Qubit-Coupled Nanomechanics

Qubit-Coupled Nanomechanics Matt LaHaye Syracuse University experiments performed at caltech with: junho suh, michael roukes - caltech keith schwab - caltech & cornell pierre echternach - j p l Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

mechanical structures in the quantum regime (Maryland) SSET/NEMS (Caltech, Cornell, JPL) NEMS/ (MIT & LIGO) (UCSB) Optomechanical Systems (Caltech, Max Planck Institute) (Oregon) m (UCSB) SET/NEMS (Delft):DC-SQUID/NEMS (Yale) (Vienna) (Cornell/Caltech) SMR/NEMS (JILA): APC/NEMS, SMR/NEMS Casimir Physics (Dartmouth/ Padova) Atoms, Ions, Spins (IBM Almaden) Nanoelectromechanical Systems (NEMS) And many others

mechanical structures in the quantum regime (Maryland) SSET/NEMS (Caltech, Cornell, JPL) NEMS/ (MIT & LIGO) (UCSB) Optomechanical Systems (Caltech, Max Planck Institute) (UCSB) SET/NEMS (Delft):DC-SQUID/NEMS (Oregon) Interesting review from a few years ago: K. Schwab and Michael Roukes, m Physics Today July 5 (Cornell/Caltech) SMR/NEMS (JILA): APC/NEMS, SMR/NEMS (Yale) (Vienna) More recently: special issue of the New Journal of Physics on mechanical systems approaching the quantum regime. September 8 Gordon Conference 8 &1: Mechanical Systems in the Quantum Regime Casimir Physics Atoms, Ions, Spins (Dartmouth/ Padova) (IBM Almaden) Nanoelectromechanical Systems (NEMS) And many others

ultimate limit of NEMS is in the quantum regime Roukes (1) Ideal characteristics: Small mass, high frequency, low dissipation Zero-point motion x / ~ 4 fm zp m Estimate for SiC resonator,.6m x.4m x.7m Mass ~ 5 fg, f ~ 17 MHz Orders of magnitude larger than gram- or kg-scale oscillators Mo Li, Hong Tang, Michael Roukes, 7 Energy-level spacing ω ο k T B For 1 GHz resonator At mk temperatures Attainable with dilution fridge. Typ. quality factors ~ 1 4-1 5, but demonstrated >1 6 May portend long coherence and relaxation times (~ sec s) Huang, Roukes, 3 Schwab 8 Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 5 Nov. 9 4

approaching the quantum limit of NEMS with an RFSET The radio-frequency single-electron transistor (RFSET) as a quantum-limited displacement detector (proposed by Blencowe and Wybourne, APL ) Demonstrated sensitivity using superconducting SET (SSET) near (~4x) the quantum limit for continuous linear detection. SSET a near-ideal linear detector: =15 / Observation of low nanoresonator thermal occupation N th = KT/ (~5). Observed SSET quantum back-action on the NEMS; measured asymmetry In SSET noise spectrum; performed back-action cooling of NEMS Potential for interesting future experiments (Ground-state cooling) A. Hopkins, K. Jacobs, S. Habib & K. Schwab, PRB (3). (Squeezing) R. Ruskov, A. Korotkov & K. Schwab, IEEE Trans. Nano., (5). (Micro-maser analog) D. Rodrigues, J. imbers & A. Armour (7). V NR M. LaHaye, O. Buu, B. Camarota, K. Schwab, Science 4 Gate of SET SSET 19.7 MHz Resonator V NR V NR NR MHz NR Gate 1m A. Naik, O. Buu, M. LaHaye, A. Armour, M. Blencowe, A. Clerk, K. Schwab, Nature 6 Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

