15 pages, dedicated to Anders Lindquist on the occasion of his 60th birthday
发表信息：
in Directions in Mathematical Systems Theory and Optimization, A. Rantzer and C. I. Byrnes Eds., Springer-Verlag, 2002, 227-238. Theory and Optimization, A. Rantzer and C. I. Byrnes Eds., Springer-Verlag, 2002, 227-238
Elaborating on M. Pavon, J.Math. Phys. 40 (1999), 5565-5577, we develop a simplified version of a variational principle within Nelson stochastic mechanics that produces the von Neumann wave packet reduction after a position measurement. This stochastic control problem parallels, with a different kinematics, the problem of the Schroedinger bridge. This gives a profound meaning to what was observed by Schroedinger in 1931 concerning Schroedinger bridges: "Merkwuerdige Analogien zur Quantenmechanik, die mir sehr des Hindenkens wert erscheinen".
A new approach to the steering problem for quantum systems relying on Nelson's stochastic mechanics and on the theory of Schroedinger bridges is presented. The method is illustrated by working out a simple Gaussian example.
Intriguing dichotomies in quantum measurement theory involving the role of the obersever, objective reality, collapse of wavefunctions and actualization of a measurement outcome are cast into a patholigical gedanken experiment involving a single electron in a double quantum dot system coupled via a weak link.
A supersymmetric technique for the solution of the effective mass Schrö% dinger equation is proposed. Exact solutions of the Schroedinger equation corresponding to a number of potentials are obtained. The potentials are fully isospectral with the original potentials. The conditions for the shape invariance of the potentials are discussed.
For the unitary operator, solution of the Schroedinger equation corresponding to a time-varying Hamiltonian, the relation between the Magnus and the product of exponentials expansions can be expressed in terms of a system of first order differential equations in the parameters of the two expansions. A method is proposed to compute such differential equations explicitly and in a closed form.
It is shown that the geometric constraint advocated in [R. S. Kaushal, Mod. Phys. Lett. A 15 (2000) 1391] is trivially satisfied. Therefore, such a constraint does not exist. We also point out another flaw in Kaushal's paper.
Spontaneous symmetry breaking (SSB) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate SSB has been independently understood for Schroedinger operators with double well potential in the classical limit (Jona-Lasinio et al, 1981; Simon, 1985) and for quantum spin systems in the thermodynamic limit (Anderson, 1952; Tasaki, 2019). We relate these to each other in the context of the Curie-Weiss model, establishing a remarkable relationship between this model (for finite N) and a discretized Schroedinger operator with double well potential.
We propose a dynamical scheme for deterministically amplifying photonic Schroedinger cat states based on a set of optimal state-transfers. The scheme can be implemented in strongly coupled qubit-cavity systems and is well suited to the capabilities of state of the art superconducting circuits. The ideal analytical scheme is compared with a full simulation of the open Jaynes-Cummings model with realistic device parameters. This amplification tool can be utilized for practical quantum information processing in non-classical continuous-variable states.
The aim of these three papers (I, II, and III) is to develop a q-deformed version of non-relativistic Schroedinger theory. Paper I introduces the fundamental mathematical and physical concepts. The braided line and the three-dimensional q-deformed Euclidean space play the role of position space. For both cases the algebraic framework is extended by a time element. A short review of the elements of q-deformed analysis on the spaces under consideration is given. The time evolution operator is introduced in a consistent way and its basic properties are discussed. These reasonings are continued by…
An interferometric experiment is described that characterizes an optical cat state in a cavity mode. Our method describes how to measure the amplitude and phase of the different coherent states that make up the cat states. We show that using a recently proposed quantum nondemolition measurement technique, we can circumvent the problem of decoherence inherent in measurement involving superposition of coherent states.
