Nonlinear Oscillators and Vacuum Squeezing

Transcription

1 Nonlinear Oscillators and Vacuum Squeezing David Haviland Nanosturcture Physics, Dept. Applied Physics, KTH, Albanova

2 Atom in a Cavity Consider only two levels of atom, with energy separation Atom drifts through electromagnetic resonant cavity with very high Q Ω Jaynes-Cummings Hamiltonian: decay rate to non - cavity modes - γ Strong Coupling limit g >> κ,γ Needed: High Q Large d

3 Spectrum Cavity photon number Vacuum Rabi flopping:,0 0 photons in cavity atom in excited state,1 atom in GS 1 photon in cavity Atom ground state Atom excited state Level splitting depends on number of photons in cavity g n +1 Blais et. al, Phys. Rev. A, 004

4 Splitting of Cavity Resonance Now consider damping: excitation is ½ photon, ½ atom decay rate: κ + γ In strong coupling limit there is a splitting of cavity resonance which can be resolved because: g = ε rms >> d κ, γ Blais et. al, Phys. Rev. A, 004

5 CP box in Microstrip line cavity Very small effective volume, ~10-5 cubic wavelengths Blais et. al, Phys. Rev. A, 004

6 Vacuum Rabi splitting n=0.06 n=0.5 Blue small T, Red large T By fitting the split cavity resonance, they can determine the mean number of thermal photons in the cavity Wallraff et. al, NATURE, 004

7 Atom Cavity detuning g Large detuning : << 1 decay rate: decay rate: Atom transition is ac Stark/Lamb shifted by ( g ) ( n + 1 ) Atom pulls cavity resonant frequency: ωr ωr ± g κγ Blais et. al, Phys. Rev. A, 004

8 Measuring qubit with caviaty Phase Probe signal at ω r, measure phase shift Blais et. al, Phys. Rev. A, 004

9 meter response after Rabi pulse π pulse π pulse 3π pulse Wallraff et. al, Phys. Rev. Lett., 005

10 Rabi oscillations with unit visibility in readout Wallraff et. al, Phys. Rev. Lett., 005

11 High Q Resonators are very important for precision measurement Crystal Oscillators Radio Receivers Quartz Crystal Microballance - zeptogram mass MEMs gyroscopes, accelerometers Gas sensors, Electronic Nose LASER cavities definition of the meter Atomic Clocks GPS (definition of the second) Superconducting Cavities qubit measurements

12 Resonance Classical Physics: The Driven Damped Harmonic Oscillator - a model to treat one eigenmode F D (t) AFM cantilever Approaching a surface z z 0 Free resonance of probe (i.e. no surface-tip force) Newtons nd Law: Cantilever force + damping force + drive force = ma dz dz k( z z0) γ m + FD ( t) = m dt dt 1 δ + δ + δ = AD ( τ ) Q dδ ω k ω0 where δ = z-z 0 δ = τ = ω0t Ω = ω0 = Q = dτ ω m γ 0

13 Phasor Notation A D In steady state, drive force and response are harmonic oscillations ( τ ) = [ ˆ ] iωτ ADe [ ˆ ] iωτ δe Re δ ( τ ) = Re where Aˆ and ˆ δ are D complex amplitudes iy 1. Equation from previous page A R θ R. Substitute in phasors for displacement and drive force divide out e iωτ. 3. phase of drive force is zero by definition A ˆ D = A D 4. Solve for displacement δˆ x ˆ δ = x + iy = A R ( iω) ˆ δ = = x ˆ δ + (1 Ω + iω Q A y R 1 δ + δ + δ = Q e iθ, A R D tanθ ( τ ) ˆ δ + ˆ δ = AD + iω / Q) R Aˆ D = y x

14 Drive is amplified at resonance by factor Q G( Ω, Q) = A A D = 1 ( 1 Ω ) + ( Ω / Q) ) θ R R tan( ) Q=10 Q=5 Q= ( Ω / Q) = 1 Ω Q=10 Q=5 Q= A R / A D Phase [0,-π] Ω Ω Tip-Surface interaction perturbs resonance of cantilever Sharper resonance (larger Q) more responsive to tip-sample forces

15 Responsivity for dissipative forces ( ) 3 θ Ω 1 Ω dg dq = Ω Ω ( 1 Ω ) + 3 Q Q d R dq = Q ( 1 Ω ) + Ω 5 A R Amplitude Responsivity da R /dq Q=5 A D =1 da R /dq 1 0 Phase responsivity dθ R /dq for Q=5 dθ R /dq 0.1 θ R Ω Ω