approaching the quantum limit of NEMS with an RFSET The radio-frequency single-electron transistor (RFSET) as a quantum-limited displacement detector (proposed by Blencowe and Wybourne, APL ) Demonstrated sensitivity using superconducting SET (SSET) near (~4x) the quantum limit for continuous linear detection. SSET a near-ideal linear detector: =15 / Observation of low nanoresonator thermal occupation N th = KT/ (~5). Observed SSET quantum back-action on the NEMS; measured asymmetry In SSET noise spectrum; performed back-action cooling of NEMS Other linear displacement detectors developed (Normal SET) R. Knobel & A. Cleland, Nature 44, 91 (3). (APC) N. Flowers-Jacobs, D. Schmidt & K. Lehnert, PRL 98, 9684 (7) (DC SQUID) S. Etaki et al., Nature Physics 4, 785 (8) V NR M. LaHaye, O. Buu, B. Camarota, K. Schwab, Science 4 Gate of SET SSET 19.7 MHz Resonator V NR V NR NR MHz NR Gate 1m A. Naik, O. Buu, M. LaHaye, A. Armour, M. Blencowe, A. Clerk, K. Schwab, Nature 6 Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

qubit-coupled nanomechanics First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 () & Physica B 316 (). Nano-electromechanical resonator Cooper-pair box () charge qubit + Cleland & Roukes, APL 69 8 Oct. 1996 Nakamura et al., Nature, 398 9 Apr. 1999 = electrostatic interaction artificial atom Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 5 Nov. 9 x > 1> > Harmonic oscillator n> resonator motion couples to charge on the qubit

qubit-coupled nanomechanics First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 () & Physica B 316 (). Nano-electromechanical resonator Cooper-pair box () charge qubit + Cleland & Roukes, APL 69 8 Oct. 1996 Nakamura et al., Nature, 398 9 Apr. 1999 electrostatic interaction n> = artificial atom x > 1> > Harmonic oscillator use qubit to prepare quantum superposition states of NEMS and study decoherence Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 5 Nov. 9 8

superconducting qubits as tools for quantum NEMS Partial list of proposals utilizing a qubit to manipulate and measure quantum states of NEMS NEMS and Cooper-pair box () entanglement to produce NEMS superposition states (Charge-state) A.D. Armour, M.P Blencowe, K.C. Schwab, PRL 88, 14831 (). (Dispersive) (1) A.D. Armour & M.P. Blencowe, New J. Phys. 1 954 (8) ()D.W. Utami, & A.A. Clerk, Phys. Rev. A 78 433 (8). (3) K. Jacobs, A.N. Jordan, & E.K. Irish, Euro. Phys. Lett. 8, 183 (8). Measurement of quantized energy spectrum of NEMS (1) E.K. Irish & K.C. Schwab, PRB 68, 155311 (3). () K. Jacobs, P. Lougovski,& M.P. Blencowe, PRB 98, 1471 (7). (3) K. Jacobs, A.N. Jordan & E.K. Irish, Euro. Phys. Lett. 8, 183 (8). (4) A.A. Clerk, & D.W. Utami, PRA 75, 43 (7). Microwave-mediated techniques (Ground-state cooling) I. Martin et al., Phys. Rev. B 69, 15339 (4). (Squeezing) P. Rabl et al., PRB 7, 534 (4). (Entanglement) L.Tian, PRB 7, 195411 (5). (Lasing) J. Hauss et al., Phys. Rev. Lett. 1, 373 (8). Many other proposals involving different types of qubits, quantum electronics Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

in the remainder of this talk Brief review of the Cooper-pair box () charge qubit, how we couple the and NEMS, dispersive interaction First experiment: observe the dispersive interaction between and NEMS and use it to perform spectroscopy of and measurement of LZ-interference effects. Parametric Amplification/(Classical)Squeezing of NEMS. Demonstrated coupling should be large enough to pursue more advanced measurement proposals, e.g. ground-state cooling, lasing, and squeezing of NEMS. Significant room for improvement to coupling strength. /NEMS entanglement experiment looks within reach. Should also be able to approach strong coupling limit, a prerequisite for NEMS number-state detection. Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

