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Macroscopic quantum phenomena

Macroscopic quantum phenomena refer to quantum mechanical effects, such as coherence, superposition, tunneling, and entanglement, that manifest in systems composed of a vast number of particles—often on scales visible to the naked eye or macroscopic instruments—rather than being confined to atomic or subatomic levels. These phenomena occur when collective degrees of freedom in the system behave as a single quantum entity, maintaining coherence against environmental decoherence, as exemplified by superfluidity in liquid helium and superconductivity in materials like niobium.^[1]^[2] Prominent examples include superfluidity, where helium-4 below 2.17 K flows without viscosity, exhibiting macroscopic quantum wave interference and quantized vortices, a direct consequence of Bose-Einstein condensation in which billions of atoms occupy the same quantum state.^[1] Superconductivity, observed in materials cooled below a critical temperature (e.g., 9.2 K for niobium), allows persistent electric currents without resistance due to paired electrons (Cooper pairs) forming a macroscopic quantum state, leading to perfect diamagnetism via the Meissner effect.^[3] The Josephson effect in superconducting junctions further demonstrates macroscopic quantum tunneling, where a supercurrent flows across a thin insulator without voltage, enabling sensitive devices like SQUIDs for magnetic field detection.^[3] A landmark advancement came in the 1980s with experiments on superconducting circuits containing Josephson junctions, revealing macroscopic quantum tunneling and energy quantization in an entire electrical circuit treated as a single quantum particle.^[4] This discovery, recognized by the 2025 Nobel Prize in Physics awarded to John Clarke, Michel H. Devoret, and John M. Martinis, showed that under low temperatures and controlled conditions, macroscopic systems can escape classical potential wells via quantum tunneling, with escape rates matching theoretical predictions from quantum mechanics.^[4]^[5] These findings extended quantum behavior to engineered macroscopic objects, such as charge and phase qubits in superconducting circuits, where coherence times have improved to milliseconds, far exceeding initial expectations.^[5]^[3] Such phenomena bridge the quantum and classical realms, providing insights into decoherence mechanisms—where environmental interactions suppress quantum superpositions—and the limits of quantum coherence in large systems.^[2] They underpin modern quantum technologies, including ultrasensitive sensors, quantum simulators, and scalable quantum computers using over 50 superconducting qubits to achieve quantum advantage.^[5] Ongoing research explores fragility measures for macroscopic quantum states and their robustness against thermal noise, essential for fault-tolerant quantum information processing.^[2]

Introduction

Definition and Scope

Macroscopic quantum phenomena refer to quantum mechanical effects that manifest in systems comprising a vast number of particles, typically on the order of Avogadro's number (approximately 10^{23}), where collective behavior emerges as these particles coherently occupy a single quantum state. This coherence leads to observable properties such as zero viscosity in superfluids or perfect diamagnetism in superconductors, distinguishing these effects from the random thermal motions typical of classical systems.^[3]^[6] These phenomena arise when quantum effects dominate over classical ones, specifically when the thermal de Broglie wavelength of the particles becomes comparable to or exceeds the average interparticle spacing, allowing wave-like interference and phase coherence to extend across macroscopic scales. In such degenerate conditions, the system's wavefunction describes the collective motion rather than individual particles, enabling quantum superposition of macroscopically distinct states.^[1] The scope encompasses processes like superfluidity, superconductivity, and Bose-Einstein condensation (BEC), where a macroscopic fraction of particles condenses into the lowest quantum state, but excludes purely microscopic quantum behaviors such as atomic spectral lines or single-particle tunneling. Fundamental prerequisites include quantum superposition, which permits the system to exist in multiple states simultaneously, and entanglement, which correlates the particles' behaviors without classical analogs.^[7]^[8]

