Physics:Quantum Relaxation and thermalization

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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.

Basic idea

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

$i\hslash \frac{\partial }{\partial t}|\psi (t)⟩=H|\psi (t)⟩,$

so the microscopic dynamics are reversible. 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.

A thermal equilibrium state is commonly represented by the density operator

${\rho }_{\mathrm{t}\mathrm{h}}=\frac{e^{-\beta H}}{Z},Z=\mathrm{T}\mathrm{r}\left(e^{-\beta H}\right),$

where $\beta =1/(k_{B}T)$ and $Z$ is the partition function.

Equilibration versus thermalization

Equilibration and thermalization are related but distinct concepts. 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.

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

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.

If the initial state is expanded as

$|\psi (0)⟩=\sum _n{c}_n|E_n⟩,$

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

$\overline{⟨A⟩}=\sum _n|c_n{|}^2⟨E_n|A|E_n⟩.$

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

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.

The GGE density operator is written as

${\rho }_{\mathrm{G}\mathrm{G}\mathrm{E}}=\frac{1}{Z_{\mathrm{G}\mathrm{G}\mathrm{E}}}\exp (-\sum _j{\lambda }_j{I}_j),$

where $I_j$ are conserved quantities and ${\lambda }_j$ are fixed by the initial state. This is one of the clearest ways in which quantum relaxation can differ from ordinary thermalization.

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.

A typical master equation has the form

$\frac{d\rho }{dt}=-\frac{i}{\hslash }[H,\rho ]+{𝒟}[\rho ],$

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

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. 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.

For spin-$\frac{1}{2}$ nuclei, the population imbalance is governed by the Boltzmann distribution,

$\frac{N_{+}}{N_{-}}=e^{-\mathrm{\Delta }E/(k_{B}T)},$

where $\mathrm{\Delta }E$ is the Zeeman energy splitting. 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.

Longitudinal and transverse relaxation

The decay of RF-induced spin polarization is described by the two characteristic times $T_1$ and $T_2$.

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

$M_z(t)=M_{z,\mathrm{e}\mathrm{q}}-[M_{z,\mathrm{e}\mathrm{q}}-M_z(0)]e^{-t/T_1}.$

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

$M_z(t)=M_{z,\mathrm{e}\mathrm{q}}(1-e^{-t/T_1}).$

In inversion recovery, with $M_z(0)=-M_{z,\mathrm{e}\mathrm{q}}$, one obtains

$M_z(t)=M_{z,\mathrm{e}\mathrm{q}}(1-2e^{-t/T_1}).$

The transverse or spin-spin relaxation time $T_2$ describes the decay of the coherent magnetization perpendicular to the field:

$M_{xy}(t)=M_{xy}(0)e^{-t/T_2}.$

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

T₂^* and field inhomogeneity

In real magnetic resonance experiments, additional dephasing is caused by magnetic field inhomogeneity. This produces an apparent decay time $T_2^{*}$, usually shorter than $T_2$, according to

$\frac{1}{T_2^{*}}=\frac{1}{T_2}+\frac{1}{T_{\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{o}\mathrm{m}}}=\frac{1}{T_2}+\gamma \mathrm{\Delta }{B}_0.$

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.

Bloch equations

A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by Felix Bloch. For the magnetization vector $\mathbf{𝐌}=(M_x,M_y,M_z)$ in a magnetic field $\mathbf{𝐁}(t)$ they are

$\frac{\partial M_x}{\partial t}=\gamma (\mathbf{𝐌}\times \mathbf{𝐁}{)}_x-\frac{M_x}{T_2},$

$\frac{\partial M_y}{\partial t}=\gamma (\mathbf{𝐌}\times \mathbf{𝐁}{)}_y-\frac{M_y}{T_2},$

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

These equations combine coherent Larmor precession with phenomenological longitudinal and transverse relaxation.

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 $I\ge 1$. Molecular reorientation modulates these couplings in time, and the resulting fluctuations drive transitions between nuclear spin states.

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

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. If the correlation function is proportional to $e^{-t/{\tau }_c}$, then for dipolar relaxation one finds

$\frac{1}{T_1}=K[\frac{{\tau }_c}{1+{\omega }_0^2{\tau }_c^2}+\frac{4{\tau }_c}{1+4{\omega }_0^2{\tau }_c^2}],$

$\frac{1}{T_2}=\frac{K}{2}[3{\tau }_c+\frac{5{\tau }_c}{1+{\omega }_0^2{\tau }_c^2}+\frac{2{\tau }_c}{1+4{\omega }_0^2{\tau }_c^2}].$

Here ${\tau }_c$ is the rotational correlation time and ${\omega }_0$ is the Larmor angular frequency. BPP theory works well for simple liquids and gives a microscopic picture of how molecular motion controls relaxation rates.

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. 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. Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization. Other unusual nonthermal behavior appears in systems with dynamical constraints, quantum scars, or special symmetry structures.

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

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

Table of content (86 articles)

Core pathway

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum mechanics measurements
4. Physics:Quantum Interpretations of quantum mechanics
5. Physics:Quantum Mathematical Foundations of Quantum Theory
6. Physics:Quantum Atomic structure and spectroscopy
7. Physics:Quantum Density matrix
8. Physics:Quantum Open systems
9. Physics:Quantum Statistical mechanics
10. Physics:Quantum Kinetic theory
11. Physics:Plasma physics (fusion context)
12. Physics:Tokamak physics
13. Physics:Tokamak edge physics and recycling asymmetries

Full contents

1. Foundations (4)

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum mechanics measurements
4. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (4)

5. Physics:Quantum Interpretations of quantum mechanics
6. Physics:Quantum A Spooky Action at a Distance
7. Physics:Quantum A Walk Through the Universe
8. Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together

3. Mathematical structure and systems (9)

9. Physics:Quantum Density matrix
10. Physics:Quantum Exactly solvable quantum systems
11. Physics:Quantum Formulas Collection
12. Physics:Quantum A Matter Of Size
13. Physics:Quantum Symmetry in quantum mechanics
14. Physics:Quantum Angular momentum operator
15. Physics:Runge–Lenz vector
16. Physics:Quantum Approximation Methods
17. Physics:Quantum Matter Elements and Particles

4. Atomic and spectroscopy (5)

18. Physics:Quantum Atomic structure and spectroscopy
19. Physics:Quantum Hydrogen atom
20. Physics:Quantum Selection rules
21. Physics:Quantum Fermi's golden rule
22. Physics:Quantum Spectral lines and series

5. Wavefunctions and modes (8)

23. Physics:Quantum Wavefunction
24. Physics:Quantum Superposition principle
25. Physics:Quantum Eigenstates and eigenvalues
26. Physics:Quantum Boundary conditions and quantization
27. Physics:Quantum Standing waves and modes
28. Physics:Quantum Normal modes and field quantization
29. Physics:Number of independent spatial modes in a spherical volume
30. Physics:Quantum Density of states

6. Quantum information and computing (6)

31. Physics:Quantum information theory
32. Physics:Quantum Qubit
33. Physics:Quantum Entanglement
34. Physics:Quantum Gates and circuits
35. Physics:Quantum Computing Algorithms in the NISQ Era
36. Physics:Quantum Noisy Qubits

7. Quantum optics and experiments (4)

37. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
38. Physics:Quantum optics beam splitter experiments
39. Physics:Quantum Ultra fast lasers
40. Physics:Quantum Experimental quantum physics
Template:Quantum optics operators

8. Open quantum systems (7)

41. Physics:Quantum Open systems
42. Physics:Quantum Master equation
43. Physics:Quantum Lindblad equation
44. Physics:Quantum Decoherence
45. Physics:Quantum Markovian dynamics
46. Physics:Quantum Non-Markovian dynamics
47. Physics:Quantum Trajectories

9. Quantum field theory (18)

Structural dependency map of quantum field theory.

48. Physics:Quantum field theory (QFT) basics
49. Physics:Quantum field theory (QFT) core
50. Physics:Quantum Fields and Particles
51. Physics:Quantum Second quantization
52. Physics:Quantum Harmonic Oscillator field modes
53. Physics:Quantum Creation and annihilation operators
54. Physics:Quantum vacuum fluctuations
55. Physics:Quantum Propagators in quantum field theory
56. Physics:Quantum Feynman diagrams
57. Physics:Quantum Path integral formulation
58. Physics:Quantum Renormalization in field theory
59. Physics:Quantum Renormalization group
60. Physics:Quantum Field Theory Gauge symmetry
61. Physics:Quantum Non-Abelian gauge theory
62. Physics:Quantum Electrodynamics (QED)
63. Physics:Quantum chromodynamics (QCD)
64. Physics:Quantum Electroweak theory
65. Physics:Quantum Standard Model

10. Statistical mechanics and kinetic theory (9)

66. Physics:Quantum Statistical mechanics
67. Physics:Quantum Partition function
68. Physics:Quantum Distribution functions
69. Physics:Quantum Liouville equation
70. Physics:Quantum Kinetic theory
71. Physics:Quantum Boltzmann equation
72. Physics:Quantum BBGKY hierarchy
73. Physics:Quantum Transport theory
74. Physics:Quantum Relaxation and thermalization

11. Plasma and fusion physics (3)

75. Physics:Plasma physics (fusion context)
76. Physics:Tokamak physics
77. Physics:Tokamak edge physics and recycling asymmetries
• Hierarchy of modern physics models showing the progression from quantum statistical mechanics to kinetic theory and plasma physics, culminating in tokamak edge transport and recycling asymmetries.

12. Timeline (2)

78. Physics:Quantum mechanics/Timeline
79. Physics:Quantum_mechanics/Timeline/Quiz/

13. Advanced and frontier topics (6)

80. Physics:Quantum Supersymmetry
81. Physics:Quantum Black hole thermodynamics
82. Physics:Quantum Holographic principle
83. Physics:Quantum gravity
84. Physics:Quantum De Sitter invariant special relativity
85. Physics:Quantum Doubly special relativity

References

Author: Harold Foppele