E (GHz) review of the Cooper-pair box Essentially it is an highly-polarizable, artificial, two-state atom layout energy bands 15 1 1 1 / 5 E J = 9 GHz -5 Nakamura et al., Nature, Vol. 398, 9 April 1999 Small capacitance yields large charging energy E c, so only two relevant charge states nˆ Cooper-pairs on box 1 1 Cooper-pair on box -1-15 1 /..4.6.8 dc gate charge n g (C g /e) 1 Hamiltonian ˆ EJ H E (1 n ) σˆ σˆ C g z x E E cos( πφ / Φ ) J J n g =C g /e - Applied gate charge = Applied flux through loop = Flux quantum Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

E (GHz) review of the Cooper-pair box Essentially it is an highly-polarizable, artificial, two-state atom layout energy bands 15 1 1 / 5-5 E J =3. GHz Nakamura et al., Nature, Vol. 398, 9 April 1999 Small capacitance yields large charging energy E c, so only two relevant charge states Hamiltonian nˆ Cooper-pairs on box 1 1 Cooper-pair on box ˆ EJ H E (1 n ) σˆ σˆ C g z x E E cos( πφ / Φ ) J J -1-15 1 /..4.6.8 dc gate charge n g (C g /e) n g =C g /e - Applied gate charge = Applied flux through loop = Flux quantum Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

review of the Cooper-pair box Essentially it is an highly-polarizable, artificial, two-state atom layout Nakamura et al., Nature, Vol. 398, 9 April 1999 Small capacitance yields large charging energy E c, so only two relevant charge states nˆ Cooper-pairs on box 1 1 Cooper-pair on box Expectation Value of Charge nˆ 1 1.5.5 1.5 Excited state nˆ 1 n g E J Quantum Capacitance Ground State nˆ 1 n g E J.5 1 1.5.5 1 dc gate charge n g (C g /e) Hamiltonian ˆ EJ H E (1 n ) σˆ σˆ C g z x E E cos( πφ / Φ ) J J n g =C g /e - Applied gate charge = Applied flux through loop = Flux quantum Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

review of the Cooper-pair box Essentially it is an highly-polarizable, artificial, two-state atom layout Gate periodicity of energy bands Excited State E/E c Nakamura et al., Nature, Vol. 398, 9 April 1999 Small capacitance yields large charging energy E c, so only two relevant charge states nˆ n n Cooper - pairs on box n 1 n 1Cooper - pairs on box Ground State dc gate charge n g (C g /e) Hamiltonian ˆ 4E ( ˆ ) C n ng H E J cosθˆ n g =C g /e - Applied gate charge Sweeping n g over many degeneracy points, Cooper-pairs tunnel to minimize electrostatic energy Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

capacitively coupled to NEMS Flexural motion of resonator couples to charge on the island V NR Resonator KNR ωnr d Spring Constant C NR Resonant Frequency Gate -NEMS Interaction Electrostatic Coupling Constant NEMS Position Operator Total Hamiltonian in energy basis (at charge degeneracy) ˆ ˆ ˆ Int ˆ X H λ a a σ E C V ω λ C NR NR NR e KNRd Charge at degeneracy (in energy basis) Mechanical quanta ˆ N aˆˆ a ˆ ˆ ˆ T NR a a 1/ H NEMS E J σ ˆ Z energy at charge degeneracy a ˆ aˆ ˆ X Interaction Similar to atom coupled to radiation field Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

Dispersive limit of -NEMS Hamiltonian E Hˆ ω Nˆ T NR 1/ J σ ˆ Z aˆ aˆ ˆ X and NEMS far-detuned for our parameters Δ E J ω N λ Direct exchange of quanta suppressed by NR Dispersive Hamiltonian EJ λ 1/ ˆ 1 Hˆ ω Nˆ σ Nˆ σˆ E disp NR Z Z J -state-dependent Frequency Shift in NEMS NEMS- Dependent shift in transition Δω NR λ E J σˆ Z λ ˆ N ΔE N 1 E J λ ~ Δ RWA ˆ ˆ 1/ E HRWA ωnr N J σ ˆ a ˆ ˆ aˆ ˆ Z Jaynes-Cummings Hamiltonian N=3 N= N=1 N= N=3 N= N=1 N=. Energy levels with Interaction. Energy levels w/o interaction ω ω ω NR E E J.. Δ N=3 N= N=1 N= N=3 N= N=1 N= N J E ω ω Δ ω, ω ω Δω NR NR NR NR Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

