1) The document discusses minimum uncertainty coherent states for a nondegenerate parametric amplifier.
2) A similarity transformation is used to map the Hamiltonian for the nondegenerate parametric amplifier to that of a two-dimensional harmonic oscillator.
3) This mapping allows the authors to obtain exact analytical solutions for the eigenfunctions and eigenvalues of the parametric amplifier Hamiltonian by using the known solutions for the two-dimensional harmonic oscillator.
The document summarizes the Buckingham π theorem, which states that any physically meaningful equation relating physical variables can be rewritten in terms of dimensionless parameters constructed from the original variables. The theorem provides a method to determine these dimensionless parameters even if the exact form of the equation is unknown. It allows identifying equivalent systems that can be compared experimentally based on having the same set of dimensionless parameters. The theorem is proved using concepts from linear algebra by representing physical dimensions as a vector space and finding the dimensionless parameters as the null space of a dimensional matrix constructed from the variables. Two examples are provided to illustrate applying the theorem.
Dimensional analysis is a mathematical technique used to establish relationships between physical quantities in fluid phenomena. It involves considering the dimensions of quantities and grouping dimensionless parameters to better understand flows. Dimensional analysis provides guidance for experimental work by indicating important influencing factors. It is applied to develop equations, convert between units, reduce experimental variables, and enable model studies through similitude. The key concepts are that theoretical equations must be dimensionally homogeneous and empirical equations have limited applications. Dimensional analysis methods include Rayleigh's method of exponential relationships and Buckingham's Π-method of grouping variables into dimensionless terms.
Second application of dimensional analysis
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The document discusses dimensional analysis, similitude, and model analysis. It provides background on how dimensional analysis and model testing are used to study fluid mechanics problems. Dimensional analysis uses the dimensions of physical quantities to determine which parameters influence a phenomenon. Model testing in a laboratory allows measurements to be applied to larger scale systems using similitude. Buckingham's π-theorem is introduced as a way to non-dimensionalize variables when there are more variables than fundamental dimensions. Rayleigh's and Buckingham's methods are demonstrated on an example of determining the resisting force on an aircraft.
This document discusses dimensional analysis, which is a mathematical technique used in fluid mechanics to reduce the number of variables in a problem by combining dimensional variables to form non-dimensional parameters. Dimensional analysis allows problems to be expressed in terms of non-dimensional parameters to show the relative significance of each parameter. It has various uses including checking dimensional homogeneity of equations, deriving equations, planning experiments, and analyzing complex flows using scale models. The Buckingham π theorem states that any relationship between physical quantities can be written as a relationship between dimensionless pi groups formed from the variables. Dimensional analysis is applied by setting up a dimensional matrix to determine the minimum number of pi groups needed to describe the relationship.
The document discusses dimensional analysis, which is a technique used to express physical quantities in terms of base quantities. It defines basic and derived quantities, and lists common base quantities like length, mass, and time. The document also shows how to use dimensional analysis to determine the units and dimensional consistency of equations. Examples are provided to illustrate determining units, checking if equations are dimensionally correct, and deriving relationships between physical quantities.
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
The document summarizes the Buckingham π theorem, which states that any physically meaningful equation relating physical variables can be rewritten in terms of dimensionless parameters constructed from the original variables. The theorem provides a method to determine these dimensionless parameters even if the exact form of the equation is unknown. It allows identifying equivalent systems that can be compared experimentally based on having the same set of dimensionless parameters. The theorem is proved using concepts from linear algebra by representing physical dimensions as a vector space and finding the dimensionless parameters as the null space of a dimensional matrix constructed from the variables. Two examples are provided to illustrate applying the theorem.
Dimensional analysis is a mathematical technique used to establish relationships between physical quantities in fluid phenomena. It involves considering the dimensions of quantities and grouping dimensionless parameters to better understand flows. Dimensional analysis provides guidance for experimental work by indicating important influencing factors. It is applied to develop equations, convert between units, reduce experimental variables, and enable model studies through similitude. The key concepts are that theoretical equations must be dimensionally homogeneous and empirical equations have limited applications. Dimensional analysis methods include Rayleigh's method of exponential relationships and Buckingham's Π-method of grouping variables into dimensionless terms.
Second application of dimensional analysis
If you liked it don't forget to follow me-
Instagram-yadavgaurav251
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The document discusses dimensional analysis, similitude, and model analysis. It provides background on how dimensional analysis and model testing are used to study fluid mechanics problems. Dimensional analysis uses the dimensions of physical quantities to determine which parameters influence a phenomenon. Model testing in a laboratory allows measurements to be applied to larger scale systems using similitude. Buckingham's π-theorem is introduced as a way to non-dimensionalize variables when there are more variables than fundamental dimensions. Rayleigh's and Buckingham's methods are demonstrated on an example of determining the resisting force on an aircraft.
This document discusses dimensional analysis, which is a mathematical technique used in fluid mechanics to reduce the number of variables in a problem by combining dimensional variables to form non-dimensional parameters. Dimensional analysis allows problems to be expressed in terms of non-dimensional parameters to show the relative significance of each parameter. It has various uses including checking dimensional homogeneity of equations, deriving equations, planning experiments, and analyzing complex flows using scale models. The Buckingham π theorem states that any relationship between physical quantities can be written as a relationship between dimensionless pi groups formed from the variables. Dimensional analysis is applied by setting up a dimensional matrix to determine the minimum number of pi groups needed to describe the relationship.
The document discusses dimensional analysis, which is a technique used to express physical quantities in terms of base quantities. It defines basic and derived quantities, and lists common base quantities like length, mass, and time. The document also shows how to use dimensional analysis to determine the units and dimensional consistency of equations. Examples are provided to illustrate determining units, checking if equations are dimensionally correct, and deriving relationships between physical quantities.
Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
This document discusses dimensional analysis and its applications. It begins with an introduction to dimensions, units, fundamental and derived dimensions. It then discusses dimensional homogeneity, methods of dimensional analysis including Rayleigh's method and Buckingham's π-theorem. The document also covers model analysis, similitude, model laws, model and prototype relations. It provides examples of applying Rayleigh's method and Buckingham's π-theorem to define relationships between variables. Finally, it discusses different types of forces acting on fluids and dimensionless numbers, and provides model laws for Reynolds, Froude, Euler and Weber numbers.
1. Dimensional analysis and the concept of similitude allow experiments using scale models to be used to study full-scale systems. Dimensional analysis uses Buckingham pi theorem to determine the minimum number of dimensionless groups needed to describe a phenomenon in terms of the variables involved.
2. For a model to accurately simulate a prototype system, the dimensionless pi groups that describe the phenomenon must be equal between the model and prototype. This establishes the modeling laws or similarity requirements that a model must satisfy.
3. Common dimensionless groups in fluid mechanics include the Reynolds number, Froude number, Strouhal number, and Weber number. These groups arise frequently in analyzing experimental data from fluid mechanics problems.
This document discusses dimensional analysis and its applications. It can be used to:
1) Derive equations by ensuring the dimensions on both sides are equal
2) Check if equations are dimensionally correct
3) Find the dimensions/units of derived quantities
Examples are provided to illustrate deriving equations based on quantities' dimensions and checking the homogeneity of equations.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Quark Model Three Body Calculations for the Hypertriton Bound StateIOSR Journals
Hyperspherical three body calculations are performed to study and review the various properties of
the hypertriton bound state nucleus
3H in the quark model using -N potentials. In these calculations we study
the different effects of the -N potentials on the hypertriton bound states as well as the separation energy B. A
combination of realistic two body N-N potentials with various - N potentials are considered. Complete
symmetric and mixed symmetric wave functions are introduced. using the renormalized Numerov method. The
agreement between the calculated
3H binding energies and the available experimental data basically depends
on the type of the -N interactions used in the calculations. It was found that the -N potentials are the most
effective part in the hypertriton binding energy as well as the separation energy B where the -N potentials is
very effective to bound or unbound the
3H hyper nucleus
Pacs numbers: 21.30. + y, 21.10.+dr,27.20.+n
This paper explores dimensional reduction of 4D general relativity to a (1+1) gravity model using a Kaluza-Klein type approach. It begins with the Einstein-Hilbert action in 4D and defines a fundamental metric that encodes the metric for a (1+1) spacetime. This allows obtaining an effective (1+1) gravity action coupled to a scalar field. The equations of motion for the metric and scalar field in the (1+1) spacetime are then derived. A solution to the metric is presented where the scalar field is the radius of a 2-sphere.
Buckingham's Π theorem is a method of dimensional analysis that arranges variables in a physical problem into dimensionless groups (Π terms) to reduce the number of variables. It works when the total number of variables (n) is greater than the number of fundamental dimensions (m) involved. The example problem applies the method to determine the drag force (FD) on a sphere moving through a fluid based on variables like sphere diameter (D), velocity (V), fluid density (ρ), and viscosity (μ). Following steps of listing variables, determining n and m, forming Π groups, and solving for exponents to satisfy dimensionality, two dimensionless groups Π1 = FD/(ρV2D2) and
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
This document provides an overview of a physics lecture on units, dimensions, and vectors. The lecture introduces students to the International System of Units (SI) and the metric system of measurement. It discusses the basic SI units of length, mass, and time. The lecture also covers dimensional analysis, which uses the dimensions of physical quantities to check the validity of equations. Vector concepts such as coordinate systems and vector components are also introduced. The document aims to equip medical sciences students with the fundamental physics concepts needed to understand measurements and quantitative relationships in physics.
1. The document analyzes the effect of radiation on flame structure and extinction in a one-dimensional planar diffusion flame. A simplified configuration is adopted where quantities depend on the flame normal direction only.
2. The governing equations are transformed to a mass-weighted coordinate system using the Howarth transformation to simplify the treatment of density. Radiation is incorporated as a small perturbation using a dimensionless parameter defined as the product of the radiation absorption coefficient and a characteristic length scale.
3. Radiation lowers the flame temperature, affecting the reaction rate. The radiation absorption term in the energy equation is obtained from the radiation transport equation, appearing as a non-local integral that is treated using Green's functions.
This example uses Buckingham's Pi theorem to relate the time taken (T) for a car to travel a distance (D) at a velocity (V). There are 3 variables (T, D, V) and 2 fundamental units (time, length), so there is 1 dimensionless parameter. Applying the theorem, it is shown that the time taken is equal to the distance divided by the velocity. So if a car travels 200 km at 100 km/hr, the time taken is 200/100 = 2 hours.
This document discusses the third law of thermodynamics. It states that the entropy of a perfectly crystalline substance is zero at absolute zero temperature. The mathematical expressions for determining absolute entropy are provided. The document also discusses Nernst's heat theorem, which states that the change in Gibbs free energy of a reaction approaches the change in enthalpy as temperature approaches absolute zero. Exceptions to the third law for certain gases with non-ordered crystal structures are also noted.
1) The document discusses model theory and similarity conditions for physical models. It defines geometric, kinematic, and dynamic similarity between models and prototypes.
2) For dynamic similarity, the Froude and Reynolds numbers must be equal between corresponding points on the model and prototype. This requires a relationship between velocity and geometric length scales.
3) For Froude models, which are used for open channel flow, the velocity scale is the square root of the length scale. Examples are provided to demonstrate calculating discharge, velocity, and force values between models and prototypes.
1+3 gr reduced_as_1+1_gravity_set_1 280521fordsplyfoxtrot jp R
1. The document describes dimensional reduction of Einstein-Hilbert action from its original 1+3 form to a 1+1 form. This is done by defining a fundamental line element that encodes lower dimensional metrics for 1+1 spacetime and a compactified 2-sphere.
2. Key steps include writing the higher dimensional metric and curvature scalar in terms of lower dimensional components, and integrating over the 2-sphere coordinates to obtain an effective 2D action.
