Transcript Dima Geshkenbian

Superconductivity
Superconductivity
Vk ' k
†
H    k ck† ck  Vk ' k ck† 'ck
c c
' k  k 
k ,
F
k
The phonon-mediated attractive electron-electron
interaction leads to the formation of Cooper-pairs


h D
V
n 1 3 = 
which undergo a k-space condensation
This condensate is a charged
quantum liquid described by a
macroscopic wave function
i r ,t
  r, t    e   .
The superconducting transition at
1 N0V
Tc  1.13 h D e
k
produces a gap
  1.764 Tc
in the electronic excitation spectrum,
thus removing all low energy excitations.
Metal

k F
kF
k
Josephson junctions
Josephson junctions,
are weak links connecting two superconducting leads/islands.
They appear in various forms, e.g., as
constrictions
l
g
l
r
g
r
V
tunnel junctions
V
Josephson relations describing the junction energy and the phase evolution
i
l  l e
l
 r   r eir


energy E ( )  Const     EJ
r l
l r
dynamic phase
d / dt  2eV


1  cos      E 1  cos  
J

r
l 
Two energy scales
e2
charging (capacitive) energy EC 
2C
charge Q Q
phase

0
current
I
I c 0
current (inductive) energy EJ 
2 c
The particle number N = Q/2e and the phase  are conjugate variables, i.e., we have a
particle  phase duality [N, ] = i (Anderson)
classical limit
EJ
EC
fixed phase 

Fock space
exp i 
exp i 
N 2
N 1
N
quantum limit
EJ
EC
N 1
N 2
fixed charge Q=2eN
Classical & quantum limits
e2
charging (capacitive) energy EC 
2C
charge Q Q
phase

0
current
I
I c 0
current (inductive) energy EJ 
2 c
Einductive  EJ 1  cos  
action
L
Ecapacitive  C V 2
2
2
 2 C 2
4e 2
2
4e2
C  2  E 1  cos  
J
2
 Einductive : I c sin  

2
Hamiltonian
2
H   EC d 2  EJ 1  cos  
d
~ EJ 2
[N, ] = i
N  i dd
Classical limit:
gauge invariance and fluxoid quantization in a loop

j  s  20   A ,
H

   ext
 self
hence


2
 unique
gauge invariant
phases 
   
0
2



and A  20  in the leads,
A  ds  20

  ds 
leads

A  ds

A  ds
junctions
 20  2n  1   2  ... 
junctions
 n 0  1  2  .....
kinetic energy
of currents ()
Free energy of a loop with inductance L:
1       ....
F  E1 1  cos   E2 1  cos 2  ....  2L
 ext 1 2 
2
.
2 junctions:
1 junction:
In a small inductance loop,
F
example :
 ext   0 2

i
 ext
i
In a large inductance loop,
0

  n 0
Quantum limit: Coulomb blockade and charge quantization on an island
2
1
electrostatic energy U   Ci Visl  Vi 
2 i
C1
V
Cn
C2
V C  Ne

C  C
i
Vn
Visl
i
i
i
i
V2
Visl U  0 & add N electrons


U  1  Ci C j Vi  Vj
2C j i

2
( Ne) 2
 2C

Quantum limit: Coulomb blockade and charge quantization on an island
=
2
1
electrostatic energy U   Ci Visl  Vi 
2 i
C1
0
Cg
C2
V C  Ne

C  C
i
Visl
Vg
Visl U  0 & add N electrons
i
i
i

i
V2 =

U  1  Ci C j Vi  Vj
2C j i
0

2
( Ne) 2
 2C

Account for the work done by the batteries
when changing the island charge N


2
1
E
C V  Ne  const.
2C g g
E
N
2
EC  e
2C
0
Vg
N
0
N 1
Vg
… in general
E
F
EC

0
EJ


8 EJ EC
EC
E


H  EC N ext  Q 2e  2J Q   Q  .
2
E
F
14
Vg
0  EJ
 ext   0 2
   p  EJ EC  e
N 1
0
EC  0
2
H  EC dd 2  EJ 1 cos 
N
flux
mixing
by E C
EC
charge
mixing
by E J
N
N 1
  EJ EC
0