The new process of quantum-injection into an optical parametric amplifier operating in entangled configuration is adopted to amplify into a large dimensionality spin 1/2 Hilbert space the quantum entanglement and superposition properties of the photon-couples generated by parametric down-conversion. The structure of the Wigner function and of the field's correlation functions shows a decoherence-free, multiphoton Schroedinger-cat behaviour of the emitted field which is largely detectable against the squeezed-vacuum noise. Furthermore, owing to its entanglement character, the system is found to…
Canonical coordinates for both the Schroedinger and the nonlinear Schroedinger equations are introduced, making more transparent their Hamiltonian structures. It is shown that the Schroedinger equation, considered as a classical field theory, shares with the nonlinear Schroedinger, and more generally with Liouville completely integrable field theories, the existence of a "recursion operator" which allows for the construction of infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clea…
Superpositions of macroscopically distinct quantum states, introduced in Schroedinger's famous Gedankenexperiment, are an epitome of quantum "strangeness" and a natural tool for determining the validity limits of quantum physics. The optical incarnation of Schroedinger's cat - the superposition of two opposite-amplitude coherent states - is also the backbone of quantum information processing in the continuous-variable domain. However, existing preparation methods limit the amplitudes of the component coherent states by about 2, which curtails the state's usefulness for fundamental and practica…
We present the quantum mechanics problem of the one-dimensional Schroedinger equation with the trigonometric Rosen-Morse potential. This potential is of possible interest to quark physics in so far as it captures the essentials of the QCD quark-gluon dynamics and (i) interpolates between a Coulomb-like potential (associated with one-gluon exchange) and the infinite wall potential (associated with asymptotic freedom), (ii) reproduces in the intermediary region the linear confinement potential (associated with multi-gluon self-interactions) as established by lattice QCD calculations of hadron pr…
This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and moment…
This is the third part of a paper about non-relativistic Schroedinger theory on q-deformed quantum spaces like the braided line or the three-dimensional q-deformed Euclidean space. Propagators for the free q-deformed particle are derived and their basic properties are discussed. A time-dependent formulation of scattering is proposed. In this respect, q-analogs of the Lippmann-Schwinger equation are given. Expressions for their iterative solutions are written down. It is shown how to calculate S-matrices and transition probabilities. Furthermore, attention is focused on the question what become…
We review the use of quantum feedback for combatting the decoherence of Schroedinger-cat-like states in electromagnetic cavities, with special emphasys on our recent proposal of an automatic mechanism based on the injection of appropriately prepared ``probe'' and ``feedback'' Rydberg atoms. In the latter scheme, the information transmission from the probe to the feedback atom is directly mediated by a second auxiliary cavity. The detection efficiency for the probe atom is no longer a critical parameter, and the decoherence time of the linear superposition state can be significantly increased u…
A seldom recognized fundamental difficulty undermines the concept of individual ``state'' in the present formulations of quantum statistical mechanics (and in its quantum information theory interpretation as well). The difficulty is an unavoidable consequence of an almost forgotten corollary proved by E. Schroedinger in 1936 and perused by J.L. Park, Am. J. Phys., Vol. 36, 211 (1968). To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss a possible resol…
We discuss two topics that are usually considered to be exclusively "quantum": the Schroedinger equation, and the uncertainty principle. We show (or rather recall) that the Schroedinger equation can be derived from Hamilton's equations using the metaplectic representation. We also show that the uncertainty principle, stated in the form of the Robertson-Schroedinger-Heisenberg inequalities can be formulated in perfectly classical terms using the topological notion of symplectic capacity.
It is shown how the essentials of quantum theory, i.e., the Schroedinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the only input is given by the assumption of fluctuations in energy and momentum to be added to the classical motion. Extending into the relativistic regime for spinless particles, this procedure leads also to a derivation of the Klein-Gordon equation. Comparing classical Hamiltonian flow with quantum theory, then, the essential difference is given by a vanishing d…
We report preparation and characterization of coherent superposition states t |0> + alpha |1> of electromagnetic field by conditional measurements on a beamsplitter. The state is generated in one of the beam splitter output channels if a coherent state and a single-photon Fock state |1> are present in the two input ports and a single photon is registered in the other beam splitter output. The single photon thus plays a role of a "catalyst": it is explicitly present in both the input and the output channels of the interaction yet facilitates generation of a nonclassical state of light.
It is often claimed that the collapse of the wave function and Born's rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schroedinger equation. Here we show that this is not true in certain models where quantum behavior can be attributed to underlying deterministic equations. It is argued that indeed the apparent spontaneous collapse of wave functions and Born's rule are features that strongly point towards determinism underlying quantum mechanics.
Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be shown that there is one-to-one physical correspondence between basic solutions (represented always by one Hamiltonian eigenfunction only) and classical ones, as the non-zero quantum potential has not any physical sense, representing only the "numerical" difference between Hamilton principal function and the phase of corresponding wave function in the case of no…
We present a tomographic method for the reconstruction of the full entangled quantum state for the cyclotron and spin degrees of freedom of an electron in a Penning trap. Numerical simulations of the reconstruction of several significant quantum states show that the method turns out to be quite accurate.
We report a refinement of Robertson-Schroedinger uncertainty relation via Wigner-Yanase skew information. Besides the well known quantum uncertainty arising from the noncommutativity of observables, there is classical uncertainty arising from the mixedness of the states that is quantified by the difference between the variance and the skew information. Our refined uncertainty relation for canonical observables is saturated by all the Gaussian states, pure or mixed, and thus provides an alternative measure for the non-Gaussianity of quantum states. Generalizations to the case of metric adjusted…
In this paper we re-investigate the core of Schroedinger's 'cat paradox'. We argue that one has to distinguish clearly between superpositions of macroscopic cat states and superpositions of entangled states which comprise both the state of the cat. It is shown, that in the first instance recurrence to decoherence or other mechanisms is not necessary in this special case in order to explain the absence of macroscopic superpositions. Additionally, we present modified versions of two quantum optical experiments as experimenta crucis. Applied rigorously, quantum mechanical formalism reduces the pr…
After a careful analysis of the feedback model recently proposed by Slosser and Milburn [Phys. Rev. Lett. 75, 418 (1995)], we are led to the conclusion that---under realistic conditions---their scheme is not significantly more effective in the production of linear superpositions of macroscopically distinguishable quantum states than the usual quantum-optical Kerr effect.
Recently, Frauchiger and Renner proposed a Gedankenexperiment, which was claimed to be able to prove that quantum theory cannot consistently describe the use of itself. Here we show that the conclusions of Frauchiger and Renner actually came from their incorrect description of some quantum states. With the correct description there will be no inconsistent results, no matter which quantum interpretation theory is used. Especially, the Copenhagen interpretation can satisfy all the three assumptions (C), (Q), and (S) of Frauchiger and Renner simultaneously, thus it has no problem consistently des…
Contractive states for a free quantum particle were introduced by Yuen [Yuen H P 1983 Phys. Rev. Lett. 51, 719] in an attempt to evade the standard quantum limit for repeated position measurements. We show how appropriate families of two- and three component ``Schroedinger cat states'' are able to support non-trivial correlations between the position and momentum observables leading to contractive behavior. The existence of contractive Schroedinger cat states is suggestive of potential novel roles of non-classical states for precision measurement schemes.
The idea of equivalence of the free electromagnetic phase and quantum-mechanical one is investigated in an attempt to seek modifications of Schrödinger's equation that could realize it. It is assumed that physically valid realizations are compatibile with the U(1)-gauge and Galilean invariance. It is shown that such extensions of the Schrödinger equation do not exist, which also means that despite their apparent similarity the quantum-mechanical phase is essentially different from the electromagnetic one.
The connection between the strictly isospectral construction in supersymmetric quantum mechanics and the general zero mode solutions of the Schroedinger equation is explained by introducing slightly generalized first-order intertwining operators. We also present a multiple-parameter generalization of the strictly isospectral construction in the same perspective
This note shows how one can be led from considerations of quantum steering to Bell's theorem. The point is that steering remote systems by choosing between two measurements can be described in a local theory if we take quantum states to be associated many-to-one with the underlying "real states" of the world. Once one adds a third measurement this is no longer possible. Historically this is not how Bell's theorem arose - there are slight and subtle differences in the arguments - but it could have been.
In the Entropic Dynamics (ED) derivation of the Schroedinger equation the physical input is introduced through constraints that are implemented using Lagrange multipliers. There is one constraint involving a "drift" potential that correlates the motions of different particles and is ultimately responsible for entanglement. The purpose of this work is to deepen our understanding of the corresponding multiplier. Its main effect is to control the strength of the drift relative to the fluctuations. We show that ED exhibits a symmetry: models with different values of the multiplier can lead to the …
The Schroedinger equation with the nonlinear term is derived by the natural generalization of the hydrodynamical model of quantum mechanics. The nonlinear term appears to be logically necessary because it enables explanation of the classical limit of the wave function, the collaps of the wave function and solves the Schroedinger cat paradox.
We are presenting an internal trajectory model for a quantum particle in the Schroedinger non-relativistic approximation. This model is based on two new mathematical concepts: a complex analytical mechanics in Minplus complex analysis and a periodical non random process which gives a complex Ito formula. This model naturally generates a concept of spin or isospin and the Heisenberg inequalities, and leads to the Schroedinger equation using a generalization of the least action principle adapted to the trajectories of this type.
