We study Wigner function of a system describing entanglement of two cat-states. Quantum interferece arising due to entanglement is shown to produce sub-Planck structures in the phase-space plots of the Wigner function. Origin of these structures in our case depends on entanglement unlike those in Zurek \cite{Zurek}. It is argued that the entangled cat-states are better suited for carrying out precision measurements.
We discuss how relaxing the requirement of locality for quantum fields can equip the Hilbert space of the theory with a richer structure in its multi-particle sector. A physical consequence is the emergence of a "planckian" mode-entanglement, invisible to an observer that cannot probe the Planck scale. To the same observer, certain unitary processes would appear non-unitary. We show how entanglement transfer to the additional degrees of freedom can provide a potential way out of the black hole information paradox.
We calculate the atomic (spin) Wigner function for the single mode Dicke model in the regime of large number of two-level atoms. The dynamics of this quasi-probability function on the Bloch sphere allows us to visualize the consequences of the entanglement process between the boson and the spin subsystems. Such investigation shows a distinct localization behavior of the spin state with respect to the polar and azimuthal Bloch sphere angles. A complete {\em breakdown of reflection symmetry} in the azimuthal angle is shown in the non-integrable case, even at short evolution times. Also, in the c…
An exactly soluble non-linear interaction Hamiltonian is proposed to study fundamental properties of the entanglement dynamics for a coupled non-linear oscillators. The time-evolved state is obtained analytically for initial products of two coherent and two number states and relevant informations are extracted from the dynamics of various quantities like subsystem linear and Von Neumann entropies, quadrature mean values, variances and Q-functions. We determined the re-coherence time scales and found among the interaction terms present in the Hamiltonian the one responsible for the entanglement…
We present a new interpretation of the terms superposition, entanglement, and measurement that appear in quantum mechanics. We hypothesize that the structure of the wave function for a quantum system at the sub-Planck scale has a deterministic cyclic structure. Each cycle comprises a sequential succession of the eigenstates that comprise a given wave function. Between unitary operations or measurements on the wave function, the sequential arrangement of the current eigenstates chosen by the system is immaterial, but once chosen it remains fixed until another unitary operation or measurement ch…
The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron and nuclear spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, whose two-qubit concurrence and quantum coherence can be …
We illustrate a procedure to generate a bipartite, entangled compass state, which shows sub-Planck structure. The proposed method uses the interaction of a standing wave laser field, with two, two-level atoms and relies on the ability of this system to choose certain mesoscopic bipartite states to couple with the internal degrees of freedom. An appropriate measurement on the internal degrees of freedom then leads to the entangled state, which shows sub-Planck structures, desired for quantum metrology.
We study the effect of decoherence on the sub-Planck scale structures of the vibrational wave packet of a molecule. The time evolution of these wave packets is investigated under the influence of a photonic or phononic environment. We determine the master equation describing the reduced dynamics of the wave-packet and analyze the sensitivity of the sub-Planck structures against decoherence in the case of a hydrogen iodide (HI) molecule.
We propose a method using the dispersive interaction between atoms and a high quality cavity to realize the mesoscopic superposition of coherent states which would exhibit sub-Planck structures in phase space. In particular we focus on a superposition involving four coherent states. We show interesting interferences in the conditional measurements involving two atoms.
It is conjectured that the spatial structure of quantum field states is influenced by a new kind of directional indeterminacy of quantum geometry set by the Planck length, $l_P$, that does not occur in a classical background geometry. Entanglement of fields with geometry modifies the transverse phase of field states at wavelength $λ$ and propagation distance $cτ$ by about $ Δϕ\approx \sqrt{l_Pτ}/λ$. The new effect is not detectable in measurements of propagating states that depend only on longitudinal coordinates. The reduced information content of fields in large systems is consistent with ho…
We demonstrate the possibility of realizing sub-Planck-scale structures in the mesoscopic superposition of molecular wave packets involving vibrational levels. The time evolution of the wave packet, taken here as the SU(2) coherent state of the Morse potential describing hydrogen iodide molecules, produces macroscopicquantum- superposition-like states, responsible for the above phenomenon. We investigate the phase-space dynamics of the coherent state through the Wigner function approach and identify the interference phenomena behind the sub-Planck-scale structures. The optimal parameter ranges…
Heisenberg's principle$^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…
We investigate the sub-Planck-scale structures associated with the SU(1,1) group by establishing that the Planck scale on the hyperbolic plane can be considered as the inverse of the Bargmann index $k$. Our discussion involves SU(1,1) versions of Wigner functions, and the quantum-interference effect is easily visualized through plots of these Wigner functions. Specifically, the superpositions of four Perelomov SU(1,1) coherent states (compass state) yield nearly isotropic sub-Planck structures in phase space scaling as $\frac1{k}$ compared with $\frac1{\sqrt{k}}$ scaling for individual SU(1,1)…
The persistence of sub-Planck structure in phase space with loss of coherence is demonstrated in a mixed state, which comprises two terms in the density matrix. Its utility in carrying out Heisenberg-limited measurement and quantum parameter estimation have been shown. It is also shown that the mixed state performs equally well as the compass state for carrying out precision measurements. The advantage of using mixed state relies on the fact that such a state can be easier to prepare and may appear from pure states after partial loss of coherence. We explicate the effect of environment on thes…
We investigate the connection between sub-Planck structure in the Wigner function and the output fidelity of continuous-variable teleportation protocols. When the teleporting parties share a two-mode squeezed state as an entangled resource, high fidelity in the output state requires a squeezing large enough that the smallest sub-Planck structures in an input pure state are teleported faithfully. We formulate this relationship, which leads to an explicit relation between the fine-scale structure in the Wigner function and large-scale extent of the Wigner function, and we treat specific examples…
We find the existence of sub-Planck scale structures in the P{ö}schl-Teller potential, which is an exactly solvable potential with both symmetric and asymmetric features. We analyze these structures in both cases by looking at the Wigner distribution of the state evolved from an initial coherent state up to various fractional revival times. We also investigate the sensitivity to perturbations of the P{ö}schl-Teller potential and we verify that, similar to the harmonic oscillator, the presence of sub-Planck structure in phase space is responsible for a high sensitivity to phase-space displaceme…
We show how sub-Planck phase-space structures in the Wigner function can be used to achieve Heisenberg-limited sensitivity in weak force measurements. Nonclassical states of harmonic oscillators, consisting of superpositions of coherent states, are shown to be useful for the measurement of weak forces that cause translations or rotations in phase space, which is done by entangling the quantum oscillator with a two-level system. Implementations of this strategy in cavity QED and ion traps are described.
We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results in [G. Gour, arxiv.org:0704.1521 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures…
The time evolution of a particle, caught in an infinitely deep square well, displays unexpected features, when one includes tiny relativistic effects. Indeed, even the smallest corrections to the non-relativistic quadratic spectrum manifest themselves in a dramatic way. Our theoretical analysis brings to light a completely new time scale, at which the system exhibits surprisingly perfect revivals. This longer time scale rules the system dynamics and replaces the original revival time of the unperturbed system. The early manifestation of such phenomenon is captured by the sensitivity of sub-Pla…
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of large number of states. The asymptotic expression for the corresponding state overlap function, controlled by the sub-Planck structures arising from phase space interference, is obtained exactly. The validity of large phase space support, in which context the asymptotic limit is achieved, is discussed in detail. For large number of coherent states, uniformly located on a circle, it identically matches with the diffraction pattern for a circular ring with…
The orthogonality of cat and displaced cat states, underlying Heisenberg limited measurement in quantum metrology, is studied in the limit of large number of states. The asymptotic expression for the corresponding state overlap function, controlled by the sub-Planck structures arising from phase space interference, is obtained exactly. For large number of coherent states, uniformly located on a circle, it identically matches with the diffraction pattern for a circular ring with uniform angular source strength. This is in accordence with the van Cittert-Zernike theorem, where the overlap functi…
In 2001, Zurek derived the generic minimum scale $a_{Z}$ for the area of structures of Wigner's quantum phase distribution. Here we show by construction, using superoscillatory functions, that the Wigner distribution can locally show regular spotty structures on scales much below Zurek's scale $a_{Z}$. The price to pay for the presence of such structures is their exponential smallness. For the case we construct there is no increased interferometric sensitivity from the presence of patches with superoscillatory structure in phase-space.
In this contribution, we investigate the entanglement behavior of a composite system consists of two different dimensional subsystems in non-inertial frames. In particular, we consider a composite system of qubit(two-dimensional) subsystem, and qutrit, (three-dimensional) subsystem. The degree of entanglement is quantified for different cases, where it is assumed that the two-subsystems are simultaneously or non- simultaneously accelerated. The entanglement decays as the acceleration of any subsystem increases.In general, the decay rate of entanglement increases as the dimensional of the accel…
Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated to a harmonic oscillator, and extend it to find analogous results for the $\mathfrak{su}(2)$ algebra, typically associate…
Heisenberg-limited and weak measurements are the two intriguing notions, used in recent times for enhancing the sensitivity of measurements in quantum metrology. Using a quantum cat state, endowed with sub-Planck structure, we connect these two novel concepts. It is demonstrated that these two phenomena manifest in complementary regimes, depending upon the degree of overlap between the mesoscopic states constituting the cat state under consideration. In particular, we find that when sub-Planck structure manifests, the imaginary weak value is obscured and vice-versa.
We present experimental data of the frequency resolved optical gating (FROG) measurements of light pulses revealing interference features corresponding to sub-Planck structures in phase space. For superpositions of pulses a small, sub-Fourier shift in the carrier frequency leads to a state orthogonal to the initial one, although in the representation of standard time-frequency distributions these states seem to have a nonvanishing overlap.
Exploiting the correspondence between Wigner distribution function and a frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate existence of the chessboard-like interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase space representation of compass states. Using superpositions of four electric pulses as realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exac…
In this article we report that the entanglement produced by single photon subtraction is maximum, by studying the entanglement of multi photon subtraction from two mode squeezed states. We argue that the single photon subtraction produces maximum entanglement as it is more non-classical in nature than many photon subtractions
Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than $\hbar$ [W.H. Zurek, Nature {\bf 412}, 712 (2001)]. We investigate the influence of those sub-Planck scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wavepacket, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interferenc…
We explore a minimal paradigm for thermalization, consisting of two weakly-coupled, low dimensional, non-integrable subsystems. As demonstrated for Bose-Hubbard trimers, chaotic ergodicity results in a diffusive response of each subsystem, insensitive to the details of the drive exerted on it by the other. This supports the hypothesis that thermalization can be described by a Fokker Plank equation. We also observe, however, that Levy-flight type anomalies may arise in mesoscopic systems, due to the wide range of time scales that characterize `sticky' dynamics.
We investigate the superposition of four different quantum states based on the $q$-oscillator. These quantum states are expressed by means of Rogers-Szegö polynomials. We show that such a superposition has the properties of the quantum harmonic oscillator when $q\to 1$, and those of a compass state with the appearance of chessboard-type interference patterns when $q \to 0$.
The Entanglement contour function quantifies the contribution from each degree of freedom in a region $\mathcal{A}$ to the entanglement entropy $S_{\mathcal{A}}$. Recently in \cite{Wen:2018whg} the author gave two proposals for the entanglement contour in two-dimensional theories. The first proposal is a fine structure analysis of the entanglement wedge which applies to holographic theories. The second proposal is a claim that for general two-dimensional theories the partial entanglement entropy is given by a linear combination of entanglement entropies of relevant subsets inside $\mathcal{A}$…