3. Varying the effective action with respect to the metric and scalar field derives the equations of motion in 1+1 dimensions, identifying an effective energy-momentum tensor and equation of motion for the dilaton field.
This document discusses translations, glide reflections, and compositions of transformations in a plane. It defines translations as transformations that move a figure without changing its size, angles, or orientation. Glide reflections are performed by first translating and then reflecting a figure. Compositions involve combining two or more transformations, like rotating and then reflecting a figure, with the order of operations changing the final result. Vectors are introduced as a way to represent translations by their horizontal and vertical components. Examples demonstrate identifying and performing translations, glide reflections, and compositions on geometric figures.
This document discusses using a master equation approach to simulate electron spin resonance (ESR) spectral lineshapes. It compares using 6-state and 48-state stochastic models to represent rotational diffusion, an important relaxation process in ESR. Simulated spectra from both models capture the main spectral features but the 48-state model provides more detail, especially at higher dispersion. The results help establish criteria for selecting appropriate models to faithfully reproduce ESR lineshapes over a wide range of transition rates.
Calculando o tensor de condutividade em materiais topológicosVtonetto
This document describes a new efficient numerical method to calculate the longitudinal and transverse conductivity tensors in solids using the Kubo-Bastin formula. The method expands Green's functions in terms of Chebyshev polynomials, allowing both diagonal and off-diagonal conductivities to be computed for large systems in a single step at any temperature or chemical potential. The method is applied to calculate the conductivity tensor for the quantum Hall effect in disordered graphene and a Chern insulator in Haldane's model on a honeycomb lattice.
Minimum uncertainty coherent states attached to nondegenerate parametric ampl...Sergio Floquet
The document summarizes research on minimum uncertainty coherent states for a nondegenerate parametric amplifier. It begins by introducing the Hamiltonian for a nondegenerate parametric amplifier and shows its equivalence to a two-dimensional harmonic oscillator Hamiltonian through a similarity transformation. It then discusses how the eigenstates of the two-dimensional harmonic oscillator Hamiltonian can be used to obtain new eigenstates for the parametric amplifier Hamiltonian. These new eigenstates are called displaced SU(2)-Perelomov coherent states and satisfy the eigenvalue equation for the parametric amplifier Hamiltonian. They form a complete set of eigenbasis vectors with the property of minimum uncertainty.
Qualitative measurement of Klauder coherent states using Bohmian machanics, C...Sanjib Dey
1. The document discusses drawing the trajectories of coherent states by applying Bohmian mechanics.
2. Bohmian mechanics provides an alternative trajectory-based interpretation of quantum mechanics by solving equations of motion for particle trajectories guided by the wave function.
3. To measure the quality of coherent states precisely, the document proposes drawing the classical trajectories using canonical equations and drawing the dynamics of coherent states by applying the Bohmian scheme, then comparing the two sets of trajectories.
This document discusses dimensional analysis and its applications. It begins with an introduction to dimensions, units, fundamental and derived dimensions. It then discusses dimensional homogeneity, methods of dimensional analysis including Rayleigh's method and Buckingham's π-theorem. The document also covers model analysis, similitude, model laws, model and prototype relations. It provides examples of applying Rayleigh's method and Buckingham's π-theorem to define relationships between variables. Finally, it discusses different types of forces acting on fluids and dimensionless numbers, and provides model laws for Reynolds, Froude, Euler and Weber numbers.
1. Dimensional analysis and the concept of similitude allow experiments using scale models to be used to study full-scale systems. Dimensional analysis uses Buckingham pi theorem to determine the minimum number of dimensionless groups needed to describe a phenomenon in terms of the variables involved.
2. For a model to accurately simulate a prototype system, the dimensionless pi groups that describe the phenomenon must be equal between the model and prototype. This establishes the modeling laws or similarity requirements that a model must satisfy.
3. Common dimensionless groups in fluid mechanics include the Reynolds number, Froude number, Strouhal number, and Weber number. These groups arise frequently in analyzing experimental data from fluid mechanics problems.
This document discusses dimensional analysis and its applications. It can be used to:
1) Derive equations by ensuring the dimensions on both sides are equal
2) Check if equations are dimensionally correct
3) Find the dimensions/units of derived quantities
Examples are provided to illustrate deriving equations based on quantities' dimensions and checking the homogeneity of equations.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Quark Model Three Body Calculations for the Hypertriton Bound StateIOSR Journals
Hyperspherical three body calculations are performed to study and review the various properties of
the hypertriton bound state nucleus
3H in the quark model using -N potentials. In these calculations we study
the different effects of the -N potentials on the hypertriton bound states as well as the separation energy B. A
combination of realistic two body N-N potentials with various - N potentials are considered. Complete
symmetric and mixed symmetric wave functions are introduced. using the renormalized Numerov method. The
agreement between the calculated
3H binding energies and the available experimental data basically depends
on the type of the -N interactions used in the calculations. It was found that the -N potentials are the most
effective part in the hypertriton binding energy as well as the separation energy B where the -N potentials is
very effective to bound or unbound the
3H hyper nucleus
Pacs numbers: 21.30. + y, 21.10.+dr,27.20.+n
This paper explores dimensional reduction of 4D general relativity to a (1+1) gravity model using a Kaluza-Klein type approach. It begins with the Einstein-Hilbert action in 4D and defines a fundamental metric that encodes the metric for a (1+1) spacetime. This allows obtaining an effective (1+1) gravity action coupled to a scalar field. The equations of motion for the metric and scalar field in the (1+1) spacetime are then derived. A solution to the metric is presented where the scalar field is the radius of a 2-sphere.
