance spectroscopy]] (NMR), where an observable nuclear spin polarization is created by an external magnetic field and then perturbed by resonant radiofrequency pulses.<ref name="Rinck">{{cite book
A concrete and experimentally important example of quantum relaxation occurs in magnetic resonance imaging (MRI) and nuclear magnetic resonance spectroscopy (NMR), where an observable nuclear spin polarization is created by an external magnetic field and then perturbed by resonant radiofrequency pulses.<ref name="Rinck">{{cite book
  |last1=Rinck
  |last1=Rinck
  |first1=Peter A.
  |first1=Peter A.
Line 287: Line 283:


==Bloch equations==
==Bloch equations==
A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by [[Felix Bloch]].<ref name="Bloch1946">{{cite journal
A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by Felix Bloch.<ref name="Bloch1946">{{cite journal
  |last=Bloch
  |last=Bloch
  |first=F.
  |first=F.
Line 348: Line 344:


==Systems resisting thermalization==
==Systems resisting thermalization==
Not all systems thermalize rapidly or at all. Integrable systems relax to GGEs rather than conventional Gibbs states.<ref name="RigolIntegrable2007"/> Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization.<ref name="NandkishoreHuse2015"/> Other unusual nonthermal behavior appears in systems with dynamical constraints, quantum scars, or special symmetry structures.<ref name="NandkishoreHuse2015"/><ref name="DAlessio2016"/>
Not all systems thermalize rapidly or at all. Integrable systems relax to GGEs rather than conventional Gibbs states.<ref name="RigolIntegrable2007"/> Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization.<ref name="NandkishoreHuse2015">{{cite journal
|last1=Nandkishore
|first1=Rahul
|last2=Huse
|first2=David A.
|title=Many-Body Localization and Thermalization in Quantum Statistical Mechanics
|journal=Annual Review of Condensed Matter Physics
|year=2015
|volume=6
|pages=15–38
|doi=10.1146/annurev-conmatphys-031214-014726
|url=https://arxiv.org/abs/1404.0686
}}</ref> Other unusual nonthermal behavior appears in systems with dynamical constraints, quantum scars, or special symmetry structures.<ref name="NandkishoreHuse2015"/><ref name="DAlessio2016"/>


==Physical interpretation==
==Physical interpretation==
Line 362: Line 370:
Quantum relaxation and thermalization are important in:
Quantum relaxation and thermalization are important in:


* [[Physics:Magnetic resonance imaging|magnetic resonance imaging]] and [[Physics:Nuclear magnetic resonance spectroscopy|nuclear magnetic resonance spectroscopy]]  
* magnetic resonance imaging and nuclear magnetic resonance spectroscopy   
* ultracold atomic gases after quenches   
* ultracold atomic gases after quenches   
* condensed-matter systems out of equilibrium   
* condensed-matter systems out of equilibrium   
* quantum simulators and quantum information devices   
* quantum simulators and quantum information devices   
* open quantum optical systems   
* open quantum optical systems   
* studies of ergodicity, integrability, and [[Physics:Many-body localization|many-body localization]]<ref name="Rinck"/><ref name="Polkovnikov2011"/><ref name="GogolinEisert2016"/><ref name="NandkishoreHuse2015"/>
* studies of ergodicity, integrability, and many-body localization<ref name="Rinck"/><ref name="Polkovnikov2011"/><ref name="GogolinEisert2016"/><ref name="NandkishoreHuse2015"/>


==See also==
==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}


==References==
= References =
{{reflist|3}}
{{reflist|3}}
{{Author|Harold Foppele}}
{{Author|Harold Foppele}}
Line 378: Line 386:
[[Category:Statistical mechanics]]
[[Category:Statistical mechanics]]


{{Sourceattribution|Quantum relaxation and thermalization|1}}
{{Sourceattribution|Physics:Quantum Relaxation and thermalization|1}}

Latest revision as of 00:31, 24 May 2026

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Relaxation and thermalization quantum relaxation and thermalization describe how a system prepared out of equilibrium evolves toward a stationary state and, under suitable conditions, toward a state consistent with quantum statistical mechanics. Relaxation refers to the approach of observables toward long-time stationary values, while thermalization means that those values are described by thermal ensembles such as the canonical or microcanonical ensemble. In general, the natural tendency of many-body systems is toward thermal equilibrium, increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems. Relaxation and thermalization in quantum systems describe how nonequilibrium states evolve toward stationary behavior through dephasing, interactions, and coupling to an environment. For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,

Quantum Relaxation and thermalization.

