Decoherence of superconducting flux qubits
Adrian Lupascu
Institute for Quantum Computing, Department of Physics and Astronomy,
University of Waterloo
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Advanced many-body and statistical methods in mesoscopic systems II,
Brasov, Sep 1-5, 2014
Ingredients of superconducting qubits
Josephson tunneling
Charging energy
Magnetic flux (for circuits with loops): Aharonov-Bohm phase
Charging and Josephson, magnetic flux
S S I
L R
HT;e® = ¡EJ
X
nL;nR
(jnL ¡ 1; nR + 1ihnL; nRj+ jnL + 1; nR ¡ 1ihnL; nRj)
n: # Cooper pairs which crossed the junction
EC =(2e)2
2CLR
100 nm
HJ = ¡EJ
X
nL;nR
(jnL ¡ 1; nR +1ihnL; nRj+ jnL + 1; nR ¡ 1ihnL; nRj)
HC = Ec
X
nL;nR
(nL ¡ nR)2jnL; nRihnL; nRj
jnL¡ 1; nR+1ihnL; nRj ! jnL¡ 1; nR+1ihnL; nRjei2¼f
Different types of superconducting qubits
Phase qubits
Charge qubits/transmon
Flux qubits
Steffen et al. (2006)
Nakamura et al. (2006) Houck et al. (2009)
van der Wal et al. (2001)
The flux qubit (persistent current qubit)
(states with persistent current +Ip, -Ip)
In basis: { , }
0,495 0,500 0,505
-5
0
5
qb
(0)
Eg,
e (G
Hz) Ee
Eg
D
0.45 0.50 0.55-300
-250
-200
qb (0)
E (
GH
z)
D
xzqbpI
hH ˆˆ
22
2ˆ 0 D Ip qb
Ip , D: design parameters
qb : control parameter
Mooij et al., Science 285 1036 (1999)
©qb
H =¡h²2
¾z ¡ h¢2
¾x
² = 2Ip¡©qb ¡ ©0
2
¢
Recent advances on superconducting qubits
Devoret and Schoelkop, Science 339, 1169 (2013)
Why are we still interested in flux qubits?
Leggett and Garg (1985)
Zhu et al. (2011) Bal et al. (2012) De Groot et al. (2012)
Peropadre et al. (2013) Sabin et al. (2012)
Outline
Setup
Circuit QED with flux qubits
Readout
Measurements of decoherence
Energy relaxation
Pure dephasing
Discussion
Quasiparticles
Magnetometry
Cavity/circuit QED
Brune et al (1996) Wallraff et al. (2004)
𝐻 = ℏ𝜔𝑟𝑎†𝑎 + ℏ𝜔𝑎/2𝜎𝑧+ℏ𝑔 𝜎+𝑎 + 𝜎−𝑎†
Jaynes-Cummings Hamiltonian
cavity atom interaction
Cavity QED Circuit QED
Device
• High resistivity silicon
• Two Al evaporation steps, liftoff,
e-beam evaporation
• Argon ion milling cleaning of the
surface
Abdumalikov et al., PRB 78, 180502(R), (2008)
Oelsner et al. PRB 81, 172505 (2010).
Niemczyk et al., Nat. Phys. 6, 772, (2010)
Jerger et al., EPL, 96, 40012, (2011)
Orgiazzi et al. arXiv:1407.1346
Qubit state readout
Readout protocol Homodyne voltage histograms
Relatively high readout contrast (70 - 80%), limited primarily by energy relaxation
Improvement over other experiments with flux qubits, where lower contrast was attributed to spurious two level systems
timereadoutpreparation
reset
repeated sequence
Spectroscopy
Qubit 1
Ip, ¢
Single qubit control
Microwave resonant driving
jÃi = cos µ2jgi+ sin µ
2eiÁjei
Dynamics in a rotating frame
• Control
• Detuning
rotations around any axis
²(t) = ²0 +Acos(!dt+Á)!d¡!01
x
y
z
µ
Á
½ = r sinµ cosÁ¾x+ r sinµ sinÁ¾y + r cosµ¾z
H =¡h²2
¾z ¡ h¢2
¾x
Energy relaxation measurements
time
readout
delaytime
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
I quad
ratu
re (
A.U
)
(s)
qubit 2 : T1 = 9.6 s ± 0.199 s
Measurement sequence Relaxation for qb 1
At the symmetry point (²=0)
T1 = 10 (6) ¹s for qb 1 (2)
T1 > 5 ¹s obtained in two other devices
Energy relaxation measurements around the
symmetry point
Qubit 1 spectroscopy
Qubit 1 T1 vs flux, span 36 MHz Qubit 2 T1 vs flux, span 25 MHz
²=p
²2 +¢2 ²=p
²2 +¢2
Energy relaxation measurement over a broad
range
¢1 ¢2ºres
Increase of the relaxation rate around the cavity
frequency: Purcell effect
Qb 2
Qb 1
Energy relaxation: discussion
Intrinsic rates (excluding the Purcell rate over
multiple modes and the control line induced energy relaxation): 7 ¹s and 20 ¹s for qubits 1 and 2
Possible sources
Tunneling of quasiparticles: measured rates can be
explained by nonequilibrium quasiparticle densities of 0.12 and 0.04 ¹m-3
.
