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124-457: The Hong–Ou–Mandel effect is a two- photon interference effect in quantum optics that was demonstrated in 1987 by three physicists from the University of Rochester : Chung Ki Hong (홍정기), Zheyu Ou (欧哲宇), and Leonard Mandel . The effect occurs when two identical single photons enter a 1:1 beam splitter , one in each input port. When the temporal overlap of the photons on the beam splitter

248-402: A Hermitian operator . In 1924, Satyendra Nath Bose derived Planck's law of black-body radiation without using any electromagnetism, but rather by using a modification of coarse-grained counting of phase space . Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction", now understood as

372-921: A fermionic system are, where {   ,   } {\displaystyle {\{\ ,\ \}}} is the anticommutator and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta . These anticommutation relations can be used to show antisymmetric behaviour of Fermionic Fock states . Number operators N k l ^ {\textstyle {\widehat {N_{{\mathbf {k} }_{l}}}}} for Fermions are given by N k l ^ = c k l † . c k l {\displaystyle {\widehat {N_{{\mathbf {k} }_{l}}}}=c_{{\mathbf {k} }_{l}}^{\dagger }.c_{{\mathbf {k} }_{l}}} . The action of

496-521: A point-like particle since it is absorbed or emitted as a whole by arbitrarily small systems, including systems much smaller than its wavelength, such as an atomic nucleus (≈10 m across) or even the point-like electron . While many introductory texts treat photons using the mathematical techniques of non-relativistic quantum mechanics, this is in some ways an awkward oversimplification, as photons are by nature intrinsically relativistic. Because photons have zero rest mass , no wave function defined for

620-419: A 1:1 beam splitter. There are four possibilities regarding how the photons will behave: We assume now that the two photons are identical in their physical properties (i.e., polarization , spatio-temporal mode structure, and frequency ). Since the state of the beam splitter does not "record" which of the four possibilities actually happens, Feynman rules dictates that we have to add all four possibilities at

744-4371: A Fock state are given by the following two equations: The bosonic Fock state creation and annihilation operators are not Hermitian operators . For a Fock state, | n k 1 , n k 2 , n k 3 … n k l , … ⟩ {\displaystyle |n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}},\dots \rangle } , ⟨ n k 1 , n k 2 , n k 3 … n k l − 1 , … | b k l | n k 1 , n k 2 , n k 3 … n k l , … ⟩ = n k l ⟨ n k 1 , n k 2 , n k 3 … n k l − 1 , … | n k 1 , n k 2 , n k 3 … n k l − 1 , … ⟩ ( ⟨ n k 1 , n k 2 , n k 3 … n k l , … | b k l | n k 1 , n k 2 , n k 3 … n k l − 1 , … ⟩ ) ∗ = ⟨ n k 1 , n k 2 , n k 3 … n k l − 1 … | b k l † | n k 1 , n k 2 , n k 3 … n k l , … ⟩ = n k l + 1 ⟨ n k 1 , n k 2 , n k 3 … n k l − 1 … | n k 1 , n k 2 , n k 3 … n k l + 1 … ⟩ {\displaystyle {\begin{aligned}\left\langle n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1,\dots \left|b_{\mathbf {k} _{l}}\right|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}},\dots \right\rangle &={\sqrt {n_{\mathbf {k} _{l}}}}\left\langle n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1,\dots |n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1,\dots \right\rangle \\[6pt]\left(\left\langle n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}},\dots \left|b_{\mathbf {k} _{l}}\right|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1,\dots \right\rangle \right)^{*}&=\left\langle n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1\dots \left|b_{\mathbf {k} _{l}}^{\dagger }\right|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}},\dots \right\rangle \\&={\sqrt {n_{\mathbf {k} _{l}}+1}}\left\langle n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}-1\dots |n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}\dots n_{\mathbf {k} _{l}}+1\dots \right\rangle \end{aligned}}} Therefore, it

868-744: A certain symmetry at every point in spacetime . The intrinsic properties of particles, such as charge , mass , and spin , are determined by gauge symmetry . The photon concept has led to momentous advances in experimental and theoretical physics, including lasers , Bose–Einstein condensation , quantum field theory , and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry , high-resolution microscopy , and measurements of molecular distances . Moreover, photons have been studied as elements of quantum computers , and for applications in optical imaging and optical communication such as quantum cryptography . The word quanta (singular quantum, Latin for how much )

992-594: A certain threshold; light of frequency lower than the threshold, no matter how intense, does not initiate the reaction. Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the photoelectric effect ); the energy of the ejected electron is related only to the light's frequency, not to its intensity. At the same time, investigations of black-body radiation carried out over four decades (1860–1900) by various researchers culminated in Max Planck 's hypothesis that

1116-513: A charge is accelerated it emits synchrotron radiation . During a molecular , atomic or nuclear transition to a lower energy level , photons of various energy will be emitted, ranging from radio waves to gamma rays . Photons can also be emitted when a particle and its corresponding antiparticle are annihilated (for example, electron–positron annihilation ). In empty space, the photon moves at c (the speed of light ) and its energy and momentum are related by E = pc , where p

1240-467: A choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle. According to the uncertainty principle, no matter how the particle is prepared, it is not possible to make a precise prediction for both of the two alternative measurements: if the outcome of the position measurement is made more certain, the outcome of the momentum measurement becomes less so, and vice versa. A coherent state minimizes

1364-502: A gauge boson , below.) Einstein's 1905 predictions were verified experimentally in several ways in the first two decades of the 20th century, as recounted in Robert Millikan 's Nobel lecture. However, before Compton's experiment showed that photons carried momentum proportional to their wave number (1922), most physicists were reluctant to believe that electromagnetic radiation itself might be particulate. (See, for example,

