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arXiv:2410.07152 · Low Temperature Physics

Creation of two-dimensional circular motion of charge qubit

L. Y. Gorelik, S. I. Kulinich, Robert I Shekhter, Danko Radić

Published Dec 1, 2024• 1 citation
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Abstract

We suggest a nanoelectromechanical setup which generates a particular type of motion—the circular motion of mesoscopic superconducting grain, where motion is described by entangled nanomechanical coherent states. The setup is based on mesoscopic terminal utilizing the ac Josephson effect between the superconducting electrodes and the grain, operating in the regime of the Cooper pair box controlled by the gate voltage. The grain is placed on the free end of the suspended cantilever, performing controlled two-dimensional mechanical vibrations. Required functionality is achieved by operating two external parameters, bias voltage between the superconducting electrodes and voltage between gate electrodes, by which the nanomechanical coherent states are formed and organized in a pair of entangled cat-states in two perpendicular spatial directions, which evolve in time in the way to provide a circular motion.

DOI
10.1063/10.0034369

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We suggest a nanoelectromechanical setup which generates a particular type of motion—the circular motion of mesoscopic superconducting grain, where motion is described by entangled nanomechanical coherent states. The setup is based on mesoscopic terminal utilizing the ac Josephson effect between the superconducting electrodes and the grain, operating in the regime of the Cooper pair box controlled by the gate voltage. The grain is placed on the free end of the suspended cantilever, performing controlled two-dimensional mechanical vibrations. Required functionality is achieved by operating two external parameters, bias voltage between the superconducting electrodes and voltage between gate electrodes, by which the nanomechanical coherent states are formed and organized in a pair of entangled cat-states in two perpendicular spatial directions, which evolve in time in the way to provide a circular motion.
We suggest a nanoelectromechanical setup for designing a quantum-nanomechanical motion of a charge qubit in two-dimensional space. The quantum character of such motion comes through the fact that states of the qubit are entangled with nanomechanical cat states. The setup is based on mesoscopic terminal utilizing the ac Josephson effect between voltage-biased superconducting electrodes and mesoscopic superconducting grain—the charge qubit, vibrating in plane fixed on a nanomechanical cantilever. Required functionality is achieved by operating two external parameters: bias voltage between superconducting electrodes and controlling voltage applied over the gate electrodes. While the resonant tunneling of Cooper pairs between the superconductors and charge qubit builds nanomechanical coherent states comprising the cat state, gate voltage has a specific role of tuning their two-dimensional character in the form of the in-plane linear motion under the voltage-controlled direction. Controlling the spatial motion of a charge qubit opens a possibility of quantum communication between different terminals located in space. Directional control of nanomechanical motion entangled with charge qubit states we call the “quantum switch”.
We present a detailed mathematical description, both an analytical model and a numerical simulation, of a physical system based on a superconducting nanoelectromechanical setup that generates nanomechanical cat-states entangled with charge qubit states. The system consists of a superconducting grain in a regime of the Cooper pair box (the charge qubit) that performs mechanical vibrations between two bulk superconductors. Operation of the device is based on the AC Josephson effect, i.e., the phase difference between superconducting electrodes is controlled by a DC bias voltage following the operational switch on/off protocol. We compare an analytical idealised solution with numerical simulation using experimentally feasible parameters, different decoherence processes, as well as imperfections of experimental procedures such as time-control of the bias voltage, to get insight into how they influence the time-evolution of the realistic system, deteriorate the quantum coherence, and affect the formation of the cat-states.
We considered a nanoelectromechanical system consisting of a movable Cooper-pair box qubit, which is subject to an electrostatic field, and coupled to the two bulk superconductors via tunneling processes. We suggest that qubit dynamics is related to that of a quantum oscillator and demonstrate that a bias voltage applied between superconductors generates states represented by the entanglement of qubit states and coherent states of the oscillator if certain resonant conditions are fulfilled. It is shown that a structure of this entanglement may be controlled by the bias voltage in a way that gives rise to the entanglement incorporating so-called cat-states—the superposition of coherent states. We characterize the formation and development of such states analyzing the entropy of entanglement and corresponding Wigner function. The experimentally feasible detection of the effect by measuring the average current is also considered.
We suggest a nanoelectromechanical setup that generates properly entangled ancillary ("ancilla") qubits for error correction algorithms in quantum computing, demonstrated as an encoder for the three-qubit bit flip code. The setup is based on mesoscopic terminal utilizing the AC Josephson effect between voltage biased superconducting electrodes and mechanically vibrating mesoscopic superconducting grain in the regime of the Cooper pair box, controlled by the gate voltage. Required functionality is achieved by specifically tailored time-protocol of operating two external parameters: bias voltage and gate voltage. The superconducting grain is fixed on the free end of a cantilever, performing controlled in-plane mechanical vibrations, generating the nanomechanical coherent states organised in a pair of entangled cat-states in two perpendicular spatial directions. Cooper pair box and nanomechanical coherent states become three entangled qubits in a particular way: quantum information, initially encoded in superposition of the Cooper pair box states, is transduced into quantum superposition of two special 3-qubit entangled states, ∣↑+ +⟩\vert \uparrow + \, + \rangle∣↑++⟩ and ∣↓− −⟩\vert \downarrow - \, - \rangle∣↓−−⟩. It constitutes the basic input state for the three-qubit bit flip code, used in quantum computation mainly for error correction, "installed" on a single physical object in which the last two ancilla qubits are generated by the nanoelectromechanical setup.
Abstract We study a nanoelectromechanical system consisting of a Cooper pair box qubit performing nanomechanical vibrations between two bulk superconductors. We demonstrate that a bias voltage applied to the superconductors may generate states represented by entanglement between qubit states and quantum ’cat states’, i.e. a superposition of the coherent states of the nanomechanical oscillator. We characterize the formation and development of such states in terms of the corresponding Wigner function and entropy of entanglement. Also, we propose an experimentally feasible detection scheme for the effect, in which the average current that attains the specific features created by the entanglement is measured.
Abstract Superconducting circuits that incorporate Josephson junctions are of considerable experimental and theoretical interest, particularly in the context of quantum computing1, 2, 3, 4, 5. A nanometre-sized superconducting grain (commonly referred to as a Cooper-pair box2) connected to a reservoir by a Josephson junction is an important example of such a system. Although the grain contains a large number of electrons, it has been experimentally demonstrated6 that its states are given by a superposition of only two charge states (differing by 2e, where e is the electronic charge). Coupling between charge transfer and mechanical motion in nanometre-sized structures has also received considerable attention7, 8, 9, 10. Here we demonstrate theoretically that a movable Cooper-pair box oscillating periodically between two remote superconducting electrodes can serve as a mediator of Josephson coupling, leading to coherent transfer of Cooper pairs between the electrodes. Both the magnitude and the direction of the resulting Josephson current can be controlled by externally applied electrostatic fields.