0
Qext  2e 2
Vg
RCSJ model, adding dissipation
RQ
C
CRQ  R   2Ice sin   2Ie ,
with the quantum resistance
Ic
I
RQ 
4e2
additional shunt resistor, e.g., accounting
for quasi-particle tunneling.
R
Effective action describing the Resistively and Capacitively Shunted Josephson junction:
2

 

     ' 
RQ
2
2
 

S   d 16 E   EJ 1  cos    4 R  d ' 
2


C
  '



ohmic dissipation (Caldeira-Leggett)
.
Schmid transition:
T=0 quantum phase transition, driven by the environment
H

  0 2
F
small inductance loop:
two well potential

2

0
  2 RQ R = 1, weak dissipation limit
F
large   inductance loop:
particle in periodic potential
with delocalized phase 

0
  2 RQ R  1, strong dissipation limit
with localized phase  -- superconducting junction
F
F
(Leggett-Chakravarti)
0

0

classical computing
Mechanics
First `Programmer’
and
Inventor of the
Difference Engine
1834
Ada Byron,
Lady Lovelace
1815-1852
The `full’ version
of this machine
was built in1991
by the Science
Museum,London
Enigma,
cracked by
Alan Turing
with help of
COLOSSUS
Charles Babbage
1791-1871
Electronics
The ENIAC
(Electronic Numerical
Integrator and Computer)
computer
was built in 1946
Built at University of Pennsylvania,
it included 18’000 tubes,
weighed 30 tons,
required 6 operators,
and 160 m2 of space.
Classical computer
Hardware
Capacitors:
Bits
0 : V  0,
1-bit gate: NOT
Vsd
R
input
Vg
output
1 : V  0.
2-bit gate: AND
Transistors:
Gates
Vg  0, closed,
Vg  0, open.
Vsd
Vg1
input
Vg2
output
Si-wafer
R
Pentium Processor,
1997, Intel
Transistors
in 50
Years
from 1 to 107
transistors
First Integrated Circuit, 1958
Jack Kilby, Texas Instruments
First Transistor, 1947
Bell Laboratories
Bardeen, Brattain,
& Shockley
Packed Device
Nanoscale Technology
V
insulator
gate
drain
source
Si-wafer
PTB
channel
gate
2 mm
box
Ultra-short channel Si-MOSFET,
IBM
0.5 mm wide, 0.1 mm channel
Switch a MOSFET with 1000 electrons,
while a SET requires only one!
source
drain
Single Electron Transistor (SET), Al
Applications
Your track control
in the car
Classical computers solve any computational task …..
Your bank account
Your washing machine
Your agenda
Your science
…. but some are really hard !
Computational Complexity
An input x is quantified via its information content L = log2 x.
A calculation is characterized by the number s of steps (logical gates) involved.
A problem is class P (efficiently solvable) if s is polynomial in L,
A problem is deemed `hard’ (not in P) if s scales exponentially in L,
m
s~L 
s ~ exp L
A `classic’ hard problem is that of prime factorization:
given a non-prime number N, find its factors;
the best known algorithm scales as s ~ exp (2 L1/3 (lnL)2/3).
A modern computer can factor a 130-decimal-digits number (L = 300) in a few weeks  days;
1827365426354265930284950398726453672819048374987653426354857645283905612849667483920396069782635471628694637109586756325221365901
doubling L would take millions of years to carry out this calculation.
A quantum computer would do the job within minutes
Public Key Encryption
(Rivest, Shamir & Adleman, 1978)
Encoding
Decoding
M  message
s, N  p  q, (non-) public
key
s
E  M mod N
t  s , p ,q 
ME
mod N
ts  1 mod( p  1)(q  1)
A quantum computer would crack this encryption scheme
Quantum computing
``…nature isn’t classical, dammit, and if you want to make a
simulation of nature, you’d better make it quantum mechanical…”
R. Feynman
L
If one has N quantum two level systems (e.g. L spins) they can have 2 different
states. To describe such a system in classical computer one needs to have 2 L
complex numbers, that requires exponentially large computational resources. Thus
modeling even small quantum system on a classical computer is practically impossible
task. But since Nature does it very efficiently one can try to use its ability to deal with
quantum systems and to apply it also for computational problem.
Bits and Qubits
A quantum bit (qubit) is the quantum mechanical generalization
spin
language
of a classical bit, a two-level system such as a spin, the
polarization of a photon, or ring currents in a superconductor.
Classical bit
Quantum bit
i
a,    0  a e
0, 1
0
spins polarizations ring-currents
Q
V
V
1
1 a
Physical realization via a
quantum two-level system
Physical realization via a
charged/uncharged capacitor
Q
1 
2
Classical & quantum gates I
The possibilities to manipulate a classical bit are quite
limited:
The NOT-gate simply interchanges the
two values 0 and 1 of the classical bit.
i
f
0
1
1
0
On the other hand, manipulation of a quantum bit is much richer! For a spin / twolevel system we can perform rotations around the x -, y -, and z - axis; placing the
`spin’ S (with magnetic moment m) into a magnetic field H, the Hamiltonian
   m S  H
with
produces the desired rotation. E.g.,….
H = Hz we obtain the time evolution
 e im H z t / 2
0 
U 
,
im H z t / 2 
e
 0