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1488-517: A geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of black-body radiation, which were derived by Einstein in 1909. In 1925, Born , Heisenberg and Jordan reinterpreted Debye's concept in a key way. As may be shown classically, the Fourier modes of the electromagnetic field —a complete set of electromagnetic plane waves indexed by their wave vector k and polarization state—are equivalent to

1612-475: A higher energy E i {\displaystyle E_{i}} is proportional to the number N j {\displaystyle N_{j}} of atoms with energy E j {\displaystyle E_{j}} and to the energy density ρ ( ν ) {\displaystyle \rho (\nu )} of ambient photons of that frequency, where B j i {\displaystyle B_{ji}}

1736-504: A higher energy E i {\displaystyle E_{i}} to a lower energy E j {\displaystyle E_{j}} is where A i j {\displaystyle A_{ij}} is the rate constant for emitting a photon spontaneously , and B i j {\displaystyle B_{ij}} is the rate constant for emissions in response to ambient photons ( induced or stimulated emission ). In thermodynamic equilibrium,

1860-455: A light beam may have mixtures of these two values; a linearly polarized light beam will act as if it were composed of equal numbers of the two possible angular momenta. The spin angular momentum of light does not depend on its frequency, and was experimentally verified by C. V. Raman and S. Bhagavantam in 1931. The collision of a particle with its antiparticle can create photons. In free space at least two photons must be created since, in

1984-511: A new notation, the occupancy number notation, by specifying the number of particles in each possible one-particle state. Let { k i } i ∈ I {\textstyle \left\{\mathbf {k} _{i}\right\}_{i\in I}} be an orthonormal basis of states in the underlying one-particle Hilbert space. This induces a corresponding basis of the Fock space called

2108-600: A paper in which he proposed that many light-related phenomena—including black-body radiation and the photoelectric effect —would be better explained by modelling electromagnetic waves as consisting of spatially localized, discrete energy quanta. He called these a light quantum (German: ein Lichtquant ). The name photon derives from the Greek word for light, φῶς (transliterated phôs ). Arthur Compton used photon in 1928, referring to Gilbert N. Lewis , who coined

2232-399: A particle from state k l {\displaystyle k_{l}} to state k m {\displaystyle k_{m}} , then we operate the Fock state by b k m † b k l {\displaystyle b_{\mathbf {k} _{m}}^{\dagger }b_{\mathbf {k} _{l}}} in the following way: Using

2356-405: A particle from state k l {\displaystyle k_{l}} to state k m {\displaystyle k_{m}} , then we operate the Fock state by c k m † . c k l {\displaystyle c_{\mathbf {k} _{m}}^{\dagger }.c_{\mathbf {k} _{l}}} in the following way: Using

2480-459: A photon can have all the properties familiar from wave functions in non-relativistic quantum mechanics. In order to avoid these difficulties, physicists employ the second-quantized theory of photons described below, quantum electrodynamics , in which photons are quantized excitations of electromagnetic modes. Another difficulty is finding the proper analogue for the uncertainty principle , an idea frequently attributed to Heisenberg, who introduced

2604-463: A photon in b . Therefore The relative minus sign appears because the classical lossless beam splitter produces a unitary transformation . This can be seen most clearly when we write the two-mode beam splitter transformation in matrix form: Similar transformations hold for the creation operators. Unitarity of the transformation implies unitarity of the matrix. Physically, this beam splitter transformation means that reflection from one surface induces

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2728-399: A photon is calculated by equations that describe waves. This combination of aspects is known as wave–particle duality . For example, the probability distribution for the location at which a photon might be detected displays clearly wave-like phenomena such as diffraction and interference . A single photon passing through a double slit has its energy received at a point on the screen with

2852-518: A photonic chip. Topological photonics have intrinsically high-coherence, and unlike other quantum processor approaches, do not require strong magnetic fields and operate at room temperature. Three-photon interference effect has been identified in experiments. Photon As with other elementary particles, photons are best explained by quantum mechanics and exhibit wave–particle duality , their behavior featuring properties of both waves and particles . The modern photon concept originated during

2976-402: A probability distribution given by its interference pattern determined by Maxwell's wave equations . However, experiments confirm that the photon is not a short pulse of electromagnetic radiation; a photon's Maxwell waves will diffract, but photon energy does not spread out as it propagates, nor does this energy divide when it encounters a beam splitter . Rather, the received photon acts like

3100-790: A relative phase shift of π, corresponding to a factor of −1, with respect to reflection from the other side of the beam splitter (see the Physical description above). When two photons enter the beam splitter, one on each side, the state of the two modes becomes where we used c ^ † 2 | 0 , 0 ⟩ c d = c ^ † | 1 , 0 ⟩ c d = 2 | 2 , 0 ⟩ c d {\displaystyle {\hat {c}}^{\dagger 2}|0,0\rangle _{cd}={\hat {c}}^{\dagger }|1,0\rangle _{cd}={\sqrt {2}}|2,0\rangle _{cd}} etc. Since

3224-410: A relatively simple assumption. He decomposed the electromagnetic field in a cavity into its Fourier modes , and assumed that the energy in any mode was an integer multiple of h ν {\displaystyle h\nu } , where ν {\displaystyle \nu } is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as

3348-497: A semi-classical, statistical treatment of photons and atoms, which implies a link between the rates at which atoms emit and absorb photons. The condition follows from the assumption that functions of the emission and absorption of radiation by the atoms are independent of each other, and that thermal equilibrium is made by way of the radiation's interaction with the atoms. Consider a cavity in thermal equilibrium with all parts of itself and filled with electromagnetic radiation and that

3472-450: A semiclassical approach, and, in 1927, succeeded in deriving all the rate constants from first principles within the framework of quantum theory. Dirac's work was the foundation of quantum electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called second quantization or quantum field theory ; earlier quantum mechanical treatments only treat material particles as quantum mechanical, not

