Synthesizing Arbitrary Photon States in a Superconducting Resonator John Martinis UC Santa Barbara

Synthesizing Arbitrary Photon States in a Superconducting Resonator John Martinis UC Santa Barbara Quantum Integrated Circuits Quantum currents & voltages Microfabricated atoms Digital to Analog Converter Classical: V = a[0] 1 + a[1] 2 + a[2] 4 + a[9] 512 Quantum: Ψ = α 0 0 + α 1 1 + α 2 2 + α 9 9

Outline Qubit coupled to resonator Fock states Arbitrary states Wigner tomography Metrology: Accurate control of qubits Qubit coupled to mechanical resonator Potential application Violation of Bell s Inequality

Josephson Qubits & Nonlinearity LC oscillator (linear): no qubit possible U(Φ) 1 0 Josephson junction: non-linear inductance with 1 photon (low loss) I 1.7 nm Al AlOx Al 2 1 0 Phase qubit - tunable with I

Josephson Phase Qubit (1) State Preparation Wait for decay to g (20 mk ; kt << hω10 ; 6 GHz) I I 0 (2) Qubit logic with current bias I DC current shifts energy; σ z Microwaves for transition; σ x, σ y time (3) State Measurement: pulse I Fast single shot high fidelity e : tunnel (96%) g : no tunnel (2%) few ns current pulse e g

Qubit Coupled to Photons (Harmonic Oscillator) Qubit-resonator first demonstrated in Saclay (1987), GHz coupling easy in phase qubits Design follows cqed (Yale) Readout through qubit (Saclay87, NIST), high fidelity Similar to ion traps (NIST), Rydberg (ENS) Qubit control Qubit C C resonator control (λ/2 total) Resonator e g qubit stimulated emission 5Ω 4Ω 3Ω 2Ω 5 4 3 2 1 Ω 0 resonator (H.O.)

Calibration with Qubit Spectroscopy I saturate meas. Coupling turned on & off via qubit bias = 40 MHz Current bias

Generating Fock States: Pumping Photons One by One 4 3 2 1 0 g e 4 3 2 1 0

Fock States: Swap oscillations depend on n Stimulated Emission

Fock States: Swap frequency scales as n1/2

Fock States: Photon number distribution for 1

Fock States: Photon number distribution for 2

Fock States: Photon number distribution for 3

Fock States: Photon number distribution for 4

Fock States: Photon number distribution for 5

Coherent states coherent state State preparation: Qubit Res. g 0 Gaussian pulse, vary ampl. α swap time τ meas. vary swap time to probe resonator state Gaussian pulse amp. Poisson distribution swap time τ (ns)

Generating Arbitrary States Desired final state ψ = 0.577( 0 + 1 + i 2 ) g Law and Eberly, PRL (1996) Reverse-engineer final state by building pulse sequence backwards arrow indicates complex amplitude of corresponding basis state 2g 1g 0g υχσβ πηασε θυβιτ γρουπ 1e 0e zero amplitude Energy level representation of desired state υχσβ

Generating Arbitrary States Desired final state ψ = 0.577( 0 + 1 + i 2 ) g res. 0 qubit g 0.566 0g (0.000 + 0.517i) 0e + 0.256 1g + 0.577 1e 2g 1g 1e 0e Qubit and resonator into resonance Transfer 2g amplitude into 1e 2g amplitude now zeroed 2g 0g 1e π

Generating Arbitrary States Desired final state ψ = 0.577( 0 + 1 + i 2 ) g res. 0 qubit g 0.566 0g (0.000 + 0.517i) 0e + 0.256 1g + 0.577 1e ( 0.234 0.473i) 0g + ( 0.528 + 0.210i) 0e + 0.632 1g 2g 1g 1e 0e e g 1e 1g Microwaves to transfer to Transfer amplitude into 1e amplitude now zeroed 1e 0g 1g Ampl. phase

Generating Arbitrary States Desired final state ψ = 0.577( 0 + 1 + i 2 ) g res. 0 qubit g 0.566 0g (0.000 + 0.517i) 0e + 0.256 1g + 0.577 1e ( 0.234 0.473i) 0g + ( 0.528 + 0.210i) 0e + 0.632 1g ( 0.234 0.473i) 0g + (0.000 + 0.568i) 0e + 0.632 1g ( 0.234 0.473i) 0g + (0.000 + 0.849i) 0e 2g 1e Transfer 1g amplitude into 0e 1g 0g 0e 1g 0e