  t   eim H z t / 2 0  a eim H zt / 2 1 
phase
shifter
1  a2 .
H = Hx we obtain the time evolution
 cos m H x t i sin m H xt 
U 
,
i
sin
m
H
t
cos
m
H
t
x
x 

  t   cos m H x t
0
amplitude
shifter (a = 0)
 i tan m H x t 1 .
Classical & quantum gates II
NOT
The combination of the classical
gates allows us to construct all
manipulations on classical bits.
i f
0 1
1 0
Is there a set of universal quantum gates ?
How does such a set look like ?
AND
i
0
0
1
1
i
0
1
0
1
 cos  2 
iei sin  2  
U   i
 ,
cos  2  
 ie sin  2 
Hadamard (basis change):
H
1 1
2  1
1
 :
1
f
0
0
0
1
i
0
0
1
1
i
0
1
0
1
f
0
1
1
1
The target flips if the control is on 1
1
0
U 
0

0
phase
shifter
   0  a ei 1 
OR
Twoqubit gate:
XOR (CNOT)
Singlequbit gates:
Rotations
amplitude
shifter
irreversible
1  a2 .
H
 0  1
0 



 
1 
    0  1 
control
2,
2.
target
0
0
0
1
0
0


0 0 1

0 1 0
i
0
0
1
1
i
0
1
0
1
f
0
1
1
0
Entangling two
qubits
C
0
H 0   0  1 
put control qubit C into superposition state,
then future gates act on two states
simultaneously
H
00  11
T
2
maximally entangled
Bell state
0
XOR H 0  0    00  11 
2
i.e., target qubit T gets flipped AND non-flipped
And subsequently:
flipping a qubit in an entangled state modifies all its components
Quantum Algorithms
Quantum algorithms
Shor’s Factorization algorithm (1994)
finds prime factors in polynomial, rather
than exponential time.
Grover’s Search algorithm
(1997)
searches unstructured database (e.g. telephone
book) of N entries by
N steps.
Although Grover’s algorithm doesn’t change complexity class it is not less
fundamental, than Shor’s algorithm.
(i) It is not hard to prove, that classical algorithms can do no better, than just straight
search through the list, requiring on average N/2 steps
(ii) The `quantum speed-up’ ~
Shor’s factorization algorithm
N
is greater than that achieved by
 ~ exp(2(ln N )1/3)
(iii) One can show, that no quantum algorithm can do better, than O
thus it is optimal!
 N ,
Grover’s Search algorithm
Task: Given an unstructured set of elements, find the one, that corresponds to the
answer of some question.
The Algorithm
Assume that the database contains N elements, N is some power of 2. Let there be
only one solution, that we are looking for x0.
We start with a homogeneous superposition of all basis states
N 1
x

