We reelaborate on a general method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter (usually related to the dispersive limit) becomes small, we immediately get a diagonal effective Hamiltonian that represents correctly the dynamics for arbitrary states and long times. We apply the same technique to obtain how the noise terms in the original model transform under this scheme, providing a systematic way of including damping effects in processes described in terms of effective Hamiltonians.
A mean field argument is used to derive a master equation for systems simultaneously interacting with external fields and coupled environmental degrees of freedom. We prove that this master equation preserves positivity of the reduced density matrix. Solutions of the master equation are compared with exact solutions for a system consisting of three spins which is manipulated with a sequence of laser pulses while interacting with a spin-bath. Exact solutions appear to converge to the master equation result as the number of bath spins increases.
A method of exactly solving the master equation is presented in this letter. The explicit form of the solution is determined by the time evolution of a composite system including an auxiliary system and the open system in question. The effective Hamiltonian governing the time evolution of the composed system are derived from the master equation. Two examples, the dissipative two-level system and the damped harmonic oscillator, are presented to illustrate the solving procedure. PACS number(s): 05.30.-d, 05.40.+j, 42.50.Ct
The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the high-temperature one. We numerically analyse the accuracy of the corresponding expressions and show that the precision of the Bloch-Redfield equantum master equation can be improved if we replace the original system Hamiltonian by the Hamiltonian of me…
This paper derives master equations for an atomic two-level system for a large set of unitarily equivalent Hamiltonians without employing the rotating wave and certain Markovian approximations. Each Hamiltonian refers to physically different components as representing the "atom" and as representing the "field" and hence results in a different master equation, when assuming a photon-absorbing environment. It is shown that the master equations associated with the minimal coupling and the multipolar Hamiltonians predict enormous stationary state narrowband photon emission rates, even in the absen…
Time-local master equations are more generally applicable than is often recognised, but at first sight it would seem that they can only safely be used in time intervals where the time evolution is invertible. Using the Jaynes-Cummings model, we here construct an explicit example where two different Hamiltonians, corresponding to two different non-invertible and non-Markovian time evolutions, will lead to arbitrarily similar time-local master equations. This illustrates how the time-local master equation on its own in this case does not uniquely determine the time evolution. The example is neve…
We consider a pair of dipoles for which direct electrostatic dipole-dipole interactions may be significantly larger than the coupling to transverse radiation. We derive a master equation using the Coulomb gauge, which naturally enables us to include the inter-dipole Coulomb energy within the system Hamiltonian rather than the interaction. In contrast, the standard master equation for a two- dipole system, which depends entirely on well-known gauge-invariant S-matrix elements, is usually derived using the multipolar gauge, wherein there is no explicit inter-dipole Coulomb interaction. We show u…
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to…
These lecture notes from the 2019 Les Houches Summer School on Quantum Information Machines provides a pedagogical introduction to the theory of non-reciprocal quantum interactions and devices. The goal is connect various approaches and concepts, including Hamiltonians encoding synthetic gauge fields, scattering descriptions, quantum master equations, and non-Hermitian Hamiltonians. The importance of having both non-trivial synthetic gauge fields and dissipation for obtaining non-reciprocal interactions is stressed. Connections to broader topics such as quantum reservoir engineering and the qu…
By using the effective Hamiltonian approach, we present a self-consistent framework for the analysis of geometric phases and dynamically stable decoherence-free subspaces in open systems. Comparisons to the earlier works are made. This effective Hamiltonian approach is then extended to a non-Markovian case with the generalized Lindblad master equation. Based on this extended effective Hamiltonian approach, the non-Markovian master equation describing a dissipative two-level system is solved, an adiabatic evolution is defined and the corresponding adiabatic condition is given.
