Free Particle States Research Articles

Design of Quantum Gates Using Quantum Scattering Theory

Given two Hamiltonians \(H_0,H_1=H_0+V\) corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time \(t=-\infty\), entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time \(t\rightarrow\infty\) to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.

Gravitation-induced temperature of single atoms on atomic and cosmological scales

The experimental findings on the rate of transfer of a single Xe atom from the sample surface towards the tip in the low temperature STM, induced by voltage pulses, are understood by assuming a vibrational temperature for a single Xe atom. But temperature is introduced in standard quantum physics as a parameter in the Boltzmann distribution of a statistical ensemble consisting of large number of non-interacting exact copies of the dynamical system under investigation. This represents a problem, if the system under investigation is to be described by a single pure quantum state being a solution of Schrödinger’s equation. In the framework of a quantum field theory based on Einstein-Hilbert action in high dimensions (EHHD) we demonstrate that temperature for a single atom emerges if the atom is beabled in a warp resonance due to entanglement with local gravitons (gravonons), living in high spacetime dimensions (11D). Desorbing Xe atom in a limbo state in the tunnel gap, due to destruction of the beable via inelastic tunnelling, is in Boltzmann distribution over plane waves {| k〉} since the squared overlap of the warp resonance with free plane wave states varies like exp(–Ek/kBTvib ) where Ek is the energy of the free particle state and Tvib is the vibrational temperature of Xe calculated from the shape and the extension of the warp resonance. The nature of the warp resonance varies with the tip-sample voltage (becoming more contracted at higher tunnelling currents, as the tip approaches the sample) and raises its energy to higher values. More contracted warps at higher tunnelling currents imply stronger overlap with free states of higher energy, described by Boltzmann factors of higher temperature. The mechanism generating single atom temperature generalizes to other dynamical processes where temperature is completely determined by the nature of the warp resonance in the beables. For homogeneous and isotropic systems this implies a universal value of the temperature everywhere in the system. With these considerations the mechanism is also applied to intergalactic hydrogen-atom clusters, where it explains the observed temperature of the CMB radiation and the absence of the horizon problem.

Squeezed and Correlated States of Parametric Oscillator and Free Particle in the Probability Representation of Quantum Mechanics

Squeezed and Correlated States of Parametric Oscillator and Free Particle in the Probability Representation of Quantum Mechanics

Probability backflow for correlated quantum states

In its original formulation, quantum backflow (QB) is an interference effect that manifests itself as a negative probability transfer for free-particle states comprised of plane waves with only positive momenta. Quantum reentry (QR) is another interference effect in which a wave packet expanding from a spatial region of its initial confinement partially returns to the region in the absence of any external forces. Here we show that both QB and QR are special cases of a more general classically-forbidden probability flow for quantum states with certain position-momentum correlations. We further demonstrate that it is possible to construct correlated quantum states for which the amount of probability transferred in the "wrong" (classically impossible) direction exceeds the least upper bound on the corresponding probability transfer in the QB and QR problems, known as the Bracken-Melloy constant.

Exact overlaps in the Lieb-Liniger model from coordinate Bethe ansatz

In the paper [1], the authors developed a new method to compute the exact overlap formulas between integrable boundary states and on-shell Bethe states in integrable spin chains. This method utilizes the coordinate Bethe ansatz representation of wave functions and singularity property of the off-shell overlaps. In this paper, we use this new method to derive the formula for overlaps between the Lieb-Liniger Bethe states and the Bose-Einstein condensate (BEC) state. As a simple application this method, we obtained the overlaps between the Lieb-Liniger eigenstates and the free particle states with pair structure.

Phase protection of Fano-Feshbach resonances

Decay of bound states due to coupling with free particle states is a general phenomenon occurring at energy scales from MeV in nuclear physics to peV in ultracold atomic gases. Such a coupling gives rise to Fano-Feshbach resonances (FFR) that have become key to understanding and controlling interactions—in ultracold atomic gases, but also between quasiparticles, such as microcavity polaritons. Their energy positions were shown to follow quantum chaotic statistics. In contrast, their lifetimes have so far escaped a similarly comprehensive understanding. Here, we show that bound states, despite being resonantly coupled to a scattering state, become protected from decay whenever the relative phase is a multiple of π. We observe this phenomenon by measuring lifetimes spanning four orders of magnitude for FFR of spin–orbit excited molecular ions with merged beam and electrostatic trap experiments. Our results provide a blueprint for identifying naturally long-lived states in a decaying quantum system.

Electroweak symmetric dark matter balls

In the simple Higgs-portal dark matter model with a conserved dark matter number, we show that there exists a non-topological soliton state of dark matter. This state has smaller energy per dark matter number than a free particle state and has its interior in the electroweak symmetric vacuum. It could be produced in the early universe from first-order electroweak phase transition and contribute most of dark matter. This electroweak symmetric dark matter ball is a novel macroscopic dark matter candidate with an energy density of the electroweak scale and a mass of 1 gram or above. Because of its electroweak-symmetric interior, the dark matter ball has a large geometric scattering cross section off a nucleon or a nucleus. Dark matter and neutrino experiments with a large-size detector like Xenon1T, BOREXINO and JUNO have great potential to discover electroweak symmetric dark matter balls. We also discuss the formation of bound states of a dark matter ball and ordinary matter.

Spatiotemporal dynamics of Coulomb-correlated carriers in semiconductors

When the excitation of carriers in real space is focused down to the nanometer scale, the carrier system can no longer be viewed as homogeneous and ultrafast transport of the excited carrier wave packets occurs. In state-of-the-art semiconductor structures like low-dimensional heterostructures or monolayers of transition metal dichalcogenides, the Coulomb interaction between excited carriers becomes stronger due to confinement or reduced screening. This demands a fundamental understanding of strongly interacting electrons and holes and the influence of Coulomb correlations. To study the corresponding particle dynamics in a controlled way we consider a system of up to two electron-hole pairs exactly within a wave function approach. We show that the excited wave packets contain a non-trivial mixture of free particle and excitonic states. We further scrutinize the influence of Coulomb interaction on the wave packet dynamics revealing its different role for below and above band-gap excitation.

Hartree-Fock treatment of Fermi polarons using the Lee-Low-Pine transformation

We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with non-interacting host fermions. We approach the problem starting with a Frohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Frohlich Hamiltonian into the fermionic LLP Hamiltonian which describes a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions with momenta $\mathbf{k}$ and $\mathbf{k}'$ interact with a potential proportional to $\mathbf{k}\cdot\mathbf{k}'$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions

Atom-optics approach to studying transport phenomena

We present a simple experimental scheme, based on standard atom optics techniques, to design highly versatile model systems for the study of single particle quantum transport phenomena. The scheme is based on a discrete set of free-particle momentum states that are coupled via momentum-changing two-photon Bragg transitions, driven by pairs of interfering laser beams. In the effective lattice models that are accessible, this scheme allows for single-site detection, as well as site-resolved and dynamical control over all system parameters. We discuss two possible implementations, based on state-preserving Bragg transitions and on state-changing Raman transitions, which respectively allow for the study of nearly arbitrary single particle Abelian U(1) and non-Abelian U(2) lattice models.

Relativistic dynamical collapse model

A model is discussed where all operators are constructed from a quantum scalar field whose energy spectrum takes on all real values. The Schr\"odinger picture wave function depends upon space and time coordinates for each particle, as well as an inexorably increasing evolution parameter $s$ which labels a foliation of space-like hypersurfaces. The model is constructed to be manifestly Lorentz invariant in the interaction picture. Free particle states and interactions are discussed in this framework. Then, the formalism of the CSL (Continuous Spontaneous Localization) theory of dynamical collapse is applied. The collapse-generating operator is chosen to to be the particle number space-time density. Unlike previous relativistically invariant models, the vacuum state is not excited. The collapse dynamics depends upon two parameters, a parameter $\Lambda$ which represents the collapse rate/volume and a scale factor $\ell$. A common example of collapse dynamics, involving a clump of matter in a superposition of two locations, is analyzed. The collapse rate is shown to be identical to that of non-relativistic CSL when the GRW-CSL choice of $\ell=a=10^{-5}$cm, is made, along with $\Lambda=\lambda/a^{3}$ (GRW-CSL choice $\lambda=10^{-16}s^{-1}$). However, it is also shown that the change of mass of a nucleon over the age of the universe is then unacceptably large. The case where $\ell$ is the size of the universe is then considered. It is shown that the collapse behavior is satisfactory and the change of mass over the age of the universe is acceptably small, when $\Lambda= \lambda/\ell a^{2}$.

Coherent States of Quantum Free Particle on the Spherical Space

In this paper, we study the quantum free particle on the spherical space by applying da costa approach for quantum particle on the curved space. We obtain the discrete energy eigenvalues and associated normalized eigenfunctions of the free particle on the sphere. In addition, we introduce the Gazeau-Klauder coherent states of free particle on the sphere. Then, the Gaussian coherent states is defined, which is used to describe the localized particle on the spherical space. Finally, we study the relation between the f-deformed coherent states and Gazeau-Klauder ones for this system.