Buckingham's Π theorem is a method of dimensional analysis that arranges variables in a physical problem into dimensionless groups (Π terms) to reduce the number of variables. It works when the total number of variables (n) is greater than the number of fundamental dimensions (m) involved. The example problem applies the method to determine the drag force (FD) on a sphere moving through a fluid based on variables like sphere diameter (D), velocity (V), fluid density (ρ), and viscosity (μ). Following steps of listing variables, determining n and m, forming Π groups, and solving for exponents to satisfy dimensionality, two dimensionless groups Π1 = FD/(ρV2D2) and
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
This document provides an overview of a physics lecture on units, dimensions, and vectors. The lecture introduces students to the International System of Units (SI) and the metric system of measurement. It discusses the basic SI units of length, mass, and time. The lecture also covers dimensional analysis, which uses the dimensions of physical quantities to check the validity of equations. Vector concepts such as coordinate systems and vector components are also introduced. The document aims to equip medical sciences students with the fundamental physics concepts needed to understand measurements and quantitative relationships in physics.
1. The document analyzes the effect of radiation on flame structure and extinction in a one-dimensional planar diffusion flame. A simplified configuration is adopted where quantities depend on the flame normal direction only.
2. The governing equations are transformed to a mass-weighted coordinate system using the Howarth transformation to simplify the treatment of density. Radiation is incorporated as a small perturbation using a dimensionless parameter defined as the product of the radiation absorption coefficient and a characteristic length scale.
3. Radiation lowers the flame temperature, affecting the reaction rate. The radiation absorption term in the energy equation is obtained from the radiation transport equation, appearing as a non-local integral that is treated using Green's functions.
This example uses Buckingham's Pi theorem to relate the time taken (T) for a car to travel a distance (D) at a velocity (V). There are 3 variables (T, D, V) and 2 fundamental units (time, length), so there is 1 dimensionless parameter. Applying the theorem, it is shown that the time taken is equal to the distance divided by the velocity. So if a car travels 200 km at 100 km/hr, the time taken is 200/100 = 2 hours.
This document discusses the third law of thermodynamics. It states that the entropy of a perfectly crystalline substance is zero at absolute zero temperature. The mathematical expressions for determining absolute entropy are provided. The document also discusses Nernst's heat theorem, which states that the change in Gibbs free energy of a reaction approaches the change in enthalpy as temperature approaches absolute zero. Exceptions to the third law for certain gases with non-ordered crystal structures are also noted.
1) The document discusses model theory and similarity conditions for physical models. It defines geometric, kinematic, and dynamic similarity between models and prototypes.
2) For dynamic similarity, the Froude and Reynolds numbers must be equal between corresponding points on the model and prototype. This requires a relationship between velocity and geometric length scales.
3) For Froude models, which are used for open channel flow, the velocity scale is the square root of the length scale. Examples are provided to demonstrate calculating discharge, velocity, and force values between models and prototypes.
1+3 gr reduced_as_1+1_gravity_set_1 280521fordsplyfoxtrot jp R
1. The document describes dimensional reduction of Einstein-Hilbert action from its original 1+3 form to a 1+1 form. This is done by defining a fundamental line element that encodes lower dimensional metrics for 1+1 spacetime and a compactified 2-sphere.
2. Key steps include writing the higher dimensional metric and curvature scalar in terms of lower dimensional components, and integrating over the 2-sphere coordinates to obtain an effective 2D action.
3. Varying the effective action with respect to the metric and scalar field derives the equations of motion in 1+1 dimensions, identifying an effective energy-momentum tensor and equation of motion for the dilaton field.
This document discusses translations, glide reflections, and compositions of transformations in a plane. It defines translations as transformations that move a figure without changing its size, angles, or orientation. Glide reflections are performed by first translating and then reflecting a figure. Compositions involve combining two or more transformations, like rotating and then reflecting a figure, with the order of operations changing the final result. Vectors are introduced as a way to represent translations by their horizontal and vertical components. Examples demonstrate identifying and performing translations, glide reflections, and compositions on geometric figures.
This document discusses using a master equation approach to simulate electron spin resonance (ESR) spectral lineshapes. It compares using 6-state and 48-state stochastic models to represent rotational diffusion, an important relaxation process in ESR. Simulated spectra from both models capture the main spectral features but the 48-state model provides more detail, especially at higher dispersion. The results help establish criteria for selecting appropriate models to faithfully reproduce ESR lineshapes over a wide range of transition rates.
Calculando o tensor de condutividade em materiais topológicosVtonetto
This document describes a new efficient numerical method to calculate the longitudinal and transverse conductivity tensors in solids using the Kubo-Bastin formula. The method expands Green's functions in terms of Chebyshev polynomials, allowing both diagonal and off-diagonal conductivities to be computed for large systems in a single step at any temperature or chemical potential. The method is applied to calculate the conductivity tensor for the quantum Hall effect in disordered graphene and a Chern insulator in Haldane's model on a honeycomb lattice.
Minimum uncertainty coherent states attached to nondegenerate parametric ampl...Sergio Floquet
The document summarizes research on minimum uncertainty coherent states for a nondegenerate parametric amplifier. It begins by introducing the Hamiltonian for a nondegenerate parametric amplifier and shows its equivalence to a two-dimensional harmonic oscillator Hamiltonian through a similarity transformation. It then discusses how the eigenstates of the two-dimensional harmonic oscillator Hamiltonian can be used to obtain new eigenstates for the parametric amplifier Hamiltonian. These new eigenstates are called displaced SU(2)-Perelomov coherent states and satisfy the eigenvalue equation for the parametric amplifier Hamiltonian. They form a complete set of eigenbasis vectors with the property of minimum uncertainty.
Qualitative measurement of Klauder coherent states using Bohmian machanics, C...Sanjib Dey
1. The document discusses drawing the trajectories of coherent states by applying Bohmian mechanics.
2. Bohmian mechanics provides an alternative trajectory-based interpretation of quantum mechanics by solving equations of motion for particle trajectories guided by the wave function.
3. To measure the quality of coherent states precisely, the document proposes drawing the classical trajectories using canonical equations and drawing the dynamics of coherent states by applying the Bohmian scheme, then comparing the two sets of trajectories.
