Fresnel tomography: a novel approach to the wave function reconstruction based on Fresnel representation of tomograms
S. De Nicola, R. Fedele, M. A. Man'ko, V. I. Man'ko
备注：
12 pages, 2 figures, talk of M.A. Man'ko at the III International Workshop ``Nonlinear Physics. Theory and Experiment'' (Gallipoli, Italy, 24 June - 3 July 2004), to appear in Theor. Math. Phys
New type of tomographic probability distribution, which contains complete information on the density matrix (wave function) related to the Fresnel transform of the complex wave function, is introduced. Relation to symplectic tomographic probability distribution is elucidated. Multimode generalization of the Fresnel tomography is presented. Examples of applications of the present approach are given.
Tomograms and quasi-distribution functions like Wigner, Glauber - Sudarshan $P$- and Husimi $Q$- functions that violate the standard normalization condition are considered. Conditions under which a reconstruction of the density matrix using these tomograms and quasi-distribution functions is possible are obtained. Three different examples of states like the de Broglie plane wave, the Moschinsky shutter problem and the stationary state of the charged particle in the uniform and constant electric field are studied. Their tomograms and quasi-distribution functions expressed in terms of the Dirac …
We address the problem of information completeness of quantum measuremets in connection to quantum state tomography and with particular concern to quantum symplectic tomography. We put forward some non-trivial situations where informationally incomplete set of tomograms allows as well the state reconstruction provided to have some a priori information on the state or its dynamics. We then introduce a measure of information completeness and apply it to symplectic quantum tomograms.
Tomographic probability representation of multimode electromagnetic field states in the scheme of center-of-mass tomography is reviewed. Both connection of the field state Wigner function and observable Weyl symbols with the center-of-mass tomograms as well as connection of Grönewold kernel with the center-of-mass tomographic kernel determining the noncommutative product of the tomograms are obtained. The dual center-of-mass tomogram of the photon states are constructed and the dual tomographic kernel is obtained. The models of other generalised center-of-mass tomographies are discussed. Examp…
For one-mode and multimode light, the photon-number tomograms of Gaussian quantum states are explicitly calculated in terms of multivariable Hermite polynomials. Positivity of the tomograms is shown to be necessary condition for positivity of the density matrix.
The symlectic quantum tomography for the general linear quantization is introduced. Using the approach based upon the Wigner function techniques the evolution equation of quantum tomograms is derived for a parametric driven oscillator.
Similar in spirit to the preceding work [Opt. Commun. 282 (2009) 3734] where the relation between optical Fresnel transformation and quantum tomography is revealed, we study this kind of relationship in the two-mode entangled case. We show that under the two-mode Fresnel transformation the bipartite entangled state density |eta><eta| becomes density operator F_2|eta><eta|F_2 ^{dag}=|eta>_{r,s}<eta|, which is just the Radon transform of the two-mode Wigner operator (sigma,gama) in entangled form, where F_2 is an two-mode Fresnel operator in quantum optics, and s,r are the complex-value expressi…
A review of probability representation of quantum states in given for optical and photon number tomography approaches. Explicit connection of photon number tomogram with measurable by homodyne detector optical tomogram is obtained. New integral relations connecting Hermite polynomials of two variables with Laguerre polynomials are found. Examples of generic Gaussian photon states (squeezed and correlated states) are studied in detail.
Corresponding to optical Fresnel transformation characteristic of ray transfer matrix elements (A;B;C;D); AD-BC = 1, there exists Fresnel operator F(A;B;C;D) in quantum optics, we show that under the Fresnel transformation the pure position density |x><x| becomes the tomographic density |x>_rs,rs_<x|, which is just the Radon transform of the Wigner operator, i.e., F|x><x|F^(+) = |x>_rs,rs_<x|= \int dx'dp'delta[x-(Dx'-Bp')]*Wigner operator where s, r are the complex-value expression of (A;B;C;D). So the probability distribution for the Fresnel quadrature phase is the tomography (Radon transform…
Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography related to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the representation for spin tomography using the Cayley-Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions…
We scrutinize the effects of non-ideal data acquisition on the homodyne tomograms of photon quantum states. The presence of a weight function, schematizing the effects of the finite thickness of the probing beam or equivalently noise, only affects the state reconstruction procedure by a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is robust under incomplete and approximate knowledge of tomograms.
We consider optical tomography of photon Fock state superpositions in connection with recent experimental achievements. The emphasis is put on the fact that it suffices to represent the measured tomogram as a main result of the experiment. We suggest a test for checking the correctness of experimental data. Explicit expressions for optical tomograms of Fock state superpositions are given in terms of Hermite polynomials. Particular cases of vacuum and low photon-number state superposition are considered as well as influence of thermal noise on state purity is studied.
The systems with multimode nonstationary Hamiltonians quadratic in position and momentum operators are reviewed. The tomographic probability distributions (tomograms) for the Fock states and Gaussian states of the quadratic systems are discussed. The tomograms for the Fock states are expressed in terms of multivariable Hermite polynomials. Using the obvious physical relations some new formulas for multivariable Hermite polynomials are found. Examples of oscillator and charge moving in electromagnetic field are presented.
