Table of Contents
- Introduction
- What Are Coherent States?
- The Harmonic Oscillator Framework
- Definition via Displacement Operator
- Definition as Eigenstates of the Annihilation Operator
- Properties of Coherent States
- Uncertainty Minimization and Gaussian Form
- Phase Space Representation
- Time Evolution of Coherent States
- Overlap and Non-Orthogonality
- Fock Basis Expansion
- Wigner Function and Quasi-Probability Distributions
- Coherent States in Quantum Optics
- Schrödinger’s Cat and Superposition of Coherent States
- Applications in Quantum Technologies
- Conclusion
1. Introduction
Coherent states are special quantum states that most closely resemble classical oscillatory motion. First introduced by Schrödinger and extensively developed in quantum optics, coherent states form the cornerstone of many semiclassical approximations and quantum technologies.
2. What Are Coherent States?
Coherent states are defined as quantum states of the harmonic oscillator that:
- Minimize the Heisenberg uncertainty principle
- Exhibit classical-like sinusoidal motion in expectation values
- Maintain their shape during time evolution
3. The Harmonic Oscillator Framework
In quantum mechanics, the harmonic oscillator uses the ladder operators:
\[
\hat{a} = \frac{1}{\sqrt{2\hbar m\omega}}(m\omega \hat{x} + i\hat{p}), \quad \hat{a}^\dagger = \frac{1}{\sqrt{2\hbar m\omega}}(m\omega \hat{x} – i\hat{p})
\]
These satisfy:
\[
[\hat{a}, \hat{a}^\dagger] = 1
\]
4. Definition via Displacement Operator
The displacement operator is:
\[
\hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger – \alpha^* \hat{a})
\]
A coherent state is then:
\[
|\alpha\rangle = \hat{D}(\alpha)|0\rangle
\]
Where \( |0\rangle \) is the vacuum (ground) state and \( \alpha \in \mathbb{C} \).
5. Definition as Eigenstates of the Annihilation Operator
Alternatively, coherent states satisfy:
\[
\hat{a}|\alpha\rangle = \alpha |\alpha\rangle
\]
This definition highlights their role as eigenstates of a non-Hermitian operator — a rare property in quantum mechanics.
6. Properties of Coherent States
- Not orthogonal: \( \langle \alpha | \beta \rangle \ne 0 \)
- Overcomplete: they form an overcomplete basis in Hilbert space
- Saturate uncertainty:
\[
\Delta x \Delta p = \frac{\hbar}{2}
\]
7. Uncertainty Minimization and Gaussian Form
Position representation of \( |\alpha\rangle \):
\[
\psi_\alpha(x) = \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \exp\left[ -\frac{m\omega}{2\hbar}(x – x_0)^2 + i p_0 x/\hbar \right]
\]
Where \( x_0 \) and \( p_0 \) are determined by \( \alpha \).
These Gaussian wave packets do not spread during evolution, preserving their minimum uncertainty.
8. Phase Space Representation
Each coherent state corresponds to a point in phase space:
\[
\alpha = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega x_0 + ip_0)
\]
Evolution follows a circular trajectory:
\[
\alpha(t) = \alpha(0) e^{-i\omega t}
\]
9. Time Evolution of Coherent States
Coherent states evolve under harmonic oscillator Hamiltonian as:
\[
|\alpha(t)\rangle = e^{-i\omega t/2} |\alpha(0)e^{-i\omega t}\rangle
\]
The state remains coherent, and expectation values trace classical motion.
10. Overlap and Non-Orthogonality
\[
\langle \alpha | \beta \rangle = \exp\left( -\frac{1}{2}|\alpha|^2 – \frac{1}{2}|\beta|^2 + \alpha^* \beta \right)
\]
This non-zero overlap leads to interference and quasi-classical behavior.
11. Fock Basis Expansion
\[
|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle
\]
This shows coherent states as superpositions of number states with Poisson distribution:
\[
P(n) = |\langle n|\alpha\rangle|^2 = \frac{|\alpha|^{2n}}{n!} e^{-|\alpha|^2}
\]
12. Wigner Function and Quasi-Probability Distributions
Coherent states have positive-definite Wigner functions:
\[
W(x, p) = \frac{1}{\pi \hbar} \exp\left( -\frac{(x – x_0)^2}{\sigma_x^2} – \frac{(p – p_0)^2}{\sigma_p^2} \right)
\]
Indicating their quasi-classical nature.
13. Coherent States in Quantum Optics
- Describe laser light
- Basis for quantum states of the electromagnetic field
- Used in optical coherence tomography, quantum metrology, and squeezing
14. Schrödinger’s Cat and Superposition of Coherent States
Superpositions like:
\[
|\psi\rangle = \frac{1}{\sqrt{2}} (|\alpha\rangle + |-\alpha\rangle)
\]
Represent macroscopic quantum superpositions, or “cat states”, with interference in phase space.
15. Applications in Quantum Technologies
- Quantum communication
- Quantum cryptography
- Bosonic quantum error correction
- Continuous-variable quantum computing
Coherent states are essential for continuous-variable encodings and optical implementations.
16. Conclusion
Coherent states elegantly blend quantum and classical behavior. They provide insight into wavepacket dynamics, laser theory, and field quantization, while serving as a key resource in quantum optics and information. Their mathematical richness and physical realism make them indispensable in both theory and application.