Loss due to surfaces and interfaces
Bal et al., arXiv:1406.7350
Loss of coherence without loss of energy
Ramsey measurement
Preparation by first pulse
Evolution at repetition I
Averaged measurement
Dephasing
readout
reset to groundstate
x(t)
tititime
repetition i
Ái =R ti+¿ti
dt@!01@»
»(t)
jÃi = 12(jgi+ jei)
jÃi = 12(jgi+ eiÁijei)
CR(¿) = heiÁi(t)
Dephasing
Spin-echo measurement
readout
reset to groundstate
x(t)
ti/2titime
repetition i
ti
Preparation by first pulse
Evolution at repetition I
Averaged measurement
Ái =³R ti+¿=2
ti¡R ti+¿ti+¿=2
´dt@!01
@»»(t)
jÃi = 12(jgi+ jei)
jÃi = 12(jgi+ eiÁijei)
CSE(¿) = heiÁi(t)
Measurements away from the symmetry point
0 500 10009.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
Qubit e
xcited
sta
te p
rob
ab
ility
(a
.u.)
Time (ns)
0 100 200 300
8
10
12
Qu
bit e
xcite
d s
tate
pro
ba
bili
ty (
a.u
.)
Time (ns)
Ramsey Spin-echo
Dephasing rate versus flux
² =Ipe(f ¡ 1=2) f = ©
©0a = ²p
²2+¢2
Yoshihara et al., PRL 97, 167001 (2006)
Dephasing with 1/f noise
Transition frequency
Low frequency noise
Spin echo coherence decay
(Similar expression for Ramsey decay)
Extracted flux noise levels: A= 2.6 (2.7) 10-6 for qubit
1(2)
!01 = 2¼p
²2 +¢2
@!01@f
= 2¼Ipe
a
Sf(!) =Aj!j
CSE(¿) = e¡(¡SE¿)2 ¡SE(a) =
Ipe
pA ln 2a
Dephasing around the symmetry point
Decay curve for different values of coupling to flux
Crossover from exponential to Gaussian
Nearly bias independent source causing exponential
decay at the symmetry point
CSE(¿) = e¡¡0¿e¡(¡(a)¿)2
Possible sources of dephasing at the symmetry
point
Measurements
Fluctuations of the photon number in the cavity
Dephasing due to flux noise coupled quadratically
Decoherence due to charge noise
¡phÁ = ·nph
Qubit 1 Qubit 2
Ramsey 1.3 MHz 1.11 MHz
Spin echo 0.97 MHz 0.54 MHz
¡quad °uxÁ = ¼
¢(Ip=e)
2A2
< 18 kHz
30 (24) kHz for qubit 1(2)
Sears et al., PRB 86, 180504 (2012)
Makhlin and Shnirman., PRL 92, 178301 (2004)
Decoherence due to charge noise
Background charge fluctuations M
1 2 3 M
n̂i ! n̂i ¡ngi
Quasiparticle tunneling
1e -1e
Decoherence due to charge noise
A change in the offset charge (ngi) induces a change in energy, which leads to dephasing
Background charges slow continuous changes
1/f spectrum, expecting stronger cancellation by spin echo
Quansiparticles Discrete change
Speed may be comparable with amplitude of energy change
Amplitude of change depends exponentially on EJ/Ec, estimating numbers in the kHz to MHz range given fabrication errors
Spectroscopic doublet lines
Spectroscopic doublets in a flux qubit
26
Doublet structure:
• Depends on flux bias
• Stable over long times (days)
• Abrupt changes in splitting 10.1 10.20.3
0.4
0.5
VH (
mV
)
Frequency (GHz)
1.498 1.500 1.502
5
6
7
f mw (
GH
z)
/0
Lupascu et al., PRB 80, 175206 (2009)
0.496 0.500 0.504
10.2
10.5
10.8
ge (
GH
z)
Magnetic Flux (0)
Microscopic quantum TLS This experiment
Working model
Two-state fluctuator two different qubit energies
TSF – random telegraph noise, rate °
Based on spectroscopic measurements
Quantitative characterization of the transition time?