Historical Development

The discovery of macroscopic quantum phenomena began with the liquefaction of helium by Heike Kamerlingh Onnes in 1908, which enabled experiments at temperatures approaching absolute zero. Three years later, in 1911, Onnes observed superconductivity in mercury, where electrical resistance abruptly vanished below 4.2 K, marking the first evidence of quantum effects manifesting on a macroscopic scale in solids. This breakthrough, initially puzzling and unexplained, laid the groundwork for exploring similar behaviors in other materials.^[9] In the 1930s, phenomenological descriptions emerged to interpret these observations, notably through the work of brothers Fritz and Heinz London, who in 1935 proposed equations capturing the electromagnetic response of superconductors, including the expulsion of magnetic fields known as the Meissner effect.^[10] Paralleling these advances, superfluidity in liquid helium was independently discovered in 1938 by Pyotr Kapitza in Moscow and by John F. Allen and Donald Misener in Cambridge; they reported a phase transition at the lambda point of 2.17 K, below which helium-4 exhibited zero viscosity and frictionless flow, another hallmark of macroscopic quantum coherence.^[11] These findings shifted focus from isolated anomalies to a broader class of quantum behaviors at low temperatures. The mid-20th century brought microscopic theories, culminating in 1957 with the Bardeen-Cooper-Schrieffer (BCS) theory, which explained superconductivity as arising from electron pairing mediated by lattice vibrations (phonons), enabling zero-resistance current flow.^[12] This marked a transition from empirical models of the 1930s–1950s to quantum mechanical explanations in the 1950s–1960s. Recognition followed with the 1972 Nobel Prize in Physics awarded to John Bardeen, Leon Cooper, and John Robert Schrieffer for their theory. Similarly, Kapitza received the 1978 Nobel Prize for his discoveries in low-temperature physics, including superfluid helium. Post-1990s developments extended these phenomena to ultracold atomic gases, with the first experimental realization of Bose-Einstein condensation (BEC) in 1995 by Eric Cornell and Carl Wieman using rubidium atoms, followed closely by Wolfgang Ketterle's work with sodium. In BEC, a macroscopic number of bosons occupy the lowest quantum state, directly demonstrating quantum occupation on a visible scale and earning the 2001 Nobel Prize in Physics for Cornell, Wieman, and Ketterle. This era solidified the understanding of macroscopic quantum effects beyond condensed matter, bridging early cryogenic discoveries with modern quantum technologies. In the 21st century, experiments on superconducting circuits demonstrated macroscopic quantum superposition, tunneling, and coherence in engineered systems, extending quantum behavior to electrical circuits treated as single quantum entities. These advancements were recognized by the 2025 Nobel Prize in Physics, awarded to John Clarke, Michel H. Devoret, and John M. Martinis for their foundational work on macroscopic quantum phenomena in superconducting junctions and circuits.^[4]

Theoretical Foundations

Macroscopic Occupation of Quantum States

Macroscopic occupation of quantum states arises in systems of bosons due to the Bose-Einstein statistics, which permit an unlimited number of identical particles to occupy the same single-particle quantum state. Unlike classical Maxwell-Boltzmann statistics, this quantum distribution function allows for the accumulation of a large number of particles in low-energy states at sufficiently low temperatures. The average occupation number n_k for a quantum state with energy \epsilon_k is given by

n_k = \frac{1}{e^{(\epsilon_k - \mu)/k_B T} - 1},

where \mu is the chemical potential, k_B is Boltzmann's constant, and T is the temperature. This formula, derived from the quantum statistical treatment of indistinguishable bosons, ensures that \mu < \min(\epsilon_k) to avoid negative occupation numbers, with \mu approaching zero from below as the system cools.^[13] In an ideal Bose gas, Bose-Einstein condensation occurs below a critical temperature T_c, where a macroscopic fraction of particles occupies the ground state. The critical temperature is determined by the condition that the maximum number of particles in excited states equals the total particle number, leading to

k_B T_c = \frac{h^2}{2\pi m} \left( \frac{n}{\zeta(3/2)} \right)^{2/3},

with n the particle number density, m the particle mass, h Planck's constant, and \zeta(3/2) \approx 2.612 the value of the Riemann zeta function. Below T_c, the chemical potential saturates at \mu = 0, and the ground-state occupation becomes N_0 / N \approx 1 - (T / T_c)^{3/2}, representing a macroscopic number of particles in a single quantum state. This transition marks the point where quantum statistics dominate over thermal disorder.^[13] The macroscopic occupation suppresses thermal fluctuations in the population of low-lying excited states, as most particles reside in the ground state, reducing the excitation of higher modes and enabling the establishment of long-range order. This order is quantified by off-diagonal long-range order (ODLRO) in the one-particle reduced density matrix, where the expectation value \langle \psi^\dagger(\mathbf{r}) \psi(0) \rangle remains finite as |\mathbf{r}| \to \infty, signifying coherent correlations across the system. In contrast, fermionic systems governed by Fermi-Dirac statistics face the Pauli exclusion principle, which restricts each state to at most one particle per spin state, preventing direct macroscopic occupation and requiring the formation of paired composite bosons—such as Cooper pairs in superconductors—to mimic bosonic behavior and achieve analogous quantum coherence. This statistical mechanism provides the foundational prerequisite for macroscopic quantum phenomena, as the coherent occupation of a single state by a large particle ensemble underlies the emergence of collective quantum behaviors without invoking specific interactions or system details. For instance, the theory has been applied to realize Bose-Einstein condensation in ultracold dilute atomic gases.