This document summarizes the activities of the Center for Physical Sciences and Technology in Lithuania in 2015. In 2015, the Center celebrated its 5-year anniversary and moved to new buildings equipped with modern scientific infrastructure. The Center focuses its research on topics such as optoelectronics, laser technologies, materials science, and applications of these areas. It aims to advance scientific research and support the development of high-tech business and knowledge-based economy in Lithuania through interdisciplinary collaboration and international partnerships.
This document summarizes a PhD thesis talk on strongly interacting fermions in optical lattices. The outline discusses chapters on the Hubbard and Heisenberg models, cooling into the Néel state, and imbalanced antiferromagnets. The introduction provides background on realizing the Hubbard model experimentally and the goal of achieving the Néel state. It also discusses imbalanced Fermi gases and how imbalance may affect the Néel state.
"Squeezed States in Bose-Einstein Condensate"Chad Orzel
1. The document discusses the formation of squeezed quantum states in Bose-Einstein condensates trapped in optical lattices. By slowly ramping up the depth of the optical lattice, the atoms can be prepared in a number-squeezed state.
2. Releasing the atoms from the lattice allows their wavefunctions to overlap and interfere, providing a way to probe the quantum phase state of the atoms. Number-squeezed states are observed to produce interference patterns with higher contrast than coherent states.
3. Variational calculations of the quantum state dynamics during lattice ramping and dephasing agree qualitatively with experimental observations of the transition between coherent and squeezed states.
"When the top is not single: a theory overview from monotop to multitops" to...Rene Kotze
This document discusses potential deviations from the standard model in top quark pair production (ttbar) due to beyond standard model (BSM) physics. It summarizes that ttbar production is well measured but sensitive to BSM effects like resonant contributions from new particles that decay to top quark pairs. Non-resonant effects are also possible and can be parameterized using effective field theory operators. The document provides examples of limits set on specific BSM models like Z' bosons by the CMS experiment through analyses of the ttbar invariant mass spectrum and other observables.
"Quantum nanophotonics"
Abstract: Quantum nanophotonics is a rapidly growing field of research that involves the study of the quantum properties of light and its interaction with matter at the nanoscale. Here, surface plasmons – electromagnetic excitations coupled to electron charge density waves on metal-dielectric interfaces or localized on metallic nanostructures – enable the confinement of light to scales far below that of conventional optics. I will review recent progress in the theoretical investigation of the quantum properties of surface plasmons, their role in controlling light-matter interactions at the quantum level and potential applications in quantum information science.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
Gradient echo pulse sequence and its applicationJayanti Gyawali
Gradient echo pulse sequences use gradient coils to alter the magnetic field strength along different axes within the MRI scanner. There are three main gradient coils that can selectively modify the field in the x, y, and z directions. Gradient echo sequences like FLASH, SPGR, and FFE are used to generate T1-weighted images and can be applied as breath-hold 3D scans for dynamic studies. Balanced steady-state free precession sequences like TRUFISP, FIESTA, and b-FFE have very high signal-to-noise ratio and are used for cardiac, fetal, and bowel imaging but lack spatial resolution.
Classical mechanics fails to explain several experimental observations such as:
1) Black-body radiation spectrum
2) Photoelectric effect
3) Compton scattering
4) Spectrum of hydrogen emissions
Quantum mechanics was developed to account for these phenomena by treating electrons as both particles and waves. Max Planck proposed quanta to explain black-body radiation, while Albert Einstein and Niels Bohr used quanta to explain the photoelectric effect and hydrogen spectrum respectively. Arthur Compton also explained Compton scattering using photons colliding with electrons.
A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.
This document provides information about lasers, including:
1. Lasers produce coherent light through stimulated emission of radiation.
2. Coherent light is uniform in frequency, amplitude, and phase, unlike incoherent light from other sources.
3. Key atomic interactions that enable laser action are induced absorption, spontaneous emission, and stimulated emission, which allows for population inversion and optical pumping to produce more atoms in the excited state.
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Minimum uncertainty coherent states attached to nondegenerate parametric amplifiers
1. Brazilian Journal of Physics
ISSN: 0103-9733
luizno.bjp@gmail.com
Sociedade Brasileira de Física
Brasil
Dehghani, A.; Mojaveri, B.
Minimum Uncertainty Coherent States Attached to Nondegenerate Parametric Amplifiers
Brazilian Journal of Physics, vol. 45, núm. 3, junio, 2015, pp. 265-271
Sociedade Brasileira de Física
Sâo Paulo, Brasil
Available in: http://www.redalyc.org/articulo.oa?id=46439436001
How to cite
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Journal's homepage in redalyc.org
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Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal
Non-profit academic project, developed under the open access initiative
3. 266 Braz J Phys (2015) 45:265–271
[20–32]). The equivalence of a nondegenerate parametric
amplifier with two parallel degenerate parametric amplifiers
has been also exploited experimentally to obtain a matched
local oscillator for the detection of quadrature squeezing
[33] and has been extended to all the stages of the com-
munication channel [34]. The nondegenerate parametric
amplifier has attracted much attention because of its impor-
tant role in quantum optics. For instance, squeezed states
can be generated in nonlinear optical process such as non-
degenerate parametric amplification, in which one photon
of a large frequency is turned in the medium into two pho-
tons of equal or nonequal frequencies [35–38]. A method
for producing light field Fock state based on nondegener-
ate parametric amplifier model was proposed by Bj¨ork [39].
By using Lie algebra representation theory, Z-Jie Wang [40]
has solved the master equation for the nondegenerate para-
metric amplifier in a thermal reservoir. Applying a series
of transformations, they show that the master equation has
multiple commuted SU(1, 1) Lie algebra structures. The
explicit solution to the master equation has been obtained.
Also, the EPR paradox was demonstrated via quadrature
phase measurements performed on the two output beams of
a nondegenerate parametric amplifier [41]. Entangled states
of light field and entanglement swapping have been well
studied using nondegenerate parametric amplifier method
[42, 43].
