We analyze some solvable models of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. We apply Nakajima-Zwanzig's projection method by choosing a reference state of the reservoir endowed with the mixing property. In van Hove's limit, the dynamics is described in terms of a master equation. We observe that Markovianity becomes a valid approximation for timescales that depend both on the form factors of the interaction and on the observables of the reservoir that can be measured.
A generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in Quantum Mechanics. The method is applied in the study of radial oscillator, Morse and Coulomb potentials to obtain a wide set of raising and lowering operators, and to show clearly the connection that link these systems.
A set of factorization energies is introduced, giving rise to a generalization of the Schrödinger (or Infeld and Hull) factorization for the radial hydrogen-like Hamiltonian. An algebraic intertwining technique involving such factorization energies leads to derive $n$-parametric families of potentials in general almost-isospectral to the hydrogen-like radial Hamiltonians. The construction of SUSY partner Hamiltonians with ground state energies greater than the corresponding ground state energy of the initial Hamiltonian is also explicitly performed.
We analyze the dynamics of a quantum mechanical system in interaction with a reservoir when the initial state is not factorized. In the weak-coupling (van Hove) limit, the dynamics can be properly described in terms of a master equation, but a consistent application of Nakajima-Zwanzig's projection method requires that the reference (not necessarily equilibrium) state of the reservoir be endowed with the mixing property.
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found,there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.
Exact solutions of the Caldeira-Leggett Master equation for the reduced density matrix for a free particle and for a harmonic oscillator system coupled to a heat bath of oscillators are obtained for arbitrary initial conditions. The solutions prove that the Fourier transform of the density matrix at time t with respect to (x + x')/2, where x and x' are the initial and final coordinates, factorizes exactly into a part depending linearly on the initial density matrix and a part independent of it. The theorem yields the exact initial state dependence of the density operator at time t and its even…
This paper gives a new perspective on how to solve the second-order linear differential equation written in normal form. Extending the argument of the potential to a complex number leads to solving exactly the Schrödinger equation when the potential is complex using the factorization method. This method leads to solving two Riccati nonlinear equations and by constructing the only possible superpotential, the factorization method gives the eigenvalues and eigenfunctions in closed form for potentials satisfying the shape invariance property. Extending the potential to the complex argument has le…
New exactly solvable problems have already been studied by using a modification of the factorization method introduced by Mielnik. We review this method and its connection with the traditional factorization method. The survey includes the discussion on a generalization of the factorization energies used in the traditional Infeld and Hull method.
An easily solvable quantum master equation has long been sought that takes into account memory effects induced on the system by the bath, i.e., non-Markovian effects. We briefly review the Post-Markovian master equation (PMME), which is relatively easy to solve, and analyze a simple example where solutions obtained exhibit non-Markovianity. We apply the distinguishability measure introduced by Breuer et al., and we also explicitly analyze the divisibility of the associated quantum dynamical maps. We give a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divi…
The interrelationship between the non-Markovian stochastic Schrödinger equations and the corresponding non-Markovian master equations is investigated in the finite temperature regimes. We show that the general finite temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite temperature reservoir in the rotating wave approximation. The exact convolutionless master equation for the damped harmonic os…
A huge family of solvable potentials can be generated by systematically exploiting the factorization (Darboux) method. Starting from the free case, a large class of the known solvable families is thus reproduced, together with new ones. We explicitly find and solve several new singular potentials obtained by iteration from the V=0 case; some of them have an E=0 bound state and constant phase shift without being explicitly scale invariant. The new potentials are rational functions, and can be related to rational solutions of the KdV family.
