Collège de France abroad Lectures
Quantum information with real or
artificial atoms and photons in cavities
Serge Haroche,
Collège de France & Ecole Normale Supérieure, Paris
www.college-de-france.fr
A series of six lectures at NUS in the Course «!QT5201E Special Topics in
quantum information: advanced quantum optics!»
Goal of lectures
Manipulating states of simple quantum systems has become an important
field in quantum optics and in mesoscopic physics, in the context of
quantum information science. Various methods for state preparation,
reconstruction and control have been recently demonstrated or proposed.
Two-level systems (qubits) and quantum harmonic oscillators play an
important role in this physics. The qubits are information carriers and the
oscillators act as memories or quantum bus linking the qubits together.
Coupling qubits to oscillators is the domain of Cavity Quantum
Electrodynamics (CQED) and Circuit Quantum Electrodynamics (CircuitQED). In microwave CQED, the qubits are Rydberg atoms and the
oscillator is a mode of a high Q cavity while in Circuit QED, Josephson
junctions act as artificial atoms playing the role of qubits and the
oscillator is a mode of an LC radiofrequency resonator.
The goal of these lectures is to analyze various ways to synthesise nonclassical states of qubits or quantum oscillators, to reconstruct these
states and to protect them against decoherence. Experiments
demonstrating these procedures will be described, with examples from
both CQED and Circuit-QED physics. These lectures will give us an
opportunity to review basic concepts of measurement theory in quantum
physics and their links with classical estimation theory.
Outline of lectures
• Lecture 1 (February 6th):
Introduction to Cavity QED with Rydberg atoms
interacting with microwave fields stored in a high Q superconducting resonator.
• Lecture 2 (February 8th):
Review of measurement theory illustrated by the
description of quantum non-demolition (QND) photon counting in Cavity QED
•Lecture 3 (February 10th): Estimation and reconstruction of quantum states
in Cavity QED experiments; the cases of Fock and Schrödinger cat stats.
•Lecture 4 (February 13th): Quantum feedback experiments in Cavity QED
preparing and protecting against decoherence non-classical states of a radiation
field.
• Lecture 5 (February 15th): An introduction to Circuit-QED describing
Josephson junctions as qubits and radiofrequency resonators as quantum
oscillators.
•Lecture 6 (February 17th): Description of Circuit-QED experiments
synthesizing arbitrary states of a field oscillator.
I-A
The basic ingredients of Cavity QED:
qubits and oscillators
Two-level system
(qubit)
Field mode
(harmonic oscillator)
Description of a qubit (or spin 1/2)
|0>
!
Any pure state of a qubit (0/1) is parametrized
by two polar angles !," and is represented by a
point on the Bloch sphere :
!
!
! , " = cos e#i" /2 0 + sin ei" /2 1
2
2
"
P
! = "qubit "qubit
( pure state)
(i )
(i )
! = # pi "qubit
"qubit
|1>
i
(# pi = 1) (mixture)
i
which can be expanded on Pauli matrices:
"0 1%
" 0 (i %
"1 0 %
!x =$
!
=
!
=
;
;
' y $
' z $
'
#1 0&
#i 0 &
# 0 (1&
! i2 = I
A statistical mixture is represented
by a density operator:
(i = x, y, z)
! !
1
! = I + P. "
2
(
)
;
!
P #1
! hermitian, with unit trace and " 0 eigenvalues
!
!
!
P = 1 : pure state ; P < 1 : mixture ; P > 1 : non physical (negative eigenvalue)
A qubit quantum state (pure or mixture) is fully determined by its Bloch vector
Description of a qubit (cont’d)
The Bloch vector components are the expectation values of the Pauli operators:
Pi = Tr!" i = " i
'
1$
(i = x, y, z) ; ! = & I + # " i " i )
2%
(
i
(Tr! !
i
j
= 2"ij )
The qubit state is determined by performing averages on an ensemble of
realizations: the concept of quantum state is statistical.
Calling pi+ et pi- the probabilities to find the qubit in the eigenstates of "i with
eigenvalues ±1, we have also:
Pi = pi+ ! pi!