N x0
 s  1
This can be achieved, e.g., by starting with n qubits in the state
Hadamard transform
Hn
0  H    H 0  s 
1
2
n

0 1
The goal is to to increase the amplitude of the

n
 s
x0
component
0
and applying the
H
1 1
2
1
1
 :
1
The algorithm will iterate the Grover rotation G a certain number of times to obtain
state very close to 0 . The Grover rotation consists of two parts: an oracle call and a
reflection about s .
x
The oracle call is just quantum implementation of searching. There must exist unitary
operator
O x0   x0 ,
O x  x for x  x0
x0
s
O s
Each Grover rotation rotates our state by an angle
2
towards
x0
.
x0
G s
2
s

O s
We want
Gt s  x0 ,

Thus we should stop and measure after
sin(  2t )  1
 N 1
t
4
1
 steps ( 
)
2
N
Quantum Hardware
Physical
implementation
All hardware implementations of
quantum computers have to deal with the
conflicting requirements of
controllability
while minimizing the coupling
to the environment in order to
avoid decoherence.
Solid state implementations
enjoy good scalability & variability
but require careful designs in order to avoid decoherence
when trying to build Schrödinger cats
Network model of quantum computing
(David Deutsch, 1985)
initial state
• each qubit can be prepared in some known state, 00000K 0000 .
• each qubit can be measured in a basis, 01100K 1010 .
• the qubits can be manipulated through quantum gates
final
state
• the qubits are protected from decoherence
f
i
Parallel evolution
providing the
quantum speedup.
Perturbations
from the environment
destroy the parallel evolution of the computation
Physical implementations
Quantum optics, NMR-schemes
Good decoupling & precision:
• trapped atoms (Cirac & Zoller)
• photons in QED cavities (Monroe ea, Turchette ea)
• molecular NMR (Gershenfeld & Chuang)
• 31P in silicon (Kane)
Solid state implementations
Good scalability & variability:
• spins on quantum dots (Loss & DiVincenzo)
• 31P in silicon (Kane)
• Josephson junctions, charge (Schön ea, Averin)
phase (Bocko ea, Mooij ea)
All hardware implementations of
quantum computers have to deal with the
conflicting requirements of controllability
while minimizing the coupling
to the environment in order
to avoid decoherence.
Have to deal with individual
atoms, photons, spins,……
Problems with control,
interconnections,
measurements.
Have to deal with
many degrees of freedom.
Problems with decoherence.
The rules of the game
achievements
Find a system which emulates a spin / quantum two-level system and
• which remains coherent
Quantum optics
not a problem
Condensed Matter
up to Q ~ 104
• which can be manipulated (rotations)
• which can be interconnected and entangled with other qubits
Condensed Matter
2 charge qubits, interacting
Quantum optics
4 9Be ions, deterministic
• which can be projected (measured)
Quantum optics
• which carries out an algorithm (e.g., Shor’s prime factorization)
Bell inequailty checks
NMR
15 = 3 * 5
Superconducting quantum bits
Superconducting qubits
Josephson junctions
/2
Currents in a superconducting ring
Charges on a superconducting island
V
Superconducting quantum bits
Loop
vs
Island
In a superconducting ring,
the wave function
 x   x exp  i x 
Im
satisfies periodic
boundary conditions.
0
zero flux
state
Re

A finite bias draws a
Cooper-pair
onto the island
The macroscopic
wave function winds once
around the ring; the ring
carries a current

