For a particle in a box, the operator $- i \partial_x$ is not Hermitean. We provide an alternative construction of a momentum operator $p = p_R + i p_I$, which has a Hermitean component $p_R$ that can be extended to a self-adjoint operator, as well as an anti-Hermitean component $i p_I$. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box.
In this paper we show that the point-group (geometrical) symmetry is insufficient to account for the degeneracy of the energy levels of the particle in a cubic box. The discrepancy is due to hidden (dynamical symmetry). We obtain the operators that commute with the Hamiltonian one and connect eigenfunctions of different symmetries. We also show that the addition of a suitable potential inside the box breaks the dynamical symmetry but preserves the geometrical one.The resulting degeneracy is that predicted by point-group symmetry.
Several mechanical realizations of PR-Boxes are discussed. Apart from superluminal correlations, which are known not to be realizable by classical devices, the requirements (1) PR-condition on input-output relations, (2) non-signaling, and (3) allowing for independent measurements are discussed separately. The examples show that internal signaling (inside the box) does not invariably imply the possibility of external signaling and may externally be attributed to contextuality. The proposed model requires two classical bits to be exchanged internally in order to realize the defined requirements…
In this paper we present a solution of the Einstein's boxes paradox by modern Quantum Mechanics in which a notion of density matrix is equivalent to a notion of a quantum state of a system. We use a secondary quantization formalism in the attempt to make a description particularly clear. The aim of this paper is to provide pedagogical help to the students of quantum mechanics.
We estimate the correction to the Cooper-pair box energy level splitting due to the quantum motion of a coupled micromechanical gate electrode. While the correction due to zero-point motion is very small, it should be possible to observe thermal motion-induced corrections to the photon-assisted tunneling current.
It is commonly believed that the most general type of a quantum-mechanical measurement is one described by a positive-operator valued measure (POVM). In the present paper, this statement is proven for any measurements on quantum systems with a finite-dimensional state space. The proof of POVM nature of an arbitrary measurement is carried out using a purely operational approach, which is fully ignorant about what is inside a measurement apparatus. The suggested approach gives also an opportunity to derive the Born rule.
We consider quantum ensembles which are determined by pre- and post-selection. Unlike the case of only pre-selected ensembles, we show that in this case the probabilities for measurement outcomes at intermediate times satisfy causality only rarely; such ensembles can in general be used to signal between causally disconnected regions. We show that under restrictive conditions, there are certain non-trivial bi-partite ensembles which do satisfy causality. These ensembles give rise to a violation of the CHSH inequality, which exceeds the maximal quantum violation given by Tsirelson's bound, $B_{\…
We propose a scheme for enhancing the optomechanical coupling between microwave and mechanical resonators by up to seven orders of magnitude to the ultrastrong coupling limit in a circuit optomechanical setting. The tripartite system considered here consists of a Josephson junction Cooper-pair box that mediates the coupling between the microwave cavity and the mechanical resonator. The optomechanical coupling can be modified by tuning the gate charge and the magnetic flux bias of the Cooper-pair box which in turn affect the Josephson capacitance of the Cooper-pair box. We additionally show tha…
We show that the propositional system of a many-box model is always a set-representable effect algebra. In particular cases of 2-box and 1-box models it is an orthomodular poset and an orthomodular lattice respectively. We discuss the relation of the obtained results with the so-called Local Orthogonality principle. We argue that non-classical properties of box models are the result of a dual enrichment of the set of states caused by the impoverishment of the set of propositions. On the other hand, quantum mechanical models always have more propositions as well as more states than the classica…
In the quantum logic framework we show that the no-signaling box model is a particular type of tensor product of the logics of single boxes. Such notion of tensor product is too strong to apply in the category of logics of quantum mechanical systems. Consequently, we show that the no-signaling box models cannot be considered as generalizations of quantum mechanical models.