3596-460: A set of uncoupled simple harmonic oscillators . Treated quantum mechanically, the energy levels of such oscillators are known to be E = n h ν {\displaystyle E=nh\nu } , where ν {\displaystyle \nu } is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy E = n h ν {\displaystyle E=nh\nu } as

3720-399: A state with n {\displaystyle n} photons, each of energy h ν {\displaystyle h\nu } . This approach gives the correct energy fluctuation formula. Dirac took this one step further. He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing

3844-716: A two particle system in the tensor product representation we have P ^ | x 1 , x 2 ⟩ = | x 2 , x 1 ⟩ {\displaystyle {\hat {P}}\left|x_{1},x_{2}\right\rangle =\left|x_{2},x_{1}\right\rangle } . We should be able to express the same symmetric property in this new Fock space representation. For this we introduce non-Hermitian bosonic creation and annihilation operators , denoted by b † {\displaystyle b^{\dagger }} and b {\displaystyle b} respectively. The action of these operators on

Hong–Ou–Mandel effect - Misplaced Pages Continue

3968-414: A unit related to the illumination of the eye and the resulting sensation of light and was used later in a physiological context. Although Wolfers's and Lewis's theories were contradicted by many experiments and never accepted, the new name was adopted by most physicists very soon after Compton used it. In physics, a photon is usually denoted by the symbol γ (the Greek letter gamma ). This symbol for

4092-410: Is clear that adjoint of creation (annihilation) operator doesn't go into itself. Hence, they are not Hermitian operators. But adjoint of creation (annihilation) operator is annihilation (creation) operator. The commutation relations of creation and annihilation operators in a bosonic system are where [     ,     ] {\displaystyle [\ \ ,\ \ ]}

4216-551: Is done by annihilating one particle in one state and creating one in other. If we start with a Fock state | ψ ⟩ = | n k 1 , n k 2 , . . . n k m . . . n k l . . . ⟩ {\displaystyle |\psi \rangle =\left|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},...n_{\mathbf {k} _{m}}...n_{\mathbf {k} _{l}}...\right\rangle } and want to shift

4340-404: Is non-classical: a classical light wave entering a classical beam splitter with the same transfer matrix would always exit in arm c due to destructive interference in arm d , whereas the quantum result is random. Changing the beam splitter phases can change the classical result to arm d or a mixture of both, but the quantum result is independent of these phases. For a more general treatment of

4464-483: Is not quantized, but matter appears to obey the laws of quantum mechanics . Although the evidence from chemical and physical experiments for the existence of photons was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, and a sufficiently complete theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in

4588-590: Is obtained by applying a certain sum of permutation operators to the tensor product of eigenkets as follows: This determinant is called the Slater determinant . If any of the single particle states are the same, two rows of the Slater determinant would be the same and hence the determinant would be zero. Hence, two identical fermions must not occupy the same state (a statement of the Pauli exclusion principle ). Therefore,

4712-421: Is perfect, the two photons will always exit the beam splitter together in the same output mode, meaning that there is zero chance that they will exit separately with one photon in each of the two outputs giving a coincidence event. The photons have a 50:50 chance of exiting (together) in either output mode. If they become more distinguishable (e.g. because they arrive at different times or with different wavelength),

4836-1555: Is the commutator and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta . The number operators N k l ^ {\textstyle {\widehat {N_{{\mathbf {k} }_{l}}}}} for a bosonic system are given by N k l ^ = b k l † b k l {\displaystyle {\widehat {N_{{\mathbf {k} }_{l}}}}=b_{{\mathbf {k} }_{l}}^{\dagger }b_{{\mathbf {k} }_{l}}} , where N k l ^ | n k 1 , n k 2 , n k 3 . . . n k l . . . ⟩ = n k l | n k 1 , n k 2 , n k 3 . . . n k l . . . ⟩ {\displaystyle {\widehat {N_{{\mathbf {k} }_{l}}}}|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle =n_{{\mathbf {k} }_{l}}|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}}...\rangle } Number operators are Hermitian operators. The commutation relations of

4960-458: Is the gauge boson for electromagnetism , and therefore all other quantum numbers of the photon (such as lepton number , baryon number , and flavour quantum numbers ) are zero. Also, the photon obeys Bose–Einstein statistics , and not Fermi–Dirac statistics . That is, they do not obey the Pauli exclusion principle and more than one can occupy the same bound quantum state. Photons are emitted in many natural processes. For example, when

5084-414: Is the magnitude of the momentum vector p . This derives from the following relativistic relation, with m = 0 : The energy and momentum of a photon depend only on its frequency ( ν {\displaystyle \nu } ) or inversely, its wavelength ( λ ): where k is the wave vector , where Since p {\displaystyle {\boldsymbol {p}}} points in

Hong–Ou–Mandel effect - Misplaced Pages Continue

5208-512: Is the rate constant for absorption. For the reverse process, there are two possibilities: spontaneous emission of a photon, or the emission of a photon initiated by the interaction of the atom with a passing photon and the return of the atom to the lower-energy state. Following Einstein's approach, the corresponding rate R i j {\displaystyle R_{ij}} for the emission of photons of frequency ν {\displaystyle \nu } and transition from

5332-427: Is the photon's frequency . The photon has no electric charge , is generally considered to have zero rest mass and is a stable particle . The experimental upper limit on the photon mass is very small, on the order of 10 kg; its lifetime would be more than 10 years. For comparison the age of the universe is about 1.38 × 10 years. In a vacuum, a photon has two possible polarization states. The photon

