Phonon Operators Research Articles

A new boson expansion theory utilizing a norm operator

Abstract We propose a new boson expansion method using a norm operator. The small parameter expansion, in which the boson approximation becomes the zeroth-order approximation, requires the double commutation relations between phonon operators that are not closed between the phonon excitation modes adopted as boson excitations. This results in an infinite expansion regardless of whether the type of the boson expansion is Hermitian or non-Hermitian. The small parameter expansion does not hold when the commutation relations are closed. The norm operator is expressed as a function of the number operator in the physical subspace, which enables us to obtain substantially a finite boson expansion regardless of the Hermitian or non-Hermitian type. We also point out the problems of the conventional boson expansion methods. The normal-ordered linked-cluster expansion theory has failed to refute Marshalek’s claim that KT-1 and KT-2 are of chimerical boson expansion. The Dyson boson expansion theory does not have exceptional superiority over other types. Previous studies using the boson expansion methods should be re-examined.

Quadrupole phonon excitations and transition probabilities for low-lying states of neutron-rich Cd isotopes

In this paper we study the low-lying states of neutron-rich $^{122,124,126,128}\mathrm{Cd}$ and $^{130,132,134,136,138}\mathrm{Cd}$ within the nucleon-pair approximation of the shell model. We adopt the phenomenological Hamiltonian for $^{122,124,126,128}\mathrm{Cd}$, and the shell-model effective interaction $jj46$ for $^{130,132,134,136,138}\mathrm{Cd}$. The available experimental excitation energies and quadrupole transition probabilities are well reproduced by our calculation, and we also make predictions for very neutron-rich Cd nuclei. Based on our calculation, the $B(E2;{0}_{\mathrm{g}.\mathrm{s}.}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})$ values exhibit an asymmetric feature with respect to the $N=82$ shell closure, which mainly comes from the contributions of the proton transition matrix elements. We also investigate for the eight open-shell Cd nuclei, whether two low-lying yrast states with spins differing by 2 can be connected by a quadrupole-phonon excitation, and here we take the proton and neutron quadrupole operators multiplied by $1/{r}_{0}^{2}$ as our quadrupole phonon operators. We calculate explicit overlaps between the low-lying yrast states and the constructed quadrupole-phonon states based on the low-lying yrast states with lower spins, and the results indicate that, for all eight open-shell Cd nuclei, $|{2}_{1}^{+}\ensuremath{\rangle}$ and $|{4}_{1}^{+}\ensuremath{\rangle}$ can be well described to be the states constructed by coupling the proton or neutron phonon to $|{0}_{\mathrm{g}.\mathrm{s}.}^{+}\ensuremath{\rangle}$ and $|{2}_{1}^{+}\ensuremath{\rangle}$, respectively. Very interestingly, for $^{126,124,122}\mathrm{Cd}$ and $^{136,138}\mathrm{Cd}$, the ${2}_{1}^{+}$ state can be well described to be both the proton phonon state and the neutron phonon state, which indicates a nonorthogonal feature of these two phonon states. We further present an analytic relation for the overlap between these two phonon states, which implies that the proton and neutron phonon states constructed using the quadrupole operators and the ${0}_{\mathrm{g}.\mathrm{s}.}^{+}$ state in an open-shell nucleus are almost impossibly orthogonal.

On the absence of reverse running waves in general displacement of lattice vibration in popular books on solid state theory

In this article we point out that there is a deficiency in the presentation of the general solution of harmonic lattice vibration, the omission of half of the allowed running waves, in many popular textbooks published since 1940, e.g. O Madelung’s 1978 Introduction to Solid-State Theory and J Solyom’s 2007 Fundamentals of the Physics of Solids, vol 1. So we provide a revised presentation, which gives a complete general solution and demonstrates clearly that the conventional complex normal coordinate should be a superposition of two coordinates (multiplied by a factor ) of running waves travelling oppositely along and not only a coordinate of a unidirectional running wave as many books considered. It is noticed that the book, Quantum Theory of the Solid State: An Introduction, by L Kantorovich, published in 2004, and the review article, ‘Phonons in perfect crystals’ by W Cochran and R A Cowly, published in 1967, for a one-dimensional single-atom chain gave correct (but not normalized) formulae for the general solution of lattice vibration and the normal coordinate. However, both of them stated still that each normal coordinate describes an independent mode of vibration, which in our opinion needs to be further discussed. Moreover, in books such as Fundamentals of the Physics of Solids, vol 1, by J Solyom, and The Physics and Chemistry of Solids, by S R Elliott, published in 2006 and 2007, respectively, the reverse waves were still lost. Hence, we also discuss a few related topics. In quantization of the lattice vibration, the introduction of the conventional two (not one) independent phonon operators in a normal coordinate is closely related to the ‘independence’ of the two constituent waves mentioned above, and we propose a simple propositional relation between the phonon operator and the corresponding running wave coordinate. Moreover, only the coordinate of the superposition wave (not the running wave), as the normal coordinate can give the correct quantization commutation relations. In addition, there is an interference between the direct and reverse running waves in kinetic and potential energies, which also questions the popular term ‘normal mode’ for a running wave mode. Therefore we have made a few suggestions and discuss the terms of relative quantities.

