Decoherence of superconducting flux qubits - Horia Hulubei · Adrian Lupascu Institute for Quantum Computing, Department of Physics and Astronomy, University of Waterloo TexPoint

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