V
CP

h  * x  cc 
j x ~ 2mi


flux one
state
These two states
are degenerate
at
half-flux frustration
These two states
are degenerate
at
half-Cooper-pair frustration
Superconducting quantum bits
Frustrate a ring with
half-flux and obtain
two degenerate
flux states
OR
Produce a weak spot to
flip between flux states:
Josephson junction.
EJ
Frustrate an island with
half-Cooper pair and
obtain two degenerate
charge states
CP/2
0/2
Connect the box to
allow charge hopping:
Josephson junction.
EC
EC
EJ
The Josephson junctions are key ingredients in any superconducting qubit design.
Or, in other words,
the Josephson junctions introduce the quantum dynamics into the superconducting structure.
e2
EC 
2C
superconductor
phase
L
2
16EC
0
 2  EJ 1  cos  
superconductor

EJ 
Ic  0
2c
2
H   EC d 2  EJ 1  cos  
d
Three types
Charge
Schön et al.
Averin
1997
EC
Flux/Phase
C
Bocko et al.
1997
EJ
Vg
EC
n 1
n
0
2 
EJ
C

Ioffe et al.
Orlando et al.
1999
Ig
n  n
Josephson
superconductor

phase 0

EC

charge
mixing
by E J

phase
states
V
2E J
EJ
EC
0
Vg


2
EJ
0

2

1/ 4
 ~  p  EC EJ  exp   EJ EC 
h p
EJ

 p  8EJ EC
Manipulation
Manipulation
Charge
V  Vg
Phase
Ig
C
EC
EJ
V 0
N
N

 EJ
I J , CJ
n
EC
Vg
C
E
2E J
E
N
EC
N 1
charge
mixing
by E J
EJ
+ ac-microwave
voltage / current
induces transitions
across the gap
OR
+ fast (non-adiabatic)
0
Vg
phase 
amplitude 
shifter
shifter
switching induces
(incomplete)
Zener tunneling.

 2
Ig

2
0

flux mixing by E C
tunneling gap
 ~ p
 
EC
EJ
1/ 4
e

8 EJ
EC
one-parameter (q)
qubit
Manipulation (general)
ac-microwave
E
induced
transitions
(NMR scheme)
two-parameter (q,Q)
qubit
E
mixing

Q
trivial
idle state
phase shift
q
q
phase shift
fast non-adiabatic
switching, amplitude shift
decoupled
states
q
Q
coupled
states
potential or dynamical
About the NMR scheme:
With Nqu qubits the distance between resonances
is ~  / Nqu. The transition time of the k-th qubit
is related to the ac-signal V via
top ~  Vk .
Other qubits nearby are excited with
2
2
probability
2
Vk
   Vk
and a precise addressing requires
long times
top  N qu   .
Coherent
devices
Charge (Nakamura et al., 1999)
SQUIDloop
detector
detector
box
1mm
pulse
gate

0
2
Q
Q
t
fast gate
voltage pulse
time
Coherent
devices
Charge (Nakamura et al., 1999)
SQUIDloop
detector
detector
box
1mm
pulse
gate

0
2
Q
Q
t
This time domain
time
fast gate
experiment
shows
voltage
pulse
coherent
charge
oscillations of
50  100 ps duration
during a total
coherence time of 2 ns.
Second generation
Charge

  pulse
2
Balanced
energy
scales
EJ ~ E C
Phase

104 coherent
charge oscillations
observed via
Ramsey fringes,
Vion et al.,2002

Ramsey
interference

Box
V


free
evolution
Im

  pulse
2


~100 coherent oscillations
observed via Ramsey
interference


Chiorescu et al., 2003
Charge-qubits
Qubit duet,
entanglement
Spectral analysis
of interacting
J-qubits
Berkley et al., 2003
q0
Spectral analysis of
interacting q-qubits
Pashkin et al., 2003
q1
q
Josephson-qubits
Two-qubit gate
CNOT (XOR)
Device
Result
1

0

U
0

0
0
0
0
1
0
0


0 0 1

0 1 0 
Puls sequence
Yamamoto et al., 2003
The End