Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
The description of photon polarization contains many of the physical concepts and much of the mathematical machinery of more involved quantum descriptions, such as the quantum mechanics of an electron in a potential well. Polarization is an example of a qubit degree of freedom, which forms a fundamental basis for an understanding of more complicated quantum phenomena. Much of the mathematical machinery of quantum mechanics, such as state vectors, probability amplitudes, unitary operators, and Hermitian operators, emerge naturally from the classical Maxwell's equations in the description. The quantum polarization state vector for the photon, for instance, is identical with the Jones vector, usually used to describe the polarization of a classical wave. Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave. Hermitian operators then follow for infinitesimal transformations of a classical polarization state.
Many of the implications of the mathematical machinery are easily verified experimentally. In fact, many of the experiments can be performed with polaroid sunglass lenses.
The connection with quantum mechanics is made through the identification of a minimum packet size, called a photon, for energy in the electromagnetic field. The identification is based on the theories of Planck and the interpretation of those theories by Einstein. The correspondence principle then allows the identification of momentum and angular momentum (called spin), as well as energy, with the photon.
This section duplicates the scope of other articles, specifically Sinusoidal plane-wave solutions of the electromagnetic wave equation.(July 2014) |
The wave is linearly polarized (or plane polarized) when the phase angles are equal,
This represents a wave with phase polarized at an angle with respect to the x axis. In this case the Jones vector can be written with a single phase:
The state vectors for linear polarization in x or y are special cases of this state vector.
If unit vectors are defined such that and then the linearly polarized polarization state can be written in the "x–y basis" as
If the phase angles and differ by exactly and the x amplitude equals the y amplitude the wave is circularly polarized. The Jones vector then becomes where the plus sign indicates left circular polarization and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x–y plane.
If unit vectors are defined such that and then an arbitrary polarization state can be written in the "R–L basis" as where and
We can see that
The general case in which the electric field rotates in the x–y plane and has variable magnitude is called elliptical polarization. The state vector is given by
To get an understanding of what a polarization state looks like, one can observe the orbit that is made if the polarization state is multiplied by a phase factor of and then having the real parts of its components interpreted as x and y coordinates respectively. That is:
If only the traced out shape and the direction of the rotation of (x(t), y(t)) is considered when interpreting the polarization state, i.e. only (where x(t) and y(t) are defined as above) and whether it is overall more right circularly or left circularly polarized (i.e. whether |ψR| > |ψL| or vice versa), it can be seen that the physical interpretation will be the same even if the state is multiplied by an arbitrary phase factor, since and the direction of rotation will remain the same. In other words, there is no physical difference between two polarization states and , between which only a phase factor differs.
It can be seen that for a linearly polarized state, M will be a line in the xy plane, with length 2 and its middle in the origin, and whose slope equals to tan(θ). For a circularly polarized state, M will be a circle with radius 1/√2 and with the middle in the origin.
The energy per unit volume in classical electromagnetic fields is (cgs units) and also Planck units:
For a plane wave, this becomes: where the energy has been averaged over a wavelength of the wave.
The fraction of energy in the x component of the plane wave is with a similar expression for the y component resulting in .
The fraction in both components is
The momentum density is given by the Poynting vector
For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:
The momentum density has been averaged over a wavelength.
Electromagnetic waves can have both orbital and spin angular momentum. [1] The total angular momentum density is
For a sinusoidal plane wave propagating along axis the orbital angular momentum density vanishes. The spin angular momentum density is in the direction and is given by where again the density is averaged over a wavelength.
A linear filter transmits one component of a plane wave and absorbs the perpendicular component. In that case, if the filter is polarized in the x direction, the fraction of energy passing through the filter is
An ideal birefringent crystal transforms the polarization state of an electromagnetic wave without loss of wave energy. Birefringent crystals therefore provide an ideal test bed for examining the conservative transformation of polarization states. Even though this treatment is still purely classical, standard quantum tools such as unitary and Hermitian operators that evolve the state in time naturally emerge.
A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light polarized perpendicular to the axis are called "ordinary rays" or "ordinary photons". If a linearly polarized wave impinges on the crystal, the extraordinary component of the wave will emerge from the crystal with a different phase than the ordinary component. In mathematical language, if the incident wave is linearly polarized at an angle with respect to the optic axis, the incident state vector can be written and the state vector for the emerging wave can be written
While the initial state was linearly polarized, the final state is elliptically polarized. The birefringent crystal alters the character of the polarization.
The initial polarization state is transformed into the final state with the operator U. The dual of the final state is given by where is the adjoint of U, the complex conjugate transpose of the matrix.
The fraction of energy that emerges from the crystal is
In this ideal case, all the energy impinging on the crystal emerges from the crystal. An operator U with the property that where I is the identity operator and U is called a unitary operator. The unitary property is necessary to ensure energy conservation in state transformations.
If the crystal is very thin, the final state will be only slightly different from the initial state. The unitary operator will be close to the identity operator. We can define the operator H by and the adjoint by
Energy conservation then requires
This requires that
Operators like this that are equal to their adjoints are called Hermitian or self-adjoint.
The infinitesimal transition of the polarization state is
Thus, energy conservation requires that infinitesimal transformations of a polarization state occur through the action of a Hermitian operator.
