josephson qubit circuits and their readout · 2020-06-17 · qbit space = hilbert space of 0 or 1...
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Michel DEVORET, Applied Physics and Physics, Yale University
JOSEPHSON QUBIT CIRCUITSAND THEIR READOUT
2009 APS March Meeting Pittsburgh, PA
W.M. KECK
Final version of this presentation available at http://qulab.eng.yale.edu/archives.htm
Sponsors:
• OPTICAL PHOTONS
• NUCLEAR SPINS
• IONS
• ATOMS
• MOLECULES
• QUANTUM DOTS
• SUPERCONDUCTING JOSEPHSONTUNNEL CIRCUITS
QUANTUM INFORMATION PROCESSING
micro
MACRO
couplingwith envt.
# of atoms
10µm
50µm
from A. O. Niskanen et al. Science 316, 723 (2007) courtesy of J. Martinis, 2009 from Metcalfe et al., 2007
103 106 109 1012 1015 1018 1020100
106 103 100 10-3 10-6 10-9 10-12109
IR UV RXμW γRF /
ELECTROMAGNETIC SPECTRUM
VIDEOAUDIO
ELECTRICITY OPTICS
f (Hz)
λ (m)
10 GHz ~ 0.5K
1 bit = RF signal with 0/1 photon?
MICROFABRICATION L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz
SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT
HOW CAN A SUPERCONDUCTING CIRCUITBEHAVE LIKE AN ATOM?
ELECTRONIC FLUID SLOSHES BACK AND FORTHFROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN,
BEHAVES AS A SINGLE CHARGE CARRIER
Example of H atom with largeprincipal quantum number
SuperconductingLC oscillator
L C
velocity of electron → voltage across capacitorforce on electron → current through inductor
DEGREE OF FREEDOM IN ATOM vs CIRCUIT
+Qφ
-Q
FLUX AND CHARGE DO NOT COMMUTE
V
I
ˆ ˆ, iQφ⎡ ⎤⎣ =⎦
LIφ = Q VC=
+Qφ
-Q
φ
E
LC CIRCUIT AS QUANTUMHARMONIC OSCILLATOR
rωh
( )†
†
ˆ 1ˆ ˆ 2ˆ ˆ ˆ ˆ
ˆ ˆ;
2
2
r
r r r r
r r
r r
H a a
Q Qa i a iQ Q
L
Q C
ω
φ φφ φ
φ ω
ω
= +
= + = −
=
=
annihilation and creation operatorsfor mesoscopic excitation of circuit
φ
φ
WAVEFUNCTIONS OF LC CIRCUIT
rωhI
φ
E
Ψ(φ)Ψ0
0Ψ1
In every energy eigenstate,(photon state)
current flows in opposite directions simultaneously!
2 rφ
φ
φ
EFFECT OF DAMPING
important: as littledissipation as possible dissipation broadens energy levels
E
112 2n r
r
iE n
RC
ω
ω
⎡ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=
φ
CAN PLACE CIRCUIT IN ITS GROUND STATE
Br k Tω10 GHz 25mK
rωh
E
residual dissipationprovides
reset of circuit
φ
φ
E
PB: ALL TRANSITIONS ARE DEGENERATE!
CANNOT STEER THE SYSTEM TO AN ARBITRARY STATEIF PERFECTLY LINEAR
rωh
Potential energy
Position coordinate
NEED NON-LINEARITY TO FULLYREVEAL QUANTUM MECHANICS
JOSEPHSON TUNNEL JUNCTIONPROVIDES A NON-LINEAR INDUCTOR
WITH NO DISSIPATION
1nm SI
S
superconductor-insulator-
superconductortunnel junction
φ
ΙΙ = φ / LJ
( )0 0sin /I I φ φ=
CJLJ
Ι
( )' 't
V t dtφ−∞
= ∫
0 2eφ =
20 0
0J
J
LE Iφ φ
= =
JOSEPHSON TUNNEL JUNCTIONPROVIDES A NON-LINEAR INDUCTOR
WITH NO DISSIPATION
1nm SI
S
superconductor-insulator-
superconductortunnel junction
φ
( )0cos /JU E φ φ= −
CJLJ
Ι
( )' 't
V t dtφ−∞
= ∫
0 2eφ =
20 0
0J
J
LE Iφ φ
= =
bare Josephson potential
0
2 2ˆ
ˆ2ˆ 1
41
e
er
pm
AH eπε
⎛ ⎞= − −⎜ ⎟⎜ ⎟
⎝ ⎠
RESTOF
CIRCUIT(CONTROL
& READOUT)
ˆextq
THE Hamiltonian:(we mean it!)
( )21 2coˆ s2 ext
JJJ
CH EQ q eφ
= − −
COUPLING PARAMETERSOF THE JOSEPHSON "ATOM"
Comparable with lowest ordermodel for hydrogen atom
Josephson tunnel junctioncontribution to total circuit
Hamiltonian
TWO ENERGY SCALESTwo dimensionlessvariables:
ˆˆ
ˆˆ
2
2QN
e
e
φϕ =
=
ˆˆ, iNϕ⎡ ⎤ =⎣ ⎦
( )2
co8 ˆs2
ext
JCJ
NE EH
Nϕ
−= −
2
2CJ
E eC
=
2x
exte tqNe
=
Hamiltonian becomes :
Coulomb charging energy for 1e
18JE = ΔNT
Josephson energy
gap
# condion channels
barrier transpcy
reduced offset charge
valid foropaque barrier
HARMONIC APPROXIMATION
( )2
co8 ˆs2
ext
JCJ
NE EH
Nϕ
−= −
( )22
,2ˆ
82
e
CJ h J
xtNE
NEH ϕ−
= +
8 CP
JE Eω =
Josephson"plasma" frequency:
Spectrum independent of DC value of Next
( )28
2J
J
CZe
EE
=
JosephsonRF impedance:
SOME JOSEPHSONTUNNEL JUNCTIONS
IN REAL LIFE
100nm
EJ ~ 50K
EJ ~ 0.5K
ωp ~ 30-40GHz
credit L. Frunzio and D. Schuster
credit I. Siddiqi and F.Pierre
RF CONTROL & BIAS vs SENSITIVITY TO NOISE
"charge" qubit "flux" qubit
+_
"phase" qubit
off 0 / 2Φ < Φ
0off 2
ΦΦ =
0off 2
ΦΦ >
"phase"
"flux"
"charge"
ΔΦoff ΔEJΔQoffflavors
noises
- +-
charged impurityunstable tunnel channel
loose elec. dipoles
see R. McDermot's tutorial
island
a.k.a. Cooper Pair Box
Devoret and Martinis, Quant. Inf. Proc. 2004
THE MONSTERS WE FEAR:
freak electron spin
EFFECTIVE POTENTIALOF 3 MAIN BIAS SCHEMES
"phase" bias
"flux" bias
"charge" bias
UC Berkeley, NIST, UCSB,U. Maryland, I. Neel Grenoble...
TU Delft, NEC, NTT, IBM,MIT, UC Berkeley, SUNY, IPHT Jena ....
CEA Saclay, NEC, YaleChalmers, JPL, ...
22
eh
ϕπ
φ=
2ehφ
2ehφ
12
− 12
+
e giN
see also proposals fortopologically protected
qubits, for exampleFeigelman et al. PRL 92,
098301 (2004)
HOW DO WE FIND THE HAMILTONIANOF AN ARBITRARY CIRCUIT?
Yurke B. and Denker J.S., Phys. Rev. A 29, 1419 (1984)
Devoret M. H. in "Quantum Fluctuations", S. Reynaud, E. Giacobino,J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385
G. Burkard, R. H. Koch, and D. P. DiVincenzo, Phys. Rev. B 69, 064503 (2004)
L CCJLJ
Cc
Example: 2 Josephson junctions capacitively coupled to 1 resonator mode
C'JL'J
Cc
U
Φ U
ΦgCisland
COOPER PAIR "BOX"
Bouchiat PhD Thesis 97, et al. Physica Scripta '98Nakamura, Pashkin & Tsai, Nature '99
qbit space = Hilbert space of 0 or 1quanta in this non-linear oscillator
Φ controls EJ, CgU/2e controls Ng
Review by Blais et al., Phys. Rev. A 75, 032329 (2007)
SCHEMATIC OF COOPER PAIR BOXESIN A MICROWAVE CAVITY
TWO-QUBIT QUANTUM PROCESSORslide courtesy of
L. DiCarlo & Rob Schoelkopf
V17.6 Thu. March 19
see also 1 qubit and 2 cavities: B. Johnson et al. V17.4
small junction
array junctions
50 Ω port 1
50 Ω port 2resonator
"FLUXONIUM QUBIT"V. Manucharyan et al. Q17.5 Wednesday
A FEW USEFUL IDEAS FOR CIRCUITHAMILTONIANS.....
BRANCH VARIABLES
Vβ
Iβ
restof
circuit
banch βIntroduce branch flux and charge
position variable: φ Xmomentum variable: Q Pgeneralized force : Ι fgeneralized velocity: V V
( ) ( )
( ) ( )
' '
' '
t
tQ t
t dt
t dI
V
t
t
ββ
ββφ−∞
−∞
=
=
∫∫
Vβ
Iβ
,ˆ ˆ iQβ βφ⎡ ⎤⎦ =⎣
For every branch β in the circuit:
restof
circuit
banch βIntroduce branch flux and charge
( ) ( )
( ) ( )
' '
' '
t
tQ t
t dt
t dI
V
t
t
ββ
ββφ−∞
−∞
=
=
∫∫
BRANCH VARIABLES
PROBLEM: NOT ALL BRANCHVARIABLES ARE INDEPENDENT
b bd
branchesaround loop
0Vλ
λ =∑branchestied to node
0Iν
ν =∑
KIRCHHOFF'S LAWS:
b
c
b
a
d
a
c
IMPOSE CONSTRAINTS ON BRANCH VARIABLES
TWO METHODS FOR DEFINING A COMPLETESET OF INDEPENDENT VARIABLES
Method of nodes
Method of loops
08-II-11a
Defines loop charges
Defines node fluxes
Method of nodes
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)
Φ1
Φ2 Φ3 Φ7
Φ5
φg
φe
Φ 7 = φd + φe + φgexample:
4) Node flux Φ n is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)
φh
φh = Φ 7 - Φ 6 + cst
EXAMPLE OF A COMPLETESET OF INDEPENDENT VARIABLES
φd
φf
( ) ( ) cstn nγγ γφ+ −
Φ Φ= − +
tree branchesleading to
n
nβ
βφΦ = ∑
Φ4
Φ6
INDUCTIVE vs CAPACITIVE ELEMENTS
( )0
' ' 't
E I Vdt I dφ
φ φ−∞
= ⋅ =∫ ∫I
φ
Inductance : current I function only of flux φ
Electrical equivalent of spring: φ ↔ X ; I ↔ f
( )0
' ' 't Q
E V Idt V Q dQ−∞
= ⋅ =∫ ∫Q
V
Capacitance : voltage V function only of charge Q
Electrical equivalent of mass: Q ↔ P ; V ↔ V
NODE CHARGES
The conjugate coordinates of node fluxesare node charges: they are the sum of all the charges
going into capacitances linked to this node.
This can be demonstrated by writing the Lagrangianof the circuit from its dynamical equations
and performing a Legendre transform to obtain the Hamiltonian
Dynamicalequations
Lagrangian conjugatecoordinates Hamiltonian
capacitive branchesexiting from
n
n
Q Qβ= ∑ n
( ), ,L x x t ( ), ,H x p tLp x∂= ∂
[ ], nn iQΦ =for eachnode:
HAMILTONIAN OF TWO CAPACITIVELYCOUPLED RESONATOR MODES
LbCaLa
Cc
Cb
Φ1 Φ2
( )2 21 2
1 1 2 2
2 21 2 1 2
1 2 3
ˆ ˆˆ ˆˆ ˆ ˆ, ; ,2 2
ˆ ˆ ˆ ˆ
2 2
a b
H Q QL L
Q Q Q QC C C
Φ ΦΦ Φ = +
+ + +
Reduced capacitance matrix:
a c c
c b c
C C CC C C+ −⎡ ⎤
= ⎢ ⎥− +⎣ ⎦C
Inverse of reduced capacitance matrix:
1 1 b c c
c a ca b a c b c
C C CC C CC C C C C C
− +⎡ ⎤= ⎢ ⎥++ + ⎣ ⎦
C
ANHARMONICITY vs CHARGE SENSITIVITYCooper pair box levels are "exactly soluble" (A. Cottet, PhD thesis, Orsay, 2002)
290 μm
ANHARMONICITY vs CHARGE SENSITIVITYIN THE LIMIT EJ /EC >> 1
peak-to-peak charge modulation amplitude of level m:
anharmonicity:
TRANSMON: SHUNT CPB JUNCTION WITH CAPACITANCE
Courtesy of J. Schreier and R. Schoelkopf
J. Koch et al. Phys. Rev. A '07( )
12 01
12 01 / 2 8C
J
EE
ω ωω ω
−→
+
ω01
ω12ω02 2
Sufficient to control the artificial atom as a two level system: Qubit
SPECTROSCOPY OF A JOSEPHSON ATOM
Anharmonicity:
01 12 455MHz CEω ω− =
ω01
ω02 2
ω12
J. Schreier et al., Phys. Rev. B '08
Slide courtesy of J. Schreier and R. Schoelkopf
THE MEMORY READOUT PROBLEM
QUBIT READOUTON
OFF0 1
QUBIT READOUTON
OFF
1) SWITCH WITH ON/OFF RATIO AS LARGE AS POSSIBLE2) READOUT WITH F AS CLOSE TO 1 AS POSSIBLEWANT:
0 1
1
0
QUBIT READOUTON
OFF0 1
0 1
0 10 11F ε ε= − −
1ε0ε
FIDELITY:
or
STATE DECAY STRATEGY
1
0
1
0
Martinis, Devoret and Clarke, PRL 55 (1985)Martinis, Nam, Aumentado and Urbina, PRL 89 (2002)
QUBITCIRCUIT
10
or
or
rf signal in rf signal out
A) FILTER OUT EVERYTHING ELSE THAN READOUT RFB) REPEAT WITH ENOUGH PHOTONS TO BEAT
NOISE : USE THE BEST AMPLIFIER AS POSSIBLE
DISPERSIVE READOUT STRATEGY
01ω ω≠
QUBIT STATEENCODED IN PHASE
OF OUTGOING SIGNAL,NO ENERGY DISSIPATED
ON-CHIP
Blais et al. PRA 2004, Walraff et al., Nature 2004
(see whole sessions J3 & L17 + Q17.6& Y34.4)
Si or Al2O3
Al or Nb
SUPERCONDUCTING MICROWAVE RESONATOR: ANALOG OF FABRY-PEROT CAVITY FILTER
~
can get Qup to 106
( in bulk:Q ~ 1010)
ADC
length ~ 1cm, but photon decay length ~ 10km!
IN OUT
1/2 photon: ~1nA~100nV
ν ~ 10GHz
CHIP
w~20μm
VERY SMALL MODE VOLUME
3mm
10mm
100μm
Nb
SUPERCONDUCTING CAVITY = FABRY-PEROT
qubit goes here (see slide 2, lower right, in this talk)
OFF-RESONANT NMR-TYPEPULSE SEQUENCE
FOR QUBIT MANIPULATION
EXAMPLE OF QUANTRONIUM IN MICROWAVE CAVITY
|0>
|1>
READOUTPROBING
PULSE
QUBIT STATEENCODED IN PHASE
OF TRANSMITTED PULSE
or
QUANTRONIUM IN MICROWAVE CAVITY
|0>
|1>
QUANTRONIUM IN MICROWAVE CAVITY
~ ADCIN OUT
AWG
40ns40ns200ns
t t
Metcalfe et al.Phys. Rev. B6174516 (2007)
latching effectdue to bifurcation
of cavity mode
50Ω 50Ω
Cin~200aF Cout~20fFCcouple~6fF
qubit cavity readout cavity
qubitreadoutjunction
IN-LINE TRANSMON WITH BIFURCATING MICROWAVE CAVITY READOUT
See V17.6 M. Brink et al. Thursday morning, and also recent Saclay group results(in preparation)
Acknowledgements: Circuit Quantum Electrodynamics GroupsDepts. Applied Physics and Physics, Yale
P.I.'s
Grads
Post-Docs
Res. Sc.Undrgrds
Collab.
M. D.R. VIJAY (UCB)C. RIGETTI M. METCALFE (NIST)V. MANUCHARIANF. SCHAKERTN. MASLUKA. KAMALI. SIDDIQI (UCB)C. WILSON (Chalmers)E. BOAKNIN (McK)N. BERGEAL (ESPCI)M. BRINKA. MARBLESTONED. ESTEVE et coll.(Saclay)B. HUARD (LPA/ENS)
S. GIRVINT. YUL. BISHOP
J. KOCHJ. GAMBETTA(U. Waterloo)E. GINOSSARA. NUNNENKAMP
F. MARQUARDT(Munich)A. BLAIS(Sherbrooke)A. CLERK (McGill)
R. SCHOELKOPFJ. CHOWB. TUREK (MIT)J. SCHREIERB. JOHNSONA. SEARSM. READA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCK (Princeton)D. SCHUSTERL. DiCARLOL. FRUNZIOJ. SCHWEDEP. ZOLLER(Innsbruck)
W.M. KECK
Acknowledgements: Circuit Quantum Electrodynamics GroupsDepts. Applied Physics and Physics, Yale
P.I.'s
Grads
Post-Docs
Res. Sc.Undrgrds
Collab.
M. D.R. VIJAY (UCB)C. RIGETTI M. METCALFE (NIST)V. MANUCHARIANF. SCHAKERTN. MASLUKA. KAMALI. SIDDIQI (UCB)C. WILSON (Chalmers)E. BOAKNIN (McK)N. BERGEAL (ESPCI)M. BRINKA. MARBLESTONED. ESTEVE et coll.(Saclay)B. HUARD (LPA/ENS)
S. GIRVINT. YUL. BISHOP
J. KOCHJ. GAMBETTA(U. Waterloo)E. GINOSSARA. NUNNENKAMP
F. MARQUARDT(Munich)A. BLAIS(Sherbrooke)A. CLERK (McGill)
R. SCHOELKOPF(Keithley Award, T3)J. CHOWB. TUREK (MIT)J. SCHREIERB. JOHNSONA. SEARSM. READA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCK (Princeton)D. SCHUSTERL. DiCARLOL. FRUNZIOJ. SCHWEDEP. ZOLLER(Innsbruck)
W.M. KECK
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