Quantum Coherence and Phase Locking

In macroscopic quantum systems, such as superfluids and superconductors, the collective behavior of many particles is described by a macroscopic wavefunction \psi(\mathbf{r}, t) = \sqrt{N} \exp(i \phi), where N represents the large number of particles occupying the lowest quantum state, and \phi is a global phase that varies slowly over macroscopic distances. This wavefunction acts as an order parameter, signaling the spontaneous breaking of U(1) phase symmetry, which distinguishes the quantum coherent phase from the normal state above the critical temperature T_c. Phase locking arises from the synchronization of the phases of individual particle wavefunctions, establishing a rigid, unified phase across the entire system. This coherence suppresses classical dissipative processes, such as viscosity or resistance, by ensuring that perturbations do not cause random phase slips; instead, the system responds with a collective, reversible adjustment of the global phase. The resulting phase rigidity manifests in quantized circulation, where the superfluid velocity \mathbf{v}_s = (\hbar / m) \nabla \phi leads to circulation \oint \mathbf{v}_s \cdot d\mathbf{l} = (2\pi \hbar / m) n around closed paths, with integer n, preventing fractional or dissipative flow. Macroscopic quantum interference effects emerge from this phase coherence, analogous to Aharonov-Bohm phase shifts, where the global phase accumulates shifts due to enclosed flux or vector potentials, even in regions inaccessible to particles.^[14] In these systems, the coherence length \xi, which sets the scale over which the order parameter varies, diverges as \xi \propto (T_c - T)^{-1/2} when approaching T_c from below, allowing interference patterns to extend over increasingly large scales near the transition. The mathematical foundation for these dynamics in weakly interacting systems is provided by the mean-field Gross-Pitaevskii equation, which governs the evolution of the macroscopic wavefunction:

i \hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) + g |\psi|^2 \right] \psi,

where V(\mathbf{r}) is an external potential, m is the particle mass, and g parameterizes the particle interactions. This equation captures the essential balance between kinetic energy, trapping, and nonlinear self-interaction, enabling simulations of coherent flow and phase evolution. This phase coherence underpins manifestations such as quantized vortex circulation in superfluids.

Superfluidity

Liquid Helium Superfluids

Liquid helium exhibits superfluidity as a macroscopic quantum phenomenon, manifesting distinct behaviors in its isotopes due to their quantum statistics. Helium-4 (^4He), a boson with zero spin, undergoes a phase transition to a superfluid state below the lambda point at approximately 2.17 K, marking the boundary between normal helium I and superfluid helium II. In contrast, helium-3 (^3He), a fermion with half-integer spin, remains liquid down to absolute zero under its own vapor pressure but achieves superfluidity at much lower temperatures, around 2.5 mK through Cooper pairing of fermions analogous to superconductivity, discovered in 1972 by Douglas Osheroff, David Lee, and Robert Richardson.^[15] The focus here is on ^4He, where superfluidity was first discovered in 1938 by Pyotr Kapitza, and independently by John F. Allen and Donald Misener, through observations of anomalous flow properties.^[16] The hallmark of superfluid ^4He is its vanishingly small viscosity, enabling frictionless flow through narrow channels. This was demonstrated in Poiseuille flow experiments where helium II flowed through fine capillaries with no measurable pressure drop, indicating a viscosity less than 10^{-7} poise, far below that of normal liquids.^[17] Related thermomechanical effects include the fountain effect, where heating one side of a superfluid-filled capillary causes helium to emerge as a jet from the other side due to a pressure gradient arising from the superfluid's zero entropy. Another striking property is the formation of thin Rollin films, coherent superfluid layers about 100–300 nm thick that creep along surfaces against gravity, allowing helium to flow out of open containers without spilling. Persistent currents, sustained circulatory flows without dissipation, further exemplify this zero-viscosity behavior, persisting for hours in annular channels. Key experiments elucidated these properties and revealed the two-fluid nature of superfluid helium. Pyotr Kapitza observed anomalously high thermal boundary resistance between helium II and solids, known as Kapitza resistance, where heat transfer is limited by phonon mismatch at the interface rather than bulk conductivity. Efraim Andronikashvili's torsion oscillator experiments, using stacked disks immersed in helium, showed a gradual decoupling of the superfluid component below the lambda point: the oscillator's moment of inertia decreased as the superfluid fraction, which does not entrain with the oscillating disks, became dominant, supporting the two-fluid model with a viscous normal fluid and an inviscid superfluid component. In rotating superfluid helium, quantized vortices provide direct evidence of macroscopic quantum coherence. When a bucket of helium II is rotated above a critical angular velocity, the superfluid forms an array of vortex lines, each with circulation quantized as \oint \mathbf{v} \cdot d\mathbf{l} = n \frac{h}{m}, where n is an integer, h is Planck's constant, and m is the mass of a ^4He atom. These vortices were observed through ion trapping and second sound attenuation in rotating containers, confirming the quantum circulation and distinguishing superfluid rotation from classical solid-body motion.

