Planck constant

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Planck_constant 

 

Plaque at the Humboldt University of Berlin: "In this house taught Max Planck, the discoverer of the elementary quantum of action ${\displaystyle h}$, from 1889 to 1928."

The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant is a fundamental physical constant denoted as ${\displaystyle h}$, and of fundamental importance in quantum mechanics. In metrology it is used to define the kilogram in SI units.

The Planck constant is defined to have the exact value ${\displaystyle h=}$ 6.62607015×10⁻³⁴ J⋅s in SI units.

At the end of the 19th century, accurate measurements of the spectrum of black body radiation existed, but predictions of the frequency distribution of the radiation by then-existing theories diverged significantly at higher frequencies. In 1900, Max Planck empirically derived a formula for the observed spectrum. He assumed a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment, ${\displaystyle E}$, that was proportional to the frequency of its associated electromagnetic wave. He was able to calculate the proportionality constant, ${\displaystyle h}$, from the experimental measurements, and that constant is named in his honor. In 1905, the value ${\displaystyle E}$ was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle. It was eventually called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

Confusion can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI. In the language of quantity calculus, the expression for the value of the Planck constant, or a frequency, is the product of a numerical value and a unit of measurement. The symbol f (or ν), when used for the value of a frequency, implies cycles per second or hertz as the unit. When the symbol ω is used for the frequency's value it implies radians per second as the unit. The numerical values of these two ways of expressing the frequency have a ratio of 2π.  Omitting the units of angular measure "cycle" and "radian" can lead to an error of 2π. A similar state of affairs occurs for the Planck constant. The symbol h is used to express the value of the Planck constant in J⋅s/cycle, and the symbol ħ ("h-bar") is used to express its value in J⋅s/radian. Both represent the value of the Planck constant, but, as discussed below, their numerical values have a ratio of 2π. In this Wikipedia article the word "value" as used in the tables means "numerical value", and the equations involving the Planck constant and/or frequency actually involve their numerical values using the appropriate implied units. The distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file.

Since energy and mass are equivalent, the Planck constant also relates mass to frequency.

Origin of the constant

Intensity of light emitted from a black body. Each curve represents behavior at different body temperatures. Max Planck was the first to explain the shape of these curves.

Planck's constant was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution of thermal radiation from a closed furnace (black-body radiation). This mathematical expression is now known as Planck's law.

In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by Kirchhoff some 40 years earlier. Every physical body spontaneously and continuously emits electromagnetic radiation. There was no expression or explanation for the overall shape of the observed emission spectrum. At the time, Wien's law fit the data for short wavelengths and high temperatures, but failed for long wavelengths. Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, now known as the Rayleigh–Jeans law, that could reasonably predict long wavelengths but failed dramatically at short wavelengths.

Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic oscillators, one for each possible frequency. He examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for the black-body spectrum, which gave a simple empirical formula for long wavelengths.

Planck tried to find a mathematical expression that could reproduce Wien's law (for short wavelengths) and the empirical formula (for long wavelengths). This expression included a constant, ${\displaystyle h}$, which subsequently became known as the Planck Constant. The expression formulated by Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by

${\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle h}$ is the Planck constant, and ${\displaystyle c}$ is the speed of light in the medium, whether material or vacuum.

The spectral radiance of a body, ${\displaystyle B_{\nu }}$, describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency. The spectral radiance can also be expressed per unit wavelength ${\displaystyle \lambda }$ instead of per unit frequency. In this case, it is given by

${\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}},}$

showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.

Planck's law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI units of ${\displaystyle B_{\nu }}$ are W·sr⁻¹·m⁻²·Hz⁻¹, while those of ${\displaystyle B_{\lambda }}$ are W·sr⁻¹·m⁻³.

Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics." One of his new boundary conditions was

to interpret U_N [the vibrational energy of N oscillators] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element ε;

— Planck, On the Law of Distribution of Energy in the Normal Spectrum

With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his own words, but one that would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now sometimes termed the "Planck–Einstein relation":

${\displaystyle E=hf.}$

Planck was able to calculate the value of ${\displaystyle h}$ from experimental data on black-body radiation: his result, 6.55×10⁻³⁴ J⋅s, is within 1.2% of the currently accepted value. He also made the first determination of the Boltzmann constant ${\displaystyle k_{B}}$ from the same data and theory.

The divergence of the theoretical Rayleigh-Jeans (black) curve from the observed Planck curves at different temperatures.

Development and application

The black-body problem was revisited in 1905, when Rayleigh and Jeans (on the one hand) and Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "ultraviolet catastrophe", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".

Photoelectric effect

The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by Alexandre Edmond Becquerel in 1839, although credit is usually reserved for Heinrich Hertz, who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard in 1902. Einstein's 1905 paper discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921, after his predictions had been confirmed by the experimental work of Robert Andrews Millikan. The Nobel committee awarded the prize for his work on the photo-electric effect, rather than relativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, and dissent amongst its members as to the actual proof that relativity was real.