Major efforts in these documents, are focused on the
study of nonclassical properties of the parametric ampli-
fier [44–48]. Hence, our main motivation in this paper
will be concentrated on the classical one. For this rea-
son, we will consider a new technique of preparation of
the minimum-uncertainty states corresponding to a nonde-
generate parametric amplifier. A method of characterizing
these wave packets, which are based on dynamical sym-
metry of the Hamiltonian is analyzed. The basic idea of
the construction of these states is closely related to that
of coherent states introduced by the pioneering works of
Glauber, Klauder, and Sudarshan [49–56]. Such quantum
states which fulfill this requirement will be constructed
as solutions of an eigenvalue problem of some symme-
try operators lowering the energy, as well as an orbit of
states generated by a chosen group element from a fixed
state and as minimum-uncertainty states for some physically
significant operators.
This paper is organized as follows: taking advantage of
quantum analysis of the two coupled harmonic oscillators,
then by using a similarity transformation, we obtain the non-
degenerate parametric amplifier Hamiltonian in Section 2.
Accurate analysis of nondegenerate parametric amplifier,
its eigenvalues, and eigenvectors are given in Section 2.1.
Section 3, is devoted to introduce an approach which results
in a new kind of minimum-uncertainty states, corresponding
to the parametric amplifier. Also, basic statistical proper-
ties of these states are studied in more details. Finally, we
conclude in Section 4.
2 Nondegenerate Parametric Amplifier
and it’s Connection with 2D Harmonic Oscillator
Here, we make brief reviews of two-mode nondegenerate
and stationary parametric amplifier Hamiltonian
H = Ha + Hb + Hab (1)
where
Ha = ω a†
a +
1
2
(2)
Hb = ω b†
b +
1
2
(3)
in natural units ( = c = 1). The interaction term between
the modes a and b is chosen to be of the form
Hab = i gab − g∗
a†
b†
, g = |g|ei
∈ C (4)
which describes a two-mode nondegenerate parametric
down-conversion process. The time-independent pump
parameter |g| denotes an arbitrary stationary and classical
pump field. Here, |g| is proportional to the second-order sus-
ceptibility of the medium and amplitude of the pump, is
the phase of the pump field, and ω is frequency [57–60]. It
is straightforward that the Hamiltonian (1) can be rewritten
in the following form
H = 2ωK0 + igK− − ig∗
K+, (5)
where we have used the two-mode representation [61] of the
Lie algebra su(1, 1)1:
K+ = a†
b†
, K− = ab, K0 =
1
2
(a†
a + b†
b + 1), (6)
[K+, K−] = −2K0, [K0, K±] = ±K±. (7)
Along with the application of a similarity transformation
D(ξ)HD†
(ξ),
with D(ξ) as Klauder’s displacement operator, we have
D(ξ) = eξK+−ξK− ,
ξ := |ξ|eiϕ
, 0 ≤ ϕ ≤ 2π (8)
then, by choosing
ϕ = − + n +
1
2
π, n ∈ {1, 2, 3, . . .} (9)
tanh(|ξ|) = γ − γ 2 − 1 , γ =
ω
|g|
≥ 1. (10)
1Here, a(a†) and b(b†) are the annihilation (creation) operators corre-
sponding to the two hermitian harmonic oscillators, respectively.
4. Braz J Phys (2015) 45:265–271 267
An effective Hamiltonian representing a two-dimensional
harmonic oscillator can be obtained2
D(ξ)HD†
(ξ) = 2|g|f (γ ) K0
= |g|f (γ ) (a†
a + b†
b + 1), (13)
where f (γ ) can be calculated to be the following positive
definite function
f (γ ) =
γ 2 − 1
1 − γ 2 + γ γ 2 − 1
γ − γ 2 − 1 . (14)
Obviously, (13) shows that the Hamiltonian D(ξ)HD†(ξ)
is “really” proportional to the two-dimensional harmonic
oscillator Hamiltonian Hho, i.e.,
D(ξ)HD†
(ξ) =
f (γ )
γ
Hho, (15)
and
Hho := ω(a†
a + b†
b + 1). (16)
Then, the eigenfunctions of the Hamiltonian D(ξ)HD†(ξ)
are those of the Hamiltonian Hho [62]. Along with the
energy eigenvalues and the eigenstates of the transferred
Hamiltonian Hho, we try to get the analogous ones for the
original Hamiltonian H.
Because of the fact that the 2D Harmonic Oscillator
includes su(2) Lie algebra as dynamical symmetry [63],
so it enables us to introduce a new class of eigenstates
corresponding to the Hamiltonian Hho. Indeed, they are
associated with the “su(2)”-Perelomov coherent states and
provide new eigenvectors for the Hamiltonian H, that will
be introduced later as the displaced su(2)-Perelomov coher-
ent states.
