The Many Faces of Coherence in Physics (and Beyond)

The term coherence has multiple meanings across physics and philosophy, all centered on an underlying idea of parts sticking together or acting in unison. In everyday language and philosophy, coherence usually means logical consistency and intelligibility - a coherent argument is one whose parts fit together without contradiction. In physics, coherence more specifically describes correlated behavior of waves or quantum states. For example, coherent waves are in phase with each other (maintaining a fixed phase relationship), and quantum coherence refers to the definite phase relationships in a quantum superposition. Despite the varied contexts, these meanings share the notion of a unified, orderly relationship among components (whether phases of waves, quantum amplitudes, or propositions in an argument). In the sections below, I survey the diverse meanings of “coherence”, focusing primarily on physics (classical waves and quantum mechanics) and touching on philosophical usage.

• Historical Evolution of the Concept
• Coherence in Classical Wave Physics
• Quantum Coherence and Superposition
  • Coherent States in Quantum Mechanics
  • Quantum Decoherence
  • Coherent Control and Manipulation in Quantum Systems
• Coherence in Philosophy and Logic
• Conclusion

Historical Evolution of the Concept

The concept of coherence in physics emerged from 19th century studies of wave interference. Thomas Youngs famous double-slit experiment back in 1801 implicitly required coherent light - using a single light source split into two paths - to produce stable interference fringes. Young and other physicists at that time recognized, that two independent light sources - for example the sun and a lamp - generally do not form visible interference because they lack fixed phase relations. In 1819, Fresnel and Arago formulated laws of interference, effectively noting conditions under which light waves cohere (e.g. same frequency and polarization) to produce fringes. By the late 19th century, techniques like Michelsons interferometry further quantified coherence: observers noticed that white light interference disappeared beyond a certain path difference, hinting at a finite coherence length.

In the early 20th century, coherence was formalized in statistical optics. Pioneering work by Fritz Zernike in 1938 introduced the degree of coherence as a quantitative measure by measuring the fringe visibility between two points in a wavefield. Zernikes work and the van Cittert–Zernike theorem showed how a sources size and spectral bandwidth determine partial coherence of light. The invention of the laser in 1960 provided a source of nearly fully coherent, highly monochromatic and phase-stable light, which revolutionized optics and validated these theories. Llaser light can have coherence lengths of kilometers, whereas sunlight’s coherence length is only a few microns.

The quantum era brought new facets to coherence. In 1963, Roy Glauber developed the quantum theory of optical coherence, introducing coherent states of the electromagnetic field and correlation functions to describe photon statistics. Glaubers work - awarded with a Nobel Prize 2005 - established how classical coherence concepts extend to quantum light. Meanwhile, physicists like Erich Joos, Dieter Zeh and Wojciech Zurek in the 1970s - 1990s studied quantum decoherence - how interactions with the environment destroy coherence and make quantum systems appear classical. By the 21st century, quantum coherence became recognized as a resource for technologies like quantum computing, requiring careful preservation and manipulation.

In philosophy, on the other hand, the notion of coherence has an older pedigree in theories of truth and knowledge. Coherence as a criterion of truth was advanced by 19th-20th century idealist philosophers (e.g. Hegel, Bradley) and later formalized as the coherence theory of truth. According to this view, a proposition is true if it coheres (i.e. is consistent or entailed by) a set of other accepted propositions. Early versions simply equated coherence with logical consistency, though more refined versions involve mutual explanatory support. Thus, the idea of “coherence” as internal consistency in a system of ideas has a long history in epistemology alongside its development in physics.

Coherence in Classical Wave Physics

In classical optics and wave physics, coherence describes the ability of waves to exhibit stable interference due to fixed phase relationships. Optical coherence specifically refers to the capacity of a light wave (or two waves) to produce an interference pattern of alternating constructive and destructive fringes. If two light beams show no interference (no stable bright/dark pattern), they are said to be incoherent with each other; if they produce clear, high-contrast fringes (including complete destructive cancellation at some points), they are fully coherent. Intermediate cases (partial fringe visibility) indicate partial coherence.