Theoretical Models and Explanations

The two-fluid model, developed by Lev Landau in 1941, provides a phenomenological framework for understanding superfluidity in liquid helium by treating it as a superposition of two interpenetrating fluids: a viscous normal fluid with density \rho_n(T) that carries all the entropy and viscosity, and an inviscid superfluid component with density \rho_s(T) that flows without dissipation.^[18] The total mass density is \rho = \rho_s + \rho_n, with \rho_s = \rho and \rho_n = 0 at absolute zero temperature, and both components vary continuously with temperature up to the lambda transition where \rho_s = 0.^[18] The superfluid velocity \mathbf{v}_s obeys the relation \mathbf{v}_s \cdot \nabla s = 0, where s is the specific entropy, ensuring that the superfluid component transports no entropy and thus experiences no dissipative forces.^[18] A microscopic foundation for superfluidity in bosonic systems like ^4He was established by Nikolai Bogoliubov in 1947 through a canonical transformation that diagonalizes the Hamiltonian of a weakly interacting Bose gas, revealing the spectrum of elementary excitations.^[19] The resulting Bogoliubov dispersion relation is given by

E(k) = \sqrt{\varepsilon_k (\varepsilon_k + 2\mu)},

where \varepsilon_k = \hbar^2 k^2 / 2m is the free-particle energy and \mu is the chemical potential related to the condensate density.^[19] For small momenta k, this spectrum approaches a linear phonon-like form E(k) \approx c k, with sound velocity c = \sqrt{\mu / m}, explaining the gapless excitations necessary for superfluid flow without energy dissipation.^[19] Richard Feynman advanced the theoretical description of liquid helium in 1953 by employing path integral formulations to treat it as a nearly ideal Bose gas with weak interatomic interactions, capturing the role of quantum fluctuations in the ground state and excitations.^[20] This approach highlights how correlations beyond mean-field approximations are essential, particularly for ^3He superfluidity, where fermionic atoms pair in a p-wave state due to attractive interactions in higher angular momentum channels, as proposed by Philip W. Anderson and Pierre Morel in 1961. In ^3He, the p-wave pairing leads to anisotropic superfluid phases with spin-triplet symmetry, contrasting with the s-wave pairing in ^4He. Despite these models, theoretical treatments of superfluid helium face significant challenges due to the strong, short-range interactions among atoms, which preclude exact solutions and necessitate approximations like the Bogoliubov transformation or path integral Monte Carlo methods.^[21] Liquid ^4He, in particular, behaves as a dense quantum fluid where hard-sphere correlations dominate, making it deviate from the dilute Bose gas idealization and requiring advanced numerical simulations to quantify properties like the superfluid density.^[21] For ^3He, the fermionic nature and p-wave pairing further complicate the many-body problem, relying on generalized BCS theory with realistic potentials.

Superconductivity

Basic Phenomenology

Superconductivity manifests as a macroscopic quantum phenomenon in which certain materials exhibit zero direct current (DC) electrical resistance below a critical temperature T_c. This striking property was first observed in 1911 by Heike Kamerlingh Onnes during experiments on mercury cooled to approximately 4.2 K using liquid helium, where the resistance abruptly dropped to undetectable levels.^[22] In superconducting loops, this enables persistent currents to circulate indefinitely without dissipation, a direct consequence of the zero-resistance state, provided the current density remains below a critical value J_c that depends on material properties and temperature.^[22] A defining feature of superconductivity is the Meissner-Ochsenfeld effect, discovered in 1933, wherein a superconductor expels nearly all magnetic flux from its interior upon entering the superconducting state below T_c, resulting in perfect diamagnetism with internal magnetic field B = 0.^[23] This expulsion occurs even for fields applied before cooling, distinguishing superconductivity from simple perfect conductivity and highlighting its quantum nature. The effect is not absolute in all cases but characterizes bulk type-I superconductors, where magnetic fields are completely excluded from the material's core. To describe these observations phenomenologically, Fritz and Heinz London introduced the London equations in 1935, which relate the supercurrent density \mathbf{j} to the magnetic field \mathbf{B}. The second London equation, \nabla \times \mathbf{j} = -\frac{n_s e^2}{m} \mathbf{B}, implies that currents arise to oppose applied fields, leading to the Meissner effect.^[24] Combined with Maxwell's equations, this yields a characteristic penetration depth \lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}, over which magnetic fields decay exponentially inside the superconductor, with n_s denoting the density of superconducting electrons and other symbols having standard meanings.^[24] This length scale, typically on the order of tens to hundreds of nanometers near T_c, quantifies the spatial extent of the Meissner effect. Further evidence for the role of lattice vibrations in superconductivity emerged from the isotope effect, independently observed in 1950 by Reynolds, Serin, Wright, and Nesbitt in mercury isotopes, and by Maxwell.^[25]^[26] Measurements showed that the critical temperature scales as T_c \propto M^{-\alpha} with \alpha \approx 0.5, where M is the ionic mass, indicating that electron-phonon interactions mediate the pairing responsible for the superconducting state.^[25]^[26] This phenomenological insight, analogous to the zero-viscosity flow in neutral superfluids like liquid helium, underscores the charged analog of macroscopic quantum coherence in superconductors.^[25]

Flux Quantization in Superconducting Rings

Flux quantization in superconducting rings arises from the requirement that the superconducting wave function remains single-valued around a closed loop, a direct consequence of the macroscopic quantum nature of superconductivity. This phenomenon was first predicted by Fritz London in his phenomenological theory, where he anticipated that magnetic flux through a multiply connected superconductor would be quantized to avoid discontinuities in the order parameter. Experimental confirmation came in 1961 through measurements on hollow superconducting cylinders, demonstrating that trapped flux occurs in discrete units of the flux quantum \Phi_0 = h / (2e) \approx 2.07 \times 10^{-15} Wb, where h is Planck's constant and e is the elementary charge. The underlying principle is encapsulated in the quantization of the fluxoid, a generalized flux that accounts for both the magnetic flux and the contribution from supercurrents in the superconductor. For a closed contour C encircling the ring, the line integral condition derived from the single-valuedness of the wave function \psi is

\oint_C \left( m \mathbf{v}_s + \frac{q}{c} \mathbf{A} \right) \cdot d\mathbf{l} = n \frac{h}{q},

where m is the Cooper pair mass, \mathbf{v}_s is the superfluid velocity, \mathbf{A} is the vector potential, q = 2e is the charge of a Cooper pair, c is the speed of light (in cgs units), and n is an integer. This leads to the fluxoid \Phi' = \Phi + (m/q) \oint \lambda_L^2 \mathbf{j}_s \cdot d\mathbf{l}, where \Phi = \oint \mathbf{A} \cdot d\mathbf{l} is the magnetic flux, \lambda_L is the London penetration depth, and \mathbf{j}_s is the supercurrent density; the fluxoid is quantized in units of \Phi_0. In the limit of negligible supercurrent (wide or thick rings), the fluxoid reduces to the pure magnetic flux \Phi, which is then trapped in integer multiples of \Phi_0. In thick superconducting rings, where the ring width significantly exceeds \lambda_L, the Meissner effect expels magnetic fields from the bulk during cooling below the critical temperature T_c, but any applied field present during the transition becomes trapped upon entering the superconducting state. This trapped flux adjusts to the nearest multiple of \Phi_0 to satisfy the quantization condition, with minimal perturbation to the interior field if the ring dimensions are much larger than \lambda_L. The process involves discrete flux jumps, as continuous variation would require phase slips in the order parameter \psi, which are energetically forbidden in the absence of dissipation; instead, the system undergoes abrupt transitions to maintain the single-valuedness of \psi, with Cooper pairs (charge q = 2e) ensuring the flux quantum is \Phi_0 = h/(2e) rather than h/e. A key experimental demonstration is the Little-Parks effect, observed in thin-walled superconducting cylinders, where the critical temperature T_c oscillates periodically with applied magnetic flux through the ring, with a period corresponding to \Phi_0. This oscillation arises because the persistent supercurrent required to screen non-integer flux shifts the boundary conditions for the order parameter, effectively modulating the kinetic energy and thus T_c. For flux \Phi = (n + f) \Phi_0 with fractional part f, the shift in T_c is \Delta T_c / T_c \propto (f - 1/2)^2, reaching maxima at integer flux and minima at half-integer, confirming the role of phase coherence around the loop.

