Dissipative Landau-Zener level crossing subject to continuous measurement: Excitation despite decay

The Landau-Zener formula provides an analytical expression for the final excitation of a quantum system after passage of an avoided crossing of two energy levels. If the two levels correspond to a ground state and to an excited state which is subject to radiative decay, the probability of exciting the system by adiabatic passage of the level crossing is reduced. In this article we use a stochastic master equation to study the level crossing dynamics when the system is subject to continuous probing of the emitted radiation. The measurement back action on the system associated with the fluctuating homodyne detection record alters the level crossing dynamics, leading to significant excitation in spite of decay and imperfect transfer.

Figure 1

The left-hand side depicts the implementation of the Landau-Zener model by the driving of a two-level system between states $|g\rangle$ and $|e\rangle$ with a field of constant amplitude $\Omega$ and with a detuning $\Delta$ that varies linearly with time. The dotted lines show the level crossing with $\Omega =0$, and the solid lines show the adiabatic, time-dependent eigenenergies of the field dressed states. Spontaneous decay from the excited to the ground state with rate $\Gamma$ causes emission of radiation, and on the right-hand side we illustrate the mixing of the signal on a beam splitter (BS) with a local oscillator (LO) with phase $\varphi$. The resulting signal is detected in a photodetector (D) with efficiency $\eta$.

Figure 2

Excited-state probabilities $P_e\left(t^{\prime }\right)$ as a function of $t^{\prime }=\Gamma t$ for $\alpha ={10}^4{\Gamma }^2$, $\Omega =100\Gamma$, and $\varphi =0$. (a) Solution of the unitary LZ model. (b) A sample of individual trajectories (gray lines) and the average value over 6000 trajectories (black line). (c) Two individual trajectories with exceptionally high [red (light gray) curve] and low [blue (dark gray) curve] final excited-state probabilities.

Figure 3

Histograms showing the distribution of the values of $P_e\left(t^{\prime }\right)$ for the simulated sample of trajectories at different times $t^{\prime }=\Gamma t$. The model parameters are the same as in Fig. 2.

Figure 4

The fraction of trajectories that exceed excitation probability $C$ at some point during their evolution for different values of the detector efficiency $\eta$. The model parameters are the same as in Fig. 2.

Figure 5

The maximal [upper red (gray) lines] and average (lower black lines) fraction of the total sweep time $t^{\prime }=\Gamma \tau \in [-1,1]$ spent by quantum trajectories with excitation probabilities $P_e\left(t^{\prime }\right)\ge C$ for different values of the detector efficiency $\eta$. The model parameters are the same as in Fig. 2.

Figure 6

Bloch sphere picture of individual quantum trajectories according to the stochastic master equation (7). Results are shown for the same parameters as in Fig. 2 for three different values of the detection efficiency $\eta$. In (a) $\eta =1$, and the trajectory stays on the surface of the Bloch sphere. In (b) $\eta =0$, corresponding to the trajectory shown by the black line in Fig. 2. In (c) $\eta =0.5$, and as in (a), we observe strong angular precession and diffusion, but with a shorter Bloch vector, thus preventing perfect excitation of the emitter.

Figure 7

Fraction of trajectories (black lines) that exceed excitation probability $C$ at some point during their evolution for different values of the adiabaticity parameter $\gamma$. The blue (gray) vertical lines represent the maximum value during a unitary LZ sweep for the respective values of $\gamma$. All parameter values are the same as in Fig. 2, except for the Rabi frequency $\Omega$, which has been modified to correspond to each value of $\gamma$.

Figure 8

The excitation probability $P_e\left(t^{\prime }\right)$ for a single trajectory [top red (dark gray) curve], for unconditioned dynamics (bottom black curve), and for a unitary LZ sweep (middle gray curve). The parameter values are the same as for the intermediate speed sweep with $\gamma =0.44$ in Fig. 7.