Когерентные и полуклассические состояния свободной частицы

Когерентные и полуклассические состояния свободной частицы

Comment on the Uncertainty Relation with Periodic Boundary Conditions

The Kennard-type uncertainty relation $\Delta x\Delta p >\frac{\hbar}{2}$ is formulated for a free particle with given momentum < \hat{p}>$ inside a box with periodic boundary conditions in the large box limit. Our construction of a free particle state is analogous to that of the Bloch wave in a periodic potential. A simple Robertson-type relation, which minimizes the effect of the box boundary and may be useful in some practical applications, is also presented.

Dirac oscillator with minimal lengths and free particle on AdS2 and S2

We use two different types of laddering equations which are realized by the associated Gegenbauer functions. We show that the Dirac oscillator equation with minimal lengths can be written to form associated Gegenbauer functions. This leads us to write the Dirac oscillator equation in terms of first order equations. By changing some variable in two types of equations, we have some u(1,1) and u(2) algebras. These algebras correspond to AdS2 and S2. Finally, we conclude that all quantum states from the Dirac oscillator (presence of minimal lengths) correspond to the quantum states of free particle on AdS2 and S2.

Quantum-mechanical time evolution and uniform forces

The time evolution of an arbitrary one-dimensional initial quantum state, subject to a time-independent, spatially uniform classical force, is shown to be that of a free-particle state, plus an overall motion arising from the classical force. Bohmian mechanics is then used to extend this result to an arbitrarily time-dependent, uniform force in three dimensions. The resulting solution is completed and confirmed using conventional quantum mechanics.

Photoionization cross section of hydrogenic impurities in spherical quantum dots

We have used a variational wave function for hydrogenic impurities in spherical quantum dots to calculate the dependence of the photoionization cross section of such impurities on the energy and polarization of the photon. The calculation has been performed using both the infinite and finite confining potential well models. The results are presented for various dot radii and for different locations of the impurity in the dot. We find that for hydrogenic impurities at the center of a spherical quantum dot, the transitions take place between the impurity level associated with the ground subband and the free particle state in the second subband and the cross section is independent of the polarization of the photons. For off center impurities, the transitions take place between the impurity level associated with the ground subband and the free particle state in the ground subband when the photons are polarized along the direction connecting the impurity with the center of the dot while for photons polarized perpendicular to this direction, we find that the transitions take place between the impurity level associated with the ground subband and the free particle state in the second subband. Therefore the threshold energy for the photoionization cross section is lower for off center impurities when the photons are polarized along the direction connecting the impurity with the center of the dot than it is for on center impurities or for off center impurities for photons polarized perpendicular to this direction.

Asymptotic freedom for nonrelativistic confinement

Some aspects of asymptotic freedom are discussed in the context of a simple two-particle nonrelativistic confining potential model. In this model, asymptotic freedom follows from the similarity of the free-particle and bound state radial wave functions at small distances and for the same angular momentum and the same large energy. This similarity, which can be understood using simple quantum mechanical arguments, can be used to show that the exact response function approaches that obtained when final state interactions are ignored. A method of calculating corrections to this limit is given, and explicit examples are given for the case of a harmonic oscillator.

Principles of maximally classical and maximally realistic quantum mechanics

Recently Auberson, Mahoux, Roy and Singh have proved a long standing conjecture of Roy and Singh: In 2N-dimensional phase space, a maximally realistic quantum mechanics can have quantum probabilities of no more than N+1 complete commuting cets (CCS) of observables coexisting as marginals of one positive phase space density. Here I formulate a stationary principle which gives a nonperturbative definition of a maximally classical as well as maximally realistic phase space density. I show that the maximally classical trajectories are in fact exactly classical in the simple examples of coherent states and bound states of an oscillator and Gaussian free particle states. In contrast, it is known that the de Broglie-Bohm realistic theory gives highly nonclassical trajectories.

The partial averaging Fourier path Integral approach based on the harmonic reference path

The Monte Carlo Fourier path integral approach has proved to be quite useful in calculating equilibrium thermodynamic properties. One of its advantages is that it can be formulated in such a way as to include higher order terms using the partial averaging technique, which includes the contribution from higher terms usually neglected by the discretized path integral approach. In the original approach, the Feynman path integral is evaluated via a free-particle reference state. Here, using a new expression for the Feynman paths expanded around a harmonic reference path, we derive an alternative formulation for the density matrix element and its corresponding partial averaging expression.