Before Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterize different types of radiation. The energy transferred by a wave in a given time is called its intensity. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the color of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their intensity. However, the energy account of the photoelectric effect didn't seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light, but depends linearly on the frequency; and if the frequency is too low (corresponding to a photon energy that is less than the work function of the material), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect). Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy.

Einstein's explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" of energy, which would later be named photons, was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation:

${\displaystyle E=hf.}$

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light ${\displaystyle f}$ and the kinetic energy of photoelectrons ${\displaystyle E}$ was shown to be equal to the Planck constant ${\displaystyle h}$.

Atomic structure

A schematization of the Bohr model of the hydrogen atom. The transition shown from the n = 3 level to the n = 2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts.

Niels Bohr introduced the first quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of Rutherford's classical model. In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a nucleus, the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck's work: an electron in a Bohr atom could only have certain defined energies ${\displaystyle E_{n}}$

${\displaystyle E_{n}=-{\frac {hcR_{\infty }}{n^{2}}},}$

where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle R_{\infty }}$ is an experimentally determined constant (the Rydberg constant) and ${\displaystyle n\in \{1,2,3,...\}}$. Once the electron reached the lowest energy level (${\displaystyle n=1}$), it could not get any closer to the nucleus (lower energy). This approach also allowed Bohr to account for the Rydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant ${\displaystyle R_{\infty }}$ in terms of other fundamental constants.

Bohr also introduced the quantity ${\displaystyle {\frac {h}{2\pi }}}$, now known as the reduced Planck constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg's matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if ${\displaystyle J}$ is the total angular momentum of a system with rotational invariance, and ${\displaystyle J_{z}}$ the angular momentum measured along any given direction, these quantities can only take on the values

${\displaystyle {\begin{aligned}J^{2}=j(j+1)\hbar ^{2},\qquad &j=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots ,\\J_{z}=m\hbar ,\qquad \qquad \quad &m=-j,-j+1,\ldots ,j.\end{aligned}}}$

Uncertainty principle

The Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given numerous particles prepared in the same state, the uncertainty in their position, ${\displaystyle \Delta x}$, and the uncertainty in their momentum, ${\displaystyle \Delta p_{x}}$, obey

${\displaystyle \Delta x\,\Delta p_{x}\geq {\frac {\hbar }{2}},}$

where the uncertainty is given as the standard deviation of the measured value from its expected value. There are several other such pairs of physically measurable conjugate variables which obey a similar rule. One example is time vs. energy. The inverse relationship between the uncertainty of the two conjugate variables forces a tradeoff in quantum experiments, as measuring one quantity more precisely results in the other quantity becoming imprecise.

In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between the position operator ${\displaystyle {\hat {x}}}$ and the momentum operator ${\displaystyle {\hat {p}}}$:

${\displaystyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij},}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

Photon energy

The Planck–Einstein relation connects the particular photon energy E with its associated wave frequency f:

${\displaystyle E=hf}$

This energy is extremely small in terms of ordinarily perceived everyday objects.

Since the frequency f, wavelength λ, and speed of light c are related by ${\displaystyle f={\frac {c}{\lambda }}}$, the relation can also be expressed as

${\displaystyle E={\frac {hc}{\lambda }}.}$

The de Broglie wavelength λ of the particle is given by

${\displaystyle \lambda ={\frac {h}{p}}}$

where p denotes the linear momentum of a particle, such as a photon, or any other elementary particle.

In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant. It is equal to the Planck constant divided by 2π, and is denoted ħ (pronounced "h-bar"):

${\displaystyle \hbar ={\frac {h}{2\pi }}.}$

The energy of a photon with angular frequency ω = 2πf is given by

${\displaystyle E=\hbar \omega ,}$

while its linear momentum relates to

${\displaystyle p=\hbar k,}$

where k is an angular wavenumber. In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterward. This holds throughout the quantum theory, including electrodynamics.

These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors.

${\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}$

Classical statistical mechanics requires the existence of h (but does not define its value). Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some integer multiple of a very small quantity, the "quantum of action", now called the reduced Planck constant or the natural unit of action. This is the so-called "old quantum theory" developed by Bohr and Sommerfeld, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time. Thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of classical particle motion.

In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.

Value

The Planck constant has dimensions of physical action; i.e., energy multiplied by time, or momentum multiplied by distance, or angular momentum. In SI units, the Planck constant is expressed in joule-seconds (J⋅s or N⋅m⋅s or kg⋅m²⋅s⁻¹). Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the hertz, represents one complete cycle, 360 degrees or 2π radians, per second. An angular frequency in radians per second is often more natural in mathematics and physics and many formulas use a reduced Planck constant (pronounced h-bar)

${\displaystyle h=6.626\ 070\ 15\times 10^{-34}\ {\text{J}}{\cdot }{\text{s}}}$

${\displaystyle \hbar ={{h} \over {2\pi }}=1.054\ 571\ 817...\times 10^{-34}\ {\text{J}}{\cdot }{\text{s}}=6.582\ 119\ 569...\times 10^{-16}\ {\text{eV}}{\cdot }{\text{s}}}$

The above values are recommended by 2018 CODATA.