In Physics one distinguishes two aspects of coherence for waves:

• Temporal Coherence: Correlation of a wave with itself at different times. This relates to the waves monochromaticity (single-frequency purity). A perfectly monochromatic (single-frequency) wave has infinite temporal coherence - its phase is predictable for any time shift, yielding interference even between widely separated time samples. In practice, real sources have finite bandwidth, so the phase drifts over time. The coherence time $\Delta \tau_c$ is defined as the maximum time interval over which the waves phase is predictable or correlated with itself. Equivalently, it is the time over which the field “looks” approximately sinusoidal with a stable phase. Beyond $\tau_c$, phase relationships effectively randomize and interference visibility drops to zero. The coherence length $L_c = c \Delta \tau_c$ is the propagation distance corresponding to the coherence time. For example, a stabilized laser with extremely narrow linewidth might have $\tau_c \sim 10^{-4}$ s and $L_c$ on the order of $30$ km, whereas broadband sunlight has $\tau_c$ on the order of $10^{-12}$ s and $L_c$ of a few micrometers. Temporal coherence is what a Michelson interferometer measures by varying path delay: fringes contrast diminishes as the delay exceeds the source’s coherence time. In summary, temporal coherence quantifies how well a wave maintains a stable phase over time – the narrower the spectrum (smaller bandwidth), the longer the coherence time.
• Spatial Coherence: Correlation of a waves phase at different points in space across an extended wavefront or between separate beams. This is related to the spatial uniformity of phase across the wave. Spatial coherence determines whether two points on a wavefront can form interference fringes when combined. A wave is spatially coherent across a region if any two points within that region emit waves with a fixed phase relation. Youngs double-slit experiment is a test of spatial coherence: if the two pinholes (separated by some distance $d$) are illuminated by a source, clear interference fringes appear on a screen only if $d$ is within the sources transverse coherence length. If the pinholes are too far apart, each sees a different phase of the source at a given time, and the fringe visibility degrades to nothing. The coherence area is defined as the area over which the light field is spatially coherent (e.g. for filtered sunlight it might be ~$4\times10^{-3}$ mm$^2$, meaning pinholes must lie within a ~0.06 mm distance to observe interference). In general, smaller or distant sources (like a distant star approximating a point source) have higher spatial coherence than extended sources (like the sun’s disk). The van Cittert–Zernike theorem provides a quantitative link: the spatial coherence function across a plane is essentially the Fourier transform of the sources intensity distribution - a large angular source yields low spatial coherence, and a point-like source yields high coherence.

In practice most light fields are neither fully coherent nor fully incoherent, but somewhere in between. The degree of coherence (first-order coherence) can be quantified by a complex coherence function or correlation coefficient $\gamma_{12}$ between two points or times. This is essentially the normalized cross-correlation of the wave’s electric field at the two points/times. If $\mid\gamma_{12}\mid = 1$, the fields are perfectly coherent (phase of one completely predicts the phase of the other) and interference contrast is maximal. If $\mid\gamma_{12}\mid = 0$, they are completely uncorrelated (incoherent) and no clear interference appears. Partially coherent light yields an intermediate $0<\mid\gamma\mid<1$, producing fringes of reduced contrast (washed-out interference). In summary, coherence in classical waves refers to the presence of stable correlations (in phase and amplitude) either over time or across space, enabling observable interference effects. Techniques like holography, interferometry, and optical coherence tomography all rely on manipulating and measuring these coherence properties of light.

In other wave phenomena beyond optics, coherence has analogous meanings. For example, in acoustics two sound waves are coherent if they have a constant phase difference; in radio communications, a coherent receiver maintains a reference phase to interfere the incoming signal. Coherence is a unifying wave concept signifying the presence of an underlying order or correlation in the wavefield.

Quantum Coherence and Superposition

In quantum physics, coherence refers to the existence of definite phase relationships between quantum states. A quantum state that is a superposition of two or more basis states is coherent if the relative phases are well-defined and stable, allowing for interference effects at the quantum level. In contrast, if those phase relationships are randomized or unknown (as in a statistical mixture), the superposition is incoherent. Quantum coherence is what differentiates a pure quantum superposition from a mere classical probabilistic mixture.

For example, an electron in a superposition $\frac{1}{\sqrt{2}}(\mid\uparrow> + e^{i\phi}\mid\downarrow>)$ has coherence between the spin-up and spin-down components - the phase $e^{i\phi}$ will lead to interference effects in experiments. If that phase is completely random or the state is an incoherent mixture $\tfrac{1}{2}(\mid\uparrow>\langle\uparrow\mid + \mid\downarrow>\langle\downarrow\mid)$, no single-particle interference can occur between the $\mid\uparrow>$ and $\mid\downarrow>$ outcomes. Thus, quantum coherence is essential for phenomena like single-particle interference (an electron or photon interfering with itself in a double-slit experiment) and is a prerequisite for entangled correlations between particles.

