Synthesizing arbitrary photon states in a superconducting resonator Max Hofheinz, Haohua Wang, Markus Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O Connell, D. Sank, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland Department of Physics, University of California, Santa Barbara Collège de France, March
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator
Generation of quantum states in a resonator Classical drive on qubit and resonator arbitrary states only coherent states
Generation of quantum states in a resonator drive the qubit couple it to the harmonic oscillator arbitrary states arbitrary states
Device die size: 6 6mm
Coplanar waveguide resonator U V ν x very small capacitors are open ends each mode forms a quantum harmonic oscillator we use fundamental mode at frequency ν = c eff L make out of superconductor for high quality factor Q Performance: Q... 5 5 or T.5... µs
Superconducting qubits level system hard to make in a circuit very strong nonlinearity is enough allows to address transition exclusively we use Josephson junction as nonlinear circuit element
Josephson junction superconductor superconductor phase φ L tunnel barrier phase φ R Dynamics are determined by: I = I sinδ V = δ e t δ = φ L φ R phase difference I critical current of junction V DC Josephson relation AC Josephson relation
Josephson junction V Dynamics are determined by: I = I sinδ V = δ e t δ = φ L φ R phase difference I critical current of junction DC Josephson relation AC Josephson relation
Superconducting phase qubit U Φ bias k s δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei
Superconducting phase qubit U Φ bias k s e g δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei
Superconducting phase qubit U Φ bias k s e g δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei
Superconducting phase qubit U Φ bias k e g s δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei
Superconducting phase qubit U e δ Φ bias e g Γ e = 5Γ g Γ g readout squid δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei
UCSB phase qubit readout flux bias microwaves energy relaxation time T 6 ns quality factor Q 4 phase coherence time T 5 ns nonlinearity MHz limits gate time ns readout visibility P e g P g e.9
Coupling qubit and resonator H = hν q σ + σ + Ω ( ( ) aσ+ + a σ + hνr a a + )
Spectroscopy of qubit-resonator coupling 6.6 4 Frequency (GHz) ν r 6.5 6.4 6.3 Ω/π ν q 35 3 5 5 Excited state probability Pe (%) high probability indicates eigenfrequencies of qubit-resonator system 4 6 8 3 3 34 Qubit flux bias (mφ ) 5
Spectroscopy of qubit-resonator coupling coupling 6.6 on off 4 Frequency (GHz) ν r 6.5 6.4 6.3 Ω/π ν q 35 3 5 5 Excited state probability Pe (%) high probability indicates eigenfrequencies of qubit-resonator system Qubit and resonator only interact when ν q ν r Coupling can be turned on and off 4 6 8 3 3 34 Qubit flux bias (mφ ) 5
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one Ω H int = Ω ( ) aσ+ + a σ total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one 3Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one 3Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity
Fock states: Clear Rabi oscillations resonator qubit preparation measurement τ Excited state probability Pe (%) 5 5 5 5 9 5 8 5 7 5 6 5 5 5 4 5 3 5 5 5 5 Interaction time τ (ns) Prepared state
Fock states: Rabi frequencies scale as n Prepared state 9 8 7 6 5 4 3 3 4 5 6 7 8 Rabi frequency (MHz)..8.6.4.. Normalized Fourier amplitude
Fock states: Photon number distribution for.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 3.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 4.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 5.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 6.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 7.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 8.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for 9.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for.8 probability.6.4.. 4 6 8 photon number
Fock states: Photon number distribution for.8 probability.6.4.. 4 6 8 photon number
Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ
Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ
Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ
Directly driving the resonator creates a coherent state nω Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ
Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ
Coherent state: Non-periodic time traces resonator qubit τ Excited state probability (%) 5 3.6 5 3.3 5 3. 5.7 5.4 5. 5.8 5.5 5. 5.9 5.6.3 5 3 4 5 Interaction time τ (ns) Displacement α
Coherent state: Superposition of Fock states Displacement α 3.5 3..5..5..5. 4 6 8 Rabi frequency (MHz)
Coherent state: Superposition of Fock states Displacement α 3.5 3..5..5..5 Probability..8.6.4.. 4 6 8 Rabi frequency (MHz). 4 6 8 Photon number
Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind
Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind
Generalizing the pump sequence to arbitrary states Ω only perform partial state transfers leave some probability behind
Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind
Generalizing the pump sequence to arbitrary states Ω, Ω only perform partial state transfers leave some probability behind
Generalizing the pump sequence to arbitrary states qubit and resonator entangled during sequence phases matter swaps act on several photon numbers
Algorithm for arbitrary states: Reverse Engineering ψ = 3 ( + + i ) g g e g e g
Algorithm for arbitrary states: Reverse Engineering ψ = 3 ( + + i ) g g g g e e Time-reversed problem easier: start with the desired state eliminate states from top to bottom Law, Eberly, PRL 76, 55 (996) resonator qubit
Algorithm for arbitrary states: Reverse Engineering g e g e g resonator qubit
Algorithm for arbitrary states: Reverse Engineering ψ = (.577 +.56 ) g + (.57i.577 ) e g e g e g resonator qubit
Algorithm for arbitrary states: Reverse Engineering g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase g resonator qubit
Algorithm for arbitrary states: Reverse Engineering ψ = ((.34.473i) +.63 ) g + (.58 +.i) e g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase g resonator qubit
Algorithm for arbitrary states: Reverse Engineering g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase g resonator qubit
Algorithm for arbitrary states: Reverse Engineering ψ = ((.34.473i) +.63 ) g + (.568i ) e g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase g resonator qubit
Algorithm for arbitrary states: Reverse Engineering g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit
Algorithm for arbitrary states: Reverse Engineering ψ = ((.34.473i) ) g + (.849i ) e g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit
Algorithm for arbitrary states: Reverse Engineering g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit
Algorithm for arbitrary states: Reverse Engineering ψ = (.444.896i) g g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit
Calibrations impulse response (sampling scope) imperfections of IQ-mixer (spectrum analyzer) cable response (qubit as sampling scope) various parameters for readout SQUID π pulses (amplitude and frequency) measure pulse amplitude swap pulse amplitude swap pulse time for each photon number first order correction for finite rise time of swap pulses qubit/resonator dephasing rate when off resonance resonator displacement/drive amplitude ratio resonator drive phase readout visibility BUT: no per-state calibrations; just run the calculated sequence
A superposition of and 3 ψ = + 3 Pe Probability (a.u.)..8.6.4.. 5 5 5 3 35 time (ns)..8.6.4.. 3 4 5 6 7 8 Frequency (MHz) Probability..8.6.4.. 3 4 Photon number
A superposition of and 3 ψ = + 3 and φ = + i 3 look the same! Pe Probability (a.u.)..8.6.4.. 5 5 5 3 35 time (ns)..8.6.4.. 3 4 5 6 7 8 Frequency (MHz) Probability..8.6.4.. 3 4 Photon number
Full description of a resonator state Example: ψ = + 3 Density matrix: ρ = ψ ψ Wigner function: W (α) = π ψ D ( α)πd( α) ψ m 3 i ρ mn Im(α) + π + π π W (α) 3 n Re(α) π
Full description of a resonator state Example: ψ = + i 3 Density matrix: ρ = ψ ψ Wigner function: W (α) = π ψ D ( α)πd( α) ψ m 3 i ρ mn Im(α) + π + π π W (α) 3 n Re(α) π
Wigner tomography: measurement protocol resonator qubit Preparation of state ψ (Law & Eberly protocol) ρ = F ψ ψ +... τ State reconstruction displacement: ρ ρ α = D( α)ρd(α) qubit resonator interaction for time τ qubit measurement Pe..8.6.4.. 5 5 interaction time τ Pn.5..5..5. 4 6 8 Photon number n W (α) = π Tr (ρ απ) = π Im(α) Re(α) ( ) n P n n= + π + π π π W (α)
Controlling photon number ψ = 7 + π + π Im(α) W (α) π Re(α) π 8 6 m 4 Fidelity: F = ψ ρ ψ =.75 ρmn i 4 6 8 n 4 6 8
Controlling photon number ψ = 8 + π + π Im(α) W (α) π Re(α) π 8 6 m 4 Fidelity: F = ψ ρ ψ =.7 ρmn i 4 6 8 n 4 6 8
Controlling superpositions ψ = + + π + π Im(α) W (α) π Re(α) π 5 4 3 m ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 n 3 4 5 i
