Conditioned quantum motion of an atom in a continuously monitored one-dimensional lattice

We consider a quantum particle on a one-dimensional lattice subject to weak local measurements and study its stochastic dynamics conditioned on the measurement outcomes. Depending on the measurement strength our analysis of the quantum trajectories reveals dynamical regimes ranging from quasicoherent wave-packet oscillations to a Zeno-type dynamics. We analyze how these dynamical regimes are directly reflected in the spectral properties of the noisy measurement records.

Figure 1

Time evolution of the probability $p_n\left(t\right)$ for the particle to be on site $n=11$ for a lattice of size $N=21$, with the particle initially in its ground state. The results are obtained by simulations of the probing of the population with strength (a) $k=0.1J$, (b) $k=1J$, (c) $k=10J$. The inset of (a) sketches the $N=21$ sites with arrow indicating the probing of the middle site (i.e., a measurement of the observable ${\mathrm{\Pi }}_{11}$).

Figure 2

The power spectrum $P_n\left(\omega \right)$ for $T=100/J$ averaged over 200 time evolutions (blue $\times$) and the steady-state spectrum $S_n\left(\omega \right)$ (green solid) for $k=0.1J$ for different probed sites $n$: (a) $n=1$, (b) $n=2$, (c) $n=3$. The observed frequencies correspond to transition frequencies between different system eigenstates (black dashed). The insets sketch the $N=5$ sites with different local probing.

Figure 3

Time evolutions for initial state $\left|1\right.〉\left.〈1\right|, N=5$ sites and probing on site $n=3$. The black thin line shows the probability for the particle the be on the probed site $p_n\left(t\right)$, while the red and green lines show the probabilities $p_{\mathrm{od}}$ and $p_{\mathrm{ev}}$ to be in the odd and even subspaces, respectively, Despite identical initial conditions, in the upper plot the random measurement back-action drives the system into the odd subspace, while in the lower case the system converges into the even subspace.

Figure 4

Time evolution for an initial thermally mixed state $\rho \left(0\right)=1/Zexp(-\beta H), \beta =1/J$, on $N=5$ sites, where the total population ${\mathrm{\Pi }}_{2,4}={\mathrm{\Pi }}_2+{\mathrm{\Pi }}_4$ of sites $n=2$ and $n=4$ is probed. Probing of two-site populations can, for example, be realized using a reflecting mirror which sends the probe beam twice through the system (see inset). The red and green lines show the occupation of the lowest- and highest-energy eigenstates $\left.〈w_1\right|\rho \left|w_1\right.〉$ and $\left.〈w_5\right|\rho \left|w_5\right.〉$, both converging to $0.5$ in the simulation. The blue line depicts the purity of the state $\mathcal{P}=\mathrm{tr}\left[{\rho }^2\right]$; its convergence to unity shows that the continuous probing drives the system to a coherent superposition of $\left|w_1\right.〉$ and $\left|w_5\right.〉$ in accordance with the full contrast sinusoidal oscillations of the expectation value $〈{\mathrm{\Pi }}_{2,4}〉$ with frequency $(e_5-e_1)/\hslash$ (black line).

Figure 5

The power spectrum $P_n\left(\omega \right)$ for $T=100/J$ averaged over 200 time evolutions (blue $\times$) and the steady-state spectrum $S_n\left(\omega \right)$ (green solid) for $k=J$ and different numbers of sites $N$: (a) $N=7$, (b) $N=13$, (c) $N=19$, where the middle site $n=N/2+1$ is probed. In all plots a dominant low frequency peak is emerging. Moreover the peaks are shifted away from their expected position at the transition frequencies between (even) eigenstates (black dashed) towards higher frequencies.

Figure 6

Lowest-frequency spectral peak in the measurement signal as function of the number of lattice sites $N$. For $k=0.1$, this frequency decreases like $1/N^2$, reflecting the lowest Bohr excitation frequency between the eigenstates. For $k=1$, it scales like $1/N$ as expected for a classical particle bouncing back and forth in a lattice of length $N$. The free parameters were chosen as $a\approx 8.66$ and $b\approx 43.3$.

Figure 7

Time evolution between two subsequent peaks in Fig. 1. The measurements localize the particle on the measured site and thus creates wave packets propagating along the chain. After reflection the wave packets cross the probed site again where the measurement back-action refocuses the dispersed wave packets.

Figure 8

Top: Steady-state spectrum in the Zeno regime with $N=9$ and $n=5$. Bottom: Eigenvalues of the Liouvillian operator $\mathcal{L}\left[{\mathrm{\Pi }}_n\right]$ in the complex plan. The eigenvalues appear in groups with $\mathrm{Re}{\lambda }_i=-k$ and $\mathrm{Re}{\lambda }_i=0$.

Figure 9

Two effective eigenfunctions $|\langle i\mid {\tilde{w}}_l\rangle {|}^2$ defined via Eq. (17) for $N=21, n=8$ and different measurement strength $k$. For $k\to 0$, the eigenfunction in the lower plot coincides with the ground state, while the eigenfunction in the upper corresponds to a highly excited energy. When $k$ is increased these eigenstates get distorted: the eigenfunction in the upper plot localizes on the probed site, while the one in the lower plot develops a node on that site.