Electrical quantum engineering - Université Paris Saclay · Electrical quantum engineering . ... 0...
Transcript of 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
0 100 200 300 4000.0
0.2
0.4
0.6
0.8
1.0
11
00
01
switc
hing
pro
babi
lity
swap duration (ns)
10
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
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 for quantum information and quantum optics on a chip
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 for quantum information and quantum optics on a chip
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 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
Josephson AC relation : 0d QVdt Cθϕ= =
Josephson DC relation : sinCI I 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
−∞= ∫
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= − = −
Josephson AC relation : 0d QVdt Cθϕ= =
Josephson DC relation : sinCI I 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 of an arbitrary circuit
=
HAMILTONIAN ???
=
1) Identify the relevant independent circuit variables
2) Write the circuit Lagrangian
3) Determine the canonical conjugate variables and the Hamiltonian
Hamiltonian of an arbitrary circuit
=
=
1) Choose reference node (ground)
branch node
Identifying the relevant independent circuit variables
Hamiltonian of an arbitrary circuit
=
=
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
Identifying the relevant independent circuit variables
Hamiltonian of an arbitrary circuit
=
=
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
3) Define « tree branch fluxes »
φa
φc φd
φe φb
( ) ( ') 't
i t V t dtφ−∞
= ∫
Identifying the relevant independent circuit variables
Hamiltonian of an arbitrary circuit
=
=
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
3) Define « tree branch fluxes »
4) Define node fluxes
Φ4= φb+ φd
Φ5 Φ1
Φ3 Φ2
Identifying the relevant independent circuit variables
φa
φc φd
φe φb
( ) ( ') 't
i t V t dtφ−∞
= ∫
branches leadingto nn β
β
φ=Φ ∑
Hamiltonian of an arbitrary circuit
=
=
Φ4= φb+ φd
Φ5 Φ1
Φ3 Φ2
φa
φc φd
φe φb
5) What about « closure branches » ??
φf
Hamiltonian of an arbitrary circuit
=
=
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
Hamiltonian of an arbitrary circuit
=
=
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
Hamiltonian of an arbitrary circuit
=
=
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
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 )
I.2) Cooper-Pair Box
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)
Two simple limits : (1) 𝐸𝐸𝐽𝐽 Φ ≪ 𝐸𝐸𝐶𝐶
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
Two simple limits : (1) 𝐸𝐸𝐽𝐽 Φ ≪ 𝐸𝐸𝐶𝐶
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
I.2) Cooper-Pair Box
(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
I.2) Cooper-Pair Box
01( )2 z
ω σΦ= − 01( )
2 zω σΦ
= −Two-level
approximation
ℏΩ𝑅𝑅 = 2𝐸𝐸𝐶𝐶 0 𝑁𝑁� 1 Δ𝑁𝑁𝑔𝑔
I.2) Cooper-Pair Box
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)
I.2) Cooper-Pair Box
One-qubit gates
Φ
TRANSMON QUBIT
φ0 φ0+φ
Rotation : Xφ
|0>
|1>
X
Z
Y φ Xφ
X
ω=ω01
I.2) Cooper-Pair Box
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)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
I.3) Decoherence
Decoherence
Φ+dΦ(t)
Ng+dNg(t)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
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)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
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)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
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)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
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