Energy (GHz) estimates for the nanomechanical frequency shift Frequency shift depends on state, and magnitude proportional to energy band curvature*: Δf NEMS λ EJ π ((4 E (1 n )) E C g J E E cos( πφ / Φ ) J J,max Expect frequency shift of 1 s ppm at charge degneracy 3/ NEMS frequency detection schemes can routinely achieve better than ppm sensitivity σˆ Z Energy and NEMS frequency shift vs n g Excited State Ground State Gate Charge, n g (e) Parameters *This is the quantum capacitance effect measured via LC resonator in Sillanpaa et al., PRL 95 686 (5) and Duty et al., PRL 95 687 (5) Δf NEMS / f NEMS C NR ~ 5 af d ~ 3 nm V NR ~ 1 V f NEMS = o / ~ 6 MHz ~ 1 5 Gate Charge, n g (e) K ~ 6 N/m /~. MHz E C ~14 GHz E J ~ 13 GHz NEMS Frequency Shift (Hz) Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

device layout fabrication at JPL and Caltech Flux Bias loop Reservoir Gate NEMS NEMS Gate Silicon Nitride Aluminum Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9 18

B V NR Amplitude (V) Phase (Rad) measurement layout NEMS response with biased off charge degeneracy 1 1 8 NEMS response at T mc ~ 1 mk Q ~ 5, -1 - -3-4 -5 58.4 58.45 58.43 58.435 58.44 Frequency (MHz) 6 V NR = 1 V Drive Force (V NR - nr )V drive 4 58.4 58.45 58.43 58.435 58.44 Frequency (MHz) ELECTROMECHANICAL IMPEDANCE L T C T 5 LNA rf if lo On resonance Z M = R m ~ M s L m C m Rm C gnr nr V drive REFLECTOMETRY TO MEASURE Z M Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

B V NR Amplitude (V) Phase (Rad) measurement layout NEMS response on and off a charge degeneracy Drive Force (V NR - nr )V drive 1 1 8 6 4 Off Degeneracy On Degeneracy - f NEMS ~ 6 Hz 58.44 58.46 58.48 58.43 58.43 58.434 58.436 V NR = 1 V 58.44 58.46 58.48 58.43 58.43 58.434 58.436 Frequency (MHz) -1 - -3-4 Frequency (MHz) T mc ~ 1 mk ELECTROMECHANICAL IMPEDANCE L T C T 5 LNA rf if lo On resonance Z M = R m ~ M s L m C m Rm C gnr nr V drive REFLECTOMETRY TO MEASURE Z M Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

f NEMS (Hz) f NEMS (Hz) (mv) n g (e) dispersive interaction: measurement vs. model From M.D. LaHaye et al., Nature 459, 96 (9). -15-1 -5 5 1 15 Measurement: V NR = 7. V, T mc ~ 1 mk -1-5 -5 6 -.5-1. -.5..5 1. Applied Magnetic Field (A.U.) 15 1 Flux Periodicity: Note: Magnetic field applied on top of ~ 1 G E E cosπφ/ Φ 3 1 4-1 -5 5 1 (mv) 15-1 -15 - -5 J fnems (Hz) J,max Model: /= 1.4 MHz, T =1mK E J,max /h= 13. GHz, E C /h= 14. GHz Notes: Model convolved with.1 CP rms charge noise, and includes thermal population of excited state Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9..5 ο -1-6 3 4 -.5..5 5 Flux ( o ) Model Exp. 6-1. -.5..5 1. Flux (A.U.) -5-1 -15 - fnems (Hz)