In non relativistic quantum mechanics time enters as a parameter in the Schroedinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable - i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admit a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the origina…
The main distinction between classical mechanics and quantum mechanics is the lack in the latter of a full mechanical determinism: different final states can arise from the same physical state, after the measurement. No hidden variable is supposed to exist, nothing can discriminate two apparently identical states even if they give a different result. In this paper we try to put the basis for a more fundamental theory that (approximately) coincides with quantum mechanics when comparing statistics, but it is more fundamental, since it mathematically describes measurement processes giving an expl…
The Schroedinger- and Klein-Gordon equations are directly derived from classical Lagrangians. The only inputs are given by the discreteness of energy (E=hbar.w) and momentum (p=hbar.k), respectively, as well as the assumed existence of a space-pervading field of "zero-point energy" (E_0=hbar.w/2 per spatial dimension) associated to each particle of energy E. The latter leads to an additional kinetic energy term in the classical Lagrangian, which alone suffices to pass from classical to quantum mechanics. Moreover, Heisenberg's uncertainty relations are also derived within this framework, i.e.,…
The hydrodynamical model of quantum mechanics based on the Schroedinger equation is combined with the magnetohydrodynamical term to form so called quantum magnetohydrodynamic equation. It is shown that the quantum correction to the Alfven waves follows from this new equation. The possible generalization is considered for the so called nonlinear Schroedinger equation and for the situation where dissipation is described by the Navier-Stokes equation.
Schroedinger's first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schroedinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully this theory is seen to be an empirically adequate many-worlds theory and not an empirically inadequate theory describing a single world. Moreover, this formulation--Schroedi…
The assumptions added by Bohr and concerning the Hilbert space (formed by all solutions of Schroedinger equation) changed fundamentally the original physical interpretation of these solutions proposed earlier by Schroedinger. This new alternative was refused by Einstein on the basis of the EPR Gedankenexperiment, but accepted fully for microscopic reality by scientific community. Both the quantum alternatives were discussed, however, again later. Bell tried to find a possibility how to decide between them; he generalized Einstein's Gedankenexperiment assuming that also spins of two detected pa…
The well known interpretational difficulties with nonlinear Schrödinger and von Neumann equations can be reduced to the problem of computing multiple-time correlation functions in the absence of Heisenberg picture. Having no Heisenberg picture one often resorts to Zeno-type reasoning which explicitly involves the projection postulate as a means of computing conditional and joint probabilities. Although the method works well in linear quantum mechanics, it completely fails for nonlinear evolutions. We propose an alternative way of performing the same task in linear quantum mechanics and show th…
Here I show that a classical or quantum bit state plus one simple operation, an action, are sufficient ingredients to derive a quantum dynamical equation that rules the sequential changes of the state. Then, by assuming that a freely moving massive particle is the qubit carrier, it is found that both, the particle position in physical space and the qubit state, change in time according to the Pauli-Schroedinger equation. So, this approach suggests the following conjecture: because it carries one qubit of information the particle motion has its description enslaved by the very existence of the …
We propose a measure for the "size" of a Schroedinger cat state, i.e. a quantum superposition of two many-body states with (supposedly) macroscopically distinct properties, by counting how many single-particle operations are needed to map one state onto the other. This definition gives sensible results for simple, analytically tractable cases and is consistent with a previous definition restricted to Greenberger-Horne-Zeilinger-like states. We apply our measure to the experimentally relevant, nontrivial example of a superconducting three-junction flux qubit put into a superposition of left- an…
Schroedinger's Equation at 100: The Wave Picture That Helped and Possibly Hurt
Institute for Quantum Optics and Quantum Information - Vienna, Austrian Academy of Sciences（奥地利科学院维也纳量子光学与量子信息研究所）；Faculty of Physics, University of Vienna（维也纳大学物理学院）
We discuss Staruszkiewicz's nonlinear modification of the Schrödinger equation. It is pointed out that the expression for the energy functional for this modification is not unique as the field-theoretical definition of energy does not coincide with the quantum-mechanical one. As a result, this modification can be formulated in three different ways depending on which physically relevant properties one aims to preserve. Some nonstationary one-dimensional solutions for suitably chosen potentials, including a KdV soliton, are presented, and the question of finding stationary solutions is also disc…
Heisenberg's principle$^1$ states that the product of uncertainties of position and momentum should be no less than Planck's constant $\hbar$. This is usually taken to imply that phase space structures associated with sub-Planck ($\ll \hbar$) scales do not exist, or, at the very least, that they do not matter. I show that this deeply ingrained prejudice is false: Non-local "Schrödinger cat" states of quantum systems confined to phase space volume characterized by `the classical action' $A \gg \hbar$ develop spotty structure on scales corresponding to sub-Planck $a = \hbar^2 / A \ll \hbar$. Suc…
It is shown that the eigenvalue problem for the Hamiltonians of the standard form, $H=p^2/(2m)+V(x)$, is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the solution of the dynamical equations of a class of time-dependent harmonic oscillators is also pointed out.