Basic idea

For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,

iℏ∂∂t|ψ(t)⟩=H|ψ(t)⟩,

so the microscopic dynamics are reversible.[1] Yet experimentally relevant observables may still relax because of dephasing between many energy eigenstates, redistribution of correlations, and the practical inaccessibility of detailed phase information in local measurements.[2][1]

A thermal equilibrium state is commonly represented by the density operator

ρth=e−βHZ,Z=Tr(e−βH),

where β=1/(kBT) and Z is the partition function.[3]

Equilibration versus thermalization

Equilibration and thermalization are related but distinct concepts.[2] A system equilibrates when expectation values of observables become nearly time-independent for most late times, even if the exact state continues to evolve unitarily. It thermalizes when those stationary values coincide with predictions from an appropriate thermal ensemble determined by conserved quantities such as energy.[4]

Thus a system can relax without becoming thermal in the ordinary Gibbs sense. This distinction is central in modern nonequilibrium quantum theory.[3][2]

Eigenstate thermalization

For generic nonintegrable many-body systems, thermalization is commonly explained by the eigenstate thermalization hypothesis (ETH). ETH states that expectation values of simple observables in individual many-body energy eigenstates are smooth functions of energy, so a single eigenstate already exhibits thermal properties for such observables.[4][3]

If the initial state is expanded as

|ψ(0)⟩=∑ncn|En⟩,

then the long-time average of an observable A is approximately determined by diagonal matrix elements,

⟨A⟩‾=∑n|cn|2⟨En|A|En⟩.

Under ETH, the dependence of ⟨En|A|En⟩ on energy is smooth, so the long-time value agrees with the thermodynamic prediction for a narrow energy window.[4][3]

Integrable systems and generalized Gibbs ensembles

Integrable systems possess an extensive number of conserved quantities, so ordinary thermalization may fail. Instead, such systems can relax to a generalized Gibbs ensemble (GGE), which includes all relevant conserved charges.[5]

The GGE density operator is written as

ρGGE=1ZGGEexp⁡(−∑jλjIj),

where Ij are conserved quantities and λj are fixed by the initial state.[5] This is one of the clearest ways in which quantum relaxation can differ from ordinary thermalization.[3]

Open systems and dissipative relaxation

Real systems are rarely perfectly isolated. When coupled to an environment, their reduced dynamics are described by density matrices and quantum master equations, so decoherence and dissipation contribute directly to relaxation.[6]

A typical master equation has the form

dρdt=−iℏ[H,ρ]+𝒟[ρ],

where the dissipator 𝒟[ρ] encodes coupling to the surroundings.[6] In weak-coupling and approximately Markovian regimes, this often drives the system toward a stationary or thermal state.[6]

Magnetic resonance as a relaxation example

A concrete and experimentally important example of quantum relaxation occurs in magnetic resonance imaging (MRI) and nuclear magnetic resonance spectroscopy (NMR), where an observable nuclear spin polarization is created by an external magnetic field and then perturbed by resonant radiofrequency pulses.[7] The return of the longitudinal magnetization to equilibrium is called spin-lattice relaxation, while the loss of transverse phase coherence is called spin-spin relaxation.[8][9]

For spin-12 nuclei, the population imbalance is governed by the Boltzmann distribution,

N+N−=e−ΔE/(kBT),

where ΔE is the Zeeman energy splitting.[8] Because this energy gap is extremely small at ordinary magnetic fields, spontaneous radiative emission is negligible, and relaxation instead occurs through fluctuating local fields generated by surrounding molecules, nuclei, or electrons.[8][9]

Longitudinal and transverse relaxation

The decay of RF-induced spin polarization is described by the two characteristic times T1 and T2.[7][10]

The longitudinal or spin-lattice relaxation time T1 governs the return of the magnetization component parallel to the static field:

Mz(t)=Mz,eq−[Mz,eq−Mz(0)]e−t/T1.

