Non-linear driving and Entanglement of a quantum bit with a quantum readout Irinel Chiorescu Delft University of Technology
Quantum Transport group Prof. J.E. Mooij Kees Harmans Flux-qubit team visitors Yasunobu Nakamura (NEC Japan, 2001-2002) Kouichi Semba (NTT Japan, 2002-2003) PhD students Alexander ter Haar Adrian Lupascu Jelle Plantenberg postdocs Patrice Bertet Irinel Chiorescu students technical staff collaborations NTT, NEC, MIT, TU Delft (theory), U Munich acknowledgements FOM (NL), IST (EU), ARO (US)
Outline basics about the flux-qubit qubit initialization, operation & readout Rabi oscillations, Ramsey fringes present status - extreme stability during qubit operation - strong microwave driving multi-photon induced coherent oscillations experimental demonstration of entanglement quantum bit quantum readout (squid) conclusions
3 Josephson-junctions Quantum Bit J.E. Mooij et al, Science, 285, 1036 (1999) superconducting loop, with 3 Josephson junctions 2 are identical and the 3rd is smaller (α=0.6 0.8) Josephson Potential: U=ΣE J I u = U/E J u = 2 + α - cosγ 1 - cosγ 2 - αcos(γ 2 - γ 1 + 2πf) φ 1 = (γ 1 - γ 2 )/2, φ 2 = (γ 1 + γ 2 )/2 u = 2(1 - cosφ 1 cosφ 2 ) + 2αsin 2 (φ 1 - πf)
Josephson potential - phase space 2 wells separated by a barrier for f=0.5, symmetric barrier α=0.8, f=0.5 T in T out T out
Flux Qubit two level system C. van der Wal et al, Science, 290, 773 (2000) Exact diagonalisation: two levels at the bottom of the spectra Two wells separated by a barrier Persistent currents of opposite direction and SQUID critical current qubit persistent current Microwave induced excitation level structure see also, J. Friedman et al, Nature, 6, 43 (2000)
Coherent oscillations Magnetic resonance with a single, macroscopic quasi-spin Bloch sphere Ψ>=α >+β > Rabi oscillations microwave excitation with frequency ω and amplitude A coherent rotations with Ω Rabi A A MW pulse ω = E Ω Rabi A e> g>
Qubit operated at the magic point Hamiltonian and eigenstates H = -ε/2 σz /2 σx tan2θ = / ε 0 = cosθ + sinθ 1 = -sinθ + cosθ Initialization, ε = 0 Q = 0 = ( + )/ 2 Operation, ε = 0 Q = α 0 + β 1 Q 0 MW pulse ON (rotating frame) <σx> = α 2 - β 2 1 Q MW pulse OFF (lab frame) 1 0 shift Readout, ε > 0 Q = α 0 + β 1
Switching event measurements Device qubit merged with the SQUID strong coupling L I pulse ~ns rise/fall time t Readout bias current to switch the SQUID ramping generates the shift (preserving the qubit information) switching current depends on qubit state (spin up or down) pulse height: I sw0 < I b < I sw1 shift
Single shot resolution (in an ideal sample) switching probability (%) 100 80 60 20 0 ground state excited state 2.7 2.8 2.9 3.0 pulse height @ AW generator (V)
Sample E J /E C = 34.65 E C = 7.36 GHz α = 0.8 = 3.4 GHz I p = 3 na large junctions I c = 2 µa strong coupling L=10 ph shunt capacitance C = 10 pf bias line R b = 150 Ω voltage line R v = 1 kω
Cavity, wiring
Qubit spectroscopy Energy (GHz) 20 0-20 - 0.46 0.48 0.50 0.52 0.54 total flux (Φ 0 ) (I sw - I bg ) / I ctr (%) F (GHz) 2 1 0 15 10 5 0 16 GHz 16 GHz 0.008 π = 3.4 GHz -0.005 0.000 0.005 Φ ext / Φ 0
Rabi: pulse scheme RF line: one microwave pulse with varying length bias line: Ib pulse trigger MW pulse operation Ib pulse read-out time voltage line: detection of the switching pulse
Rabi coherent oscillations decay time 150 ns 80 0.6 switching probability (%) 60 80 60 80 A = 0 dbm A = -6 dbm 0.5 0.4 0.3 0.2 Rabi frequency (GHz) 60 0.1 0.0 0 10 20 50 60 70 80 90 100 0.0 0.5 1.0 1.5 2.0 F Larmor = 6.6 GHz pulse length (ns) A = -12 dbm MW amplitude 10^(A/20) (a.u.) I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, J.E. Mooij, Science, 299, 1869 (2003)
