Heisenberg Picture Research Articles

Surface plasmaritons: Wave-mechanical and second-quantized theories

In the wake of recently established quantum theories for bulk and surface plasmons, and for bulk plasmaritons [Jung and Keller, Phys. Rev. A 103, 063501 (2021); Phys. Rev. A 104, 053508 (2021)] wave-mechanical and second-quantized (QED) theories for surface plasmaritons are developed. Our photon wave-mechanical theory is based on the covariant four-potential in the Lorenz gauge, where dynamical equations for surface plasmariton quantum particles consisting of scalar ($S$), longitudinal ($L$), and transverse ($T1$, $T2$) photons and driven by a surface current density sheet are obtained. The second quantized QED theory of surface plasmaritons is formulated on the basis of the Heisenberg equations of motion for the contravariant four-potential annihilation operators. The connection of the surface plasmariton quantum theory to the quasiparticle picture of bulk plasmaritons is elucidated by studying the $T$-photon exponential decay lengths appearing in the description. The shortest decay length (equal on both sides of the surface) characterizes the one-dimensional spatial confinement of the $T$-photon source. Starting from a reexamination (and correction) of the dynamic boundary conditions for a flat jellium-vacuum interface carrying a surface current density, the dispersion relation for surface plasmaritons is obtained. The explicit form of the dispersion relation is derived upon a study of the local field in the selvedge region of the boundary. Fundamental aspects of the dispersion relation are pointed out, modeling the electron dynamics by the microscopic electrodynamics of a quantum well. Remarks on (i) our covariant surface plasmariton quantum theory seen in a broader perspective, (ii) suggestions for applications of the theory, and (iii) a comparison of the key ingredients in our trio of papers on plasmon and plasmariton quantum physics are presented.

Four‐mode Squeezed States in de Sitter Space: A Study With Two Field Interacting Quantum System

AbstractIn this paper we study the application of four‐mode squeezed states in the cosmological context, studying two weakly coupled scalar fields in the planar patch of the de Sitter space. We construct the four‐mode squeezed state formalism and connect this concept with the Hamiltonian of the two coupled inverted harmonic oscillators having a time‐dependent effective frequency in the planar patch of the de Sitter space. Further, the corresponding evolution operator for the quantum Euclidean vacuum state has been constructed, which captures its dynamics. Using the Heisenberg picture coupled differential equations describing the time evolution for all squeezing parameters (amplitude, phase and angle) have been obtained, for the weakly coupled two scalar field model. With the help of these evolutions for the coupled system, we simulate the dynamics of the squeezing parameters in terms of conformal time. From our analysis, we observe interesting dynamics, which helps us to explore various underlying physical implications of the weakly coupled two scalar field system in the planar patch of the de Sitter cosmological background.

Operative approach to quantum electrodynamics in dispersive dielectric objects based on a polarization-mode expansion

In this paper, we deal with the macroscopic electromagnetic response of a finite size dispersive dielectric object, in unbounded space, in the framework of quantum electrodynamics, using the Heisenberg picture. We keep the polarization and the electromagnetic field distinct to enable the treatment of the polarization and electromagnetic fluctuations on equal footing in a self-consistent QED Hamiltonian. We apply a Hopfield type scheme to account for the dispersion and dissipation of the matter. We provide a general expression of the time evolution of the polarization density field observable as function of the initial conditions of the matter field observables and of the electromagnetic field observables. It is an integral operator whose kernel is a linear combination of the impulse responses of the dielectric object that we obtain within the framework of classical electrodynamics. The electric field observable is expressed in terms of the polarization density field observable by means of the full wave dyadic Green's function for the free space. The statistical functions of the observables of the problem can be expressed through integral operators of the statistics of the initial conditions of the matter field observables and of the electromagnetic field observables, whose kernels are linear or multilinear expressions of the impulse responses of the dielectric object. We expand the polarization density field observable in terms of the static longitudinal and transverse modes of the object to diagonalize the Coulomb and Ampere interaction energy terms of the Hamiltonian in the Coulomb gauge. Few static longitudinal and transverse modes are needed to calculate each element of the impulse response matrix for dielectric objects with sizes of the order up to $\underset{\ensuremath{\omega}}{min}{{c}_{0}/[\ensuremath{\omega}\sqrt{|\ensuremath{\chi}(\ensuremath{\omega})|}]}$, where $\ensuremath{\chi}(\ensuremath{\omega})$ is the susceptibility of the dielectric. We apply the proposed approach to different scenarios describing the dielectric susceptibility by the Drude-Lorentz model.

