Geometric phases in the presence of a composite environment

We compute the geometric phase for a spin-1/2 particle under the presence of a composite environment, composed of an external bath (modeled by an infinite set of harmonic oscillators) and another spin-1/2 particle. We consider both cases: an initial entanglement between the spin-1/2 particles and an initial product state in order to see if the initial entanglement has an enhancement effect on the geometric phase of one of the spins. We follow the nonunitary evolution of the reduced density matrix and evaluate the geometric phase for a single two-level system. We also show that the initial entanglement enhances the sturdiness of the geometric phase under the presence of an external composite environment.

Figure 1

Geometric phase ${\Phi }_E$ as a function of the concurrence $\mathcal{C}$ and the angle ${\theta }_0$ for the uncoupled case $\chi =0$ and ${\gamma }_0=0$. The phase is measured in units of $\pi$.

Figure 2

Geometric phase of one spin initially set up in an entangled bipartite state for the isolated case ${\gamma }_0=0=\chi$ as a function of ${\theta }_0$. Red solid curve for $\mathcal{C}=1$ and purple dashed curve for $\mathcal{C}=0.06$. The phase is measured in units of $\pi$.

Figure 3

The evolution of the system can be illustrated by the path traversed in the Bloch sphere in the case that ${\gamma }_0=0$ and $\chi =0$. Red curve for $\mathcal{C}=1$, pink curve for $\mathcal{C}=0.95$, purple curve for $\mathcal{C}=0.8$, and black curve $\mathcal{C}=0.43$. Time is measured in units of ${\Omega }_1$.

Figure 4

Geometric phase ${\Phi }_E$ as a function of the initial degree the entanglement through $\mathcal{C}$ and the angle ${\theta }_0$ for ${\gamma }_0=0$ and $\chi =0.1{\Omega }_1$. The phase is normalized by $\pi$.

Figure 5

Geometric phase ${\Phi }_E$ as a function of $\mathcal{C}$ and ${\theta }_0$. The value of the geometric phase is measured in units of $\pi$. Parameters used are as follows: $\chi =0.1{\Omega }_1$, ${\gamma }_0=0.02$, and $\Lambda =20{\Omega }_1$. The choice of the parameters used is for graphical reasons in order to compare with those plots of the preceding section.

Figure 6

The geometric phase for the entangled initial state ${\Phi }_E$ as a function of time. The dashed pink line represents the initial state with ${\lambda }_0=0.2$; the dotted-dashed purple line ${\lambda }_0=0.1$, and the black dotted-dashed line is for ${\lambda }_0=0.01$, all curves for ${\theta }_0=\pi /5$. Parameters used are as follows: $\Lambda =20$, ${\gamma }_0=0.02$, and $\chi =0.1$. The geometric phase is normalized with $\pi$.

Figure 7

The evolution of the system can be illustrated by the path traversed in the Bloch sphere. Pink curve for $\mathcal{C}=0.91$, purple curve for $\mathcal{C}=0.71$, black curve for $\mathcal{C}=0.43$. Parameters used are as follows: $\Lambda =20{\Omega }_1$, ${\gamma }_0=0.02$, $\chi =0.1{\Omega }_1$, and ${\theta }_0=\pi /5$.

Figure 8

The geometric phase, measured in units of $\pi$, as a function of the coupling constant to the bosonic bath ${\gamma }_0$, and the concurrence $\mathcal{C}$. Parameters used are as follows: $\Lambda =20{\Omega }_1$, $\chi =0$, ${\theta }_0=\pi /3$, and $\tau =2\pi /{\Omega }_1$.

Figure 9

Geometric phase for the product initial state ${\Phi }_p$ as a function of $q$ (${\lambda }_0$) and ${\theta }_0$, for $\Lambda =20{\Omega }_1$, $\chi =0$, and ${\gamma }_0=0$.

Figure 10

Geometric phase as a function of time for different initial states. Dashed lines (superposed) represent different values of $q$ and ${\theta }_0=\pi /5$ and the solid line is for $q=0.05$ and ${\theta }_0=\pi /3$. Parameters used are as follows: $\chi =0.1{\Omega }_1$, $\Lambda =20{\Omega }_1$, ${\gamma }_0=0.02$. The choice of the parameters used is for graphical reasons in order to compare with those plots of the preceding section.

Figure 11

Trajectories in the Bloch sphere for the same initial states of Fig. 7. Parameters used are as follows: $\chi =0.1{\Omega }_1$, $\Lambda =20{\Omega }_1$, ${\gamma }_0=0.02$.

Figure 12

The geometric phase, measured in units of $\pi$, for different initial states as a function of the dimensionless coupling constant to the bosonic bath ${\gamma }_0$. Pink lines for $q=0.4$, $\theta =\pi /3$; black lines for $q=0.01$, $\theta =\pi /3$. Dotted and dashed lines imply that $\chi =0.1{\Omega }_1$ while solid lines imply $\chi =0$. Parameters used are as follows: $\Lambda =20{\Omega }_1$ and $\tau =2\pi /{\Omega }_1$.