Formally, quantum coherence can be defined in terms of the density matrix of a system. In a chosen reference basis, the off-diagonal elements of the density matrix measure the coherence between the corresponding basis states. An incoherent state is one whose density matrix is diagonal in the reference basis (no superposition terms). Any state with nonzero off-diagonal entries is a coherent superposition in that basis. It possesses quantum coherence as a resource. For example, if we take the computational basis ${\mid 0>,\mid 1>}$ for a qubit, an incoherent state would be of the form $\rho = p\mid 0>\langle 0\mid + (1-p)\mid 1>\langle 1\mid$ (diagonal), whereas a state like $\mid\psi> = \alpha\mid 0> + \beta\mid 1>$ has off-diagonal terms $\alpha\beta^*$ in its density matrix and hence is coherent. The magnitude of those off-diagonals relates to the visibility of interference one could observe between the states $\mid 0>$ and $\mid 1>$.

In modern quantum information science, coherence is treated as a quantifiable resource (much like entanglement). There are measures, coherence monotones, that assign a number to how much coherence a given state has relative to a specified basis. Intuitively, this corresponds to how well the state can produce interference or be used in quantum algorithms. Coherence is “consumed” or degraded by interactions that cause decoherence and it can be partially converted into other quantum resources like entanglement under the right operations. Quantum computing relies on maintaining coherence in qubits throughout computational gate operations. The superposition of $\mid 0>$ and $\mid 1>$ in each qubit (and across multiple qubits) must remain coherent long enough to perform interference-based algorithms. If qubits lose coherence too quickly, quantum computation reverts to classical outcomes.

At a fundamental level, quantum coherence underlies phenomena like quantum interference (e.g. electron diffraction patterns require the electron wavefunction to remain coherent across the paths) and is linked to entanglement. Entanglement can be viewed as a kind of coherence between subsystems: an entangled pair of particles has no local coherence (each reduced state may be mixed) but has joint coherence in the global state. In summary, quantum coherence captures the wavelike aspect of quantum states - the ability of probability amplitudes to superpose and interfere.

Coherent States in Quantum Mechanics

In quantum mechanics, the phrase coherent state has a specific technical meaning beyond just “state with coherence”. Coherent states typically refer to a special set of quantum states of a harmonic oscillator or fields, that most closely resemble classical oscillations. The canonical example is the Glauber–Sudarshan coherent state of the electromagnetic field - the quantum state of light that a stabilized laser outputs.

Mathematically, a coherent state $\mid \alpha>$ (for a harmonic oscillator or single mode of the field) is defined as the eigenstate of the annihilation (lowering) operator $\hat{a}$. These $\mid \alpha>$ states form an overcomplete, non-orthogonal basis of the oscillators Hilbert space. They are often written as displaced vacuum states and have the minimum uncertainty allowed by quantum mechanics (equal uncertainties in $x$ and $p$), which is why they are sometimes called “minimum uncertainty wavepackets”. A coherent state exhibits Poissonian number statistics and, in many respects, behaves like a classical sinusoidal oscillation with amplitude $\mid \alpha\mid$ and phase $\arg(\alpha)$. For instance, the electric field expectation oscillates in time as a classical field would, and the probability of finding $n$ photons in $\mid \alpha>$ is $P(n)=e^{-\mid\alpha\mid^2}\mid\alpha\mid^{2n}/n!$ (a Poisson distribution).

$ \begin{aligned} \hat{a} \mid\alpha> &= \alpha \mid\alpha> \end{aligned} $

Historically, coherent states were studied by Schrödinger (1926) as Gaussian wavepackets that remain localized in a harmonic potential without spreading. In the 1960s, Glauber and Sudarshan formally introduced them in the context of quantum optics to describe the output of a laser and to define what classical light means quantum-mechanically. In optical coherence theory, these coherent states are considered the most classical states of the field - any state that is a statistical mixture of coherent states is regarded as a classical light field, whereas states that cannot be expressed as such a mixture (e.g. squeezed states, Fock states) are nonclassical. In fact, one can define optical nonclassicality as the presence of quantum coherence that cannot be accounted for by a random classical field. Coherent states thus straddle the line between quantum and classical: they are quantum states with maximal coherence (in the sense that they saturate certain coherence measures) but they produce dynamics and statistics reminiscent of classical waves.

