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Physical Review A

Generalized coupling system between a superconducting qubit and two nanomechanical resonators

Feng‐Yang Zhang, Wei‐Bin Yan, Chui‐Ping Yang

Published Oct 24, 2018• 13 citations
Publisher

Abstract

We study a generalized coupling system between a superconducting qubit and two nanomechanical resonators. When the eigenfrequency of the superconducting qubit is twice the frequency of the resonators, we show that one qubit is able to excite simultaneously two resonators. Moreover, tripartite and bipartite macroscopic entangled states are generated, respectively. We give conditions under which general tripartite and bipartite entangled nonorthogonal states become maximally entangled states. Under a large detuning condition, the coupling between two resonators is induced by a superconducting qubit. This coupling strength can be enhanced via a squeezing transformation.

DOI
10.1103/physreva.98.042331

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We study a generalized coupling system between a superconducting qubit and two nanomechanical resonators. When the eigenfrequency of the superconducting qubit is twice the frequency of the resonators, we show that one qubit is able to excite simultaneously two resonators. Moreover, tripartite and bipartite macroscopic entangled states are generated, respectively. We give conditions under which general tripartite and bipartite entangled nonorthogonal states become maximally entangled states. Under a large detuning condition, the coupling between two resonators is induced by a superconducting qubit. This coupling strength can be enhanced via a squeezing transformation.
The coupled dynamics of a nanomechanical resonator and superconducting quantum circuits are studied in three experiments, in the context of studying the quantum limit for force detection and quantum physics of macroscopic objects. In the first experiment, the dispersive mechanical resonance shift from the interaction with a Cooper-pair box qubit is studied. The measured coupling strength is large enough to satisfy one of the conditions required to perform many of the proposed quantum nanomechanical measurements. The resonance shift also probes the microwave-driven response of the qubit, showing Rabi oscillation and Landau-Zener tunneling, proving the coherent dynamics of the qubit. Second, the parametric excitation of nanomechanical motion is studied via experiments with a driven qubit. Degenerate parametric amplification and oscillation are demonstrated, with a new observation of nonlinear dissipation. The squeezing of the back-action noise from the detection amplifier is also observed, up to 4dB. It is the first demonstration using a qubit as an auxiliary system to modify the nanomechanical dynamics, showing a possible route for generation of nanomechanical quantum states. Finally, back-action cooling of nanomechanical motion has been investigated, which is implemented by capacitively coupling a high-Q coplanar waveguide microwave resonator to a nanomechanical resonator. The thermal state with 7.5 mechanical quanta on average is reached, a result that is ultimately limited due to increased bath heating with microwave power. The heating is consistent with a model based on two-level systems resonantly coupled to the nanomechanical mode. This additional heating suggests future efforts to improve coupling and for reducing two-level system density in materials employed to reach the motional ground state via back-action cooling.
We propose a hybrid system to realize long information storage and fast quantum operation via Rydberg atoms and superconducting qubits (SCQs). The internal states of a Rydberg atom are coupled to a nanomechanical resonator. The atom-resonator coupling is achieved via an electric field generated by the quantized motions of the resonator and temporarily exciting the atom to Rydberg states. The coupling can be made large enough to allow quantum manipulation and measurement of the resonator via the atom. The resonator can also be coupled to a SCQ, and therefore the tripartite system can provide a quantum interface between the SCQ and the atom via the nanomechanical resonator.
Time-dependent linear coupling between macroscopic quantum resonator modes generates both a parametric amplification also known as a 'squeezing operation' and a beam splitter operation, analogous to quantum optical systems. These operations, when applied properly, can robustly generate entanglement and squeezing for the quantum resonator modes. Here, we present such coupling schemes between a nanomechanical resonator and a superconducting electrical resonator using applied microwave voltages as well as between two superconducting lumped-element electrical resonators using a rf SQUID-mediated tunable coupler. By calculating the logarithmic negativity of the partially transposed density matrix, we quantitatively study the entanglement generated at finite temperatures. We also show that characterization of the nanomechanical resonator state after the quantum operations can be achieved by detecting the electrical resonator only. Thus, one of the electrical resonator modes can act as a probe to measure the entanglement of the coupled systems and the degree of squeezing for the other resonator mode.
We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semiclassical approximation of this system has an interesting fixed-point bifurcation structure. In the semiclassical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positive to negative detuning. We show that signatures of this bifurcation structure are also present in the full dissipative quantum system and further show that the bifurcation structure leads to mixed-state entanglement between the nanomechanical resonator and the microwave cavity in the dissipative quantum system that is a maximum close to the semiclassical bifurcation. Quantum signatures of the semiclassical limit cycles are presented.
The mechanical and quantum properties of a nanomechanical resonator can be improved by connecting it to a superconducting resonator in a way that the resonator exhibits new phenomena that are possibly available to novel quantum technologies. The quantum characteristics of a nanomechanical resonator coupled to a superconducting resonator have been investigated on the basis of rigorous quantum solutions of the combined system. The solutions of the Schrödinger equation for the coupled system have been derived using the unitary transformation approach. The analytic formula of the wave functions has been obtained by applying the adiabatic condition for time evolution of the coupling parameter. The behavior of the quantum wave functions has been analyzed for several different values of parameters. The probability densities depicted in the plane of the two resonator coordinates are distorted and rotated due to the coupling between the resonators. In addition, we have shown that there are squeezing effects in the wave packet along one of the two resonator coordinates or along both the two depending on the magnitude of several parameters, such as mass, inductance, and angular frequencies.
We propose a scheme for generating squeezed states based on a superconducting hybrid system. Our system consists of a nanomechanical resonator, a superconducting flux qubit, and a superconducting transmission line resonator. Using our proposal, one can easily generate the squeezed states of the nanomechanical resonator. In our scheme, the nonlinear interaction between the nanomechanical resonator and the superconducting transmission line resonator can be implemented by the flux qubit as 'nonlinear media' with a tunable Josephson energy. The realization of the nonlinearity does not need any operations on the flux qubit and just needs to adiabatically keep it at the ground state, which can greatly decrease the effect of the decoherence of the flux qubit on the squeezed efficiency.