Quantum trajectories and quantum measurement process by a single-electron transistor

Alexander Shnirman, Yuriy Makhlin, Gerd Sehön

W. W. Xue, Zhongqing Ji, Feng Pan +3 more

Yin Ye, Jing Ping, HuJun Jiao +2 more

Hsi‐Sheng Goan

We derive the quantum trajectory or stochastic (conditional) master equation for a single superconducting Cooper-pair box (SCB) charge qubit measured by a single-electron transistor (SET) detector. This stochastic master equation describes the random evolution of the measured SCB qubit density matrix which both conditions and is conditioned on a particular realization of the measured electron tunneling events through the SET junctions. Hence it can be regarded as a Monte Carlo method that allows us to simulate the continuous quantum measurement process. We show that the master equation for the ``partially'' reduced density matrix [Y. Makhlin et al., Phys. Rev. Lett. 85, 4578 (2000)] can be obtained when a ``partial'' average is taken on the stochastic master equation over the fine grained measurement records of the tunneling events in the SET. Finally, we present some Monte Carlo simulation results for the SCB/SET measurement process. We also analyze the probability distribution $P(m,t)$ of finding $m$ electrons that have tunneled into the drain of the SET in time $t$ to demonstrate the connection between the quantum trajectory approach and the ``partially'' reduced density matrix approach.

Michel Devoret, Robert J. Schoelkopf

A. Alarcón, Guillermo Albareda, Fabio L. Traversa +1 more

Albert Benseny,a Joan Baguda`,a Xavier Oriols,b Gerhard Birkl,c Atomtronics is one of the foremost bottom-top approaches to quan- tum information science, where neutral atoms in optical, magnetic, and electric potentials form the building blocks of coherent diode and transistor-like devices. In this context, we discuss in this chapter an efficient and robust technique to coherently transport a single neutral atom, a single hole, or even a Bose-Einstein condensate (BEC) between the two extreme traps of a triple-well optical potential. Solving the corresponding quantum dynamics in terms of Bohmian trajectories, we investigate this adiabatic transport process and present a very counterintuitive effect: by slowing down the total time duration of the transport process, it is possible to achieve sudden particle accelerations yielding ultrahigh atomic velocities.a 3.1 Introduction 191 3.1.1 Atomtronics 191 3.1.2 Three-level atom optics 192 3.1.3 Adiabatic transport with trajectories 195 3.2 Physical System: Neutral Atoms in Optical Microtraps 198 3.2.1 One-dimensional Hamiltonian 200 3.3 Adiabatic Transport of a Single Atom 201 3.3.1 The matter wave STIRAP paradox with Bohmian trajectories 201 3.3.2 Velocities and accelerations of Bohmian trajectories 202 3.4 Adiabatic Transport of a Single Hole 206 3.4.1 Hole transfer as an array-cleaning technique 206 3.4.2 Adiabatic transport of a hole in an array of three traps 208 3.4.2.1 Three-level approximation description 208 3.4.2.2 Numerical simulations 210 3.4.3 Hole transport fidelity 212 3.4.4 Bohmian trajectories for the hole transport 213 3.4.5 Atomtronics with holes 215 3.4.5.1 Single hole diode 215 3.4.5.2 Single-hole transistor 217 3.5 Adiabatic Transport of a Bose-Einstein Condensate 219 3.5.1 Madelung hydrodynamic formulation 222 3.5.2 Numerical simulations 223 3.6 Conclusions 227 Electronics is the branch of science and technology that deals with the controlled flow of electrons. Most of today’s electronic devices use semiconductor diodes and transistors to manipulate electronic current. Notably, the miniaturization of these devices is getting closer and closer to the single atom scale, reaching the limit of Moore’s law [1].a Nowadays, the electronics top-bottom approach is crossing the frontier between the classical and the quantum realm, where quantum effects such as quantum tunneling and quantum correlations, that is, entanglement, are unavoidable. In particular, the transition from the classical to the quantum world constitutes one of the physical scenarios where the Bohmian formulation can give more insight into the underlying physics. See chapter 1 for a detailed discussion of these aspects and chapter 6 for a particular application of Bohmian trajectories to quantum electronic devices.