Introduction%20to%20Nanoelectromechanical%20Systems - PowerPoint PPT Presentation

Title: Introduction%20to%20Nanoelectromechanical%20Systems


1
Introduction to Nanoelectromechanical Systems

• Andrew Armour
• University of Nottingham

2

• Outline
• Classical mechanical resonators
• Seminar Dynamics of nanomechanical
  single-electron transistors
• Cooling and lasing with mesoscopic circuits
• Towards the quantum regime
• Quantum vs. classical oscillators
• Macroscopic superpositions motivation
• Using a Cooper-pair box to produce mechanical
  superpositions
• Practical issues alternative approaches

3
Interaction with environment plays a key role
decoherence What effect does size have? Is
moving the border up in scale limited by
practical problems or are there fundamental
constraints?
4
Interaction with environment plays a key role
decoherence What effect does size have? Is
moving the border up in scale limited by
practical problems or are there fundamental
constraints?
5
Interaction with environment plays a key role
decoherence What effect does size have? Is
moving the border up in scale limited by
practical problems or are there fundamental
constraints?
6
Superconducting circuits

• Prescription for macroscopic quantum approach
• Leggett J. Phys. Cond. Matt. 14 R415
• Start from classical dynamical equations of
  motion for superconducting phase
• Drop dissipative terms and derive Lagrangian
• Identify canonical momentum, write down
  Hamiltonian
• Treat phase and canonical momentum as canonical
  quantum operators
• Include dissipation through coupling to a bath
• Later experimental results proved consistent
• Important steps towards increasing scale of
  quantum superpositions
• Superpositions of counter-propagating currents in
  SQUID devices of few ?A Freidman et al., Nature
  406 43, Chiorescu et al., Science 299 1869
• Suggest a path for nanomechanical systems
  justification for quantising collective modes

7
Quantum vs Classical dynamics

• Master equation for oscillator coupled to
  oscillator bath
• Wigner function description
• How and when are quantum classical dynamics
  different?
• Near T0
• Non-linear dynamics contribution to resonator
  potential at third order or higher
• Initial conditions Can start with a
  superposition state (negative regions of W), but
  not a delta function
• Can drive the system into a non-classical state
  by including an auxiliary quantum system

8
Non-linear dynamics

• Difference in regions of phase space accessible
  to quantum and classical systems
• E.g. Katz et al., PRL 99 040404 2007
• Stringent conditions on temperature, mass of
  resonator etc.
• Driven non-linear resonator
• Investigation of quantum and classical versions
• Resonances that occur only in quantum case exist
• Peano and Thorwart PRB 70 235401

Katz et al., PRL 99 040404
9
Superpositions

• Feynman on two-slit interference (1963)
• We choose to examine a phenomenon which is
  impossible, absolutely impossible, to explain in
  any classical way, and which has in it the heart
  of quantum mechanics. In reality, it contains the
  only mystery
• Superpositions of quantum states are needed to
  describe two-slit interference patterns
• Is there a limit to the size of systems that
  support superpositions of spatially separated
  states?

x
Z0
z
10
Molecule interference

• Can perform two-slit interference experiments
  with relatively large molecules
• What is the environment?
• Presence of small molecules, photons etc which
  scatter off molecules (no vacuum is perfect)
• Light scattering particles are deflected by large
  molecules in principle they measure molecular
  position
• Scattering by lighter particles long thought of
  as reason why large particles dont show
  interference effects Joos and Zeh 85
• ? depends on number, speed, cross-section of
  scatterers For air molecules acting on dust
  grain (r?10-5m) ?1040m-2s-1
• Experiments on molecules allowed these ideas to
  be tested

11
Experiment C70 Molecules

• Hornberger et al.,
• PRL 90 160401 (2003)
• Controlled degredation of the vacuum by allowing
  in CH4 progressively destroys fringes

12
Harmonic oscillator decoherence

• Consider initial superposition of
  minimum-uncertainty Gaussian states centred at x
  and x
• Evolution of master equation including coupling
  to thermal bath means decay into mixture of
  states
• over time scale
• Where initial separation larger than thermal de
  Broglie wave-length
• and

t
Zurek RMP 75 715
13
Non-linear potential

• Why not simply compress the beam to generate a
  non-linear potential?
• Cf SQUID system
• To get measurable tunnelling need very careful
  control over potential
• Investigated by
• Carr et al., PRB 64 220101, conclusion
  challenging
• Savelev et al., PRB 75 165417 More optimistic
• Effect of environment not studied carefully

14
Auxiliary system

• Use a Cooper pair box prepared in superposition
  state to generate one in mechanical system
• Canonical Hamiltonian
• Superpositions in CPB involve charge
• Should be easy to couple to a resonator
• Can reveal quantum coherence and measure
  decoherence/dephasing rate through Ramsey
  interference

15
Probing quantum coherence

• For simplicity assume ?0gtgt?, so 0gt,1gt are
  eigenstates to very good approximation
• Can control magnitude of ?0 through Vg can change
  Hamiltonian for certain periods to
  
• Start with CPB in ground state (N0)
• 1. Apply pulse to set ?0 to zero for time ?h/8?
  