f NEMS (Hz) f NEMS (Hz) (mv) n g (e) dispersive interaction: measurement vs. model From M.D. LaHaye et al., Nature 459, 96 (9). -15-1 -5 5 1 15 Measurement: V NR = 7. V, T mc ~ 1 mk -1-5 -5 6 -.5-1. -.5..5 1. Note: Magnetic field applied on top of ~ 1 G E E cosπφ/ Φ Model: /= 1.4 MHz, T =1mK E J,max /h= 13. GHz, E C /h= 14. GHz With coupling strength, proposals Flux Periodicity: suggest that it should Flux be (possible o ) to Applied Magnetic Field (A.U.) 15 1 Notes: Model convolved with.1 CP rms charge noise, and includes thermal population of excited state J J,max ο implement single qubit lasing, ground-state cooling, squeezing of NEMS, 3 4-1 -5 5 1 (mv) 15-1 -15 - -5 fnems (Hz) -1. -.5..5 1. Flux (A.U.) Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9..5-6 3 4 -.5..5 (Lasing) J. Hauss,, A. Federov, C. Hutter, A. Shnirman, G. Schon, PRL. 1, 373 (8) (Ground-state Cooling) I. Martin, A. Shnirman, L. Tian, 1-1 P. Zoller, Phys. Rev. B 69, 15339 (4). 5 (Squeezing) P. Rabl,, A. Shnirman, P. Zoller. Phys. Rev. B 7, 534 (4). Model Exp. 6-5 -1-15 - fnems (Hz)

E cpb /h(ghz) f NEMS (Hz) NEMS-based spectroscopy of 1 V NR Resonator d 3 C + (E J,N g ) d /=13 GHz DEVICE SCHEMATIC C NR (t)= +Vcos d t d =(E/) ENERGY BAND DIAGRAM E J /h= 13 GHz Gate 13 GHz (t) C - (E J,N g )..3.4.5.6.7.8 n g (e) EXPECTED NEMS FREQUENCY SHIFT E J /h = 13. GHz p - =p + 13 GHz applied..4.6.8 1 APPLY MICROWAVES THAT ARE RESONANT WITH SPLITTING. Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9-1 - -3-4 n g (e) no microwaves AVERAGE NEMS FREQUENCY SHIFT Δf p Δf p Δf NEMS NEMS NEMS Δf NEMS Δf NEMS p - = p + as given by Bloch equations

E cpb /h(ghz) f NEMS (Hz) NEMS-based spectroscopy of 3 C + (E J,N g ) 1 V NR Resonator d d /=13 GHz DEVICE SCHEMATIC C NR (t)= +Vcos d t d =(E/) ENERGY BAND DIAGRAM E J /h= 9 GHz Gate (t) 13 GHZ C - (E J,N g )..3.4.5.6.7.8 n g (e)..4.6.8 1 Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9-1 - -3-4 -5-6 EXPECTED NEMS FREQUENCY SHIFT E J /h = 9. GHz n g (e) p - =p + no microwaves 9 GHz applied APPLY MICROWAVES THAT ARE RESONANT WITH SPLITTING. AVERAGE NEMS FREQUENCY SHIFT Δf p Δf p Δf NEMS NEMS NEMS Δf NEMS Δf NEMS p - = p + as given by Bloch equations

E cpb /h(ghz) n g (e) f NEMS (Hz) NEMS-based spectroscopy of V NR DEVICE SCHEMATIC (t)= +Vcos d t d =(E/) d C NR Resonator Gate (t) -1 - -3-4 -5 EXPECTED NEMS FREQUENCY SHIFT p - =p + no microwaves E J /h = 9. GHz 9 GHz applied ENERGY BAND DIAGRAM -6..4.6.8 1 n g (e) 3 C + (E J,N g ) E J /h= 9 GHz EXPECTED NEMS FREQUENCY SHIFT f NEMS (Hz) 1 d /=13 GHz 13 GHZ..4 - -4 C - (E J,N g )..3.4.5.6.7.8 n g (e) 1-1 -.5.5 Quantum Measurement and Metrology with Solid State Devices PBH, ( o ) Germany 5 Nov. 9.6.8 Max E J -6-8

