Macroscopic quantum phenomena are processes showing
quantum behavior at the
macroscopic scale
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena a ...
, rather than at the
atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena are
superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
and
superconductivity; other examples include the
quantum Hall effect
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exh ...
and
topological order
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ...
. Since 2000 there has been extensive experimental work on quantum gases, particularly
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.6 ...
s.
Between 1996 and 2016 six
Nobel Prizes were given for work related to macroscopic quantum phenomena. Macroscopic quantum phenomena can be observed in
superfluid helium and in
superconductors
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, but also in dilute quantum gases,
dressed photons such as
polaritons
In physics, polaritons are quasiparticles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. They are an expression of the common quantum phenomenon known as level repulsion, also ...
and in
laser light. Although these media are very different, they are all similar in that they show macroscopic quantum behavior, and in this respect they all can be referred to as
quantum fluid A quantum fluid refers to any system that exhibits quantum mechanical effects at the macroscopic level such as superfluids, superconductors, ultracold atoms, etc. Typically, quantum fluids arise in situations where both quantum mechanical effects ...
s.
Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (of the order of the
Avogadro number
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining co ...
) or the quantum states involved are macroscopic in size (up to kilometer-sized in
superconducting
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
wires).
Consequences of the macroscopic occupation
The concept of macroscopically-occupied quantum states is introduced by
Fritz London
Fritz Wolfgang London (March 7, 1900 – March 30, 1954) was a German physicist and professor at Duke University. His fundamental contributions to the theories of chemical bonding and of intermolecular forces ( London dispersion forces) are today ...
. In this section it will be explained what it means if a single state is occupied by a very large number of particles. We start with the wave function of the state written as
with Ψ
0 the amplitude and
the phase. The wave function is normalized so that
The physical interpretation of the quantity
depends on the number of particles. Fig. 1 represents a container with a certain number of particles with a small control volume Δ''V'' inside. We check from time to time how many particles are in the control box. We distinguish three cases:
# There is only one particle. In this case the control volume is empty most of the time. However, there is a certain chance to find the particle in it given by Eq. (). The probability is proportional to Δ''V''. The factor ΨΨ
∗ is called the chance density.
# If the number of particles is a bit larger there are usually some particles inside the box. We can define an average, but the actual number of particles in the box has relatively large fluctuations around this average.
# In the case of a very large number of particles there will always be a lot of particles in the small box. The number will fluctuate but the fluctuations around the average are relatively small. The average number is proportional to Δ''V'' and ΨΨ
∗ is now interpreted as the particle density.
In quantum mechanics the particle probability flow density ''J''
p (unit: particles per second per m
2), also called
probability current
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is th ...
, can be derived from the
Schrödinger equation to be
with ''q'' the charge of the particle and
the vector potential; cc stands for the complex conjugate of the other term inside the brackets. For neutral particles , for superconductors (with ''e'' the elementary charge) the charge of Cooper pairs. With Eq. ()
If the wave function is macroscopically occupied the particle probability flow density becomes a particle flow density. We introduce the fluid velocity ''v''
s via the mass flow density
The density (mass per volume) is
so Eq. () results in
This important relation connects the velocity, a classical concept, of the condensate with the phase of the wave function, a quantum-mechanical concept.
Superfluidity
At temperatures below the
lambda point
The lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). The lowest pressure at which He-I and He-II can coexist is the vapor−He-I−He-II ...
, helium shows the unique property of
superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
. The fraction of the liquid that forms the superfluid component is a macroscopic
quantum fluid A quantum fluid refers to any system that exhibits quantum mechanical effects at the macroscopic level such as superfluids, superconductors, ultracold atoms, etc. Typically, quantum fluids arise in situations where both quantum mechanical effects ...
. The helium atom is a
neutral particle
In physics, a neutral particle is a particle with no electric charge, such as a neutron.
The term ''neutral particles'' should not be confused with '' truly neutral particles'', the subclass of neutral particles that are also identical to their o ...
, so . Furthermore, when considering
helium-4, the relevant particle mass is , so Eq. () reduces to
For an arbitrary loop in the liquid, this gives
Due to the single-valued nature of the wave function
with integer, we have
The quantity
is the quantum of circulation. For a circular motion with radius ''r''
In case of a single quantum ()
When superfluid helium is put in rotation, Eq. () will not be satisfied for all loops inside the liquid unless the rotation is organized around vortex lines (as depicted in Fig. 2). These lines have a vacuum core with a diameter of about 1 Å (which is smaller than the average particle distance). The superfluid helium rotates around the core with very high speeds. Just outside the core (''r'' = 1 Å), the velocity is as large as 160 m/s. The cores of the vortex lines and the container rotate as a solid body around the rotation axes with the same angular velocity. The number of vortex lines increases with the angular velocity (as shown in the upper half of the figure). Note that the two right figures both contain six vortex lines, but the lines are organized in different stable patterns.
Superconductivity
In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending
on the energy of the interface between the normal and superconducting states. The
Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In
Type I superconductor
The interior of a bulk superconductor cannot be penetrated by a weak magnetic field, a phenomenon known as the Meissner effect. When the applied magnetic field becomes too large, superconductivity breaks down. Superconductors can be divided into ...
s, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value ''H
c''. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a baroque pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In
Type II superconductors, raising the applied field past a critical value ''H''
''c''1 leads to a mixed state (also known as the vortex state) in which an increasing amount of
magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength ''H''
''c''2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called
fluxons because the flux carried by these vortices is
quantized. Most pure
elemental
An elemental is a mythic being that is described in occult and alchemical works from around the time of the European Renaissance, and particularly elaborated in the 16th century works of Paracelsus. According to Paracelsus and his subsequent fo ...
superconductors, except
niobium and
carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.
The most important finding from
Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenol ...
was made by
Alexei Abrikosov in 1957.
He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux
vortices.
Fluxoid quantization
For
superconductors
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
the bosons involved are the so-called
Cooper pairs
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
which are
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
s formed by two electrons. Hence ''m'' = 2''m''
e and ''q'' = −2''e'' where ''m''
e and ''e'' are the mass of an electron and the elementary charge. It follows from Eq. () that
Integrating Eq. () over a closed loop gives
As in the case of helium we define the vortex strength
and use the general relation
where Φ is the magnetic flux enclosed by the loop. The so-called
fluxoid is defined by
In general the values of ''κ'' and Φ depend on the choice of the loop. Due to the single-valued nature of the wave function and Eq. () the fluxoid is quantized
The unit of quantization is called the
flux quantum
The flux quantum plays a very important role in superconductivity. The earth magnetic field is very small (about 50 μT), but it generates one flux quantum in an area of 6 μm by 6 μm. So, the flux quantum is very small. Yet it was measured to an accuracy of 9 digits as shown in Eq. (). Nowadays the value given by Eq. () is exact by definition.
In Fig. 3 two situations are depicted of superconducting rings in an external magnetic field. One case is a thick-walled ring and in the other case the ring is also thick-walled, but is interrupted by a weak link. In the latter case we will meet the famous
Josephson relations. In both cases we consider a loop inside the material. In general a superconducting circulation current will flow in the material. The total magnetic flux in the loop is the sum of the applied flux Φ
a and the self-induced flux Φ
s induced by the circulation current
Thick ring
The first case is a thick ring in an external magnetic field (Fig. 3a). The currents in a superconductor only flow in a thin layer at the surface. The thickness of this layer is determined by the so-called
London penetration depth
In superconductors, the London penetration depth (usually denoted as \lambda or \lambda_L) characterizes the distance to which a magnetic field penetrates into a superconductor and becomes equal to e^ times that of the magnetic field at the surface ...
. It is of μm size or less. We consider a loop far away from the surface so that ''v''
s = 0 everywhere so ''κ'' = 0. In that case the fluxoid is equal to the magnetic flux (Φ
v = Φ). If ''v''
s = 0 Eq. () reduces to
Taking the rotation gives
Using the well-known relations
and
shows that the magnetic field in the bulk of the superconductor is zero as well. So, for thick rings, the total magnetic flux in the loop is quantized according to
Interrupted ring, weak links
Weak links play a very important role in modern superconductivity. In most cases weak links are oxide barriers between two superconducting thin films, but it can also be a crystal boundary (in the case of
high-Tc superconductors). A schematic representation is given in Fig. 4. Now consider the ring which is thick everywhere except for a small section where the ring is closed via a weak link (Fig. 3b). The velocity is zero except near the weak link. In these regions the velocity contribution to the total phase change in the loop is given by (with Eq. ())
The line integral is over the contact from one side to the other in such a way that the end points of the line are well inside the bulk of the superconductor where . So the value of the line integral is well-defined (e.g. independent of the choice of the end points). With Eqs. (), (), and ()
Without proof we state that the supercurrent through the weak link is given by the so-called DC
Josephson relation
In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mec ...
The voltage over the contact is given by the AC Josephson relation
The names of these relations (DC and AC relations) are misleading since they both hold in DC and AC situations. In the steady state (constant
) Eq. () shows that ''V''=0 while a nonzero current flows through the junction. In the case of a constant applied voltage (voltage bias) Eq. () can be integrated easily and gives
Substitution in Eq. () gives
This is an AC current. The frequency
is called the Josephson frequency. One μV gives a frequency of about 500 MHz. By using Eq. () the flux quantum is determined with the high precision as given in Eq. ().
The energy difference of a Cooper pair, moving from one side of the contact to the other, is . With this expression Eq. () can be written as which is the relation for the energy of a photon with frequency ''ν''.
:The AC Josephson relation (Eq. ()) can be easily understood in terms of Newton's law, (or from one of the
London equation's). We start with Newton's law
:Substituting the expression for the
Lorentz force and using the general expression for the co-moving time derivative
gives
:Eq. () gives
so
:Take the line integral of this expression. In the end points the velocities are zero so the ∇''v''
2 term gives no contribution. Using
and Eq. (), with and , gives Eq. ().
DC SQUID
Fig. 5 shows a so-called DC
SQUID. It consists of two superconductors connected by two weak links. The fluxoid quantization of a loop through the two bulk superconductors and the two weak links demands
If the self-inductance of the loop can be neglected the magnetic flux in the loop Φ is equal to the applied flux
with ''B'' the magnetic field, applied perpendicular to the surface, and ''A'' the surface area of the loop. The total supercurrent is given by
Substitution of Eq() in () gives
Using a well known geometrical formula we get
Since the sin-function can vary only between −1 and +1 a steady solution is only possible if the applied current is below a critical current given by
Note that the critical current is periodic in the applied flux with period . The dependence of the critical current on the applied flux is depicted in Fig. 6. It has a strong resemblance with the interference pattern generated by a laser beam behind a double slit. In practice the critical current is not zero at half integer values of the flux quantum of the applied flux. This is due to the fact that the self-inductance of the loop cannot be neglected.
Type II superconductivity
Type-II superconductivity is characterized by two critical fields called ''B''
c1 and ''B''
c2. At a magnetic field ''B''
c1 the applied magnetic field starts to penetrate the sample, but the sample is still superconducting. Only at a field of ''B''
c2 the sample is completely normal. For fields in between ''B''
c1 and ''B''
c2 magnetic flux penetrates the superconductor in well-organized patterns, the so-called
Abrikosov vortex
In superconductivity, fluxon (also called a Abrikosov vortex and quantum vortex) is a vortex of supercurrent in a type-II superconductor, used by Alexei Abrikosov to explain magnetic behavior of type-II superconductors. Abrikosov vortices occur ...
lattice similar to the pattern shown in Fig. 2. A cross section of the superconducting plate is given in Fig. 7. Far away from the plate the field is homogeneous, but in the material superconducting currents flow which squeeze the field in bundles of exactly one flux quantum. The typical field in the core is as big as 1 tesla. The currents around the vortex core flow in a layer of about 50 nm with current densities on the order of 15 A/m
2. That corresponds with 15 million ampère in a wire of one mm
2.
Dilute quantum gases
The classical types of quantum systems, superconductors and superfluid helium, were discovered in the beginning of the 20th century. Near the end of the 20th century, scientists discovered how to create very dilute atomic or molecular gases, cooled first by
laser cooling
Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) a ...
and then by
evaporative cooling. They are trapped using magnetic fields or optical dipole potentials in ultrahigh vacuum chambers. Isotopes which have been used include rubidium (Rb-87 and Rb-85), strontium (Sr-87, Sr-86, and Sr-84) potassium (K-39 and K-40), sodium (Na-23), lithium (Li-7 and Li-6), and hydrogen (H-1). The temperatures to which they can be cooled are as low as a few nanokelvin. The developments have been very fast in the past few years. A team of NIST and the University of Colorado has succeeded in creating and observing vortex quantization in these systems.
The concentration of vortices increases with the angular velocity of the rotation, similar to the case of superfluid helium and superconductivity.
See also
*
Charge density wave
*
Chiral magnetic effect
*
Domain wall (magnetism)
A domain wall is a term used in physics which can have similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously ...
*
Flux pinning
*
Flux quantization
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
*
Ginzburg–Landau theory
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenol ...
*
Husimi Q representation
*
Josephson effect
In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mec ...
*
Magnetic flux quantum
*
Meissner effect
The Meissner effect (or Meissner–Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state when it is cooled below the critical temperature. This expulsion will repel a ne ...
*
N-slit interferometric equation
Quantum mechanics was first applied to optics, and interference in particular, by Paul Dirac. Richard Feynman, in his Lectures on Physics, uses Dirac's notation to describe thought experiments on double-slit interference of electrons. Feynman' ...
*
Quantum boomerang effect
*
Quantum turbulence
*
Quantum vortex
In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was ...
*
Schrödinger's cat paradox
*
Second sound
Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Its presence leads to a very high thermal conductivity. It is known as "second sound" because t ...
*
SQUID
*
Superconductivity
*
Topological defect
A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...
*
Type-I superconductor
*
Type-II superconductor
References and footnotes
{{Reflist, 35em
Atomic, molecular, and optical physics
Condensed matter physics
Exotic matter
Phases of matter
Quantum phases