In Hartree atomic units,

${\displaystyle h=2\pi {\text{ a.u. of action}}}$

${\displaystyle \hbar =~~1{\text{ a.u. of action}}}$

Understanding the 'fixing' of the value of h

Since 2019, the numerical value of the Planck constant has been fixed, with finite significant figures. Under the present definition of the kilogram, which states that "The kilogram [...] is defined by taking the fixed numerical value of h to be 6.62607015×10⁻³⁴ when expressed in the unit J⋅s, which is equal to kg⋅m²⋅s⁻¹, where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed cesium-133 atom Δν_Cs." This implies that mass metrology is now aimed to find the value of one kilogram, and thus it is kilogram which is compensating. Every experiment aiming to measure the kilogram (such as the Kibble balance and the X-ray crystal density method), will essentially refine the value of a kilogram.

As an illustration of this, suppose the decision of making h to be exact was taken in 2010, when its measured value was 6.62606957×10⁻³⁴ J⋅s, thus the present definition of kilogram was also enforced. In future, the value of one kilogram must have become refined to 6.62607015/6.62606957 ≈ 1.0000001 times the mass of the International Prototype of the Kilogram (IPK), neglecting the metre and second units' share, for sake of simplicity.

Significance of the value

The Planck constant is related to the quantization of light and matter. It can be seen as a subatomic-scale constant. In a unit system adapted to subatomic scales, the electronvolt is the appropriate unit of energy and the petahertz the appropriate unit of frequency. Atomic unit systems are based (in part) on the Planck constant. The physical meaning of the Planck constant could suggest some basic features of our physical world. These basic features include the properties of the vacuum constants ${\displaystyle \mu _{0}}$ and ${\displaystyle \epsilon _{0}}$. The Planck constant can be identified as

${\displaystyle h=13.1Q{\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}\zeta ^{2}}$,

where Q is the quality factor and ${\displaystyle \zeta }$ is the integrated area of the vector potential at the center of the wave packet representing a particle. 

The Planck constant is one of the smallest constants used in physics. This reflects the fact that on a scale adapted to humans, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant (the quantum of action) is very small. One can regard the Planck constant to be only relevant to the microscopic scale instead of the macroscopic scale in our everyday experience.

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, green light with a wavelength of 555 nanometres (a wavelength that can be perceived by the human eye to be green) has a frequency of 540 THz (540×10¹² Hz). Each photon has an energy E = hf = 3.58×10⁻¹⁹ J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light more typical in everyday experience (though much larger than the smallest amount perceivable by the human eye) is the energy of one mole of photons; its energy can be computed by multiplying the photon energy by the Avogadro constant, N_A = 6.02214076×10²³ mol⁻¹, with the result of 216 kJ/mol, about the food energy in three apples.

Determination

In principle, the Planck constant can be determined by examining the spectrum of a black-body radiator or the kinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. In practice, these are no longer the most accurate methods.

Since the value of the Planck constant is fixed now, it is no longer determined or calculated in laboratories. Some of the practices given below to determine the Planck constant are now used to determine the mass of the kilogram. The below-given methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect.

Josephson constant

The Josephson constant K_J relates the potential difference U generated by the Josephson effect at a "Josephson junction" with the frequency ν of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that K_J = 2e/h.

${\displaystyle K_{\rm {J}}={\frac {\nu }{U}}={\frac {2e}{h}}\,}$

The Josephson constant may be measured by comparing the potential difference generated by an array of Josephson junctions with a potential difference which is known in SI volts. The measurement of the potential difference in SI units is done by allowing an electrostatic force to cancel out a measurable gravitational force, in a Kibble balance. Assuming the validity of the theoretical treatment of the Josephson effect, K_J is related to the Planck constant by

${\displaystyle h={\frac {8\alpha }{\mu _{0}c_{0}K_{\rm {J}}^{2}}}.}$

Kibble balance

A Kibble balance (formerly known as a watt balance) is an instrument for comparing two powers, one of which is measured in SI watts and the other of which is measured in conventional electrical units. 

From the definition of the conventional watt W₉₀, this gives a measure of the product K_J²R_K in SI units, where R_K is the von Klitzing constant which appears in the quantum Hall effect. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particular assuming that R_K = h/e², the measurement of K_J²R_K is a direct determination of the Planck constant.