Fast oscillations Switching probability (%) 100 90 80 70 60 50 Psw (%) Psw (%) 90 60 62 60 58 0 10 20 RF pulse length (ns) RF pulse length (ns) 20 500 510 520 5 0 50 100 150 200 250 0 350 0 450 500 RF pulse length (ns)
Ramsey interference Ramsey: two π/2 pulses with varying time in between trigger π/2 free run π/2 Ib pulse time operation read-out
Ramsey fringes 0 MHz F L = 5.61 GHz detuning P SW (%) 90 60 0 5 10 15 20 25 time between two π/2 pulses (ns) 310 MHz
Ramsey interference Ramsey: decoherence time τ φ 20 ns 80 P SW (%) 70 60 50 π/2 π/2 0 5 10 15 20 25 distance between two p/2 pulses (ns) F L = 5.7 GHz, df= 220 MHz, TRamsey: 4.5 ns
Relaxation measurements one π pulse and read-out pulse delayed trigger π delay time Ib pulse time operation read-out 100 switching probability (%) 90 80 70 60 50 0 1 2 3 4 5 6 7 8 9 10 delay time (µs) 8.3 ns, A=-12dBm 6 ns, A=-9dBm 4.5 ns, A=-6dBm 3.2 ns, A=-3dBm 2.445 ns, A=0 dbm exp fit of A=-12dBm τ φ = 870 ns
quasi-particle traps strong coupling with the MW line heat sinks on the current and voltage lines current injection: high frequency noise ground via the shunt capacitance Sample (2003) qp traps V heat sink I b
Spectroscopy Larmor frequency (GHz) Resonant frequencies (GHz) 12 10 8 6 4 2 = 5.866 GHz I q = 272 na spectroscopy peaks fit: E J /E C =.834 E C =7.281 GHz α=0.76 0 0.500 0.502 0.504 0.506 F/F 0 12 Q + ω 10 8 6 4 2 0 0.000 0.002 0.004 0.006 DF/F 0 Q Q - ω (Q + ω)/2 Q/2 ω switching probability (%) level repulsion 5.866 GHz persistent current 272 na spectroscopy peaks: Q qubit ω plasma frequency 2.91GHz Q+/-ω sidebands 2-, 3-photon peaks 50 45 35 25 20 15 Q/3 3 (Q+3) /2 Q/2 Q-3 1 2 3 4 5 6 7 8 9 10 11 12 frequency (GHz) Q Q+3
Rabi oscillations at the magic point low coherence time, but extreme stability of the qubit energy levels switching probability (%) 35 25 distance between pulses Rabi oscillations: F mw = Rabi oscillations: F mw = + F Rabi 20 Hadamard gate Ramsey with π pulses (Hadamard) 0 5 10 15 20 25 35 45 50 pulse length (ns)
Ramsey fringes at the magic point coherence time ~15-20 ns (mostly limited by the relaxation time) 6.1 P sw (%) 52.00 6.0 45.00 Frequency (GHz) 5.9 5.8 5.7 = 5.856 GHz 38.00 31.00 24.00 5.6 5 10 15 20 25 35 distance between two p/2 pulses (ns)
Coherence time at the magic point coherence time ~20 ns (mostly limited by the relaxation time) τ φ and τ r (ns) 100 90 80 70 60 50 20 10 7.5 7.0 6.5 6.0 5.5 Larmor frequency (GHz) 85-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Φ-Φ 0 /2 (mφ 0 ) when optimizing the readout τ φ ~120 ns switching probability (%) 80 75 70 Ib=2.841µA Ib=2.976µA Ib=2.565µA 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 delay between two p/2 pulses (microseconds)
60 55 50 45 35 Multi-photon processes ONE-PHOTON F mw =7.16GHz 60 55 50 45 35 TWO-PHOTON F mw =3.62 GHz Switching probability (%) 25 20 60 55 50 45 35 25 20 60 55 A = -14 dbm A = -18 dbm 25 20 60 55 50 45 35 25 20 60 55 A = -15 dbm A = -17 dbm 50 50 45 45 35 35 25 A = -22 dbm 20 0 5 10 15 20 25 35 25 pulse length (ns) A = -19 dbm 20 0 5 10 15 20 25 35
Multi-photon processes One-photon Rabi frequency (GHz) 3.0 2.5 2.0 1.5 1.0 0.5 one-photon Rabi frequency J 1 (b10 A/20 ) with b=0.92, =5.344 GHz 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 10 A/20 (a.u.) power calibration (check the b fit parameter) Two-photon Rabi frequency (GHz) 0. 0.25 0.20 0.15 0.10 0.05 Rabi frequency: n = J n (ε mw /F L ) can be renormalized ~ by noise ( < ) two-photon Rabi frequency J 2 (b10 A/20 ) with =5.344 GHz, b=5.15 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 10 A/20 (a.u.)
Coherent rotations in the non-linear regime several peaks in the Fourier transform of the oscillations Rabi frequencies higher than the Larmor frequency 7 6 Rabi frequency (FFT) (GHz) 5 4 3 2 1 =5.03 GHz b=1.41 J 1 (b10 A/20 ) 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 10^(A/20)
Peaks in FFT of the Rabi oscillations (GHz) 14 12 10 8 Numerical simulations H/h=ω 0 σ z /2+ω x σ x /2+(ω 1 σ x cosωt)/2 ~12.25 GHz 6 ω 0 4 2 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 w 1 (GHz) ω x =0.1 GHz 2ω 0
Qubit entangled with a quantum readout QUBIT, two-level system hf L SQUID, harmonic oscillator 0>, 1> 0>, 1>,..., N> MI q I circ... hω p microwave field 11> 10> 12>... F L 00> ω p 01> 02>
Coherent oscillations of the coupled system qubit Larmor frequency 7.16 GHz plasma frequency : 2.91 GHz coupled system at 10.15 GHz 10> blue-side band 00> 11> 01> switching probability (%) 38 36 34 32 28 Rabi oscillations F=10.15 GHz, A=-5dBm 26 qubit: F 24 L =7.16 GHz squid: ω pl =2.91 GHz 22 0 2 4 6 8 10 12 14 16 pulse length (ns) switching probability (%) 26 24 22 20 18 Rabi oscillations F=10.15 GHz, A=3dBm qubit: F L =7.16 GHz squid: ω pl =2.91 GHz 0 2 4 6 8 10 12 14 16 pulse length (ns)
Blue-side band qubit Larmor frequency 6.43 GHz, plasma frequency : 2.91 GHz coupled system at 9.38 GHz Rabi oscillations at F L =6.43 GHz switching probability (%) 35 25 20 15 coherent oscillations 01> 11> coherent oscillations 00> 10> 0 10 20 pulse length (ns) 10> 11> 00> 01> π pulse
switching probability (%) 35 25 20 15 qubit Larmor frequency 6.43 GHz plasma frequency : 2.91 GHz coupled system at 9.38 GHz p pulse Rabi oscillations at F L =6.43 GHz Red-side band switching probability (%) 28 26 24 22 20 red-side band: coherent oscillations 01> <10 switching probability (%) 25 20 15 0 10 20 pulse length (ns) red-side band 3.52 GHz F L = 6.43 GHz blue-side band 9.38 GHz +10 db 3 4 5 6 7 8 9 10 MW frequency (GHz) after π after 2π 0 5 10 15 20 25 pulse length (ns) 11> 10> π 2π 00> 01>
Conclusion entanglement of the qubit with its quantum readout multi-photon induced coherent oscillations very strong (non-linear) qubit driving, F Rabi >F L qubit operated at the magic point extreme stability of the qubit operation τ rel 1 µs, τ Rabi 150 ns Ramsey interference: decoherence time 20 ns