(mv) (mv) (mv) (mv) (mv) (mv) (mv) (mv) increasing microwave frequency f (Hz) NEMS f (Hz) NEMS Microwave Frequency: 1.5 GHz f (Hz) Microwave Frequency: 13.5 GHz f (Hz) NEMS NEMS -4-4 -8-6 MW OFF J,max 1.5 GHz 1.5 GHz 13.5 GHz - E J = E J, - -6-4 -1-1 -1-1 - - -4 - - - - -18-18 - - -3-3 -3-16 -16 - -3-4 -4-4 -14 4-14 -18-4 -5-5 -5 6-1 -1-16 -6-6 -5-6 8-1 -1-14 -5.73-5.7-5.71-5.7-5.69-5.68-5.67-5.73-5.7-5.71-5.7-5.69-5.68-5.67-5.73-5.7-5.71-5.7-5.69-5.68-5.73-5.7-5.71-5.7-5.69-5.68-5.67 Flux (A.U.) Flux (A.U.) Flux (A.U.) Flux (A.U.) Microwave Frequency: 14.5 GHz f Microwave Frequency: 16 GHz f (Hz) Microwave Frequency: 17 GHz f NEMS (Hz) NEMS (Hz) NEMS Microwave Frequency: GHz f (Hz) -1-1 NEMS -4 14.5 GHz 16. GHz -1 17. GHz -1-1 1. GHz - -1-1 -1-8 -8-1 - - - -8-1 - -6-3 -6-18 -3-6 - -3-4 -4-4 -16-4 -3-4 -4 - -5 - -4-14 -5-5 -6 - -5-1 -6-6 -7-6 -5.73-5.7-5.71-5.7-5.69-5.68-5.74-5.73-5.7-5.71-5.7-5.69-5.68-5.76-5.75-5.74-5.73-5.7-5.71-5.7-1 -5.79-5.78-5.77-5.76-5.75-5.74-5.73 Flux (A.U.) Flux (A.U.) Flux (A.U.) Flux (A.U..) Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

Microwave Frequency (GHz) (mv) (mv) (mv) (mv) (mv) (mv) (mv) (mv) increasing microwave frequency f (Hz) NEMS -4 - MW OFF E J = E J J,max, -1 - - -18-3 -16-4 -14-5 -1-6 -1-5.73-5.7-5.71-5.7-5.69-5.68-5.67 Flux (A.U.) f (Hz) NEMS Microwave Frequency: 1.5 GHz f (Hz) Microwave Frequency: 13.5 GHz f (Hz) NEMS NEMS -4-8 -6 1.5 GHz 1.5 GHz 13.5 GHz - -6-4 -1-1 -1 - - V -4 g - - -18 - - -3-3 -16 - -3-4 -4 4-14 -18-4 -5-5 6-1 -16-6 -5-6 8-1 -14-5.73-5.7-5.71-5.7-5.69-5.68-5.67-5.73-5.7-5.71-5.7-5.69-5.68-5.73-5.7-5.71-5.7-5.69-5.68-5.67 Flux (A.U.) Flux (A.U.) Flux (A.U.) -1-1 -8-6 -4 - Microwave Frequency: 14.5 GHz f Microwave Frequency: 16 GHz f (Hz) Microwave Frequency: 17 GHz f NEMS (Hz) NEMS (Hz) NEMS Microwave Frequency: GHz f (Hz) -1 NEMS -4 14.5 GHz 16. GHz -1 17. GHz -1 1. GHz - -1-1 -1-8 -1 - - - -8-1 - -3-6 -18-3 -6 - -3-4 -4-16 -4-3 -4-5 -4 - -4-14 -5-5 -6 - -5-1 -6-6 -7-6 -5.73-5.7-5.71-5.7-5.69-5.68-5.74-5.73-5.7-5.71-5.7-5.69-5.68-5.76-5.75-5.74-5.73-5.7-5.71-5.7-1 -5.79-5.78-5.77-5.76-5.75-5.74-5.73 Flux (A.U.) Flux (A.U.) Flux (A.U.) Flux (A.U..) 15 1 5..4.8.1.16. /18.7 (e) For each value of E J Fit the data to hf Δ E (8E Δ n ) E μ C g J Where Δ ng ng.5 and EJ EJmax cos( πφ / Φ ) E C / h 13 14 GHz and E J / h ~ [,9,1] GHz E J max / h ~ 1.5 13.5GHz From M.D. LaHaye et al., Nature 459, 96 (9). Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