5456-774: Is therefore When the two modes a and b are mixed in a 1:1 beam splitter, they produce output modes c and d . Inserting a photon in a produces a superposition state of the outputs: if the beam splitter is 50:50 then the probabilities of each output are equal, i.e. a ^ † | 0 ⟩ a → 1 2 ( c ^ † + d ^ † ) | 00 ⟩ c d {\displaystyle {\hat {a}}^{\dagger }|0\rangle _{a}\to {\frac {1}{\sqrt {2}}}\left({\hat {c}}^{\dagger }+{\hat {d}}^{\dagger }\right)|00\rangle _{cd}} , and similarly for inserting

5580-535: Is written as a Fock state , a tensor product of the states for each electromagnetic mode Fock state In quantum mechanics , a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta ). These states are named after the Soviet physicist Vladimir Fock . Fock states play an important role in the second quantization formulation of quantum mechanics. The particle representation

5704-406: The c ^ † 2 {\displaystyle {\hat {c}}^{\dagger 2}} and d ^ † 2 {\displaystyle {\hat {d}}^{\dagger 2}} terms. Therefore, when two identical photons enter a 1:1 beam splitter, they will always exit the beam splitter in the same (but random) output mode. The result

5828-537: The Fock states , so, for example | 0 ⟩ a {\displaystyle |0\rangle _{a}} corresponds to mode a empty (the vacuum state), and inserting one photon into a corresponds to | 1 ⟩ a = a ^ † | 0 ⟩ a {\displaystyle |1\rangle _{a}={\hat {a}}^{\dagger }|0\rangle _{a}} , etc. A photon in each input mode

5952-565: The Hamiltonian density function is given by The total Hamiltonian is given by In free Schrödinger theory, and and where a n {\displaystyle a_{n}} is the annihilation operator. Only for non-interacting particles do H {\displaystyle {\mathfrak {H}}} and a n {\displaystyle a_{n}} commute; in general they do not commute. For non-interacting particles, If they do not commute,

6076-499: The center of momentum frame , the colliding antiparticles have no net momentum, whereas a single photon always has momentum (determined by the photon's frequency or wavelength, which cannot be zero). Hence, conservation of momentum (or equivalently, translational invariance ) requires that at least two photons are created, with zero net momentum. The energy of the two photons, or, equivalently, their frequency, may be determined from conservation of four-momentum . Seen another way,

6200-463: The degeneracy of the state i {\displaystyle i} and that of j {\displaystyle j} , respectively, E i {\displaystyle E_{i}} and E j {\displaystyle E_{j}} their energies, k {\displaystyle k} the Boltzmann constant and T {\displaystyle T}

6324-423: The energy of a Maxwellian light wave were localized into point-like quanta that move independently of one another, even if the wave itself is spread continuously over space. In 1909 and 1916, Einstein showed that, if Planck's law regarding black-body radiation is accepted, the energy quanta must also carry momentum p = ⁠ h  / λ  ⁠ , making them full-fledged particles. This photon momentum

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6448-663: The i -th elementary state k i . The corresponding eigenvalue gives the number of particles in the state. This criterion nearly defines the Fock states (one must in addition select a phase factor). A given Fock state is denoted by | n k 1 , n k 2 , . . n k i . . . ⟩ {\displaystyle |n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},..n_{{\mathbf {k} }_{i}}...\rangle } . In this expression, n k i {\displaystyle n_{{\mathbf {k} }_{i}}} denotes

6572-483: The photoelectric effect , Einstein introduced the idea that light itself is made of discrete units of energy. In 1926, Gilbert N. Lewis popularized the term photon for these energy units. Subsequently, many other experiments validated Einstein's approach. In the Standard Model of particle physics , photons and other elementary particles are described as a necessary consequence of physical laws having

6696-454: The power spectrum of the single-photon wave packet and is therefore determined by the physical process of the source. Common shapes of the HOM dip are Gaussian and Lorentzian . A classical analogue to the HOM effect occurs when two coherent states (e.g. laser beams) interfere at the beamsplitter. If the states have a rapidly varying phase difference (i.e. faster than the integration time of

6820-571: The probability amplitude level. In addition, reflection from the bottom side of the beam splitter introduces a relative phase shift of π, corresponding to a factor of −1 in the associated term in the superposition. This sign is required by the reversibility (or unitarity of the quantum evolution) of the beam splitter. Since the two photons are identical, we cannot distinguish between the output states of possibilities 2 and 3, and their relative minus sign ensures that these two terms cancel. This cancelation can be interpreted as destructive interference of

6944-407: The probability amplitude of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy E = p c {\displaystyle E=pc} , and may have extra polarization states; depending on the gauge used, virtual photons may have three or four polarization states, instead of

7068-419: The "occupancy number basis". A quantum state in the Fock space is called a Fock state if it is an element of the occupancy number basis. A Fock state satisfies an important criterion: for each i , the state is an eigenstate of the particle number operator N k i ^ {\displaystyle {\widehat {N_{{\mathbf {k} }_{i}}}}} corresponding to

7192-415: The 1970s and 1980s by photon-correlation experiments. Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven. Photons obey the laws of quantum mechanics, and so their behavior has both wave-like and particle-like aspects. When a photon is detected by a measuring instrument, it is registered as a single, particulate unit. However, the probability of detecting

7316-536: The Hamiltonian will not have the above expression. Therefore, in general, Fock states are not energy eigenstates of a system. The vacuum state or | 0 ⟩ {\displaystyle |0\rangle } is the state of lowest energy and the expectation values of a {\displaystyle a} and a † {\displaystyle a^{\dagger }} vanish in this state: The electrical and magnetic fields and

7440-548: The Maxwell theory of light allows for all possible energies of electromagnetic radiation, most physicists assumed initially that the energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. In 1905, Einstein was the first to propose that energy quantization was a property of electromagnetic radiation itself. Although he accepted the validity of Maxwell's theory, Einstein pointed out that many anomalous experiments could be explained if

7564-470: The Nobel lectures of Wien , Planck and Millikan.) Instead, there was a widespread belief that energy quantization resulted from some unknown constraint on the matter that absorbed or emitted radiation. Attitudes changed over time. In part, the change can be traced to experiments such as those revealing Compton scattering , where it was much more difficult not to ascribe quantization to light itself to explain

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7688-487: The ability to register single photons as bright spots clearly distinguished from the low-noise background. In the figure above, the pairs of photons are registered in the middle of the Hong–Ou–Mandel dip. In most cases, they appear grouped in pairs either on the left or right side, corresponding to two output ports of a beam splitter. Occasionally a coincidence event occurs, manifesting a residual distinguishability between

7812-428: The action of measurement). When a photon enters a beam splitter, there are two possibilities: it will either be reflected or transmitted. The relative probabilities of transmission and reflection are determined by the reflectivity of the beam splitter. Here, we assume a 1:1 beam splitter, in which a photon has equal probability of being reflected and transmitted. Next, consider two photons, one in each input mode of

7936-1888: The anticommutation relation we have but, c k l . c k m † | n k 1 , n k 2 , . . . . n k m . . . n k l . . . ⟩ = − c k m † . c k l | n k 1 , n k 2 , . . . . n k m . . . n k l . . . ⟩ = − n k m + 1 n k l | n k 1 , n k 2 , . . . . n k m + 1... n k l − 1... ⟩ {\displaystyle {\begin{aligned}&c_{{\mathbf {k} }_{l}}.c_{{\mathbf {k} }_{m}}^{\dagger }|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle \\={}-&c_{{\mathbf {k} }_{m}}^{\dagger }.c_{{\mathbf {k} }_{l}}|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}...n_{{\mathbf {k} }_{l}}...\rangle \\={}-&{\sqrt {n_{{\mathbf {k} }_{m}}+1}}{\sqrt {n_{{\mathbf {k} }_{l}}}}|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},....n_{{\mathbf {k} }_{m}}+1...n_{{\mathbf {k} }_{l}}-1...\rangle \end{aligned}}} Thus, fermionic Fock states are antisymmetric under operation by particle exchange operators. In second quantization theory,

8060-757: The antisymmetric behaviour of fermions , for Fermionic Fock states we introduce non-Hermitian fermion creation and annihilation operators, defined for a Fermionic Fock state | ψ ⟩ = | n k 1 , n k 2 , n k 3 . . . n k l , . . . ⟩ {\displaystyle |\psi \rangle =|n_{{\mathbf {k} }_{1}},n_{{\mathbf {k} }_{2}},n_{{\mathbf {k} }_{3}}...n_{{\mathbf {k} }_{l}},...\rangle } as: These two actions are done antisymmetrically, which we shall discuss later. The anticommutation relations of creation and annihilation operators in

8184-438: The atoms can emit and absorb that radiation. Thermal equilibrium requires that the energy density ρ ( ν ) {\displaystyle \rho (\nu )} of photons with frequency ν {\displaystyle \nu } (which is proportional to their number density ) is, on average, constant in time; hence, the rate at which photons of any particular frequency are emitted must equal

8308-530: The average across many interactions between matter and radiation. However, refined Compton experiments showed that the conservation laws hold for individual interactions. Accordingly, Bohr and his co-workers gave their model "as honorable a funeral as possible". Nevertheless, the failures of the BKS model inspired Werner Heisenberg in his development of matrix mechanics . A few physicists persisted in developing semiclassical models in which electromagnetic radiation

8432-495: The basic entangling mechanism in linear optical quantum computing , and the two-photon quantum state | 2 , 0 ⟩ + | 0 , 2 ⟩ {\displaystyle |2,0\rangle +|0,2\rangle } that leads to the HOM dip is the simplest non-trivial state in a class called NOON states . In 2015 the Hong–Ou–Mandel effect for photons was directly observed with spatial resolution using an sCMOS camera with an image intensifier. Also in 2015

8556-402: The beam splitter with arbitrary reflection/transmission coefficients, and arbitrary numbers of input photons, see the general quantum mechanical treatment of a beamsplitter for the resulting output Fock state. Customarily the Hong–Ou–Mandel effect is observed using two photodetectors monitoring the output modes of the beam splitter. The coincidence rate of the detectors will drop to zero when

8680-486: The coefficients A i j {\displaystyle A_{ij}} , B j i {\displaystyle B_{ji}} and B i j {\displaystyle B_{ij}} once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis". Not long thereafter, in 1926, Paul Dirac derived the B i j {\displaystyle B_{ij}} rate constants by using

8804-542: The commutation relation we have, b k m † . b k l = b k l . b k m † {\displaystyle b_{\mathbf {k} _{m}}^{\dagger }.b_{\mathbf {k} _{l}}=b_{\mathbf {k} _{l}}.b_{\mathbf {k} _{m}}^{\dagger }} So, the Bosonic Fock state behaves to be symmetric under operation by Exchange operator. To be able to retain

8928-399: The commutator of the two creation operators c ^ † {\displaystyle {\hat {c}}^{\dagger }} and d ^ † {\displaystyle {\hat {d}}^{\dagger }} is zero because they operate on different spaces, the product term vanishes. The surviving terms in the superposition are only

9052-418: The concept in analyzing a thought experiment involving an electron and a high-energy photon . However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position–momentum uncertainty principle is due to Kennard , Pauli , and Weyl . The uncertainty principle applies to situations where an experimenter has

9176-768: The creation and annihilation operators ensure that the bosonic Fock states have the appropriate symmetric behaviour under particle exchange. Here, exchange of particles between two states (say, l and m ) is done by annihilating a particle in state l and creating one in state m . If we start with a Fock state | ψ ⟩ = | n k 1 , n k 2 , . . . . n k m . . . n k l . . . ⟩ {\displaystyle |\psi \rangle =\left|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},....n_{\mathbf {k} _{m}}...n_{\mathbf {k} _{l}}...\right\rangle } , and want to shift

9300-810: The definition of Fock state ensures that the variance of measurement Var ⁡ ( N ^ ) = 0 {\displaystyle \operatorname {Var} \left({\widehat {N}}\right)=0} , i.e., measuring the number of particles in a Fock state always returns a definite value with no fluctuation. For any final state | f ⟩ {\displaystyle |f\rangle } , any Fock state of two identical particles given by | 1 k 1 , 1 k 2 ⟩ {\displaystyle |1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\rangle } , and any operator O ^ {\displaystyle {\widehat {\mathbb {O} }}} , we have

9424-523: The detectors) then a dip will be observed in the coincidence rate equal to one half the average coincidence count at long delays. (Nevertheless, it can be further reduced with a proper discriminating trigger level applied to the signal.) Consequently, to prove that destructive interference is two-photon quantum interference rather than a classical effect, the HOM dip must be lower than one half. The Hong–Ou–Mandel effect can be directly observed using single-photon-sensitive intensified cameras. Such cameras have

9548-466: The direction of the photon's propagation, the magnitude of its momentum is The photon also carries spin angular momentum , which is related to photon polarization . (Beams of light also exhibit properties described as orbital angular momentum of light ). The angular momentum of the photon has two possible values, either +ħ or −ħ . These two possible values correspond to the two possible pure states of circular polarization . Collections of photons in

9672-400: The effect was observed with helium-4 atoms. The HOM effect can be used to measure the biphoton wave function from a spontaneous four-wave mixing process. In 2016 a frequency converter for photons demonstrated the Hong–Ou–Mandel effect with different-color photons. In 2018, HOM interference was used to demonstrate high-fidelity quantum interference between topologically protected states on

9796-484: The electric field of an atomic nucleus. The classical formulae for the energy and momentum of electromagnetic radiation can be re-expressed in terms of photon events. For example, the pressure of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change in momentum per unit time. Current commonly accepted physical theories imply or assume

9920-450: The electromagnetic field. Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the direction of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in

10044-487: The electromagnetic wave, Δ N {\displaystyle \Delta N} , and the uncertainty in the phase of the wave, Δ ϕ {\displaystyle \Delta \phi } . However, this cannot be an uncertainty relation of the Kennard–Pauli–Weyl type, since unlike position and momentum, the phase ϕ {\displaystyle \phi } cannot be represented by

10168-441: The energy of any system that absorbs or emits electromagnetic radiation of frequency ν is an integer multiple of an energy quantum E = hν . As shown by Albert Einstein , some form of energy quantization must be assumed to account for the thermal equilibrium observed between matter and electromagnetic radiation ; for this explanation of the photoelectric effect, Einstein received the 1921 Nobel Prize in physics. Since

10292-501: The final blow to particle models of light. The Maxwell wave theory , however, does not account for all properties of light. The Maxwell theory predicts that the energy of a light wave depends only on its intensity , not on its frequency ; nevertheless, several independent types of experiments show that the energy imparted by light to atoms depends only on the light's frequency, not on its intensity. For example, some chemical reactions are provoked only by light of frequency higher than

10416-411: The first two decades of the 20th century with the work of Albert Einstein , who built upon the research of Max Planck . While Planck was trying to explain how matter and electromagnetic radiation could be in thermal equilibrium with one another, he proposed that the energy stored within a material object should be regarded as composed of an integer number of discrete, equal-sized parts. To explain

10540-1123: The following condition for indistinguishability : So, we must have ⟨ f | O ^ | 1 k 1 , 1 k 2 ⟩ = e i δ ⟨ f | O ^ | 1 k 2 , 1 k 1 ⟩ {\displaystyle \left\langle f\left|{\widehat {\mathbb {O} }}\right|1_{\mathbf {k} _{1}},1_{\mathbf {k} _{2}}\right\rangle =e^{i\delta }\left\langle f\left|{\widehat {\mathbb {O} }}\right|1_{\mathbf {k} _{2}},1_{\mathbf {k} _{1}}\right\rangle } where e i δ = + 1 {\displaystyle e^{i\delta }=+1} for bosons and − 1 {\displaystyle -1} for fermions . Since ⟨ f | {\displaystyle \langle f|} and O ^ {\displaystyle {\widehat {\mathbb {O} }}} are arbitrary, we can say, Note that

10664-418: The galactic vector potential . Although the galactic vector potential is large because the galactic magnetic field exists on great length scales, only the magnetic field would be observable if the photon is massless. In the case that the photon has mass, the mass term ⁠ 1 / 2 ⁠ m A μ A would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on

10788-661: The galactic vector potential have been shown to be model-dependent. If the photon mass is generated via the Higgs mechanism then the upper limit of m ≲ 10  eV/ c from the test of Coulomb's law is valid. In most theories up to the eighteenth century, light was pictured as being made of particles. Since particle models cannot easily account for the refraction , diffraction and birefringence of light, wave theories of light were proposed by René Descartes (1637), Robert Hooke (1665), and Christiaan Huygens (1678); however, particle models remained dominant, chiefly due to

10912-459: The identical input photons overlap perfectly in time. This is called the Hong–Ou–Mandel dip , or HOM dip. The coincidence count reaches a minimum, indicated by the dotted line. The minimum drops to zero when the two photons are perfectly identical in all properties. When the two photons are perfectly distinguishable, the dip completely disappears. The precise shape of the dip is directly related to

11036-404: The influence of Isaac Newton . In the early 19th century, Thomas Young and August Fresnel clearly demonstrated the interference and diffraction of light, and by 1850 wave models were generally accepted. James Clerk Maxwell 's 1865 prediction that light was an electromagnetic wave – which was confirmed experimentally in 1888 by Heinrich Hertz 's detection of radio waves – seemed to be