Polaronic quasiparticle picture for generation dynamics of coherent phonons in semiconductors: Transient and nonlinear Fano resonance

We examine generation dynamics of coherent phonons (CPs) in both of polar and non-polar semiconductors -- such as GaAs and Si -- based on a polaronic-quasiparticle (PQ) model. In the model concerned, the PQ operator is composed of two kinds of operators. One is a quasiboson operator -- defined as a linear combination of a set of pairs of electron operators -- and the other is a longitudinal optical (LO) phonon operator. The problem of transient and non-linear Fano resonance (FR) is tackled in particular; the vestige of this quantum interference effect was observed exclusively in lightly $n$-doped Si immediately after carriers were excited by an ultrashort pulse-laser [M. Hase et. al., Nature 426, 51 (2003)], though not observed yet in GaAs. It is shown that the phonon energy state is embedded in a continuum state formed by a set of adiabatic eigenstates of the quasiboson. This result implies the possibility of manifestation of the transient FR in the present optically-non-linear system.

Temporal evolution of instantaneous phonons in time-dependent harmonic oscillators

We study a time-dependent harmonic oscillator based on the dynamics of instantaneous phonons, which have obvious physical meaning and direct experimental relevance. We find simple analytic solutions for an important class of evolution and identify two parameter-changing-rate regimes with qualitatively different oscillator behaviors. We show that rapid adiabatic processes are possible if the frequency and the mass of the oscillator change in opposite directions. The state vector in the Schrödinger picture is handily achieved by use of the eigenstates of the instantaneous phonon operators that are analytically known for arbitrary frequency and mass values.

Excitonic polarons in low-dimensional transition metal dichalcogenides

We examine the excitonic polaron properties of common monolayer transition metal dichalcogenides (MoS2, MoSe2, WS2 and WSe2). The excitonic polaron is formed when excitons interact with acoustic or optical phonons via coupling to the deformation potentials associated with the conduction and valence bands. A unitary transformation which performs an approximate diagonalization of the exciton–phonon operator is used to evaluate the ground state energy of the excitonic polaron. We derive analytical expressions of the changes in the excitonic polaron energy and mass at small exciton wavevectors involving the deformation potential due to optical phonons. The polaronic effect of the monolayer transition metal dichalcogenides is examined by comparing changes in the energy gap shift and effective masses based on known deformation potential constants for carrier–phonon interactions. Our results indicate the occurrence of comparable energy shifts when the ground state exciton interacts with optical or acoustic phonons. We extend our calculations to explore the influence of exciton–lattice interactions on the binding energies and the self-trapping of excitons in two-dimensional layers of transition metal dichalcogenides.

The influence of atomic collisions on the spectrum of light scattered from an f-deformed Bose–Einstein condensate

In this paper, we investigate the spectrum of light scattered from a Bose–Einstein condensate (BEC) in the framework of an f-deformed boson model. We use an f-deformed quantum model in which Gardiner's phonon operators for the BEC are deformed by an operator-valued function, , of the particle-number operator . We also consider the collisions between the atoms as a special kind of f-deformation where the collision rate κ is regarded as the corresponding deformation parameter. By applying the small fluctuation approximation, we obtain the spectrum of light scattered from the f-deformed BEC. By analysing the scattering spectrum we find that by increasing the values of the deformation parameters κ and (N is the total number of condensate atoms) the spectrum shows deviation from the spectrum associated with the non-deformed Bose–Einstein condensate.

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ , as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ , it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ .

Energy and charge-transfer dynamics using projected modes

For electron-phonon Hamiltonians with the couplings linear in the phonon operators, we construct a class of unitary transformations that separate the normal modes into two groups. The modes in the first group interact with the electronic degrees of freedom directly. The modes in the second group interact directly only with the modes in the first group but not with the electronic system. These transformations can be carried out independently for different types of phonon modes, e.g., high- versus low-frequency phonon bands. This construction generalizes recently introduced transformations for systems exhibiting a conical intersection topology. The separation of the normal modes into several groups allows one to develop new approximation schemes. We apply one of such schemes to study electronic relaxation at a semiconducting polymer interface.

Constants of motion in an interacting electron-lattice system

The electron-lattice interaction term of the Bloch-FrOhlich Hamiltonian, written in the occupation number representation, is bilinear in the electron creation and annihilation operators but linear in the phonon raising and lowering operators. This asymmetry leads to the non-commutability of the phonon number operator and the interaction term of the Hamiltonian. It is shown that by introducing a bilinear form for each phonon operator the commutability of these two terms can be achieved. Formally, the new lattice operators arise from the quantum field-theoretical treatment of the normal modes. Specifically, in the harmonic approximation and coordinate representation of the lattice Hamiltonian in phonon space, this is achieved by the replacement of wave functions by a coordinate-dependent operator field. One obtains then such a field for every normal mode of the lattice. The operator field expansion coefficients are the lattice vibrational mode creation and annihilation operators which appear only in a bilinear combination throughout the theory. To obtain an equivalence relation between the field-theoretical and wave-mechanical formulations, one requires that only one lattice quasiparticle per lattice vibrational mode be allowed. For this case, the relation between the lattice field operators and the conventional phonon operators assumes a simple form. Finally, the entire Bloch-FrOhlich Hamiltonian is rewritten in terms of the new set of lattice operators and the conservation of the new lattice quasiparticles is demonstrated. Thus, both the number of electrons and the number of lattice quasiparticles are shown to be constants of motion of an interacting electron-lattice system.

Controlling subluminal to superluminal behavior of group velocity in an f-deformed Bose–Einstein condensate beyond the rotating wave approximation

In this paper, we investigate tunable control of the group velocity of a weak probe field propagating through an f-deformed Bose–Einstein condensate of Λ-type three-level atoms beyond the rotating wave approximation. For this purpose, we use an f-deformed generalization of an effective two-level quantum model of the three-level Λ-configuration without the rotating wave approximation in which the Gardiner’s phonon operators for Bose–Einstein condensate are deformed by an operator-valued function, f ( n ˆ ) , of the particle-number operator n ˆ . We consider the collisions between the atoms as a special kind of f-deformation where the collision rate κ is regarded as the deformation parameter. We demonstrate the enhanced effect of subluminal and superluminal propagation based on electromagnetically induced transparency and electromagnetically induced absorption, respectively. In particular, we find that (i) the absorptive and dispersive properties of the deformed condensate can be controlled effectively in the absence of the rotating wave approximation by changing the deformation parameter κ, the total number of atoms N ^ and the counter-rotating terms parameter λ, (ii) by increasing the values of λ, κ and η = 1/ N, the group velocity of the probe pulse changes, from subluminal to superluminal and (iii) beyond the rotating wave approximation, the subluminal and superluminal behaviors of the probe field are enhanced.

Study of Squeezed Excitons in Polar Semiconductors

Some properties of excitons in polar semiconductors are studied theoretically by means of squeezed state variational approach. This method makes it possible to consider bilinear terms of the phonon operators as well as linear terms arising from the Lee–Low–Pines (LLP)-like transformation. The exciton ground state energy and binding energy are calculated numerically. It is shown that the squeezing effect is significant in the case of strong exciton-phonon coupling region.

The influence of atomic collisions on collective spontaneous emission from an f-deformed Bose–Einstein condensate

In this paper, we investigate the spontaneous emission of an f-deformed Bose–Einstein condensate of a gas with N identical two-level atoms immersed in a single-mode ideal cavity with s atoms initially excited. We apply an f-deformed quantum model in which Gardiner's phonon operators are deformed by an operator-valued function of the particle number operator . We consider the collisions between the atoms as a special kind of f-deformation where the collision rate κ is regarded as a corresponding deformation parameter. The time evolution of the expectation value of the atomic inversion is presented, the phenomenon of collective collapses and revivals is shown and the effects of deformation on the cooperative behaviour of the system are discussed.