NEMS coupled to strongly-driven Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (6), Oliver et al., Science 31 (5), Izmalkov et al., PRL (8), Sun et al., APL (9). Similar multi-photon transitions: see Wilson et al., PRL 98 (7) Apply periodic modulation n g to gate large enough to sweep through charge degeneracy ENERGY BANDS IN n g -SPACE ω RF E J n g (t) = n go + n RF sin( RF t) E slope~ 8E n C g E J n RF n RF E n g Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

NEMS coupled to strongly-driven Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (6), Oliver et al., Science 31 (5), Izmalkov et al., PRL (8), Sun et al., APL (9). Similar multi-photon transitions: see Wilson et al., PRL 98 (7) Starting in ground state, as approach degeneracy, probability P LZ for to tunnel from to ENERGY BANDS IN n g -SPACE E n g (t) = n go + n RF sin( RF t) P LZ πej exp( ) ν Energy Variation rate ν ~ 8E n ω C RF RF E J n RF n RF E n g Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

NEMS coupled to strongly-driven Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (6), Oliver et al., Science 31 (5), Izmalkov et al., PRL (8), Sun et al., APL (9). Similar multi-photon transitions: see Wilson et al., PRL 98 (7) After crossing degeneracy, time-dependent phase (t) develops in wave function between and ENERGY BANDS IN n g -SPACE E Probability Amplitudes After tunneling Ψ Ψ ic P LZ C 1 P LZ P LZ πej exp( ) ν Ψ Ψ Ψ(t) Wave Function Ψ 1 φ(t) t e -iφ(t) dt' ΔE Ψ (ng(t' )) n RF n RF E n g ΔE E E Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

NEMS coupled to strongly-driven Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (6), Oliver et al., Science 31 (5), Izmalkov et al., PRL (8), Sun et al., APL (9). Similar multi-photon transitions: see Wilson et al., PRL 98 (7) Return swing: degeneracy crossed, probability for LZ tunneling to occur, interference between tunneling events ENERGY BANDS IN n g -SPACE Wave Function Amplitudes πej exp( ) ν E PLZ Phase-developed between e -i φ / P LZ cos( φ/) first and second LZ events t 1 i PLZ (1 PLZ ) cos( φ/ ) φ dt' ΔE (ng(t' )) n RF n RF E n g ΔE E E Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

NEMS coupled to strongly-driven Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (6), Oliver et al., Science 31 (5), Izmalkov et al., PRL (8), Sun et al., APL (9). Similar multi-photon transitions: see Wilson et al., PRL 98 (7) After full cycle: if coherence time is longer than cycle period, oscillations in excited state probability with ENERGY BANDS IN n g -SPACE Probability to be in P (1 P )(1 cos( φ)) LZ LZ n RF n RF E Phase-developed between first and second LZ events t 1 φ dt' ΔE (ng(t' E )) n g (t) = n go + n RF sin( RF t) n g ΔE E E Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

V RF V (V) (Volts) NEMS as a probe of LZ interferometry Nanomechanical measurement of LZ interference From M.D. LaHaye et al., Nature 459, 96 (9). NR / (Hz).5-4 - 4 6 Modulate the gate with large RF excitation V RF, and track NEMS frequency shift as a function of and V RF. 1.5 1. Excited state becomes populated, changing sign of NEMS frequency shift.5-1. -8. -6. -4. -. V cpb (mv) / π 4. GHz (mv) ω RF φ 1 t dt' ΔE (ng(t' )) Function of V, g V RF,ω RF Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

