This paper is dedicated to finding the quadrature operator eigenstates and wavefunctions of the most general $f$-deformed oscillators. A definition for quadrature operator for deformed algebra is derived to obtain the quadrature operator eigenstates. A new set of polynomials are obtained using this quadrature operator and these polynomials are used to find explicitly the wavefunctions of the deformed oscillators. We have plotted wavefunctions for three different types of deformations and compared it with the wavefunctions of the non-deformed oscillator. Our result will immensely help the research groups working in the quantum state reconstruction and quantum information theory of deformed states.
The spectrum and eigenstates of any field quadrature operator restricted to a finite number $N$ of photons are studied, in terms of the Hermite polynomials. By (naturally) defining \textit{approximate} eigenstates, which represent highly localized wavefunctions with up to $N$ photons, one can arrive at an appropriate notion of limit for the spectrum of the quadrature as $N$ goes to infinity, in the sense that the limit coincides with the spectrum of the infinite-dimensional quadrature operator. In particular, this notion allows the spectra of truncated phase operators to tend to the complete u…
Orbital angular momentum eigenfunctions are readily understood in terms of spherical harmonic wavefunctions. However, the quantum mechanical phenomenon of spin is often said to be mysterious and hard to visualize, with no classical analogue. Many textbooks give a heuristic and somewhat unsatisfying picture of a precessing spin vector. Here we advocate for the "spin wavefunction" in the spin coherent state representation as a striking, elegant, and mathematically meaningful visual tool. We also demonstrate that cartographic projections such as the Hammer projection are useful for visualizing wa…
In this letter, we define the homodyne $q$-deformed quadrature operator. Analytic expression for the wavefunctions of $q$-deformed oscillator in the quadrature basis are found. Furthermore, we compute the explicit analytical expression for the tomogram of the $q$-deformed coherent states by finding the eigenstates of the $q$-deformed quadrature operator.
We present a systematic analysis on coherent states of composite bosons consisting of two distinguishable particles. By defining an effective composite boson (coboson) annihilation operator, we derive its eigenstate and commutator. Depending on the elementary particles comprising the composite particles, we gauge the resemblance between this eigenstate and traditional coherent states through typical measures of nonclassicality, such as quadrature variances and Mandel's Q parameter. Furthermore, we show that the eigenstate of the coboson annihilation operator is useful in estimating the maximum…
A quantum mechanical description of vortices in two dimensional superfluid $^4$He flims is presented. Single vortex creation and annihilation operators are defined and wavefunctions for these states are explicitly constructed. A hamitonian for these states is then proposed.
We suggest an interferometric scheme assisted by squeezing and linear feedback to realize the whole class of field-quadrature quantum nondemolition measurements, from Von Neumann projective measurement to fully non-destructive non-informative one. In our setup, the signal under investigation is mixed with a squeezed probe in an interferometer and, at the output, one of the two modes is revealed through homodyne detection. The second beam is then amplitude-modulated according to the outcome of the measurement, and finally squeezed according to the transmittivity of the interferometer. Using str…
Variational quantum eigensolver~(VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its applications to many problems may involve a group of Hamiltonians, e.g., Hamiltonian of a molecule is a function of nuclear configurations. In this paper, we incorporate derivatives of Hamiltonian into VQE and develop some hybrid quantum-classical algorithms, which explores both Hamiltonian and wavefunction spaces for optimization. Aiming for solving quantum chemistry problems more efficiently, we first propose mutual gradient descent algorithm …
In this paper we calculate the position and momentum space information entropies for the quantum states associated with a particular physical system, i.e. the isotonic oscillator Hamiltonian. We present our results for its ground states, as well as for its excited states. We observe that the lower bound of the sum of the position and momentum entropies expressed by the Beckner, Bialynicki-Birula and Mycielski (BBM) inequality is satisfied. Moreover, there exist eigenstates that exhibit squeezing in the position information entropy. In fact, entropy squeezing, which occurs in position, will be …
We present a wavefunction methodology to account for finite temperature initial conditions in the quantum Rabi model. The approach is based on the Davydov-Ansatz together with a statistical sampling of the canonical harmonic oscillator initial density matrix. Equations of motion are gained from a variational principle and numerical results are compared to those of the thermal Hamiltonian approach. For a system consisting of a single spin and a single oscillator and for moderate coupling strength, we compare our new results with full quantum ones as well as with other Davydov-type results based…
An interpretation of the probability flux is given, based on a derivation of its eigenstates and relating them to coherent state projections on a quantum wavefunction. An extended definition of the flux operator is obtained using coherent states. We present a "processed Husimi" representation, which makes decisions using many Husimi projections at each location. The processed Husimi representation reverse engineers or deconstructs the wavefunction, yielding the underlying classical ray structure. Our approach makes possible interpreting the dynamics of systems where the probability flux is uni…
Motivated by the existence of exact many-body quantum scars in the AKLT chain, we explore the connection between Matrix Product State (MPS) wavefunctions and many-body quantum scarred Hamiltonians. We provide a method to systematically search for and construct parent Hamiltonians with towers of exact eigenstates composed of quasiparticles on top of an MPS wavefunction. These exact eigenstates have low entanglement in spite of being in the middle of the spectrum, thus violating the strong Eigenstate Thermalization Hypothesis (ETH). Using our approach, we recover the AKLT chain starting from the…