° À (Nrep £Trep)¡1
° ¿¢º
27
Li et al., Nature Comm. 4, 1420 (2013)
Rabi oscillations
28
S1
S2
TSF state
g
e
Qubit state
1
-1
Readout result
Conditional
excitation Readout
cj =1
N ¡ j
N¡jX
i=1
riri+jAuto correlation
Measurement of autocorrelation function for
different excitation values
¡R1 = ¡R2 = °S1!S2+ °S2!S1
29
Variation with EJ/EC
4 junction PCQ model
30
EC =(2e)2
C
EJ = Á0Ic
Charge modulation
ng1
ng2 -0.5
-0.5
0.5
0.5
Calculation of transition rates
31
Rate for tunneling through a single junction
Selection rules for the operators
L 1, 2 31 2e
g
L
1
3
2
L
¡i!fa!b =
4¼
~
ZdEb
ZdEaDqp;b(Eb)Dqp;a(Ea)±(Eb + ~!if ¡Ea)
£ fa(Ea) (1¡ fb(Eb)) jMa!bi!f j2
Lutchyn, PhD thesis; Catelani et al.
Leppäkangas and Marthaler, PRB 85, 144503 (2012)
Non equilibrium quasiparticles
32
Measured and calculated thermal equilibrium rate
Quasiparticle density
From T1= 5 ¹s, nqp < 0.7/¹m3
Martinis et al, PRL 103, 097002 (2009)
Riste et al, Nature Comm 4, 1913 (2013)
Bal et al., arXiv:1406.7350
Quasiparticles in flux qubits
Effect of quasiparticles in flux qubits is very strong
High conductance tunnel junctions
Further work is needed to understand the details of
the multiple island circuit
Improvements
Infrared/microwave shielding, traps, gap engineering,
vortices
Improved designs to reduce effect of quasiparticles
Other recent work on decoherence in flux qubits
CEA Saclay
Yale (fluxonium)
Stern et al, arxiv:1403.3871 (2014)
Pop et al, Nature 508, 369 (2014)
Spin-echo with an applied AC field
Budkert and Romalis,
Nature Physics 3, 227 (2007)
Il’ichev and Greenberg,
EPL 77, 58005 (2007)
Field detection sensitivity
• 𝑐𝑖 = 𝑟𝑖𝑟𝑖−1
• No qubit reset is necessary
• It requires single-shot and projective
measurement
𝑆𝛷 =𝑆𝑟
𝜕 𝑟𝜕𝛷
2 𝑆𝛷1 2 = 3.9 × 10−8 Φ0 Hz
𝑆𝐵1 2 = 3.3 pT Hz
M. Bal et al., Nature Comm. 3, 1324 (2012)
Trep = 1 ¹s
¿ = 100 ns
Field frequency: 10 MHz
Detection bandwidth
A peak is observed at ν𝑠 − 𝜏−1 in the band (𝜏−1−1
2𝑇rep , 𝜏−1 +
1
2𝑇rep)
BW = 1¿
BWred =1
Trep
M. Bal et al., Nature Comm. 3, 1324 (2012)
𝑆𝛷1 2 Φ0 Hz 𝑆𝜖 = 𝑆𝛷 2𝐿
This Work 3.9 × 10−8 1.1ℏ
D. D. Awschalom et al., APL 53, 2108 (1988) 8.4 × 10−8 1.4ℏ
F. C. Wellstood et al., IEEE Tran. Magn. 25, 1001
(1989) 3.5 × 10−7 5ℏ
Relevant figure of merit for comparison: 𝛿𝐵min 𝑇 𝑉
𝛿𝐵min 𝑇 ~ 0.1 − 1 fT Hz for 𝑉 ~ cm3 • I. Kominis et al., Nature 422, 596 (2003)
• H. B. Dang et al., APL 97, 151110 (2010)
Theoretical limit to sensitivity:
𝛿𝐵min 𝑇 ~ 1 pT Hz for 𝑉 ~ μm3 • V. Shah et al., Nature Photonics 1, 649 (2007)
Relevant figure of merit for comparison: 𝑆𝜖 = 𝑆𝛷 2𝐿
Comparison to other magnetometers
Comparison to DC-SQUIDs
Comparison to atomic magnetometers
Limit to sensitivity
Theory curve
Perfect fidelity
State preparation
and measurement
short compared to
evolution
1/f noise spectrum
Conclusions
Made progress in understanding and improving
coherence times in superconducting qubits
CQED implementation
Planar, local control
ideal platform to study decoherence
Magnetometry, two qubit gates
Quasiparticles – important role
Acknowledgement
SQD group Jean-Luc Orgiazzi
Chunqing Deng
Marty Otto
Ali Yurtalan
Feiruo Shen
Nicolas David Gonzalez
Pol Forn Diaz
Alumni (these experiments) Mustafa Bal
Florian Ong
41