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Squeezing Operator
In quantum physics, the squeeze operator for a single mode of the electromagnetic field is :\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^ where the operators inside the exponential are the ladder operators. It is a unitary operator and therefore obeys S(z)\,S^\dagger (z) = S^\dagger (z)\,S(z) = \hat 1, where \hat 1 is the identity operator. Its action on the annihilation and creation operators produces :\hat^(z) \, \hat \, \hat(z) = \hat\cosh r - e^ \hat^ \sinh r \qquad\text\qquad \hat^(z) \, \hat^ \, \hat(z) = \hat^\cosh r - e^ \hat \sinh r The squeeze operator is ubiquitous in quantum optics and can operate on any state. For example, when acting upon the vacuum, the squeezing operator produces the squeezed vacuum state. The squeezing operator can also act on coherent states In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (mathematics)
In mathematics, an operator is generally a Map (mathematics), mapping or function (mathematics), function that acts on elements of a space (mathematics), space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the domain of a function, domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from \mathbb^n to \mathbb^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladder Operators
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising and lowering operators are commonly known as the creation and annihilation operators, respectively. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Terminology There is a relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory which lies in representation theory. The creation operator ''a''''i''† increments the number of particles in state ''i'', while the corresponding annihilation operator ''ai'' decrements the number of particles in state ''i''. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Optics
Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction with, for instance, atoms and molecules. Photons have been used to test many of the counter-intuitive predictions of quantum mechanics, such as entanglement and teleportation, and are a useful resource for quantum information processing. History Light propagating in a restricted volume of space has its energy and momentum quantized according to an integer number of particles known as photons. Quantum optics studies the nature and effects of light as quantized photons. The first major development leading to that understanding was the correct modeling of the blackbody radiation spectrum by Max Planck in 1899 under the hypothesis of light being emitted in discrete units of energy. The photoelectric effect was further evidence of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent States
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squeezed Coherent State
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) electric field in the amplitude X (phase 0) and in the mode Y (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle: :\Delta x \Delta p \geq \frac2\; and \;\Delta X \Delta Y \geq \frac4 , respectively. Trivial examples, which are in fact not squeezed, are the ground state , 0\rangle of the quantum harmonic oscillator and the family of coherent states , \alpha\rangle. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with \Delta x_g = \Delta p_g in "natural oscillator units" and \Delta X_g = \Delta Y_g = 1/2. The term squeezed state is actually used for states with a standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Operator
In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics, :\hat(\alpha)=\exp \left ( \alpha \hat^\dagger - \alpha^\ast \hat \right ) , where \alpha is the amount of displacement in optical phase space, \alpha^* is the complex conjugate of that displacement, and \hat and \hat^\dagger are the lowering and raising operators, respectively. The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude \alpha. It may also act on the vacuum state by displacing it into a coherent state. Specifically, \hat(\alpha), 0\rangle=, \alpha\rangle where , \alpha\rangle is a coherent state, which is an eigenstate of the annihilation (lowering) operator. This operator was introduced independently by Richard Feynman and Roy J. Glauber in 1951. Properties The displacement operator is a unitary operator, and therefore obeys \hat(\alpha)\hat^\dagger(\alpha)=\hat^\dagger( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squeezed Coherent State
In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x and momentum p of a particle, and the (dimension-less) electric field in the amplitude X (phase 0) and in the mode Y (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle: :\Delta x \Delta p \geq \frac2\; and \;\Delta X \Delta Y \geq \frac4 , respectively. Trivial examples, which are in fact not squeezed, are the ground state , 0\rangle of the quantum harmonic oscillator and the family of coherent states , \alpha\rangle. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with \Delta x_g = \Delta p_g in "natural oscillator units" and \Delta X_g = \Delta Y_g = 1/2. The term squeezed state is actually used for states with a standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker–Campbell–Hausdorff Formula
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X and Y and iterated commutators thereof. The first few terms of this series are: Z = X + Y + \frac ,Y+ \frac ,[X,Y + \frac [Y,[Y,X + \cdots\,, where "\cdots" indicates terms involving higher Commutator#Identities_(ring_theory)">commutators of X and Y. If X and Y are sufficiently small elements of the Lie algebra \mathfrak g of a Lie group G, the series is convergent. Meanwhile, every element g sufficiently close to the identity in G can be expressed as g = e^X for a small X in \mathfrak g. Thus, we can say that ''near the identity'' the group multiplication in G—written as e^X e^Y = e^Z—can be expressed in purely Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