When the wavefunction of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops "branches": a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structur…
Ab initio electronic excited state calculations are necessary for the quantitative study of photochemical reactions, but their accurate computation on classical computers is plagued by prohibitive scaling. The Variational Quantum Deflation (VQD) is an extension of the Variational Quantum Eigensolver (VQE) for calculating electronic excited state energies, and has the potential to address some of these scaling challenges using quantum computers. However, quantum computers available in the near term can only support a limited number of circuit operations, so reducing the quantum computational co…
If the initial quantum state of the universe is a multiverse superposition over many different sets of values of the effective coupling "constants" of physics, and if this quantum state collapses to an eigenstate of the set of coupling "constants" with a probability purely proportional to the absolute square of the amplitude (with no additional factor for something like life or consciousness), then one should not expect that the coupling `constants' would be so biophilic as they are observed to be. Therefore, the observed biophilic values (apparent fine tuning) of the coupling "constants" is s…
In fast forward quantum shortcuts to adiabaticity, a designed potential $U_{FF}(q,t)$ steers a wavefunction to evolve from the $n$'th eigenstate of an initial Hamiltonian $\hat H(0)$ to the $n$'th eigenstate of a final Hamiltonian $\hat H(τ)$, in finite time $τ$. Previously proposed strategies for constructing $U_{FF}$ are (in the absence of special symmetries) limited to the ground state, $n=0$. We develop a method that overcomes this limitation, thereby substantially expanding the applicability of this shortcut to adiabaticity, and we illustrate its effectiveness with numerical simulations. …
In this short communication, I gave a generalization of measurement postulate in quantum mechanics. It is regarding the case with partial measurement, namely, measurement on only part of a wave function. Upon a partial measurement, the wavefunction will either collapse in one of the eigenstate covered by the measured partial wavefunction; or collapses out of the measured part and shifts to the unmeasured part. An explanation in the WISE (Wavefunction Is System Entity) interpretation is given.
The auxiliary field method is a powerful technique to obtain approximate closed-form energy formulas for eigenequations in quantum mechanics. Very good results can be obtained for Schrödinger and semirelativistic Hamiltonians with various potentials, even in the case of many-body problems. This method can also provide approximate eigenstates in terms of well known wavefunctions, for instance harmonic oscillator or hydrogen-like states, but with a characteristic size which depends on quantum numbers. In this paper, we consider two-body Schrödinger equations with linear, logarithmic and exponent…
Variational quantum algorithms (VQA) are considered as some of the most promising methods to determine the properties of complex strongly correlated quantum many-body systems, especially from the perspective of devices available in the near term. In this context, the development of efficient quantum circuit ansatze to encode a many-body wavefunction is one of the keys for the success of a VQA. Great efforts have been invested to study the potential of current quantum devices to encode the eigenstates of fermionic systems, but little is known about the encoding of bosonic systems. In this work,…
We study the eigenstates of quantum systems with large Hilbert spaces, via their distribution of wavefunction amplitudes in a real-space basis. For single-particle 'quantum billiards', these real-space amplitudes are known to have Gaussian distribution for chaotic systems. In this work, we formulate and address the corresponding question for many-body lattice quantum systems. For integrable many-body systems, we examine the deviation from Gaussianity and provide evidence that the distribution generically tends toward power-law behavior in the limit of large sizes. We relate the deviation from …
Using the kinematic constraints of classical bodies we construct the allowable wavefunctions corresponding to classical solids. These are shown to be long lived metastable states that are qualitatively far from eigenstates of the true Hamiltonian. Extensions of this give an explicit description of phonon oscillations in terms of the wavefunction itself and some consequences for the general validity of the quasiparticle picture are presented. An intrinsic theory of quantum measurement naturally arises based on Schrödinger evolution that is local, consistent with relativity and extends to the ca…
Two-photon photoassociation of hot magnesium atoms by femtosecond laser pulses, creating electronically excited magnesium dimer molecules, is studied from first principles, combining \textit{ab initio} quantum chemistry and molecular quantum dynamics. This theoretical framework allows for rationalizing the generation of molecular rovibrational coherence from thermally hot atoms [L. Rybak \textit{et al.}, Phys. Rev. Lett. {\bf 107}, 273001 (2011)]. Random phase thermal wave functions are employed to model the thermal ensemble of hot colliding atoms. Comparing two different choices of basis func…
Parity describing the symmetry of quantum mechanics wavefunction under space inversion transformation not only plays an essential role in solving quantum systems but also can be used to manipulate and measure the motional quantum states of such hybrid quantum systems as quantum Rabi model(QRM) and/or its variants through parity measurements. Here we address an exotic parity behavior of the QRM in its superradiant phase by numerical exact diagonalization, namely, the parities of eigenstates of the QRM behave irregular in the strong coupling regime but the sum of parities for each pair of eigens…
We propose the algorithm for determining vibrational quantum eigenstates of periodic linear chain of atoms coupled by harmonic and third order anharmonic interactions (Fermi-Ulam-Pasta $α$ problem) in the long wavelength limit within the resonant approach. Eigenstates can be encoded by the sequence of integer numbers determining their energies and wavefunctions. Using these eigenstates we described a single phonon state time evolution showing coherent oscillations. The applications of theory to vibrational energy transport and quantum informatics are discussed.
