Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor
description
Transcript of Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor
![Page 1: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/1.jpg)
Dynamics of a Nanomechanical Resonator Coupled to a Single
Electron Transistor
Miles Blencowe
Dartmouth College
![Page 2: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/2.jpg)
Collaborators: Theory: Andrew Armour (Nottingham, U.K.), Aash Clerk (McGill)
Expt: Keith Schwab, Olivier Buu, Matt Lahaye, Akshay Naik (LPS-Maryland)
Funding: NSF
![Page 3: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/3.jpg)
Single electron transistor (SET) as ultrasensitive displacement detector [M.B. & M. Wybourne, Appl. Phys. Lett.
77, 3845 (2000)].
![Page 4: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/4.jpg)
Experimental demonstration of SET displacement detector[M. LaHaye et al., Science 304, 74 (2004);M.B., Science (Perspective) 304, 56 (2004)].
~20 MHz mechanical resonator
SET
![Page 5: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/5.jpg)
Predicted shot-noise limited sensitivity Actual measured sensitivity
X/
XZ
P
![Page 6: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/6.jpg)
[Base temperature of cryostat was 35 mK, ~20 mK lower than mechanical resonator temperature. Suggests SET is heating the resonator: electrons tunneling into drain electrode relax, emitting phonons.]
Sample data: direct measurement of temperature of mechanical resonator through detection of thermal Brownian motion.
![Page 7: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/7.jpg)
Second generation device: stronger coupling between nanoresonator and SET allows to observe backaction.
![Page 8: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/8.jpg)
What kind of coupled SET-resonator dynamics arises when tunneling particles exert significant back-action on resonator?
Theory for normal state SET: A.D. Armour, M.P.B, & Y. Zhang, Phys. Rev. B 69, 125313 (2004).
Theory for superconducting SET: M.P.B., J. Imbers, & A.D. Armour, New J. Phys. 7, 236 (2005); A.A. Clerk & S. Bennett, New J. Phys. 7, 238 (2005).
Experiment: A. Naik, O. Buu, M.D. LaHaye, A.D. Armour, A.A. Clerk, M.P.B., & K.C. Schwab (submitted). (See next talk by Schwab)
![Page 9: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/9.jpg)
€
IDS vs VDS vs VG
![Page 10: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/10.jpg)
![Page 11: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/11.jpg)
How to describe this coupled system dynamically?
Minimal set of coordinates:
mechanical resonator’s centre-of-mass position x, velocity v, and SET’s island excess electron number N.
State described by a probability density function N(x,v,t).
Require Boltzmann-like master equation for N(x,v,t):
![Page 12: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/12.jpg)
JQP master equation:
![Page 13: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/13.jpg)
Rewrite master equation in concise, dimensionless 55 matrix form:
![Page 14: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/14.jpg)
![Page 15: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/15.jpg)
What is the effective dynamics of the mechanical resonator PHO(x,v,t)= N(x,v,t)+ N+1(x,v,t)+ N+2(x,v,t)?
Solve master equation approximately using self-consistent Born approx. (weak SET-oscillator coupling), followed by Markov approx. (wide separation of SET and oscillator timescales).
Recover Fokker-Planck equation:
mechanical resonator undergoes thermal Brownian motion; perceives the SSET as a thermal reservoir.
![Page 16: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/16.jpg)
![Page 17: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/17.jpg)
For superconducting SET about the JQP resonance line:
Compare with laser Doppler cooling of trapped, two-level atoms [J. Javanainen & S. Stenholm, Appl. Phys. 21, 283 (1980)]:
![Page 18: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/18.jpg)
Backaction and cooling of nanoresonator by SET:
State of resonator cooled to : “Cooper-pair molasses”
If TSET< TBath and SET Bath, then SET can cool the resonator
For experimental device, minimum predicted 220mK (determined by ).
![Page 19: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/19.jpg)
€
IDS vs VDS vs VG
![Page 20: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/20.jpg)
Note, can have . E.g., sweeping through the JQP resonance:
€
TSET < 0 (and γSET < 0)
![Page 21: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/21.jpg)
-SET
![Page 22: Dynamics of a Nanomechanical Resonator Coupled to a Single Electron Transistor](https://reader035.fdocuments.net/reader035/viewer/2022081503/568145ea550346895db2edff/html5/thumbnails/22.jpg)
In progress: Role of two level systems for damping/dephasing of nanomechanical resonators at low temperatures.
Earlier related work for bulk amorphous insulating solids: “Resonant decay of quasimonochromatic acoustic phonons in glasses below 1K,” B. Golding et al., Phys. Rev. B 14, 1660 (1975).
“Phonon echoes in a glass at low temperatures,” J. Graebner and B. Golding, Phys. Rev. B 19, 964 (1979).