${\displaystyle h={\frac {4}{K_{\rm {J}}^{2}R_{\rm {K}}}}.}$

Magnetic resonance

The gyromagnetic ratio γ is the constant of proportionality between the frequency ν of nuclear magnetic resonance (or electron paramagnetic resonance for electrons) and the applied magnetic field B: ν = γB. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in water at 25 °C is known to better than one part per million. The protons are said to be "shielded" from the applied magnetic field by the electrons in the water molecule, the same effect that gives rise to chemical shift in NMR spectroscopy, and this is indicated by a prime on the symbol for the gyromagnetic ratio, γ′_p. The gyromagnetic ratio is related to the shielded proton magnetic moment μ′_p, the spin number I (I = ​¹⁄₂ for protons) and the reduced Planck constant.

${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{I\hbar }}={\frac {2\mu _{\rm {p}}^{\prime }}{\hbar }}}$

The ratio of the shielded proton magnetic moment μ′_p to the electron magnetic moment μ_e can be measured separately and to high precision, as the imprecisely known value of the applied magnetic field cancels itself out in taking the ratio. The value of μ_e in Bohr magnetons is also known: it is half the electron g-factor g_e. Hence

${\displaystyle \mu _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{2}}}$

${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}{\frac {g_{\rm {e}}\mu _{\rm {B}}}{\hbar }}.}$

A further complication is that the measurement of γ′_p involves the measurement of an electric current: this is invariably measured in conventional amperes rather than in SI amperes, so a conversion factor is required. The symbol Γ′_p-90 is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there are two methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors are different in the two cases. Only the high-field value Γ′_p-90(hi) is of interest in determining the Planck constant.

${\displaystyle \gamma _{\rm {p}}^{\prime }={\frac {K_{\rm {J-90}}R_{\rm {K-90}}}{K_{\rm {J}}R_{\rm {K}}}}\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})={\frac {K_{\rm {J-90}}R_{\rm {K-90}}e}{2}}\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})}$

Substitution gives the expression for the Planck constant in terms of Γ′_p-90(hi):

${\displaystyle h={\frac {c_{0}\alpha ^{2}g_{\rm {e}}}{2K_{\rm {J-90}}R_{\rm {K-90}}R_{\infty }\Gamma _{\rm {p-90}}^{\prime }({\rm {hi}})}}{\frac {\mu _{\rm {p}}^{\prime }}{\mu _{\rm {e}}}}.}$

Faraday constant

The Faraday constant F is the charge of one mole of electrons, equal to the Avogadro constant N_A multiplied by the elementary charge e. It can be determined by careful electrolysis experiments, measuring the amount of silver dissolved from an electrode in a given time and for a given electric current. In practice, it is measured in conventional electrical units, and so given the symbol F₉₀. Substituting the definitions of N_A and e, and converting from conventional electrical units to SI units, gives the relation to the Planck constant.

${\displaystyle h={\frac {c_{0}M_{\rm {u}}A_{\rm {r}}({\rm {e}})\alpha ^{2}}{R_{\infty }}}{\frac {1}{K_{\rm {J-90}}R_{\rm {K-90}}F_{90}}}}$

X-ray crystal density

The X-ray crystal density method is primarily a method for determining the Avogadro constant N_A but as the Avogadro constant is related to the Planck constant it also determines a value for h. The principle behind the method is to determine N_A as the ratio between the volume of the unit cell of a crystal, measured by X-ray crystallography, and the molar volume of the substance. Crystals of silicon are used, as they are available in high quality and purity by the technology developed for the semiconductor industry. The unit cell volume is calculated from the spacing between two crystal planes referred to as d₂₂₀. The molar volume V_m(Si) requires a knowledge of the density of the crystal and the atomic weight of the silicon used. The Planck constant is given by

${\displaystyle h={\frac {M_{\rm {u}}A_{\rm {r}}({\rm {e}})c_{0}\alpha ^{2}}{R_{\infty }}}{\frac {{\sqrt {2}}\ d_{220}^{3}}{V_{\rm {m}}({\rm {Si}})}}.}$

Particle accelerator

The experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in 2011. The study called PCC using a giant particle accelerator helped to better understand the relationships between the Planck constant and measuring distances in space.

Molecular vibration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Molecular_vibration 

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies, range from less than 10¹³ Hz to approximately 10¹⁴ Hz, corresponding to wavenumbers of approximately 300 to 3000 cm⁻¹.

In general, a non-linear molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 modes, because rotation about the molecular axis cannot be observed. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond. Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule.

A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is Planck's constant. A fundamental vibration is evoked when one such quantum of energy is absorbed by the molecule in its ground state. When multiple quanta are absorbed, the first and possibly higher overtones are excited.

To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, because the potential energy of the molecule is more like a Morse potential or more accurately, a Morse/Long-range potential.

The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state.

Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Number of vibrational modes

For a molecule with N atoms, the positions of all N nuclei depend on a total of 3N coordinates, so that the molecule has 3N degrees of freedom including translation, rotation and vibration. Translation corresponds to movement of the center of mass whose position can be described by 3 cartesian coordinates.

