Einselection from incompatible decoherence channels

Decoherence of quantum systems from entanglement with an unmonitored environment is, to date, the most compelling explanation of the emergence of a classical picture from a quantum world. While it is well understood for a single Lindblad operator, the role in the einselection process of a complex system-environment interaction remains to be clarified. In this paper, we analyze an open quantum dynamics inspired by cavity QED experiments with two noncommuting Lindblad operators modeling decoherence in the number basis and dissipative decoherence in the coherent-state basis. We study and solve exactly the problem using quantum trajectories and phase-space techniques. The einselection optimization problem, which we consider to be about finding states that minimize the variation of some entanglement witness at a given energy, is studied numerically. We show that Fock states remain the most robust states to decoherence up to a critical coupling.

Figure 1

Dynamics of the cavity. The cavity is coupled to its own electromagnetic environment and to an atom, which is itself coupled to its own electromagnetic environment. We suppose that the two electromagnetic baths are not coupled. Eventually, the coupling of the cavity to its bath leads to photon losses. The coupling with the atom, in the dispersive regime, leads to decoherence on the photon basis.

Figure 2

Parametrization of a quantum trajectory built from two types of quantum jumps, $L_a=\sqrt{{\kappa }_a}a$ and $L_n=\sqrt{{\kappa }_n}{a}^{†}a$. The times ${\tau }_{\sigma }$ correspond to the $L_a$ jumps and the quantum state between each jump is above each slice.

Figure 3

Evolution of a superposition of coherent states from $t=0$, subject to the $L_n$ channel only (${\kappa }_a=0$) in the Wigner function representation. The Gaussian spots are spread over a timescale given by ${\kappa }_n{\tau }_s=2/\langle N\rangle$. For the chosen parameters $\alpha =\sqrt{40}$, it is given by ${\kappa }_n{\tau }_s=1/20$. At longer times ${\kappa }_{n}t=2$, the superposition evolves toward a statistical mixture of Fock states with the characteristic rotation invariance in phase space.

Figure 4

Depending on the preparation, with, for instance, here ${\alpha }_{+}=\sqrt{40}$ and ${\alpha }_{-}=-\sqrt{10}$, the interference spot is totally washed out by the $L_n$ channel (${\kappa }_a=0$) after the spreading timescale ${\kappa }_{n}t=1/10$, leading at longer times ${\kappa }_{n}t=2$ to a mixture of Fock states.

Figure 5

Summary of different evolution and timescales of a superposition of coherent states under the effects of two decoherence channels: the $L_n$ channel induces a spreading of the wave packet, while the $L_a$ channel destroys interferences and makes the system relax to the vacuum.

Figure 6

Coherent-state dynamics for $\alpha =\sqrt{40}\gg {\kappa }_n/{\kappa }_a$. Depending on the relative ratio of the coupling constants ${\kappa }_n/{\kappa }_a$, we either see an emergent classical picture based solely on coherent states ${\kappa }_n/{\kappa }_a<1$ or both coherent and then Fock states (${\kappa }_n/{\kappa }_a>1$).

Figure 7

Wave packet that minimizes the loss of purity at initial time, for an average energy of 20 photons, obtained by numerical optimization techniques. The optimal (with respect to the purity) state ${\dot{\gamma }}_{\text{opt}}$ interpolates between Fock states and coherent states, depending on the relative value ${\kappa }_a/({\kappa }_a+{\kappa }_n)$ of the coupling constants.

Figure 8

Evolution of the approximate pointer states for several $({\kappa }_a,{\kappa }_n)$ parameter regimes, for an initial energy of $n_0=20$ photons.

Figure 9

Purity and scalar product of the approximate pointer state with Fock states for energies of $n_0=1,2,5,10$ photons. We see that in the low-${\kappa }_a$ regime, the approximate pointer state $|{\psi }_{\text{opt}}\rangle$ is still a Fock state, until the critical value ${\kappa }_n/{\kappa }_a=n_0+\sqrt{n_0(n_0+1)}+1/2$. As expected, the derivative of the purity linearly decreases on this plateau.

Figure 10

Purity variation, and scalar product of the approximate pointer state with the approximate pointer states when one of the coupling constants is zero. The average energy corresponds to ${\overline{n}}_0=1.5,2.5,5.5,10.5$ photons. In this case, the optimal state when ${\kappa }_a=0$ is $|{\psi }_{\text{sf}}\rangle =(|\lfloor {\overline{n}}_0\rfloor \rangle +|\lceil {\overline{n}}_0\rceil \rangle )/\sqrt{2}$. Conversely, when ${\kappa }_n=0$, we have a coherent state whose average energy is ${\overline{n}}_0$. Contrary to the integer case, there is no plateau appearing in the low-${\kappa }_a$ regime.

Figure 11

Evolution of the purity for the coherent state, the Fock state, and the approximate pointer state for the energy $n_0=20$ photons.

Figure 12

Radial part of the Wigner function for different initial Fock states $|n_0\rangle$.