Vectorization

In this module, integration of ordinary and stochastic master equations is performed on density operators parametrized by \(d^2\) real numbers, where \(d\) is the dimension of the system Hilbert space. These are the components of the density operator as a vector in a basis that is Hermitian and, excepting the identity, traceless. Since the ordinary and stochastic master equations under consideration are trace preserving, one could neglect the basis element corresponding to the identity, but as the module currently stands it is included to simplify some expressions and provide a simple test to make sure calculations are proceeding as they ought to.

The preliminary basis having these properties that is used consists of the generalized Gell–Mann matrices:

\[\begin{split}\Lambda^{jk}=\begin{cases} |j\rangle\langle k|+|k\rangle\langle j|, & 1\leq k<j\leq d \\ \\ -i|j\rangle\langle k|+i|k\rangle\langle j|, & 1\leq j<k\leq d \\ \\ \sqrt{\frac{2}{k(k+1)}}\left(\sum_{l=1}^k|l\rangle\langle l|- |k\rangle\langle k|\right), & 1\leq j=k<d \\ \\ I, & j=k=d \end{cases}\end{split}\]

I have toyed around with building a custom basis to make the coupling operator sparse by applying orthogonal transformations to the normalized version of this basis, but since that appears to have little effect I believe I will simply use this basis for the time being. This basis as I have written it is orthogonal, but not normalized:

\[\operatorname{Tr}[\Lambda^{jk}\Lambda^{mn}]=\delta_{jm}\delta_{kn}\big(2+ \delta_{jd}\delta_{kd}(d-2)\big)\]

The density operator and coupling operator are vectorized in the following manner:

\[\begin{split}\begin{align} \rho &=\sum_{j,k}\rho_{jk}\Lambda^{jk}, & \rho_{jk} &\in\mathbb{R} \\ c &=\sum_{j,k}c_{jk}\Lambda^{jk}, & c_{jk} &\in\mathbb{C} \end{align}\end{split}\]

Matrix representations of superoperators

We can write the unconditional vacuum master equation \(d\rho/dt=c\rho c^\dagger-\frac{1}{2}(c^\dagger c\rho+\rho c^\dagger c)\) as a system of coupled first-order ordinary differential equations:

\[\begin{split}\begin{align} \operatorname{Tr}[\Lambda^{jk}\Lambda^{jk}]\frac{\mathrm{d}\rho_{jk}} {\mathrm{d}t} &=\sum_{p,q}\rho_{pq}\left(\sum_{m,n}|c_{mn}|^2 \operatorname{Tr} \left[\Lambda^{jk}\left(\Lambda^{mn}\Lambda^{pq}\Lambda^{mn}- \frac{1}{2}(\Lambda^{mn}\Lambda^{mn}\Lambda^{pq}+\Lambda^{pq}\Lambda^{mn} \Lambda^{mn})\right)\right]+\right. \\ & \quad\left.\sum_{dm+n<dr+s}2\Re\left\{c_{mn}c_{rs}^* \operatorname{Tr}\left[\Lambda^{jk}\left(\Lambda^{mn}\Lambda^{pq} \Lambda^{rs}-\frac{1}{2}(\Lambda^{rs}\Lambda^{mn}\Lambda^{pq}+ \Lambda^{pq}\Lambda^{rs}\Lambda^{mn})\right)\right]\right\}\right) \end{align}\end{split}\]

This means I can write the vectorized version of the equation, using single indices \(w=dr+s\), \(x=dj+k\), \(y=dp+q\), and \(z=dm+n\) for \(\vec{\rho}\):

\[\frac{d\vec{\rho}}{dt}=D(\vec{c})\vec{\rho}\]

The matrix \(D(\vec{c})\) has entries:

\[\begin{split}\begin{align} D_{xy}(\vec{c}) &=(\operatorname{Tr}[\Lambda^x\Lambda^x])^{-1}\left( \sum_z|c_z|^2\operatorname{Tr}[\Lambda^x(\Lambda^z\Lambda^y\Lambda^z- \frac{1}{2}(\Lambda^z\Lambda^z\Lambda^y+ \Lambda^y\Lambda^z\Lambda^z))]+\right. \\ & \quad\left.\sum_{z>w}2\Re\left\{c_z c_w^*\operatorname{Tr}[\Lambda^x( \Lambda^z\Lambda^y\Lambda^w-\frac{1}{2}(\Lambda^w\Lambda^z\Lambda^y+ \Lambda^y\Lambda^w\Lambda^z))]\right\}\right) \end{align}\end{split}\]

In a similar way we can calculate:

\[\rho^\prime=\frac{M}{2}[c,[c,\rho]]+\frac{M^*}{2}[c^\dagger,[c^\dagger,\rho]]\]

in the vectorized form:

\[\vec{\rho}^\prime=E(M,\vec{c})\vec{\rho}\]

where \(E(M,\vec{c})\) has entries:

\[\begin{split}\begin{align} E_{xy}(M,\vec{c})&=2(\operatorname{Tr}[\Lambda^x\Lambda^x])^{-1} \left(\sum_{w<z}\Re\{M^*c_wc_z\}\Re\{ \operatorname{Tr}[\Lambda^x(\Lambda^w\Lambda^z\Lambda^y+ \Lambda^y\Lambda^w\Lambda^z)- 2\Lambda^x\Lambda^w\Lambda^y\Lambda^z]\}+\right. \\ &\quad\left.\sum_w\Re\{M^*c_w^2\}(\Re\{ \operatorname{Tr}[\Lambda^x\Lambda^w\Lambda^w\Lambda^y]\}- \operatorname{Tr}[\Lambda^x\Lambda^w\Lambda^y\Lambda^w])\right) \end{align}\end{split}\]

If I vectorize the plant Hamiltonian:

\[\begin{split}\begin{align} H&=\sum_zh_z\Lambda^z,&h_z&\in\mathbb{R} \end{align}\end{split}\]

I can then calculate:

\[\frac{d\rho}{dt}=-i[H,\rho]\]

in the vectorized form:

\[\frac{d\vec{\rho}}{dt}=F(\vec{h})\vec{\rho}\]

where \(F(\vec{h})\) has entries:

\[F_{x,y}(\vec{h})=(\operatorname{Tr}[\Lambda^x\Lambda^x])^{-1}\sum_zh_z\, \Im\left\{\operatorname{Tr}[\Lambda^x(\Lambda^z\Lambda^y- \Lambda^y\Lambda^z)]\right\}\]

Nonlinear superoperator representation

The stochastic expression:

\[d\rho=dW\,(c\rho+\rho c^\dagger-\rho\operatorname{Tr}[(c+c^\dagger)\rho])\]

can be calculated:

\[d\vec{\rho}=dW(G+\vec{k}\cdot\vec{\rho})\vec{\rho}\]

where we define:

\[\begin{split}\begin{align} G_{x,y}&=2\left(\operatorname{Tr}[\Lambda^x\Lambda^x]\right)^{-1}\sum_z \Re\left\{c_z\operatorname{Tr}[\Lambda^x\Lambda^z\Lambda^y]\right\} \\ k_x&=-2\Re\{c_x\}\operatorname{Tr}[\Lambda^x\Lambda^x] \end{align}\end{split}\]