Transcript Life after Charge Noise: An Introduction to the Transmon Qubit

Departments of Physics
and Applied Physics,
Yale University
Chalmers University of Technology,
Feb. 2009
What's super about
superconducting qubits?
Jens Koch
Outline
charge qubit - Chalmers
Introduction
Superconducting qubits
► overview, challenges
► circuit quantization
► the Cooper pair box
next lecture:
flux qubit - Delft
phase qubit
- UCSB
Transmon qubit
► from the CPB to the transmon
► advantages of the transmon
► experimental confirmation
Circuit QED with the transmon:
examples
Quantum Bits and all that jazz
2-level quantum system
(two distinct states
)
computational speedup
P.W. Shor, SIAM J. Comp. 26,
1484 (1997)
quantum cryptography
can exist in an infinite number
of physical states intermediate
between
and .
state
N. Gisin et al., RMP 74, 145 (2002)
fundamental questions
What makes quantum information more
powerful than classical information?
Entanglement – how to create it?
How to quantify it?
Mechanisms of decoherence?
superposition
of
AND
state
Measurement theory, evolution under
continuous measurement
…
2-level systems
Nature provides a few
true 2-level systems:
Spin-1/2 systems,
e.g. electron (→ Loss-DiVincenzo proposal)
nuclei (→ NMR)
Polarization of
electromagnetic waves
(→ linear optics
quantum computing)
2-level systems
…
Using multi-level systems
as 2-level systems
Requirements:
• anharmonicity
…
• long-lived states
• good coupling to EM field
• preparation, trapping etc.
e.g. atoms and molecules
(→ cavity QED,
→ trapped ions
→ liquid-state NMR)
R. Schoelkopf
artificial atoms: superconducting qubits, quantum dots
(→ cavity QED,
→ circuit QED…)
C. Schönenberger
The crux of designing qubits
environment
environment
qubit
control
measurement
►need good coupling!
protection against
decoherence
►need to be uncoupled!
Relaxation and dephasing
relaxation – time scale T1
dephasing – time scale T2
qubit
► fast parameter changes:
sudden approx, transitions
transition
► random switching
► slow parameter changes:
adiabatic approx, energy modulation
► phase randomization
Bringing the  into electrical circuits
Idea of superconducting qubits:
Electrical circuits can behave quantum mechanically!
Bringing the  into electrical circuits
Idea of superconducting qubits:
Electrical circuits can behave quantum mechanically!
What's good about circuits?
• Circuits are like LEGOs!
a few elementary building blocks,
gazillions of possibilities!
Bringing the  into electrical circuits
Idea of superconducting qubits:
Electrical circuits can behave quantum mechanically!
What's good about circuits?
• Circuits are like LEGOs!
a few elementary building blocks,
gazillions of possibilities!
• Chip fabrication:
well-established techniques
hope: possibility of scaling
Why use superconductors?
Wanted:
► electrical circuit as artificial atom
superconductor
► atom should not spontaneously lose energy
► anharmonic spectrum
E
“forest” of states
Superconductor
2D ~ 1 meV
► dissipationless!
► provides nonlinearity via Josephson effect
► can use dirty materials for superconductors
superconducting gap
Building Quantum Electrical Circuits
circuit elements
(
)
SC qubits:
macroscopic articifical atoms
ingredients:
• nonlinearities
• low temperatures
• small dissipation
Two-level system:
fake spin 1/2
Review: Josephson Tunneling
• couple two superconductors
via oxide layer → acts as tunneling barrier
• superconducting gap inhibits e- tunneling
• Cooper pairs CAN tunnel!
► Josephson tunneling
(2nd order with virtual intermediate state)
Tunneling operator for Cooper pairs:
normal state
conductance
SC gap
Josephson energy
Tight binding model: hopping on a 1D lattice!
Review: Josephson Tunneling II
…
…
Tight binding model:
Diagonalization:
‘position’
‘wave vector’
(compact!)
‘plane wave eigenstate’
Junction capacitance: charging energy
Transfer of Cooper pairs across junction
+2en
-2en
charging of SCs
► junction also acts as capacitor!
charging energy
with
quadratic in n
Circuit quantization
Best reference that I know:
(beware of a few typos though)
Circuit quantization – a quick survival guide
► Step 1: set up Lagrangian
- determine the circuit's independent coordinates
branch
node
► use generalized node fluxes
also:
ideal current sources, ideal voltage sources,
resistors
as position variables
Circuit quantization – a quick survival guide
► Step 1: set up Lagrangian
S capacitive energies S inductive energies
► Step 2: Legendre transform  Hamiltonian
conjugate momenta: charges
Circuit quantization – a quick survival guide
► Final Step 3: canonical quantization
Canonical quantization makes NO statement about boundary conditions!
Usually, assume
Works if each node is connected to an inductor ( confining potential).
This does NOT work if
SC islands are present!
• charge transfer between island
and rest of circuit:
only whole Cooper pairs!
• canonical quantization
is blind to the
quantization of electric charge!
Circuit quantization – a quick survival guide
► Final Step 3: quantization in the presence of SC islands
island charge operator has discrete spectrum:
charge basis
position
momentum
Peierls:
leads to contradiction -phase operator is ill-defined!
?
Circuit quantization – a quick survival guide
► Final Step 3: quantization in the presence of SC islands
Have already defined charge operator
What about
?
► should define this in phase basis!
usually:
now:
►
►
lives on circle!
is periodic!
Different types of SC qubits
► Nonlinearity from Josephson junctions
NEC, Chalmers, charge
Saclay, Yale
qubit
EJ = EC,
EJ =50EC
TU Delft,UCB
flux
qubit
EJ = 40-100EC
NIST,UCSB
phase
qubit
EJ = 10,000EC
Nakamura et al., NEC Labs
Vion et al., Saclay
Devoret et al., Schoelkopf et al., Yale,
Delsing et al., Chalmers
Lukens et al., SUNY
Mooij et al., Delft
Orlando et al., MIT
Clarke, UC Berkeley
Martinis et al., UCSB
Simmonds et al., NIST
Wellstood et al., U Maryland
Reviews:
Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)
M. H. Devoret, A. Wallraff and J. M. Martinis, cond-mat/0411172 (2004)
J. Q. You and F. Nori, Phys. Today, Nov. 2005, 42
J. Clarke, F. K. Wilhelm, Nature 453, 1031 (2008)
CPB Hamiltonian
3 parameters:
offset charge
(tunable by gate)
Josephson energy
charging energy
(fixed by geometry)
charge basis:
numerical
diagonalization
phase basis:
exact solution with
Mathieu functions
CPB as a charge qubit
Charge limit:
big
small perturbation
CPB as a charge qubit
Charge limit:
big
small perturbation
Next lecture: from the charge regime to the transmon regime