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Dimensional transformation of defect-induced noise, dissipation, and nonlinearity

R. O. Behunin, F. Intravaia, and P. T. Rakich
Phys. Rev. B 93, 224110 – Published 28 June 2016
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  • INTRODUCTION
  • MATERIAL-INDUCED NOISE AND DISSIPATION
  • THEORY OF DEFECT-PHONON/DEFECT-PHOTON…
  • DEFECT-PHONON AND DEFECT-PHOTON…
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    In recent years, material-induced noise arising from defects has emerged as an impediment to quantum-limited measurement in systems ranging from microwave qubits to gravity-wave interferometers. As experimental systems push to ever smaller dimensions, extrinsic system properties can affect its internal material dynamics. In this paper, we identify intriguing regimes of material physics (defect-phonon and defect-defect dynamics) that are produced by dimensional confinement. Our models show that a range of tell-tale signatures, encoded in the characteristics of defect-induced noise, dissipation, and nonlinearity, are profoundly altered by geometry. Building on this insight, we demonstrate that the magnitude and character of this material-induced noise is transformed in microscale systems, providing an opportunity to improve the fidelity of quantum measurements. Moreover, we show that many emerging nanoelectromechanical, cavity optomechanical, and superconducting resonator systems are poised to probe these regimes of dynamics, in both high- and low-field limits, providing a way to explore the fundamental tenets of glass physics.

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    • Received 12 January 2016

    DOI:https://doi.org/10.1103/PhysRevB.93.224110

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    DefectsFluctuation theoremsLocal density of statesMesoscopicsPhonons
    1. Physical Systems
    Nonlinear waves
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    R. O. Behunin1, F. Intravaia2, and P. T. Rakich1

    • 1Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA
    • 2Max-Born-Institut, 12489 Berlin, Germany

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    Vol. 93, Iss. 22 — 1 June 2016

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    Images

    • Figure 1
      Figure 1

      Systems where defect-phonon and/or defect-photon interactions may be dimensionally reduced: (a) silica microtoroids, (b) silicon optomechanical systems, (c) superconducting qubits, and (d) nanoelectromechanical systems

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    • Figure 2
      Figure 2

      (a) Double-well potential of asymmetry Δ for a tunneling state defect. (b) Excited e and ground g eigenstates are gapped by energy E.

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    • Figure 3
      Figure 3

      Illustration of system Hamiltonian: (a) interaction of jth defect with qth phonon mode, (b) interaction of jth defect with the EM field, (c) defect-defect interactions, and (d) illustration of total coupled system.

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    • Figure 4
      Figure 4

      Illustration of defect-strain coupling mechanisms. (a) Arbitrarily oriented defect in an undeformed elastic body. (b) Defect in elastic body undergoing compressional motion, induced defect elastic dipole proportional to γℓ(n̂). (c) Defect in an elastic body undergoing shear motion, induced defect elastic dipole proportional to γt(n̂) [see Eq. (30)].

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    • Figure 5
      Figure 5

      Conditions for dimensional reduction: (a) relevant phonon frequencies less than cutoff Ωco (yellow region), e.g., E/ℏ, frequency of emitted phonons in defect decay, and/or ωth=kBT/ℏ thermal frequency, (b) mean separation between thermally active defects Λ greater than one or more system dimension.

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    • Figure 6
      Figure 6

      (Left) Dispersion relations for compressional (red), torsional (blue) and flexural (black) phonon modes in a cylinder. Excitations with zero group velocity are indicated by red points a–g. The branches with points a and c and e and f are two examples of modes that have wave-vector regions with negative group velocity. The dashed gray line indicates a phonon energy supporting two defect decay channels at the gray points h and i for a single mode. (Inset) System geometry and four fundamental modes without cutoff (two degenerate flexural modes). (Right) Phonon density of states in a cylinder (gray) and idealized 1D system (gray dashed). Red arrows indicate frequencies supporting zero group velocity excitations.

