Quantization of the damped harmonic oscillator is taken as leitmotiv to gently introduce elements of quantum probability theory for physicists. To this end, we take (graduate) students in physics as entry level and explain the physical intuition and motivation behind the, sometimes overwhelming, math machinery of quantum probability theory.
The main text starts with the quantization of the (undamped) harmonic oscillator from the Heisenberg and Schroedinger point of view. We show how both treatments are special instances of a quantum probabilistic quantization procedure: the second quantization functor. We then apply the second quantization functor to the damped harmonic oscillator and interpret the quantum dynamics of the position and energy operator as stochastic processes.
In this introductory course we sketch the framework of quantum probability in order to discuss open quantum systems, in particular the damped harmonic oscillator.
Quantum dissipation is studied within two model oscillators, the Caldirola-Kanai (CK) oscillator as an open system with one degree of freedom and the Bateman-Feshbach-Tikochinsky (BFT) oscillator as a closed system with two degrees of freedom. Though these oscillators describe the same classical damped motion, the CK oscillator retains the quantum coherence, whereas the damped subsystem of the BFT oscillator exhibits both quantum decoherence and classical correlation. Furthermore the amplified subsystem of the BFT oscillator shows the same degree of quantum decohernce and classical correlation…
The interaction of a quantum deformed oscillator with the environment is studied deriving a master equation whose form strongly depends on the type of deformation.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a dam…
We solve the optimal quantum limit of probing a classical force exactly by a damped oscillator initially prepared in the factorized squeezed state. The memory effects of the thermal bath on the oscillator evolution are investigated. We show that the optimal force sensitivity obtained by the quantum estimation theory approaches to zero for the non-Markovian bath, whereas approaches to a finite non-zero value for the Markovian bath as the energy of the damped oscillator goes to infinity.
We analyse the properties of a strongly-damped quantum harmonic oscillator by means of an exact diagonalisation of the full Hamiltonian, including both the oscillator and the reservoir degrees of freedom to which it is coupled. Many of the properties of the oscillator, including its steady-state properties and entanglement with the reservoir can be understood and quantified in terms of a simple probability density, which we may associate with the ground-state frequency spectrum of the oscillator.
By taking a Klein-Gordon field as the environment of an harmonic oscillator and using a new method for dealing with quantum dissipative systems (minimal coupling method), the quantum dynamics and radiation reaction for a quantum damped harmonic oscillator investigated. Applying perturbation method, some transition probabilities indicating the way energy flows between oscillator, reservoir and quantum vacuum, obtained
Quantization of a damped harmonic oscillator leads to so called Bateman's dual system. The corresponding Bateman's Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.
We analyze the new equation of motion for the damped oscillator. It differs from the standard one by a damping term which is nonlocal in time and hence it gives rise to a system with memory. Both classical and quantum analysis is performed. The characteristic feature of this nonlocal system is that it breaks local composition low for the classical Hamiltonian dynamics and the corresponding quantum propagator.
The phase-space path-integral approach to the damped harmonic oscillator is analyzed beyond the Markovian approximation. It is found that pairs of nonclassical trajectories contribute to the path-integral representation of the Wigner propagating function. Due to the linearity of the problem, the sum coordinate of a pair still satisfies the classical equation of motion. Furthermore, it is shown that the broadening of the Wigner propagating function of the damped oscillator arises due to the time-nonlocal interaction mediated by the heat bath.
The quantum dynamics of a damped harmonic oscillator is investigated in the presence of an anisotropic heat bath. The medium is modeled by a continuum of three dimensional harmonic oscillators and anisotropic coupling is treated by introducing tensor coupling functions. Starting from a classical Lagrangian, the total system is quantized in the framework of the canonical quantization. Following Fano technique, Hamiltonian of the system is diagonalized in terms of creation and annihilation operators that are linear combinations of the basic dynamical variables. Using the diagonalized Hamiltonian…
The expression for the density operator of the damped harmonic oscillator is derived from the master equation in the framework of the Lindblad theory for open quantum systems. Then the von Neumann entropy and effective temperature of the system are obtained. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function.