A nonlinear molecule can rotate about any of three mutually perpendicular axes and therefore has 3 rotational degrees of freedom. For a linear molecule, rotation about the molecular axis does not involve movement of any atomic nucleus, so there are only 2 rotational degrees of freedom which can vary the atomic coordinates.

An equivalent argument is that the rotation of a linear molecule changes the direction of the molecular axis in space, which can be described by 2 coordinates corresponding to latitude and longitude. For a nonlinear molecule, the direction of one axis is described by these two coordinates, and the orientation of the molecule about this axis provides a third rotational coordinate.

The number of vibrational modes is therefore 3N minus the number of translational and rotational degrees of freedom, or 3N–5 for linear and 3N–6 for nonlinear molecules.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C–H or C–C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule.
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule,
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups.
• Out–of–plane: a change in the angle between any one of the C–H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethene there are 12 internal coordinates: 4 C–H stretching, 1 C–C stretching, 2 H–C–H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H–C–C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

Vibrations of a methylene group (–CH₂–) in a molecule for illustration

The atoms in a CH₂ group, commonly found in organic compounds, can vibrate in six different ways: symmetric and asymmetric stretching, scissoring, rocking, wagging and twisting as shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry–adapted coordinates

Symmetry–adapted coordinates may be created by applying a projection operator to a set of internal coordinates. The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four (un–normalized) C–H stretching coordinates of the molecule ethene are given by

${\displaystyle Q_{s1}=q_{1}+q_{2}+q_{3}+q_{4}\!}$

${\displaystyle Q_{s2}=q_{1}+q_{2}-q_{3}-q_{4}\!}$

${\displaystyle Q_{s3}=q_{1}-q_{2}+q_{3}-q_{4}\!}$

${\displaystyle Q_{s4}=q_{1}-q_{2}-q_{3}+q_{4}\!}$

where ${\displaystyle q_{1}-q_{4}}$ are the internal coordinates for stretching of each of the four C–H bonds.

Illustrations of symmetry–adapted coordinates for most small molecules can be found in Nakamoto.^[6]

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. If the molecule possesses symmetries, the normal modes "transform as" an irreducible representation under its point group. The normal modes are determined by applying group theory, and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch:

• symmetric stretching: the sum of the two C–O stretching coordinates; the two C–O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C–O stretching coordinates; one C–O bond length increases while the other decreases. Q = q₁ - q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

• principally C–H stretching with a little C–N stretching; Q₁ = q₁ + a q₂ (a << 1)
• principally C–N stretching with a little C–H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere.

${\displaystyle \mathrm {Force} =-kQ\!}$

By Newton's second law of motion this force is also equal to a reduced mass, μ, times acceleration.

${\displaystyle \mathrm {Force} =\mu {\frac {d^{2}Q}{dt^{2}}}}$

Since this is one and the same force the ordinary differential equation follows.

${\displaystyle \mu {\frac {d^{2}Q}{dt^{2}}}+kQ=0}$

The solution to this equation of simple harmonic motion is

${\displaystyle Q(t)=A\cos(2\pi \nu t);\ \ \nu ={1 \over {2\pi }}{\sqrt {k \over \mu }}.\!}$

A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as

${\displaystyle {\frac {1}{\mu }}={\frac {1}{m_{A}}}+{\frac {1}{m_{B}}}.}$

The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.

${\displaystyle k={\frac {\partial ^{2}V}{\partial Q^{2}}}}$

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,ν_i are obtained from the eigenvalues,λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule. F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

${\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}}\!}$,

where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.

The difference in energy when n (or v) changes by 1 is therefore equal to ${\displaystyle h\nu }$, the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency ${\displaystyle \nu }$ (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

${\displaystyle \Delta n=\pm 1}$

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate. Likewise, the intensity of Raman bands depends on the derivative of polarizability with respect to the normal coordinate. There is also a dependence on the fourth-power of the wavelength of the laser used.

Quantum network

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_network 

 

This article is about the implementation and operation of quantum networks. For a mathematical description of quantum communication channels, see Quantum channel.

Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum processors. A quantum processor is a small quantum computer being able to perform quantum logic gates on a certain number of qubits. Quantum networks work in a similar way to classical networks. The main difference is that quantum networking, like quantum computing, is better at solving certain problems, such as modeling quantum systems.

Basics

Quantum networks for computation

Networked quantum computing or distributed quantum computing works by linking multiple quantum processors through a quantum network by sending qubits in-between them. Doing this creates a quantum computing cluster and therefore creates more computing potential. Less powerful computers can be linked in this way to create one more powerful processor. This is analogous to connecting several classical computers to form a computer cluster in classical computing. Like classical computing this system is scale-able by adding more and more quantum computers to the network. Currently quantum processors are only separated by short distances.

