Recent changes to wikihttp://sourceforge.net/p/wavepacket/wiki/Recent changes to wikienThu, 10 Jul 2014 07:19:14 -0000Numerics.LvNE modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.LvNE/<div class="markdown_content"><pre>--- v40
+++ v41
@@ -24,6 +24,8 @@
[[img src="Lindblad_1.gif" alt="\hat C_{ij}=\sqrt{\Gamma_{i\leftarrow j}}|i\rangle\langle j|"]]
with rate constants Γ<sub>ij</sub>=1/T<sub>ij</sub> (inverse times). With this choice (positive prefactors!) the above evolution is trace-preserving, i.e. the sum of populations remains constant, and completely positive, i. e., the indidual populations remain positive. The summation in the above equation extends over all dissipation/decoherence channels.
+
+In conclusion, the above Ansatz of the quantum-master equation in Born-Markov approximation (weak couling, no memory) provides a description useful in many cases ranging from NMR over quantum optics to molecular/chemical applications. However, it has to be noted that this method is not applicable in many problems in solid state physics at low temperatures.
Dissipation models
------------------
</pre>
</div>Burkhard SchmidtThu, 10 Jul 2014 07:19:14 -0000http://sourceforge.neta7a35be73d25977750aa76de36ac92770e524a5dNumerics.LvNE modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.LvNE/<div class="markdown_content"><pre>--- v39
+++ v40
@@ -19,7 +19,7 @@
[[img src="Lindblad_0.gif" alt="{\hat{\hat{\mathcal L}}}_D\hat\rho=-\sum_k\left \{\hat C_k\hat\rho\hat C_k^\dagger-\frac{1}{2}\left\[\hat C_k^\dagger \hat C_k,\hat\rho\right\]_+\right\}"]]
-where the operators C describe the effect of the environment on the system in Born-Markov approximation. In most applications, they are simply chosen to be projectors
+where the operators C describe the effect of the environment on the system in Born-Markov approximation (weak couling, no memory). In most applications, they are simply chosen to be projectors
[[img src="Lindblad_1.gif" alt="\hat C_{ij}=\sqrt{\Gamma_{i\leftarrow j}}|i\rangle\langle j|"]]
</pre>
</div>Burkhard SchmidtThu, 10 Jul 2014 07:14:34 -0000http://sourceforge.net2afadb342f654014d9584c5e2287b7ae5df66728Numerics.LvNE modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.LvNE/<div class="markdown_content"><pre>--- v38
+++ v39
@@ -5,7 +5,7 @@
[[img src="LvNE_0.gif" alt="\frac{\partial}{\partial t}\hat{\rho}={\hat{\hat{\mathcal L}}}_H\hat\rho+{\hat{\hat{\mathcal L}}}_D\hat\rho"]]
-where the first term of the r.h.s. represents the closed system Liouvillian (Hermitian/Hamiltonian part)
+where the first term of the r.h.s. represents the closed system dynamics in terms of a Liouvillian for the Hermitian(aka Hamiltonian) part
[[img src="LvNE_1.gif" alt="{\hat{\hat{\mathcal L}}}_H\hat\rho=-\frac{i}{\hbar}\left\[\hat H,\hat\rho \right\]_-"]]
</pre>
</div>Burkhard SchmidtThu, 10 Jul 2014 07:04:05 -0000http://sourceforge.net19e4a3e8d4e6ad6c397039a4d68c9f95bcd46abaNumerics.LvNE modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.LvNE/<div class="markdown_content"><pre>--- v37
+++ v38
@@ -19,7 +19,7 @@
[[img src="Lindblad_0.gif" alt="{\hat{\hat{\mathcal L}}}_D\hat\rho=-\sum_k\left \{\hat C_k\hat\rho\hat C_k^\dagger-\frac{1}{2}\left\[\hat C_k^\dagger \hat C_k,\hat\rho\right\]_+\right\}"]]
-where the operators C are simply given by
+where the operators C describe the effect of the environment on the system in Born-Markov approximation. In most applications, they are simply chosen to be projectors
[[img src="Lindblad_1.gif" alt="\hat C_{ij}=\sqrt{\Gamma_{i\leftarrow j}}|i\rangle\langle j|"]]
</pre>
</div>Burkhard SchmidtThu, 10 Jul 2014 07:02:13 -0000http://sourceforge.net4670923a83227270af381aa910157a8eef7f6f21Numerics.LvNE modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.LvNE/<div class="markdown_content"><pre>--- v36
+++ v37
@@ -15,7 +15,7 @@
for a quantum system interacting with electrical fields within the semiclassical dipole approximation where H<sub>0</sub>=T+V is a diagonal Hamiltonian for the unperturbed system (in energy representation), D is a Hermitian matrix with dipole moments and F(t) denotes the electric field.
-The second term on the r.h.s. of the first equation represents the coupling to the environment which is often described in terms of a [Lindblad Liouvillian](http://en.wikipedia.org/wiki/Lindblad_equation) accounting for dissipation and/or dephasing
+The second term on the r.h.s. of the first equation represents the coupling to the environment accounting for time-directed irreversibility, i. e., dissipation and/or dephasing. The most general form preserving complete positivity of the reduced density operator is the [Lindblad Liouvillian form](http://en.wikipedia.org/wiki/Lindblad_equation)
[[img src="Lindblad_0.gif" alt="{\hat{\hat{\mathcal L}}}_D\hat\rho=-\sum_k\left \{\hat C_k\hat\rho\hat C_k^\dagger-\frac{1}{2}\left\[\hat C_k^\dagger \hat C_k,\hat\rho\right\]_+\right\}"]]
</pre>
</div>Burkhard SchmidtThu, 10 Jul 2014 06:56:39 -0000http://sourceforge.net8cd71b14c159b88a322d1a603f3446d6936871d7Numerics.Control modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.Control/<div class="markdown_content"><pre>--- v28
+++ v29
@@ -14,7 +14,7 @@
[[img src="Control_1a.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m F_d(t)N_d\right)x(t),\,x(0)=x_0"]]
-with matrices N<sub>l</sub>∈R<sup>n×n</sup>, 1≤d≤m where the control term is bilinear, i. e., linear both in the field(s) u(t) and the state vector x(t). Upon shifting the latter one, x(t) → x(t) - x<sub>e</sub> by the equilibrium density A x<sub>e</sub> = 0 (eigenvector of A corresponding to the eigenvalue 0), we end up with
+with matrices N<sub>d</sub>∈R<sup>n×n</sup>, 1≤d≤m where the control term is bilinear, i. e., linear both in the field components F<sub>d</sub>(t) and the state vector x(t). Upon shifting the latter one, x(t) → x(t) - x<sub>e</sub> by the equilibrium density A x<sub>e</sub> = 0 (eigenvector of A corresponding to the eigenvalue 0), we end up with
[[img src="Control_2.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m u_d(t)N_d\right)x(t)+iBu(t),\,x(0)=x_0"]]
</pre>
</div>Burkhard SchmidtThu, 03 Jul 2014 09:13:16 -0000http://sourceforge.netc8f734a0b654ed4bee92e19042af4a1fcd19185cNumerics.Control modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.Control/<div class="markdown_content"><pre></pre>
</div>Burkhard SchmidtThu, 03 Jul 2014 09:11:09 -0000http://sourceforge.net5d75f6e762bf8fc13c8343361600f52880dabb20Numerics.Control modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.Control/<div class="markdown_content"><pre></pre>
</div>Burkhard SchmidtThu, 03 Jul 2014 09:10:29 -0000http://sourceforge.net99dbcf3daaa41f980ad4d56c7cd2002a1f5ca569Numerics.Control modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.Control/<div class="markdown_content"><pre>--- v25
+++ v26
@@ -10,15 +10,15 @@
[[img src="Control_1.gif" alt="\dot{x}(t)=Ax(t)+iBu(t),\,x(0)=x_0"]]
-In case of quantum dynamics driven by electric and/or magnetic fields (see below), however, this has to be replaced by the following bi-linear version of the input equation
+In case of quantum dynamics driven by electric and/or magnetic fields (see below), however, we are dealing with the following bi-linear version of the input equation
-[[img src="Control_1a.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m u_d(t)N_d\right)x(t),\,x(0)=x_0"]]
+[[img src="Control_1a.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m F_d(t)N_d\right)x(t),\,x(0)=x_0"]]
with matrices N<sub>l</sub>∈R<sup>n×n</sup>, 1≤d≤m where the control term is bilinear, i. e., linear both in the field(s) u(t) and the state vector x(t). Upon shifting the latter one, x(t) → x(t) - x<sub>e</sub> by the equilibrium density A x<sub>e</sub> = 0 (eigenvector of A corresponding to the eigenvalue 0), we end up with
[[img src="Control_2.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m u_d(t)N_d\right)x(t)+iBu(t),\,x(0)=x_0"]]
-which we will have to solve numerically. On the one hand, this shift transforms a homogeneous equation to an inhomogeneous one and seems to complicate things. On the other hand, the shift establishes the basis for the balancing method by setting x(0) = 0 for an equilibrium initial condition. Note, however, that while most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.
+where we have renamed u(t) = F(t) and where we have set B = N x<sub>e</sub>. On the one hand, this shift transforms a homogeneous equation to an inhomogeneous one and seems to complicate things. On the other hand, the shift establishes the basis for the balancing method by setting x(0) = 0 for an equilibrium initial condition. Note, however, that while most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.
The **output equation** defines the *low-dimensional* observables y(t)∈R<sup>k</sup>, k<<n in terms of output matrix C ∈R<sup>k×n</sup>.
</pre>
</div>Burkhard SchmidtThu, 03 Jul 2014 09:09:57 -0000http://sourceforge.net633af4978c63d599de60b67da77211515de44706Numerics.Control modified by Burkhard Schmidthttp://sourceforge.net/p/wavepacket/wiki/Numerics.Control/<div class="markdown_content"><pre>--- v24
+++ v25
@@ -6,19 +6,19 @@
General concepts: input/output
------------------------------
-In linear control theory, the **input equation** describes the evolution of a *high-dimensional* system in terms of its state vector x(t)∈C<sup>n</sup>. The field-free evolution is described by linear ODEs with real symmetric matrix A∈R<sup>n×n</sup> (or complex Hermitian matrix A∈C<sup>n×n</sup>) with simple eigenvalue 0. The (linear!) interaction with a *low-dimensional* control u(t)∈R<sup>m</sup>, m<<n is given by input matrix B∈R<sup>n×m</sup>.
+In standard linear control theory, the **input equation** describes the evolution of a *high-dimensional* system in terms of its state vector x(t)∈C<sup>n</sup>. The field-free evolution is described by linear ODEs with real symmetric matrix A∈R<sup>n×n</sup> (or complex Hermitian matrix A∈C<sup>n×n</sup>) with simple eigenvalue 0. The (linear!) interaction with a *low-dimensional* control u(t)∈R<sup>m</sup>, m<<n is given by input matrix B∈R<sup>n×m</sup>.
[[img src="Control_1.gif" alt="\dot{x}(t)=Ax(t)+iBu(t),\,x(0)=x_0"]]
In case of quantum dynamics driven by electric and/or magnetic fields (see below), however, this has to be replaced by the following bi-linear version of the input equation
-[[img src="Control_1a.gif" alt="\dot{x}(t)=\left( A+i\sum_{l=1}^m u_l(t)N_l\right)x(t),\,x(0)=x_0"]]
+[[img src="Control_1a.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m u_d(t)N_d\right)x(t),\,x(0)=x_0"]]
-with matrices N<sub>l</sub>∈R<sup>n×n</sup>, 1≤l≤m where the control term is bilinear, i. e., linear both in the field(s) u(t) and the state vector x(t). Upon shifting vector x(t) → x - x<sub>e</sub> by the equilibrium density A x<sub>e</sub> = 0 (eigenvector of A corresponding to the eigenvalue 0), we end up with
+with matrices N<sub>l</sub>∈R<sup>n×n</sup>, 1≤d≤m where the control term is bilinear, i. e., linear both in the field(s) u(t) and the state vector x(t). Upon shifting the latter one, x(t) → x(t) - x<sub>e</sub> by the equilibrium density A x<sub>e</sub> = 0 (eigenvector of A corresponding to the eigenvalue 0), we end up with
-[[img src="Control_2.gif" alt="\dot{x}(t)=\left( A+i\sum_{l=1}^m u_l(t)N_l\right)x(t)+iBu(t),\,x(0)=x_0"]]
+[[img src="Control_2.gif" alt="\dot{x}(t)=\left( A+i\sum_{d=1}^m u_d(t)N_d\right)x(t)+iBu(t),\,x(0)=x_0"]]
-Note, however, that while most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.
+which we will have to solve numerically. On the one hand, this shift transforms a homogeneous equation to an inhomogeneous one and seems to complicate things. On the other hand, the shift establishes the basis for the balancing method by setting x(0) = 0 for an equilibrium initial condition. Note, however, that while most of control theory considers systems with linear input equations, there is only little theory for this bilinear setting.
The **output equation** defines the *low-dimensional* observables y(t)∈R<sup>k</sup>, k<<n in terms of output matrix C ∈R<sup>k×n</sup>.
</pre>
</div>Burkhard SchmidtThu, 03 Jul 2014 09:05:13 -0000http://sourceforge.net7ba8bf68d306d16558c74eaea10ccbe4e7775422