18 votes
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Lindblad equation for Heisenberg operators?

$\def\dd{{\rm d}} \def\LL{\mathcal{L}} \def\ii{{\rm i}} \def\ee{{\rm e}}$ The trick here is very simple and physically motivated. You simply demand that the expectation value of an operator $A$ is the ...
Mark Mitchison's user avatar
13 votes

Ohmic spectral density

I like to think of the spectral density as a filter for the bosonic field frequencies, it tells you "how much" of each frequency there is. In this case, if $S=1$ you have a linear increase ...
kl0z's user avatar
  • 466
11 votes
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Why should the dynamics of open quantum systems be always linear?

First of all, let me point out that there are theories that propose nonlinear extensions to quantum mechanics (for instance Weinberg's nonlinear quantum mechanics). But there are very strong arguments ...
Martin's user avatar
  • 15.5k
11 votes
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Fastest numerical method to solve Lindblad Master Equation?

Are you familiar with the Stochastic simulation method (quantum trajectories)? It reduces the cost from evolving and $N\times N$ density matrix, to wavefunctions $|\psi\rangle$ of only $N$ elements. ...
Wouter's user avatar
  • 1,545
10 votes
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Hermitian and non-Hermitian jump operators in Lindblad master equation

Writing the Lindblad equation in non-diagonal form with Hermitian operators is always possible. This follows from the simple observation that an arbitrary operator can be expanded in a Hermitian ...
Mark Mitchison's user avatar
9 votes

What is an open quantum system?

The examples you give are of two particle systems. The open quantum system presupposes a many body state. The underlying nature of reality is quantum mechanical , thus all particles in the universe, ...
anna v's user avatar
  • 233k
9 votes
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Pure dephasing $\gamma_\phi$ in a master equation and noise power spectral densities

$\def\ii{{\rm i}} \def\dd{{\rm d}} \def\ee{{\rm e}} $ It turns out that the case of pure dephasing is exactly solvable, and one can obtain nice solutions under certain conditions. In particular, I ...
Mark Mitchison's user avatar
8 votes
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Is the Heisenberg picture of an open-system very different than that of a closed one?

Indeed, for a product operator $\hat{C} = \hat{A}\hat{B}$, it is not true that $\hat{C}(t) = \hat{A}(t) \hat{B}(t)$ for a general (i.e. non-unitary) evolution in the Heisenberg picture. It is ...
Mark Mitchison's user avatar
8 votes

Lindblad and Input-Output Formalism in Quantum Optics

There is already a nice answer but I feel that some important aspects deserve additional attention. My answer is simply a list of observations: Master equations involve approximations: It is ...
Wolpertinger's user avatar
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8 votes
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Born-Markov Approximation: Why is $\rho_{I}(s) \to \rho_{I}(t)$ taken, and not $\rho(s) \to \rho(t)$?

Roughly speaking, it is the Schrödinger-picture density operator which has rapidly oscillating phase factors. Transforming to the interaction picture removes these phase factors. The residual time ...
Mark Mitchison's user avatar
8 votes

Why do time evolution semigroups have to be contracting?

I do not know the general context, but I see that the book deals with open systems. The probability is not conserved, in general, for open systems. However it cannot increase (it remains a probability)...
Valter Moretti's user avatar
8 votes
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Consistency of Lindblad-type operator evolution equations

The adjoint Liouvillian generates a perfectly acceptable operator evolution from a quantum-mechanical point of view. However, the Leibniz rule no longer applies with respect to the operator product, ...
Mark Mitchison's user avatar
7 votes
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Classical approximation of coupling to a quantum bath

$\def\ii{{\rm i}} \def\dd{{\rm d}} \def\ee{{\rm e}} \def\Tr{{\rm Tr}} $As far as I know and would expect, the replacement of a quantum heat reservoir with a noisy classical field cannot be rigorously ...
Mark Mitchison's user avatar
7 votes

In what sense is a quantum damped harmonic oscillator dissipative?

I'll use the convention of writing the exponent as $\gamma t / m$ rather than $\gamma t$. The actual energy of the HO is $$ E = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}m\omega^2x^2 = \frac{1}{2m}p^2\mathrm{...
ACuriousMind's user avatar
  • 124k
7 votes
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How is it acceptable that the Lindblad equation typically depends on a Cauchy principal value?

From a mathematical perspective, ostensibly dodgy distributional issues often arise because the familiar formula for the Fourier transform is not the whole story. If a function (lets say of one ...
J. Murray's user avatar
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6 votes
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Lindblad equation derivation

The statement is almost true, although one has to redefine both $H_S$ and $H_{SB}$ to make it work. You need to also make use of the standard assumption of factorising initial conditions, $$ \rho(0) = ...
Mark Mitchison's user avatar
6 votes

What is an open quantum system?

The distinction between an open and a closed quantum system is mostly about whether information about the system is copied into the outside world, not about interaction per se. Suppose, for example, ...
alanf's user avatar
  • 7,114
6 votes
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Lindbladian and Dynamical semigroups

I have done more reading and see an important point I missed above. In my last line for $\dot{\rho_A(t)}$ I made the step $$ \text{Tr}_B(\mathcal{L}[\rho(t)]) = \mathcal{L}_A[\rho_A(t)] $$ This step ...
Jagerber48's user avatar
  • 13.6k
6 votes

Correspondence between ground state and steady state in quantum systems

The system + environment will not go into its ground state if it is isolated. In fact, it will not even go to an equilibrium state but keep evolving under the unitary dynamics govered by the total ...
Norbert Schuch's user avatar
6 votes
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Why do physical quantum maps need to be completely positive?

First, I assume you are fine with requiring positivity, since if the map is not positive then you will get negative probabilities for some initial states. If a map $\Phi_s$ in your Hilbert space $\...
peep's user avatar
  • 725
5 votes
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Book recommendations for learning about open quantum systems

A well-known reference for this is the book by Breuer and Petruccione: The theory of Open Quantum Systems (Oxford University Press; 1 edition (August 29, 2002)). It seems to largely overlap with what ...
5 votes

What are the open problems in the field of quantum thermodynamics?

This is an active field of research with very many open problems, a summary of which would constitute a partially subjective list and thus be inappropriate for this site. You might find the following ...
5 votes

Understanding quantum stochastic master equations

The Belavkin equation and other stochastic master equations describe the evolution of a system which is being continuously measured. Since quantum measurement is inherently stochastic, this is what ...
asph's user avatar
  • 533
5 votes

Modelling excitation loss in a quantum network: non-Hermitian Hamiltonian vs. Lindblad formulation

Excitation loss can be modelled using a Lindblad equation (with a Hermitian Hamiltonian) which is guaranteed to be trace-preserving. However, one must work in a slightly larger Hilbert space. In ...
Mark Mitchison's user avatar
5 votes

Relationship between the Lindblad Equation and Redfield Equation

Usually it is the Redfield equation that is converted to Lindblad form. Doing the reverse depends strongly on the Lindblad form and the available interpretation of the Lindblad operators. The ...
udrv's user avatar
  • 10.4k
5 votes
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Numerical Simulation of Stochastic Master Equation using Stochastic Schrödinger Equation (Wave Function Monte Carlo)

Before tackling the OP's questions, let us first quickly establish some conventions and notation. A stochastic master equation (SME) describing quantum-jump trajectories can be written in the form $$ ...
Mark Mitchison's user avatar
5 votes

Complete positivity: why is the condition sufficient for quantum maps?

Let me start by asking the question: What do you think the definition of complete positivity should be instead? You want it to "ensure that I will never find a non positive global transformation", but ...
Noiralef's user avatar
  • 7,163
5 votes
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Examples of non-Hermitian Hamiltonians in open systems?

A decaying particle has this. We are only looking at part of the total hamiltonian (we drop the terms that describe the decay products). Our particle has sates $i$ that have their own energy and decay ...
TEH's user avatar
  • 986
5 votes
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Lindblad from infinitesimal Kraus sum representation

I think that your notes want to show that any (time-independent) Markovian master equation is written in the Gorini-Kossakowski-Sudarshan-Lindblad (GKLS) form. My feeling is that they are ignoring ...
Goffredo_Gretzky's user avatar

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