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← Part VI/Adiabatic Approximation

7. Adiabatic Approximation

Reading time: ~28 minutes | Pages: 7

Slowly varying Hamiltonians: system tracks instantaneous eigenstates.

The Adiabatic Theorem

If Hamiltonian changes slowly enough:

$$\hat{H}(t)|\psi_n(t)\rangle = E_n(t)|\psi_n(t)\rangle$$

Then system initially in $|\psi_n(0)\rangle$ remains in $|\psi_n(t)\rangle$:

$$|\Psi(t)\rangle = e^{i\theta_n(t)}|\psi_n(t)\rangle$$

Instantaneous eigenstate with phase factor

Dynamic Phase

Accumulated phase from energy evolution:

$$\theta_n(t) = -\frac{1}{\hbar}\int_0^t E_n(t')dt'$$

Standard Schrödinger phase - same as for time-independent case

Geometric (Berry) Phase

Additional phase for cyclic evolution $\hat{H}(T) = \hat{H}(0)$:

$$\gamma_n = i\oint \langle\psi_n(t)|\frac{\partial}{\partial t}|\psi_n(t)\rangle dt$$

Geometric phase - depends only on path in parameter space, not on rate

Total phase after cycle:

$$\phi_{total} = \theta_n(T) + \gamma_n$$

Adiabaticity Condition

Change must be slow compared to level spacing:

$$\left|\langle\psi_m|\frac{\partial\hat{H}}{\partial t}|\psi_n\rangle\right| \ll |E_n - E_m|^2/\hbar$$

Transition amplitudes to other states remain negligible

Alternatively, characteristic time $\tau$ of change:

$$\tau \gg \frac{\hbar}{|E_n - E_m|}$$

Example: Particle in Expanding Box

Box length increases slowly: $L(t)$

Instantaneous energy:

$$E_n(t) = \frac{n^2\pi^2\hbar^2}{2mL(t)^2}$$

If expansion is adiabatic:

  • Quantum number $n$ remains constant
  • Energy decreases as $L^{-2}$
  • Wave function adjusts: $\psi_n(x,t) = \sqrt{2/L(t)}\sin(n\pi x/L(t))$

Example: Spin in Rotating Field

Magnetic field $\vec{B}(t)$ rotates slowly

Hamiltonian:

$$\hat{H}(t) = -\gamma\vec{B}(t)\cdot\vec{S}$$

If rotation is adiabatic:

  • Spin remains aligned (or anti-aligned) with field
  • Picks up geometric Berry phase
  • For spin-1/2: $\gamma = \Omega(1 - \cos\theta)/2$ where $\Omega$ is solid angle traced

Landau-Zener Transitions

When levels approach crossing:

$$E_1(t) = \alpha t, \quad E_2(t) = -\alpha t + \Delta$$

Transition probability (Landau-Zener formula):

$$P_{1\to 2} = e^{-2\pi\Delta^2/(\hbar\alpha)}$$

Slow passage ($\alpha \to 0$): adiabatic, $P \to 0$

Fast passage ($\alpha \to \infty$): diabatic, $P \to 1$

Born-Oppenheimer Approximation

Molecular physics application:

  • Nuclei move slowly ($m_n \gg m_e$)
  • Electrons adjust instantaneously to nuclear positions
  • Separate electronic and nuclear motion

Electronic Hamiltonian at fixed nuclear positions $\vec{R}$:

$$\hat{H}_{el}(\vec{R})|\psi_{el}(\vec{r};\vec{R})\rangle = E_{el}(\vec{R})|\psi_{el}(\vec{r};\vec{R})\rangle$$

Nuclear motion on potential energy surface $E_{el}(\vec{R})$

Quantum Annealing

Computational method based on adiabatic evolution:

Step 1: Start with simple Hamiltonian $\hat{H}_0$ (easy ground state)

Step 2: Evolve adiabatically to problem Hamiltonian $\hat{H}_p$:

$$\hat{H}(s) = (1-s)\hat{H}_0 + s\hat{H}_p, \quad s: 0 \to 1$$

Step 3: System ends in ground state of $\hat{H}_p$ (solution)

Used in D-Wave quantum computers for optimization problems

Adiabatic Quantum Computation

Alternative to gate-based quantum computing:

  • Encode problem in Hamiltonian
  • Solution = ground state
  • Adiabatic evolution finds solution
  • Provably equivalent to gate model (with polynomial overhead)

Key challenge: maintaining adiabaticity while avoiding exponentially small gaps

Shortcuts to Adiabaticity

Modern techniques to speed up adiabatic processes:

  • Counterdiabatic driving: Add compensating terms to Hamiltonian
  • Optimal control: Design time-dependent parameters for fast transfer
  • Transitionless driving: Engineer Hamiltonian to suppress transitions

Goal: achieve adiabatic-like results in shorter time

Applications

  • Molecular dynamics: Born-Oppenheimer separation
  • Cold atoms: State preparation in optical lattices
  • Quantum computing: Adiabatic algorithms, quantum annealing
  • Topological phases: Berry phase in band structure
  • Quantum control: STIRAP (stimulated Raman adiabatic passage)
  • Cosmology: Particle production in expanding universe
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