If the magnetization starts in the transverse plane so that Mz(0)=0, this reduces to

Mz(t)=Mz,eq(1−e−t/T1).

In inversion recovery, with Mz(0)=−Mz,eq, one obtains

Mz(t)=Mz,eq(1−2e−t/T1).

The transverse or spin-spin relaxation time T2 describes the decay of the coherent magnetization perpendicular to the field:

Mxy(t)=Mxy(0)e−t/T2.

This is fundamentally a decoherence process produced by fluctuating local precession frequencies and progressive phase scrambling of the spins.[10][7]

T2* and field inhomogeneity

In real magnetic resonance experiments, additional dephasing is caused by magnetic field inhomogeneity. This produces an apparent decay time T2*, usually shorter than T2, according to

1T2*=1T2+1Tinhom=1T2+γΔB0.

Unlike true spin-spin relaxation, inhomogeneity-induced dephasing can often be refocused by a spin-echo sequence, so it is not an irreversible relaxation mechanism in the strict microscopic sense.[11]

Bloch equations

A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by Felix Bloch.[12] For the magnetization vector 𝐌=(Mx,My,Mz) in a magnetic field 𝐁(t) they are

∂Mx∂t=γ(𝐌×𝐁)x−MxT2,

∂My∂t=γ(𝐌×𝐁)y−MyT2,

∂Mz∂t=γ(𝐌×𝐁)z−Mz−M0T1.

These equations combine coherent Larmor precession with phenomenological longitudinal and transverse relaxation.[12][10]

Microscopic relaxation mechanisms

Microscopically, relaxation requires couplings that allow the spin system to exchange energy or phase information with its surroundings. In NMR the dominant mechanisms often include magnetic dipole-dipole interactions, chemical shift anisotropy, spin-rotation coupling, and quadrupolar interactions for nuclei with I≥1.[8] Molecular reorientation modulates these couplings in time, and the resulting fluctuations drive transitions between nuclear spin states.[8]

Within time-dependent perturbation theory, relaxation rates are determined by spectral density functions, which are Fourier transforms of autocorrelation functions of these fluctuating interactions.[8]

BPP theory

A classic microscopic model is the Bloembergen-Purcell-Pound (BPP) theory, which explains nuclear relaxation in terms of molecular tumbling and an exponentially decaying autocorrelation function.[13] If the correlation function is proportional to e−t/τc, then for dipolar relaxation one finds

1T1=K[τc1+ω02τc2+4τc1+4ω02τc2],

1T2=K2[3τc+5τc1+ω02τc2+2τc1+4ω02τc2].

Here τc is the rotational correlation time and ω0 is the Larmor angular frequency.[13] BPP theory works well for simple liquids and gives a microscopic picture of how molecular motion controls relaxation rates.[8]

Prethermalization and slow relaxation

Some nearly integrable or weakly perturbed quantum systems show prethermalization: observables first relax to a long-lived quasistationary state and only much later drift toward full thermal equilibrium.[1][3] This reflects the presence of approximately conserved quantities and separated timescales in the dynamics.

Systems resisting thermalization

Not all systems thermalize rapidly or at all. Integrable systems relax to GGEs rather than conventional Gibbs states.[5] Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization.[14] Other unusual nonthermal behavior appears in systems with dynamical constraints, quantum scars, or special symmetry structures.[14][3]

Physical interpretation

Quantum relaxation and thermalization connect reversible microscopic laws with irreversible macroscopic behavior:

  • dephasing suppresses coherent oscillations in observables
  • interactions redistribute energy and correlations
  • conserved quantities constrain the final stationary state
  • coupling to an environment introduces dissipation and decoherence
  • coarse-grained measurements reveal equilibration even when the full state remains pure and unitary[1][6][2]

Applications

Quantum relaxation and thermalization are important in:

  • magnetic resonance imaging and nuclear magnetic resonance spectroscopy
  • ultracold atomic gases after quenches
  • condensed-matter systems out of equilibrium
  • quantum simulators and quantum information devices
  • open quantum optical systems
  • studies of ergodicity, integrability, and many-body localization[7][1][2][14]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Polkovnikov, Anatoli; Sengupta, Krishnendu; Silva, Alessandro; Vengalattore, Mukund (2011). "Colloquium: Nonequilibrium dynamics of closed interacting quantum systems". Reviews of Modern Physics 83 (3): 863–883. doi:10.1103/RevModPhys.83.863. https://link.aps.org/doi/10.1103/RevModPhys.83.863. 
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Gogolin, Christian; Eisert, Jens (2016). "Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems". Reports on Progress in Physics 79 (5): 056001. doi:10.1088/0034-4885/79/5/056001. https://arxiv.org/abs/1503.07538. 
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 D'Alessio, Luca; Kafri, Yariv; Polkovnikov, Anatoli; Rigol, Marcos (2016). "From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics". Advances in Physics 65 (3): 239–362. doi:10.1080/00018732.2016.1198134. https://arxiv.org/abs/1509.06411. 
  4. ↑ 4.0 4.1 4.2 Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim (2008). "Thermalization and its mechanism for generic isolated quantum systems". Nature 452: 854–858. doi:10.1038/nature06838. https://www.nature.com/articles/nature06838. 
  5. ↑ 5.0 5.1 5.2 Rigol, Marcos; Dunjko, Vanja; Yurovsky, Vladimir; Olshanii, Maxim (2007). "Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of lattice hard-core bosons". Physical Review Letters 98 (5): 050405. doi:10.1103/PhysRevLett.98.050405. https://link.aps.org/doi/10.1103/PhysRevLett.98.050405. 
  6. ↑ 6.0 6.1 6.2 6.3 Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press. ISBN 9780198520634. https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780198520634. 
  7. ↑ 7.0 7.1 7.2 7.3 Rinck, Peter A. (2022). Relaxation Times and Basic Pulse Sequences in MR Imaging. in: Magnetic Resonance in Medicine. A Critical Introduction. 12th edition. pp. 65-92.. Offprint to download: TRTF - The Round Table Foundation / EMRF - European Magnetic Resonance Forum. ISBN 978-3-7460-9518-9. http://trtf.eu/textbook.htm. 
  8. ↑ 8.0 8.1 8.2 8.3 8.4 8.5 8.6 Abragam, A. (1961). "VII Thermal Relaxation in Liquids and Gases". Principles of Nuclear Magnetism. Oxford University Press. p. 264. ISBN 019852014X. 
  9. ↑ 9.0 9.1 Hoult, D. I.; Bahkar, B. (1998). "NMR Signal Reception: Virtual Photons and Coherent Spontaneous Emission". Concepts in Magnetic Resonance 9 (5): 277–297. doi:10.1002/(SICI)1099-0534(1997)9:5<277::AID-CMR1>3.0.CO;2-W. 
  10. ↑ 10.0 10.1 10.2 Levitt, Malcolm H. (2008). Spin Dynamics: Basics of Nuclear Magnetic Resonance (2 ed.). John Wiley & Sons. ISBN 9780470511176. 
  11. ↑ Chavhan, Govind B.; Babyn, Paul S.; Thomas, Bejoy; Shroff, Manohar M.; Haacke, E. Mark (2009). "Principles, Techniques, and Applications of T2*-based MR Imaging and its Special Applications". Radiographics 29 (5): 1433–1449. doi:10.1148/rg.295095034. PMID 19755604. 
  12. ↑ 12.0 12.1 Bloch, F. (1946). "Nuclear Induction". Physical Review 70: 460–473. doi:10.1103/PhysRev.70.460. 
  13. ↑ 13.0 13.1 Bloembergen, N.; Purcell, E. M.; Pound, R. V. (1948). "Relaxation Effects in Nuclear Magnetic Resonance Absorption". Physical Review 73 (7): 679–712. doi:10.1103/PhysRev.73.679. 
  14. ↑ 14.0 14.1 14.2 Nandkishore, Rahul; Huse, David A. (2015). "Many-Body Localization and Thermalization in Quantum Statistical Mechanics". Annual Review of Condensed Matter Physics 6: 15–38. doi:10.1146/annurev-conmatphys-031214-014726. https://arxiv.org/abs/1404.0686. 
Author: Harold Foppele

Source attribution: Physics:Quantum Relaxation and thermalization

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