Generating macroscopic superposition states in nanomechanical graphene resonators

Abstract

A theoretical study of the quantum dynamics of a symmetric nanomechanical graphene resonator with degenerate flexural modes is presented. Applying voltage pulses to two back gates, flexural vibrations of the membrane can be selectively actuated and manipulated. For graphene, nonlinear response becomes important for amplitudes comparable to the magnitude of zero-point fluctuations. We show, using analytical and numerical methods, that this allows for creation of catlike superpositions of coherent states as well as superpositions of coherent catlike nonproduct states. DOI: 10.1103/PhysRevB.85.205415 PACS number(s): 85.85.+j, 42.50.Dv, 73.50.Fq Coherent superposition of states are characteristic traits of quantum mechanics. These phenomena have already been realized in many-particle contexts such as trapped ultracold atoms, 1 superconductors, 2 and photonic systems. 3 A current challenge is to observe these effects for collective degrees of freedom in a macroscopic context in, e.g., mechanical resonators. 4 Recent advances in cooling mechanical resonators and sensitive displacement detection have allowed reaching the motional ground state and observing zero point fluctuations of center of mass. 5,6 Active manipulation and characterization of the quantum state of these systems, as already achieved with photons, 7 seem to be within reach. For a mechanical system, a desirable state to generate is a "cat" state. This is a coherent superposition of two minimum uncertainty wave packets separated by more than their individual quantum fluctuations.For the harmonic oscillator, a minimum uncertainty wave packet is a coherent state |α = exp[αa † − α * a]|0 generated by displacing the oscillator ground state. 8 As shown by Yurke and Stoler, 9 for a nonlinear oscillator H =hω 0n +h n 2 , an initial coherent state |α will after a time t = π/(2 ) evolve into the cat state (1/ √ 2)[e −iπ/4 |α + e iπ/4 |−α ] with the maximum spatial separation = 2|α|.Nanoelectromechanical resonators are typically intrinsically nonlinear. 10 Recent theoretical studies of their quantum dynamics show that it differs from the classical motion 11,12 and the nonlinearity can be exploited in the detection of the quantum signatures. The amplitudes needed to observe nonlinear effects are often orders of magnitude larger than the quantum zero-point fluctuations x 0 = √h /mω 0 . A cat state obtained due to this nonlinearity would have a separation 1. As the decoherence rate scales as 2 (see Refs. 13 and 14), this has, until now, been unfeasible. Instead, coupling to auxiliary quantum systems has been proposed to engineer the nonlinearity. 15,16 We show how, in the limit k B T hω 0 , the intrinsic nonlinearity of a graphene membrane resonator can be used to prepare cat states by applying voltage pulses to local backgates. The reason for using graphene is the ultralow mass of the graphene sheet, which leads to a large x 0 , and an onset of nonlinear response at small amplitudes. 17,18 This implies that cat states with moderate can be constructed without the...

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Negative nonlinear damping of a multilayer graphene mechanical resonator

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“…A quadratic interaction between spins [7][8][9]15,17,[23][24][25][26][27][28][29], the QND interaction [30,31], and the dispersive Tavis-Cummings interaction [32,33] generate these spin cat states, whereas a series of controlled-NOT gates [10,16,34] or a sequence of spin measurements [35][36][37][38] have been proposed. The quadratic interaction, essentially equivalent to the sequence of the controlled-NOT gates [39,40], shows better scalability with respect to the number of spins.…”