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arXiv:quant-ph/0504056 · Physical Review A

Quantum transducers: Integrating transmission lines and nanomechanical resonators via charge qubits

C. P. Sun, L. F. Wei, Yu-xi Liu, Franco Nori

Published Feb 10, 2006• 91 citations
arXivHTMLPDF

Abstract

We propose a mechanism to interface a transmission line resonator (TLR) with a nanomechanical resonator (NAMR) by commonly coupling them to a charge qubit, a Cooper-pair box with a controllable gate voltage. Integrated in this quantum transducer or simple quantum network, the charge qubit plays the role of a controllable quantum node coherently exchanging quantum information between the TLR and NAMR. With such an interface, a quasiclassical state of the NAMR can be created by controlling a single-mode classical current in the TLR. Alternatively, a ``Cooper pair'' coherent output through the transmission line can be driven by a single-mode classical oscillation of the NAMR.

DOI
10.1103/physreva.73.022318

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We propose a mechanism to interface a transmission line resonator (TLR) with a nanomechanical resonator (NAMR) by commonly coupling them to a charge qubit, a Cooper-pair box with a controllable gate voltage. Integrated in this quantum transducer or simple quantum network, the charge qubit plays the role of a controllable quantum node coherently exchanging quantum information between the TLR and NAMR. With such an interface, a quasiclassical state of the NAMR can be created by controlling a single-mode classical current in the TLR. Alternatively, a ``Cooper pair'' coherent output through the transmission line can be driven by a single-mode classical oscillation of the NAMR.
We propose a spectroscopic approach to probe tiny vibrations of a nanomechanical resonator (NAMR), which may reveal classical or quantum behavior depending on the decoherence-inducing environment. Our proposal is based on the detection of the voltage-fluctuation spectrum in a superconducting transmission line resonator (TLR), which is indirectly coupled to the NAMR via a controllable Josephson qubit acting as a quantum transducer. The classical (quantum mechanical) vibrations of the NAMR induce symmetric (asymmetric) Stark shifts of the qubit levels, which can be measured by the voltage fluctuations in the TLR. Thus, the motion of the NAMR, including if it is quantum mechanical or not, could be probed by detecting the voltage-fluctuation spectrum of the TLR.
We propose a scheme for generating squeezed states of a nanomechanical resonator. The scheme is based on a hybrid system consisting of two NAMRs and a superconducting charge qubit (SCCQ). The nonlinear interaction between the two nanomechanical resonators (NAMRs) can be implemented by setting the external biased flux of the SCCQ at some certain values, which plays the role of “nonlinear media”. The noise in the SCCQ does not need to be considered, since we can adiabatically keep the SCCQ at the ground state. In addition, the squeezing parameters can also be adjusted by changing the external driving voltage.
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”.
A scheme for generating the Schrödinger cat state based on geometric operations by a nanomechanical resonator coupled to a superconducting charge qubit is proposed. The charge qubit, driven by two strong classical fields, interacts with a high‐frequency phonon mode of the nanomechanical resonator. During the operation, the charge qubit undergoes no real transitions, while the phonon mode of the nanomechanical resonator is displaced along different paths in the phase space, dependent on the states of the charge qubit. This generates the entangled cat state between the NAMR and charge qubit, and the NAMR cat state can be achieved after some operations applied on this entangled cat state. The robustness of the scheme is justified by considering noise from environment, and the feasibility of the scheme is discussed.
We propose a scheme for generating squeezed states in solid state circuits consisting of a nanomechanical resonator (NMR), a superconducting Cooper-pair box (CPB) and a superconducting transmission line resonator (STLR). The nonlinear interaction between the NMR and the STLR can be implemented by setting the external biased flux of the CPB at certain values. The interaction Hamiltonian between the NMR and the STLR is derived by performing Fro¨\rm\ddot oo¨hlich transformation on the total Hamiltonian of the combined system. Just by adiabatically keeping the CPB at the ground state, we get the standard parametric down-conversion Hamiltonian. The CPB plays the role of ``nonlinear media", and the squeezed states of the NMR can be easily generated in a manner similar to the three-wave mixing in quantum optics. This is the three-wave mixing in a solid-state circuit.
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.