(
1%
; " = ' I + $ ( pi+ ! pi! ) # i *
2&
)
i
A useful formula: overlap of two qubit states described by their Bloch vectors:
1" ! ! $
S12 = Tr {!1! 2 } = #1+ P1 .P2 %
2
Manipulation and measurement of qubits: Qubit rotations are realized by
applying resonant pulses whose frequency, phase and durations are
controlled. In general, it is easy to measure the qubit in its energy basis
("z component). To measure an arbitrary component, one starts by
performing a rotation which maps its Bloch vector along 0z, and then one
measures the energy.
Qubit rotation induced by microwave pulse
Coupling of atomic qubit with a classical field:
"# "#
!eg
" z ; H int (t) = #D.E mw (t)
H = H at + H int (t) ; H at = !
2
Microwave electric field linearly polarized with controlled phase #0:
E
Emw (t) = E0 cos(! mwt " # 0 ) = 0 [ exp i(! mwt " # O ) + exp" i(! mwt " # O )]
2
Qubit electric-dipole operator component along field direction is off-diagonal and
real in the qubit basis (without loss of generality):
Dalong field = d! x
(d
real)
Hence the Hamiltonian:
H =!
!eg
2
"z #!
$ mw
" x [ exp i(! mwt # % O ) + exp# i(! mwt # % O )] ; $ mw = d.E0 / !
2
and in frame rotating at frequency $mw around Oz:
i!
d !
dt
=H !
(#
H" = !
eg
+ # mw )
2
;
" = exp $&i " z# mw t ') !
!
%
(
2
i
,
" z + ! mw e
2
" z # mw t
2
" xe
+i
" z # mw t
2
* i!
"
d !
dt
"
= H" !
./ei(#mw t +-O ) + e+i(#mw t +-O ) 01
;
Bloch vector rotation (ctn’d)
(!
H! = "
" ! mw )
i
$
# z " " mw e
2
# z ! mw t
2
"i
# z ! mw t
2
&'ei(!mw t "%O ) + e"i(!mw t "%O ) ()
2
! " t
! " t
$ ! x + i! y ' i"mw t $ ! x # i! y ' #i"mw t
i z mw
#i z mw
2
2
e
! xe
= ! x cos " mwt # ! y sin " mwt = &
+&
)e
)e
2
% 2
(
%
(
The rotating wave approximation (rwa) neglects terms evolving at frequency ±2$mw:
eg
# xe
(!eg " !mw ) # " " $mw # ei%0 + # e"i%0 ; # = (# ± i# ) / 2
H! rwa = "
( +
)
z
"
±
x
y
2
2
The rwa hamiltonian is t-independant. At resonance ($eg=$mw), it simplifies as:
"
"
H! rwa = !" mw (# + ei$ 0 + # ! e!i$ 0 ) = !" mw (# x cos $ 0 ! # y sin $ 0 )
2
2
A resonant mw pulse of length % and phase #0 rotates Bloch vector by angle &mw%
around direction Ou in Bloch sphere equatorial plane making angle -#0 with Ox:
! ! ) = exp("iH! rwa! / ") = exp("i # mw! $ u )
U(
2
$ u = $ x cos % 0 " $ y sin % 0
A method to prepare arbitrary pure qubit state from
state |e> or |g>. By applying a convenient pulse prior to
detection in qubit basis, one can also detect qubit
state along arbitrary direction on Bloch sphere.
Rotation angle
control by pulse
length
Rotation
axis
control by
pulse
phase
Description of harmonic oscillator
(phonons or photons)
p (E )
X=
!"#$%&#
n
Gaussian x
Hermite
Pol.
Gaussian
0
x
Particle in a parabolic potential or
field mode in a cavity
2
Mechanical or
electromagnetic
oscillation.
Coupling qubits to
oscillators is an
important ingredient
in quantum
information.
; P=
[ X, P ] =
a ! a†
2i
i
I
2
x (E1)
Phase space (with conjugate
coordinates x,p or E1,E2)
Basic formulae with photon annihilation and
creation operators
!"a, a † #$ = I ; a n = n n %1 ; a † n = n +1 n +1
a + a†
2
;
n =
a† n
0
n!
P(n)
N = a †a ; H field = !& N ; eiN&t ae%iN&t = e%i&t a
Displacement operator : D(' ) = e' a
Coherent state : ' = D(' ) 0 = e
†
%'
n
%' * a
2
/2
(
n
'n
n
n!
Photon (or phonon) number
distribution in a coherent state:
Poisson law
Coherent state
Coupling of cavity mode with a small resonant classical antenna located at r=0:
! !
V = ! J (t).A(0) ; J(t) " cos "t ; A(0) " a + a †
Hence, the hamiltonian for the quantum field mode fed by the classical source:
H Q = !!a †a + V = !!a †a + " ( ei!t + e#i!t ) ( a + a † )
(" : constant proportional to current amplitude in antenna)
Interaction representation:
d !! field
†
!
! field = exp(i"a at) ! field # i"
= H! Q !! field with H! Q = $ ( ei"t + e%i"t ) ei" a at (a + a † )e%i" a at
dt
Rotating wave approximation (keep only time independent terms):
H! Q (rwa) = ! ( a + a † )
Field evolution in cavity starting from vacuum at t=0:
†
( #$
†& +
†
*
".. * /2 . a† ". * a
!
! field (t) = exp *"i %a + a ' t - 0 = exp (. a " . a ) 0 = e
e e
0
) "
,
†
(. = "i#t / ")
We have used Glauber formula to split the exponential of the sum in last
expression. Expanding exp('a†) in power series, we get the field in Fock state basis:
!! field (t) = e
2
" # /2
$
n
#n
n
n!
Coupling field mode to classical source generates coherent state whose
amplitude increases linearly with time.
Coherent state (ctn’d)
Produced by
coupling the
cavity to a
source (classical
oscillating
current)
A Poissonian superposition of
photon number states:
! = # Cn n
, Cn = e " !
2
/2
n
!n
,
n!
!n = " = n
n
n
n= !
, P(n) = Cn
2
n=0
n
=e
n!
"n
Complex plane representation
Uncertainty
Im(# )
|
#
e|
The larger n,
the more
classical the
field is
2
p
m
A
d
u
t
li
circle
(«!width!» of
Wigner functionsee next page)
" = " 0$ % c t
Re(#)
Representations of a field state
Density operator for a field mode:
! = " field " field
)
)
( pure state) ; ! = # pi " (ifield
" (ifield
i
(# pi = 1) (mixture)
i
Matrix elements of ! can be discrete (!nn’ in Fock state basis) or continuous
(!xx’ in quadrature basis where |x> are the eigenstates of a+a†). Going from one
representation to the other is easy knowing the amplitudes <x|n> expressing the
oscillator energy eigenstates in the x basis (Hermite polynomial multiplied by
gaussian functions).
Representation in phase space: the Wigner function:
1
du exp("2ipu) x + u / 2 $ x " u / 2
#
!
W is a real distribution in (x,p) space (equivalently, in complex plane), whose
knowledge is equivalent to that of !. The «shadow» of W on p plane yields the
x- distribution in the state.
W (x, p) =
Gaussian
function
Ground state
'
Coherent state
Gaussian
function
translated
Positive and
negative
rings
Fock state
W<0 :
non-classical
I-B
Coupling a qubit to a quantized field mode:
the Jaynes-Cummings Hamiltonian
Matter
Semiclassical
Classical current
Quantum (qubit)
Fully Quantum
quantum
(qubit)
Radiation
Quantum field mode
Classical field
Quantum field mode
Coherent
states
Qubit
rotations
Cavity
QED
Qubit-quantum field mode coupling
Coupling qubit dipole to quantum electric field operator:
! !
V = ! D. EQ (0) ; EQ (0) = iE0 ( a ! a † )
V = !idE0" x (a ! a † ) = !idE0 (" + + " ! ) (a ! a † )
Absorption and emission
of photons while qubit
jumps between its
energy states.
Rotating wave approximation:
Vrwa = !i
with:
!+ =
! x + i! y
2
!"0
# + a ! # ! a † ) ; "0 = 2d.E0 / !
(
2
"0 1%
=$
'= e g
#0 0&
; !( =
! x ( i! y
2
"0 0%
=$
'= g e
#1 0&
Vacuum Rabi frequency in cavity mode of volume V:
0 EQ2 (0) 0 = E02 0 aa † +a †a 0 = E02
! 0V E02 =
!"
2
# E0 =
2d 2"
!0 =
# 0 !V
!"
2! 0V
Requires large dipole
d and small cavity
volume V
Qubit-oscillateur coupling:
the Jaynes-Cummings Hamiltonian
H = !!qb
"z
!$ %
†'
+ !!oh a †a # i
"
a
#
"
a
#
& +
(
2
2
g, n +1
! = "qb # "oh
&
e, n
g, 3
e,2
g,2
e,1
g,1
e,0
! n +1
±, n
! 3
±, 2
! 2
!
g, 0
g,0
g, n +1
e, n
uncoupled states
+, n
(()0)
!, n
g, n +1
e, n
0
E±,n = ( n +1 / 2 ) !!oh ±
(
!
" 2 + #2 ( n +1)
2
Anticrossing of dressed qubit
Coupled states (dressed qubit)
at resonance ((=0)
1
±, n =
e, n ± i g, n +1 )
(
2
! n +1
Light shift
±,1
±, 0
# n +1t
# n +1t
e, n + sin
g, n +1
2
2
# n +1t
# n +1t
g, n +1 !!
" $sin
e, n + cos
g, n +1
2
2
e, n
!!
" cos
Non-Resonant coupling: light shifts in CQED
Second order perturbation
theory:
g, n +1
e, n
+, n
Energy shift due to off-resonant
coupling (distance between
energy level and asymptot):
! n +1
!, n
g, n +1
e, n
(
0
E±,n = ( n +1 / 2 ) !!oh ±
! 2
" + #2 ( n +1)
2
Anticrossing of dressed qubit
E±,n
% # $2 (n +1) (
! ( n +1 / 2 ) !"C ± ! ' +
*
4# )
&2
#2 (n +1)
E+,n ! Ee,n " !
+ ... ;
4$
#2 (n +1)
+ ...
E!,n ! Eg,n+1 " !!
4$
Vacuum shift (Lamb-shift):
!g,0 = 0
; !e,0 = E+,0 " Ee,0
Light shift induced on eg transition by n-photons:
#2
#2
#2t
n + ... % & ("eg ) =
n ; '0 =
E+,n ! E!,n!1 = !"eg + !
2$
2$
2$
$2
#!
4%
#0=Phase shift per
photon accumulated
during time t
Measuring qubit phase shift in CQED:
the Ramsey interferometer
C
R1
R2
z
z
x
Qubit
initially
in e
R1 pulse
rotates Bloch
vector by */2
around Ox
$ " '
exp &!i # x )
% 4 (
Qubit phase shift
#c during C crossing
amounts to Bloch
vector rotation
around Oz
$ "
'
exp &!i C # z )
(
% 2
R2 pulse realizes */2
rotation around Ou
whose direction
depends on R2-R1
relative phase #r
$ " '
exp &!i # u ) =
% 4 (
$ "
'
exp &!i +,# x cos * r ! # y sin * r -.)
% 4
(
Ramsey interferometer (ctn’d)
A useful formula:
exp(!i"# u ) = cos " I ! i sin " # u
( for any Pauli operator )
Hence the rotation induced by Ramsey interferometer:
$ " '
$ *
'
$ " '
R = exp &!i # u ) exp &!i C # z ) exp &!i # x )
% 4 (
% 2
(
% 4 (
Rotation
induced by R2
Cavity phase-shift
1$ 1
= & !i*
2 %!ie r
!iei* r ' $ e !i*C /2
)&
1 (% 0
Rotation
induced by R1
0 ' $ 1 !i '
)&
)
ei*C /2 ( %!i 1 (
$ i* r /2 $ * c + * r '
$* + *r ''
i* /2
sin &
) e r cos & c
))
&e
%
(
%
()
2
2
= !i &
& !i* /2
$ * c + * r ' !i* r /2 $ * c + * r ')
r
cos
sin &
e
&
) e
))
&
% 2 (
% 2 ((
%
and the evolution of |e> and |g> states:
#" + "r &
#" + "r &
!i" /2
R e = !iei" r /2 sin % c
( e ! ie r cos % c
(g
$ 2 '
$ 2 '
#" + "r &
#"c + "r &
!i" r /2
R g = !iei" r /2 cos % c
!
ie
sin
e
(
%
(g
$ 2 '
$ 2 '
#" + "r &
Pe!g = cos 2 % c
(
$ 2 '
Ramsey fringes detected by
sweeping #r. Fringe-phase used
to measure cavity shift #C.