Controlling superpositions ψ = + + π + π Im(α) W (α) π Re(α) π 5 4 3 m ρmn Fidelity: F = ψ ρ ψ =.89 3 4 5 n 3 4 5 i
Controlling superpositions ψ = + 3 + π + π Im(α) W (α) π Re(α) π 5 4 3 m ρmn Fidelity: F = ψ ρ ψ =.88 3 4 5 n 3 4 5 i
Controlling superpositions ψ = + 4 + π + π Im(α) W (α) π Re(α) π 5 4 3 m ρmn Fidelity: F = ψ ρ ψ =.94 3 4 5 n 3 4 5 i
Controlling superpositions ψ = + 5 + π + π Im(α) W (α) π Re(α) π 5 4 3 m ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 n 3 4 5 i
Controlling phase ψ = +e 8 iπ 3 + 6 + π + π Im(α) W (α) π Re(α) π 6 5 4 m 3 ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 6 n 3 4 5 6 i
Controlling phase ψ = +e 8 iπ 3 + 6 + π + π Im(α) W (α) π Re(α) π 6 5 4 m 3 ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 6 n 3 4 5 6 i
Controlling phase ψ = +e 8 iπ 3 + 6 + π + π Im(α) W (α) π Re(α) π 6 5 4 m 3 ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 6 n 3 4 5 6 i
Controlling phase ψ = +e 3 8 iπ 3 + 6 + π + π Im(α) W (α) π Re(α) π 6 5 4 m 3 ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 6 n 3 4 5 6 i
Controlling phase ψ = +e 4 8 iπ 3 + 6 + π + π Im(α) W (α) π Re(α) π 6 5 4 m 3 ρmn Fidelity: F = ψ ρ ψ =.9 3 4 5 6 n 3 4 5 6 i
Voodoo cat ψ k = e i π 3 k 3 dead + π + π Im(α) alive W (α) zombie π 3 3 3 3 3 Re(α) π 8 m 6 4 ρmn i 4 6 8 n 4 6 8
Voodoo cat ψ = k=,, n=,3,6,9... e i π 3 k n n! n Im(α) 3 3 3 3 3 3 Re(α) + π + π π π W (α) 8 6 m 4 ρmn Fidelity: F = ψ ρ ψ =.83 4 6 8 n 4 6 8 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.5 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.35 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.5 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.65 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.8 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.95 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.4 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.7 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.3 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.9 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 3.5 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 4.4 µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 6. µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 9. µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = µs 3 4 n 3 4 i
Decay of resonator states (Haohua Wang) ψ = + 3 + π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 5 µs 3 4 n 3 4 i
Conclusions Full control over qubit state extended to resonator: arbitrary states (N < due to coherence) deterministic generation full characterization Max Hofheinz, H. Wang, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O Connell, D. Sank, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland Fock states: Nature 454, 3 34 (8) decoherence: H. Wang, PRL, 44 (8) Arbitrary states: Nature 459, 546 549 (9) decoherence: H. Wang, PRL 3, 44 (9) Funding: IARPA, ARO, NSF
Superconducting circuits: Microwave quantum optics atom Josephson qubit optical cavity LC or wave-guide resonator light microwaves (ν 7 GHz) Advantages: Qubits and cavities can be coupled at will Microwave signal waveforms can be fully controlled Drawbacks: Low photon energy requires low temperatures T < 5 mk Large solid state system limits coherence times
Calibrations impulse response (sampling scope) imperfections of IQ-mixer (spectrum analyzer) cable response (qubit as sampling scope) various parameters for readout SQUID π pulses (amplitude and frequency) measure pulse amplitude swap pulse amplitude swap pulse time for each photon number first order correction for finite rise time of swap pulses qubit/resonator dephasing rate when off resonance resonator displacement/drive amplitude ratio resonator drive phase readout visibility BUT: no per-state calibrations; just run the calculated sequence
Calibrating the swap pulse length Calibrate with the pulse lengths from the Fock state tune-up. 5 programmed pulse length (ns) 5 5...4.6.8. / n
Detuning time calibration: Qubit resonator Ramsey fringes resonator qubit π π..8.6 Pe.4.. 8 9 3 4 5 Time between pulses (ns)
Custom microwave electronics
Sub-ns timing resolution with a GHz DAC deconvolved signal 7 6 5 DAC 4 bit @ GHz Gaussian lowpass 3dB @ MHz Signal (mv) 4 3 98 4 6 Time (ns)
Sub-ns timing resolution with a GHz DAC deconvolved signal 7 6 5 DAC 4 bit @ GHz Gaussian lowpass 3dB @ MHz Signal (mv) 4 3 4 6 Time from programmed edge (ns)