Quantum networks for communication

In the realm of quantum communication, one wants to send qubits from one quantum processor to another over long distances. This way local quantum networks can be intra connected into a quantum internet. A quantum internet supports many applications, which derive their power from the fact that by creating quantum entangled qubits, information can be transmitted between the remote quantum processors. Most applications of a quantum internet require only very modest quantum processors. For most quantum internet protocols, such as quantum key distribution in quantum cryptography, it is sufficient if these processors are capable of preparing and measuring only a single qubit at a time. This is in contrast to quantum computing where interesting applications can only be realized if the (combined) quantum processors can easily simulate more qubits than a classical computer (around 60). Quantum internet applications require only small quantum processors, often just a single qubit, because quantum entanglement can already be realized between just two qubits. A simulation of an entangled quantum system on a classical computer cannot simultaneously provide the same security and speed.

Overview of the elements of a quantum network

The basic structure of a quantum network and more generally a quantum internet is analogous to a classical network. First, we have end nodes on which applications are ultimately run. These end nodes are quantum processors of at least one qubit. Some applications of a quantum internet require quantum processors of several qubits as well as a quantum memory at the end nodes.

Second, to transport qubits from one node to another, we need communication lines. For the purpose of quantum communication, standard telecom fibers can be used. For networked quantum computing, in which quantum processors are linked at short distances, different wavelengths are chosen depending on the exact hardware platform of the quantum processor.

Third, to make maximum use of communication infrastructure, one requires optical switches capable of delivering qubits to the intended quantum processor. These switches need to preserve quantum coherence, which makes them more challenging to realize than standard optical switches.

Finally, one requires a quantum repeater to transport qubits over long distances. Repeaters appear in-between end nodes. Since qubits cannot be copied, classical signal amplification is not possible. By necessity, a quantum repeater works in a fundamentally different way than a classical repeater.

Elements of a quantum network

End nodes: quantum processors

End nodes can both receive and emit information. Telecommunication lasers and parametric down-conversion combined with photodetectors can be used for quantum key distribution. In this case, the end nodes can in many cases be very simple devices consisting only of beamsplitters and photodetectors.

However, for many protocols more sophisticated end nodes are desirable. These systems provide advanced processing capabilities and can also be used as quantum repeaters. Their chief advantage is that they can store and retransmit quantum information without disrupting the underlying quantum state. The quantum state being stored can either be the relative spin of an electron in a magnetic field or the energy state of an electron. They can also perform quantum logic gates.

One way of realizing such end nodes is by using color centers in diamond, such as the nitrogen-vacancy center. This system forms a small quantum processor featuring several qubits. NV centers can be utilized at room temperatures. Small scale quantum algorithms and quantum error correction has already been demonstrated in this system, as well as the ability to entangle two remote quantum processors, and perform deterministic quantum teleportation.

Another possible platform are quantum processors based on Ion traps, which utilize radio-frequency magnetic fields and lasers. In a multispecies trapped-ion node network, photons entangled with a parent atom are used to entangle different nodes. Also, cavity quantum electrodynamics (Cavity QED) is one possible method of doing this. In Cavity QED, photonic quantum states can be transferred to and from atomic quantum states stored in single atoms contained in optical cavities. This allows for the transfer of quantum states between single atoms using optical fiber in addition to the creation of remote entanglement between distant atoms.

Communication lines: physical layer

Over long distances, the primary method of operating quantum networks is to use optical networks and photon-based qubits. This is due to optical networks having a reduced chance of decoherence. Optical networks have the advantage of being able to re-use existing optical fiber. Alternately, free space networks can be implemented that transmit quantum information through the atmosphere or through a vacuum.

Fiber optic networks

Optical networks using existing telecommunication fiber can be implemented using hardware similar to existing telecommunication equipment. This fiber can be either single-mode or multi-mode, with multi-mode allowing for more precise communication. At the sender, a single photon source can be created by heavily attenuating a standard telecommunication laser such that the mean number of photons per pulse is less than 1. For receiving, an avalanche photodetector can be used. Various methods of phase or polarization control can be used such as interferometers and beam splitters. In the case of entanglement based protocols, entangled photons can be generated through spontaneous parametric down-conversion. In both cases, the telecom fiber can be multiplexed to send non-quantum timing and control signals.

Free space networks

Free space quantum networks operate similar to fiber optic networks but rely on line of sight between the communicating parties instead of using a fiber optic connection. Free space networks can typically support higher transmission rates than fiber optic networks and do not have to account for polarization scrambling caused by optical fiber. However, over long distances, free space communication is subject to an increased chance of environmental disturbance on the photons.

Importantly, free space communication is also possible from a satellite to the ground. A quantum satellite capable of entanglement distribution over a distance of 1,203 km has been demonstrated. The experimental exchange of single photons from a global navigation satellite system at a slant distance of 20,000 km has also been reported. These satellites can play an important role in linking smaller ground-based networks over larger distances.

