Energy Uncertainty Relation Research Articles

Tunneling time in attosecond experiments, intrinsic-type of time. Keldysh, and Mandelstam–Tamm time

Tunneling time in attosecond and strong-field experiments is one of the most controversial issues in current research, because of its importance to the theory of time, the time operator and the time–energy uncertainty relation in quantum mechanics. In Kullie (2015 Phys. Rev. A 92 052118) we derived an estimation of the (real) tunneling time, which shows an excellent agreement with the time measured in attosecond experiments, our derivation is found by utilizing the time–energy uncertainty relation, and it represents a quantum clock. In this work, we show different aspects of the tunneling time in attosecond experiments, we discuss and compare the different views and approaches, which are used to calculate the tunneling time, i.e. Keldysh time (as a real or imaginary quantity), Mandelstam–Tamm time, the classical view of the time measurement and our tunneling time relation(s). We draw some conclusions concerning the validity and the relation between the different types of the tunneling time with the hope that they will help to answer the question put forward by Orlando et al (2014 J. Phys. B 47 204002, 2014 Phys. Rev. A 89 014102): tunneling time, what does it mean? However, as we will see, the important question is a more general one: how to understand the time and the measurement of the time of a quantum system? In respect to our result, the time in quantum mechanics can be, in more general fashion, classified in two types, intrinsic dynamically connected, and external dynamically not connected to the system, and consequently (perhaps only) classical Newtonian time remains as a parametric type of time.

Quantum Correlations of Photons and Qudit States

We present briefly the historical aspects of studying the properties of light including both the classical theory of light based on the electromagnetic wave picture and the quantum corpuscular interpretation of light based on the notion of photons. We consider vibrations of electric and magnetic fields in light waves and quantum correlations described by the Heisenberg and Schrodinger–Robertson uncertainty relations valid for any vibrations in connection with specific statistical properties of coherent light and squeezed light. We review the qubit and qudit states and the new uncertainty relation for entropy and energy for qudit systems. We discuss the dynamical (nonstationary) Casimir effect providing a vacuum generator of squeezed photons based on the technique of superconducting circuits realized by Josephson junctions. We demonstrate new entropic inequalities for two qubits (polarization states of two photons) and hidden correlations of noncomposite quantum systems (quantum roulette).

Excitation of time-dependent quantum systems: An application of time–energy uncertainty relations

The conditions under which time–energy uncertainty relations derived by Deffner and Lutz [10] for time-dependent quantum systems minimize the time necessary to excite such systems from their ground state to excited states are examined. The generalized Margolus–Levitin and Mandelstam–Tamm inequalities are worked out for specific fermionic and bosonic systems.

Position-dependent effective mass system in a variable potential: displacement operator method

Within the framework of the translation operator for a quantum system with position-dependent mass, we examine the quantum state of a position-dependent mass system in a variable potential. By imposing conditions of resolvability, we arrive at a potential with a quartic and a quadratic term. It emerges naturally that the energy eigen states of the system are negative. We have found the quantum mechanical quantities: energy spectrum, eigen functions and uncertainty relation. These quantities depend on the parameters of the potential.

EPR before EPR: a 1930 Einstein–Bohr thought experiment revisited

In 1930, Einstein argued against the consistency of the time–energy uncertainty relation by discussing a thought experiment involving a measurement of the mass of the box which emitted a photon. Bohr seemingly prevailed over Einstein by arguing that Einstein's own general theory of relativity saves the consistency of quantum mechanics. We revisit this thought experiment from a modern point of view at a level suitable for an undergraduate readership and find that neither Einstein nor Bohr was correct. Instead, this thought experiment should be thought of as an early example of a system demonstrating nonlocal ‘EPR’ quantum correlations, five years before the famous Einstein–Podolsky–Rosen paper.

Exact energy–time uncertainty relation for arrival time by absorption

We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle’s arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wavefunction, we show that and , where 〈T〉 denotes the mean arrival time and p is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.

Neutrino oscillations and uncertainty relations

We show that coherent flavor neutrino states are produced (and detected) due to the momentum-coordinate Heisenberg uncertainty relation. The Mandelstam–Tamm time–energy uncertainty relation requires non-stationary neutrino states for oscillations to happen and determines the time interval (propagation length) which is necessary for that. We compare different approaches to neutrino oscillations which are based on different physical assumptions but lead to the same expression for the neutrino transition probability in standard neutrino oscillation experiments. We show that a Mössbauer neutrino experiment could allow us to distinguish different approaches and we present arguments in favor of the 163Ho–163Dy system for such an experiment.

Reply to the Comment on ‘Time–energy uncertainty relations for neutrino oscillations and the Mössbauer neutrino experiment’ by E Kh Akhmedov, J Kopp and M Lindner

We clarify our derivation of the Mandelstam–Tamm time–energy uncertainty relation for neutrino oscillations presented in Bilenky et al (2008 J. Phys. G: Nucl. Part. Phys. 35 095003) and refute critiques of our paper given in the contribution by Akhmedov et al (2009 J. Phys. G: Nucl. Part. Phys. 36 078001). We demonstrate that the inequality ΔE ⩾ |E3 − E2| obtained by Akhmedov et al (2009 J. Phys. G: Nucl. Part. Phys. 36 078001) cannot be interpreted as a time–energy uncertainty relation.

Comment on ‘Time–energy uncertainty relations for neutrino oscillations and the Mössbauer neutrino experiment’

We discuss the implications of the time–energy uncertainty relation to recoillessly emitted and captured neutrinos (Mössbauer neutrinos) and show that it does not preclude oscillations of these neutrinos, contrary to a recent claim (J. Phys. G: Nucl. Part. Phys. 35 (2008) 095003, arXiv:0803.0527).

Deformed time–energy uncertainty in string theory

The space-time extension of strings is studied in terms of the ordinary time–energy uncertainty relation. Based on the minimal length uncertainty relation, a deformed time–energy uncertainty relation is obtained.

A note on the innocuous implications of a minimum length in quantum gravity

A modification to the time–energy uncertainty relation in quantum gravity has been interpreted as increasing the duration of fluctuations producing virtual black holes with masses greater than the Planck mass. I point out that such virtual black holes have an exponential factor arising from the action such that their contribution to proton decay is suppressed, rather than enhanced, relative to Planck-mass black holes.

Time–energy uncertainty relations for neutrino oscillations and the Mössbauer neutrino experiment

Using the Mandelstam–Tamm method we derive time–energy uncertainty relations for neutrino oscillations. We demonstrate that the small energy uncertainty of antineutrinos in a recently considered experiment with recoilless resonant (Mössbauer) production and absorption of tritium antineutrinos is in conflict with the energy uncertainty which, according to the time–energy uncertainty relation, is necessary for neutrino oscillations to happen. Oscillations of Mössbauer neutrinos would indicate a stationary phenomenon where the evolution of the neutrino state occurs in space rather than in time. A Mössbauer neutrino experiment could provide a unique possibility to reveal the true nature of neutrino oscillations.

Almost-periodic time observables for bound quantum systems

It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalizing the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalizes the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.

Modulation of supercontinuum generation and formation of an attosecond pulse from a generalized two-level medium

The propagation of ultrashort pulses in a generalized two-level system, consisting of permanent dipole moments, is simulated by solving the full Maxwell–Bloch equations. Special attention has been paid to the supercontinuum generation of spectra and the formation of attosecond (as) pulses. It is found that the supercontinuum generation is strongly modulated by both area and width of the pulse, resulting from the interference between the splitting pulses in the time domain and the implication of the time–energy uncertainty relation. The effect of the permanent dipole moment on the supercontinuum generation is discussed in detail. Calculations show that a well-shaped 132 as pulse can be generated from a 2 fs incoming pulse under the condition where the permanent dipole moment difference between two levels is equal to the transition dipole moment between them. Influences of carrier-envelope phase and time-dependent ionization on the spectral and temporal evolution of the ultrashort pulses are also discussed at length.

Time operator and quantum projection evolution

In this paper, we consider time as a dynamical variable. In particular, we present the explicit realization of the time operator within four-dimensional nonrelativistic spacetime. The approach assumes including events as a part of the evolution. The evolution is not driven by the physical time, but it is based on the causally related physical events. The usual Schrodinger unitary evolution can be easily derived as a special case of the three-dimensional projection onto the space of simultaneous events. Also the time—energy uncertainty relation makes clear and mathematically rigorous interpretation.

On neutrino oscillations and time–energy uncertainty relation

We consider neutrino oscillations as a non-stationary phenomenon. We show that the time–energy uncertainty relation plays a crucial role in neutrino oscillations.

On survival probability of quantum states

The survival probability of a quantum state encodes essential information concerning the decay rate of quantum particles and is the primary object for investigating the time–energy uncertainty relations and the occurrence of the quantum Zeno effect. The purpose of this article is to uncover some curious properties concerning the relations between the values of the survival probability of a quantum state at different times. These relations put surprising restrictions on the evolution pictures of quantum states, and also illustrate their peculiar intricacies.

Toward Heisenberg-limited spectroscopy with multiparticle entangled states.

The precision in spectroscopy of any quantum system is fundamentally limited by the Heisenberg uncertainty relation for energy and time. For N systems, this limit requires that they be in a quantum-mechanically entangled state. We describe a scalable method of spectroscopy that can potentially take full advantage of entanglement to reach the Heisenberg limit and has the practical advantage that the spectroscopic information is transferred to states with optimal protection against readout noise. We demonstrate our method experimentally with three beryllium ions. The spectroscopic sensitivity attained is 1.45(2) times as high as that of a perfect experiment with three non-entangled particles.

How fast can a quantum state evolve into a target state?

Given a starting quantum state and a target quantum state, and a driving Hamiltonian, how fast can the starting state evolve into the target state according to the Schrödinger dynamics? This problem arises in a variety of contexts such as quantum computation, quantum control, and in particular, the problems of maximum information processing rate of quantum systems. We will establish a new time–energy uncertainty relation, which provides a general upper bound estimation of the evolution speed. A naive yet fundamental example which has interesting implications for quantum search is worked out explicitly.

Time in quantum mechanics and quantum field theory

W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥμ (with Ĥ0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrodinger equation i∂μψ = Ĥμψ and Heisenberg equation d/dxμ = ∂μ + i[Ĥμ, ] are obtained, from which we have: (1) t is to Ĥ0 as xj is to Ĥj (j = 1, 2, 3); likewise, t is to i∂0 as xj is to i∂j; (2) the proposed time operator is canonically conjugate to i∂0 rather than to Ĥ0, therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual ΔxμΔpμ ≥ 1/2 and the Mandelstam–Tamm treatment ΔxμΔHμ ≥ 1/2, have been formulated.