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Study and generation of nonclassical states in nanocircuits: properties and aplications

Clodoaldo Valverde

Published Jan 1, 2012• 0 citations

Abstract

In this work we used an arrangement consisting of Cooper pairs (Cooper Pair Box, CPB) interacting with a nanomechanical Resonator (NR) for various studies: to produce controlable holes in the statistical distribution of excitations of the NR, for resonant and non-resonant cases; to study the evolution of the entropy and the inversion of excitations, including losses in the CPB; and to study the evolution of the Wigner function under the infuence of a reservoir. We have obtained the exact solution of the master equation describing the non-degenerate parametric amplifier interacting with a generalized model of linear phase sensitive reservoir, having the form of the equation equivalent to that of the Fokker-Planck propagator for the Wigner function. We have calculated the Wigner function describing the temporal evolution of a state initially in a superposition of two coherent states, a kind of "Schr?odinger cat", and finally, we have also studied the temporal evolution of the negativity of this function, one of the indicators of nonclassicality of vibrational states of the NR.

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In this work we used an arrangement consisting of Cooper pairs (Cooper Pair Box, CPB) interacting with a nanomechanical Resonator (NR) for various studies: to produce controlable holes in the statistical distribution of excitations of the NR, for resonant and non-resonant cases; to study the evolution of the entropy and the inversion of excitations, including losses in the CPB; and to study the evolution of the Wigner function under the infuence of a reservoir. We have obtained the exact solution of the master equation describing the non-degenerate parametric amplifier interacting with a generalized model of linear phase sensitive reservoir, having the form of the equation equivalent to that of the Fokker-Planck propagator for the Wigner function. We have calculated the Wigner function describing the temporal evolution of a state initially in a superposition of two coherent states, a kind of "Schr?odinger cat", and finally, we have also studied the temporal evolution of the negativity of this function, one of the indicators of nonclassicality of vibrational states of the NR.

The rapid experimental progress in the field of superconducting nanocircuits gives rise to an increasing quest for advanced quantum-control techniques for these macroscopically coherent systems. Here we demonstrate theoretically that stimulated Raman adiabatic passage should be possible with the quantronium setup of a Cooper-pair box. The scheme appears to be robust against decoherence and should be realizable even with the existing technology. As an application we present a method to generate single-phonon states of a nanomechanical resonator by vacuum-stimulated adiabatic passage with the superconducting nanocircuit coupled to the resonator.

We show two effects as a result of considering the second-order correction to the spectrum of a nanomechanical resonator electrostatically coupled to a Cooper-pair box. The spectrum of the Cooper-pair box is modified in a way which depends on the Fock state of the resonator. Similarly, the frequency of the resonator becomes dependent upon the state of the Cooper-pair box. We consider whether these frequency shifts could be utilized to prepare the nanomechanical resonator in a Fock state, to perform a quantum non-demolition measurement of the resonator Fock state, and to distinguish the phase states of the Cooper-pair box.

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 Fr

\rm\ddot o

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.

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.