A challenge of modern physics is to investigate the quantum behavior of a bulk material object, for instance a mechanical oscillator. We have earlier demonstrated that by coupling a mechanical oscillator to the energy levels of embedded rare-earth ion dopants, it is possible to prepare such a resonator in a low phonon number state. Here, we describe how to extend this protocol in order to prepare momentum- and position squeezed states, and we analyze how the obtainable degree of squeezing depends on the initial conditions and on the coupling of the oscillator to its thermal environment.
We propose to use an acoustic surface wave as a memory for a photon carrying orbital angular momentum. We clarify the physical mechanism that enables the transfer of information, derive the angular momentum selection rule that must be obeyed in the process, and show how to optimize the optoacoustic coupling. We theoretically demonstrate that high fidelities can be achieved, using realistic parameters, for the transfer of a coherent optical Laguerre-Gaussian state, associated with large angular momentum, to a mechanical shear mode. Our results add a significant possibility to the ongoing effort…
We show that the quantum mechanical momentum and angular momentum operators are fixed by the Noether theorem for the classical Hamiltonian field theory we proposed.
We give a modified Hamiltonian for a particle in a box with infinite potential walls that takes into account wall effects. The Hamiltonian is expressed in both the position and momentum representation. In the momentum representation the eigenvalue problem for energy is an integral equation.
We show that for a generic quantum mechanical system with more than one open scattering channel, it is not possible to fully reconstruct the theory's S-matrix from spectral information obtained in large finite volumes with periodic boundary conditions. Physically distinct S-matrices can have identical finite-volume spectra for large finite boxes of arbitrary sizes. If the theory is not time-reversal symmetric, there exists an uncountably infinite set of distinct S-matrices with the same spectra. If the theory respects time-reversal symmetry there exists a discrete set of S-matrices with identi…
As it is well-known one of the most fascinating examples in remarkable discussion between Einstein and Bohr on the conceptual foundation of the quantum mechanics (Heisenberg energy-time uncertainty relation especially) was an experimental device representing a box hanged on an elastic spring. The pair of similar devices is used in this work for formulation of a classical and (implicitly) quantum mechanical analog of the famous two capacitors paradox. It admits a simple solution of the paradox since energy difference or seeming paradoxical "loss" can be explained by work of the elastic force fo…
At the 1927 Solvay conference, Einstein presented a thought experiment intended to demonstrate the incompleteness of the quantum mechanical description of reality. In the following years, the thought experiment was picked up and modified by Einstein, de Broglie, and several other commentators into a simple scenario involving the splitting in half of the wave function of a single particle in a box. In this paper we collect together several formulations of this thought experiment from the existing literature; analyze and assess it from the point of view of the Einstein-Bohr debates, the EPR dile…
We describe a nano-electromechnical system that exhibits the "retroactive" quantum jumps discovered by Mabuchi and Wiseman [Phys. Rev. Lett. 81, 4620 (1998)]. This system consists of a Cooper-pair box coupled to a nano-mechanical resonator, in which the latter is continuously monitored by a single-electron transistor or quantum point-contact. Further, we show that these kinds of jumps, and the jumps that emerge in a continuous quantum non-demolition measurement are one and the same phenomena. We also consider manipulating the jumps by applying feedback control to the Cooper-pair box.
We use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are ${\rm SU}(2)$ and its quotient ${\rm SO}(3,\mathbb{R})$. The scheme developed is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. The oth…
We present a graphical approach to understanding the degeneracy, density of states, and cumulative state number for some simple quantum systems. By taking advantage of basic computing operations we define a straightforward procedure for determining the relationship between discrete quantum energy levels and the corresponding density of states and cumulative level number. The density of states for a particle in a rigid box of various shapes and dimensions is examined and graphed. It is seen that the dimension of the box, rather than its shape, is the most important feature. In addition, we look…
A model calculation using the Darwin Lagrangian is carried out for a magnet consisting of two current-carrying charges constrained by centripetal forces to move in a circular path in the presence of the electric field from a distant external point charge. In the limit that the magnet's two charges are non-interacting, the calculation recovers the only valid calculation for hidden mechanical momentum. However, if the magnet's charges are mutually interacting, then there is internal electromagnetic linear momentum associated with the perturbed magnet's electrostatic charge distribution and the m…
Quantizing the transfer of energy and momentum between interacting particles, we obtain a quantum impulse equation and relations that the corresponding mechanical power, force and torque satisfy. In addition to the energy-frequency and momentum-wavelength relations, we introduce the power-frequency and force-wavelength analogs, respectively. Further, we obtain an operator representation for the mechanical power and impact force in the position space and discuss their correspondence with the relevant momentum operator. The position representation of the torque operator and its relation to the o…
We show that the ability to make direct measurements of momentum, in addition to the usual direct measurements of position, allows a simple configuration of two identical mechanical oscillators to be used for broadband back-action-free force metrology. This would eliminate the need for an optical reference oscillator in the scheme of Tsang and Caves [Phys. Rev. Lett. 105, 123601 (2010)], along with its associated disadvantages. We also show that if one is restricted to position measurements alone then two copies of the same two-oscillator configuration can be used for narrow-band back-action-f…
A formalism is developed to obtain the energy eigenvalues of spatially confined quantum mechanical systems in the framework of The usual WKB and MAF methods. The technique is applied to three different cases,viz one dimensional Harmonic Oscillators,Quartic Oscillators and a boxed-in charged particle in electric field.
It has been accepted that the polarization of the photon in vector beams is entangled with its momentum. Here a quantum description is advanced for the polarization that shows entanglement with the momentum. This is done by showing that the Jones vector at each value of the momentum plays the role of the polarization wavefunction in the sense that the Pauli matrices represent the Cartesian components of the polarization in the local reference system with respect to which the Jones vector is defined. The unit vector that the constraint of transversality condition requires to specify the local r…
The main purpose of this paper is to demonstrate and illustrate, once again, the potency of the variational technique as an approximation procedure for the quantization of quantum mechanical systems. By choosing particle-in-a-box wavefunctions as trial wavefunctions, with the size of the box as the variation parameter, approximate eigenenergies and the corresponding eigenfunctions are obtained for the one dimensional free harmonic oscillator.
Einstein's objection against both the completeness claim of the orthodox version and the Bohmian interpretation of quantum theory, using the example of a 'particle in a box', is reiterated and resolved. This is done by proving that the corresponding quantum mechanical states exactly match classical analogues. The latter are shown to result from the recently elaborated physics of diffusion waves.
We study the properties of an atomic Bose--Einstein condensate produced in an optical-box potential, using high-resolution Bragg spectroscopy. For a range of box sizes, up to $70~μ$m, we directly observe Heisenberg-limited momentum uncertainty of the condensed atoms. We measure the condensate interaction energy with a precision of $k_B \times 100$ pK and study, both experimentally and numerically, the dynamics of its free expansion upon release from the box potential. All our measurements are in good agreement with theoretical expectations for a perfectly homogeneous condensate of spatial exte…
We give an explicit prescription for experimentally determining the evolution operators which completely describe the dynamics of a quantum mechanical black box -- an arbitrary open quantum system. We show necessary and sufficient conditions for this to be possible, and illustrate the general theory by considering specifically one and two quantum bit systems. These procedures may be useful in the comparative evaluation of experimental quantum measurement, communication, and computation systems.
The compressibility of a medium, quantifying its response to mechanical perturbations, is a fundamental property determined by the equation of state. For gases of material particles, studies of the mechanical response are well established, in fields from classical thermodynamics to cold atomic quantum gases. Here we demonstrate a measurement of the compressibility of a two-dimensional quantum gas of light in a box potential and obtain the equation of state for the optical medium. The experiment is carried out in a nanostructured dye-filled optical microcavity. We observe signatures of Bose-Ein…
We theoretically investigate the mechanical effect of the light-induced dipole-dipole interaction potential on the atoms in a Bose-Einstein condensate. We present numerical calculations on the magnitude and shape of the induced potentials for different experimentally accessible geometries. It is shown that the mechanical effect can be distinguished from the effect of incoherent scattering for an experimentally feasible setting.
Complex and spinorial techniques of general relativity are used to determine all the states of the $SU(2)$ invariant quantum mechanical systems in which the equality holds in the uncertainty relations for the components of the angular momentum vector operator in two given directions. The expectation values depend on a discrete quantum number and two parameters, one of them is the angle between the two angular momentum components and the other is the quotient of the two standard deviations. Allowing the angle between the two angular momentum components to be arbitrary, \emph{a new genuine quant…
It is known that there exist non-local correlations that respect no-signaling criterion, but violate Bell-type inequalities more than quantum-mechanical correlations. Such super quantum correlations were introduced as the Popescu-Rohrlich (PR) box. We consider such non-local boxes with two/three inputs and two/three outputs. We show that these super quantum correlations can lead to signaling when at least one of the input bit has access to a word line along a closed time-like curve.
A unified vision of the symmetric coupling of angular momenta and of the quantum mechanical volume operator is illustrated. The focus is on the quantum mechanical angular momentum theory of Wigner's 6j symbols and on the volume operator of the symmetric coupling in spin network approaches: here, crucial to our presentation are an appreciation of the role of the Racah sum rule and the simplification arising from the use of Regge symmetry. The projective geometry approach permits the introduction of a symmetric representation of a network of seven spins or angular momenta. Results of extensive c…
The induced fractional zero-point canonical angular momentum on charged particles by the Aharonov - Bohm (AB) vector potential is realized via modified combined traps. It explores new features for this type of quantum effects: In a limit of vanishing mechanical kinetic energy the AB vector potential alone cannot induce a fractional zero-point canonical angular momentum on charged particles at the quantum mechanical level in the AB magnetic field-free region; But for the case of the AB vector potential with another one of a "spectator" magnetic field the AB vector potential induces a fractional…
Bessel beams are studied within the general framework of quantum optics. The two modes of the electromagnetic field are quantized and the basic dynamical operators are identified. The algebra of these operators is analyzed in detail; it is shown that the operators that are usually associated to linear momentum, orbital angular momentum and spin do not satisfy the algebra of the translation and rotation group. In particular, what seems to be the spin is more similar to the helicity. Some physical consequences of these results are examined.
We show that a continuous quantum non-demolition measurement of the energy of a nanomechanical resonator can be achieved by monitoring the resonator with a quantum point contact via a Cooper-pair box. This technique can further be used to prepare highly non-classical states of two resonators, such as canonical phase-reference states, and so-called "noon" states.
Propagation effects are analyzed for electromagnetic (EM) waves which satisfy the one-soliton non-linear Schrodinger (NLS) equation in a dispersive wave guide. The coupling between momentum and frequencies due to dispersion relation is treated by a coupled Hamiltonian-Momentum operator with equal-space commutation relations(CR). Kerr interactions in the soliton are treated. The classical solution of the one-soliton with possible quantum effects are analyzed. The integrability of nonlinear equations is related to compatibility-condition for scattering matrices and such relation is applied to th…
The non-commutativity of position and momentum observables is a hallmark feature of quantum physics. However this incompatibility does not extend to observables which are periodic in these base variables. Such modular-variable observables have been suggested as tools for fault-tolerant quantum computing and enhanced quantum sensing. Here we implement sequential measurements of modular variables in the oscillatory motion of a single trapped ion, using state-dependent displacements and a heralded non-destructive readout. We investigate the commutative nature of modular variable observables by de…
A so called 'weak value' of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes (e.g., the so called Three Box paradox and Hardy s paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in c…
An impact of integration over the paths of the Levy flights on the quantum mechanical kernel has been studied. Analytical expression for a free particle kernel has been obtained in terms of the Fox H-function. A new equation for the kernel of a partical in the box has been found. New general results include the well known quantum formulae for a free particle kernel and particle in box kernel.
Motivated by recent progress in the generation of optical spatiotemporal vortex pulses (STVPs), there is a theoretical discussion about the transverse orbital angular momentum (OAM) carried by such pulses. Two recent works [K. Y. Bliokh, Phys. Rev. Lett. 126, 243601 (2021)] and [S. W. Hancock et al., Phys. Rev. Lett. 127, 193901 (2021)] claimed the OAM values which differ by a factor of 2 for circular STVPs. Here we resolve this controversy by showing that the result by Hancock et al. is correct for the total OAM, while the result by Bliokh describes the suitably defined intrinsic part of the …
We investigate the effects of the chiral MIT boundary conditions for a Dirac particle in a 1D box. We show how the boundary condition affects the discrete momentum, energy level, transition frequency, and spin state in a box. The effects of the chiral MIT boundary conditions on the probability and scalar densities are also demonstrated. The results show that an asymmetric distribution appears in the box depending on the parameters of the spin orientation and chiral angle.
Hybrid quantum systems exhibiting coupled optical, spin, and mechanical degrees of freedom can serve as a platform for sensing, or as a bus to mediate interactions between qubits with disparate energy scales. These systems are also creating opportunities to test foundational ideas in quantum mechanics, including direct observations of the quantum regime in macroscopic objects. Here, we make use of angular momentum conservation to study the dynamics of a pair of paramagnetic centers featuring different spin numbers in the presence of a properly tuned external magnetic field. We examine the inte…
After briefly reviewing the definitions of classical probability densities for position, $P_{CL}(x)$, and for momentum, $P_{CL}(p)$, we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, $V(x) = F|x|$, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent…
It is a maxim often stated, yet rarely illustrated, that the combination of special relativity and quantum mechanics necessarily leads to quantum field theory. An elementary illustration is provided, using the familiar particle in a box, boosted to relativistic speeds. It is shown that quantum fluctuations of momentum lead to energy fluctuations, that are inexplicable without a framework that endows the vacuum with dynamical degrees of freedom and allows particle creation/annihilation.
Single-atom detection with a quantum-controlled mechanical oscillator
ETH Zürich（苏黎世联邦理工学院）；Photonics Laboratory, ETH Zürich（苏黎世联邦理工学院光子学实验室）；Quantum Center, ETH Zürich（苏黎世联邦理工学院量子中心）；Laboratory of Physical Chemistry, ETH Zürich（苏黎世联邦理工学院物理化学实验室）；Kavli Institute of Nanoscience, Department of Quantum Nanoscience, TU Delft（代尔夫特理工大学量子纳米科学系卡弗里纳米科学研究所）
In the paper, it is argued that the phenomenon known as the quantum pigeonhole principle (namely, three quantum particles are put in two boxes, yet no two particles are in the same box) can be explained not as a violation of Dirichlet's box principle in the case of quantum particles but as a nonvalidness of a bivalent logic for describing not-yet verified propositions relating to quantum mechanical experiments.
An induced fractional zero-point angular momentum of charged particles by the Bohm-Aharonov (B-A) vector potential is realized via a modified combined trap. It explores a "spectator" mechanism in this type of quantum effects: In the limit of the kinetic energy approaching one of its eigenvalues the B-A vector potential alone cannot induce a fractional zero-point angular momentum at quantum mechanical level in the B-A magnetic field-free region; But when there is a "spectator" magnetic field the B-A vector potential induces a fractional zero-point angular momentum. The "spectator" does not cont…
Initial momenta of de Broglie-Bohm trajectories generally do not obey quantum mechanical momentum distributions. The solution to this problem presented in the following leads to an extended hydrodynamic interpretation of quantum mechanics. To demonstrate the new formalism, it is applied to the double-slit experiment. From this example follows a proposal for experimental verification of trajectory formulations of quantum mechanics.
We study the quantum mechanical generalization of force or pressure, and then we extend the classical thermodynamic isobaric process to quantum mechanical systems. Based on these efforts, we are able to study the quantum version of thermodynamic cycles that consist of quantum isobaric process, such as quantum Brayton cycle and quantum Diesel cycle. We also consider the implementation of quantum Brayton cycle and quantum Diesel cycle with some model systems, such as single particle in 1D box and single-mode radiation field in a cavity. These studies lay the microscopic (quantum mechanical) foun…
