Table of Contents
- Introduction
- Motivation for the Interaction Picture
- Schrödinger vs Heisenberg vs Interaction Picture
- Definition of the Interaction Picture
- Time Evolution in the Interaction Picture
- State Vector Evolution
- Operator Evolution
- The Role of the Interaction Hamiltonian
- Dyson Series and Time-Ordered Exponentials
- Application to Time-Dependent Perturbation Theory
- Example: Two-Level System Driven by a Field
- Relation to the Dirac Picture
- Use in Quantum Field Theory
- Advantages and Limitations
- Interaction Picture in Quantum Optics and Computing
- Conclusion
1. Introduction
The interaction picture, also called the Dirac picture, is an intermediate representation in quantum mechanics. It combines elements of both the Schrödinger and Heisenberg pictures and is particularly useful in time-dependent perturbation theory and quantum field theory.
2. Motivation for the Interaction Picture
When dealing with a time-dependent Hamiltonian that has both solvable and perturbative components, the interaction picture simplifies calculations:
- The free part of the Hamiltonian governs operator evolution
- The interaction part governs state evolution
This separation is ideal for perturbative expansions.
3. Schrödinger vs Heisenberg vs Interaction Picture
| Picture | State Evolution | Operator Evolution |
|---|---|---|
| Schrödinger | \( |\psi(t)\rangle \) evolves | Operators fixed |
| Heisenberg | State fixed | \( \hat{A}(t) \) evolves |
| Interaction | \( |\psi_I(t)\rangle \) evolves | \( \hat{A}_I(t) \) evolves |
The interaction picture allows splitting time evolution across both states and operators.
4. Definition of the Interaction Picture
Let the full Hamiltonian be:
- \( \hat{H}_0 \): free Hamiltonian (solvable)
- \( \hat{H}_{\text{int}}(t) \): interaction Hamiltonian (perturbative)
The state in the interaction picture is:
\[
|\psi_I(t)\rangle = e^{i\hat{H}_0 t/\hbar} |\psi_S(t)\rangle
\]
The operator in the interaction picture is:
5. Time Evolution in the Interaction Picture
Time evolution is governed by:
Where:
This makes the interaction picture ideal for time-dependent perturbation theory.
6. State Vector Evolution
The evolution of the state is driven by the interaction Hamiltonian:
\[
|\psi_I(t)\rangle = \hat{U}_I(t, t_0) |\psi_I(t_0)\rangle
\]
Where \( \hat{U}_I \) satisfies:
\[
i\hbar \frac{d}{dt} \hat{U}_I(t, t_0) = \hat{H}_I(t) \hat{U}_I(t, t_0), \quad \hat{U}_I(t_0, t_0) = \hat{I}
\]
7. Operator Evolution
Operators evolve using only the free Hamiltonian:
\[
\hat{A}_I(t) = e^{i\hat{H}_0 t/\hbar} \hat{A}_S e^{-i\hat{H}_0 t/\hbar}
\]
8. The Role of the Interaction Hamiltonian
The interaction Hamiltonian \( \hat{H}_{\text{int}} \):
- Drives transitions between unperturbed eigenstates
- Encodes external fields, coupling, and perturbations
- Is assumed to be small compared to \( \hat{H}_0 \)
9. Dyson Series and Time-Ordered Exponentials
The solution to the state evolution is given by the Dyson series:
Where \( \mathcal{T} \) is the time-ordering operator, ensuring proper ordering of non-commuting terms.
10. Application to Time-Dependent Perturbation Theory
Used to compute transition probabilities:
\[
P_{i \to f}(t) = |\langle f | \hat{U}_I(t, t_0) | i \rangle|^2
\]
Expanding \( \hat{U}_I \) in a series yields successive orders of perturbation, widely used in quantum optics and spectroscopy.
11. Example: Two-Level System Driven by a Field
Consider:
\[
\hat{H}(t) = \hat{H}_0 + V \cos(\omega t) \hat{\sigma}_x
\]
Interaction picture separates:
- Free evolution from \( \hat{H}_0 \)
- Coupling from \( V \cos(\omega t) \hat{\sigma}_x \)
Used to study Rabi oscillations and quantum gates.
12. Relation to the Dirac Picture
The interaction picture is often referred to as the Dirac picture, especially in field theory. It’s the standard choice in scattering theory, S-matrix formulation, and perturbative QFT.
13. Use in Quantum Field Theory
- Fields evolve like in Heisenberg picture
- States evolve with interaction Hamiltonian
- Wick’s theorem and Feynman diagrams derive from this framework
Essential for computing scattering amplitudes.
14. Advantages and Limitations
Advantages:
- Ideal for perturbation theory
- Clean separation of solvable and interaction parts
- Adaptable to numerical methods
Limitations:
- Requires \( \hat{H}_0 \) to be exactly solvable
- Breaks down for strong interactions
15. Interaction Picture in Quantum Optics and Computing
- Describes laser-atom interactions
- Used in Jaynes–Cummings model
- Basis for interaction-frame Hamiltonians in quantum control
- Underlies pulse shaping in qubit manipulation
16. Conclusion
The interaction picture bridges the dynamics of the Schrödinger and Heisenberg pictures, providing a flexible and powerful framework for handling time-dependent problems in quantum mechanics. It’s essential for perturbative approaches and underpins key calculations in quantum optics, field theory, and quantum information science.