Weak Links and Josephson Effects

Weak links in superconductors refer to narrow interruptions or constrictions in otherwise continuous superconducting structures, such as point contacts, thin insulating barriers, or dayem bridges, where the superconducting order parameter exhibits a phase difference φ across the link due to quantum mechanical tunneling of Cooper pairs.^[27] These structures enable macroscopic quantum coherence between the superconducting regions on either side, manifesting as dissipationless current flow without an applied voltage.^[28] The DC Josephson effect describes the supercurrent that flows across such a weak link at zero voltage bias, given by the relation I = I_c \sin \phi, where I_c is the critical current, the maximum supercurrent sustainable before the phase slips and a voltage appears.^[27] This current arises from the coherent tunneling of Cooper pairs, predicted theoretically in 1962 and experimentally verified shortly thereafter.^[27] The critical current I_c for a superconductor-insulator-superconductor (SIS) junction at low temperatures is determined by the Ambegaokar-Baratoff formula, I_c = \frac{\pi \Delta}{2 e R_n}, where \Delta is the superconducting energy gap, e is the electron charge, and R_n is the normal-state resistance of the junction; at finite temperatures, it includes a thermal factor \tanh(\Delta / 2kT).^[28] When a finite voltage V is applied across the junction, the AC Josephson effect occurs, where the phase difference evolves dynamically as V = \frac{\hbar}{2e} \frac{d\phi}{dt}, leading to an alternating supercurrent at frequency f = \frac{2eV}{h}.^[27] Irradiation of the junction with microwaves at frequency f synchronizes this oscillation, producing constant-voltage Shapiro steps in the current-voltage characteristics at voltages V_n = n \frac{h f}{2e} (for integer n), first observed in 1963. These steps demonstrate the quantum nature of the tunneling process and serve as a precise frequency-to-voltage converter. Experimental confirmation of superconducting energy gaps, foundational to understanding weak-link tunneling, came from Giaever's electron tunneling measurements between a normal metal and a superconductor, revealing a gap of approximately 3.5 meV in aluminum at low temperatures.^[29] In fully superconducting junctions, this evolved into observations of the Josephson effects, highlighting the role of phase coherence across barriers. Applications of weak links and Josephson effects include radio-frequency superconducting quantum interference devices (RF SQUIDs), which exploit the phase sensitivity to detect magnetic flux changes as small as $10^{-6} \Phi_0 (where \Phi_0 = h/2e is the flux quantum), enabling ultrasensitive magnetometry in fields like biomagnetism and geophysics. In these devices, the weak link in a superconducting loop maintains phase coherence, allowing interference patterns that amplify flux signals.

Type II Superconductors and Vortices

Type II superconductors are distinguished from type I materials by the Ginzburg-Landau parameter \kappa = \lambda / \xi > 1/\sqrt{2}, where \lambda is the London penetration depth and \xi is the coherence length; this condition enables magnetic flux penetration without the intermediate state observed in type I superconductors, leading to a mixed state where superconductivity coexists with quantized magnetic flux structures.^[30] In this regime, unlike the complete expulsion of magnetic fields in the Meissner state of type I materials, type II superconductors allow partial flux entry above the lower critical field H_{c1}.^[31] The hallmark of the mixed state in type II superconductors is the formation of the Abrikosov vortex lattice, consisting of discrete flux tubes or vortices, each carrying a quantized magnetic flux \Phi_0 = h/(2e) \approx 2.07 \times 10^{-15} Wb, arranged in a triangular lattice to minimize magnetic energy.^[31] Predicted theoretically by Alexei Abrikosov in 1957, these vortices feature a normal-conducting core of radius approximately \xi, where the superconducting order parameter vanishes, surrounded by a region where the magnetic field decays over a distance \lambda. The lattice spacing depends on the applied field, with vortices interacting via repulsive forces that stabilize the ordered structure, observable through techniques like small-angle neutron scattering. The boundaries of the mixed state are defined by two critical fields: the lower critical field H_{c1} \approx (\Phi_0 / 4\pi \lambda^2) \ln(\kappa), below which the superconductor remains fully Meissner-like, and the upper critical field H_{c2} = \Phi_0 / (2\pi \xi^2), above which superconductivity is destroyed as the vortex density becomes too high.^[30] Vortex motion under currents can generate dissipation, but pinning centers—such as impurities or defects—immobilize them, enabling high critical current densities J_c essential for practical devices.^[31] Type II superconductors underpin key applications, notably in generating high magnetic fields for medical imaging; niobium-titanium (NbTi) alloys, with \kappa \approx 100-300, form the basis of superconducting magnets in MRI systems, achieving fields up to 3 T while maintaining zero resistance.^[32] Enhanced pinning in these materials allows sustained operation at currents exceeding 100 A/mm², far surpassing type I limits.^[33] In high-temperature cuprate superconductors, such as YBa₂Cu₃O₇, the d-wave pairing symmetry leads to shorter coherence lengths (\xi \sim 1-2 nm) and more complex vortex dynamics compared to s-wave conventional superconductors, yet the Abrikosov lattice persists, manifesting macroscopic quantum coherence up to critical temperatures around 90 K.^[34] The nodal structure of d-wave order parameters results in extended quasiparticle states around vortex cores, influencing pinning and flux flow, but the quantized flux tubes remain a fundamental quantum feature.^[35]

Ultracold Dilute Gases

Bose-Einstein Condensates

Bose-Einstein condensates (BECs) in trapped ultracold atomic gases represent a dilute-gas realization of macroscopic quantum occupation, where a significant fraction of bosons occupy the system's ground state at temperatures near absolute zero. The achievement of the first such BEC occurred in 1995 using rubidium-87 atoms, cooled via laser cooling followed by evaporative cooling in a magnetic trap to temperatures around 170 nK, with a peak density on the order of 10^{13} cm^{-3} and a critical temperature T_c \approx 170 nK.^[36] This milestone, accomplished by the JILA team led by Eric A. Cornell and Carl E. Wieman (who shared the 2001 Nobel Prize in Physics for this achievement),^[37] involved reducing the temperature of a few thousand atoms to achieve a condensate fraction of up to 40%, marking the first experimental observation of BEC in a weakly interacting dilute gas.^[36] In these experiments, atoms are confined using magnetic traps, which exploit the Zeeman effect to create harmonic potentials for low-field-seeking states, or optical traps formed by laser beams via dipole forces. The equilibrium density profile of the condensate in the Thomas-Fermi regime, valid for large atom numbers where interactions dominate over kinetic energy, is given by

n(\mathbf{r}) = \frac{\mu - V(\mathbf{r})}{g},

where \mu is the chemical potential, V(\mathbf{r}) is the trapping potential (typically harmonic, V(\mathbf{r}) = \frac{1}{2} m (\omega_x^2 x^2 + \omega_y^2 y^2 + \omega_z^2 z^2)), and g = 4\pi \hbar^2 a / m is the interaction parameter with s-wave scattering length a and atomic mass m; the density vanishes outside the Thomas-Fermi radius R_{TF}, defined by \mu = V(R_{TF}). This parabolic profile, with R_{TF} \propto (\mu / m \omega^2)^{1/2} for isotropic traps, has been directly imaged and matches theoretical predictions for interacting BECs. The macroscopic occupation and coherence of BECs are confirmed through time-of-flight expansion, where the trap is turned off and the cloud expands freely for milliseconds, revealing the momentum distribution via absorption imaging. In the initial Rb-87 experiments, this technique showed a narrow, high-amplitude peak at zero velocity, indicating coherent occupation of the ground state, distinct from the broader thermal distribution.^[36] Further evidence of phase coherence comes from matter-wave interferometry, such as splitting a single BEC into two spatially separated components and observing high-contrast interference fringes upon expansion, demonstrating phase locking over distances of several microns.

Superfluidity in Ultracold Gases

Superfluidity in ultracold atomic gases manifests through collective quantum behaviors such as quantized flows and persistent currents, distinct from the denser liquid helium systems due to the dilute nature and tunable interactions of these gases. In Bose-Einstein condensates (BECs) of bosonic atoms, superfluidity arises from macroscopic occupation of the ground state, enabling irrotational flow with quantized circulation. A hallmark is the formation of quantized vortices, where the circulation around a vortex core is \Gamma = \frac{2\pi \hbar}{m}, with m the atomic mass, reflecting the single-valuedness of the order parameter wavefunction. These vortices have been observed in stirred BECs of rubidium atoms, appearing as density depressions in the expanded cloud after time-of-flight imaging. The size of the vortex core is set by the healing length \xi = \frac{\hbar}{\sqrt{2 m \mu}}, where \mu is the chemical potential, derived from the Gross-Pitaevskii equation describing the BEC dynamics. This length scale, typically on the order of 1 \mum in experiments, determines the region over which the superfluid density recovers from zero at the core. Vortices are also visualized through interference imaging, where matter-wave interference between two BECs reveals phase singularities as dislocations in the fringe pattern, confirming the \pi phase winding around each vortex. In rotating traps, multiple vortices arrange into lattices, mimicking Abrikosov lattices in type-II superconductors, with up to hundreds observed under controlled stirring. Fermionic superfluidity emerges in paired ultracold gases, such as ^6Li atoms in two hyperfine states, tuned across the BCS-BEC crossover via magnetic Feshbach resonances that adjust the s-wave scattering length a from negative (weakly attractive, BCS-like) to positive (strongly bound molecules, BEC-like). At unitarity ($1/(k_F a) = 0, where k_F is the Fermi wavevector), interactions reach the strong-coupling limit, yielding universal thermodynamics independent of microscopic details. Superfluidity is evidenced below critical temperatures around 200 nK, with direct observation of the phase transition via time-of-flight expansion showing paired fermions maintaining coherence post-release. Vortex lattices have also been imaged in these fermionic systems, confirming quantized circulation analogous to bosonic cases but with pairing gaps up to \Delta \sim 0.5 E_F. Unlike fixed-interaction superfluid helium, ultracold gases allow precise tuning of interactions through Feshbach resonances, enabling exploration of the BCS-BEC crossover regimes inaccessible in liquid systems. This tunability facilitates studies of pairing mechanisms and critical phenomena. Interference experiments in double-well potentials demonstrate Josephson-like dynamics, where population imbalances between wells oscillate at the plasma frequency \omega_J = \sqrt{8 E_C E_J}/\hbar, with E_C the charging energy and E_J the Josephson coupling energy. These oscillations, observed in sodium BECs, persist for hundreds of cycles, evidencing macroscopic quantum coherence and self-trapping transitions at high imbalances.