We show that quantum Bateman's system which arises in the quantization of a damped harmonic oscillator is equivalent to a quantum problem with 2D parabolic potential barrier known also as 2D inverted isotropic oscillator. It turns out that this system displays the family of complex eigenvalues corresponding to the poles of analytical continuation of the resolvent operator to the complex energy plane. It is shown that this representation is more suitable than the hyperbolic one used recently by Blasone and Jizba.
The quantum dynamics of a damped and forced harmonic oscillator is investigated in terms of a Lindblad master equation. Elementary algebraic techniques are employed allowing for example to analyze the long time behavior, i.e. the quantum limit cycle. The time evolution of various expectation values is obtained in closed form as well as the entropy and the Husimi phase space distribution. We also discuss the related description in terms of a non-Hermitian Hamiltonian.
In this letter we show that it is not possible to set up a canonical quantization for the damped harmonic oscillator using the Bateman lagrangian. In particular, we prove that no square integrable vacuum exists for the {\em natural} ladder operators of the system, and that the only vacua can be found as distributions. This implies that the procedure proposed by some authors is only formally correct, and requires a much deeper analysis to be made rigorous.
Extended phase space (EPS) formulation of quantum statistical mechanics treats the ordinary phase space coordinates on the same footing and thereby permits the definite the canonical momenta conjugate to these coordinates . The extended lagrangian and extended hamiltonian are defined in EPS by the same procedure as one does for ordinary lagrangian and hamiltonian. The combination of ordinary phase space and their conjugate momenta exhibits the evolution of particles and their mirror images together. The resultant evolution equation in EPS for a damped harmonic oscillator, is such that the ener…
We obtain the tomogram of squeezed correlated states of a quantum parametric damped oscillator in an explicit form. We study the damping within the framework of the Caldirola--Kanai model and chose the parametric excitation in the form of a very short pulse simulated by a delta-kick of frequency; the squeezing phenomenon is reviewed. The cases of strong and weak damping are investigated.
We investigate the exact dynamics of the damped quantum harmonic oscillator under the (un)correlated initial conditions. The master equation is generalized to the cases of the arbitrary factorized state and/or Gaussian state. We show that the variances of the factorized Gaussian state do not sensitively depend on the initial oscillator-bath correlation, which however can remarkably affect the mean values even at high temperature. We also illustrate that the correlations among the factorized states still give rise to the initial dips during the purity evolutions, which can be smoothed out by in…
We consider the Fermi quantization of the classical damped harmonic oscillator (dho). In past work on the subject, authors double the phase space of the dho in order to close the system at each moment in time. For an infinite-dimensional phase space, this method requires one to construct a representation of the CAR algebra for each time. We show that unitary dilation of the contraction semigroup governing the dynamics of the system is a logical extension of the doubling procedure, and it allows one to avoid the mathematical difficulties encountered with the previous method.
We use the exact solution for the damped harmonic oscillator to discuss some relevant aspects of its open dynamics often mislead or misunderstood. We compare two different approximations both referred to as Rotating Wave Approximation. Using a specific example, we clarify some issues related to non--Markovian dynamics, non--Lindblad type dynamics, and positivity of the density matrix.
The investigation of quantum-classical correspondence may lead to gain a deeper understanding of the classical limit of quantum theory. We develop a quantum formalism on the basis of a linear-invariant theorem, which gives an exact quantum-classical correspondence for damped oscillatory systems that are perturbed by an arbitrary force. Within our formalism, the quantum trajectory and expectation values of quantum observables are precisely coincide with their classical counterparts in the case where we remove the global quantum constant h from their quantum results. In particular, we illustrate…
Intracavity and external third order correlations in the damped nondegenerate parametric oscillator are calculated for quantum mechanics and stochastic electrodynamics (SED), a semiclassical theory. The two theories yield greatly different results, with the correlations of quantum mechanics being cubic in the system's nonlinear coupling constant and those of SED being linear in the same constant. In particular, differences between the two theories are present in at least a mesoscopic regime. They also exist when realistic damping is included. Such differences illustrate distinctions between qu…
In the framework of the Lindblad theory for open quantum systems, we determine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath. It is found that the system manifests a quantum decoherence which is more and more significant in time. We also calculate the decoherence time and show that it has the same scale as the time after which thermal fluctuations become comparable with quantum fluctuations.
Nowadays, two of the most prospering fields of physics are quantum computing and spintronics. In both, the loss of information and dissipation plays a crucial role. In the present work we formulate the quantization of the dissipative oscillator, which aids understanding of the above mentioned, and creates a theoretical frame to overcome these issues in the future. Based on the Lagrangian framework of the damped spring system, the canonically conjugated pairs and the Hamiltonian of the system are obtained, by which the quantization procedure can be started and consistently applied. As a result,…
We consider a driven damped anharmonic oscillator which classically leads to a bistable steady state and to hysteresis. The quantum counterpart for this system has an exact analytical solution in the steady state which does not display any bistability or hysteresis. We use quantum state diffusion theory to describe this system and to provide a new perspective on the lack of hysteresis in the quantum regime so as to study in detail the quantum to classical transition. The analysis is also relevant to measurements of a single periodically driven electron in a Penning trap where hysteresis has be…
The damped harmonic oscillator is a workhorse for the study of dissipation in quantum mechanics. However, despite its simplicity, this system has given rise to some approximations whose validity and relation to more refined descriptions deserve a thorough investigation. In this work, we apply a method that allows us to diagonalize exactly the dissipative Hamiltonians that are frequently adopted in the literature. Using this method we derive the conditions of validity of the rotating-wave approximation (RWA) and show how this approximate description relates to more general ones. We also show th…
We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D {\bf 45}, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the…
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these…
We propose a new approach to the quantization of the damped harmonic oscillator in the framework of deformation quantization. The quantization is performed in the Schrödinger picture by a star-product induced by a modified "Poisson bracket". We determine the eigenstates in the damped regime and compute the transition probability between states of the undamped harmonic oscillator after the system was submitted to dissipation.
We revisit quantum dynamics of the damped and driven nonlinear oscillator. In the classical case this system has two stationary solutions (the limit cycles) in the certain parameter region, which is the origin of the celebrated bistability phenomenon. The quantum-classical correspondence for the oscillator dynamics is discussed in details.
We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under such standard settings, we discuss and report on $n$-dimensional PDM damped harmonic oscillators (DHO). We use some $n$-dimensional point canonical transformation to facilitate the linearizability of their $n$-PDM dynamical equations into some $n$-linear DHOs' dynamical equations for constant mass set…
By definition, the Kraus representation of a harmonic oscillator suffering from the environment effect, modeled as the amplitude damping or the phase damping, is directly given by a simple operator algebra solution. As examples and applications, we first give a Kraus representation of a single qubit whose computational basis states are defined as bosonic vacuum and single particle number states. We further discuss the environment effect on qubits whose computational basis states are defined as the bosonic odd and even coherent states. The environment effects on entangled qubits defined by two …
A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we ex…
In the framework of the Lindblad theory for open quantum systems, a master equation for the quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived for the case when the interaction is based on deformed algebra. The equations of motion for observables strongly depend on the deformation function. The expectation values of the number operator and squared number operator are calculated in the limit of a small deformation parameter for the case of zero temperature of the thermal bath. The steady state solution of the equation for the de…
In the framework of the Lindblad theory for open quantum systems, expressions for the density operator, von Neumann entropy and effective temperature of the damped harmonic oscillator are obtained. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function. We give a series of inequalities, relating uncertainty to von Neumann entropy and linear entropy. We analyze the conditions for purity of states and show that for a special choice of the diffusion coefficients, the correlated coherent…
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes - grand canonical ensemble, number states and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and …
We model a quantum system coupled to an environment of damped harmonic oscillators by following the approach of Caldeira-Leggett and adopting the Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master equation of the quantum system of interest (a particle in a general potential), we show that the potential is modified non-trivially by a new inverted harmonic oscillator term, induced by the damping of the bath oscillators. We analyze numerically the case of a particle in a double-well potential, and find that this modification changes both the rate of decoherence at short t…
The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with…
The dynamics of the next quantum jump for a qubit [two level system] coupled to a readout resonator [damped driven harmonic oscillator] is calculated. A quantum mechanical treatment of readout resonator reveals non exponential short time behavior which could facilitate detection of the state of the qubit faster than the resonator lifetime.
