INDEPENDENT ELECTRON INTERFERENCES. DIRECTLY ... ARBITRARILY LARGE ATOM-FIELD COUPLING. ARTIFICIAL ATOMS PRONE TO VARIAT
INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 1 QUAN
ELEC
UM – MECHANICAL
RONICS LAB
Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN
N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT)
N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET
Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK
Rutgers Dec. 09
MESOSCOPIC ELECTRONS vs PHOTONS INDEPENDENT ELECTRON INTERFERENCES DIRECTLY REVEAL QUANTUM MECHANICS Schrödinger's equation
Example: Sharvin-Sharvin & A.B. effects Φ
2 ⎡ 2⎛ ⎤ eA ⎞ ∂ ∇ − + Ψ = Ψ i eU i ⎢ ⎜ ⎥ ⎟ ∂t ⎠ ⎢⎣ m ⎝ ⎥⎦
Interactions provide further quantum effects
INDEPENDENT PHOTON INTERFERENCES DO NOT DIRECTLY REVEAL QUANTUM MECHANICS Maxwell's equations
(
)
(
∂ c∇ × E + icB = i E + icB ∂t Planck's constant drops out! Need ph-ph interactions....
)
QUANTUM OPTICS
QUANTUM RF CIRCUITS
FIBERS, BEAMS
TRANSM. LINES, WIRES
BEAM-SPLITTERS
COUPLERS
MIRRORS
CAPACITORS
LASERS
~
GENERATORS
PHOTODETECTORS
AMPLIFIERS
ATOMS
JOSEPHSON JUNCTIONS
ADVANTAGES OF CIRCUITS: - PARALLEL FABRICATION METHODS - LEGO BLOCK CONSTRUCTION OF HAMILTONIAN - ARBITRARILY LARGE ATOM-FIELD COUPLING
DRAWBACKS OF CIRCUITS:
ARTIFICIAL ATOMS PRONE TO VARIATIONS
SELECTED BIBLIOGRAPHY ON QUANTUM CIRCUITS Michel Devoret, in "Quantum Fluctuations", S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385 Michel Devoret & John Martinis, Quant. Inf. Proc., 3, 351-380 (2004) Robert J. Schoelkopf & Steve M. Girvin, Nature 451, 664-669 (2008)
John Clarke & Frank K. Wilhelm, Nature 453, 1031-1042 (2008)
OUTLINE TODAY:
TOMORROW:
PRESENT BASIC CIRCUIT ELEMENTS
COMPARE QUBIT CIRCUITS
SUPERCONDUCTING CIRCUIT ELEMENTS
CAPACITANCE
INDUCTANCE
RESISTANCE
SUPERCONDUCTING CIRCUIT ELEMENTS e2 EC = 2C
CAPACITANCE
EL =
2 2
4e L
EJ =
INDUCTANCE
RESISTANCE
2
4e 2 L J
TRANSMISSION LINES ARE JUST L's AND C's
L
L C
L C
L C
L C
LIKEWISE, EVERY LINEAR PASSIVE RECIPROCAL RF COMPONENT IS BUILT FROM INDUCTANCES AND CAPACITANCES RESISTORS CAN BE THOUGHT OF AS INFINITE TRANSMISSION LINES (NYQUIST, CALDEIRA & LEGGETT)
DYNAMICAL VARIABLES OF CIRCUIT CIRCUIT : ARBITRARY NETWORK OF ELECTRICAL ELEMENTS node n
element
Vnp
branch np
loop
Inp node p
TWO DYNAMICAL VARIABLES CHARACTERIZE THE STATE OF EACH DIPOLE ELEMENT AT EVERY INSTANT: Voltage across the element:
Vnp ( t ) = ∫ E ⋅ d p
n
Current through the element:
I np ( t ) = ∫∫ j ⋅ dσ np
Signals: any linear combination of these variables
CONSTITUTIVE RELATIONS OF CIRCUIT ELEMENTS linear inductance:
V = L dI/dt non-linear inductance:
linear capacitance:
I = C dV/dt
E = LI2/2= Φ2/2L E = f(Φ) f non-quadratic
E = CV2/2= Q2/2L
Inductances and capacitances are "bottles" for electric and magnetic fields
KIRCHHOFF’S LAWS b
c a b
b d
∑ λ Vλ = 0
branches around loop
c
d a b
∑ν
branches tied to node
Iν = 0
CONSTITUTIVE RELATIONS + KIRCHHOFF'S LAWS
MAXWELL'S EQUATIONS ON A NETWORK
QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit
branch β
Vβ
Need to take branch flux and branch charge as basic variables:
φβ ( t ) = ∫ Vβ ( t ')dt ' t
−∞
Qβ ( t ) = ∫ I β ( t ')dt ' t
−∞
For every branch β in the circuit:
⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
1) Choose a reference electrode (ground)
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop)
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop)
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop)
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
φb
φc φa
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) 3) Select tree branch fluxes (closure branches left out)
φd
φg
φf φe
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes
Φ3
Φ7
Φ1 Φ6
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) 3) Select tree branch fluxes (closure branches left out) 4) Node flux is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes)
Φ4
Φ5
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes
Φ3
φh
Φ1 φf
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) 3) Select tree branch fluxes (closure branches left out) 4) Node flux is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes)
φd
Φ4
example:
Φ7 φg Φ6 φe
Φ5
Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst
FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes
Φ3
φh
Φ1 φf
1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop) 3) Select tree branch fluxes (closure branches left out) 4) Node flux is sum of branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes)
φd
φg Φ6 φe
Φ4
example:
Φ7
Φ5
Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst
Φn =
∑
tree branches β leading to n
φβ
φγ = Φ n (γ ) − Φ n (γ ) + cst +
−
TWO METHODS FOR DEFINING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes
Method of loops
Defines loop charges
HAMILTONIAN OF CIRCUIT mechanical analog world
electrical world
φ
I
X +Q
-Q
V
M
Q (φ − φ0 ) H= + 2C 2L 2
φ=
∂H Q = ∂Q C
∂H − (φ − φ0 ) Q=− = L ∂φ
ωr =
f
2
1 LC
k
k ( X − X0 ) P H= + 2M 2 2
∂H P = X= ∂P M ∂H P=− = −k ( X − X 0 ) ∂X
2
ωr =
k M
A SUPERCONDUCTING CIRCUIT BEHAVING LIKE AN ATOM?
SIMPLEST EXAMPLE: SUPERCONDUCTING
MICROFABRICATION
LC OSCILLATOR
CIRCUIT
L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz
ELECTRONIC FLUID SLOSHES BACK AND FORTH FROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN BEHAVES AS A SINGLE CHARGE CARRIER
DEGREE OF FREEDOM IN ATOM vs CIRCUIT Superconducting LC oscillator
Example of Rydberg atom
L
velocity of electron → force on electron →
voltage across capacitor current through inductor
C
LC CIRCUIT AS A QUANTUM HARMONIC OSCILLATOR E
φ +Q -Q
hω r
φ
φˆ Qˆ φˆ Qˆ † aˆ = ; aˆ = +i −i φZPF QZPF φZPF QZPF φZPF = 2 ωr L
(
⎡⎣ aˆ , aˆ † ⎤⎦ = 1 trapped photons!
QZPF = 2 ωr C Hˆ = ωr aˆ † aˆ + 1
annihilation and creation operators
2
)
ALL TRANSITIONS BETWEEN QUANTUM LEVELS ARE DEGENERATE IN PURELY LINEAR CIRCUITS! E
φ
hω r
φ CANNOT STEER THE SYSTEM TO AN ARBITRARY STATE IF PERFECTLY LINEAR
NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy
Position coordinate
JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S
1nm
I S
Ι
Ι
LJ
superconductorinsulatorsuperconductor tunnel junction
Ι = φ / LJ
CJ
LJ =
φ0 I0
I0
φ
φ = ∫−∞ V ( t ')dt ' t
I = I 0 sin (φ / φ0 )
φ0 =
2e
JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S
1nm
I S
superconductorinsulatorsuperconductor tunnel junction
LJ
CJ
LJ =
U = − EJ cos (φ / φ0 )
Ι
φ = ∫−∞ V ( t ')dt ' t
φ02 EJ
φ
2 EJ
bare Josephson potential
φ0 =
2e
TRANSMISSION LINE AS 1D BOSON FIELD L
n-1
C
L Vn −1 I n −1 C
n
L
L
n+1
Vn I n C
n+2
L
Vn +1 I n +1 C
a Dynamical equations:
d Vn − Vn +1 = L I n dt d I n −1 − I n = C Vn dt
Continuum limit:
Field equations:
Vn +1 − Vn ∂V → ∂x a I n +1 − I n ∂I → ∂x a C L →C ; → L a a
∂V = −L ∂x ∂I = −C ∂x
∂I ∂t ∂V ∂t
CHARGE AND FLUX IN CONTINUUM LIMIT Φ ( x) Lδx
Π ( x)δ x
Φ ( x + δ x) Lδx
C δx
Lδx
C δx
Π ( x + δ x)δ x
δx ⎡⎣Φ ( x ) , Π ( x ) δ x ⎤⎦ = i ⎡⎣Φ ( x ) , Π ( x + δ x ) δ x ⎤⎦ = 0
δx→0
ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣
Hamiltonian : energy density as a function of field and conjugate momentum: 2 2⎫ +∞ ⎧ 1 ˆ 1 ˆ ˆ ⎡ ⎤ ⎡ ⎤ H = ∫ dx ⎨ Π ( x )⎦ + ∇Φ ( x ) ⎦ ⎬ ⎣ ⎣ −∞ 2L ⎩ 2C ⎭
COMPATIBILITY WITH STANDARD QED For simplest geometry, consider stripline waveguide:
w
y
h
x
z B ( x1 , y1 , z1 )
E ( x2 , y2 , z2 )
Flux between strips: x
yb + h
−∞
yb
ˆ ( x ) = 1 dx Φ 1 ∫ ∫
Strip charge per unit length:
dy Bˆ z ( x, y, z1 )
ˆ (x ) = ε Π 2 0∫
z f +w
zf
dz Eˆ y ( x2 , y2 , z )
Commutation relations between field operators in standard QED:
i ∂ ⎡ Bˆ z ( x1 , y1 , z1 ) , Eˆ y ( x2 , y2 , z2 ) ⎤ = ⎣ ⎦ ε ∂x δ ( x1 − x2 ) δ ( y1 − y2 ) δ ( z1 − z2 ) 0 1 ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣
OK!
END OF 1ST LECTURE
INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 2 QUAN
ELEC
UM – MECHANICAL
RONICS LAB
Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN
N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT)
N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET
Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK
Rutgers Dec. 09
OUTLINE YESTERDAY:
TODAY:
PRESENT BASIC CIRCUIT ELEMENTS
COMPARE QUBIT CIRCUITS
QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit
branch β
Branch flux and branch charge are primary variables:
φβ ( t ) = ∫ Vβ ( t ')dt ' t
−∞
Vβ
Qβ ( t ) = ∫ I β ( t ')dt ' t
−∞
For every branch β in the circuit:
⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦
IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy
2Δ rate
before SIS TUNNEL JUNCTION
after
IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy
2Δ rate
before SIS TUNNEL JUNCTION
after
2e CP tunneling quasiparticle energy
matrix element
DYNAMICS OF JOSEPHSON ELEMENT FROM CHARGE POINT OF VIEW (IN ABSENCE OF Q.P. TUNNELING) integer Hopping
N0 − 1 N0 N0 + 1
Q =N 2e
1 h / e2 M = EJ = Gt Δ 2 16 Charge states:
Nˆ N = N N
Josephson tunneling hamiltonian in charge basis:
EJ ˆ HJ = − 2
∑( N N
N +1 + N +1 N
)
FROM NUMBER TO "PHASE" REPRESENTATION
ϕˆ = 2eφˆ /
⎡ϕˆ , Nˆ ⎤ = i ⎣ ⎦
eiϕˆ Nˆ e − iϕˆ = Nˆ − 1
EJ ˆ HJ = − N N +1 + N +1 N ( ∑ 2 N EJ iϕˆ (from expression of − iϕˆ = − ( e + e ) translation operator) 2 = − EJ cos ϕˆ
(
= − EJ cos φˆ / φ0 Josephson's
ϕ
)
) ϕ
Φ0 φ0 = = 2e 2π
runs on a line but tunnel hamiltonian is periodic
ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT
REST OF CIRCUIT
Z (ω ) CJ
EJ
U(t)
2
φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t
1
N.B. The electric field E encompasses here all contributions of the force on the electrons doing work, including those usually called chemical potential effects.
ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT t
Qext = ∫ Idτ −∞
Q = 2eN REST OF CIRCUIT
EJ
QC
Z (ω )
CJ
U(t)
2
φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t
1
Equation of motion:
∂ CJ φ + ∂φ
⎡ ⎛ φ ⎞⎤ ⎢ − EJ cos ⎜ ⎟ ⎥ = I φ , φ ,.... ⎝ φ0 ⎠ ⎦ ⎣
(
)
Can be obtained in general from a Lagrangian:
d ∂L ∂L − =0 dt ∂φ ∂φ
CJ 2 φ L = L J +L ext = φ + EJ cos + L ext φ , φ ,... 2 φ0
(
)
TWO CHARACTERISTIC ENERGIES OF ENVIRONMENT Total environment admittance:
Effective shunt capacitance of junction:
Effective shunt inductance of junction
Ytot (ω ) = iCJ ω + Z −1 (ω )
Im ⎡⎣Ytot (ω ) ⎤⎦
CΣ = lim
ω
ω →0
Leff
⎧⎪ ⎡ ⎤ ⎫⎪ 1 = lim ⎨Im ⎢ ⎥⎬ ω →0 ⎩⎪ ⎣ ωYtot (ω ) ⎦ ⎭⎪ (electron charge appears here instead of Cooper pair charge for convenience in some formulas)
2
Coulomb charging energy
Inductive energy
e EC = 2CΣ
EL
h / 2e ) ( = Leff
2 (form chosen for easy comparison with Josephson energy)
TWO BASIC SUPERCONDUCTING "ATOMS" flux
charge superconducting island
LJ
superconducting loop
ϕ∈
LJ
N∈
CJ
Φext
L > LJ
Cg
CJ U
I "HYSTERETIC RF SQUID"
E J > EL
EC
Δϕ 2π
Friedman et al. (2000)
"COOPER PAIR BOX"
1
EL = 0; EJ
EC
ΔN < 1
Bouchiat et al. (1998), Nakamura, Pashkin, Tsai (1999)
PHASE POTENTIALS OF SQUID AND BOX E J > EL
EL = 0; EJ
EC
EC
E SQUID
BOX
max. curvature
= EL − E J 2EJ −1
2eΦ ext /
0
ϕ 2π
+1
+2
−
ϕ 2π
1 2
e
+
1 2
iπ C gU / e
RICH LEVEL STRUCTURE, BUT EXTREME SENSITIVITY TO NOISE 09-III-10
PHASE-CHARGE DUALITY E J > EL
EL = 0; EJ
EC
EC
E SQUID
BOX
8 EC N g −
2π EL ϕext − π
−1
0
ϕ 2π
+1
+2
1 2
−1
+1
0
N
+2
CHARGE QUBIT FAMILY
THE SINGLE COOPER PAIR BOX ARTIFICIAL ATOM EJ
Bouchiat et al. (1998) Nakamura, Tsai and Pashkin (1999)
0.5EC U
Φ U
Q=2Ne Φ Cg
EJ
4 EC
QUANTRONIUM Vion et al. (2001)
U
Φ
Q=2Ne Φ
U
Cg
TRANSMON COOPER PAIR BOX J. Koch et al. (2007) A. Houck et al. (2007) J. Schreier et al. (2007)
U
Φ
EJ
50 EC
U
Q=2Ne
Φ Cg
Cottet et al. (2002)
Koch et al. (2007)
ANHARMONICITY vs OFFSET CHARGE INSENSITIVITY IN COOPER PAIR BOX
gap EJ
"C.P. BOX"
"QUANTRONIUM"
"TRANSMON"
FLUX QUBIT FAMILY
EXAMPLES OF QUANTUM CIRCUITS BELONGING TO THE RF-SQUID TYPE Readout Readout
Friedman, Patel, Chen, Tolpygo and J. E. Lukens, Nature 406, 43 (2000).
Chiorescu, Nakamura, Harmans & Mooij, Science 299, 1869 (2003).
"flux" qubit
"phase" qubit
Readout
Steffen et al., Phys. Rev. Lett. 97, 050502 (2006)
FLUXONS PLAY WITH FLUX QUBITS A ROLE SIMILAR TO WHAT COOPER PAIRS PLAY WITH CHARGE QUBITS Inductive energy
-1
0
1
Φ Φ0 ⎛ EJ ⎞ ~ exp ⎜⎜ −α ⎟⎟ E C ⎠ ⎝
OUR RECENT CHILD IN FLUX QUBIT FAMILY
NOISE IN THE TWO BASIC QUBITS
superconducting loop
flux
charge
LJ
LJ ϕ
ΦN Φext flux
superconducting island
CJ
L ∼ LJ
noise!
N
Cg
CJ
UN
U
charge noise!
I "RF SQUID"
Δϕ 2π
1
"COOPER PAIR BOX"
ΔN < 1
FLUXONIUM IDEA: SHUNT A COOPER PAIR BOX AT DC, LEAVE IT UNSHUNTED AT RF L
LJ LJ
Cg
CJ UN U
Manucharyan. et al., Science, 326, 113; arxiv 0906.0831 Koch. et al., Phys. Rev. Lett, to appear; arxiv 0902.2980 Manucharyan. et al., submitted to Nature, arxiv 0910.3039
PRACTICAL IMPLEMENTATION
write / d a e r
Small junction:
EJ/EC = 3.6 Array junctions (N =43):
EJA/ECA = 28 Readout f0 = 8 GHz; Q=400 Every island is shunted by at least one large junction: array performs as inductor
CHIP AND MODEL “Atom” EC = ½ e2/(CJ+Cc)=2.5 GHz EJ = ½ (Φ0/2π)2/LJ=8.9GHz EL = ½ (Φ0/2π)2/L=0.52GHz “Cavity” ωR=(LRCR)-1/2/2π = 8.2GHz ZR=(LR/CR)1/2 = 82Ω Coupling constant g = ωR(ZR/2RQ)1/2 /(1+CJ/Cc) = 135MHz (RQ=1 kΩ) CJ/Cc=11
PARAMETERS OF THE ARRAY EJA=22.5GHz
ECA=0.8GHz
Cg~CJ/2000
N =43 72 hours w/o jumps or drifts! NO OFFSET CHARGE
THY vs EXP CONFIRMS SINGLE COOPER PAIR REGIME
V.Manucharyan, J.Koch, L.Glazman, M.Devoret, Science, 326, 113; arxiv 0906.0831
LOCATING FLUXONIUM ON THE SUPERCONDUCTING QUBIT MAP
A QUBIT OPERATING FOR ALL VALUES OF FLUX ~350MHz Inductive energy
9GHz
Potential energy
-1
1
0 Φext =1/4 Φ0
Φext =1/2 Φ0
Φ Φ0 Φext = 0
~2EJ
2π
2π2EL
2π
4π
ϕ
Φ0 TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX Φ ≈ 2
A 370MHz ATOM MEASURED THROUGH A 8.2GHz CAVITY?
atom transition 0-1: store/manipulate atom transition 1-2: couple to readout
g12 = g ~ 100MHz ω12 - ωR ~ 1GHz χ= g122/(ω12 - ωR) ~10MHz At the same time ω01