Physics:Quantum Adiabatic theorem



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This page is about the quantum mechanics concept. For the thermodynamics concept, see Adiabatic process.

Adiabatic theorem the adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. In simpler terms, a quantum-mechanical system subjected to gradually changing external conditions adapts its functional form, whereas under rapidly varying conditions there is insufficient time for the state to adapt, so the spatial probability density remains unchanged. The adiabatic theorem is a concept in quantum mechanics stating that a physical system remains in its instantaneous eigenstate if a given perturbation acts on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. The theorem in its original form was given by Max Born and Vladimir Fock in 1928.

Quantum Adiabatic theorem.

Introduction

The theorem in its original form was given by Max Born and Vladimir Fock in 1928. It is one of the central results of quantum mechanics because it explains how quantum states behave when the Hamiltonian changes slowly in time. If the evolution is sufficiently gradual and the relevant eigenvalue remains separated from the rest of the spectrum, the system stays in the corresponding instantaneous eigenstate, acquiring only phase factors.

The theorem is closely related to the notion of adiabatic invariant in the old quantum theory, but its meaning in quantum mechanics differs from the thermodynamic meaning of the word adiabatic. In quantum mechanics, adiabatic generally means slow variation compared with the system’s internal timescales.

Adiabatic pendulum

At the 1911 Solvay conference, Albert Einstein discussed the quantum hypothesis that for atomic oscillators $E=nh\nu$. After Einstein's lecture, Hendrik Lorentz pointed out that if a classical pendulum is shortened gradually, its energy appears to change smoothly. Einstein replied that although both the energy $E$ and the frequency $\nu$ vary, their ratio $\frac{E}{\nu }$ remains conserved, preserving the quantum hypothesis.

Before the conference Einstein had read a paper by Paul Ehrenfest on the adiabatic hypothesis, and later referred to it in correspondence with Michele Besso.

Diabatic vs. adiabatic processes

Rapidly changing conditions prevent the system from adapting its configuration during the process, so the spatial probability density remains unchanged. The system then generally ends in a linear superposition of eigenstates of the final Hamiltonian. By contrast, gradually changing conditions allow the probability density to change continuously, and if the system begins in an eigenstate of the initial Hamiltonian it ends in the corresponding eigenstate of the final Hamiltonian.

At an initial time $t_0$ a quantum-mechanical system has Hamiltonian $\hat{H}(t_0)$ and is in an eigenstate $\psi (x,t_0)$. A continuous change in conditions modifies the Hamiltonian to $\hat{H}(t_1)$ at time $t_1$. The theorem states that the behavior depends critically on the interval $\tau =t_1-t_0$ over which the change occurs.

For a truly adiabatic process, $\tau \to \mathrm{\infty }$, and the final state becomes an eigenstate of the final Hamiltonian with a modified spatial form: $$|\psi (x,t_1){|}^2\ne |\psi (x,t_0){|}^2.$$

Conversely, in the limit $\tau \to 0$, corresponding to infinitely rapid or diabatic passage, the configuration remains unchanged: $$|\psi (x,t_1){|}^2=|\psi (x,t_0){|}^2.$$

The approximation depends on both the energy gap separating the relevant state from nearby states and the ratio of the process duration $\tau$ to the internal timescale ${\tau }_{\text{int}}=2\pi \hslash /E_0$.

The original theorem assumed a discrete and nondegenerate spectrum, so that one can identify which final eigenstate corresponds to the initial one. Later work generalized the theorem to cases without a strict gap condition.

Comparison with the adiabatic concept in thermodynamics

In thermodynamics, an adiabatic process is one in which there is no exchange of heat between system and environment. In practice, such processes are often relatively fast compared with the timescale of heat transfer.

In classical and quantum mechanics, however, adiabatic means something closer to a quasistatic process: a process slow enough that the system remains near equilibrium or stays in the corresponding instantaneous eigenstate.

Thus, in quantum mechanics the adiabatic theorem states that quantum jumps are suppressed when the Hamiltonian changes sufficiently slowly, and the system tends to preserve its state label and quantum numbers.

Example systems

Simple pendulum

A useful classical analogy is a pendulum whose support is moved slowly. If the change is gradual enough, the mode of oscillation relative to the support remains essentially unchanged, showing how a slow external variation allows the system to adapt continuously.

Quantum harmonic oscillator

A more specifically quantum example is the quantum harmonic oscillator with increasing spring constant $k$. If $k$ increases adiabatically, the system remains in the corresponding instantaneous eigenstate of the current Hamiltonian. For a state initially in the ground state, the wavefunction narrows as the potential steepens, but the quantum number remains unchanged.

If $k$ increases rapidly, the process becomes diabatic. The system then has no time to adapt to the changing potential and the final state becomes a superposition of many eigenstates of the new Hamiltonian, whose combination reproduces the original probability density.

