Simulation of open-quantum-system dynamics using the quantum Zeno effect

We suggest a quantum simulator that allows to study the role of memory effects in the dynamics of open quantum systems. A particular feature of our simulator is the ability to engineer both Markovian and non-Markovian dynamics by means of quantum measurements and the quantum Zeno dynamics induced by them. The simulator is realized by two subsystems of a bipartite quantum system or two subspaces of a single system which can be identified as system and meter. Exploiting the analogy between dissipation and quantum measurements, the interaction between system and meter gives rise to quantum Zeno dynamics, and the dissipation strength experienced by the system can be tuned by changing the parameters of the measurement, i.e., the interaction with the meter. Our proposal can readily be realized with existing experimental technology, such as cavity- or circuit-QED platforms or ultracold atoms.

Figure 1

(a) Analogy between indirect measurement of a system by a meter (left) and an open system (right). (b) Energy level diagram of a bipartite system consisting of a harmonic oscillator and a few-level system, interacting with three external fields—a drive $D$ of the harmonic oscillator (red) and two Zeno pulses $Z_1$ (green, open notched arrowhead) and $Z_2$ (blue, notched arrowhead) addressing the levels $|z=1\rangle$ and $|z=2\rangle$. Arrows with solid (dashed) lines indicate (off-)resonant transitions.

Figure 2

Infidelity $P_{\overline{Z}}$ of the quantum Zeno dynamics (a), dissipation $S_L$ (b), and non-Markovianity measure ${\mathcal{N}}_{\text{BLP}}$ (c) as a function of the displacement $\beta$ due to the drive $D$ and the Rabi angle ${\varphi }_2$ accumulated due to the Zeno pulse $Z_2$. (Inset) ${\mathcal{N}}_{\text{BLP}}$ as a function of ${\varphi }_2$ for $\beta =0.025$ (blue dotted, circles, highest peaks), 0.050 (red dashed, crosses), and 0.075 (green dash-dotted, squares, smallest peaks) with linearly increasing peak heights. The data points which generate the linear functions were calculated analytically.

Figure 3

Attainable combinations of dissipation $S_L$ and non-Markovianity ${\mathcal{N}}_{\text{BLP}}$ using two Zeno pulses $Z_2$ and $Z_1$ with $-i\alpha T=10\pi$ (dark grey dots). The colored lines highlight the combinations of $S_L$ and ${\mathcal{N}}_{\text{BLP}}$ for a Rabi angle ${\varphi }_2=4\pi$ with different values of displacement $\beta$ and Rabi angle ${\varphi }_1\le \pi$. The light gray shading highlights the area which is attainable by an extended sampling range.

Figure 4

Dependency of the dissipation $S_L$ (a) and the non-Markovianity ${\mathcal{N}}_{\text{BLP}}$ (b) on the Rabi angle ${\varphi }_2$ for ${\varphi }_1=0.05\pi$. $\beta$ is fixed as given by the key. The horizontal gray lines in (a) indicate a linear entropy of $1/2$ and $2/3$, respectively. The vertical gray line indicates the value of ${\varphi }_1$ shown in Fig. 5. The total duration $T$ of each propagation was set to fulfill $-i\alpha T=10\pi$.

Figure 5

Same as Fig. 4 but for variation of the Rabi angle ${\varphi }_1$ with ${\varphi }_2=4\pi$. The vertical gray line indicates the value of ${\varphi }_1$ shown in Fig. 4.

Figure 6

Energy level diagram of the bipartite system indicating different subspaces: the Zeno subspace ${\mathcal{H}}_Z$ (red shaded), the Zeno level $|z\rangle$ (blue shaded), the extended Zeno subspace ${\mathcal{H}}_Z^{+}$ (black dashed box), and the subspace with purely unitary dynamics (grey shaded).

Figure 7

Evolution of the trace distance for two different initial state pairs with Rabi angle ${\varphi }_2=4\pi$ and a displacement $\beta =0.025$. The state pairs are $|{\mathrm{\Psi }}_1\rangle =|\psi (\vartheta =0,\phi =0)\rangle =|0\rangle$ (red) and $|{\mathrm{\Psi }}_1\rangle =|\psi (\vartheta =\frac{\pi }{2},\phi =\frac{\pi }{2})\rangle =\frac{1}{\sqrt{2}}\left(|0\rangle +i|1\rangle \right)$ (blue) which is an eigenstate of $U_{z=2}\left(t\right)$, paired with the Zeno level $|{\mathrm{\Psi }}_2\rangle =|2\rangle$. The dark lines show the result of the analytical calculation of the lower envelope, the light lines show the numerical calculations. The inset shows the fast oscillation of the trace distance due to the Zeno pulse $Z_2$ acting on $|{\mathrm{\Psi }}_2\rangle$. The vertical black dashed lines show the positions of the meter's measurement.

Figure 8

BLP measure ${\mathcal{N}}_{\text{BLP}}$ for different initial states $|{\mathrm{\Psi }}_1\rangle =|\psi (\vartheta ,\phi )\rangle$ on the Bloch sphere spanned by the Zeno subspace $\{|0\rangle ,|1\rangle \}$. The second state of the state pair is the Zeno level $|{\mathrm{\Psi }}_2\rangle =|2\rangle$ outside of this subspace. The parameters are $\beta =0.025$ and ${\varphi }_2=4\pi$. The data was obtained numerically but the analytical expression in Eq. (A7) gives the same results.

Figure 9

Difference obtained when solving the master equation and using the piecewise dynamics for the Zeno infidelity $P_{\overline{Z}}$ (a) and the dissipation $S_L$ (b) as a function of the displacement $\beta$ due to the drive $D$ and the Rabi angle ${\varphi }_2$ accumulated due to the Zeno pulse $Z_2$.

Figure 10

Decay rates, ${\gamma }_A$ and ${\gamma }_{\mathrm{\Pi }}$, and their ratio for the off-resonant case coupling $|h,z\rangle \leftrightarrow |e,n=z-1\rangle$ (solid lines) and the resonant case coupling $|h,z\rangle \leftrightarrow |+,n=z\rangle$ (dashed lines).

Figure 11

Decay channels in the resonant case. The straight arrows in the middle of each panel indicate the driven transitions, the snaked arrows indicate the effective decay channels in the master equation.