Coherent states play a role in coherent state quantization and path integrals - they provide a convenient basis to represent quantum dynamics (especially in quantum optics and many-body theory). In summary, a coherent state in quantum mechanics is a specific type of quantum state (especially of oscillators/fields) characterized by classical-like behavior and defined by eigenstate relations like $\hat{a}\mid\alpha> = \alpha\mid\alpha>$. It should not be confused with the broader notion of “a state having coherence”; rather, it is a term for these minimum-uncertainty wavepackets that remain as coherent as possible.

Quantum Decoherence

Quantum decoherence is the process by which quantum coherence is lost - or apparently lost - due to a systems interaction with its environment. When a quantum system is not perfectly isolated, the phase information that defines its coherent superposition can become entangled with environmental degrees of freedom. From the perspective of the system alone, the coherent superposition then appears to collapse into a mixture, as the relative phases are no longer observable (having “leaked” into the environment). Decoherence is the mechanism that explains how classical behavior emerges from quantum systems: it suppresses interference between a systems quantum states by irreversibly correlating those states with different states of the environment.

“Quantum decoherence is the loss of quantum coherence”, typically through loss of information from the system to the environment. During decoherence, no fundamental wavefunction collapse is assumed to occur; rather, the system-plus-environment evolves unitarily, but the system’s reduced state transitions from pure to mixed. The off-diagonal elements of the system’s density matrix decay towards zero as coherence is delocalized into the environment. A well-known analogy is friction: just as mechanical energy dissipates into environmental heat, quantum phase information dissipates into environmental degrees of freedom. The coherence is not destroyed per se but becomes inaccessible - stored as correlations between the system and environment that are practically irretrievable.

Decoherence theory, developed by Zeh, Zurek and others, provides a resolution to the quantum measurement paradox in that it explains the apparent collapse of the wavefunction. For example, a state describing Schrödingers cat $\frac{1}{\sqrt{2}}(\mid\text{alive}> + \mid\text{dead}>)$ rapidly decoheres due to environmental interactions (air molecules, photons, etc.) that measure the cat’s state. The environment gains information about which branch (alive or dead) occurred, and interference between the branches becomes unobservable - the cat is effectively in a classical probabilistic mixture from any local standpoint. Decoherence does not produce a true wavefunction collapse on its own, but it makes interference between macroscopically distinct states vanish extraordinarily quickly, yielding the appearance of a definite outcome.

Quantitatively, decoherence can be characterized - analogous to coherence time - by a decoherence time over which off-diagonals decay. This can be exceedingly short for macroscopic differences - e.g. a dust particle can decohere in $10^{-20}$ seconds when hit by air molecules. Decoherence ties into the concept of pointer states: the basis in which the system remains robust (diagonal) under environmental interaction. Those pointer states (like alive or dead cat) are the classical-like states that do not themselves get blurred by interference because the environment continually monitors them.

For emerging quantum technologies, decoherence is a critical enemy. Qubits in a quantum computer must be isolated from noise sources because any unmonitored interaction (thermal fluctuations, stray fields, etc.) can entangle with the qubit and collapse superpositions, thereby ruining quantum computations. Quantum error-correcting codes and decoherence-free subspaces are being developed to counteract this.

Decoherence is the process that destroys coherence: a coherent quantum superposition evolves into an incoherent mixture when the system’s phase information leaks into the environment. It bridges quantum and classical physics by explaining why large, open systems don’t usually display quantum interference, and it highlights why maintaining coherence (isolation or error correction) is essential in quantum experimentation and technology.

Coherent Control and Manipulation in Quantum Systems

The term coherent is also used in the context of controlling physical systems, especially in quantum mechanics, to imply control that preserves phase relations. Coherent manipulation (or coherent control) refers to using interactions (like laser pulses, microwave fields, etc.) to steer a quantum systems state in a deterministic, phase-preserving way. The system’s evolution is unitary and maintains quantum coherence throughout the process, as opposed to incoherent processes (like measurements or random thermal kicks) that induce decoherence.

For example, one can apply a sequence of precisely timed magnetic resonance pulses to a spin qubit system to rotate the spin state on the Bloch sphere. If these operations are done faster than the decoherence time and with precise phase reference, the spin is undergoing coherent manipulation. The ability to perform such operations is a prerequisite to any scalable quantum information platform - you must be able to apply coherent control to qubits in order to do quantum gates without losing the information to decoherence. Coherent manipulation typically implies that the system remains in a pure state during the operation.

In a broader sense, coherent control is a subfield of physics and chemistry where interference of quantum amplitudes is used to direct outcomes. For instance, in photochemistry, carefully shaped laser pulses have been used to coherently control reaction pathways by interfering different excitation pathways. The coherence here implies that the driving fields and the systems response maintain a fixed phase relationship, producing constructive or destructive interference in the quantum transition amplitudes to favor a desired product.