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Reviewing macroscopic quantum phenomena and quantum dissipation, from the phenomenology of magnetism and superconductivity to the presentation of ...
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A century of Bose-Einstein condensation | Communications Physics
Jul 1, 2025 · The remarkable features of quantum gases, fluids, and superconductors are related to the coherent macroscopic wavefunction with a unique phase, ...
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This Month in Physics History | American Physical Society
However, the story of superconductivity begins in 1911, when Heike Kamerlingh Onnes first discovered the phenomenon. Kamerlingh Onnes was born on September 21, ...
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LONDON THEORY | SpringerLink
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In December 1937, J. F. Allen and A. D. Misener in Cambridge and simultaneously P. Kapitza in Moscow discovered the superfluidity of liquid helium. In March ...
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Theory of Superconductivity | Phys. Rev.
A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive.
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Apr 12, 2001 · Abstract: The paper addresses the problem of the transverse force (Magnus force) on a vortex in a Galilean invariant quantum Bose liquid.
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This paper presents an attempt of explaining the phenomenon of superfluidity on the basis of the theory of degeneracy of a non-perfect Bose-Einstein gas.
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The properties of liquid helium at very low temperatures (below 0.5°K) are discussed from the atomic point of view.
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Apr 1, 1995 · One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of ...
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[PDF] Onnes 1911 - Physics
128 (June 1911) of the cryostat which, by allowing the contained liquid to be stirred, enabled me to keep resistances at uniform well-defined temperatures; and ...
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[PDF] Meissner and Ochsenfeld revisited - Physics Courses
The paper by Meissner and Ochsenfeld which ap- pears below in translation was first published in Die. Naturwissenschaften in November 1933. The dis- covery ...
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The electromagnetic equations of the supraconductor - Journals
Electric currents are commonly believed to persist in a supra-conductor without being maintained by an electromagnetic field.
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View PDF; Download full issue. Search ScienceDirect. Elsevier · Physics Letters ... Possible new effects in superconductive tunnelling☆. Author links open ...
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Tunneling Between Superconductors | Phys. Rev. Lett.
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If a small potential difference is applied between two metals separated by a thin insulating film, a current will flow due to the quantum mechanical tunnel ...
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Ginzburg-Landau theory of type II superconductors in magnetic field
Jan 26, 2010 · The type II superconductivity refers to materials in which the ratio κ = λ ∕ ξ is larger than κ c = 1 ∕ 2 ( Abrikosov, 1957 ). The vortices ...
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[PDF] Alexei A. Abrikosov - Nobel Lecture
After the work on films I decided to look what are the magnetic properties of bulk type II superconductors. It was definite that the transition to the nor ...
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Superconducting materials: Challenges and opportunities for large ...
Jun 25, 2021 · The important application areas of Nb3Sn superconductors include MRI systems, NMR devices, particle accelerators, tokamak fusion devices, and ...
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Superconducting Magnets - HyperPhysics Concepts
Type II superconductors such as niobium-tin and niobium-titanium are used to make the coil windings for superconducting magnets.
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Phys. Rev. B 97, 174511 (2018) - Pair-density waves, charge ...
May 14, 2018 · We study the charge-density wave structures near the vortex core in these models. We emphasize the importance of the phase winding of the d-wave ...
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Magnetic field–induced pair density wave state in the cuprate vortex ...
High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave (DW) state coexisting with unexplained quantum oscillations.
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Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor
A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled.