Capturing an Electron in the Virtual State

Abstract

We address a foundational question in quantum mechanics: Can a particle be directly found in a classically forbidden virtual state? We instantiate this conceptual question by investigating the traversal of electrons through a tunnel barrier, which we define in a triple quantum dot (TQD) system where the occupation of the central dot is energetically avoided. The motivation behind this setup is to answer whether the central dot is occupied or not during a virtual transition when it is being explicitly monitored. We investigate this problem in two different limits of continuous measurements: the stochastic quantum diffusion and the quantum jump. We find that, even though individual trajectories differ considerably across these limits, measuring leads to a higher occupation in the central dot on average. Our results demonstrate that the act of observation fundamentally reshapes tunneling dynamics, resolving the seeming paradox of detecting a particle in a classically forbidden region: weak measurements partially localize the particle, while strong measurements enforce a discontinuous either/or detection or no detection outcome.

Figures (5)

• Figure 1: Scheme of the triple quantum dot coupled to two terminals ($l={\rm L,R}$) via tunneling rates $\Gamma_l$. The energies of the singly-occupied quantum dots are represented: the central dot is split by $\Delta$ with respect to the other two, at $\varepsilon$. The nearest neighbor hopping is $\Omega$. The charge of the central dot is monitored by a coupled QPC measuring with a rate $\gamma$.
• Figure 2: (a) Stochastic evolution of the central dot occupation plotted vs time. The black curve shows the ensemble average, while the colored curves show time-averaged measurement realizations. Time averaging is done over a rectangular window of $0.1~\Omega^{-1}$. Initial state is taken to be ${\rm |L\rangle\langle L|}$, for $\Delta=10~\Omega,$$\gamma=10~\Omega$, and $dt=10^{-4}~\Omega^{-1}$. (b, c) Steady state, ensemble averaged central dot occupation ($\rho_{\rm CC}$) and current through the TQD ($I_{\rm T}$) tuning $\Delta$ and $\gamma$. Other parameters common to (a, b, c): $\Gamma_{\rm L}=10~\Omega \text{, and } \Gamma_{\rm R}=8~\Omega$.
• Figure 3: Zero frequency cross correlation (a), and Pearson coefficient (b) between the current through the TQD and the detector as a function of measurement strength $\gamma$, for $\Delta = 10~\Omega,~14~\Omega \text{ and } 20~\Omega$, $\Gamma_{\rm L}=20~\Omega$ and $\Gamma_{\rm R} = 16~\Omega$. Some low $\gamma$ points in the plots for $\Delta =10~\Omega,~20~\Omega$ are omitted because of high noise.
• Figure 4: (a,b) Central dot occupation ($\rho_{\rm CC}$) (left axis) and number of electrons (cyan, right axis) collected at the QPC detector ($N$) as a function of time. Figure (a) contains both the numerical (red, solid) as well as an approximate analytic result (black, dashed). $\gamma$ is taken to be 0.5$\Omega$ in (a) and 5$\Omega$ in (b). Rest of the parameters are common: $\Delta=20\Omega$, $\Gamma_{\rm L}=20\Omega,~\Gamma_{\rm R}=16\Omega$, and $dt=10^{-4}\Omega^{-1}$. Initial state is taken to be the pure state ${\rm |L\rangle \langle L|}$.
• Figure 5: Ensemble-averaged dwell time ($e\,\rho_{\rm CC}/I_{\rm T}$) for different values of $\Delta$ and $\gamma$. Parameters: $\Gamma_{\rm L}=10~\Omega \text{, and } \Gamma_{\rm R}=8~\Omega$.