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.
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…
The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck's constant, hbar/2, as demonstrated by Heisenberg's thought experiment using a gamma-ray microscope. Here I show that this common assumption is false: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise…
Sub-Planck phase-space structures in the Wigner function of the motional degree of freedom of a trapped ion can be used to perform weak force measurements with Heisenberg-limited sensitivity. We propose methods to engineer the Hamiltonian of the trapped ion to generate states with such small scale structures, and we show how to use them in quantum metrology applications.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relat…
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…
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…
We review Heisenberg's uncertainty principle for the orbital angular momentum (OAM) of light. By taking into account the largest and smallest scales present in nature, such as the the Hubble radius and the Planck length, we have found that there exist upper and lower physical limits to the determination of the OAM of a photon.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the ac…
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…
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.
The noncommutative space provides a framework to understand phenomena in Planck scale physics. However, there is no any direct experimental evidence to demonstrate the existence of noncommutative space. We propose an experimental scheme based on the Aharonov-Bohm effect in the nano-scale quantum mechanics to probe the phenomena in the noncommutative phase space. By the Seiberg-Witten map, the free electrons of the nano-scale ring in noncommutative phase space can be mapped equivalently to the quantum mechanical (Heisenberg's algebra) phase space with an extra effective magnetic flux. We introd…
The Generalized Uncertainty Principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of Quantum Gravity. It consists of a modified commutator between position and momentum. In this work we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent and squeezed states in Quantum Mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We com…
Position-deformed Heisenberg algebra with maximal length uncertainty has recently been proven to induce strong quantum gravitational fields at the Planck scale (2022 J. Phys. A: Math. Theor.55 105303). In the present study, we use the position space representation on the one hand and the Fourier transform and its inverse representations on the other to construct propagators of path integrals within this deformed algebra. The propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. Since the effects of quantum gravity are strong in…
In interferometry, sub-Heisenberg strategies claim to achieve a phase estimation error smaller than the inverse of the mean number of photons employed (Heisenberg bound). Here we show that one can achieve a comparable precision without performing any measurement, just using the large prior information that sub-Heisenberg strategies require. For uniform prior (i.e. no prior information), we prove that these strategies cannot achieve more than a fixed gain of about 1.73 over Heisenberg-limited interferometry. Analogous results hold for arbitrary single-mode prior distributions. These results ext…
We provide evidence that the uncertainty in detection of small and deterministic phase-shift deviations from a working point can be lower than the Heisenberg bound, for fixed finite mean number of photons. We achieve that by exploiting non-linearity of estimators and coherence with the vacuum.
The non zero value of Planck constant $h$ underlies the emergence of several inequalities that must be satisfied in the quantum realm, the most prominent one being Heisenberg Uncertainty Principle. Among these inequalities, Bekenstein bound provides a universal limit on the entropy that can be contained in a localized quantum system of given size and total energy. In this Letter, we explore how Bekenstein bound is affected when Heisenberg uncertainty relation is deformed so as to accommodate gravitational effects close to Planck scale (Generalized Uncertainty Principle). By resorting to genera…
Quantum sensitivity is an important issue in the field of quantum metrology where sub-Planck scale structures play crucial role in the Heisenberg limited measurement. We investigate the mesoscopic superposition structures, particularly for well-known cat-like and compass-like states, in the rotating Morse system where sub-Planck scale structures originate in the dynamics of a suitably constructed SU(2) coherent state. A detail study of the sensitivity analysis reveals that rotational coupling in the vibrational wave packet can be used as a probe to enhance the sensitivity limit in a diatomic m…
We propose a Heisenberg-limited quantum interferometer whose input is twin optical beams from which one or more photons have been indistinguishably subtracted. Such an interferometer can yield Heisenberg-limited performance while at the same time giving a direct fringe reading, unlike for the twin-beam input of the Holland-Burnett interferometer. We propose a feasible experimental realization using a photon-number correlated source, such as non-degenerate parametric down-conversion, and perform realistic analyses of performance in the presence of loss and detector inefficiency.
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.
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…
It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was discovered within the framework of using pretty-good measurements for obtaining optimal measurements in the hidden subgroup problem. Here we show how to solve the Heisenberg hidden subgroup problem using arguments based instead on the symmetry of certain hidden subgroup states. The symmetry we consider leads naturally to a unitary transform known as the Clebsch-Gordan transform over the Heisenberg g…
Within the Heisenberg's uncertainty principle it is explicitly discussed the impact of these inequalities on the theory of integrated photonics at sub-wavelength regime. More especially, the uncertainty of the effective index values in nanophotonics at sub-wavelength regime, which is defined as the eigenvalue of the overall opto-geometric problems in integrated photonics, appears directly stemming from Heisenberg's uncertainty. An apt formula is obtained allowing us to assume that the incertitude and the notion of eigenvalue called effective optical index or propagation constant is inversely p…
Several models of quantum gravity predict the emergence of a minimal length at Planck scale. This is commonly taken into consideration by modifying the Heisenberg Uncertainty Principle into the Generalized Uncertainty Principle. In this work, we study the implications of a polynomial Generalized Uncertainty Principle on the harmonic oscillator. We revisit both the analytic and algebraic methods, deriving the exact form of the generalized Heisenberg algebra in terms of the new position and momentum operators. We show that the energy spectrum and eigenfunctions are affected in a non-trivial way.…
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.
We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn to be handy on this occasion, this provides a further illustration to Similarity and Correspondence Principle. Keywords: Heisenberg group, Kirillov's method of orbits, geometric quantisation, quantum mechanics, classical mechanics, P…