V RF V (V) (Volts) NEMS as a probe of LZ interferometry Nanomechanical measurement of LZ interference From M.D. LaHaye et al., Nature 459, 96 (9). NR / (Hz).5-4 - 4 6 Modulate the gate with large RF excitation V RF, and track NEMS frequency shift as a function of and V RF. 1.5 1. -4-3 -3-4 Excited state becomes populated, changing sign of NEMS frequency shift.5-1. -8. -6. -4. -. V cpb (mv) / π 4. GHz (mv) ω RF Constructive interference occurs at where = n (intersection of black lines in plot ). P P (1 P )(1 cos( φ)) LZ LZ φ 1 t dt' ΔE (ng(t' )) Function of V, g V RF,ω RF Parameters used for contour overlay: Ec = 15 GHz, Ej=13 GHz Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

n g (e) V RF V (Volts) (V) f NR (Hz) NEMS coupled to strongly-driven Estimate of E C from LZ interference LZ Fringes at constant V RF NR / (Hz) -5 5 1 15 5 6.5 GHz.5 4 3 n g V cpb 5.66 GHz 1.5 1-4 -3-3 -4 1 4.83 GHz /=4. GHz.5-8. -6. -4. -.. n g (e) -8. -6. -4. -.. cpb (mv) ω RF n (mv) 6.5 GHz g conversion: 18.7 mv per e π Expected Fringe spacing: Δn g 4E C ω RF From fit E C /h = 14.9.6 GHz 4. 5. 6. 7. Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9.1.1.8.6 Fit to straight Line thru origin RF / (GHz)

Amplitude (V) Phase (Rad.) prospects for strong dispersive coupling limit Definition of strong coupling limit: Dispersive interaction exceeds qubit and NEMS linewidth Δf NEMS Δ E N (N 1) h λ γnems γ [, ] πe π π J Demonstrated f NEMS NEMS / 35 3 15 1 5 Present Sample: NEMS Linewidth Off Degeneracy - 5 On Degeneracy 58.47 58.475 58.48 Frequency (MHz) 58.47 58.475 58.48 58.485 Frequency (MHz) 1-1 1.6 khz V NR = 15 V T ~ 13 mk With conservative improvements to sample geometry, should achieve f NEMS ~ 1 s khz Present Sample: Linewidth Present sample: /f NEMS However, there is significant room to improve, e.g. in circuit QED, / 1 MHz e.g. see Wallraff et al., Nature 431 (4) Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9

prospects for dispersive -NEMS entangled states Proposals: D.W. Utami, & A.A. Clerk,, Phys. Rev. A 78 433 (8) A.D. Armour & M.P. Blencowe, New J. Phys. 1 954 (8) General idea: (1) With and NEMS uncoupled, prepare in superposition of energy eigenstates and nanoresonator in displaced thermal state () Dispersively couple and NEMS After time t: Envelope of oscillations after -pulse Initial state: Ψ( t) 1 ( α( t) i α( t) ) α ω NR ΔωNR α ω Δω Nanoresonator Is in a superposition of states winding at frequencies dressed by the 1 Ψ() ( i ) α() qubit state Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9 NR resonantor state 1 (1 exp[ () (1 cos(δ NR ))]) P env α ω t NR Using qubit echo method recoherences should be visible for similar device, e.g. λ 1 MHz ω / π 5 MHz NR Δ ω / π 4 khz Qubit recoherences: signature of entanglement T NR T ~1' s n sec 5 mk

conclusions New era of experiments studying the quantum properties of mechanical structures Superconducting qubits should serve as viable tools to manipulate and measure quantum states of NEMS We have demonstrated the first coupling between a superconducting qubit and NEMS - use dispersive interaction to perform spectroscopy and read-out quantum interference in the, parametric amplification/squeezing - with realistic improvements to devices, experiments with quantum NEMS, even entanglement of NEMS and, are within reach Thanks to Gerard Milburn, Andrew Doherty, Katya Babourina, Aash Clerk, Andrew Armour, Miles Blencowe, Christopher Wilson, and Tim Duty for helpful insight and advice. Quantum Measurement and Metrology with Solid State Devices PBH, Germany 5 Nov. 9