Quantum operations are usually defined as completely positive (CP), trace preserving (TP) maps on quantum states, and can be represented by operator-sum or Kraus representations. In this paper, we calculate operator-sum representation and master equation of an exactly solvable dynamic of one-qubit open system in layered environment . On the other hand, we obtain exact Nakajima-Zwanzig (NZ) and time-convolutionless (TCL) master equation from the maps. Finally, we study a simple example to consider the relation between CP maps and initial quantum correlation and show that vanishing initial quant…
The "standard" Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schroedinger equation (SSE), closely analogous to Langevin's equation, that has no equivalent Markovian master equation. Considering a spec…
A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determin…
When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed \textit{locally} on each component, i.e. ignoring the coherent couplings, which are later added `by hand'. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent \textit{global} master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node `quantum wire' connected to two heat baths. The stationary solution of the LME and …
We study a class of multipartite open quantum dynamics for systems of arbitrary number of qubits. The non-Markovian quantum master equation can involve arbitrary single or multipartite and time-dependent dissipative coupling mechanisms, expressed in terms of strings of Pauli operators. We formulate the general constraints that guarantee the complete positivity of this dynamics. We characterize in detail underlying mechanisms that lead to memory effects, together with properties of the dynamics encoded in the associated system rates. We specifically derive multipartite "eternal" non-Markovian m…
The stationary Schroedinger equation of the harmonic oscillator is deformed by a Darboux transformation to construct time-dependent potentials with the oscillator profile. The Darboux (supersymmetric or factorization) method is usually developed in the spatial variables of the Schroedinger equation. Here we follow a variation introduced by Bagrov, Samsonov and Shekoyan to include the time-variable as a parameter of the transformation.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed p…
We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION, and COSET INTERSECTION. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solv…
The dissipative dynamics of many-electron systems interacting with a thermal environment has remained a long-standing challenge within time-dependent density functional theory (TDDFT). Recently, the formal foundations of open quantum systems time-dependent density functional theory (OQS-TDDFT) within the master equation approach were established. It was proven that the exact time-dependent density of a many-electron open quantum system evolving under a master equation can be reproduced with a closed (unitarily evolving) and non-interacting Kohn-Sham system. This potentially offers a great adva…
A systematic procedure to derive exact solutions of the associated Lame equation for an arbitrary value of the energy is presented. Supersymmetric transformations in which the seed solutions have factorization energies inside the gaps are used to generate new exactly solvable potentials; some of them exhibit an interesting property of periodicity defects.
The interplay between dissipation, topology and sensitivity to boundary conditions has recently attracted tremendous amounts of attention at the level of effective non-Hermitian descriptions. Here we exactly solve a quantum mechanical Lindblad master equation describing a dissipative topological Su-Schrieffer-Heeger (SSH) chain of fermions for both open boundary condition (OBC) and periodic boundary condition (PBC). We find that the extreme sensitivity on the boundary conditions associated with the non-Hermitian skin effect is directly reflected in the rapidities governing the time evolution o…
Starting from a system of $N$ radial Schrödinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N \times N$ exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on $N (N+1)/2$ unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2\times 2$ case…
The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the \emph{exact} solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the \emph{exact} c…
We introduce a general analytic approach to the study of factorization points and factorized ground states in quantum cooperative systems. The method allows to determine rigorously existence, location, and exact form of separable ground states in a large variety of, generally non-exactly solvable, spin models belonging to different universality classes. The theory applies to translationally invariant systems, irrespective of spatial dimensionality, and for spin-spin interactions of arbitrary range.
Here we present a Lindblad master equation that approximates the Redfield equation, a well known master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap…
We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo m and does not require factorization of m. For smooth m, i.e., when the prime factors of m are of size poly(log m), the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general m, the algorithm of Mosca and Zalka is available. Even for m=3 and k=1 our result appears to be new. We also present applications to compute the structure of abelian and solvable groups w…
We study a 1-dimensional XX chain under nonequilibrium driving and local dephasing described by the Lindblad master equation. The analytical solution for the nonequilibrium steady state found for particular parameters in [J.Stat.Mech., L05002 (2010)] is extended to arbitrary coupling constants, driving and homogeneous magnetic field. All one, two and three-point correlation functions are explicitly evaluated. It is shown that the nonequilibrium stationary state is not gaussian. Nevertheless, in the thermodynamic and weak-driving limit it is only weakly correlated and can be described by a matr…
An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequities between the exact and approximate rates.
We put forth a new class of quantum master equations that correctly reproduce the asymptotic state of an open quantum system beyond the infinitesimally weak system-bath coupling limit. Our method is based on incorporating the knowledge of the reduced steady state into its dynamics. The correction not only steers the reduced system towards a correct steady state but also improves the accuracy of the dynamics, thereby refining the archetypal Born-Markov weak-coupling second-order master equations. In case of equilibrium, since a closed form for the steady state exists in terms of a mean force Gi…
The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.
The so-called Lindblad equation, a typical master equation describing the dissipative quantum dynamics, is shown to be solvable for finite-level systems in a compact form without resort to writing it down as a set of equations among matrix elements. The solution is then naturally given in an operator form, known as the Kraus representation. Following a few simple examples, the general applicability of the method is clarified.
A central problem in the theory of the dynamics of open quantum systems is the derivation of a rigorous and computationally tractable master equation for the reduced system density matrix. Most generally, the evolution of an open quantum system is described by a completely positive linear map. We show how to derive a completely positive Markovian master equation (the Lindblad equation) from such a map by a coarse graining procedure. We provide a novel and explicit recipe for calculating the coefficients of the master equation, using perturbation theory in the weak-coupling limit. The only para…
We present a unified treatment of three cases of quasi-exactly solvable problems, namely, charged particle moving in Coulomb and magnetic fields, for both the Schrödinger and the Klein-Gordon case, and the relative motion of two charged particles in an external oscillator potential. We show that all these cases are reducible to the same basic equation, which is quasi-exactly solvable owing to the existence of a hidden $sl_2$ algebraic structure. A systematic and unified algebraic solution to the basic equation using the method of factorization is given. Analytic expressions of the energies and…
For a specific exactly solvable 2 by 2 matrix model with a PT-symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics and show that none of them admits a factorization CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed.
We present a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies $ω_x$ and $ω_y$. The new curved Hamiltonian ${H}_κ$ depends on the curvature $κ$ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of ${H}_κ$ relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superi…
By invoking supersymmetry, we found a condition under which the Stark effect problem for a polar and polarizable molecule subject to nonresonant electric fields becomes exactly solvable. The exact solvability condition for the interaction parameters involved yields exact wavefunction for the "stretched" states, $|J=m,m>$, and for the $|1,1>$ state in the case of a purely induced-dipole interaction. The analytic expressions for the eigenenergy, the space-fixed dipole moment, the alignment cosine, and the expectation value of the angular momentum allow to readily reverse-engineer the problem of …
We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. Then, we compare this dynamics with the exact dynamics in an exactly solvable spin-boson mode…
We present a rigorous analysis of the phenomenon of decoherence for general N-level systems coupled to reservoirs of free massless bosonic fields. We apply our general results to the specific case of the qubit. Our approach does not involve master equation approximations and applies to a wide variety of systems which are not explicitly solvable.
The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling tra…
We present a systematic approach for the separation of variables for the two-dimensional Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and p…
The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the so-called {\bf Master Constraint} which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical con…
We investigate the existence and the properties of fully separable (fully factorized) ground states in quantum spin systems. Exploiting techniques of quantum information and entanglement theory we extend a recently introduced method and construct a general, self-contained theory of ground state factorization in frustration free quantum spin models defined on lattices in any spatial dimension and for interactions of arbitrary range. We show that, quite generally, non exactly solvable translationally invariant models in presence of an external uniform magnetic field can admit exact, fully factor…
The quantum version of a non-linear oscillator, previouly analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form $m={(1+λx^2)}^{-1}$ and with a $\la$-dependent nonpolynomial rational potential. This $\la$-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for $\la\to 0$ all the characteristics of the linear oscillator are recovered. Firstly, the $\la$-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem and the $\la$-dependent eigenenergies and eig…
We apply the factorization technique developed by Kuru and Negro [Ann. Phys. 323 (2008) 413] to study complex classical systems. As an illustration we apply the technique to study the classical analogue of the exactly solvable PT symmetric Scarf II model, which exhibits the interesting phenomenon of spontaneous breakdown of PT symmetry at some critical point. As the parameters are tuned such that energy switches from real to complex conjugate pairs, the corresponding classical trajectories display a distinct characteristic feature - the closed orbits become open ones.
