Control of population and entanglement of two qubits under the action of different types of dissipative noise

We investigate the quantum control of two qubits interacting with a Markovian environment and evolving as an $X$ state. The control is implemented via a simple applied field, whose profile is determined by means of the piecewise time-independent quantum control method. The goal is to make either the population or the system concurrence to follow a specified tracking control trajectory. By considering the system under the influence of three kinds of noise—phase damping, amplitude damping, or a combination of both—we unveil the conditions (like energy balance) under which the quantum control is properly achieved and fidelity (monitored through the trace distance) is kept satisfactorily high, even if the qubits interact differently with the dissipative medium. We further find that the effects of phase damping, a type of noise which notoriously destroys coherence and entanglement, can be mitigated if during the control the system is also exposed to amplitude damping. Of potential interest for the usage of entanglement as resource for quantum information, our results indicate that distinct entanglement measures and coherence can be maintained relatively high (even at long times) as a consequence of the tracking quantum control process.

Figure 1

In the absence of an external applied field, the temporal evolution of both $\rho$ and the expected value of the system energy ${\varepsilon }_s={\varepsilon }_0$. Each column represents one type of environment noise: first, second, and third, respectively, PD, AD, and APD. Here all ${\rho }_{jk}\left(0\right)=0$ unless for ${\rho }_{11}\left(0\right)=1$. Thus, initially both qubits are in the excited state with ${\varepsilon }_s\left(0\right)=1$.

Figure 2

The same as in Fig. 1, but for all the ${\rho }_{jk}\left(0\right)$ in the $X$ structure of Eq. (7) equal to 1/4. Here ${\varepsilon }_s\left(0\right)=1/2$.

Figure 3

For the conditions and parameters of Fig. 2, measures of entanglement, negativity $E_N$, and coherence $C$, see Appendix. All these quantities go to zero, with the AD presenting the slower decay. The first, second, and third columns correspond, respectively, to the PD, AD, and APD noise.

Figure 4

Assuming a fixed applied field (main text), different quantities as function of the dimensionless time $\tau$ for $\rho \left(0\right)$ given by Eq. (13). In the first, second, and third columns, the results are shown, respectively, for the PD, AD, and APD noises. [(a)–(c)] The density matrix elements. Always ${\rho }_{23}=0$ and ${\rho }_{22}={\rho }_{33}=0$ in panel (a) and ${\rho }_{22}={\rho }_{33}$ in panels (b) and (c). [(d)–(f)] Energies' expected values. [(g)–(i)] Entanglement measures: concurrence $E_C$ and negativity $E_N$. [(j)–(l)] The $l_1$ norm coherence measure $C$.

Figure 5

The same as in Fig. 4, but for $\rho \left(0\right)=|1\rangle \langle 1|$.

Figure 6

[(a)–(c)] The obtained tracking QC of ${{\rho }_{ee}}_1={\rho }_{11}+{\rho }_{22}$ (symbols) compared to the aimed target trajectory of Eq. (15) (dashed curves) under the three types of noise, PD, AD, and APD, respectively, in the first, second, and third columns. Here $\xi =10$ and the values of $S_0$ are 0.08 (diamond), 0.16 (circle), 0.24 (square), 0.32 (triangle), and 0.40 (star). The corresponding ${\rho }_{jk}\left(\tau \right)$ for $S_0=0.32$ (d)–(f) and $S_0=0.4$ (g)–(i). [(j)–(l)] The energies ${\varepsilon }_s$ and ${\varepsilon }_0$ for the evolution in panels (a)–(c). Exactly like ${{\rho }_{ee}}_1$, the energies tend to increase with $S_0$ [so, the relative order of the curves with respect to $S_0$ is as in panels (a)–(c)]. Note that in panel (j) ${\varepsilon }_s$ always coincides with ${\varepsilon }_0$ whereas in (k) ${\varepsilon }_s=-1$ for all $S_0$.

Figure 7

The resulting entanglement and coherence evolution associated to the population control in Fig. 6. The symbols represent the values of $S_0$: 0.08 (diamond), 0.16 (circle), 0.24 (square), 0.32 (triangle), and 0.40 (star). The PD, AD, and APD noise cases are displayed, respectively, in the first, second, and third columns.

Figure 8

External field amplitude profiles for the QC in Fig. 6. The symbols represent the values of $S_0$: 0.08 (diamond), 0.16 (circle), 0.24 (square), 0.32 (triangle), and 0.40 (star). [(a)–(c)] The cases in which the control is possible along the full time range $0\le \tau \le 40$. [(d)–(f)] The situations where the control breaks down around $\tau =30$. The PD, AD, and APD noise cases are displayed, respectively, in the first, second, and third columns.

Figure 9

[(a)–(c)] The obtained tracking QC of $E_C$ (continuous line) following the target trajectory in Eq. (15) (circles). In the first, second, and third columns one has, respectively, PD and $S_0=0.009$, AD and $S_0=0.23$, and APD and $S_0=0.10$. [(d)–(f)] The associated ${\rho }_{jk}$'s. In panel (d), the inset shows $|{\rho }_{14}|$. The corresponding $E_N$ and $C$ are displayed in panels (g)–(i) and the energies ${\varepsilon }_s$ and ${\varepsilon }_0$ in panels (j)–(l). [(m)–(o)] The applied field amplitude profiles leading to the observed control.

Figure 10

(a) For the PD noise, comparison of the $E_C$ tracking QC displayed in Fig. 9 ($S_0=0.009$) with the case where $S_0=0.018$. In the latter, the control is lost around $\tau \approx 18$. (b) The behavior of ${\varepsilon }_0\left(\tau \right)$. The break down of the QC corresponds to ${\varepsilon }_0$ approaching 0.

Figure 11

The same QC as in Fig. 9, but for smaller $S_0$'s. The QC is maintained for much longer times, here up to $\tau ={10}^3$. In the first, second, and third columns, one has, respectively, PD and $S_0=0.002$, AD and $S_0=0.18$, and APD and $S_0=0.08$.

Figure 12

The trace distance $T_D$, Eq. (A9), as function of ${\mathrm{\Gamma }}_2$ and time for an applied field with $\phi =0$ and ${\mathrm{\Omega }}_R=0.5$. Here ${\mathrm{\Gamma }}_1=0.1$. The system under the AD noise (a) and APD (b). The graphs are limited to the range $0\le \tau \le 20$ just for a better visualization. The overall pattern remains the same for larger times.

Figure 13

[(a), (b)] The same QC and parameters of Figs. 6 and 6, but for $\phi =-\pi /2$ (see the main text) and a heterogeneous coupling of the qubits to the environment, where ${\mathrm{\Gamma }}_1=0.1$ and ${\mathrm{\Gamma }}_2=0.07$ (in Fig. 6, ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0.1$). The left (right) column corresponds to the AD (APD) noise. The corresponding ${\rho }_{jk}\left(t\right)$ for $S_0=0.32$ in panels (c) and (d) and for $S_0=0.4$ in panels (e) and (f). All the energies corresponding to panels (a) and (b) are given in panels (g) and (h).

Figure 14

The resulting $E_C$ [(a), (b)] and $T_D$ [(c), (d)] and the necessary external field amplitude profiles [(e)–(h)] for the QC of Fig. 13. The symbols represent the values of $S_0$: 0.08 (diamond), 0.16 (circle), 0.24 (square), 0.32 (triangle), and 0.40 (star). Similarly to the example in Fig. 6, for $S_0=0.4$ the control is attained only up to $\tau =30$. The left (right) column corresponds to the AD (APD) noise. Clearly, $T_D$ is smaller for the AD than for the APD noise.