11160-409: The integrality of the particles' spin , the tensor products must be alternating (anti-symmetric) or symmetric products of the underlying one-particle Hilbert space . Specifically: If the number of particles is variable, one constructs the Fock space as the direct sum of the tensor product Hilbert spaces for each particle number . In the Fock space, it is possible to specify the same state in

11284-455: The light particle determined which of the two paths a single photon would take. Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation from quantum mechanics. Ironically, Max Born 's probabilistic interpretation of the wave function was inspired by Einstein's later work searching for a more complete theory. In 1910, Peter Debye derived Planck's law of black-body radiation from

11408-621: The light should go to one of the outputs (the one with the positive phase). Consider two optical input modes a and b that carry annihilation and creation operators a ^ {\displaystyle {\hat {a}}} , a ^ † {\displaystyle {\hat {a}}^{\dagger }} , and b ^ {\displaystyle {\hat {b}}} , b ^ † {\displaystyle {\hat {b}}^{\dagger }} . Identical photons in different modes can be described by

11532-429: The most convenient basis of a Fock space. Elements of a Fock space that are superpositions of states of differing particle number (and thus not eigenstates of the number operator) are not Fock states. For this reason, not all elements of a Fock space are referred to as "Fock states". If we define the aggregate particle number operator N ^ {\textstyle {\widehat {N}}} as

11656-418: The number of atoms in state i {\displaystyle i} and those in state j {\displaystyle j} must, on average, be constant; hence, the rates R j i {\displaystyle R_{ji}} and R i j {\displaystyle R_{ij}} must be equal. Also, by arguments analogous to the derivation of Boltzmann statistics ,

11780-497: The number of particles in the i-th state k i , and the particle number operator for the i-th state, N k i ^ {\displaystyle {\widehat {N_{{\mathbf {k} }_{i}}}}} , acts on the Fock state in the following way: Hence the Fock state is an eigenstate of the number operator with eigenvalue n k i {\displaystyle n_{{\mathbf {k} }_{i}}} . Fock states often form

11904-722: The number operator as well as the creation and annihilation operators might seem same as the bosonic ones, but the real twist comes from the maximum occupation number of each state in the fermionic Fock state. Extending the 2-particle fermionic example above, we first must convince ourselves that a fermionic Fock state | ψ ⟩ = | n k 1 , n k 2 , n k 3 . . . n k l . . . ⟩ {\displaystyle |\psi \rangle =\left|n_{\mathbf {k} _{1}},n_{\mathbf {k} _{2}},n_{\mathbf {k} _{3}}...n_{\mathbf {k} _{l}}...\right\rangle }

12028-423: The number operator does not distinguish bosons from fermions; indeed, it just counts particles without regard to their symmetry type. To perceive any difference between them, we need other operators, namely the creation and annihilation operators . Bosons , which are particles with integer spin, follow a simple rule: their composite eigenstate is symmetric under operation by an exchange operator . For example, in

12152-509: The numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A i j {\displaystyle A_{ij}} and B i j {\displaystyle B_{ij}} coefficients from first principles, and showed that the Bose–Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in

12276-476: The observed results. Even after Compton's experiment, Niels Bohr , Hendrik Kramers and John Slater made one last attempt to preserve the Maxwellian continuous electromagnetic field model of light, the so-called BKS theory . An important feature of the BKS theory is how it treated the conservation of energy and the conservation of momentum . In the BKS theory, energy and momentum are only conserved on

12400-441: The occupation number of any single state is either 0 or 1. The eigenvalue associated to the fermionic Fock state N k l ^ {\displaystyle {\widehat {N_{{\mathbf {k} }_{l}}}}} must be either 0 or 1. Antisymmetric behaviour of Fermionic states under Exchange operator is taken care of the anticommutation relations. Here, exchange of particles between two states

12524-472: The opposite direction; he derived Planck's law of black-body radiation by assuming B–E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose–Einstein statistics. Dirac's second-order perturbation theory can involve virtual photons , transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories ,

12648-405: The overall uncertainty as far as quantum mechanics allows. Quantum optics makes use of coherent states for modes of the electromagnetic field. There is a tradeoff, reminiscent of the position–momentum uncertainty relation, between measurements of an electromagnetic wave's amplitude and its phase. This is sometimes informally expressed in terms of the uncertainty in the number of photons present in

12772-400: The photon can be considered as its own antiparticle (thus an "antiphoton" is simply a normal photon with opposite momentum, equal polarization, and 180° out of phase). The reverse process, pair production , is the dominant mechanism by which high-energy photons such as gamma rays lose energy while passing through matter. That process is the reverse of "annihilation to one photon" allowed in

12896-472: The photon mass of m < 3 × 10  eV/ c . The galactic vector potential can also be probed directly by measuring the torque exerted on a magnetized ring. Such methods were used to obtain the sharper upper limit of 1.07 × 10  eV/ c (the equivalent of 10   daltons ) given by the Particle Data Group . These sharp limits from the non-observation of the effects caused by

13020-467: The photon probably derives from gamma rays , which were discovered in 1900 by Paul Villard , named by Ernest Rutherford in 1903, and shown to be a form of electromagnetic radiation in 1914 by Rutherford and Edward Andrade . In chemistry and optical engineering , photons are usually symbolized by hν , which is the photon energy , where h is the Planck constant and the Greek letter ν ( nu )

13144-473: The photon to be strictly massless. If photons were not purely massless, their speeds would vary with frequency, with lower-energy (redder) photons moving slightly slower than higher-energy photons. Relativity would be unaffected by this; the so-called speed of light, c , would then not be the actual speed at which light moves, but a constant of nature which is the upper bound on speed that any object could theoretically attain in spacetime. Thus, it would still be