16 Responsivity for reactive forces Reactive forces cause small shift in resonant frequency ω 0 6 Amplitude Responsivity da R /dω for Q=5 A D =1 0 da R /dω A R 0 Phase Responsivity dθ R /dω for Q=5 0 θ R dθ R /dω Ω Ω

17 Nonlinearity can be used to enhance Resonant detectors Sharpen Responsivity Bifurcation, Bistability, sample-and-hold Parametric Amplification

18 Driven Damped Nonlinear Oscillator δ δ + δ + Dδ = AD Q Q=10, A D =1 sin( Ωτ ) D= A R / A D Ω

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20 Low drive power, High Q oscillator

21 Increasing drive Power Bending of resonance curve to lower frequency Bifrication at critical power Classic Duffing oscillator behavior Sample 1 s = 10µm d = 00 nm

22 Driven, damped nonlinear oscillator ) ( LI I L I U = ( ) t I I I I I Q D D ω β cos 3 = 0 b I L L = β

23 The Phase Plane Atractor

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30 Intermodulation and Parametric Amplification 3 δ 3 [ δ cos( ω t) + δ cos( ω t) ] = 6δ δ cos( ω t) + 3δ δ cos ( ω ω) t) + 3δ δ cos( ( ω + ω) ) +... = t p p s s p s s p s p Signal + Idler 1 + Idler + p s s Idlers - intermodulation products (IMPs), generated by nonlinearity Hierarchy of IMPs separated by ω

31 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A sin( Ωτ ) D=0 Q Drive A 1 =1 A =0. ω = strength of duffing term D=0 Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

32 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D= Drive A 1 =1 A =0. ω = strength of duffing term d= Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

33 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D= Drive A 1 =1 A =0. ω = strength of duffing term d= Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

34 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D= Drive A 1 =1 A =0. ω = strength of duffing term d= Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

35 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D=0.000 Drive A 1 =1 A =0. ω = strength of duffing term d=-0.00 Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

36 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D=0.005 Drive A 1 =1 A =0. ω = strength of duffing term d= Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

37 1 3 δ + δ + δ + Dδ = A1 sin( Ω1τ ) + A Q sin( Ω τ ) D= Drive A 1 =1 A =0. ω = strength of duffing term d= Quality factor Q= ω 1 - ω ω 1 ω ω - ω 1 Amplitude ω / ω 0

38 The Parametric oscillator Change parameter (moment of inertia) during oscillation. Small amplitude motion is amplified to large amplitude of swing Amplification is phase sensitive. Pumping at correct phase gives maximum amplification Pumping π/ out of phase gives maximum deamplification (squeezing).

39 Mechanical pumping, E&M oscillator

40 Hard Work!

41 Non-linear element and E&M pump

42 How does the (non-degeneate) paraamp work? 1.5 signal Pump + signal = drive applied to NL resonator 1 Bifrucation threshold time signal pump Drive Amplitude above bifrucation threshold Building up large circulating power in cavity Signal and pump in phase Drive Amplitude below bifrucation threshold Reducing to small circulating power in cavity Signal and pump out of phase Net effect: signal is amplified because large circulating power and in-phase pumping coencide.

43 CPW resonator with weak-link array Nb film, 00 nm thick, Saphire substrate,photo lithograpahy, etch channel Made by Robin Cantor: Star Cryoelectronics, FIB to make weak link E. A. Tholen et. al arxiv:

44 Signal and Idler gain E. A. Tholen et. al arxiv:

45 Measuring phase dependant gain and noise

46 Phase dependent gain and signal squeezing E. A. Tholen et. al arxiv:

47 Calibration of Noise Measurement ( ) N B T k T k e E B + = + = E Bw G P measured noise power N ω ω E. A. Tholen PhD. Thesis, KTH, 009

48 Attempt to Squeeze Vacuum Noise

49 Pump-off level

50 Recent result from K. Lehnert group 10 db of Vacuum Noise Squeezing! M. Castallanos-Beltran et. al, Nature Physics (008) See Poster by Yale group: Signal squeezing 6dB, T N =130mK

51 Back Action Evasion Yurke and Denkar, Quantum Network Theory, PRA, 9, 1419 (1984)