A classic example is coherent population trapping or STIRAP (STImulated Raman Adiabatic Passage) in atomic physics, where two coherent lasers create interference that drives an atom from state $\mid A>$ to $\mid B>$ without populating an intermediate state $\mid C>$. The success of this technique relies on maintaining phase coherence between the two laser fields and the atomic polarization. If the lasers were not phase-coherent with each other, the interference would average out and the coherent transfer would fail.

Describing an experimental technique as coherent (coherent manipulation, coherent spectroscopy, etc.) implies that phase coherence is preserved throughout the process. The systems evolution is phase-correlated with the driving fields or between its own states. This is crucial in quantum computing (for executing logic gates on superpositions), in quantum optics (such as creating entangled photons via coherent pump processes), and in quantum sensing (where a phase-coherent superposition interacts with a field and accumulates a measurable phase shift). A large effort in modern physics is to extend the duration of coherent control (increase coherence times) by improving isolation, material purity, and using dynamical decoupling or error correction.

Coherence in Philosophy and Logic

Outside of physics, coherence generally refers to a property of statements, beliefs, or arguments - essentially, a logical and orderly consistency. A coherent statement or theory is one whose parts are logically connected and free of contradictions, so that the whole sticks together conceptually. For example, we might say a scientific theory is coherent if its various hypotheses and observations support each other and form a unified explanation. In everyday terms, someone “speaking coherently” is expressing their thoughts clearly and consistently.

In epistemology and the theory of truth, coherence plays a prominent role through the coherence theory of truth and coherentism. The coherence theory of truth holds that the truth of a proposition consists in its coherence with a specified set of other propositions. In other words, a new claim is true if it fits harmoniously into a larger body of beliefs without contradiction and with mutual support. This is in contrast to the correspondence theory of truth, which says truth consists in correspondence to objective facts. Coherentist philosophers argue that an isolated proposition cannot be judged true or false except by seeing whether it coheres with an entire system of propositions believed to be true. Early versions of coherentism equated coherence with mere logical consistency, but more developed versions require a stronger entailment or explanatory relation among beliefs. For instance, one belief coheres with others not just by avoiding contradiction, but by perhaps being entailed by them or contributing to their overall support.

Coherentism in epistemology is the view that justification of beliefs lies in their coherence with all other beliefs one holds, rather than in some foundational self-evident truths. In this view, our knowledge is like a web where each strand supports the others, and the entire network is coherent if it contains no contradictions and is mutually reinforcing. A perfectly coherent belief system would be one where every belief is consistent with every other and perhaps each is derivable from the whole. While perfect coherence is an ideal, coherence theories allow that truth and justification come in degrees - a belief can be more or less coherent with the rest, and accordingly more or less justified or likely true.

To illustrate, consider a detective assembling a story of what happened at a crime scene. A coherent explanation is one where all pieces of evidence fit into a single consistent timeline and causation, with no inexplicable gaps or contradictions. If a new piece of evidence doesn’t contradict but instead is predicted by or entailed by the theory, it increases the coherence of the theory, and thus increases our confidence in its truth. If the evidence creates a contradiction, the theory becomes incoherent and must be revised or rejected.

In the realm of ideas, coherence means internal consistency and logical connectivity. A coherent argument has premises that support the conclusion in an organized way. A coherent policy plan is one where the measures align with each other and with the overall goals. The common thread is that nothing sticks out as inconsistent or unrelated - all parts contribute to a unified whole. This everyday/philosophical notion of coherence, while conceptually distinct from physical coherence, metaphorically resonates with the physics usage: just as coherent waves march in step to produce a clear signal, coherent thoughts align together to produce clear understanding.

Conclusion

Across physics and philosophy, coherence signifies a kind of unity or correlation that makes the behavior of a system (be it waves, particles, or ideas) orderly and intelligible. In physics, coherence underlies the patterns of interference and the delicate power of quantum superpositions - it marks the difference between random, uncorrelated phenomena and ones that act with a fixed relationship. In quantum systems, maintaining coherence is essential for harnessing quantum effects before decoherence sets in. In philosophy, coherence is the glue of rationality, binding beliefs and assertions into a consistent worldview. The many faces of coherence all reflect the original Latin root cohaerere, “to stick together”.

This article is tagged: Physics, Quantum mechanics, Quantum optics, Electrodynamics, Philosophy, Basics