Generating Arbitrary States Desired final state ψ = 0.577( 0 + 1 + i 2 ) g res. qubit 2g 1g 0g 0 g ψ = ( 0.444 0.896i) 0g 1e 0e 0.566 0g (0.000 + 0.517i) 0e + 0.256 1g + 0.577 1e ( 0.234 0.473i) 0g + ( 0.528 + 0.210i) 0e + 0.632 1g ( 0.234 0.473i) 0g + (0.000 + 0.568i) 0e + 0.632 1g ( 0.234 0.473i) 0g + (0.000 + 0.849i) 0e initial ground state 0g Phase of does not matter Run sequence in reverse

Generating Arbitrary States: Quantum Digital to Analog Converter ψ = 1 + 3 Now try: ψ = 1 + i 3 a b Measure using qubit and watching evolution (usual method): Fourier transform yields number composition: 1 photon 3 photons Cannot distinguish based on this measure!

State Tomography of Harmonic Oscillator

State Tomography of Harmonic Oscillator P = ( ) p n n

Wigner Tomography of Fock State 7 Theory Experiment υχσβ πηασε θυβιτ γρουπ υχσβ

Wigner Tomography, 0 + N states:

Wigner Tomography: States with phase υχσβ πηασε θυβιτ γρουπ υχσβ

Off-Diagonal Decay of Density Matrix 2 + 3 (decay cascade) 0 + N (far-element decay)

3-state Schrodinger Cat : Voodoo Cat ψ = 2e i0 Ψ dead + 2e i2π /3 Ψ alive + 2e i4π /3 Ψ zombie = ( 2!) n n n= 0,3,6,9 Digitally programmed cat! n Ψ alive Ψ dead Im(α) Ψ zombie Re(α) + 20 ps -

3-state Schrodinger Cat : Voodoo Cat ψ = 2e i0 + 2e i2π /3 + 2e i4π /3 = ( 2!) n n n= 0,3,6,9 n Ψ dead Ψ alive Ψ zombie Ψ alive Ψ dead Im(α) Ψ zombie Re(α)

Collapse of Schrodinger Cat State Quantum feature disappears first, and the classical peaks merge later

Accurate Pulse Shaping I Q Shaped µ-wave Perfect by feedback Use measured waveform to corrected digital inputs µwave amplitude [a.u] 0.4 0 Measured waveform -0.4-8 -4 0 time [ns] 8

Fast pulses: 2-state errors at 10-4 ω 21 ω 10 10 0 pulse power 8ns 4ns 10-1 10-2 ω 21 ω 10 Gaussian pulses: Minimum width in time and frequency 2 error 10-3 10-4 10-5 π pulse π-π FT theory Spectrum analyzer Quantum simulation 10-4 10-6 0 1 2 3 4 5 6 7 8 τ [ns]

SWAP pulses: Correction with qubit as sampling o-scope offset P 1 Problem: Pulses not flat Distortions going to qubit For times ~ 10 ns: D/A s: ~1% Op-amps: ~1% O-scope: ~0.5% Z X,Y τ delay π pulse P 1 [%] Qubit measures change of resonance frequency offset If design well: Measure ~2% change with exponential curve Correct with D/A waveform τ delay

High Fidelity SWAP (coupling) Operation Unoptimized pulse GRAPE optimized pulse (theory) detuning time (ns) time (ns) 95.8% fidelity 99.9999% fidelity Fidelity limited by finite risetime Fidelity restored through overshoots We used simple Gaussian overshoots (improved swapping)

Swap Calibration (S) Qubit-resonator swap (P) Qubit-resonator phase

Phase Qubit coupled to Mechanical Resonator Aaron O connell Andrew Cleland 6.24 GHz piezoelectric 13 fab layers!

