Squeezed States and Photon States in Quantum Optics: Coherence, Displacement, and Bogoliubov Transformations

MIT OpenCourseWare presents a lecture on squeezed states and photon states, building on coherent states. The talk reviews how a coherent state is generated by displacing the harmonic oscillator ground state with a unitary operator, and how the eigenvalue alpha of the annihilation operator encodes the state's average position and momentum. It then extends to squeezed states, showing how a sudden change in the oscillator parameters leads to anisotropic uncertainties in X and P and introduces Bogoliubov transformations that mix annihilation and creation operators. The squeezing operator S(gamma) produces a squeezed vacuum with reduced position uncertainty and increased momentum uncertainty, possible to combine with a coherent displacement to yield general squeezed-coherent states. The talk highlights practical applications such as precision measurements and gravitational-wave detectors, and ends with the delta-function limit and the emergence of position and momentum eigenstates.

Coherent States Revisited

The lecture begins with a recap of coherent states, formed by displacing the ground state of the harmonic oscillator using a unitary displacement operator. In this framework, alpha, a complex number, is the eigenvalue of the annihilation operator A acting on the coherent state, linking to the expectation values of position and momentum. The visualization in the X-P plane shows a Gaussian blob centered at (⟨X⟩, ⟨P⟩) with fixed uncertainties inherited from the ground state, illustrating the classical-like evolution of the state.

Time Evolution and Uncertainty

The time evolution preserves the coherent-state form, rotating alpha in the complex plane with angular velocity omega. The position and momentum uncertainties (ΔX, ΔP) remain at the ground-state values, saturating the Heisenberg bound with ΔX ΔP = ħ/2. This section connects the coherent-state picture to practical intuition about phase, amplitude, and measurement statistics.

From Coherence to Squeezing

The lecturer introduces squeezing by considering an abrupt change in the oscillator’s parameters at time t=0. The new Hamiltonian does not have the same ground state as before, so the state appears squeezed in the new basis. The uncertainties transform as ΔX → e^{-γ} ΔX0 and ΔP → e^{+γ} ΔP0, where γ is a squeezing parameter. This reveals a state with reduced position uncertainty and stretched momentum uncertainty from the viewpoint of the second Hamiltonian, a hallmark of squeezed states.

Bogoliubov Transformations

A central tool in the analysis is the Bogoliubov transformation, which mixes annihilation and creation operators between the two Hamiltonians. The old and new annihilation operators relate as A1 = cosh γ A2 + sinh γ A2† and A1† = cosh γ A2† + sinh γ A2. This mixing underpins how the original ground state decomposes into a superposition of even-number states of the second oscillator and motivates the construction of squeezed states within a single-oscillator formalism.

Squeezed Vacuum and the Squeezing Operator

The squeezed vacuum is constructed as S(γ)|0⟩ with S(γ) = exp[-(γ/2)(a†a† − aa)]. The state is normalized using overlaps with the ground state of the original oscillator, leading to a compact expression for the squeezed vacuum involving a tanh(γ) and a†a†. The squeezing operator therefore provides a unitary map that produces a Gaussian state with controllable ΔX and ΔP, suitable for reducing noise in measurements beyond what a coherent state can provide.

General Squeezed-Coherent States and Applications

By combining squeezing with a coherent displacement, one obtains general squeezed-coherent states D(α) S(γ)|0⟩, which in phase space correspond to translated and elongated Gaussians. The talk notes practical uses, including injection of squeezed vacuum in gravitational-wave detectors to stabilize mirror motion and improve measurement precision. It concludes with a discussion of extreme squeezing limits that approach position or momentum eigenstates and the conceptual link to eigenstates of X and P, illustrating the completeness and versatility of the oscillator-based formalism in quantum optics.