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arXiv:0804.0603 · New Journal of Physics

Ground-state cooling of a nanomechanical resonator via a Cooper-pair box qubit

Konstanze Jaehne, Klemens Hammerer, Margareta Wallquist

Published Sep 1, 2008• 52 citations
arXivHTMLPDF

Abstract

In this paper, we present a scheme for ground-state cooling of a flexural mode of a nanomechanical beam incorporated in a loop-shaped Cooper-pair box (CPB) circuit. Via the Lorentz force coupling of the beam motion to circulating CPB-circuit currents, energy is transferred to the CPB qubit which acts as a dissipative two-level system. The cooling process is driven by a detuned gate voltage drive acting on the CPB. We analyze the cooling force spectrum and present analytical expressions for the cooling rate and final occupation number for a wide parameter regime. In particular, we find that cooling is optimized in a strong drive regime, and we present the necessary conditions for ground-state cooling.

DOI
10.1088/1367-2630/10/9/095019

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In this paper, we present a scheme for ground-state cooling of a flexural mode of a nanomechanical beam incorporated in a loop-shaped Cooper-pair box (CPB) circuit. Via the Lorentz force coupling of the beam motion to circulating CPB-circuit currents, energy is transferred to the CPB qubit which acts as a dissipative two-level system. The cooling process is driven by a detuned gate voltage drive acting on the CPB. We analyze the cooling force spectrum and present analytical expressions for the cooling rate and final occupation number for a wide parameter regime. In particular, we find that cooling is optimized in a strong drive regime, and we present the necessary conditions for ground-state cooling.
Abstract. In this paper we present a scheme for ground state cooling of a flexural mode of a nanomechanical beam incorporated in a loop-shaped Cooper-pair box (CPB) circuit. Via the Lorentz force coupling of the beam motion to circulating CPB-circuit currents, energy is transferred to the CPB qubit which acts as a dissipative two-level system. The cooling process is driven by a detuned gate-voltage drive acting on the CPB. We analyze the cooling force spectrum and present analytical expressions for the cooling rate and final occupation number for a wide parameter regime. In particular, we find that cooling is optimized in a strong drive regime, and we present the necessary conditions for ground-state cooling. PACS numbers: 85.25.-j, 85.85.+j, 37.10.RsGround state cooling of a nanomechanical resonator via a Cooper pair box qubit 2
We propose and study an active cooling mechanism for the nanomechanical resonator (NAMR) based on periodical coupling to a Cooper pair box (CPB), which is implemented by a designed series of magnetic flux pluses threading through the CPB. When the initial phonon number of the NAMR is not too large, this cooling protocol is efficient in decreasing the phonon number by 2 to 3 orders of magnitude. Our proposal is theoretically universal in cooling various boson systems of a single mode. It can be specifically generalized to prepare the nonclassical state of the NAMR.
We propose an application of a single Cooper pair box (Josephson qubit) for active cooling of nanomechanical resonators. Latest experiments with Josephson qubits demonstrated that long coherence time of the order of microsecond can be achieved in special symmetry points. Here we show that this level of coherence is sufficient to perform an analog of the well known in quantum optics ``laser'' cooling of a nanomechanical resonator capacitively coupled to the qubit. By applying an ac driving to the qubit or the resonator, resonators with frequency of order 100 MHz and quality factors higher than 103{10}^{3}103 can be efficiently cooled down to their ground state, while lower-frequency resonators can be cooled down to micro-Kelvin temperatures. We also consider an alternative setup where dc-voltage-induced Josephson oscillations play the role of the ac driving and show that cooling is possible in this case as well.
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 superconducting nanomechanical device that can actuate and probe an oscillating flying qubit facilitating quantum information transfer over a distance. The flying qubit is formed by a movable Cooper-pair box (CPB) consisting of a superconducting dot and a bulk superconductor, which are entangled by lifting the Coulomb blockade of Cooper-pair tunneling electrostatically. We show that flying qubit states formed on a movable CPB can be observed in electron transport to a normal electrode via Andreev reflections. The charge transfer due to periodic mechanical motion of CPB leads to the occurrence of a nonzero current at zero bias voltage, and its coherence can be identified through the oscillatory current dependence of the current on a gate voltage. The proposed device provides a building block for dissipationless quantum information circuits.
We calculate the dynamics of a nanomechanical oscillator (NMO) coupled capacitively to a Cooper-pair box (CPB) by solving a stochastic Schr\"odinger equation with two Lindblad operators [Commun. Math. Phys. 48, 119 (1976).]. Both the NMO and the CPB are assumed dissipative, and the coupling is treated within the rotating wave approximation. We show numerically that, if the CPB decay time is smaller than the NMO decay time, the coupled NMO will lose energy faster and the coupled CPB more slowly than the uncoupled NMO and CPB do. The results show that the efficiency of energy loss by an NMO can be substantially increased if the NMO is coupled to a CPB.