QOptCraft: A Python package for the design and study of linear optical quantum systems

The manipulation of the quantum states of light in linear optical systems has multiple applications in quantum optics and quantum computation. The package QOptCraft gives a collection of methods to solve some of the most usual problems when designing quantum experiments with linear interferometers. The methods include functions that compute the quantum evolution matrix for n photons from the classical description of the system and inverse methods that, for any desired quantum evolution, will either give the complete description of the experimental system that realizes that unitary evolution or, when this is impossible, the complete description of the linear system which approximates the desired unitary with a locally minimal error. The functions in the package include implementations of different known decompositions that translate the classical scattering matrix of a linear system into a list of beam splitters and phase shifters and methods to compute the effective Hamiltonian that describes the quantum evolution of states with n photons. The package is completed with routines for useful tasks like generating random linear optical systems, computing matrix logarithms, and quantum state entanglement measurement via metrics such as the Schmidt rank. The routines are chosen to avoid usual numerical problems when dealing with the unitary matrices that appear in the description of linear systems. Program summaryProgram Title: QOptCraftCPC Library link to program files:https://doi.org/10.17632/r24hszggf4.1Developer's repository link:https://github.tel.uva.es/juagar/qoptcraftLicensing provisions: Apache-2.0Programming language: Python 3 v3.9.5:0a7dcbd, May 3 2021 17:27:52Supplementary material:: User's manual at https://github.tel.uva.es/juagar/qoptcraft/-/blob/main/QOptCraft_V1.1_user_guide.pdfNature of problem: The evolution of the quantum states of light in linear optical devices can be computed from the scattering matrix of the system using a few alternative points of view. Apart from being able to compute the evolution through a known optical system, it is interesting to consider the less studied inverse problem of design: finding the optical system which gives or approximates a desired evolution. Linear optical systems are limited and can only provide a small subset of all the physically possible quantum transformations on multiple photons. Choosing the best approximation for the evolutions that cannot be achieved with linear optics is not trivial.This software deals with the analysis of the quantum evolution of multiple photons in linear optical devices and the design of optical setups that achieve or approximate a desired quantum evolution.Solution method: We have automated multiple computation processes regarding quantum experiments via linear optic devices. The methods rely on the properties of the groups and algebras that describe the problem of light evolution in linear system.The library QOptCraft for Python 3 includes known numerical methods for decomposing an optical system into beam splitters and phase shifters and methods to give the quantum evolution of system classically described by a scattering matrix using either the Heisenberg picture evolution of the states, a description based on the permanents of certain matrices or the evolution from the effective Hamiltonian of the system. It also provides methods for the design of achievable evolutions, using the adjoint representation, and for approximating quantum evolutions outside the reach of linear optics with an iterative method using Toponogov's comparison theorem from differential geometry. The package is completed with useful functions that deal with systems including losses and gain, described with quasiunitary matrices, the generation of random matrices and stable implementations of the matrix logarithm.Additional comments including restrictions and unusual features: The package is designed to work with intermediate scale optical systems. Due to the combinatorial growth in the state space with the number of photons and modes involved, there is an upper limit on the efficiency of any classical calculation. QOptCraft serves as a design tool to explore the building blocks of photonic quantum computers, optical systems that generate useful quantum states of light for their use in metrology or other applications or the design of quantum optics experiments to probe the foundations of quantum mechanics.

Quantum regression theorem for multi-time correlators: A detailed analysis in the Heisenberg picture

The quantum regression theorem is one of the central results in open quantum systems and is extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schr\"odinger picture. In this paper we make use of the Heisenberg picture to derive the quantum regression theorems for multi-time correlation functions, which in the special limit reduce to the well-known two-time regression theorem. For the multi-time correlation function we find that the regression theorem takes the same form as it takes for the two-time correlation function with a mild restriction that one of the times should be greater than all other time variables. Interestingly, the Heisenberg picture also allows us to derive an analog of regression theorem for out-of-time-ordered correlators. We further extend our study for the case of non-Markovian dynamics and report the modifications to the standard quantum regression theorem. We illustrate all of the above results using the paradigmatic dissipative spin-boson model.

Ultimate speed limits to the growth of operator complexity

In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its validity in paradigmatic models of quantum chaos.

Heisenberg treatment of multiphoton pulses in waveguide QED with time-delayed feedback

The dynamics of waveguide-QED systems involving coherent time-delayed feedback give rise to a hierarchy of multitime correlations within the Heisenberg picture due to the induced non-Markovianity. We propose to perform a projection onto a complete set of states in the Hilbert space to decompose the multitime correlations into single-time matrix elements. To illustrate the procedure, we consider the paradigmatic example of a two-level system that couples to a semi-infinite waveguide and interacts with quantum light pulses. Our approach complements the range of available methods as it allows calculating the dynamics under the inclusion of additional dissipation channels in a numerically exact and efficient manner for multiphoton pulses of arbitrary shape where memory requirements are known in advance.

Is instability near a black hole key for "thermalization" of its horizon?

We put forward an attempt towards building a possible theoretical model to understand the observer dependent thermalization of black hole horizon. The near horizon Hamiltonian for a massless, chargeless particle is xp type. This is unstable in nature and so the horizon can induce instability in a system. The particle in turn finds the horizon thermal when it interacts with it. We explicitly show this in the Schrodinger as well as in Heisenberg pictures by taking into account the time evolution of the system under this Hamiltonian. Hence we postulate that existing instability near the horizon can be one of the potential candidates for explaining the black hole thermalization.

Effective Heisenberg equations for quadratic Hamiltonians

In this paper, we discuss effective quantum dynamics obtained by averaging projector with respect to free dynamics. For unitary dynamics generated by quadratic fermionic Hamiltonians, we obtain effective Heisenberg dynamics. By perturbative expansions, we obtain the correspondent effective time-local Heisenberg equations. We also discuss a similar problem for bosonic case.

Quantum nondemolition measurements of photon number in monolithic microcavities

We revisit the idea of quantum nondemolition measurement (QND) of optical quanta via a resonantly enhanced Kerr nonlinearity taking into account quantum backaction. We show that the monolithic microcavities enable QND measurement of the number of quanta in a weak signal field using a classical probe field spatially overlapping with the signal. The phase of the probe field acquires information about the signal number of quanta without altering it due to the cross-phase modulation effect. We find the exact solution to the Heisenberg equations of motion of this system and calculate the measurement error, accounting for the optical losses in the measurement path. We identify a realistic approximation to obtain the explicit form of the final conditional quantum state of the signal field, accounting for the undesirable self-phase modulation effect and designing the optimal homodyne measurement of the probe beam to evade this effect. We show that the best modern monolithic microcavities allow achieving the measurement imprecision several times better than the standard quantum limit.

Influence of a non-linear interaction and Kerr medium on the emission spectrum and entropy squeezing of Su(1,1) algebra with single qubit

The interaction of a quantum algebraic system su(1,1) interacting with a qubit is studied. A nonlinear function has been added to interaction with the presence of Kerr medium (KM). The general solution is obtained using the constants of motion and Heisenberg equation for the time-evolution operator. The effects of the nonlinear function and KM on the emission spectrum and entropy squeezing are calculated. The results showed that the emission spectrum is affected by the nonlinear function. Symmetric peaks are generated around the vertical axis with increasing influence of the nonlinear function. By activating the role of the KM, the emission spectrum loses symmetry around the vertical axis and the maximum values decrease. Moreover, the KM reduces the squeezing intervals, while the squeezing intervals are improved by increasing the effect of the nonlinear function.

Spin-wave theory in a randomly disordered lattice: A Heisenberg ferromagnet

Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson–Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averaging over all possible distributions of impurities, we express the free energy of the system as a function of the mean impurity concentration. Subsequently, we set up the double-time single particle Green function at temperature T in the momentum space in terms of magnon operators and derive the equation of motion for the Green function through the Heisenberg equation of motion and then solve the resulting equation. From this, we calculate the self-energy and then the spectral density function for the system. We apply the formalism to the case of the simple cubic lattice and compute the density of states, the spectral density function and the lifetime of the magnons as a function of energy for several values of the mean concentration of nonmagnetic ions in the ferromagnetic lattice. We calculate the magnon energy spectrum as a function of the average impurity concentration fraction c, which shows that for low lying states, the excitation energy increases continuously with c in the studied range 0.1≤c≤0.7. We also use the spectral density function to compute some thermodynamical quantities through the magnon occupation number. We have obtained closed form expressions for the configurationally averaged physical quantities of interest in a unified fashion as functions of the mean concentration of nonmagnetic impurities c to any order of c applicable below a critical percolation concentration cp. The quantities of interest comprise the thermodynamic potential (free energy), the spin-wave self-energy and the spectral density function from which other quantities can be derived.

The quantum theory of time: a calculus for q-numbers.

In quantum theory, physical systems are usually assumed to evolve relative to a c-number time. This c-number time is unphysical and has turned out to be unnecessary for explaining dynamics: in the timeless approach to quantum theory developed by Page & Wootters 1983 Phys. Rev. D 27, 2885. (doi:10.1103/PhysRevD.27.2885), subsystems of a stationary universe can instead evolve relative to a ‘clock', which is a quantum system with a q-number time observable. Page & Wootters formulated their construction in the Schrödinger picture, which left open the possibility that the c-number time still plays an explanatory role in the Heisenberg picture. I formulate their construction in the Heisenberg picture and demonstrate how to eliminate c-number time from that picture, too. When the Page–Wootters construction is formulated in the Heisenberg picture, the descriptors of physical systems are functions of the clock's q-number time, and derivatives with respect to this q-number time can be defined in terms of the clock's algebra of observables, which results in a calculus for q-numbers.

Higher Order Moments Dynamics for Some Multimode Quantum Master Equations

We derive Heisenberg equations for arbitrary high order moments of creation and annihilation operators in the case of the quantum master equation with a multimode generator which is quadratic in creation and annihilation operators and obtain their solutions. Based on them we also derive similar equations for the case of the quantum master equation, which occur after averaging the dynamics with a quadratic generator with respect to the classical Poisson process. This allows us to show that dynamics of arbitrary finite-order moments of creation and annihilation operators is fully defined by finite number of linear differential equations in this case.

Disentangling Many-Body Effects in the Coherent Optical Response of 2D Semiconductors.

In single-layer (1L) transition metal dichalcogenides, the reduced Coulomb screening results in strongly bound excitons which dominate the linear and the nonlinear optical response. Despite the large number of studies, a clear understanding on how many-body and Coulomb correlation effects affect the excitonic resonances on a femtosecond time scale is still lacking. Here, we use ultrashort laser pulses to measure the transient optical response of 1L-WS2. In order to disentangle many-body effects, we perform exciton line-shape analysis, and we study its temporal dynamics as a function of the excitation photon energy and fluence. We find that resonant photoexcitation produces a blue shift of the A exciton, while for above-resonance photoexcitation the transient response at the optical bandgap is largely determined by a reduction of the exciton oscillator strength. Microscopic calculations based on excitonic Heisenberg equations of motion quantitatively reproduce the nonlinear absorption of the material and its dependence on excitation conditions.

The Tomonaga‐Schwinger equation in flat space-time

A derivation of the Tomonaga‐Schwinger equation is presented based on defining a wave functional on a general spacelike surface such that the expectation values match those in the corresponding Heisenberg picture. This derivation is carried out in flat space-time.

Nonlinear spin excitations in a Heisenberg antiferromagnet with NNN and D–M interactions

Spin deviation vs site number in an inhomogeneous (site dependent) antiferromagnet incorporating next nearest neighbour (NNN) and Dzyaloshinski–Moriya (D–M) interactions along with nearest neighbour and anisotropic interactions is investigated. In this context, a model Hamiltonian for 1D inhomogeneous antiferromagnetic spin system with site dependent inhomogenities is constructed. Glauber coherent state representation along with H-P representation is applied to derive the Heisenberg equation of motion. The dynamics in the continuum level is investigated using the Sine–Cosine (SC) function method and the results are analysed graphically.

Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations

The Kitaev model on the honeycomb lattice, while being integrable via the spin-fermion mapping, has generally resisted an analytical treatment of the far-from-equilibrium dynamics due to the extensive number of relevant configurations of conserved charges. Here we study a close proxy of this model, the isotropic Kitaev spin-1/2 model on the Bethe lattice. Instead of relying on the spin-fermion mapping, we take a straightforward approach of solving Heisenberg equations for a tailored subset of spin operators. The simplest operator in this subset corresponds to the energy contribution of a single bond direction. As an example, we calculate the time-dependent expectation value of this observable for a factorized translation-invariant (or staggered-translation-invariant) initial state with arbitrary initial (staggered) polarization.

Perturbation approach in Heisenberg equations for lasers

Nonlinear Heisenberg-Langevin equations are solved analytically by operator Fourier-expansion for the laser in the LED regime. Fluctuations of populations of lasing levels are taken into account as perturbations. Spectra of operator products are calculated as convolutions, preserving Bose commutations for the lasing field operators. It is found that fluctuations of population significantly affect spontaneous and stimulated emissions into the lasing mode, increase the radiation rate, the number of lasing photons and broad the spectrum of a bad cavity thresholdless and the superradiant lasers. The method can be applied to various resonant systems in quantum optics.

Non‐classical Lie symmetries for nonlinear time‐fractional Heisenberg equations

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation.