Coherent States in the Harmonic Oscillator: From Displaced Vacuum to Alpha Coherent States

In this MIT OpenCourseWare lecture, the instructor builds coherent states of the harmonic oscillator by displacing the vacuum with a translation operator. After reviewing Heisenberg operators and time evolution, the talk shows how a displaced ground state retains its Gaussian shape and minimum uncertainty while oscillating like a classical particle. The session then explores the energy content by expanding the coherent state in the energy basis, revealing a Poisson distribution of n with mean related to the displacement. The discussion culminates with the general alpha coherent states, demonstrating that they are eigenstates of the annihilation operator, with real and imaginary parts of alpha corresponding to position and momentum, and shows how time evolution rotates alpha in the complex plane.

Overview and Motivation

The lecture examines coherent states as quantum states that resemble classical motion in a quantum system. Starting from the harmonic oscillator, the presenter recalls the Heisenberg picture and time evolution, and then introduces a unitary displacement operator TX0 which translates the position, effectively moving the vacuum state to a new location in coordinate space. This displaced vacuum is defined as a coherent state, denoted here as the translated ground state, and it serves as a cornerstone for understanding quantum-classical correspondence in simple systems.

Translation Operators and Coherent States

TX0 is unitary and displaces the position operator X by X0. The dagger operation shows TX0† X TX0 = X + X0, and the action on momentum is unchanged. By applying TX0 to the ground state |0>, the state |X0tilde> is obtained, which has a wavefunction translating to X0. This construction demonstrates that coherent states are not position eigenstates but Gaussian wave packets localized around the translated center, preserving unit norm under displacement.

Basic Observables in Coherent States

Calculations of expectation values in the coherent state reveal = X0 and

= 0 at time zero, signaling a state centered at X0 with no net momentum. The Hamiltonian expectation value decomposes into the zero-point energy plus a classical cost associated with the displacement, emphasizing the classical-quantum interplay: a displaced oscillator carries energy proportional to the displacement squared plus the ground-state energy.

Time Evolution and Classical-Like Dynamics

Using the Heisenberg-picture X(t) and P(t), the expectation values evolve as = X0 cos(ωt) and = -Mω X0 sin(ωt). This yields a motion that mirrors a classical oscillator, with the momentum following the position in a way that keeps the shape of the wave packet intact, i.e., the coherent state moves without spreading, an hallmark of coherence.

Uncertainty and Coherence

The uncertainties ΔX and ΔP are computed in the evolving state. Remarkably, ΔX^2 remains constant in time and ΔP^2 remains constant as well, yielding ΔX ΔP = ħ/2 at all times. This demonstrates that the coherent state is a minimum-uncertainty wave packet that preserves its Gaussian form while translating in phase space, hence the term coherence.

Energy Basis Perspective and Poisson Statistics

The coherent state can be expanded in the harmonic-oscillator energy basis, revealing a precise superposition of energy eigenstates with Poisson-distributed occupation numbers. The probability of finding the system in the nth energy eigenstate follows a Poisson law with a mean determined by the displacement, highlighting how a classically prepared state populates many energy levels in a controlled way.

Generalized Coherent States: The Alpha Parameter

To generalize, the displacement operator is extended to the complex parameter alpha, defining the most general coherent state as |α> = D(α)|0> with D(α) = exp(α a† - α* a). Coherent states satisfy a|α> = α|α>, making them eigenstates of the annihilation operator. The real part of α corresponds to and the imaginary part to

, connecting the geometry of the complex plane to the phase-space location of the state. Time evolution multiplies α by e^{-iωt}, effectively rotating the state in the complex plane and tracing a circular trajectory in the α-plane, which mirrors the classical phase-space rotation with frequency ω.

Geometric Picture and Applications

In the α-plane, each coherent state sits at a point whose coordinates encode the classical displacement and momentum. The rotation under time evolution provides a powerful geometric picture: a coherent state is a quantum state that behaves like a classical oscillator, moving in phase space without changing shape, and with minimal quantum uncertainty. This framework underpins the understanding of coherent light in lasers and has broad applications in quantum optics and quantum information science.

Conclusion

The lecture ties together displacement operators, Heisenberg dynamics, and the α-parameter generalization to present a coherent, physically intuitive account of coherent states. By showing how a displaced vacuum evolves, preserves minimal uncertainty, and can be expressed as a superposition of energy eigenstates with a Poisson distribution, the talk illuminates the deep connection between quantum states and classical trajectories in a simple, exactly solvable system.