Avoided curve crossing

Related topic: Avoided crossing A particularly important example is a two-level atom in an external magnetic field. The system may be described in terms of diabatic states $|1⟩$ and $|2⟩$ with wavefunction $$|\mathrm{\Psi }⟩=c_1(t)|1⟩+c_2(t)|2⟩.$$

If the magnetic-field dependence is linear, the Hamiltonian matrix may be written $$\mathbf{𝐇}=\left(\begin{array}{cc}\mu B(t)-\hslash {\omega }_0/2& a\\ a^{*}& \hslash {\omega }_0/2-\mu B(t)\end{array}\right)$$

with eigenvalues $$\begin{array}{rl}{\epsilon }_1(t)& =-\frac{1}{2}\sqrt{4a^2+(\hslash {\omega }_0-2\mu B(t){)}^2}\\ {\epsilon }_2(t)& =+\frac{1}{2}\sqrt{4a^2+(\hslash {\omega }_0-2\mu B(t){)}^2}.\end{array}$$

Because of the off-diagonal coupling, the eigenvalues do not cross but exhibit an avoided crossing. Under slow variation of the magnetic field, the system follows an adiabatic eigenstate. Under rapid variation it instead follows a diabatic path, giving a finite probability of transition between the two eigenstates.

Mathematical statement

Under a slowly changing Hamiltonian $H(t)$ with instantaneous eigenstates $|n(t)⟩$ and corresponding energies $E_n(t)$, a quantum system evolves from $$|\psi (0)⟩=\sum _n{c}_n(0)|n(0)⟩$$ to $$|\psi (t)⟩=\sum _n{c}_n(t)|n(t)⟩,$$ where the coefficients change only by phase factors, $$c_n(t)=c_n(0)e^{i{\theta }_n(t)}{e}^{i{\gamma }_n(t)}.$$

The dynamical phase is $${\theta }_m(t)=-\frac{1}{\hslash }\underset{0}{\overset{t}{\int }}{E}_m(t^{\prime })d{t}^{\prime }$$

and the geometric phase is $${\gamma }_m(t)=i\underset{0}{\overset{t}{\int }}⟨m(t^{\prime })|\dot{m}(t^{\prime })⟩d{t}^{\prime }.$$

Hence $|c_n(t){|}^2=|c_n(0){|}^2$. In particular, if the system starts in an eigenstate of $H(0)$, it remains in the corresponding eigenstate of $H(t)$ throughout the evolution, apart from phase changes only.

Proofs

A standard derivation expands the full state in the instantaneous eigenbasis of the time-dependent Hamiltonian and inserts that expansion into the time-dependent Schrödinger equation. This yields an exact set of coupled first-order differential equations for the coefficients. The off-diagonal couplings are proportional to matrix elements of $\dot{H}(t)$ divided by energy differences between eigenstates. If the Hamiltonian changes slowly enough and a finite gap is maintained, these off-diagonal terms can be neglected, leading directly to the adiabatic approximation and to the phase factors above.

A more detailed formulation expresses the slowly varying Hamiltonian as $\hat{H}(t/T)$ and studies the limit $T\to \mathrm{\infty }$. In this limit, oscillatory off-diagonal terms are suppressed and the exact solution approaches the adiabatic one with an error of order $O(T^{-1})$. For cyclic adiabatic evolution, the geometric phase becomes a gauge-invariant physical observable, known as the Berry phase.

A parameter-space formulation writes the Hamiltonian as $H(\overrightarrow{R}(t))$, where $\overrightarrow{R}$ varies slowly in time. In this approach the Berry phase can be written as a line integral along a path in parameter space, $${\gamma }_m(t)=i\underset{C}{\int }⟨{\psi }_m(\overrightarrow{R})|{\partial }_{\overrightarrow{R}}|{\psi }_m(\overrightarrow{R})⟩d\overrightarrow{R},$$ which highlights its geometric character and its relation to topological ideas.

Example applications

The adiabatic theorem underlies many approximation schemes in quantum physics, especially those involving a separation of fast and slow degrees of freedom. A prominent example is the Born–Oppenheimer approximation, where fast electronic motion and slow ionic motion in crystals or molecules are treated separately.