2.1 Eigenvalue Equation for the Nondegenerate Parametric
Amplifier H Based on the su(2)-Perelomov Coherent
States Attached to the Hamiltonian, Hho
It is well known that using the representation of two Hermi-
tian harmonic oscillators (a and b), i.e.,
[a, a†
] = 1, [b, b†
] = 1, [a, b] = 0, (17)
which act on the complete and ortho-normal Fock space
states, Hj = {|j, m | − j ≤ m ≤ j}, with the following
2It should be noticed that, to achieve the (13) we put
D(ξ)aD†
(ξ) = D(ξ)a†
D†
(ξ)
†
= cosh(ξ)a −
ξ sinh(ξ)
|ξ|
b†
(11)
D(ξ)bD†
(ξ) = D(ξ)b†
D†
(ξ)
†
= cosh(ξ)b −
ξ sinh(ξ)
|ξ|
a†
(12)
laddering relations:
a|j, m =
j − m
2
|j − 1, m + 1 ,
a†
|j − 1, m + 1 =
j − m
2
|j, m , (18)
and
b|j, m =
j + m
2
|j − 1, m − 1 ,
b†
|j − 1, m − 1 =
j + m
2
|j, m , (19)
the unitary and irreducible j representation of su(2) can be
attained [64]
[J+, J−] = 2J0, [J0, J±] = ±J±. (20)
It is spanned by the generators
J+ = ab†
, J− = a†
b, J0 =
bb† − aa†
2
, (21)
and realize the following positive j integer irreducible rep-
resentation of su(2) Lie algebra on the Hilbert subspaces
Hj , as [65]
J±|j, m =
j ∓ m
2
j ± m
2
+ 1 |j, m ± 2 ,
J0|j, m =
m
2
|j, m . (22)
The so-called Klauder-Perelomov coherent states for a
degenerate Hamiltonian Hho, are defined as the action of a
displacement operator on the normalized lowest (or highest)
weight vectors:
|α j := eαJ+−¯αJ− |j, −j ,
= (1 + |η|2
)
−j
2
j
m=0
(j + 1)
(m + 1) (j − m + 1)
ηm
|j,
−j + 2m , (23)
where η = α tan |α|
|α| is an arbitrary complex variable with
the polar form η = eiθ so that 0 ≤ < ∞ and
0 ≤ θ < 2π. Clearly, the positive definite and non-
oscillating measure is,
dμ(α) = 2
j + 1
(1 + 2)2
d 2
2
dθ, (24)
which satisfies the resolution of the identity condition on the
whole of the complex plane for the coherent states |α j in
the Hilbert sub-spaces Hj , i.e.,
C
|α j j α|dμ(α) = Ij . (25)
5. 268 Braz J Phys (2015) 45:265–271
Also, they satisfy the completeness as well as the orthogo-
nality relationships as follows
∞
j=0
|α j j α| =
∞
j=0
|j, −j j, −j| = I (26)
j α |α j = j , −j |j, −j = δj j , (27)
which is inherited because of the fact that the displacement
operator D(ξ) = eαJ+−¯αJ− is a unitary action.
Using (16), (17), and (21), one can show that the Hamil-
tonian Hho commutes3 with all of the generators J± and J0,
therefore the following eigenvalue equation is achieved
Hho |α j = ω (j + 1) |α j , (28)
and provides us with the following eigenvalue equation cor-
responding to the Hamiltonian H, in terms of the displaced
su(2)-coherent states, |ξ, α j := D†(ξ) |α j , as
H D†
(ξ) |α j = |g|f (γ ) (j + 1) D†
(ξ) |α j . (29)
They form a complete and orthonormal set of eigenba-
sis vectors, which is inherited from the completeness and
orthogonality relations (26) and (27), respectively; in other
words
∞
j=0
|ξ, α j j ξ, α| =
∞
j=0
|α j j α|
=
∞
j=0
|j, −j j, −j| = I, (30)
j ξ, α |ξ, α j =j α |α j = δj j . (31)
3 Representation of the Heisenberg Algebras Through
the States |α j and |ξ, α j
On one hand, the su(2)-Perelomov coherent states |α j rep-
resent the Hamiltonian Hho as well as the Hamiltonian H,
and both of them are factorized in terms of two harmonic
oscillator modes a(a†) and b(b†) (see (5) and (16)). Then a
natural question arises here: is it possible for these states to
be used for showing harmonic oscillator algebra? The rea-
son comes from (18), (19), and (23) that the operators a and
b act on the vectors |α j as the lowering operators, which
means
a |α j =
√
j
1 + |η|2
|α j−1 , b |α j =
η
√
j
1 + |η|2
|α j−1
⇓
a + ¯ηb
1 + |η|2
|α j = j |α j−1 , (32)
3In other words, the Lie algebra symmetry of su(2) is dynamical
symmetry of the Hamiltonian Hho.
and
a† + ηb†
1 + |η|2
|α j = j + 1 |α j+1 , (33)
where we have used, here, the formulas
e−αJ++¯αJ− aeαJ+−¯αJ− =
a + ¯ηb
1 + |η|2
, (34)
e−αJ++¯αJ− beαJ+−¯αJ− =
b + ηa
1 + |η|2
. (35)
Now, taking into account (32), (33), and (17), we find the
following harmonic oscillator algebra:
a + ¯ηb
1 + |η|2
,
a† + ηb†
1 + |η|2
= 1. (36)
Note that the above algebraic structure remains invariant
under the unitary transformation, D(ξ). Then, by using
the displaced su(2)-coherent states |ξ, α j , other irreps
(irreducible representations) of Weyl-Heisenberg algebra
are obtainable
A, A†
= 1, (37)
A|ξ, α j = j|ξ, α j−1,
A†
|ξ, α j = j + 1|ξ, α j+1, (38)
through the operators
A := D†
(ξ)
a + ¯ηb
1 + |η|2
D(ξ) =
cosh |ξ|
1 + |η|2
(a + ¯ηb)
+
ξ sinh |ξ|
|ξ| 1 + |η|2
(b†
+ ¯ηa†
), (39)
A†
:= D†
(ξ)
a† + ηb†
1 + |η|2
D(ξ) =
cosh |ξ|
1 + |η|2
a†
+ ηb†
+
ξ sinh |ξ|
|ξ| 1 + |η|2
(b + ηa). (40)
They will be utilized to generate a new kind of minimum
uncertainty states associated with the Hamiltonian Hho and
H, respectively.
3.1 Minimum Uncertainty Quantum States Attached
to the 2D Harmonic Oscillator Hho
Let us define following new normalized states
|β, α := e
β
√
1+|η|2
(a†+ηb†)− β
√
1+|η|2
(a+ηb)
|α 0
= e
β
√
1+|η|2
a†
e
βη
√
1+|η|2
b†
|0, 0 , (41)
which can be considered as eigenvectors of annihilation
operator a+¯ηb
√
1+|η|2
that will be written in a series form as
6. Braz J Phys (2015) 45:265–271 269
follows:
|β, α = e− |β|2
2
∞
j=0
βj
√
(j + 1)
|α j ,
= e− |β|2
2
∞
j=0
β
1 + |η|2
j
×
j
m=0
ηm
√
(m + 1) (j − m + 1)
|j,−j +2m . (42)
Resolution of the identity condition is realized for such
two-variable coherent states on complex plane C2 by the
measure 1
π d2αd2β.