Drive Qubit: Avoided Level Crossing Spectroscopy Drive Mech. Resonator: See Response weak strong

Mechanical Resonator in Quantum Ground State On resonance, no extra population from resonator

Time Dynamics Experiments Qubit Qubit + M-Res M-Res P e =3.8 ns T 1 = 17 ns good swaps; 2g= 134 MHz T 1 = 6.1 ns, T 2 ~2T 1

Drive Coherent State & Measure with Qubit experiment simulation Oscillation frequency increases with drive, as expected Coherence time too short to see frequencies in FT

Potential Application : Couple to Optical Photons Idea: Oskar Painter and coworkers, CalTech Optical Photons Raman many MHz 6 GHz Mechanical Resonator piezo 100 MHz 6 GHz Phase Qubit optical and mechanical cavity Key idea: Photons & Phonons similar λ Mode matched Strong coupling possible Possible to integrate? Many experimental hurdles Basic ideas being demonstrated

Bell Violation - Motivation Bell violation proves entangled-state correlations are quantum 01> - 10> Global test of DiVincenzo criteria (all >90% in one device) Initialization, single qubit gates, entanglement, measurement Close detector loophole (0, 1, not measured) Both states always measured, as needed for quantum computer. As 0 or 1, no correction to measurement Don t throw away (calibrate out) ANY errors. Results from other experiments (S > 2) : Altepeter Rowe Moehring Optics Express Oct 05 Nature Feb 01 PRL Aug 04 2.7252 ± 0.000585 (28sec) 2.25 ± 0.03 (80sec) 2.218 ± 0.028 (80min) First violation in solid-state

All measurements along z-axis must rotate state 01>-e iφ 10> Measuring the Inequality (CHSH) measure (a,a ) x (b,b ) directions a a a a b b b b α a z β b + - - + p ab 00> p ab 01> p ab 10> p ab 11> + - - + p a b 00> p a b 01> p a b 10> p a b 11> + - - + p ab 00> p ab 01> p ab 10> p ab 11> + - - + p a b 00> p a b 01> p a b 10> p a b 11> S = E ab + E a b - E ab + E a b HVT: QM: -2 S 2-2 2 S 2 2 FMinSearch experiment directly to find angles α, β for largest S

Simultaneous Measurement: Understanding Crosstalk C cpl I bias I bias I1> I0> I1> I0> I1> I1> Measurement crosstalk turns I01>, I10> into I11> (10% to 20% of events) introduces correlation!

Solution: Resonator Coupling Device Fabrication by Haohua Wang Photography by Erik Lucero

Negligible Measurement Crosstalk (0.005) Off resonance, Rabi only one qubit Probability of x1>, 1x> [%] 100 80 60 40 20 (94% visibility) Amplitude of x1>, 1x> [a.u.] 10 4 10 3 10 2 10 1 10 0 7091 41.9 0.59% Crosstalk 10-1 0 50 100 150 200 250 10 4 10 3 10 2 10 1 10 0 6108 0.31% Crosstalk 19.1 0 t [ns] 0 100 200 300 400 500 10-1 0 50 100 150 200 250 Frequency [MHz] 3% stray tunneling of I1>, + 0.5% crosstalk

Creating Entanglement through a Resonator 100 Probability of 01>, 10>, or 11> [%] Qubit A 0 π 80 t Reson. 0 60 Qubit B 0 swap 40 20 0 0 100 200 300 400 500 600 700 800 900 1000 Swap time t [ns]

Checking the Entanglement (15 o Z rotation for 01>-e iφ 10> ) raw data measure corrected Trace-Fidelity compared to Bell Singlet State: 88.3% (92.1%) Entanglement of Formation: 0.378 (0.449)

Sequence 000 100 + 010 100 100 + 001 Qubit A 0 π α/α Resonator 0 Qubit B 0 β/β Qubit A t i-swap Qubit B t i-swap

Results (optimized for maximum S) Axes: ab a b ab a b p 00> p 01> p 10> p 11> 41.6% 34.1% 15.8% 8.5 39.8% 35.3% 17.6% 7.3 37.0% 10.5% 13.5% 39.0% 36.1% 40.7% 11.4% 11.9% E xy : 0.515 0.502 0.521 0.536 5,010 runs each to find probabilities Repeated 6,720 times over 8 hours S = 2.0732 ± 0.0003 S = 2.355, corrected for single-qubit measure fidelities

Can the experiment be trusted? Axes: ab a b ab a b p 00> p 01> p 10> p 11> E xy : 1.0 1.0 1.0 1.0 If you turn off all control electronics and take the data you get: S = 2.0000 ± 0.0000 Gross errors with artificially introduced correlations can yield S > 2!