Repeaters

Long distance communication is hindered by the effects of signal loss and decoherence inherent to most transport mediums such as optical fiber. In classical communication, amplifiers can be used to boost the signal during transmission, but in a quantum network amplifiers cannot be used since qubits cannot be copied – known as the no-cloning theorem. That is, to implement an amplifier, the complete state of the flying qubit would need to be determined, something which is both unwanted and impossible.

Trusted repeaters

An intermediary step which allows the testing of communication infrastructure are trusted repeaters. Importantly, a trusted repeater cannot be used to transmit qubits over long distances. Instead, a trusted repeater can only be used to perform quantum key distribution with the additional assumption that the repeater is trusted. Consider two end nodes A and B, and a trusted repeater R in the middle. A and R now perform quantum key distribution to generate a key ${\displaystyle k_{AR}}$. Similarly, R and B run quantum key distribution to generate a key ${\displaystyle k_{RB}}$. A and B can now obtain a key ${\displaystyle k_{AB}}$ between themselves as follows: A sends ${\displaystyle k_{AB}}$ to R encrypted with the key ${\displaystyle k_{AR}}$. R decrypts to obtain ${\displaystyle k_{AB}}$. R then re-encrypts ${\displaystyle k_{AB}}$ using the key ${\displaystyle k_{RB}}$ and sends it to B. B decrypts to obtain ${\displaystyle k_{AB}}$. A and B now share the key ${\displaystyle k_{AB}}$. The key is secure from an outside eavesdropper, but clearly the repeater R also knows ${\displaystyle k_{AB}}$. This means that any subsequent communication between A and B does not provide end to end security, but is only secure as long as A and B trust the repeater R.

Quantum repeaters

Diagram for quantum teleportation of a photon

A true quantum repeater allows the end to end generation of quantum entanglement, and thus - by using quantum teleportation - the end to end transmission of qubits. In quantum key distribution protocols one can test for such entanglement. This means that when making encryption keys, the sender and receiver are secure even if they do not trust the quantum repeater. Any other application of a quantum internet also requires the end to end transmission of qubits, and thus a quantum repeater.

Quantum repeaters allow entanglement and can be established at distant nodes without physically sending an entangled qubit the entire distance.

In this case, the quantum network consists of many short distance links of perhaps tens or hundreds of kilometers. In the simplest case of a single repeater, two pairs of entangled qubits are established: ${\displaystyle |A\rangle }$ and ${\displaystyle |R_{a}\rangle }$ located at the sender and the repeater, and a second pair ${\displaystyle |R_{b}\rangle }$ and ${\displaystyle |B\rangle }$ located at the repeater and the receiver. These initial entangled qubits can be easily created, for example through parametric down conversion, with one qubit physically transmitted to an adjacent node. At this point, the repeater can perform a bell measurement on the qubits ${\displaystyle |R_{a}\rangle }$ and ${\displaystyle |R_{b}\rangle }$ thus teleporting the quantum state of ${\displaystyle |R_{a}\rangle }$ onto ${\displaystyle |B\rangle }$. This has the effect of "swapping" the entanglement such that ${\displaystyle |A\rangle }$ and ${\displaystyle |B\rangle }$ are now entangled at a distance twice that of the initial entangled pairs. It can be seen that a network of such repeaters can be used linearly or in a hierarchical fashion to establish entanglement over great distances.

Hardware platforms suitable as end nodes above can also function as quantum repeaters. However, there are also hardware platforms specific only to the task of acting as a repeater, without the capabilities of performing quantum gates.

Error correction

Error correction can be used in quantum repeaters. Due to technological limitations, however, the applicability is limited to very short distances as quantum error correction schemes capable of protecting qubits over long distances would require an extremely large amount of qubits and hence extremely large quantum computers.

Errors in communication can be broadly classified into two types: Loss errors (due to optical fiber/environment) and operation errors (such as depolarization, dephasing etc.). While redundancy can be used to detect and correct classical errors, redundant qubits cannot be created due to the no-cloning theorem. As a result, other types of error correction must be introduced such as the Shor code or one of a number of more general and efficient codes. All of these codes work by distributing the quantum information across multiple entangled qubits so that operation errors as well as loss errors can be corrected.

In addition to quantum error correction, classical error correction can be employed by quantum networks in special cases such as quantum key distribution. In these cases, the goal of the quantum communication is to securely transmit a string of classical bits. Traditional error correction codes such as Hamming codes can be applied to the bit string before encoding and transmission on the quantum network.

Entanglement purification

Quantum decoherence can occur when one qubit from a maximally entangled bell state is transmitted across a quantum network. Entanglement purification allows for the creation of nearly maximally entangled qubits from a large number of arbitrary weakly entangled qubits, and thus provides additional protection against errors. Entanglement purification (also known as Entanglement distillation) has already been demonstrated in Nitrogen-vacancy centers in diamond.

Applications

A quantum internet supports numerous applications, enabled by quantum entanglement. In general, quantum entanglement is well suited for tasks that require coordination, synchronization or privacy.

Examples of such applications include quantum key distribution, clock synchronization, protocols for distributed system problems such as leader election or byzantine agreement, extending the baseline of telescopes, as well as position verification, secure identification and two-party cryptography in the noisy-storage model. A quantum internet also enables secure access to a quantum computer in the cloud. Specifically, a quantum internet enables very simple quantum devices to connect to a remote quantum computer in such a way that computations can be performed there without the quantum computer finding out what this computation actually is (the input and output quantum states can not be measured without destroying the computation, but the circuit composition used for the calculation will be known).

Secure communications

When it comes to communicating in any form the largest issue has always been keeping your communications private. From when couriers were used to send letters between ancient battle commanders to secure radio communications that exist today the main purpose is to ensure that what a sender sends out to the receiver reaches the receiver unmolested. This is an area in which Quantum Networks particularly excel. By applying a quantum operator that the user selects to a system of information the information can then be sent to the receiver without a chance of an eavesdropper being able to accurately be able to record the sent information without either the sender or receiver knowing. This works because if a listener tries to listen in then they will change the information in an unintended way by listening thereby tipping their hand to the people on whom they are attacking. Secondly, without the proper quantum operator to decode the information they will corrupt the sent information without being able to use it themselves.

Current status

Quantum internet

At present, there is no network connecting quantum processors, or quantum repeaters deployed outside a lab.

Quantum key distribution networks

Several test networks have been deployed that are tailored to the task of quantum key distribution either at short distances (but connecting many users), or over larger distances by relying on trusted repeaters. These networks do not yet allow for the end to end transmission of qubits or the end to end creation of entanglement between far away nodes.

Major quantum network projects and QKD protocols implemented

Quantum network	Start	BB84	BBM92	E91	DPS	COW
DARPA Quantum Network	2001	Yes	No	No	No	No
Geneva area network (SwissQuantum)	2010	Yes	No	No	No	Yes
Hierarchical network in Wuhu, China	2009	Yes	No	No	No	No
SECOCQ QKD network in Vienna	2003	Yes	Yes	No	No	Yes
Tokyo QKD network	2009	Yes	Yes	No	Yes	No

DARPA Quantum Network

Starting in the early 2000s, DARPA began sponsorship of a quantum network development project with the aim of implementing secure communication. The DARPA Quantum Network became operational within the BBN Technologies laboratory in late 2003 and was expanded further in 2004 to include nodes at Harvard and Boston Universities. The network consists of multiple physical layers including fiber optics supporting phase-modulated lasers and entangled photons as well free-space links.

SECOQC Vienna QKD network

From 2003 to 2008 the Secure Communication based on Quantum Cryptography (SECOQC) project developed a collaborative network between a number of European institutions. The architecture chosen for the SECOQC project is a trusted repeater architecture which consists of point-to-point quantum links between devices where long distance communication is accomplished through the use of repeaters.

Chinese hierarchical network

In May 2009, a hierarchical quantum network was demonstrated in Wuhu, China. The hierarchical network consists of a backbone network of four nodes connecting a number of subnets. The backbone nodes are connected through an optical switching quantum router. Nodes within each subnet are also connected through an optical switch and are connected to the backbone network through a trusted relay.

Geneva area network (SwissQuantum)

The SwissQuantum network developed and tested between 2009 and 2011 linked facilities at CERN with the University of Geneva and hepia in Geneva. The SwissQuantum program focused on transitioning the technologies developed in the SECOQC and other research quantum networks into a production environment. In particular the integration with existing telecommunication networks, and its reliability and robustness.

Tokyo QKD network

In 2010, a number of organizations from Japan and the European Union setup and tested the Tokyo QKD network. The Tokyo network build upon existing QKD technologies and adopted a SECOQC like network architecture. For the first time, one-time-pad encryption was implemented at high enough data rates to support popular end-user application such as secure voice and video conferencing. Previous large-scale QKD networks typically used classical encryption algorithms such as AES for high-rate data transfer and use the quantum-derived keys for low rate data or for regularly re-keying the classical encryption algorithms.

Beijing-Shanghai Trunk Line

In September 2017, a 2000-km quantum key distribution network between Beijing and Shanghai, China, was officially opened. This trunk line will serve as a backbone connecting quantum networks in Beijing, Shanghai, Jinan in Shandong province and Hefei in Anhui province. During the opening ceremony, two employees from the Bank of Communications completed a transaction from Shanghai to Beijing using the network. The State Grid Corporation of China is also developing a managing application for the link. The line uses 32 trusted nodes as repeaters. A quantum telecommunication network has been also put into service in Wuhan, capital of central China's Hubei Province, which will be connected to the trunk. Other similar city quantum networks along the Yangtze River are planned to follow.^[