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arXiv:2503.10064 · New Journal of Physics

Capturing an electron in the virtual state

Alok Nath Singh, Bibek Bhandari, Rafael Sánchez, Andrew N. Jordan

Published Jun 30, 2026• 0 citations

Abstract

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 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. We also find that a suitably strong measurement can significantly boost the tunneling current—an effect we call measurement-assisted tunneling . 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.

DOI: 10.1088/1367-2630/ae846a

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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 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. We also find that a suitably strong measurement can significantly boost the tunneling current—an effect we call measurement-assisted tunneling . 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.

Quantum tunneling allows electrons to be transferred between two regions separated by an energetically forbidden barrier. Performing a position measurement that finds a particle in the barrier forces the tunneling electrons to transition from having a classically forbidden energy to an energy above the barrier height. We exploit this effect to define quantum tunneling engines that can use the unconditioned detection of virtually occupied states as a resource for power generation and cooling. Leveraging energy exchange with the detector, we show that the device can operate in a hybrid regime, enabling simultaneous cooling and power generation. Furthermore, we demonstrate measurement-assisted autonomous refrigeration and "checkpoint" cooling driven purely by a thermal bias, without the need for an applied potential. We also find a "purification-by-noise" effect when the measurement drives the system into a stationary dark state. These results underscore the intriguing dual role of measurement as a thermodynamic resource and a dark state generator.

We present the conditional quantum dynamics of an electron tunneling between two quantum dots subject to a measurement using a low transparency point contact or tunnel junction. The double dot system forms a single qubit and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dot. We illustrate the difference between conditional and unconditional dynamics of the qubit. The conditional dynamics is discussed in two regimes depending on the rate of tunneling through the point contact: quantum jumps, in which individual electron tunneling current events can be distinguished, and a diffusive dynamics in which individual events are ignored, and the time-averaged current is considered as a continuous diffusive variable. We include the effect of inefficient measurement and the influence of the relative phase between the two tunneling amplitudes of the double dot/point contact system.

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Quantum mechanics allows the existence of ``virtual states'' that have no classical analog. Such virtual states defy direct observation through strong measurement, which would destroy the volatile virtual state. Here, we show how a virtual state of an interacting many-body system can be detected employing a weak measurement protocol with post-selection. We employ this protocol for the measurement of the time it takes an electron to tunnel through a virtual state of a quantum dot (cotunneling). Contrary to classical intuition, this cotunneling time is independent of the strength of the dot-lead coupling and may deviate from that predicted by time-energy uncertainty relation. Our approach, amenable to experimental verification, may elucidate an important facet of quantum mechanics which hitherto was not accessible by direct measurements.

We describe the conditional and unconditional dynamics of two coupled quantum dots when one dot is subjected to a measurement of its occupation number by coupling it to a third readout dot via the Coulomb interaction. The readout dot is coupled to source and drain leads under weak bias, and a tunnel current flows through a single bound state when energetically allowed. The occupation of the quantum dot near the readout dot shifts the bound state of the readout dot from a low conducting state to a high conducting state. The measurement is made by continuously monitoring the tunnel current through the readout dot. We show that there is a difference between the time scale for the measurement-induced decoherence between the localized states of the dots, and the time scale on which the system becomes localized due to the measurement.