Figure 1

Path in parameter space starting and ending at P. The duration of the path segments $\mathrm{P}\to \mathrm{Q},\mathrm{Q}\to \mathrm{R}$, and $\mathrm{R}\to \mathrm{P}$ is $T_1,T_2$, and $T_3$, respectively, as described in Eq. (78). $\delta \phi$ is the opening angle in the equatorial plane. Note that for this path the enclosed solid angle equals $\delta \phi$.Reuse & Permissions

Figure 2

(Color online) The solid lines show the value of the matrix element $⟨1\mid \varrho \left(T\right)\mid 1⟩$ of the output density operator of the exact equation, as a function of $T$. The run-time $T$ is measured in units of ${\omega }^{-1}$, defined in Eq. (66). The dashed lines show the corresponding value $⟨1\mid {\varrho }^a\left(T\right)\mid 1⟩$. Counted from the top and down, the solid-dashed line pairs correspond to $\Gamma =0$, $\Gamma =0.01$, and $\Gamma =0.1$, respectively. The initial state is pure, with polar angle $x=\pi ∕5$ and azimuthal angle $y=3\pi ∕4$ on the Bloch sphere. The horizontal dashed line corresponds to the ordinary adiabatic approximation for the closed evolution case. One may note that the rate at which the exact solution approaches the approximate solution appears to be rather independent of the strength $\Gamma$ of the decoherence.Reuse & Permissions

Figure 3

(Color online) These graphs highlight another aspect of the same series of calculations as in Fig. 2. They show the error between the exact evolution and the approximate evolution, in form of the normalized fidelity $D$ defined in Eqs. (85, 86), as a function of $T$. The run-time $T$ is given in units of ${\omega }^{-1}$, defined in Eq. (66). Note the different range of the run-time compared to the other plots. Here the dotted line corresponds to $\Gamma =0$, the dashed line to $\Gamma =0.01$ and the solid line to $\Gamma =0.1$. Note that the dotted and the dashed lines almost coincide. These graphs indicate that the distance between the approximate and exact evolution, within the computational subspace, decreases with the run-time $T$ at a rate more or less independent of $\Gamma$.Reuse & Permissions

Figure 4

(Color online) The intensity losses, as defined in Eq. (84), out of the computational subspace for the same series of calculations as in Figs. 2, 3. The solid lines show the intensity losses $I$ of the exact evolution, and the dashed the intensity losses $I^a$ of the approximate evolution, as a function of $T$. The run-time $T$ is given in units of ${\omega }^{-1}$, defined in Eq. (66). Here the lowermost pair of curves correspond to $\Gamma =0$. Note that for $\Gamma =0$ the loss is identically zero for the approximate evolution. The uppermost pair of curves corresponds to $\Gamma =0.1$, and the pair in the middle corresponds to $\Gamma =0.01$.Reuse & Permissions

Figure 5

(Color online) The maximum error in the Hilbert-Schmidt norm ${\max }_{s∊[0,1]}\parallel \rho \left(s\right)-{\rho }^a\left(s\right)\parallel$ between the solution $\rho \left(s\right)$ of Eq. (88) and the solution ${\rho }^a\left(s\right)$ of the approximate equation as a function of the run-time $T$, the latter measured in arbitrary units. The plots are generated for one random instance of $H_0,Z,A$, and initial state, for a four dimensional Hilbert space. Each curve corresponds to a value of $\Gamma$, and shows the maximum error as a function of $T$. To the right of the figure the curves correspond, counted from the bottom and up, to $\Gamma =0$, 0.002, 0.004, 0.006, 0.008, 0.01. As seen, the error for the adiabatic approximation in the closed case $(\Gamma =0)$ seems to tend to zero as $T$ increases, while the other cases appear to have a minimum error for a certain $T$.Reuse & Permissions