These different regimes are obtained by tuning the interaction strength parameter in the model, which also determines whether the model is Markovian or non-Markovian. Other model parameters are also varied to estimate the degree of thermalization and relaxation during the interaction. The model is then discussed in depth in the second chapter where the Hamiltonian of the total system is presented and the picture of the interaction Hamiltonian is drawn.
The four chapters contain simulated models and a discussion of the effects and results of changing the parameters governing the interaction and thermalization of the system and the environment.
Closed Quantum Systems
A closed quantum system in an environment, the system is completely isolated from the environment. If one encounters the situation where the considered system is driven by external classical fields, e.g. electromagnetic fields, and the system's dynamics can still be formulated in terms of a time-dependent Hamiltonian generatorH(t), the system will again be said to be closed, while the term isolated is reserved to mean that the system's Hamiltonian is time-independent. If the system is in a mixed state, it is represented by a statistical operator called the density matrix⇢.
Equation (1.18) is a first-order differential equation that governs the dynamics of the density matrix, and thus the time evolution of the probability of the system occupying a set of states.
Heisenberg and Interaction Picture
Note that the time evolution of interaction image operators is only generated by the free part of the Hamiltonian H0 and not by the full Hamiltonian H. With the initial condition UI(t0, t0) = I, the time evolution operator of the interaction image is the solution of the differential equation. The interaction Hamiltonian in the interaction picture is denoted by 1.43) The Von Neumann equation in the interaction picture is of the form 1.44).
The integral form of the von Neumann equation in the interaction picture is given by.
Open Quantum Systems
The Hilbert space of the general system S+Eis is denoted by the tensor product space H =HS⌦HE. If the state of the total system is described by a density matrix⇢, then the expected values of all the observers acting in the Hilbert space of the open system are given by. This map is called a dynamic map and describes the changing state of the open system over time.
The Hilbert space of the composite system is given by the tensor product H =HS⌦HE.
The Interaction Picture Hamiltonian
The index n1 indicates the levels of the lower energy band and n2 indicates the levels of the upper energy band. 2.18).
Overview of the Approximate Analytical Solution
Therefore, we conclude that the lowest order of the TCL expansion with the projection superoperator introduced in Eq. We notice that the population dynamics!11!t" is strongly non-Markovian due to the presence of the initial condition on the right-hand side of the equation. This expression expresses a pronounced memory effect; namely, the population dynamics never forgets its initial data.
Note also that the dynamics of the reduced density matrix is not in Lindblad form, nor does it represent a semigroup. 3–5 we compare the result given by Eq.!4.51" with the HAM prediction and numerical simulations of the Schrödinger equation. The figures clearly show that even the lowest order of the TCL expansion with our projection Fig.
This is clearly the difference between TCL2 and the exact numerical solution of the Schrödinger equation. Each node in this parallel setup typically works on one part of the problem and passes its data to another node. MPI has commands that can display information such as how many processors are available, the name of the processor the process is running on, and also the processor number.
It transmits the same data from one processor to all other processors and distributes individual data from one processor to each processor. Since some processors run faster than others, the bottleneck can be used to synchronize all processors so that the rest of the process can run as data from one processor may be needed for the overall process to continue.
Simulation Code using MPI
This is because the parallel part of the code is implemented when each processor makes a time evolution of the system and then sends its data to the master processor, where an average of all the results together with errors is calculated. This part is where the program timer starts when the code is executed and also where the initial values and parameters of the model are defined. This is the main loop of the code where the time evolution takes place and this part is executed in parallel.
This tells the program that each iteration of the loop must be executed on a separate processor. The potential (V) is then defined together with the total Hamiltonian (H), which is the sum of the system and environment Hamiltonians and the potential. The initial wave function is then defined as a tensor product of the initial states of the system and the environment, and the time evolution is performed with themesolve command.
Here the 'if' statement tells the program to execute this part of the code on the master processor. This last part of the code also runs on the main processor, as it is important to display only the total time the simulation lasted in seconds, minutes or hours. At the end of the code, the file is closed and the show() command tells the program to display the simulation time on the screen once it has completed.
In this chapter, numerical solutions are performed based on the model of the full Schrödinger equation corresponding to the Hamiltonian given in equation (2.3). The other three modes are the same except for the environment mode| iis of the form.
All simulations plotted show ⇢11 versus t, where ⇢11 is the excited state population or the probability that the system is in the excited state|1i, and is the evolution time of the system during interaction with the environment. Figures 4.6–4.9 show different values of N plotted with different values of ✏where the system is initially in the excited state|1i and the lower ambient band is populated. In each case, the system starts in an excited state and through interaction with the environment converges to a stationary population⇢stat11 = 12.
As before it can be seen that as N increases, the faster the system reaches the stationary population. Again, as before, it can be observed that as ✏ increases, the longer the system has to thermalize, which means that the smaller the distance between the levels in the band, the faster the system thermalizes and relaxes after the interaction. Figure 4.10 shows N = 500 plotted with ✏= 1.0 with the system initially in the excited state|1 and the upper band of the environment populated.
Here⇢11(t) stays at one for all time and this indicates that there is not enough energy in the medium to cause any change in the system when the lower band is unpopulated. This is exactly the opposite of the case when the system was initially in the ground state with the bottom band populated. Figures 4.11-4.18 show different values of N plotted against different values of ✏where the system is initially in a superposition state12(|0i+|1i), when the upper and then the lower bands of the environment are populated respectively.
The system starts in the superposition state and after interacting with the environment has a 0.75 probability of being in the excited state with a 0.25 probability of being in the ground state when the upper band is populated (Figures 4.11-4.14) and a 0.25 probability for to be in the excited state with a probability of 0.75 to be in the ground state when the lower band is populated (Figures 4.15-4.18). Since the system is initially in a superposition state, it always interacts with the environment regardless of whether the lower or upper band is initially populated, unlike when the initial state of the system is either in the ground or excited state.
Strong Coupling ( = 0.01)
Discussion
Numerical solutions allow to investigate model regimes where analytical solutions do not predict the correct results. In this thesis, a numerical study of the dynamics of a two-level system associated with a structured environment was studied. The code used to simulate the time evolution of the two-level system when interacting with the finite-level structured environment was written in Python using the QuTiP quantum package and optimized to run in parallel using MPI.
By following the degrees of freedom of the structured environment, we obtained the density matrix of the two-level system. The probability that the system is in an excited state after the interaction (⇢11) was plotted against time. The graphs were produced using both levels of the two-level system, ground and excited, as well as the superposition state.
The parameters of the environment were also varied such as the number of levels in the bands, which band was initially populated as well as the width of the bands. These plots also consist of the two types of interactions between the two-level system and the structured environment, weak and strong coupling. It was observed that the width of the bands also affected the time taken for the system to thermalize, as the band width increased and the time taken for the system to thermalize increased.
It was then observed that the overall system can be controlled in some way, by varying the parameters and choosing which band to populate, one can predict the evolution of the two-level system after interacting with the structured environment. From the fact that ⇢11 remained at zero and one, it can be deduced that the system can be forced to remain in the ground or excited state, depending on which of the two environmental bands is populated. It was also noted that if the system is initially in a superposition state, it can be forced to have a higher probability of being in the ground or excited state depending on which of the two bands of the environment initially is populated.
Two papers have been published so far on the features of the QuTiP library and its applications [23,24].