For the generalized master equations derived by Karrlein and Grabert for the microscopic model of a damped harmonic oscillator, the conditions for purity of states are written, in particular for different initial conditions and different types of damping, including Ohmic, Drude and weak coupling cases, Agarwal and Weidlich-Haake models. It is shown that the states which remain pure are the squeezed states with constant in time variances. For pure states, the generalized nonlinear Schr\" odinger-type equations corresponding to these master equations are also obtained. Then the condition for pur…
The interaction of (two-level) Rydberg atoms with dissipative QED cavity fields can be described classically or quantum mechanically, even for very low temperatures and mean number of photons, provided the damping constant is large enough. We investigate the quantum-classical border, the entanglement and decoherence of an analytically solvable model, analog to the atom-cavity system, in which the atom (field) is represented by a (driven and damped) harmonic oscillator. The maximum value of entanglement is shown to depend on the initial state and the dissipation-rate to coupling-constant ratio.…
Quantum dynamical semigroups are applied to the study of the time evolution of harmonic oscillators, both bosonic and fermionic. Explicit expressions for the density matrices describing the states of these systems are derived using the holomorphic representation. Bosonic and fermionic degrees of freedom are then put together to form a supersymmetric oscillator; the conditions that assure supersymmetry invariance of the corresponding dynamical equations are explicitly derived.
The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.
We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier t…
In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of dynamical invariants previously proposed by the authors, in which fundamental importance is given to the linear invariants of the oscillator.
We provide a new canonical approach for studying the quantum mechanical damped harmonic oscillator based on the doubling of degrees of freedom approach. Explicit expressions for Lagrangians of the elementary modes of the problem, characterising both forward and backward time propagations are given. A Hamiltonian analysis, showing the equivalence with the Lagrangian approach, is also done. Based on this Hamiltonian analysis, the quantization of the model is discussed.
We show theoretically how a driven harmonic oscillator can be used as a quantum simulator for non-Markovian damped harmonic oscillator. In the general framework, the results demonstrate the possibility to use a closed system as a simulator for open quantum systems. The quantum simulator is based on sets of controlled drives of the closed harmonic oscillator with appropriately tailored electric field pulses. The non-Markovian dynamics of the damped harmonic oscillator is obtained by using the information about the spectral density of the open system when averaging over the drives of the closed …
We numerically study the quantum dynamics of a $\mathcal{PT}$-symmetric kicked harmonic oscillator. We observe that directed current of momentum and ballistic diffusion of energy coexist under the non-resonant conditions, whereas both the momentum and energy oscillate as damped cosine functions with identical frequencies under the resonant conditions. The research shows that the directed current of momentum and ballistic diffusion of energy arise from nearest-neighbor hopping between momentum eigenstates with the non-Hermitian driving, while the damped oscillations of momentum and energy origi…
The quadrature distribution for the quantum damped oscillator is introduced in the framework of the formulation of quantum mechanics based on the tomography scheme. The probability distribution for the coherent and Fock states of the damped oscillator is expressed explicitly in terms of Gaussian and Hermite polynomials, correspondingly.
Valtancoli in his paper entitled [P. Valtancoli, Canonical transformations, and minimal length J. Math. Phys. 56, 122107 (2015)] has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the …
A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.
We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.
In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.