This helps explain a variety of phenomena in:

• thermodynamics: temperature dependence of specific heat, thermal expansion, and melting
• transport phenomena: temperature dependence of electric resistivity in conductors, temperature dependence of electric conductivity in insulators, and properties of low-temperature superconductivity
• optics: infrared absorption in ionic crystals, Brillouin scattering, and Raman scattering

Deriving conditions for diabatic vs adiabatic passage

One way to analyze the validity of the adiabatic approximation is through the time-evolution operator $\hat{U}(t,t_0)$, defined by $$i\hslash \frac{\partial }{\partial t}\hat{U}(t,t_0)=\hat{H}(t)\hat{U}(t,t_0),$$ with $$\hat{U}(t_0,t_0)=1.$$

The system evolves as $$|\psi (t)⟩=\hat{U}(t,t_0)|\psi (t_0)⟩.$$

To test whether a process is sudden or adiabatic, one can define the probability of leaving the initial state. In the perturbative limit this leads to $$\zeta =\frac{{\tau }^2\mathrm{\Delta }{\overline{H}}^2}{{\hslash }^2},$$ where $\overline{H}$ is the Hamiltonian averaged over the interval and $\mathrm{\Delta }\overline{H}$ is its root-mean-square deviation.

The sudden approximation is valid when $$\tau \ll \frac{\hslash }{\mathrm{\Delta }\overline{H}},$$ which is a form of the time-energy uncertainty relation.

Diabatic passage

In the limit $\tau \to 0$, $$\underset{\tau \to 0}{\lim }\hat{U}(t_1,t_0)=1.$$

The functional form of the state remains unchanged: $$|⟨x|\psi (t_1)⟩{|}^2={|⟨x|\psi (t_0)⟩|}^2.$$

This is the sudden approximation, and the probability that the system remains unchanged is $$P_D=1-\zeta .$$

Adiabatic passage

In the limit $\tau \to \mathrm{\infty }$, the system has time to adapt continuously to the changing Hamiltonian: $$|⟨x|\psi (t_1)⟩{|}^2\ne |⟨x|\psi (t_0)⟩{|}^2.$$

If the system begins in an eigenstate of $\hat{H}(t_0)$, it evolves into the corresponding eigenstate of $\hat{H}(t_1)$. The probability associated with adiabatic passage is $$P_A=\zeta .$$

Calculating adiabatic passage probabilities

The Landau–Zener formula

Related topic: Landau–Zener formula For a two-level system with a linearly changing perturbation, the probability of a diabatic transition can be estimated using the Landau–Zener formula. The key quantity is the Landau–Zener velocity $$v_{\text{LZ}}=\frac{\frac{\partial }{\partial t}|E_2-E_1|}{\frac{\partial }{\partial q}|E_2-E_1|}\approx \frac{dq}{dt},$$ where $q$ is the perturbation parameter.

The diabatic transition probability is then $$\begin{array}{rl}{P}_{\mathrm{D}}& =e^{-2\pi \mathrm{\Gamma }}\\ \mathrm{\Gamma }& =\frac{a^2/\hslash }{|\frac{\partial }{\partial t}(E_2-E_1)|}=\frac{a^2/\hslash }{|\frac{dq}{dt}\frac{\partial }{\partial q}(E_2-E_1)|}\\ & =\frac{a^2}{\hslash |\alpha |}.\end{array}$$

A large sweep rate gives a large diabatic transition probability, whereas a slow sweep favors adiabatic following.

The numerical approach

For nonlinear changes in the perturbation variable or time-dependent couplings, the system generally cannot be solved analytically and one must use numerical methods for ordinary differential equations. The amplitudes satisfy $$i\hslash {\dot{\underset{\_}{c}}}^A(t)={\mathbf{𝐇}}_A(t){\underset{\_}{c}}^A(t),$$ where ${\underset{\_}{c}}^A(t)$ is the vector of adiabatic-state amplitudes and ${\mathbf{𝐇}}_A(t)$ is the adiabatic Hamiltonian.

For a two-state system beginning in state 1, the diabatic transition probability can be obtained from $$P_D=|c_2^A(t_1){|}^2.$$

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

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10. Physics:Quantum state

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12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

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24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

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38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (20) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Scattering theory

77. Physics:Quantum Scattering matrix

78. Physics:Quantum S-matrix

79. Physics:Quantum tunnelling

80. Physics:Quantum speed limit

81. Physics:Quantum revival

82. Physics:Quantum reflection

83. Physics:Quantum oscillations

84. Physics:Quantum jump

85. Physics:Quantum boomerang effect

86. Physics:Quantum chaos

7. Measurement and information (9) Back to index

87. Physics:Quantum Measurement theory

88. Physics:Quantum Measurement operators

89. Physics:Quantum Projective measurement

90. Physics:Quantum POVM

91. Physics:Quantum Weak measurement

92. Physics:Quantum Measurement collapse

93. Physics:Quantum entanglement

94. Physics:Quantum Zeno effect

95. Physics:Quantum limit

8. Quantum information and computing (10) Back to index

96. Physics:Quantum information theory

97. Physics:Quantum Qubit

98. Physics:Quantum Entanglement

99. Physics:Quantum Gates and circuits

100. Physics:Quantum Computing Algorithms in the NISQ Era

101. Physics:Quantum Noisy Qubits

102. Physics:Quantum random access code

103. Physics:Quantum pseudo-telepathy

104. Physics:Quantum network

105. Physics:Quantum money

9. Quantum optics and experiments (5) Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

106. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

107. Physics:Quantum optics beam splitter experiments

108. Physics:Quantum Ultra fast lasers

109. Physics:Quantum Experimental quantum physics

110. Physics:Quantum optics

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10. Open quantum systems (9) Back to index

