Introduction to Decoherence and How To Fight It Expt. PI’s: Andrew Houck David Schuster Rob Schoelkopf Hannes Majer Michel Devoret Jerry Chow Steven Girvin Andreas Wallraff Joseph Schreier Blake Johnson Luigi Frunzio Theory Alexandre Blais Jay Gambetta Jens Koch Lev Bishop Terri Yu David Price Why is decoherence so difficult? Because h is so small!
Charge qubits: 10 −15 eV-s = (1e)(1nV)(1µs) 1e,2e h
Flux qubits: 2 h (1 nT)(1 nA)(10 µm) (1 µs) 1 nT 1 nA
10µm SENSITIVITY OF BIAS SCHEMES TO NOISE (EXPD)
CHARGED IMPURITY junction noises TUNNEL CHANNEL
bias ∆Qoff ∆EJ ∆EC
- + - charge
flux
ELECTRIC DIPOLES phase
charge flux phase qubit qubit qubit What can we do about environmental decoherence?
1. Choose materials and fabrication techniques which minimize 1/f noise, two-level fluctuators, dielectric loss, etc.
2. Use quiet dispersive readouts that leave no energy behind and which do not heat up the dirt or the q.p.’s (And: Develop low noise pre-amps so fewer photons needed for read out.)
3. Design using symmetry principles which immunize qubits against unavoidable environmental influences (e.g. sweet spots, topological protection) ω 1 HZ=+0 σ zz()()tσσ+X()tx+Y()tσy 22
E↑ Review of NMR t language
E↓ Zt( ) causes transition frequency 111 to fluctuation in time * = + TT212 Tϕ Xt( ),Y(t) cause diabatic transitions between eigenstates 1 = Γ+↑ Γ↓ T1 ω 1 HZ=+0 σ zz()()tσσ+X()tx+Y()tσy 22
Fluctuations in transition frequency make the phase of superpositions unpredictable. ω0 + Z()t t t −idττZ() −itω ∫ Gaussian white noise leads to 00 ρρ↑↓ ()te= ↑↓ (0) e homogeneous broadening (Lorentzian line shape)
tt 1 − ddττZ()τZ(τ) 11 2 ∫∫12 1 2 = S (0) ρρ()te≈ (0) −itω00e0 T 2 ZZ ↑↓ ↑↓ ϕ 1 − [(StZZ 0)] −itω0 2 ρρ↑↓ ()te≈ ↑↓ (0) e ω 1 HZ=+0 σ zz()()tσσ+X()tx+Y()tσy 22
ω0 + Z()t t
t −id∫ ττZ() −itω00 ρρ↑↓ ()te= ↑↓ (0) e
Low frequency 1/f noise leads to inhomogeneous broadening −itω0 −itZ (gaussian line shape) ρρ↑↓ ()te≈ ↑↓ (0) e 1 − Zt22 Spin echo can help in −itω0 2 some cases. ρρ↑↓ ()te≈ ↑↓ (0) e Dephasing of CPB qubit due to gate charge n g noise ◄ charge fluctuations
ω()tZ= ω0 + ()t
2 ∂∂ZZ1 2 Zt()≈ δδngg()t++2 n()t ... energy ∂∂nngg2
ng 2 11 1⎛⎞∂Z = SS(0) ≈+⎜⎟ (0) ... ZZ ⎜⎟δδnggn Tnϕ 22⎝⎠∂ g Outsmarting noise: CPB sweet spot
◄ charge fluctuations
sweet spot only sensitive nd
energy to 2 order
energy fluctuations in gate charge!
ng (gate charge) Vion et al., Science 296, 886 (2002) ng ► Best CPB performance (Schoelkopf Lab) @ sweet spot: A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005) DISTRIBUTION OF RAMSEY RESULTS at Charge Sweet Spot
(Devoret group, fast CBA readout) Lopsided frequency fluctuations
1/f Charge noise! α 2 Se()ωα= , 1.9 10−3 Ng EJ/EC = 3.6 ω value OK 10 DISTRIBUTION OF RAMSEY RESULTS
Second order (curvature) effects limit coherence times to 500-600 ns Lopsided frequency fluctuations
1/f Charge noise! α 2 Se()ωα= , 1.9 10−3 Ng ω value OK Need larger EJ/EC for less curvature 11 Comparison of superconducting qubits
EJ/EC ~ 0.1-10 EJ/EC ~ 30-100 EJ/EC ~ 10,000
Cooper pair box Transmon Phase qubit
5 µm 60 µm 100 µm 50 µm sweet spot E E everywhere! E
Ng Ng Φ Ibias ≈ Ic Large junction
Vg
12 Transmon qubit: Sweet Spot Everywhere
EJ/EC = 1 EJ/EC = 5 EJ/EC = 10 EJ/EC = 50 Energy
Ng (gate charge)
charge dispersion Exponentially small charge dispersion!
290 µm quantum rotor (charged, in constant vector potential ) Offset charge = ‘vector potential’ T2 and Larmor Frequency Statistics 1st generation transmon with on Si
with EJ/EC ~ 50, flux sweet spot 20 hours of Ramsey experiments, no retuning
T1 =±79 3 ns ω =±7350495 90 kHz TT21=±140 10 ns =2 01
There is no significant drift or 1/f dephasing on these scales! 15 Coherence in Second Generation Transmon Rabi experiment T1 Measurement
12 ns π-pulses T1 = 1.5 µs Rabi visibility 100.4% +- 1%
Ramsey experiment 111 T * = 2 µs 2 * = + TT212 Tϕ
no echo, Tϕ = 6sµ at flux sweet spot consistent with ~ 20 kHz of
residual charge dispersion at EJ/EC = 50 * T2 =±2.05 0.1µs NO Echo: Tϕ = 6sµ T1 =1.5µs 111 consistent with ~ 20 kHz of residual charge dispersion at EJ/EC = 50 * =+ TT212 Tφ
Phase coherence time becomes exponentially larger
for only modest increase in EJ/EC
Quasiparticles plentiful but non-poisoning. See Rob Schoelkopf’s talk.
Is T1 the only remaining problem??? ω 1 HZ=+0 σ zz()()tσσ+X()tx+Y()tσy 22
1 =Γ↑↓+Γ T1 1 Γ= []SS()+ω + ()+ω emission into environment ↓ 4 XX 00YY 1 Γ= []SS()−ω + ()−ω absorption from environment ↑ 4 XX 00YY Environment = Junction Loss?? 1 Q = tanδ CC=−12jC=C1(1t−janδ )
it()
2 1 ω0 = Snii ()ωω=+h ()1 B LC R 1 1 = ωδ0Btan ()2n +1 T1 ω Reduce Junction Participation Ratio
1 Q = tanδ CS it()
C If shunt capacitance tanδ ′ = tanδ is lossless then: CCS +
UCSB: overlap shunt capacitor Nori et al. proposal: shunt capacitance in flux qubits Yale: single layer transmon shunt capacitance
Qubit is simply an anharmonic oscillator. A necessary but not sufficient condition to achieve high Q is to be able to make high Q resonators on the same substrate. Purcell Effect: Engineering Spontaneous Emission from Cavity
? Substrate losses Junction losses
? ?
Shunt capacitance Density of EM States Seen by Qubit Weakly Coupled to Cavity
Cavity filtering enhances qubit decay rate
κ Cavity filtering reduces qubit decay rate
50 ohm background decay rate ω ωcavity ωcavity This picture only valid in the ‘bad cavity’ limit: g κ = Q Strong Qubit-Cavity Coupling: ‘Good Cavity’ Limit q u b it
∆ cavity κ 2g 250 MHz cavity q u With proper engineering b it and detuning from cavity resonances, spontaneous fluorescence can be made small.
2 11⎛⎞g In limit of large detuning: ∆ g ≈+NR ⎜⎟κ TT11 ⎝⎠∆ Tϕ again...
Measurement Induced Dephasing
Photons intentionally or accidentally introduced into the cavity cause a light shift (ac Stark shift) of the qubit frequency
2 † ⎛⎞h g † Heff =+hωra a + ⎜⎟ω0′1 h a a σ z ⎝ 2 ∆ ⎠
g = atom-cavity coupling ∆ = atom-cavity detuning Measurement induced dephasing ∆ g : 2 † ⎛⎞h g † Heff =+hωra a + ⎜⎟ω0′1 h a a σ z ⎝ 2 ∆ ⎠ AC Stark shift of qubit by photons
† † aa =1 na= a= 20
δ nnRMS =
AC Stark measurements: Schuster et al., PRL 94, 123602 (2005). Phase shift induced by passage of a single photon
11+−iiθθ/2 /2 ()↑+↓ 11in → ()ee↑+ ↓ out 22
θ ⎛⎞2g 2 ==arctan ⎜⎟ κ resonator decay rate 2 ⎝⎠∆κ
For strong coupling, even a single photon measures the state of the qubit and destroys the superposition: 1 nκ Tϕ What can we do about environmental decoherence?
1. Choose materials and fabrication techniques which minimize 1/f noise, two-level fluctuators, dielectric loss, etc.
2. Use quiet dispersive readouts that leave no energy behind and which do not heat up the dirt or the q.p.’s (And: Develop low noise pre-amps so fewer photons needed for read out.)
3. Design using symmetry principles which immunize qubits against unavoidable environmental influences (e.g. sweet spots, topological protection) The End Coherence Limits for SC Qubits
Noise source transmon CPB flux qubit phase
EJ/EC=85 EJ/EC=1 qubit
dephasing 1/f amplitude T2 [ns] T2 [ns] T2 [ns] T2 [ns] charge 10-4 –10-3 e [1] 400,000 1,000* 1,500 ∞
-6 -5 flux 10 –10 Φ0 [2,3] 3,600,000* 1,000,000* 1,000 - 2,000* 300 1,000 - 10,000†
-7 -6 critical current 10 –10 I0 [4] 35,000 17,000 1,500 1,000
last year 140 > 500 [5] 4,000 [3,6] 160 [7] measured T2 [ns] this year 2,000
last year 90 ~ 7,000 [5] 2,000 - 4,000 110 [7] measured T1 [ns] this year 4,000 [3,6]
* values evaluated at sweet spots † value away from flux sweet spot at Φ0/4 [4] D. J. Van Harlingen et al., Phys. Rev. B 70, 064517 (2004). [1] A. B. Zorin et al., Phys. Rev. B 53, 13682 (1996). [5] A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005). [2] F. C. Wellstood et al., Appl. Phys. Lett. 50, 772 (1987). [6] P. Bertet et al., Phys. Rev. Lett. 95, 257002 (2005). [3] F. Yoshihara et al., Phys. Rev. Lett. 97, 16001 (2006). [7] M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006).