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Expt.Andrew HouckDavid SchusterHannes Majer
Jerry ChowAndreas WallraffJoseph SchreierBlake JohnsonLuigi Frunzio
TheoryAlexandre BlaisJay Gambetta
Jens KochLev Bishop
Terri YuDavid Price
PI’s:Rob SchoelkopfMichel Devoret Steven Girvin
Introduction to Decoherenceand
How To Fight It
Why is decoherence so difficult?
Because is so small!h
Charge qubits:1510 eV-s (1 )(1nV)(1 s)e µ− =h
1e,2e
Flux qubits: 2(1 nT)(1 nA)(10 m) (1 s)µ µh
1 nA1 nT
10 mµ
SENSITIVITY OF BIAS SCHEMES TO NOISE (EXPD)
junction noises
bias ∆Qoff ∆EJ ∆EC
charge
flux
phase
CHARGED IMPURITYTUNNEL CHANNEL
- +-
ELECTRIC DIPOLES
chargequbit
fluxqubit
phasequbit
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 energybehind 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)
( )0 1 ( ) ( ) ( )2 2
z z x yH Z t X t Y tω σ σ σ σ= + + +
t
E↓
E↑
Review of NMRlanguage
( ) causes transition frequency to fluctuation in time
( ), ( ) cause diabatic transitionsbetween eigenstates
Z t
X t Y t*
2 1
1 1 12T T Tϕ
= +
1
1T ↑ ↓= Γ +Γ
( )0 1 ( ) ( ) ( )2 2
z z x yH Z t X t Y tω σ σ σ σ= + + +
0 ( )Z tω +
t0 0
1 2 1 20 0 0
0
( )
1 ( ) ( )2
1[ (0)]2
( ) (0)
( ) (0)
( ) (0)
t
t t
ZZ
i d Zi t
d d Z Zi t
S ti t
t e e
t e e
t e e
τ τω
τ τ τ τω
ω
ρ ρ
ρ ρ
ρ ρ
−−
↑↓ ↑↓
−−
↑↓ ↑↓
−−↑↓ ↑↓
∫=
∫ ∫≈
≈
1 1 (0)2 ZZS
Tϕ=
Fluctuations in transition frequency make the phaseof superpositions unpredictable.
Gaussian white noise leads tohomogeneous broadening(Lorentzian line shape)
( )0 1 ( ) ( ) ( )2 2
z z x yH Z t X t Y tω σ σ σ σ= + + +
0 ( )Z tω +
t
0 0
0
2 2
0
( )
12
( ) (0) e
( ) (0)
( ) (0)
t
i d Zi t
i t itZ
Z ti t
t e
t e e
t e e
τ τω
ω
ω
ρ ρ
ρ ρ
ρ ρ
−−
↑↓ ↑↓
− −↑↓ ↑↓
−−↑↓ ↑↓
∫=
≈
≈
Low frequency 1/f noise leads toinhomogeneous broadening(gaussian line shape)
Spin echo can help insome cases.
Dephasing of CPB qubit due to gate charge noise
ener
gy
ng
◄ charge fluctuations
gn
22
g g2g g
1( ) ( ) ( ) ...2
Z ZZ t n t n tn n
δ δ∂ ∂≈ + +∂ ∂
0( ) ( )t Z tω ω= +
g g
2
g
1 1 1(0) (0) ...2 2ZZ n n
ZS ST n δ δϕ
⎛ ⎞∂= ≈ +⎜ ⎟⎜ ⎟∂⎝ ⎠
Outsmarting noise: CPB sweet spot
only sensitive to 2nd order fluctuations in gate charge!
ener
gy sweet spot
ng (gate charge)
ener
gy
ng
Vion et al., Science 296, 886 (2002)
◄ charge fluctuations
► Best CPB performance @ sweet spot: A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005)
(Schoelkopf Lab)
DISTRIBUTION OF RAMSEY RESULTSat Charge Sweet Spot
(Devoret group, fast CBA readout)Lopsided frequency fluctuations
10
Charge noise!1/f2
3( ) , 1.9 10gNS eαω α
ω−=
value OKEJ/EC = 3.6
DISTRIBUTION OF RAMSEY RESULTSSecond order (curvature) effects limit coherence times to 500-600 ns
Need larger EJ/EC for less curvature
Lopsided frequency fluctuations
Charge noise!1/f2
3( ) , 1.9 10gNS eαω α
ω−=
value OK11
Comparison of superconducting qubitsEJ/EC ~ 0.1-10 EJ/EC ~ 30-100 EJ/EC ~ 10,000
Cooper pair box Transmon Phase qubit
100 µm
5 µm
12
Ng
E
50 µm
60 µm
sweet spoteverywhere!E
Ng
EgV
Φbias cI I≈
Large junction
Transmon qubit: Sweet Spot EverywhereE
nerg
y
Ng (gate charge)
EJ/EC = 1 EJ/EC = 5 EJ/EC = 10 EJ/EC = 50
charge dispersion Exponentially small charge dispersion!
290 µm
quantum rotor(charged, in constantvector potential )
Offset charge = ‘vector potential’
T2 and Larmor Frequency Statistics1st generation transmon with on Si
with EJ/EC ~ 50, flux sweet spot20 hours of Ramsey experiments, no retuning
There is no significant drift or 1/f dephasing on these scales!
2 1140 10 ns 2T T= ± = 01 7350495 90 kHzω = ±1 79 3 nsT = ±
15
Coherence in Second Generation Transmon
12 ns π-pulsesRabi visibility100.4% +- 1%
T1 = 1.5 µs
T2* = 2 µs
Rabi experiment
Ramsey experiment
T1 Measurement
no echo, at flux sweet spot
6 sTϕ µ=
consistent with ~ 20 kHz ofresidual charge dispersion at EJ/EC = 50
*2 1
1 1 12T T Tϕ
= +
*2
1
2.05 0.1 s1.5 s
TT
µµ
= ±=
6 sTϕ µ=NO Echo:
consistent with ~ 20 kHz ofresidual charge dispersion at EJ/EC = 50
*2 1
1 1 12T T Tφ
= +
Phase coherence time becomes exponentially largerfor only modest increase in EJ/EC
Quasiparticles plentiful but non-poisoning. See Rob Schoelkopf’s talk.
Is T1 the only remaining problem???
( )0 1 ( ) ( ) ( )2 2
z z x yH Z t X t Y tω σ σ σ σ= + + +
[ ]
[ ]
1
0 0
0 0
1
1 ( ) ( )41 ( ) ( )4
XX YY
XX YY
T
S S
S S
ω ω
ω ω
↑ ↓
↓
↑
= Γ +Γ
Γ = + + +
Γ = − + −
emission into environment
absorption from environment
( )1 2 1 1 tanC C jC C j δ= − = −
Environment = Junction Loss?? 1tan
Qδ
=
( )i t
( )B2( ) 1iiS nR
ω ω= +h 01
1LC
ω =
ω( )0 B
1
1 tan 2 1nT
ω δ= +
Reduce Junction Participation Ratio
1tan
Qδ
=
( )i tCS
S
tan tanCC C
δ δ′ =+
If shunt capacitanceis lossless then:
UCSB: overlap shunt capacitorNori et al. proposal: shunt capacitance in flux qubitsYale: single layer transmon shunt capacitance
Qubit is simply an anharmonic oscillator. A necessary but not sufficient conditionto achieve high Q is to be able to make high Q resonators on the same substrate.
Purcell Effect:Engineering Spontaneous Emission from Cavity
?
?
?
Substrate lossesJunction losses
Shunt capacitance
Density of EM States Seen by Qubit Weakly Coupled to Cavity
cavityω ω
Cavity filtering enhances qubit decay rate
Cavity filtering reduces qubit decay rate
cavitygQ
ωκ =
κ
50 ohm background decay rate
This picture only valid in the ‘bad cavity’ limit:
cavity
qubit
qubit
cavity
2 250 MHzg
2
NR1 1
1 1 gT T
κ⎛ ⎞≈ + ⎜ ⎟∆⎝ ⎠
κ∆
Strong Qubit-Cavity Coupling:‘Good Cavity’ Limit
g∆
With proper engineeringand detuning from cavityresonances, spontaneousfluorescence can be madesmall.
In limit of large detuning:
again...TϕMeasurement Induced Dephasing
Photons intentionally or accidentally introducedinto the cavity cause a light shift (ac Stark shift) of the qubit frequency
eff 01†
2
r†
2 zg aH a aa ωω σ
⎛ ⎞′= +
∆⎜ ⎟⎝
+⎠
hhh
atom-cavity couplingatom-cavity detuning
g∆==
Measurement induced dephasing:g∆
eff 01†
2
r†
2 zg aH a aa ωω σ
⎛ ⎞′= +
∆⎜ ⎟⎝
+⎠
hhh
AC Stark shift of qubit by photons
†
1a a =†
20n a a= =
RMSn nδ =
AC Stark measurements: Schuster et al., PRL 94, 123602 (2005).
Phase shift induced by passage of a single photon
( ) ( )/ 2 / 2in out
2
1 11 12 2
2arctan resonator decay rate2
i ie e
g
θ θ
θ κκ
+ −↑ + ↓ → ↑ + ↓
⎛ ⎞= =⎜ ⎟∆⎝ ⎠
For strong coupling, even a single photon measures the state of the qubitand destroys the superposition:
1 nTϕ
κ
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 energybehind 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 QubitsNoise source transmon
EJ/EC=85CPB
EJ/EC=1flux qubit phase
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 ∞
flux 10-6–10-5 Φ0 [2,3] 3,600,000*1,000 - 10,000†
1,000,000* 1,000 - 2,000* 300
critical current 10-7 – 10-6 I0 [4] 35,000 17,000 1,500 1,000
measured T2 [ns]
last yearthis year
1402,000
> 500 [5] 4,000 [3,6] 160 [7]
measured T1[ns]
last yearthis year
904,000
~ 7,000 [5] 2,000 - 4,000[3,6]
110 [7]
[1] A. B. Zorin et al., Phys. Rev. B 53, 13682 (1996).[2] F. C. Wellstood et al., Appl. Phys. Lett. 50, 772 (1987).[3] F. Yoshihara et al., Phys. Rev. Lett. 97, 16001 (2006).
* 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).
[5] A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005).[6] P. Bertet et al., Phys. Rev. Lett. 95, 257002 (2005).[7] M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006).