112. Physics:Quantum Open systems

113. Physics:Quantum Master equation

114. Physics:Quantum Lindblad equation

115. Physics:Quantum Decoherence

116. Physics:Quantum dissipation

117. Physics:Quantum Markov semigroup

118. Physics:Quantum Markovian dynamics

119. Physics:Quantum Non-Markovian dynamics

120. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

Structural dependency map of quantum field theory.

121. Physics:Quantum field theory (QFT) basics

122. Physics:Quantum field theory (QFT) core

123. Physics:Quantum Fields and Particles

124. Physics:Quantum Second quantization

125. Physics:Quantum Fock space

126. Physics:Quantum Harmonic Oscillator field modes

127. Physics:Quantum Creation and annihilation operators

128. Physics:Quantum vacuum fluctuations

129. Physics:Quantum Casimir effect

130. Physics:Quantum Propagators in quantum field theory

131. Physics:Quantum Feynman diagrams

132. Physics:Quantum Path integral formulation

133. Physics:Quantum Renormalization in field theory

134. Physics:Quantum Renormalization group

135. Physics:Quantum Field Theory Gauge symmetry

136. Physics:Quantum Spontaneous symmetry breaking

137. Physics:Quantum Non-Abelian gauge theory

138. Physics:Quantum Electrodynamics (QED)

139. Physics:Quantum chromodynamics (QCD)

140. Physics:Quantum Electroweak theory

141. Physics:Quantum Standard Model

142. Physics:Quantum triviality

143. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

144. Physics:Quantum Statistical mechanics

145. Physics:Quantum Partition function

146. Physics:Quantum Distribution functions

147. Physics:Quantum Liouville equation

148. Physics:Quantum Kinetic theory

149. Physics:Quantum Boltzmann equation

150. Physics:Quantum BBGKY hierarchy

151. Physics:Quantum Relaxation and thermalization

152. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (15) Back to index

153. Physics:Quantum Band structure

154. Physics:Quantum Fermi surfaces

155. Physics:Quantum Landau levels

156. Physics:Quantum Semiconductor physics

157. Physics:Quantum Phonons

158. Physics:Quantum Electron-phonon interaction

159. Physics:Quantum Exchange interaction

160. Physics:Quantum Superconductivity

161. Physics:Quantum Topological phases of matter

162. Physics:Quantum well

163. Physics:Quantum spin liquid

164. Physics:Quantum spin Hall effect

165. Physics:Quantum phase transition

166. Physics:Quantum critical point

167. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

168. Physics:Quantum Fusion reactions and Lawson criterion

169. Physics:Quantum Plasma (fusion context)

170. Physics:Quantum Magnetic confinement fusion

171. Physics:Quantum Inertial confinement fusion

172. Physics:Quantum Plasma instabilities and turbulence

173. Physics:Quantum Tokamak core plasma

174. Physics:Quantum Tokamak edge physics and recycling asymmetries

175. Physics:Quantum Stellarator

15. Timeline (8) Back to index

176. Physics:Quantum mechanics/Timeline

177. Physics:Quantum mechanics/Timeline/Pre-quantum era

178. Physics:Quantum mechanics/Timeline/Old quantum theory

179. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

180. Physics:Quantum mechanics/Timeline/Quantum field theory era

181. Physics:Quantum mechanics/Timeline/Quantum information era

182. Physics:Quantum mechanics/Timeline/Quantum technology era

183. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

184. Physics:Quantum topology

185. Physics:Quantum battery

186. Physics:Quantum Supersymmetry

187. Physics:Quantum Black hole thermodynamics

188. Physics:Quantum Holographic principle

189. Physics:Quantum gravity

190. Physics:Quantum De Sitter invariant special relativity

191. Physics:Quantum Doubly special relativity

192. Physics:Quantum arithmetic geometry

193. Physics:Quantum unsolved problems

194. Physics:Quantum Yang-Mills mass gap

195. Physics:Quantum gravity problem

196. Physics:Quantum black hole information paradox

197. Physics:Quantum dark matter problem

198. Physics:Quantum neutrino mass problem

199. Physics:Quantum matter-antimatter asymmetry problem

References

Author: Harold Foppele

Source attribution: Physics:Quantum Adiabatic theorem