We present a closed form solution to the eigenvalue problem of a class of master equations that describe open quantum system with loss and dephasing but without gain. The method relies on the existence of a conserved number of excitation in the Hamiltonian part and that none of the Lindblad operators describe an excitation of the system. In the absence of dephasing Lindblad operators, the eigensystem of the Liouville operator can be constructed from the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian used in the quantum jump approach. Open versions of spin chains, the T…
Feedback control is expected to considerably protect quantum states against decoherence caused by interaction between the system and environment. Especially, Markovian feedback scheme developed by Wiseman can modify the properties of decoherence and eventually recover the purity of the steadystate of the corresponding master equation. This paper provides a condition for which the modified master equation has a pure steady state. By applying this condition to a two-qubit system, we obtain a complete parametrization of the feedback Hamiltonian such that the steady state becomes a maximally entan…
It is shown that if the decoherence matrix corresponding to a qubit master equation has a block-diagonal real part, then the evolution is determined by a one-dimensional oscillator equation. Further, when the full decoherence matrix is block-diagonal, then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator Hamiltonian or Lagrangian. When the solution of the oscillator equation is not known, an explicit sufficient condition for complete positivity can still be obtained, based on a Hamiltonian/Lagrangian inequality. A rotational…
We develop a master equation formalism to describe the evolution of the average density matrix of a closed quantum system driven by a stochastic Hamiltonian. The average over random processes generally results in decoherence effects in closed system dynamics, in addition to the usual unitary evolution. We then show that, for an important class of problems in which the Hamiltonian is proportional to a Gaussian random process, the 2nd-order master equation yields exact dynamics. The general formalism is applied to study the examples of a two-level system, two atoms in a stochastic magnetic field…
In this paper, we generalize the quantum Brownian motion to include momentum-dependent system-environment couplings. The conventional QBM model corresponds to the spacial case $W_k = V_k$. The generalized QBM is more complicated but the generalization is necessary. This is because the particle transition and the pair production between the system and the environment represent two very different physical processes, and usually cannot have the same coupling strengths. Thus, the conventional QBM model, which is well-defined at classical level, is hardly realized in real quantum physical world. We…
Conditions sufficient for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS. The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensur…
Open-system dynamics play a key role in the experimental and theoretical study of cavity optomechanical systems. In many cases, the quantum Langevin equations have enabled excellent models for optical decoherence, yet a master-equation approach to the fully nonlinear optomechanical Hamiltonian has thus far proven more elusive. To address this outstanding question and broaden the mathematical tool set available, we derive a solution to the Lindblad master equation that models optical decoherence for a system evolving with the nonlinear optomechanical Hamiltonian. The method combines a Lie-algeb…
Relying only on first principles, we derive a master equation of Lindblad form generally applicable for adiabatically time dependent systems. Our analysis shows that the much debated secular approximation can be valid for slowly time dependent Hamiltonians when performed in an appropriate basis. We apply our approach to the well known Landau-Zener problem where we find that adiabaticity is improved by coupling to a low temperature environment.
We show that the exact master equation incorporating initial correlations for open quantum systems, within the Nakajima-Zwanzig operator-projection method, is a homegenous master equation for the reduced density matrix. We also derive explicitly the exact master equation for a large class of bosonic and fermionic open quantum systems incorporating initial correlations, the resulting master equation is homogenous. We find that the effects of the initial correlations can be fully embedded into the fluctuation dynamics through the exact homogenous master equation. Also a generalized nonequilibriu…
We point out that the celebrated GRW master-equation is invariant under translations, reflecting the homogeneity of space, thus providing a particular realization of a general class of translation-covariant Markovian master-equations. Such master-equations are typically used for the description of decoherence due to momentum transfers between system and environment. Building on this analogy we show the exact relationship between the GRW master-equation and decoherence master-equations, further providing a collisional decoherence model formally equivalent to the GRW master-equation. This allows…
We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.
A mean field approximation is employed to derive a master equation suitable for self-interacting baths and strong system-bath coupling. Solutions of the master equation are compared with exact solutions for a central spin interacting with a spin-bath.
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative…
We do not present any original or new material. This is a tutorial addressed to students who need to study the microscopic derivation of the quantum-mechanical master equation encountered in many practical physical situations.
Hu, Paz and Zhang [ B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D {\bf 45} (1992) 2843] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases.
Driven two-level system is a useful model to describe many quantum objects, particularly in quantum information processing. However, the exact master equation for such a system is barely explored. Making use of the Feynman-Vernon influence functional theory, we derive an exact non-Markovian master equation for the driven two-level system and show the lost feature in the perturbative treatment for this system. The perturbative treatment leads to the time-convolutionless (TCL) and the Nakajima-Zwanzig (NZ) master equations. So to this end, we derive the time-convolutionless (TCL) and the Nakajim…
We define an ensemble of projection operators, each of which has an exact associated Nakajima-Zwanzig master equation for quantum open system evolution. A mean field approximation for the memory kernels is introduced that yields, for an optimally chosen projection operator, a completely determined inhomogeneous master equation. Previous proofs of positivity and equilibration are extended to these new inhomogeneous non-Markovian master equations. We study a nitrogen vacancy center in diamond interacting with 13C impurities to illustrate the method.
The time-convolutionless (TCL) non-Markovian master equation was generally thought to break down at finite time due to its singularity and fail to produce the asymptotic behavior in strong coupling regime. However, in this paper, we show that the singularity is not an obstacle for validity of the TCL master equation. Further, we propose a multiscale perturbative method valid for solving the TCL master equation in strong coupling regime, though the ordinary perturbative method invalidates therein.
Non-Markovian dynamics of two interacting two-level qubits coupled to a bosonic bath was previously studied using the quantum-state-diffusion (QSD) equation, where a stochastic state is used to describe the system. In this study, we provide another perspective on this system by deriving the analytic form of the master equation, which describes the system with a reduced density matrix. Then, we validate the master equation by examining entanglement generation and state purity. In addition, with the master equation, we observe the effects from first-order noise and notice that a good asymptotic …
The quantum master equation is a widespread approach to describing open quantum system dynamics. In this approach, the effect of the environment on the system evolution is entirely captured by the dynamical generator, providing a compact and versatile description. However, care needs to be taken when several noise processes act simultaneously or the Hamiltonian evolution of the system is modified. Here, we show that generators can be added at the master equation level without compromising physicality only under restrictive conditions. Moreover, even when adding generators results in legitimate…
By introducing thermal entangled state representation which can map master equations of density operator in quantum statistics as state vector evolution equations and using dissipative interaction picture we solve the master equation of J-C model with cavity damping. In addition we derive the Wigner function for density operator when the atom is initially in the up state and the cavity mode is in coherent state.
We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.
We show that integro-differential generalized Langevin and non-Markovian master equations can be transformed into larger sets of ordinary differential equations. .On the basis of this transformation we develop a numerical method for solving such integro-differential equations. Physically motivated example calculations are performed to demonstrate the accuracy and convergence of the method.
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…
We modify the Araki-Woods double Fock space construction in order to describe general squeezed Gaussian states and use this to represent squeezed quantum stochastic noise processes. Associated master equations are derived.
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipa…
High fidelity models, which support accurate device characterization and correctly account for environmental effects, are crucial to the engineering of scalable quantum technologies. As it ensures positivity of the density matrix, one preferred model for open systems describes the dynamics with a master equation in Lindblad form. The Linblad operators are rarely derived from first principles, resulting in dynamical models which miss those additional terms that must generally be added to bring the master equation into Lindblad form, together with concomitant other terms that must be assimilated…
The quantum master equation obtained by generalizing the geometric formulation of nonequilibrium thermodynamics to dissipative quantum systems is seriously nonlinear. We argue that nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature of the nonlinear contribution. The thermodynamic nonlinearity leads to proper equilibrium solutions and improves the dynamic behavior of dissipative quantum systems. We also discuss the Markovian character of the thermodynamic quantum master equation. The general ideas are illustrated for two-level systems.
We construct a quantum Markovian Master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the Master equation to be extended beyond the adiabatic limit. The Non-Adiabatic Master Equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic Master equation, the NAME leads to coupled equations of motion for the population and cohe…
By extending the mean-field Hamiltonian to include nonhermitian operators, the master equations for fermions and bosons can be derived. The derived equations reduce to the Markoff master equation in the low-density limit and to the quasiclassical master equation for homogeneuos systems.
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. Employing the exact solution for the model we determine analytical expressions for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows a detailed investigation and comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, we find that the structure of widely used phenomenological master equations with memory kernel may be incompatible with a …
It is shown that the standard quantum Brownian equation (QBE) can violate positivity not only past the thermal correlation time, but at arbitrarily long times at high system frequencies. In an effort to improve the standard QBE, exact operator solutions are provided for a class of non-autonomous master equations. These exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations.
Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when intersubsystem interactions exist. Here we develop an alternative local master equation by virtue of similar approximations used in deriving the traditional Gorini-Kossakowski-Lindbl…
In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation. In this paper, we show universal constraints on the relaxation times valid for any d-level GKLS master equations, which is a generalization of the well-known constraints for 2-level systems. Specifically, we show that any relaxation rate, the inverse-relaxation time, is not greater than half of the sum of all relaxation rates. Since the relaxation times are …
A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N-qubit system can be derived explicitly from th…
We consider open quantum systems with dynamics described by master equations that have perturbative expansions in the system-environment interaction. We show that, contrary to intuition, full-time solutions of order-2n accuracy require an order-(2n+2) master equation. We give two examples of such inaccuracies in the solutions to an order-2n master equation: order-2n inaccuracies in the steady state of the system and order-2n positivity violations, and we show how these arise in a specific example for which exact solutions are available. This result has a wide-ranging impact on the validity of …
Convolutionless and convolution master equations are the two mostly used physical descriptions of open quantum systems dynamics. We subject these equations to time deformations: local dilations and contractions of time scale. We prove that the convolutionless equation remains legitimate under any time deformation (results in a completely positive dynamical map) if and only if the original dynamics is completely positive divisible. Similarly, for a specific class of convolution master equations we show that uniform time dilations preserve positivity of the deformed map if the original map is po…
Most future quantum devices, including quantum computers, require control that is broadband, meaning that the rate of change of the time-dependent Hamiltonian is as fast or faster than the dynamics it generates. In many areas of quantum physics, including quantum technology, one must include dissipation and decoherence induced by the environment. While Markovian master equations provide the only really efficient way to model these effects, these master equations are derived for constant Hamiltonians (or those with a discrete set of well-defined frequencies). In 2006, Alicky, Lidar, and Zanardi…
We describe a new approach called Ket-Bra Entangled State (KBES) Method which enables one convert master equations into Schrödinger-like equation. In sharply contrast to the super-operator method, the KBES method is applicable for any master equation of finite-level system in theory, and the calculation can be completed by computer. With this method, we obtain the exact dynamic evolution of a radioactivity damped 2-level atom in time-dependent external field, and a 3-level atom coupled with bath; Moreover, the master equation of N-qubits Heisenberg chain each qubit coupled with a reservoir is …
We propose a new representation for several quantum master equations in so-called quasithermodynamic form. This representation (when it exists) let one to write down dynamical equations both for diagonal and non-diagonal elements of density matrix of the quantum system of interest in unified form by means of nonequilibrium potential ("entropy") that is a certain quadratic function depending on all variables describing the state. We prove that above representation exists for the general Pauli master equation and for the Lindblad master equation ( at least in simple cases ) as well. We discuss a…
We present a local Master equation for open system dynamics in two forms: Markovian and non-Markovian. Both have a wider range of validity than the Lindblad equation investigated by Davies. For low temperatures, they do not require coupling to be exponentially weak in the system size. If the state remains a low bond dimension Matrix Product State throughout the evolution, the local equation can be simulated in time polynomial in system size.
The local approach to construct master equation for a composite open system with a weak internal coupling is simple and seems reasonable. However, it is thermodynamic consistent only when the subsystems are resonantly coupled. Efforts are being made to understand the inconsistency and test the validity of the local master equation. We present a perturbative method to solve the steady-state solutions of linear local master equations, which are demonstrated by two simple models. The solving process shows the stationary state as the result of competition between incoherent operations and the unit…
The low temperature solution of the exact master equation for an oscillator coupled to a linear passive heat bath is known to give rise to serious divergences. We now show that, even in the high temperature regime, problems also exist, notably the fact that the density matrix is not necessarily positive.