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    • Figure 7
      Figure 7

      (a) Illustration of coupling/dynamics leading to defect decay in a waveguide. b) Defect decay in a silica cylinder (R=100nm) as a function of defect frequency f=E/(2πℏ) computed using the idealized 1D model (green dotted) Eq. (7), Eq. (12) for a dimensionally reduced cylinder supporting flexural modes (green dashed), and Eq. (8) including higher-order modes (full green line). Red arrows denote frequencies where modes with zero group velocity are supported (see red points of Fig. 6). The following parameters are used: ρ=2202 kg/m3,vℓ=5944 m/s, vt=3764 m/s (and throughout the remainder of the paper), T=10 mK, and (γℓ/vℓ)2≈(γt/vt)2 is assumed.

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    • Figure 8
      Figure 8

      (a) Illustration of coupling/dynamics leading to defect decay in a resonator. (b) Decay defect decay as a function of f=E/(2πℏ) in a 3D bulk (blue) and a 4−μm cubic silica resonator (red). The resonator is defined with periodic boundary conditions applied across parallel faces. The zero energy contribution to Eq. (13) has been subtracted, and T=5 mK.

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    • Figure 9
      Figure 9

      (a) Illustration of coupling/dynamics of defect-induced RF noise in a resonator. (b) Power spectrum of dipole fluctuations of a single defect on-resonance (red-dashed line) and −0.04=ΔΩ/Ω1 fractionally detuned from (red line) a 2−μm cubic silica resonator's fundamental acoustic mode at frequency Ω1=(2π)1.882 GHz. The resonator is defined by applying periodic boundary conditions to each face, the Q of the fundamental mode is taken to be 1882, and the temperature is 10 mK. The power spectrum for the same defects in 3D bulk is displayed in blue for comparison (blue-dashed, resonance frequency) and (blue line, detuned frequency). S is the trace of S and d2=|d|2.

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    • Figure 10
      Figure 10

      (a) Illustration of coupled system leading to RF noise. (b) Power spectrum for dipole fluctuations from an ensemble of defects in 1, 2, and 3D. The compact dimension(s) and the temperature are respectively taken to be 50 nm and 10 mK (ωth=208 MHz). The volume of each system is fixed to (10μm)3 so that each system possesses the same number of defects. We adopt a uniform DDOS, given in Sec. 2, and adopt the defect properties of silica to compute the ensemble average. Stot is the trace of Stot and d2=|d|2.

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    • Figure 11
      Figure 11

      Power spectrum per unit volume from a defect ensemble in a 3D bulk (blue) and resonator (red) system at 10 mK. Material properties of silica are used and periodic boundary conditions are implemented on a cube of side L=1μm to model the resonator.

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    • Figure 12
      Figure 12

      Critical intensity at 10 mK in a silica cylinder (green), resonator (red), and 3D bulk as a function of frequency computed using Eqs. (6), (33), and (17). The cylinder exhibits enhancements of the critical intensity at van Hove singularities (a) and at low frequencies (b) where T1−1 is dominated by emission into flexural modes. The critical intensity is Purcell enhanced in the resonator. Deformation potential and sound velocity for longitudinal waves and |d|=1.3 Debye and ɛ=2.08 were used in Eq. (28).

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    • Figure 13
      Figure 13

      Acoustic quality factor for the fundamental axial-radial mode of a 100 nm radius silica wire as a function of frequency (green) with 100W/m2 intensity. The wire temperature is 10 mK. For comparison, Q factor for quasi-1D bulk (green dashed) and a 3D bulk (blue) with the same parameters are displayed, as well as the low- and high-intensity limits, respectively, with solid black and gray-dashed lines. The yellow region covers frequencies below cutoff. (Inset) Q factor for frequencies above cutoff showing large changes near van Hove singularities in the phonon DOS (red arrows).

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    • Figure 14
      Figure 14

      Acoustic quality factor of the fundamental acoustic mode of a silica resonator with acoustic intensity of 1W/m2 (red-dotted), and finesse-enhanced intensity Q2π1W/m2 (red). The system temperature is taken to be 10 mK. The quality factor ceiling for the resonator, given by Eq. (40), is shown as a black-dashed line. For comparison the result for a 3D bulk system with intensity 1W/m2 is displayed (blue). The resonator is defined using periodic boundary conditions and the frequency of the fundamental shear mode is continuously varied by scaling the resonator size.

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    • Figure 15
      Figure 15

      Asymptotic limits of inverse EM and acoustic quality factor at high and low intensities. The parameter μ is set to 0.3.

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