Recent works have established universal entanglement properties and demonstrated validity of single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians. However, a common property of all quantum-chaotic quadratic Hamiltonians studied in this context so far is the presence of random terms that act as a source of disorder. Here we introduce tight-binding billiards in two dimensions, which are described by non-interacting spinless fermions on a disorder-free square lattice subject to curved open (hard-wall) boundaries. We show that many properties of tight-binding billiar…
An emergent theory of quantum measurement arises directly by considering the particular subset of many body wavefunctions that can be associated with classical condensed matter and its interaction with delocalized wavefunctions. This transfers questions of the "strangeness" of quantum mechanics from the wavefunction to the macroscopic material itself. An effectively many-worlds picture of measurement results for long times and induces a natural arrow of time. The challenging part is then justifying why our macroscopic world is dominated by such far-from-eigenstate matter. Condensing cold mesos…
The experimental demonstration that neutrons can reside in gravitational quantum stationary states formed in the gravitational field of the Earth indicates a need to examine in more detail the general theoretical properties of gravitational eigenstates. Despite the almost universal study of quantum theory applied to atomic and molecular states very little work has been done to investigate the properties of the hypothetical stationary states that should exist in similar types of gravitational central potential wells, particularly those with large quantum numbers. In this first of a series of pa…
The problems with an emergent approach to quantum statistical mechanics are discussed and shown to follow from some of the same sources as those of quantum measurement. A wavefunction of an N atom solid is described in the ground and excited eigenstates with explicit modifications for phonons. Using the particular subclass of wavefunctions that can correspond to classical solids we investigate the localization properties of atomic centers of mass motion and contrast it with more general linear combinations of phonon states. The effectively large mass of longer modes means that localization pre…
We construct a local $ψ$-epistemic hidden-variable model of Bell correlations by a retrocausal adaptation of the originally superdeterministic model given by Brans. In our model, for a pair of particles the joint quantum state $|ψ_e(t)\rangle$ as determined by preparation is epistemic. The model also assigns to the pair of particles a factorisable joint quantum state $|ψ_o(t)\rangle$ which is different from the prepared quantum state $|ψ_e(t)\rangle$ and has an ontic status. The ontic state of a single particle consists of two parts. First, a single particle ontic quantum state $χ(\vec{x},t)|i…
In quantum mechanics, measurements cause wavefunction collapse that yields precise outcomes, for non-commuting observables such as position and momentum Heisenberg's uncertainty principle limits the intrinsic precision of a state. Although theoretical work has demonstrated the possibility to perform simultaneous non-commuting measurements and has revealed the limits on measurement outcomes, only recently has the dynamics of the quantum state been discussed. To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables to a …
Abraham-Moses transformations, besides Darboux transformations, are well-known procedures to generate extensions of solvable potentials in one-dimensional quantum mechanics. Here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the Abraham-Moses transformations for typical solvable potentials, e.g. the radial oscillator, the Darboux-Pöschl-Teller and some others. These seed solutions are simple generalisations of the virtual state wavefunctions, which are obtained from the eigenfunctions by discrete symmetries of the potent…
For finite quantum many-particle systems modeled with say $m$ fermions in $N$ single particle states and interacting with $k$-body interactions ($k \leq m$), the wavefunction structure is studied using random matrix theory. Hamiltonian for the system is chosen to be $H=H_0(t) + λV(k)$ with the unperturbed $H_0(t)$ Hamiltonian being a $t$-body operator and $V(k)$ a $k$-body operator with interaction strength $λ$. Representing $H_0(t)$ and $V(k)$ by independent Gaussian orthogonal ensembles (GOE) of random matrices in $t$ and $k$ fermion spaces respectively, first four moments, in $m$-fermion sp…
The eigenstate thermalisation hypothesis resolves the paradox of emergent thermal or classical behaviour in a closed quantum system by focussing upon local observations. This permits the remainder of the system to act as a bath, thermalisation arising due to a process of de-phasing that gradually reveals the thermal nature of local observables measured in an eigenstate. This is very different from thermalisation in closed classical systems, which is driven by dynamical chaos. We show how quantum thermalisation in closed systems can be recast in a way that is directly related to classical therm…
A novel approach is suggested for the statistical description of quantum systems of interacting particles. The key point of this approach is that a typical eigenstate in the energy representation (shape of eigenstates, SE) has a well defined classical analog which can be easily obtained from the classical equations of motion. Therefore, the occupation numbers for single-particle states can be represented as a convolution of the classical SE with the quantum occupation number operator for non-interacting particles. The latter takes into account the wavefunctions symmetry and depends on the unpe…
For eleven examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials.