Checking the Experiment 1) Sequence parameters, (a,a ) & (b,b ) make sense 2.5 2) Measure phase 3) Measure delay 2.1 T1, T 2 decay 2.0 2.0 1.5 S 1.0 S 1.9 0.5 1.8 0.0 0 90 180 270 360 Phase (meas. plane) of a, a [deg] 1.7-10 0 10 20 30 Delay of (a,a ) to (b,b ) [ns]

Global Test of Independent Measurement Suggested by R. Alicki: Measurement of only qubit A should not depend on axis of B meas: ab a b ab a b P 00 0.4162 0.3978 0.1046 0.3612 P 10 0.1575 0.1759 0.3700 0.1136 P 01 0.0852 0.0731 0.3904 0.1185 P 11 0.3412 0.3531 0.1350 0.4066 + + E 0.5147 0.5019-0.5208 0.5358 S 2.0732 difference P 1x (a) 0.4987 0.5050 0.0063 P 1x (a ) 0.5290 0.5202 0.0088 P x1 (b) 0.4264 0.4262 0.0002 P x1 (b ) 0.5254 0.5251 0.0003 Of order measured x-talk

4) Error Budget (from simulation) 2 2 2.5 2.0 1.5 1.0 0.5 0.0 T 1 : Qubit relaxes back into the ground-state 2.828 2.500 for T 1 s = 392ns, 296ns, and 2.55µs T 2 : Qubit loses phase information 2.500 2.337 for T 2 s = 146ns, 135ns, and 5.26µs Visibilities: Qubit states are misidentified during tunneling 2.337 2.064 for Visibilities = 94.6% and 93.4% Experimental Result: 2.073 (0.5% better than simulation) -T 2 assumes uncorrelated dephasing - Simulation uses theoretically optimal angles Everything but coherence and measurement seems perfect! (Only problems are single qubit errors)

P 01 [%] 100 Swap Fidelity Data vs. Simulation 90 80 70 60 50 40 30 20 10 0 Error < 1% after 3+1 swaps! 0 20 40 60 80 100 t [ns] Entangler spec. s: 10 ns, on/off = 1000, ~ 1% errors

Conclusions Superconducting qubits: Accurate quantum control with electrical signals Electrical resonator: Deterministic generation of complex states Most complex of ANY experimental system Mechanical resonator: first to Cool to ground state Manipulate quantum state Bell Violation: First in solid state Close detection loophole all errors low Acknowledgements: Photons Max Hofheinz Haohua Wang Resonators Phonons Aaron O connell Andrew Cleland Bell Violation Markus Ansmann Matthew Neeley Radek Bialczak Erik Lucero James Wenner Daniel Sank Martin Weides Michael Lenander

Decay of Fock States Master Equation (no phase info.) dpi ( t) dt Pi ( t) = + T i i+ 1 ( t) P T i+ 1 Normalized fourier amplitude 1 0.5 n = 1 0 0 2 4 6 1 n = 3 0.5 0.6 0.4 0.2 3 0 0 2 4 6 n = 5 (a) 1 2 2 1 (c) (e) 1 0.5 0.6 0.4 0.2 0.4 0.2 n = 2 0 0 2 4 6 4 3 2 n = 4 0 0 2 4 6 1 n = 6 1 (b) (d) (f) 0 0 2 4 6 t d (µs) 0 0 2 4 6 t d (µs) (exp.) (thy.) n T n T 1 /n 1 2 3 4 5 6 3.25 1.65 1.02 0.79 0.65 0.64 3.25 1.63 1.08 0.81 0.65 0.54 (µs)

Density matrix and Wigner tomography State preparation Delay D(α) State readout: bring the qubit on resonance with the resonator for τ Readout qubit P1 Measure P k (α) Direct Wigner tomography τ (ns) Matrix inversion W 2 π ( α ) = ( 1) k P ( α ) 1681 pts 88.9% k Im(α ) k D. Leibfried, et al., Phys. Rev. Lett. 77, 4281(1996) 60 pts 84.7% Re(α )