electrical quantum engineering - université paris saclay · electrical quantum engineering . ... 0...

Electrical quantum engineering with superconducting circuits

P. Bertet & R. Heeres

SPEC, CEA Saclay (France), Quantronics group

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babi

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Introduction : Josephson circuits for quantum physics

M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1908 (1985)

M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1543 (1985) YES THEY CAN

Discrete energy levels

CAN MACROSCOPIC « MAN-MADE » ELECTRICAL CIRCUITS BEHAVE QUANTUM-MECHANICALLY ????

From a fundamental question (25 years ago) ….

… to genuine artificial atoms







… to genuine artificial atoms for quantum information and quantum optics on a chip







… to genuine artificial atoms for quantum information and quantum optics on a chip

QUANTUM PHYSICS QUANTUM ALGORITHMS QUANTUM SIMULATORS

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: Multi-qubit gates and algorithms

Lecture 4: Introduction to Hybrid Quantum Devices

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: Multi-qubit gates and algorithms

Lecture 4: Introduction to Hybrid Quantum Devices

1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits

Ener

gy (a

.u)

x (a.u.)

Real atoms

Hydrogen atom

2

0

( )4

eV rrπε

= − 2ˆˆ ˆ( )2pH V rm

= +

quantization [ ]ˆ ˆ,r p i=

|0>

|1> |2>

|3>

« two-level atom »

E01=E1-E0=hν01

I.1) Introduction

Real atoms

Hydrogen atom

2

0

( )4

eV rrπε

= − 2ˆˆ ˆ( )2pH V rm

= +

[ ]ˆ ˆ,r p i=

Frequency ν (a.u)

ν01

FWHM =Γ/2π

P(1

) (a.

u)

Spectroscopy (weak field)

laser 0( ) cos 2E t E tπν=

+ spontaneous emission Γ

Optical Bloch

equations

I.1) Introduction

quantization

𝐻𝐻𝐼𝐼 = 𝑒𝑒𝑟𝑟 ⋅ 𝐸𝐸(𝑡𝑡)

P(1

)

0

1

Time (a.u)

Rabi oscillations (short pulses, strong field at ν=ν01)

|0> |0>

|1>

( )1 0 12

+

Ω𝑅𝑅 = 𝑒𝑒 0 𝑟𝑟 1 ⋅ 𝐸𝐸0/ℏ

X,φ

E

0ω|0> |1>

|2> |3>

|4>

Quantum regime ??

[ ]ˆ ˆ,x p i= ˆ ˆ, iQ = Φ

Electrical harmonic oscillator

k

m

x

2 2

( , )2 2

kx pH x pm

= +

+q φ

-q

2 2

( , )2 2

H QQL C

ΦΦ = +

0km

ω = 01

LCω =

( ) ( ') 't

t V t dt−∞

Φ = ∫( ) ( ') '

tQ t i t dt

−∞= ∫

I.1) Introduction

LC oscillator in the quantum regime ?

2 conditions :

X,φ

E

|0> |1>

|2> |3>

|4> 0kT ω<<

Typic : L nH≈ C pF≈ 0 5GHzν ≈

At T=30mK : 0 8hkTν

≈

X,φ

E

|0> |1>

|2> |3>

|4> 1Q >>

OK if dissipation negligible

Superconductors at T<<Tc

I.1) Introduction

T<<Tc : dissipation negligble at GHz frequencies

Q=4 1010

at 51GHz and at 1K

Microwave superconducting resonators

Tc(Nb)=9.2K

S. Kuhr et al., APL 90, 164101 (2006)

Quantum regime

I.1) Introduction

Necessity of anharmonicity

How to prepare |1> ?

X,φ

E

0ω|0> |1>

|2> |3>

|4> +q φ

-q

Need non-linear and non dissipative element : Josephson junction

I.1) Introduction

Basics of the Josephson junction

The building block of superconducting qubits

B. Josephson, Phys. Lett. 1, 251 (1962)

P.W. Anderson & J.M. Rowell, Phys. Rev. 10, 230 (1963)

S. Shapiro, Phys. Rev. 11, 80 (1963)

I.1) Introduction

Josephson AC relation : 0d QVdt Cθϕ= =

Josephson DC relation : sinCI I θ=

( ) ( ') 't

t V t dt−∞

Φ = ∫( ) ( ') '

tQ t i t dt

−∞= ∫

N=Q/2e∈ ℕ



[ ]R L0

= mod(2 )=θ -θ 2θ 0,π πϕ

∈Φ

N=Q/2e∈

RθLθ

( ) ( ') 't

t V t dt−∞

Φ = ∫( ) ( ') '

tQ t i t dt

−∞= ∫

N=Q/2e∈ ℕ

0 /(2 )eϕ =

Basics of the Josephson junction

The building block of superconducting qubits

Classical variables ??

NON-LINEAR INDUCTANCE ( )0

2( )

1 /J

C C

L II I I

ϕ=

−

POTENTIAL ENERGY 0( ) cos cosJ C JE I Eθ ϕ θ θ= − = −



[ ]R L0

= mod(2 )=θ -θ 2θ 0,π πϕ

∈Φ

N=Q/2e∈

RθLθ

( ) ( ') 't

t V t dt−∞

Φ = ∫( ) ( ') '

tQ t i t dt

−∞= ∫

I.1) Introduction

0 /(2 )eϕ =

N=Q/2e∈ ℕ

Hamiltonian of an arbitrary circuit

=

HAMILTONIAN ???

M. H. Devoret, p. 351 in Quantum fluctuations (Les Houches 1995) G. Burkard et al., Phys. Rev. B 69, 064503 (2004)

Correct procedure described in :

G. Wendin and V. Shumeiko, cond-mat/0508729 M.H. Devoret, lectures at Collège de France (2008) accessible online

=


=

HAMILTONIAN ???

=

1) Identify the relevant independent circuit variables

2) Write the circuit Lagrangian

3) Determine the canonical conjugate variables and the Hamiltonian


=

=

1) Choose reference node (ground)

branch node

Identifying the relevant independent circuit variables


=

=


2) Choose « spanning tree » (no loop)



=

=



3) Define « tree branch fluxes »

φa

φc φd

φe φb

( ) ( ') 't

i t V t dtφ−∞

= ∫



=

=



3) Define « tree branch fluxes »

4) Define node fluxes

Φ4= φb+ φd

Φ5 Φ1

Φ3 Φ2


φa

φc φd

φe φb

( ) ( ') 't

i t V t dtφ−∞

= ∫

branches leadingto nn β

β

φ=Φ ∑


=

=

Φ4= φb+ φd

Φ5 Φ1

Φ3 Φ2

φa

φc φd

φe φb

5) What about « closure branches » ??

φf


=

=

Φ4= φb+ φd

Φ5 Φ1

Φ3 Φ2

φa

φc φd

φe φb

5) What about « closure branches » ??

φf

PHASE QUANTIZATION CONDITION (superconducting loop)

φa

φc

φb φd φa+φb+φc+φd=Φext Φext


=

=

6) Classical Lagrangian ( , ) ( ) ( )i i electrostatic i pot iL E EΦ Φ = Φ − Φ

taking into account constraints imposed by external biases (fluxes or charges)

Φ4= φb+ φd

Φ5 Φ1

Φ3 Φ2

φa

φc φd

φe φb φf


=

=

Conjugate variables : ii

LQ ∂=

∂Φ

Classical Hamiltonian ( , )i i i iH Q Q LΦ = Φ −∑

Quantum Hamiltonian ˆˆ( , )i iH QΦ With ˆˆ ,i iQ i Φ =

ˆ ˆ( , )i iH nθ ˆ ˆ,i in iθ = or with

ˆˆ / 2i in Q e=ˆ ˆ (2 / )i i eθ = Φ

Φ4= φb+ φd

Φ5 Φ1

Φ3 Φ2

φa

φc φd

φe φb φf

Different types of qubits

Cooper-pair boxes Phase qubits Flux qubits

Junctions sizes

.01 to 0.04 µm2 .04µm2 100µm2

NIST Santa Barbara

TU Delft MIT

Berkeley NEC

Saclay Chalmers

NEC Yale

ETH Zurich

Shape Of the

Potential Energy

-1 0 1

E p (a

.u)

θ/π (rad)-2 0 2

Ep

(a.u

)

γm/π

-2 0 2 4

Ep (a

.u)

θ (rad)I.2) Cooper-Pair Box

Vg

The Cooper-Pair Box

1 degree of freedom 1 knob

θ̂,N̂ = i

ˆ ˆˆ cos2gC JEH = ( -E N N ) - θ

charging energy

𝑁𝑁𝑔𝑔 = 𝐶𝐶𝑔𝑔𝑉𝑉𝑔𝑔/2𝑒𝑒

𝐸𝐸𝑐𝑐 = 2𝑒𝑒 2/2𝐶𝐶 gate charge

𝐸𝐸𝐽𝐽,𝐸𝐸𝑐𝑐

𝐶𝐶𝑔𝑔

Vg

( )2

0ˆ cosˆ

cosˆˆ ˆ

J2

gCH = ( -N ) -2

E E2L

Nϕ δδ

θΦ −

+

1θ̂ , ˆ1N = i

2 d° of freedom

1N

2N

2θ̂ , ˆ2N = i

θ1

θ2L

inductance

2 knobs

Φ

2 11 2

ˆ ˆθ -θ ˆ ˆ ˆ2

,θ̂= =N N -N

=

ior

1 21 2

ˆ ˆˆˆ ˆ ˆ=θ +θ =2

, N +NKδ

=

i

δθ

The split CPB

𝐶𝐶𝑔𝑔

small

𝐿𝐿 ≪ 𝜙𝜙02/𝐸𝐸𝐽𝐽

Vg

ˆ cos cosˆ ˆ2JgCH = (E N E-N ) -

2δ

θ

θ̂,N̂ = i1 d° of freedom

JCE ,E

N

2 knobs

0

δ=ϕΦ

tunable JE

The split CPB

Energy levels of the CPB

ˆ ( , ) ( )cosˆˆΦ Φ θC J2

g gH N = ( - EN ) -E N

Solve either in charge basis |N>

{ }( , ), ( , )k g k gE N NψΦ Φ

,k k NN

c Nψ = ∑

( )ˆg

N

J2C

EE N N N N+1 NH = ( -N ) -2

N N+1∈

+∑

... ......

...

...

...

...

2

C Jg

2

C g

2

g

J J

CJ

N=-1 N=0 N=1

... ... ...

0 ...

... ...

... 0

... ... ..

...

E (-1- N )

E (0 - N )

-E /2

-E /2 -

E (1- N

E /2

-E

.

/2 )

Diagonalize

I.2) Cooper-Pair Box

(𝑁𝑁 ∈ ℕ)

Energy levels of the CPB

ˆ ( , ) ( )cosˆˆΦ Φ θC J2

g gH N = ( - EN ) -E N

… or in phase basis |θ>

{ }( , ), ( , )k g k gE N NψΦ Φ

[ ]0, 2θ π∈( ) 2

0

( )k kdπ

ψ θψ θ θ= ∫

ˆ ( , ) ( ) cosθθ∂

∂Φ Φ2

g gC JH N = ( -N ) -E1Ei

Solve Mathieu equation

( ) cosθψ θ ψ θθ

ψ θ∂∂

ΦJk kgC k k2 ( ) (( -N ) - =E)1 E

i(E )


Two simple limits : (1) 𝐸𝐸𝐽𝐽 Φ ≪ 𝐸𝐸𝐶𝐶

0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

0 0.5 1 1.5 2 2.5 3

0

1

2EJ/Ec=0

N

0 ( )c N

N

1( )c N3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1 0

0

0.5

0 0

0.5

θ

θ

20 ( )ψ θ

21( )ψ θ

(charge regime)


0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

EJ/Ec=0.1

E0(Ng) E1(Ng)

E2(Ng)

3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1

0 ( )c N

1( )c N

0 0

0.52

0 ( )ψ θ

21( )ψ θ

0 0

0.5

Ng=0.01

(charge regime)

QUBIT

0 0

0.5


0 0.5 1 1.5 2 2.5 3

0

1

2

Ng

Ene

rgy

EJ/Ec=0.1

E0(Ng)

E1(Ng)

E2(Ng)

0 ( )c N

1( )c N

20 ( )ψ θ

21( )ψ θ

Ng=0.5

3 2 1 0 1 2 3

1

3 2 1 0 1 2 3

1

0 0

0.5

(charge regime)

QUBIT

0.0 0.2 0.4 0.6 0.8 1.01.00.5

0.00.51.01.52.0

From 𝐸𝐸𝐽𝐽 Φ ≪ 𝐸𝐸𝐶𝐶 to 𝐸𝐸𝐽𝐽 Φ ≫ 𝐸𝐸𝐶𝐶 E

nerg

y

Ng

EJ/Ec=0.5

0.0 0.2 0.4 0.6 0.8 1.02

1

0

1

2

3

Ng

EJ/Ec=2

0.0 0.2 0.4 0.6 0.8 1.0

4

2

0

2

Ene

rgy

Ng

EJ/Ec=5

0.0 0.2 0.4 0.6 0.8 1.08642

02

Ng

EJ/Ec=10

STILL A QUBIT !

Two simple limits : (2) 𝐸𝐸𝐽𝐽 Φ ≫ 𝐸𝐸𝐶𝐶


(phase regime)

0.0 0.2 0.4 0.6 0.8 1.08642

02

Ng

EJ/Ec=10

4 2 0 2 41.00.5

0.00.51.0

4 2 0 2 41.00.5

0.00.51.0

0 ( )c N

1( )c N

0 0

0.5

0 0

0.5

20 ( )ψ θ

21( )ψ θ

J. Koch et al., PRA (2008) Transmon qubit

Experimental spectrum of a transmon J. Schreier et al., PRB (2008)

Ng(t)=∆Ngcosωt

One-qubit gates

Φ

TRANSMON QUBIT

( )ˆ ˆ ˆ( ) cost θ2

C g JH = E N-N -E

ˆ ˆ ˆˆcos 2 gN ωθ ∆2C J CH = E N -E - E cos tN

transmon drive


01( )2 z

ω σΦ= − 01( )

2 zω σΦ

= −Two-level

approximation

ℏΩ𝑅𝑅 = 2𝐸𝐸𝐶𝐶 0 𝑁𝑁� 1 Δ𝑁𝑁𝑔𝑔


One-qubit gates

Φ

TRANSMON QUBIT

φ0

Rotation : X

|0>

|1>

X

Z

Y

ω=ω01

P(1)

0

1

F. Mallet et al., Nature Phys. (2009)


One-qubit gates

Φ

TRANSMON QUBIT

φ0 φ0+φ

Rotation : Xφ

|0>

|1>

X

Z

Y φ Xφ

X

ω=ω01


One-qubit gates

Φ+δΦ

TRANSMON QUBIT

φ0 φ0+φ

Rotation : Xφ

|0>

|1>

X

Z

Y φ Xφ

X

Z rotation

ω=ω01

All rotations on Bloch sphere

Fidelity ? >99.9%

R. Barends et al., Nature (2014)

Decoherence

Φ+dΦ(t)

Ng+dNg(t)

Noise in Hamiltonian parameters

DECOHERENCE

MAJOR OBSTACLE TO QUANTUM COMPUTING

I.3) Decoherence

1

0

01ω

Relaxation (Spontaneous emission)

Decoherence in superconducting qubits (Ithier et al., PRB 72, 134519, 2005)

1 21 1 , 01( )

2T D Sλ λ

π ω−⊥= Γ =

environmental density of modes at qubit frequency

Pure dephasing

( )i te ϕα β+0 1

1

0

01( )iω λ

t

λi(t)

1 12 2 2

T ϕ− Γ

= Γ = + Γ

2, (0)zD Sϕ λ λπΓ =

2( )( ) ti tf t e eϕϕ

−Γ≡ ≈

Low-frequency noise

01,zDλ

ωλ

∂=

∂

,1 0 1HDλ λ⊥

∂=

∂

I.3) Decoherence

Decoherence

Φ+dΦ(t)

Ng+dNg(t)


DECOHERENCE

Origin of the noise ???

I.3) Decoherence

Decoherence

Φ+dΦ(t)

Ng+dNg(t)


DECOHERENCE


R

R

1) ELECTROMAGNETIC

Low-frequency : Johnson-Nyquist due to thermal noise

High-frequency : spontaneous emission (quantum noise)

≈ Under control

I.3) Decoherence

Decoherence

Φ+dΦ(t)

Ng+dNg(t)


DECOHERENCE


R

R

2) MICROSCOPIC

e- Spin flips

I.3) Decoherence

Low-frequency noise

Charge noise Flux noise

𝑆𝑆𝑁𝑁𝑔𝑔(𝜔𝜔) ≈ 10−3 2/𝜔𝜔 𝑆𝑆Φ(𝜔𝜔) ≈ 10−6Φ02/𝜔𝜔

I.3) Decoherence

Decoherence

Φ+dΦ(t)

Ng+dNg(t)


DECOHERENCE


R

R

2) MICROSCOPIC

e- Spin flips

Low-frequency noise

Charge noise Flux noise

CPB in charge regime

Transmon

𝑆𝑆𝑁𝑁𝑔𝑔(𝜔𝜔) ≈ 10−3 2/𝜔𝜔 𝑆𝑆Φ(𝜔𝜔) ≈ 10−6Φ02/𝜔𝜔

𝑇𝑇2 ≈ 10 − 100ns

𝑇𝑇2 ≈ 10 − 100ms

𝑇𝑇2 ≈ 1 − 100𝜇𝜇𝜇𝜇 𝑇𝑇2 ≈ 1 − 100𝜇𝜇𝜇𝜇

Decoherence

Φ+dΦ(t)

Ng+dNg(t)


DECOHERENCE


R

R

2) MICROSCOPIC

e- Spin flips

I.3) Decoherence

High-frequency « noise » (or equivalently « losses ») responsible for finite T1

• Imperfect dielectrics, causing microwave losses • Magnetic vortices trapped in the superconducting thin films • Unpaired electrons (« Quasiparticles ») in the superconductors

Coherence times : measurements

ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET

1999 (Nakamura et al., Nature): 𝑇𝑇1 ≈ 𝑇𝑇2 ≈ 1ns

2002 : (« Quantronium experiment ») 𝑇𝑇1 = 2𝜇𝜇𝜇𝜇, 𝑇𝑇2∗=500ns

2013 (Barends et al., PRL) 𝑇𝑇1 = 30 − 100𝜇𝜇𝜇𝜇, 𝑇𝑇2 = 1 − 30𝜇𝜇𝜇𝜇

I.3) Decoherence

SiO2

PMMA-MAA PMMA

e-

O2 Al/Al2O3/Al junctions

1) e-beam patterning 2) development 3) first evaporation 4) oxidation 5) second evap. 6) lift-off 7) electrical test

small junctions Multi angle shadow evaporation

Fabrication techniques small junctions e-beam lithography

I.3) Decoherence

gate 160 x160 nm

QUANTRONIUM (Saclay group)

FLUX-QUBIT (Delft group)

I.3) Decoherence

I.3) Decoherence

40µm

2µm

TRANSMON QUBIT (Saclay group)

END OF FIRST LECTURE

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