• Produce superposition

16
Probing quantum coherence

• 2. Allow system to evolve under normal
  Hamiltonian
• After time t,
• 3. Set ?0 to zero for time ?23h/8?
• Result
• 4. Measure whether system is in state 0gt or not,
  

17
Probing quantum coherence
?h/2?0

• Ideally temporal version of two-slits
  interference pattern
• In reality energy relaxation at rate 1/T1
  produces loss of coherence over time 1/T2,
  T2T1/2
• Plus noise in effective gate voltage produces
  phase smearing during between runs

?0
?1
18
Probing quantum coherence
?h/2?0

• Ideally temporal version of two-slits
  interference pattern
• In reality interaction with environment leads to
  energy relaxation at rate 1/T1 produces loss of
  coherence over time 1/T2, T2T1/2
• Plus noise in effective gate voltage produces
  dephasing

?0
?1
19
Add mechanical resonator

• Resonator displacement from equilibrium position
  x
• In practice xltltd
• approximate

Vg
Voltage gate mechanical beam!
d
Hamiltonian now includes harmonic oscillator
which is lowest flexural mode of resonator
Resonator frequency ?
Resonator position
20
Mechanical superpositions

• Now use electromechanical coupling to turn
  charge superpositions into mechanical
  superpositions
• Start in ground state
• Produce superposition
• Let system evolve
• Resonator evolves into
  superposition
• P(0gt,t) oscillations now depend on overlap of
  oscillator states If resonator coherent,
  oscillations will decay and revive over one
  mechanical period

AA, Blencowe Schwab PRL 88 148301
21
Including environment (1)

• Resonator is coupled to a bath
• Simplest consequence expect resonator to start
  in a thermal state centred on equilibrium
  position of resonator for CPB state 0gt
• Thermal state most naturally thought of as
  mixture of coherent states (Gaussians)
• Fast oscillations of CPB with slow amplitude and
  phase oscillations superimposed due to resonator
• Separation of oscillator states
• Need to produce a true
  superposition of states

22
Recoherences (1)

• Balance between decoherence due to entanglement
  with resonator and thermal dephasing
• Higher temperatures means wider range of initial
  states in ensemble greater range of phases
  averaged
• Recoherence peaks become sharper as temperature
  increased eventually impossible to measure
  envelope!

23
Including environment (2)

• Evolution will need to include coupling to the
  bath
• So is CPB, omit this to see how CPB is affected
  by resonator bath
• Model as set of harmonic oscillators this we
  know how to treat
• For weak damping often use a simpler form
• Damping rate would be input from classical
  Q-factor measurements
• Fundamental issue is this the correct
  description of the resonators environment?

24
Including environment (2)

• Evolution can be calculated exactly!
• Slowly varying phase on top of fast CPB
  oscillations
• Amplitude periodicity is now lost!
• Irreversible loss of coherence in CPB due to
  dynamical interaction between resonator and its
  bath

25
Recoherences (2)
Q1000
Including dissipation recoherences are
progressively degraded What happens if the bath
has a fundamentally different character?
AA, Blencowe Schwab PRL 88 148301
26
Practicalities (1)

• Would like to obtain evidence for evolution of a
  coherent superposition of spatially separated
  states by observing CPB
• Practical requirements are
• Strong enough coupling to produce superposition
  of states separated by at least zero-point
  position uncertainty
• CPB decoherence due to other factors needs to be
  well controlled!
• Want temperature to be low enough (and coupling
  large enough) so that thermal smearing is not
  dominant process
• These constraints are very tough!
• Example (from 2002)
• 50MHz cantilever, Cg20aF, d0.1?m, Vggt1V
• Ideally would like coherence of CPB over 50ns

27
Practicalities (2)

• What other ways could we think of doing this
  experiment?
• Use a SQUID system instead of the CPB
  e.g. Buks Blencowe PRB 74 174504
• Work at the degeneracy point of the CPB/SQUID
  (use undriven Rabi oscillations)
• Problematic electro-mechanical coupling is
  second order and recoherences take much longer
  than mechanical period
• Apply a drive to compensate for weak coupling
  Tian, PRB 72 195411
• Use a mirror coupled to an optical cavity in a
  superposition of photon states
  Bose et al., PRA 59 3204
  Marshall et al., PRL 91 130401

28
Conclusions

• Idea that nanomechanical resonators may have
  experimentally accessible quantum regime has
  captured imagination of many theorists
• Experiment has some way to go to catch up
• Hope for good progress over the next few years

• Reviews
• Quantum electro-mechanical Systems
• Blencowe, Phys. Rep. 395 160
• Schwab and Roukes Physics Today, July 2005
• Testing Limits of Quantum Mechanics
• Leggett, J. Phys. Cond. Matt. 14 R415