The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinterpretation is based on the theories of Max Planck and the interpretation by Albert Einstein of those theories and of other experiments.[ citation needed ]
Einstein's conclusion from early experiments on the photoelectric effect is that electromagnetic radiation is composed of irreducible packets of energy, known as photons. The energy of each packet is related to the angular frequency of the wave by the relation where is an experimentally determined quantity known as the reduced Planck constant. If there are photons in a box of volume , the energy in the electromagnetic field is and the energy density is
The photon energy can be related to classical fields through the correspondence principle that states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large , the quantum energy density must be the same as the classical energy density
The number of photons in the box is then
The correspondence principle also determines the momentum and angular momentum of the photon. For momentum where is the wave number. This implies that the momentum of a photon is
Similarly for the spin angular momentum where is field strength. This implies that the spin angular momentum of the photon is the quantum interpretation of this expression is that the photon has a probability of of having a spin angular momentum of and a probability of of having a spin angular momentum of . We can therefore think of the spin angular momentum of the photon being quantized as well as the energy. The angular momentum of classical light has been verified. [2] A photon that is linearly polarized (plane polarized) is in a superposition of equal amounts of the left-handed and right-handed states.
The spin of the photon is defined as the coefficient of in the spin angular momentum calculation. A photon has spin 1 if it is in the state and −1 if it is in the state. The spin operator is defined as the outer product
The eigenvectors of the spin operator are and with eigenvalues 1 and −1, respectively.
The expected value of a spin measurement on a photon is then
An operator S has been associated with an observable quantity, the spin angular momentum. The eigenvalues of the operator are the allowed observable values. This has been demonstrated for spin angular momentum, but it is in general true for any observable quantity.
We can write the circularly polarized states as where s = 1 for and s = −1 for . An arbitrary state can be written where and are phase angles, θ is the angle by which the frame of reference is rotated, and
When the state is written in spin notation, the spin operator can be written
The eigenvectors of the differential spin operator are
To see this note
The spin angular momentum operator is
There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the double-slit experiment:
Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.
— Paul Dirac, The Principles of Quantum Mechanics, 1930, Chapter 1
The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The polarization state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of polarization. In quantum mechanics, therefore, the state or probability amplitude contains the basic probability information. In general, the rules for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1][ clarification needed ]
- The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. For example, the amplitude for the x polarized photon to be right circularly polarized and for the right circularly polarized photon to pass through the y-polaroid is the product of the individual amplitudes.
- The amplitude for a process that can take place in one of several indistinguishable ways is the sum of amplitudes for each of the individual ways. For example, the total amplitude for the x polarized photon to pass through the y-polaroid is the sum of the amplitudes for it to pass as a right circularly polarized photon, plus the amplitude for it to pass as a left circularly polarized photon,
- The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.
For any legal[ clarification needed ] operators the following inequality, a consequence of the Cauchy–Schwarz inequality, is true.
If B A ψ and A B ψ are defined, then by subtracting the means and re-inserting in the above formula, we deduce where is the operator mean of observable X in the system state ψ and
Here is called the commutator of A and B.
This is a purely mathematical result. No reference has been made to any physical quantity or principle. It simply states that the uncertainty of one operator times the uncertainty of another operator has a lower bound.
The connection to physics can be made if we identify the operators with physical operators such as the angular momentum and the polarization angle. We have then which means that angular momentum and the polarization angle cannot be measured simultaneously with infinite accuracy. (The polarization angle can be measured by checking whether the photon can pass through a polarizing filter oriented at a particular angle, or a polarizing beam splitter. This results in a yes/no answer that, if the photon was plane-polarized at some other angle, depends on the difference between the two angles.)
Much of the mathematical apparatus of quantum mechanics appears in the classical description of a polarized sinusoidal electromagnetic wave. The Jones vector for a classical wave, for instance, is identical with the quantum polarization state vector for a photon. The right and left circular components of the Jones vector can be interpreted as probability amplitudes of spin states of the photon. Energy conservation requires that the states be transformed with a unitary operation. This implies that infinitesimal transformations are transformed with a Hermitian operator. These conclusions are a natural consequence of the structure of Maxwell's equations for classical waves.
Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather than continuous. The allowed observable values are determined by the eigenvalues of the operators associated with the observable. In the case angular momentum, for instance, the allowed observable values are the eigenvalues of the spin operator.
These concepts have emerged naturally from Maxwell's equations and Planck's and Einstein's theories. They have been found to be true for many other physical systems. In fact, the typical program is to assume the concepts of this section and then to infer the unknown dynamics of a physical system. This was done, for instance, with the dynamics of electrons. In that case, working back from the principles in this section, the quantum dynamics of particles were inferred, leading to Schrödinger's equation, a departure from Newtonian mechanics. The solution of this equation for atoms led to the explanation of the Balmer series for atomic spectra and consequently formed a basis for all of atomic physics and chemistry.
This is not the only occasion[ dubious – discuss ] in which Maxwell's equations have forced a restructuring of Newtonian mechanics. Maxwell's equations are relativistically consistent. Special relativity resulted from attempts to make classical mechanics consistent with Maxwell's equations (see, for example, Moving magnet and conductor problem ).
In optics, polarized light can be described using the Jones calculus, invented by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave.
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.
In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. The term linear polarization was coined by Augustin-Jean Fresnel in 1822. See polarization and plane of polarization for more information.
In quantum mechanics, a density matrix is a matrix that describes an ensemble of physical systems as quantum states. It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles. Mixed ensembles arise in quantum mechanics in two different situations:
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density.
In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements.
In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an optical driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.
In physics, the S-matrix or scattering matrix is a matrix which relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the system. They can be determined from directly observable phenomena. The original Stokes paper was discovered independently by Francis Perrin in 1942 and by Subrahamanyan Chandrasekhar in 1947, who named it as the Stokes parameters.
Sinusoidal plane-wave solutions are particular solutions to the wave equation.
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos and quantum chaos. It describes a free rotating stick in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian
In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism.
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.