13268-459: The photons. The Hong–Ou–Mandel effect can be used to test the degree of indistinguishability of the two incoming photons. When the HOM dip reaches all the way down to zero coincident counts, the incoming photons are perfectly indistinguishable, whereas if there is no dip, the photons are distinguishable. In 2002, the Hong–Ou–Mandel effect was used to demonstrate the purity of a solid-state single-photon source by feeding two successive photons from

13392-465: The probability of them each going to a different detector will increase. In this way, the interferometer coincidence signal can accurately measure bandwidth, path lengths, and timing. Since this effect relies on the existence of photons and the second quantization it can not be fully explained by classical optics . The effect provides one of the underlying physical mechanisms for logic gates in linear optical quantum computing (the other mechanism being

13516-444: The rate at which they are absorbed . Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate R j i {\displaystyle R_{ji}} for a system to absorb a photon of frequency ν {\displaystyle \nu } and transition from a lower energy E j {\displaystyle E_{j}} to

13640-524: The ratio of N i {\displaystyle N_{i}} and N j {\displaystyle N_{j}} is g i / g j exp ⁡ ( E j − E i ) / ( k T ) , {\displaystyle g_{i}/g_{j}\exp {(E_{j}-E_{i})/(kT)},} where g i {\displaystyle g_{i}} and g j {\displaystyle g_{j}} are

13764-405: The requirement for a symmetric quantum mechanical state . This work led to the concept of coherent states and the development of the laser. In the same papers, Einstein extended Bose's formalism to material particles (bosons) and predicted that they would condense into their lowest quantum state at low enough temperatures; this Bose–Einstein condensation was observed experimentally in 1995. It

13888-433: The source into a 1:1 beam splitter. The interference visibility V of the dip is related to the states of the two photons ρ a {\displaystyle \rho _{a}} and ρ b {\displaystyle \rho _{b}} as If ρ a = ρ b = ρ {\displaystyle \rho _{a}=\rho _{b}=\rho } , then

14012-465: The speed of light. If Coulomb's law is not exactly valid, then that would allow the presence of an electric field to exist within a hollow conductor when it is subjected to an external electric field. This provides a means for precision tests of Coulomb's law . A null result of such an experiment has set a limit of m ≲ 10  eV/ c . Sharper upper limits on the mass of light have been obtained in experiments designed to detect effects caused by

14136-412: The speed of spacetime ripples ( gravitational waves and gravitons ), but it would not be the speed of photons. If a photon did have non-zero mass, there would be other effects as well. Coulomb's law would be modified and the electromagnetic field would have an extra physical degree of freedom . These effects yield more sensitive experimental probes of the photon mass than the frequency dependence of

14260-501: The summation as well; for example, two photons may interact indirectly through virtual electron – positron pairs . Such photon–photon scattering (see two-photon physics ), as well as electron–photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the International Linear Collider . In modern physics notation, the quantum state of the electromagnetic field

14384-587: The system's temperature . From this, it is readily derived that g i B i j = g j B j i {\displaystyle g_{i}B_{ij}=g_{j}B_{ji}} and The A i j {\displaystyle A_{ij}} and B i j {\displaystyle B_{ij}} are collectively known as the Einstein coefficients . Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate

14508-578: The term in a letter to Nature on 18 December 1926. The same name was used earlier but was never widely adopted before Lewis: in 1916 by the American physicist and psychologist Leonard T. Troland , in 1921 by the Irish physicist John Joly , in 1924 by the French physiologist René Wurmser (1890–1993), and in 1926 by the French physicist Frithiof Wolfers (1891–1971). The name was suggested initially as

14632-406: The transmission/transmission and reflection/reflection possibilities. If a detector is set up on each of the outputs then coincidences can never be observed, while both photons can appear together in either one of the two detectors with equal probability. A classical prediction of the intensities of the output beams for the same beam splitter and identical coherent input beams would suggest that all of

14756-423: The two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can give apparently infinite contributions to the sum. Such unphysical results are corrected for using the technique of renormalization . Other virtual particles may contribute to

14880-458: The visibility is equal to the purity P = Tr ⁡ ( ρ 2 ) {\displaystyle P=\operatorname {Tr} (\rho ^{2})} of the photons. In 2006, an experiment was performed in which two atoms independently emitted a single photon each. These photons subsequently produced the Hong–Ou–Mandel effect. Multimode Hong–Ou–Mandel interference was studied in 2003. The Hong–Ou–Mandel effect also underlies

15004-434: Was first treated in detail by Paul Dirac for bosons and by Pascual Jordan and Eugene Wigner for fermions . The Fock states of bosons and fermions obey useful relations with respect to the Fock space creation and annihilation operators . One specifies a multiparticle state of N non-interacting identical particles by writing the state as a sum of tensor products of N one-particle states. Additionally, depending on

15128-441: Was later used by Lene Hau to slow, and then completely stop, light in 1999 and 2001. The modern view on this is that photons are, by virtue of their integer spin, bosons (as opposed to fermions with half-integer spin). By the spin-statistics theorem , all bosons obey Bose–Einstein statistics (whereas all fermions obey Fermi–Dirac statistics ). In 1916, Albert Einstein showed that Planck's radiation law could be derived from

15252-499: Was observed experimentally by Arthur Compton , for which he received the Nobel Prize in 1927. The pivotal question then, was how to unify Maxwell's wave theory of light with its experimentally observed particle nature. The answer to this question occupied Albert Einstein for the rest of his life, and was solved in quantum electrodynamics and its successor, the Standard Model . (See § Quantum field theory and § As

15376-537: Was used before 1900 to mean particles or amounts of different quantities , including electricity . In 1900, the German physicist Max Planck was studying black-body radiation , and he suggested that the experimental observations, specifically at shorter wavelengths , would be explained if the energy stored within a molecule was a "discrete quantity composed of an integral number of finite equal parts", which he called "energy elements". In 1905, Albert Einstein published

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