Finally, these states are temporally stable, i.e.,
e−itHho |β, α = e−iωt− |β|2
2
∞
j=0
βe−iωt j
√
(j + 1)
|α j
= e−iωt
βe−iωt
, α . (43)
The time-dependent coherent states of the generalized time-
dependent parametric oscillator [66] will be useful for future
studies in quantum optics as well as in atomic and molecular
physics.
3.1.1 Classical Properties of the States |β, α
In order to clarify why these states can be called minimum-
uncertainty states, we now introduce two hermitian opera-
tors
q =
1
2
b + b†
− a − a†
, (44)
p =
−i
2
b − b†
− a + a†
, (45)
such that [q, p] = i, which leads to the following uncer-
tainty relation:
(= σqqσpp − σ2
qp) ≥
1
4
, (46)
where σab = 1
2 ab + ba − ab and the angu-
lar brackets denote averaging over an arbitrary normaliz-
able state for which the mean values are well defined,
a = β, α| a |β, α . Averaging over the classical-
quantum states |β, α , one finds: q2 = 1
2 + q 2,
p2 = 1
2 + p 2, and pq + qp = 2 q p , where
q =
βη + βη − β − ¯β
2 1 + |η|2
,
p = −i
βη − βη − β + ¯β
2 1 + |η|2
,
σqq = σpp =
1
2
,
σqp = 0. (47)
They lead to = 1
4 , which is the lower bound of the
Heisenberg uncertainty relation as well. In other words,
the states |β, α meet the minimal requirement of the
Heisenberg uncertainty relation.
As specific criteria to illustrate the inherited statistical
properties of these states, it is necessary to analyze the
behavior of Mandel’s Q(|β|)4 parameter, which is defined
with respect to the expectation values of the number oper-
ator5 , ˆJ, and its square in the basis of the states |β, α :
Q(|β|) = ˆJ
ˆJ2 − ˆJ
ˆJ 2
− 1 . (48)
It is worth noting that for all accessible frequencies,
the states |β, α follow the Poissonian statistics, i.e.,
Q(|β|) = 0, which represents classical effects.
3.2 Minimum Uncertainty Quantum States Attached
to the Hamiltonian H
As it was shown in the above relations (37)–(40), the
wave functions |ξ, α j reproduce an irreps corresponding
to the Heisenberg Lie algebra through the two ladder oper-
ators A, A†. Which, in turn, leads to derivation of their
corresponding three variable coherence,
|β, ξ, α := eβA†−βA
|ξ, α 0 ,
= e− |β|2
2
∞
j=0
βj
√
(j + 1)
|ξ, α j ,
= D†
(ξ) |β, α . (49)
They clearly satisfy the following eigenvalue equation
A |β, ξ, α = β |β, ξ, α . (50)
They include all of the features of true coherent states. For
instance, they admit a resolution of the identity through
positive definite measures. Also, they are temporally stable.
3.2.1 Classical Properties of the States |β, ξ, α
Here, the uncertainty condition for the variances of the
quadratures q and p, over the states |β, ξ, α will be exam-
4A state for which Q(|β|) > 0 is called super-Poissonian (bunching
effect), if Q(|β|) = 0 the state is called Poissonian, while a state for
which Q(|β|) < 0 is called sub-Poissonian (antibunching effect).
5It is well known that the number operator ˆJ is defined as the operator
which diagonalizes the basis |α j . Using (27), we obtain
j α| ˆJ |α j = jδj j .
7. 270 Braz J Phys (2015) 45:265–271
ined. For instance, the following relations can be calculated
(easily)
a = β
cosh(ξ)
1 + |η|2
− ηβ
ξ sinh(ξ)
|ξ| 1 + |η|2
, (51a)
a2
= β
cosh(ξ)
1 + |η|2
− ηβ
ξ sinh(ξ)
|ξ| 1 + |η|2
2
= a 2
, (51b)
b = βη
cosh(ξ)
1 + |η|2
− β
ξ sinh(ξ)
|ξ| 1 + |η|2
, (51c)
b2
= βη
cosh(ξ)
1 + |η|2
− β
ξ sinh(ξ)
|ξ| 1 + |η|2
2
= b 2
, (51d)
ab = β
cosh(ξ)
1 + |η|2
− ηβ
ξ sinh(ξ)
|ξ| 1 + |η|2
× βη
cosh(ξ)
1 + |η|2
− β
ξ sinh(ξ)
|ξ| 1 + |η|2
= a b (51e)
ab†
= β
cosh(ξ)
1 + |η|2
− ηβ
ξ sinh(ξ)
|ξ| 1 + |η|2
× βη
cosh(ξ)
1 + |η|2
− β
ξ sinh(ξ)
|ξ| 1 + |η|2
= a b†
.
(51f)
Our final step is to reveal that measurements of the states
|β, ξ, α come with minimum uncertainty of the field
quadrature operators q and p. In this case, uncertainty fac-
tors σqq, σpp, and σqp can be evaluated to be taken as,
respectively,
σqq = σpp =
1
2
(52a)
σqp = 0, (52b)
consequently
= σqqσpp − σqp
2
=
1
4
. (52c)
Calculation of other statistical quantities including the
second-order correlation function and Mandel’s parameter
indicates that these states follow the classical regime.
4 Conclusions
We have constructed minimum-uncertainty wave packets of
the two-dimensional harmonic oscillators as well as their
analogous corresponding to the nondegenerate parametric
amplifiers. Remarkably, these states are nonspreading wave
packets that minimize the uncertainty of the measurement
of the position and the momentum operators. We would
like to emphasize the minimum-uncertainty quantum states
which play important roles in quantum optics and mathe-
matical physics; hence, this algebraic process can be applied
to perform minimum-uncertainty coherent, squeezed and
intelligent states associated with the other physical systems.
A precise analysis of Mandel’s parameter confirms that
Poissonian statistics is achievable. Finally, we have shown
that these states are temporally stable and may be useful for
future studies in the time-dependent parametric amplifiers
[45] too.
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