Quantum Rep., Volume 1, Issue 2 (December 2019) – 16 articles

We investigate the dynamics of a few bosons in an optical lattice induced by a quantum quench of a parameter of the many-body Hamiltonian. The evolution of the many-body wave function is obtained by solving the time-dependent many-body Schrödinger equation numerically, using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We report the time evolution of three key quantities, namely, the occupations of the natural orbitals, that is, the eigenvalues of the one-body reduced density matrix, the many-body Shannon information entropy, and the quantum fidelity for a wide range of interactions. Our key motivation is to characterize relaxation processes where various observables of an isolated and interacting quantum many-body system dynamically converge to equilibrium values via the quantum fidelity and via the production of many-body entropy. The interaction, as a parameter, can induce a phase transition in the ground state of the system from a superfluid (SF) state to a Mott-insulator (MI) state. We show that, for a quench to a weak interaction, the fidelity remains close to unity and the entropy exhibits oscillations. Whereas for a quench to strong interactions (SF to MI transition), the relaxation process is characterized by the first collapse of the quantum fidelity and entropy saturation to an equilibrium value. The dip and the non-analytic nature of quantum fidelity is a hallmark of dynamical quantum phase transitions. We quantify the characteristic time at which the quantum fidelity collapses and the entropy saturates. Full article

We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations. Full article

The myelin sheath facilitates action potential conduction along the axons, however, the mechanism by which myelin maintains the spatiotemporal fidelity and limits the hyperexcitability among myelinated neurons requires further investigation. Therefore, in this study, the model of quantum tunneling of potassium ions through the closed channels is used to explore this function of myelin. According to the present calculations, when an unmyelinated neuron fires, there is a probability of $9.15\times {10}^{-4}$ that it will induce an action potential in other unmyelinated neurons, and this probability varies according to the type of channels involved, the channels density in the axonal membrane, and the surface area available for tunneling. The myelin sheath forms a thick barrier that covers the potassium channels and prevents ions from tunneling through them to induce action potential. Hence, it confines the action potentials spatiotemporally and limits the hyperexcitability. On the other hand, lack of myelin, as in unmyelinated neurons or demyelinating diseases, exposes potassium channels to tunneling by potassium ions and induces the action potential. This approach gives different perspectives to look at the interaction between neurons and explains how quantum physics might play a role in the actions occurring in the nervous system. Full article

I impose the Newtonian criteria of inertial frames on the c.o.m.trajectories of massive objects undergoing spontaneous collapse of their wave function. The corresponding modification of the so far used stochastic Schrödinger equation eliminates the Brownian motion of the c.o.m., and restores the exact inertial motion for free masses. For the collapse of Schrödinger cat states the Born rule is satisfied invariably. The proposed machinery comes from the radical assumption that, in the vicinity of the spontaneously localized mass, the stochastic fluctuations of the c.o.m.—inevitable in the collapse process—would drag the physical inertial frame with themselves. The perspective of a general theory is presented where the spontaneous-collapse-caused breakdown of local energy-momentum conservation could be remedied by altering the metric, resulting in collapse-induced curvature of the space-time. My assumption of frame-drag by quantized masses is independent of the general relativistic frame-drag by classical masses. Full article

In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the $SU\left(2\right)$ coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space. Full article

We show that sharing a quantum reference frame requires sharing measurement operators that identify the reference frame in addition to operators that measure its state. Observers restricted to finite resources cannot, in general, operationally determine that they share such operators. Uncertainty about whether system-identification operators are shared induces decoherence. Full article

Heisenberg’s uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the $Sp\left(2\right)$ group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the $O(2,1)$ group. This group has three independent generators. The one-dimensional step-up and step-down operators can be combined into one two-by-two Hermitian matrix which contains three independent operators. If we use a two-variable Heisenberg commutation relation, the two pairs of independent step-up, step-down operators can be combined into a four-by-four block-diagonal Hermitian matrix with six independent parameters. It is then possible to add one off-diagonal two-by-two matrix and its Hermitian conjugate to complete the four-by-four Hermitian matrix. This off-diagonal matrix has four independent generators. There are thus ten independent generators. It is then shown that these ten generators can be linearly combined to the ten generators for Dirac’s two oscillator system leading to the group isomorphic to the de Sitter group $O(3,2)$ , which can then be contracted to the inhomogeneous Lorentz group with four translation generators corresponding to the four-momentum in the Lorentz-covariant world. This Lorentz-covariant four-momentum is known as Einstein’s $E=m{c}^2.$ Full article

We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces. Full article

Voltage-gated channels play an essential role in action potential propagation when their closed gates open, but their role when they are closed needs to be investigated. So, in this study, a quantum mechanical approach using the idea of quantum tunneling was used to calculate the conductance of closed channels for different ions. It was found that the conductance due to quantum tunneling of ions through the closed channels does not affect the resting membrane potential. However, under different circumstances, including change in the mass or the charge of the ion and the residues of the hydrophobic gate, the model of quantum tunneling would be useful to understand and explain several actions, processes, and phenomena in the biological systems. Full article

Spatial confinements induce localization or delocalization on the electron density in atoms and molecules, and the hydrogen atom is not the exception to these results. In previous works, this system has been confined by an infinite and a finite potential where the wave-function exhibits an exact solution, and, consequently, their Shannon entropies deliver exact results. In this article, the Shannon entropy in configuration space is examined for the hydrogen atom submitted to four different potentials: (a) infinite potential; (b) Coulomb plus harmonic oscillator; (c) constant potential; and (d) dielectric continuum. For all these potentials, the Schrödinger equation admitted an exact analytic solution, and therefore the corresponding electron density has a closed-form. From the study of these confinements, we observed that the Shannon entropy in configuration space is a good indicator of localization and delocalization of the electron density for ground and excited states of the hydrogen atom confined under these circumstances. In particular, the confinement imposed by a parabolic potential induced characteristics that were not presented for other confinements; for example, the kinetic energy exhibited oscillations when the confinement radius is varied and such oscillations coincided with oscillations showed by the Shannon entropy in configuration space. This result indicates that, when the kinetic energy is increased, the Shannon entropy is decreased and vice versa. Full article

We consider a quantum charged particle moving in the $xy$ plane under the action of a time-dependent magnetic field described by means of the linear vector potential of the form $\mathbf{A}=B\left(t\right)\left[-y(1+\beta ),x(1-\beta )\right]/2$ . Such potentials with $\beta \ne 0$ exist inside infinite solenoids with non-circular cross sections. The systems with different values of $\beta$ are not equivalent for nonstationary magnetic fields or time-dependent parameters $\beta \left(t\right)$ , due to different structures of induced electric fields. Using the approximation of the stepwise variations of parameters, we obtain explicit formulas describing the change of the mean energy and magnetic moment. The generation of squeezing with respect to the relative and guiding center coordinates is also studied. The change of magnetic moment can be twice bigger for the Landau gauge than for the circular gauge, and this change can happen without any change of the angular momentum. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields. Full article

In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in ${\mathbb{R}}^4$ . The construction of this realization is explicitly shown to clarify the role of the commutation relations of the Poincaré algebra versus their description in terms of Poisson brackets in the no-interaction theorem. Moreover, a geometrical analysis of the “eleventh generator” formalism introduced by Sudarshan and Mukunda is outlined, this formalism being at the basis of many proposals which evaded the no-interaction theorem. Full article

Ideal photon-number-resolving detectors form a class of important optical components in quantum optics and quantum information theory. In this article, we theoretically investigate the potential of multiport devices having reconstruction performances approaching that of the Fock-state measurement. By recognizing that all multiport devices are minimally complete, we first provide a general analytical framework to describe the tomographic accuracy (or quality) of these devices. Next, we show that a perfect multiport device with an infinite number of output ports functions as either the Fock-state measurement when photon losses are absent or binomial mixtures of Fock-state measurements when photon losses are present and derive their respective expressions for the tomographic transfer function. This function is the scaled asymptotic mean squared error of the reconstructed photon-number distributions uniformly averaged over all distributions in the probability simplex. We then supply more general analytical formulas for the transfer function for finite numbers of output ports in both the absence and presence of photon losses. The effects of photon losses on the photon-number resolving power of both infinite- and finite-size multiport devices are also investigated. Full article

The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states. Full article