Effects of ground state correlations on the structure of odd-mass spherical nuclei

It is well known that the Pauli principle plays a substantial role at low energies because the quasiparticle and phonon operators, used to describe them, are built of fermions and as a consequence they are not ideal bosons. The correct treatment of this problem requires calculation of the exact commutators between the quasiparticle and phonon operators and in this way to take into account the Pauli principle corrections. In addition to the correlations due to the quasiparticle interaction in the ground-state influence the single-particle fragmentation as well. In this article, we generalize the basic equations of the quasiparticle-phonon nuclear model to account for both effects mentioned above. As an illustration of our approach, calculations of the structure of the low-lying states in the odd-mass nuclei $^{131\ensuremath{-}137}\mathrm{Ba}$ have been performed.

Subluminal to superluminal propagation of an optical pulse in an f-deformed Bose–Einstein condensate

In this paper, we investigate the propagation of a weak optical probe pulse in an f-deformed Bose–Einstein condensate of a gas with the Λ-type three-level atoms in the electromagnetically induced transparency regime. We use an f-deformed generalization of an effective two-level quantum model of the three-level Λ configuration in which Gardiner's phonon operators for Bose–Einstein condensates are deformed by an operator-valued function, , of the particle-number operator . By making use of the quantum approach of the angular momentum theory, we obtain the eigenvalues and eigenfunctions of the system up to a first-order approximation. We consider the collisions between the atoms as a special kind of f-deformation. The collision rate κ is regarded as the deformation parameter and light propagation in the deformed Bose–Einstein condensate is analysed. In particular, we show that the absorptive and dispersive properties of the deformed condensate can be controlled effectively by changing the deformation parameter κ and the total number of atoms. We find that by increasing the value of κ the group velocity of the probe pulse changes, through deformed condensate, from subluminal to superluminal.

Ground state correlations and structure of odd spherical nuclei

It is well known that the Pauli principle plays a substantial role at low energies because the phonon operators are not ideal boson operators. Calculating the exact commutators between the quasiparticle and phonon operators, one can take into account the Pauli principle corrections. Besides, the ground state correlations due to the quasiparticle interaction in the ground state influence the single-particle fragmentation as well. In this paper, we generalize the basic equations of the quasiparticle-phonon nuclear model to account for both effects mentioned. As an illustration of our approach, calculations on the structure of the low-lying states in 133Ba have been performed.

Particle-hole excitations within a self-consistent random-phase approximation

We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the $X$ and $Y$ amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.

Two-mode squeezed magnetopolarons in two-dimensional quantum dot

Abstract In this Letter, some properties of magnetopolarons in two-dimensional quantum dot are investigated by two-mode squeezed states transformation. This method considers linear functions, bilinear functions of the phonon operators and the correlation between two longitudinal optical (LO) phonon modes, which is based on the Lee–Low–Pines and Huybrechts (LLP–H) canonical transformations. So it can provide results not only for the ground state energy but also for the excited states energies, furthermore, it can be applied to the entire range of the electron–phonon coupling strength. Using two-mode squeezed states transformation, we have obtained more accurate results for the ground state energy, excited states energies and renormalized cyclotron masses for some possible transitions.

Influence of the phonon‐exciton interaction on exciton‐exciton quantum correlation in semiconductor microcavities

AbstractWe present an extension of the previous descriptions based on the Dynamics Controlled Truncation Scheme of light‐matter interaction beyond mean‐field, including the microscopic description of the exciton‐photon interaction. This enables the microscopic analysis of the influence of decoherence and noise on the polariton quantum correlations originating from nonlinear optical processes. We expand the operators involved in the dynamics in terms of exact eigenstates of the electron system, the photon and phonon operators and treat phonon‐assisted transitions within the Markov approximation. In particular, we present quantum Heisenberg‐Langevin equations describing light‐induced excitations in semiconductor systems interacting with the phonon bath. This theoretical framework is applied to study the influence of dephasing and noise due to photoluminescence on polariton quantum correlations generated by parametric emission in microcavities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Study of two-dimensional squeezed magnetopolarons

In this Letter, some properties of two-dimensional squeezed magnetopolarons are investigated. The Hamiltonian of magnetopolarons is dealt with by using squeezed state transformation, which is based on the Lee–Low–Pines and Huybrechts (LLP–H) canonical transformations. This method makes it possible to consider bilinear terms of the phonon operators as well as linear terms arising from the LLP–H transformations. Some exact results are obtained, such as the energies of ground and excited states for squeezed magnetopolarons and renormalized cyclotron masses for some possible transitions.