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{{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
|last1=D'Alessio
|first1=Luca
|last2=Kafri
|first2=Yariv
|last3=Polkovnikov
|first3=Anatoli
|last4=Rigol
|first4=Marcos
|title=From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics
|journal=Advances in Physics
|year=2016
|volume=65
|issue=3
|pages=239–362
|doi=10.1080/00018732.2016.1198134
|url=https://arxiv.org/abs/1509.06411
}}</ref><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> 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
|first1=Marcos
|last2=Dunjko
|first2=Vanja
|last3=Olshanii
|first3=Maxim
|title=Thermalization and its mechanism for generic isolated quantum systems
|journal=Nature
|year=2008
|volume=452
|pages=854–858
|doi=10.1038/nature06838
|url=https://www.nature.com/articles/nature06838
}}</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"/>

==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.<ref name="RigolNature2008"/><ref name="DAlessio2016"/>

If the initial state is expanded as

<math>
|\psi(0)\rangle = \sum_n c_n |E_n\rangle,
</math>

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

<math>
\overline{\langle A \rangle}
=
\sum_n |c_n|^2 \langle E_n|A|E_n\rangle.
</math>

Under ETH, the dependence of <math>\langle E_n|A|E_n\rangle</math> on energy is smooth, so the long-time value agrees with the thermodynamic prediction for a narrow energy window.<ref name="RigolNature2008"/><ref name="DAlessio2016"/>

==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.<ref name="RigolIntegrable2007">{{cite journal
|last1=Rigol
|first1=Marcos
|last2=Dunjko
|first2=Vanja
|last3=Yurovsky
|first3=Vladimir
|last4=Olshanii
|first4=Maxim
|title=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
|journal=Physical Review Letters
|year=2007
|volume=98
|issue=5
|pages=050405
|doi=10.1103/PhysRevLett.98.050405
|url=https://link.aps.org/doi/10.1103/PhysRevLett.98.050405
}}</ref>

The GGE density operator is written as

<math>
\rho_{\mathrm{GGE}} = \frac{1}{Z_{\mathrm{GGE}}}\exp\left(-\sum_j \lambda_j I_j\right),
</math>

where <math>I_j</math> are conserved quantities and <math>\lambda_j</math> are fixed by the initial state.<ref name="RigolIntegrable2007"/> This is one of the clearest ways in which quantum relaxation can differ from ordinary thermalization.<ref name="DAlessio2016"/>

==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.<ref name="BreuerPetruccione">{{cite book
|last1=Breuer
|first1=Heinz-Peter
|last2=Petruccione
|first2=Francesco
|title=The Theory of Open Quantum Systems
|publisher=Oxford University Press
|year=2002
|isbn=9780198520634
|url=https://global.oup.com/academic/product/the-theory-of-open-quantum-systems-9780198520634
}}</ref>

A typical master equation has the form

<math>
\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] + \mathcal{D}[\rho],
</math>

where the dissipator <math>\mathcal{D}[\rho]</math> encodes coupling to the surroundings.<ref name="BreuerPetruccione"/> In weak-coupling and approximately Markovian regimes, this often drives the system toward a stationary or thermal state.<ref name="BreuerPetruccione"/>

==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
|last1=Rinck
|first1=Peter A.
|year=2022
|title=Relaxation Times and Basic Pulse Sequences in MR Imaging. in: Magnetic Resonance in Medicine. A Critical Introduction. 12th edition. pp. 65-92.
|url=http://trtf.eu/textbook.htm
|location=Offprint to download
|publisher=TRTF - The Round Table Foundation / EMRF - European Magnetic Resonance Forum
|isbn=978-3-7460-9518-9
|archive-url=https://web.archive.org/web/20240613151701/https://trtf.eu/textbook.htm
|archive-date=2024-06-13
|url-status=dead
}}</ref> 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'''.<ref name="Abragam1961">{{cite book
|last=Abragam
|first=A.
|title=Principles of Nuclear Magnetism
|publisher=Oxford University Press
|year=1961
|chapter=VII Thermal Relaxation in Liquids and Gases
|page=264
|isbn=019852014X
}}</ref><ref name="HoultBahkar1998">{{cite journal
|last1=Hoult
|first1=D. I.
|last2=Bahkar
|first2=B.
|title=NMR Signal Reception: Virtual Photons and Coherent Spontaneous Emission
|journal=Concepts in Magnetic Resonance
|year=1998
|volume=9
|issue=5
|pages=277–297
|doi=10.1002/(SICI)1099-0534(1997)9:5<277::AID-CMR1>3.0.CO;2-W
}}</ref>

For spin-<math>\tfrac{1}{2}</math> nuclei, the population imbalance is governed by the Boltzmann distribution,

<math>
\frac{N_{+}}{N_{-}} = e^{-\Delta E/(k_B T)},
</math>

where <math>\Delta E</math> is the Zeeman energy splitting.<ref name="Abragam1961"/> 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.<ref name="Abragam1961"/><ref name="HoultBahkar1998"/>

==Longitudinal and transverse relaxation==
The decay of RF-induced spin polarization is described by the two characteristic times <math>T_1</math> and <math>T_2</math>.<ref name="Rinck"/><ref name="Levitt2008">{{cite book
|last=Levitt
|first=Malcolm H.
|title=Spin Dynamics: Basics of Nuclear Magnetic Resonance
|edition=2
|publisher=John Wiley & Sons
|year=2008
|isbn=9780470511176
}}</ref>

The longitudinal or spin-lattice relaxation time <math>T_1</math> governs the return of the magnetization component parallel to the static field:

<math>
M_z(t)=M_{z,\mathrm{eq}}-\left[M_{z,\mathrm{eq}}-M_z(0)\right]e^{-t/T_1}.
</math>

If the magnetization starts in the transverse plane so that <math>M_z(0)=0</math>, this reduces to

<math>
M_z(t)=M_{z,\mathrm{eq}}\left(1-e^{-t/T_1}\right).
</math>

In inversion recovery, with <math>M_z(0)=-M_{z,\mathrm{eq}}</math>, one obtains

<math>
M_z(t)=M_{z,\mathrm{eq}}\left(1-2e^{-t/T_1}\right).
</math>

The transverse or spin-spin relaxation time <math>T_2</math> describes the decay of the coherent magnetization perpendicular to the field:

<math>
M_{xy}(t)=M_{xy}(0)e^{-t/T_2}.
</math>

This is fundamentally a decoherence process produced by fluctuating local precession frequencies and progressive phase scrambling of the spins.<ref name="Levitt2008"/><ref name="Rinck"/>

==T<sub>2</sub><sup>*</sup> and field inhomogeneity==
In real magnetic resonance experiments, additional dephasing is caused by magnetic field inhomogeneity. This produces an apparent decay time <math>T_2^*</math>, usually shorter than <math>T_2</math>, according to

<math>
\frac{1}{T_2^*}=\frac{1}{T_2}+\frac{1}{T_{\mathrm{inhom}}}
=\frac{1}{T_2}+\gamma \Delta B_0.
</math>

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.<ref name="Chavhan2009">{{cite journal
|last1=Chavhan
|first1=Govind B.
|last2=Babyn
|first2=Paul S.
|last3=Thomas
|first3=Bejoy
|last4=Shroff
|first4=Manohar M.
|last5=Haacke
|first5=E. Mark
|title=Principles, Techniques, and Applications of T2*-based MR Imaging and its Special Applications
|journal=Radiographics
|year=2009
|volume=29
|issue=5
|pages=1433–1449
|doi=10.1148/rg.295095034
|pmid=19755604
|pmc=2799958
}}</ref>

==Bloch equations==
A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by [[Felix Bloch]].<ref name="Bloch1946">{{cite journal
|last=Bloch
|first=F.
|title=Nuclear Induction
|journal=Physical Review
|year=1946
|volume=70
|pages=460–473
|doi=10.1103/PhysRev.70.460
}}</ref> For the magnetization vector <math>\mathbf{M}=(M_x,M_y,M_z)</math> in a magnetic field <math>\mathbf{B}(t)</math> they are

<math>
\frac{\partial M_x}{\partial t} = \gamma (\mathbf{M}\times\mathbf{B})_x - \frac{M_x}{T_2},
</math>

<math>
\frac{\partial M_y}{\partial t} = \gamma (\mathbf{M}\times\mathbf{B})_y - \frac{M_y}{T_2},
</math>

<math>
\frac{\partial M_z}{\partial t} = \gamma (\mathbf{M}\times\mathbf{B})_z - \frac{M_z-M_0}{T_1}.
</math>

These equations combine coherent Larmor precession with phenomenological longitudinal and transverse relaxation.<ref name="Bloch1946"/><ref name="Levitt2008"/>

==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 <math>I \ge 1</math>.<ref name="Abragam1961"/> Molecular reorientation modulates these couplings in time, and the resulting fluctuations drive transitions between nuclear spin states.<ref name="Abragam1961"/>

Within time-dependent perturbation theory, relaxation rates are determined by spectral density functions, which are Fourier transforms of autocorrelation functions of these fluctuating interactions.<ref name="Abragam1961"/>

==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.<ref name="BPP1948">{{cite journal
|last1=Bloembergen
|first1=N.
|last2=Purcell
|first2=E. M.
|last3=Pound
|first3=R. V.
|title=Relaxation Effects in Nuclear Magnetic Resonance Absorption
|journal=Physical Review
|year=1948
|volume=73
|issue=7
|pages=679–712
|doi=10.1103/PhysRev.73.679
}}</ref> If the correlation function is proportional to <math>e^{-t/\tau_c}</math>, then for dipolar relaxation one finds

<math>
\frac{1}{T_1}=K\left[\frac{\tau_c}{1+\omega_0^2\tau_c^2}+\frac{4\tau_c}{1+4\omega_0^2\tau_c^2}\right],
</math>

<math>
\frac{1}{T_2}=\frac{K}{2}\left[3\tau_c+\frac{5\tau_c}{1+\omega_0^2\tau_c^2}+\frac{2\tau_c}{1+4\omega_0^2\tau_c^2}\right].
</math>

Here <math>\tau_c</math> is the rotational correlation time and <math>\omega_0</math> is the Larmor angular frequency.<ref name="BPP1948"/> BPP theory works well for simple liquids and gives a microscopic picture of how molecular motion controls relaxation rates.<ref name="Abragam1961"/>

==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.<ref name="Polkovnikov2011"/><ref name="DAlessio2016"/> 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.<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"/>

==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<ref name="Polkovnikov2011"/><ref name="BreuerPetruccione"/><ref name="GogolinEisert2016"/>

==Applications==
Quantum relaxation and thermalization are important in:

* [[Physics:Magnetic resonance imaging|magnetic resonance imaging]] and [[Physics:Nuclear magnetic resonance spectroscopy|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 [[Physics:Many-body localization|many-body localization]]<ref name="Rinck"/><ref name="Polkovnikov2011"/><ref name="GogolinEisert2016"/><ref name="NandkishoreHuse2015"/>

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

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

{{Sourceattribution|Quantum relaxation and thermalization|1}}

Physics:Quantum Relaxation and thermalization - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template