Quantum field theory is assumed to be gauge invariant. However it is well known that when certain quantities, such as the vacuum current, are calculated the results are not gauge invariant. The non-gauge invariant terms have to be removed in order to obtain a physically correct result. It has been shown in Ref. [3] and [4] that this problem may be due to a mathematical inconsistency in the canonical formulation of QFT. In this article we will review this previous work and then examine an alternative formulation of QFT called the functional Schroedinger approach. It will be shown that this appr…
We construct a world model consisting of a matter field living in 4 dimensional spacetime and a gravitational field living in 11 dimensional spacetime. The seven hidden dimensions are compactified within a radius estimated by reproducing the particle - wave characteristic of diffraction experiments. In the presence of matter fields the gravitational field develops localized modes with elementary excitations called gravonons which are induced by the sources (massive particles). The final world model treated here contains only gravonons and a scalar matter field. The solution of the Schroedinger…
A simple version of the q-deformed calculus is used to generate a pair of q-nonlocal, second-order difference operators by means of deformed counterparts of Darboux intertwining operators for zero factorization energy. These deformed non-local operators may be considered as supersymmetric partners and their structure contains contributions originating in both the Hermite operator and the quantum harmonic oscillator operator. There are also extra $\pm x$ contributions. The undeformed limit, in which all q-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanical…
The transition from quantum to classical, in the case of a quantum harmonic oscillator, is typically identified with the transition from a quantum superposition of macroscopically distinguishable states, such as the Schrödinger cat state, into the corresponding statistical mixture. This transition is commonly characterized by the asymptotic loss of the interference term in the Wigner representation of the cat state. In this paper we show that the quantum to classical transition has different dynamical features depending on the measure for nonclassicality used. Measures based on an operatorial …
We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a sufficient condition for regularity of solutions to linear stochastic Schroedinger equations driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a ne…
It is generally assumed that quantum field theory (QFT) is gauge invariant. However it is well known that non-gauge invariant terms appear in various calculations. This problem was recently examined in [9] for a "simple" field theory and it was shown that for this case QFT in the Schroedinger picture is not, in fact, gauge invariant. In order to shed further light on this problem we will examine the Heisenberg and Schroedinger formulations of QFT. It is generally assumed that these two "pictures" are equivalent; however we will show that this is not necessarily the case. We shall consider a si…
We present a highly accurate method for solving single-active-electron (SAE) atomic eigenset in momentum space. The trouble of Coulomb kernel singularity is bypassed with numerical quadrature, which is simple but effective. The complicated Lande regularization method is no longer necessary. The data of accuracy for some low-lying states of the hydrogen and SAE helium atom were tabulated. Two examples of using the generated eigenset to solve the hydrogen atom under strong-field laser pulses were shown. The momentum and the coordinate representation are complementary to each other in quantum mec…
Nuclei are rather classical systems in a sense. In the old days, their phenomena were roughly explained in classical rules such as the liquid drop model. This fact may be understood that when we see an finite quantum many body system like nucleus, though which is a group of quantum mechanical particles, its any collective degree of freedom has any classicality. Getting a classicality does not depend on spatial scales of objects. It is made by a phenomenon called Quantum Decoherence. We have studied about Quantum Decoherence in a finite system as nucleus. In this paper, at the harmonic three bo…
Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics the equation describing change of the state of quantum system with respect to energy is introduced. The operator of time appears to be the generator of the change of energy while the operator of energy, that is conjugate to the operator of time, generates time evolution in this proposal. Two examples, one with discrete time and the other with continuous one, are given and the generalization of Schroedinger equation is proposed.
