Understanding Squeezing in Quantum Optics: Quadratures, Homodyne Detection, and Teleportation

In this MIT OpenCourseWare lecture, the focus is on what squeezing means in quantum optics for a single mode of the electromagnetic field. The instructor introduces the two quadrature components, A1 and A2, which encode amplitude and phase information of a harmonic oscillator. Through phase space representations and the Heisenberg uncertainty principle, the talk explains how squeezing reduces the noise in one quadrature at the expense of the other, and why the classical visualization of squeezing in real space can be misleading. The discussion then moves to how balanced homodyne detection uses a beam splitter and a strong local oscillator to measure specific quadratures, the impact of losses on squeezing, and how squeezed light enables quantum teleportation via entangled twin beams and displacement operations at the receiver.

Introduction to Squeezing and the One-Mode Harmonic Oscillator

The lecture begins by situating squeezing within the context of Maxwell equations reduced to a set of harmonic oscillator equations, one per mode. The discussion narrows to a single mode of the electromagnetic field, which behaves as a harmonic oscillator with two canonical variables: momentum and position, here represented as the two quadrature amplitudes of the field. A1 and A2 are the quadrature operators associated with the S Omega T and C Omega T components, and they do not commute, which underpins the quantum uncertainty that makes squeezing meaningful.

Phase Space and Quasi-Probability Distributions

Using Q and Wigner distribution representations, the instructor explains how the quantum state of the single mode is visualized in phase space. The Q distribution corresponds to projections onto the coherent-state basis, while the W function offers a phase-space view that can emphasize the spatial and momentum-like aspects of the oscillator. The discussion clarifies that while these pictures help intuition, the optical field is not literally moving through real space in the way a particle does, and that careful interpretation is required for quadratures.

Classical vs Quantum Squeezing and Pendulum Analogy

The pendulum example illustrates a classical squeezing-like effect: parametric driving can amplify the cosine component while deamplifying the sine component, yielding an elongated motion along one axis and a compressed motion along the orthogonal axis. The lecturer emphasizes that in a purely classical system, there is no genuine quantum squeezing since a definite initial state cannot possess the reduced uncertainty required by quantum mechanics. He also discusses classical squeezing realized in ion traps and notes that quantum squeezing becomes apparent when one reduces noise in one quadrature below the standard quantum limit while accepting increased noise in the conjugate quadrature.

The Two Quadrature Operators and Heisenberg Uncertainty

The two quadratures A1 and A2 are introduced as noncommuting observables obeying an uncertainty relation. Coherent states minimize the uncertainty equally between A1 and A2, while squeezed states redistribute the uncertainty to make one quadrature more precise than the other. The canonical connection to position and momentum is highlighted, but the key is the interpretation in the electromagnetic field context, where the phase-sensitive measurements are implemented through optical devices rather than direct mechanical motion.

Detection: Balanced Homodyne and Phase Sensitive Measurements

The core experimental technique behind squeezing observation is balanced homodyne detection. A 50-50 beam splitter mixes the signal mode with a strong local oscillator. The resulting photocurrent difference isolates a specific quadrature component, determined by the LO phase. The setup allows measurement of either A1 or A2 by adjusting the LO phase, which is crucial for observing noise reduction in one quadrature below shot noise. The balance and the phase relationship implement a projection of the quantum state onto the desired quadrature axis.

Losses, Attenuation, and the Vacuum

The instructor explains why simple attenuation is not a unitary operation in quantum mechanics. A beam splitter modeling losses couples the signal to a vacuum mode, injecting vacuum fluctuations that tend to degrade squeezing. As squeezing becomes more pronounced, the sensitivity to even small losses increases, making it challenging to preserve nonclassical features in realistic optical channels such as fibers. This section emphasizes the practical limitations faced in experiments and the importance of minimizing losses to maintain squeezing advantages for metrology and sensing.

From Squeezed Light to Quantum Teleportation

The talk transitions from measurement to application, introducing quantum teleportation as a scheme that leverages entangled light and measurement-induced state transfer. Squeezed light, produced by parametric down-conversion or an optical parametric oscillator, serves as a form of entanglement resource. Alice performs a joint measurement on the unknown state and her share of an entangled pair using a balanced homodyne apparatus to obtain X and P values. These results are communicated classically to Bob, who applies a corresponding displacement operation to his half of the entangled state, reconstructing a displaced copy of the original quantum state without sending the actual quantum system through a quantum channel.

Two-Mode Squeezed States and Two-Mode Teleportation

Extending squeezing to two modes introduces twin beams, with one beam sent to Alice and the other to Bob. A two-mode OPO can place the two identical photons into separate modes. The measurement outcomes on the two modes, again performed via balanced homodyne detection with a strong local oscillator, enable teleportation of the unknown state using classical information plus shared entanglement. The final state at Bob is a displaced copy of the original state, with the displacement depending on the X and P results reported by Alice.

Unbalanced Beam Splitters and the Displacement Picture

The lecturer discusses unbalanced beam splitters as a way to realize displacement operations and to relate the local oscillator field to the quantum state in a controlled way. In the limit of high transmission, the beam splitter acts as a displacement operator, illustrating how the LO field can imprint the desired shift into the signal mode. This connects the experimental apparatus to the formal description of quantum state engineering and measurement, showing how simple optical components can realize sophisticated quantum operations.

Concluding Remarks: Simplicity and Fragility

The talk closes with a reflection on the elegance of using a few simple optical elements—beam splitters, local oscillators, and detectors—to implement nonclassical state generation, measurement, and state transfer. It also emphasizes the fragility of squeezing under losses and the importance of maintaining high efficiency in all optical components to harness the advantages of nonclassical light for precision measurements and quantum information tasks.