Review of the probability representation of quantum mechanics and the symplectic tomography approach are presented. The examples of Gaussian states of nanoelectric circuit, Josephson junction, and two interacting high-quality resonant circuits are considered. The Shannon entropy, quantum information, fidelity, and purity of quantum states in the tomographic representation of quantum mechanics are studied.
We introduce quantum tomography on locally compact Abelian groups $G$. A linear map from the set of quantum states on the $C^*$-algebra $A(G)$ generated by the projective unitary representation of $G$ to the space of characteristic functions is constructed. The dual map determining symbols of quantum observables from $A(G)$ is derived. Given a characteristic function of a state the quantum tomogram consisting a set of probability distributions is introduced. We provide three examples in which $G={\mathbb R}$ (the optical tomography), $G={\mathbb Z}_n$ (corresponding to measurements in mutually…
The scheme of photon-number tomography is discussed in the framework of star-product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon{number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form…
Explicit expressions for optical tomograms of the photon-added coherent states, even/odd photon-added coherent states and photon-added thermal states are given in terms of Hermite polynomials. Suggestions for experimental homodyne detection of the considered photon states are presented.
The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying the adjoint representation. Positive maps of density operator are related to random matrices. The tomographic probability description of quantum states is used to formulate the problem of separability and entanglement as the condition for joint probability distribution of several random variables represented as the convex sum of products of probabilities of r…
The protocol of measurement the probability density of a rotated quadrature of microwave mode in an arbitrary quantum state is presented and numerically investigated. The protocol is adapted to the measuring procedure used in the CQED experiments. The method is based on measurement of integrated probability of detection of atoms - probes in the top working state from atoms flying via the cavity. Connection of random variables and is found, when is a qudrature phase. Connection of the tomogram and the tomogram is found. The protocol prescribes the rule of choice of problem parameters, such as i…
The tomographic map of quantum state of a system with several degrees of freedom which depends on one random variable, analogous to center of mass considered in rotated and scaled reference frame in the phase space, is constructed. Time evolution equation of the tomogram for this map is given in explicit form. Properties of the map like the transition probabilities between the different states and relation to the star product formalism are elucidated. Example of multimode oscillator is considered in details. Symmetry properties in respect to identical particles permutations are discussed in th…
We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography.
In open quantum systems, phenomenological master equations with unknown parameters are often introduced. Here we propose a time-independent procedure based on quantum tomography to reconstruct the potentially unknown parameters of a wide class of Markovian master equations. According to our scheme, the system under investigation is initially prepared in a Gaussian state. At an arbitrary time t, in order to retrieve the unknown coefficients one needs to measure only a finite number (ten at maximum) of points along three time-independent tomograms. Due to the limited amount of measurements requi…
The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center-of-mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic symmetries like symplectic group is considered. Entropic uncertainty relations and inequalities for spin tomograms are reviewed.
Invertible maps from operators of quantum obvservables onto functions of c-number arguments and their associative products are first assessed. Different types of maps like Weyl-Wigner-Stratonovich map and s-ordered quasidistribution are discussed. The recently introduced symplectic tomography map of observables (tomograms) related to the Heisenberg-Weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c-number functions. The star-product for symbols of the quantum-observable for each one of the maps (including the tomographic map) and explicit re…
We propose a procedure based on symplectic tomography for reconstructing the unknown parameters of a convolutionless non-Markovian Gaussian noisy evolution. Whenever the time-dependent master equation coefficients are given as a function of some unknown time-independent parameters, we show that these parameters can be reconstructed by means of a finite number of tomograms. Two different approaches towards reconstruction, integral and differential, are presented and applied to a benchmark model made of a harmonic oscillator coupled to a bosonic bath. For this model the number of tomograms neede…
We obtain sufficient conditions on kernels of quantum states under which Wigner functions, optical quantum tomograms and linking their formulas are correctly defined. Our approach is based upon the Sobolev embedding theorem. The transition probability formula and the fractional Fourier transform are discussed in this framework.
The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle without any redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistributions $\vec W({\mathbf q},{\mathbf p})$, $\vec Q({\mathbf q},{\mathbf p})$, $\vec P(\vecα)$ are introduced. The evolution equations for proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. The quantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in proposed representations and evolution equati…
We present a tomographic approach to the study of quantum nonlocality in multipartite systems. Bell inequalities for tomograms belonging to a generic tomographic scheme are derived by exploiting tools from convex geometry. Then, possible violations of these inequalities are discussed in specific tomographic realizations providing some explicit examples.
We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position a…
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture,which has evolved in the $C^\star-$algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept. The equivalence between these two pictures amounts essentially, to the Gelfand-N…
The tomographic description of a quantum state is formulated in an abstract infinite dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitr…
Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. Time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. Entanglement phenomenon of a bipartite muon-electron system is investigated via tomographic analogues of Bell number and positive partial transpose (PPT) criterion. Reconstruction of the muon-electron spin s…
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme …
