Shot Noise and Squeezed Light in Quantum Optics: An MIT OpenCourseWare Lecture

In this MIT OpenCourseWare lecture, the instructor builds intuition for shot noise in optical fields starting from coherent states, then examines how beam splitters and detectors reveal noise properties. The talk moves from foundational definitions to the role of vacuum ports, and finally demonstrates how squeezed light and displaced squeezed vacuum can suppress noise, enabling more sensitive measurements in quantum optics. Throughout, the emphasis is on physical insight over formalism, showing how quantum field fluctuations, not just measurement, govern noise phenomena.

Overview of Shot Noise and Coherent States

The lecture begins by posing a straw-poll question about the origin of shot noise when counting photons from a laser beam in coherent states. The instructor clarifies that for a coherent state with amplitude α, the detected photon number fluctuates with variance equal to n, where n ≈ |α|^2, illustrating Ponian statistics. Coherent states are defined as eigenstates of the annihilation operator, and their quadratures satisfy the minimum uncertainty relation ΔQΔP = ħ/2. The discussion covers how the standard deviation scales as the square root of the mean photon number, linking to the intuitive square-root behavior of shot noise. Key definitions are established in units aligned with photon counting, so the photocurrent corresponds to photon number in a given time window, and variance is discussed in terms of photon-number statistics rather than raw current amplitude.

Beam Splitters and Noise Splitting

With a laser beam passed through a 50:50 beam splitter, the coherent state splits into two identical coherent states, each with half the intensity and the same Poissonian noise scaling. The lecture emphasizes that beam splitters distribute both the mean field and the fluctuations, so each output beam carries half the photons and half the shot noise. A subtlety is highlighted: the open port of the beam splitter introduces vacuum fluctuations, which contribute to the total noise in each output when measured individually. Later, the instructor notes that summing the outputs preserves the original noise, while subtracting them allows common-mode noise to cancel and reveals the underlying quantum fluctuations more cleanly.

Vacuum Ports, Interference, and Homodyne Detection

The discussion then explores the impact of the vacuum port on measurements, showing that the beam splitter not only splits the signal but also introduces vacuum noise that is carried through the interferometer. When the two photocurrents are added, the vacuum port contributions cancel out in a balanced detection scheme under certain conventions. When the current difference (I1 − I2) is considered, the shot noise can be suppressed in the presence of strong squeezing, illustrating how correlated noise terms emerge from beam splitter transformations. The instructor stresses the importance of phase conventions and the role of the local oscillator in homodyne detection, where the detector is effectively projecting onto a particular quadrature of the field.

Squeezed Vacuum, Quadrature Noise Reduction, and Local Oscillator Dominance

Introducing squeezed vacuum into the open port of the beam splitter changes the noise landscape. The speaker explains squeezing as reducing the uncertainty in one quadrature while increasing it in the orthogonal quadrature, with the local oscillator selecting the quadrature component that matters for the photocurrent measurement. In the strong local oscillator limit, fluctuations in the cosine quadrature (the measured quadrature) of the squeezed vacuum can be damped significantly, potentially reducing the measured variance below the Poissonian level. The visual intuition is that the noise circle associated with the quantum fluctuations is elongated along one axis and compressed along the other, and the measurement couples primarily to the axis aligned with the local oscillator, where squeezing reduces fluctuations. The mathematical sketch highlights how the transmitted coherent-noise is affected by the beam splitter, while the vacuum fluctuations and squeezing combine to yield a reduced overall variance in the detected intensity.

Displaced Squeezed Vacuum and Practical Applications

The instructor then combines squeezing with a displacement operation to create a displaced squeezed vacuum. This state can be realized by sending a highly transmitting beam into a beam splitter in a regime where the reflected portion remains weak but carries the squeezed fluctuations, while the strong local oscillator carries the mean field. In this setup, the variance of the detected photons can be pushed toward zero in the appropriate quadrature, limited only by the squeezing strength ε. The talk mentions practical applications such as high-sensitivity spectroscopy, for instance in Caesium spectroscopy, where reducing quantum noise enhances measurement precision. The session closes this section with an invitation to calculate noise terms beyond leading order and to connect operator-based calculations with the intuitive beam-splitter picture.

Operator Picture, Intuition, and the Debate on Noise Origin

Toward the end, the instructor emphasizes a field-based view: shot noise originates from the quantum fluctuations of the electromagnetic field itself, which can be split and interfered before any measurement. The measurement process, described via photodetection as essentially E†E, reveals a noise floor set by the field’s intrinsic fluctuations rather than by the measurement step alone. The strong local oscillator assumption is used to simplify the interpretation, focusing on the quadrature that couples to the detected intensity. A final discussion argues that in these experiments the quantum-field origin of shot noise is more compelling than a purely measurement-based explanation, particularly when many photons participate and the local oscillator is dominant.

Takeaways and Outlook

The lecture ends with a broader perspective on how beam splitters and squeezing can be used as conceptual tools to model coherence, noise, and quantum-enhanced measurements. The presenter connects these ideas to quantum information concepts and hints at more advanced topics such as displaced squeezed states and their relevance to precision metrology, spectroscopy, and quantum-optical experiments. The overall message is that understanding noise requires both a quantum-field viewpoint and an appreciation for how optical components transform signals and fluctuations alike.