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Publication# The Hamiltonian of the harmonic oscillator with an attractive delta '-interaction centred at the origin as approximated by the one with a triple of attractive delta-interactions

Abstract

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction, of strength beta, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a 'coupling constant renormalisation'. Here we get the Hamiltonian as a norm resolvent limit of the harmonic oscillator Hamiltonian perturbed by a triple of attractive delta-interactions, thus extending the Cheon-Shigehara approximation to the case in which a confining harmonic potential is present.

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Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at th

Self-adjoint operator

In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-d

In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive delta-interaction centered at the origin or by a pair of identical attractive delta-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Bruning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive delta'-interaction in one dimension, fully investigated in [2, 3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of "positional disorder", introduced in [1] in the case of a quantum dot with a single delta-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single delta-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive delta-interactions symmetrically situated at the same distance from the origin.

In quantum mechanics, the Heisenberg uncertainty principle places a fundamental limit in the measurement precision for certain pairs of physical quantities, such as position and momentum, time and energy or amplitude and phase. Due to the Heisenberg uncertainty principle, any attempt to extract certain information from a quantum object would inevitably perturbitinanunpredictableway. This raises one question,"What is the precision limit in such quantum measurements?" The answer, standard quantum limit (SQL), has been obtained by Braginsky to figure out the fundamental quantum limits of displacement measurement in the context of gravitational wave detection. To circumvent the unavoidable quantum back-action from the priori measurement, quantum non-demolition measurement (QND) methods were introduced by Braginsky and Thorne. To surpass the SQL of the displacement measurement in an interferometer, one can measure only one quadrature of the mechanical motion while give up the information about the other canonically conjugated quadrature. Such measurements can be performed by periodic driving the mechanical oscillator, i.e. the back-action evading (BAE) measurement.
Cavity optomechanics provides an ideal table-top platform for the testing of the quantum measurement theory. Mechanical oscillator is coupled to electromagnetic field via radiation pressure, which is enhanced by an optical micro-cavity. Over the last decade, laser cooling has enabled the preparation of mechanical oscillator in the ground state in both optical and microwave systems. BAE measurements of mechanical motion have been allowed in the microwave electromechanical systems, which led to the observations of mechanical squeezing and entanglement. However, despite the theoretical proposal almost 40 years ago, the sub-SQL measurements still remain elusive. This thesis reports our efforts approach the sub-SQL with a highly sideband-resolved silicon optomechanical crystal (OMC) in a 3He buffer gas environment at 2K. The OMC couples an optical mode at telecommunication wavelengths and a colocalized mechanical mode at GHz frequencies. The Helium3 buffer gas environment allows sufficient thermalization of the OMC despite the drastically decreased silicon thermal conductivity. We observe Floquet dynamics in motional sideband asymmetry measurement when employing multiple probe tones. The Floquet dynamics arises due to presence of Kerr-type nonlinearities and gives rise to an artificially modified motional sideband asymmetry, resulting from a synthetic gauge field among the Fourier modes. We demonstrate the first optical continuous two-tone backaction-evading measurement of a localized GHz frequency mechanical mode of silicon OMC close to the ground state by showing the transition from conventional sideband asymmetry to backaction-evading measurement. We discover a fundamental two-tone optomechanical instability and demonstrate its implications on the back-action evading measurement. Such instability imposes a fundamental limitation on other two-tone schemes, such as dissipative quantum mechanical squeezing. We demonstrate state-of-art laser sideband cooling of the mechanical motion to a mean thermal occupancy of 0.09 quantum, which is 7.4dB of the oscillator's zero-point energy and corresponds to 92% ground state probability. This also enables us to observe the dissipative mechanical squeezing below the zero-point motion for the first time with laser light.