General scheme of cavity QED experiments
Microwave pulses preparing (left) the atomic qubit and rotating it
(right) in the detection direction
Atom
preparation
Detector of
states e and g
Rydberg
atoms High-Q-cavity storing
the trapped field
ENS experiments:
Rydberg atoms in states e
and g behave as qubits.
They are prepared in B in
state e and cross one at a
time the high-Q cavity C
where they are coupled to a
field mode. The atom-field
system evolution is ruled by
the Jaynes-Cummings
hamiltonian. A microwave
pulse applied in R1 prepares
each atom in a superposition
of e and g. After C, a second pulse, applied in R2, maps the measurement direction
of the qubit along the Oz axis of the Bloch sphere, before detection of the qubit
by selective field ionization in an electric field (in D). The R1-R2 combination
constitutes a Ramsey interferometer. This set-up has been used to entangle atoms,
realize quantum gates, count photons non-destructively, reconstruct non classical
states of the field and demonstrate quantum feedback procedure (lectures 1 to 4).
Analogy between Ramsey and Mach Zehnder
interferometers
In Ramsey interferometer, two resonant pulses
split and recombine atomic state in Hilbert space.
Qubit follows two pathes between R1 and R2 along
which it undergoes different phase-shifts. By
finally detecting qubit in e or g and sweeping
phase, one get fringes which informs about
differential phase shifts between the states.
e
In Mach-Zehnder, the
splitting and recombination
occurs on particle
trajectories. Beam splitters
replace Ramsey pulses.
Fringes inform about
differential phase shifts
induced on the pathes
De
#e
R1
B1
R2
g
#g
b
Dg
D
M'
Pg , Pe
0
2!
M
a
B2
!r
Detectors in the two outgoing channels detect fringes with opposite phases
I-C
A special system: circular Rydberg atoms
coupled to a superconducting Fabry-Perot
cavity
A qubit extremely sensitive to
microwaves:the circular Rydberg atom
nh
n!
= 2 " rR # rR =
;
mv
mv
mv 2
q2
q 2 mv
=
=
E =
2
8 "$ 0 rR 8 "$ 0 n!
n!dB =
#v=
Rydberg
formulae
Atom in circular Rydberg state:
!
2
rR = n
= n a0
electron on giant orbit
! fs mc
(tenth of a micron diameter)
q2 c
1 c
1
c
= % fs &
# E = 2 mc 2% 2fs
2n
n 137 n
4 "$ 0 !c n
Atom in ground state:
electron on 10-10 m diametre
orbit
Very long radiative lifetime TR
!n,n"1 ! n "3 ; Prad =
2
"!n,n"1
2
! (!n,n"1
rR ) ! n "8 # TR ! n 5
TR
2
Delocalized electron wave
e
(n=51)
g
(n=50)
Electron is localised on orbit by a
microwave pulse preparing
superposition of two adjacent
Rydberg states: |e> & |e> + |g>
The localized wave packet packet revolves around nucleus
at the transition frequency (51 GHz) between the two
states like a clock’s hand on a dial.
The electric dipole is proportional to the qubit Bloch vector
Complex
in preparation
the equatorialwith
planelasers
of the and
Blochradiofrequency
sphere.
The path towards circular states: an adiabatic process
involving 53 photons
Controling the atom-cavity interaction time by selecting the atom’s
velocity via Doppler effect sensitive optical pumping
Rubidium level scheme with transitions
implied in the selective depumping and
repumping of one velocity class in the
F=3 hyperfine state
In green, velocity distribution before
pumping, in red velocity distribution of
atoms pumped in F=3, before they are
excited in circular Rydberg state
Detecting circular Rydberg states by
selective field ionization
n=51
n=50
Adjusting the ionizing
field allows us to
discriminate two
adjacent circular
states. The global
detection efficiency
can be > 80%.
The ENS photon box (latest version)
In its latest version,
the cavity has a
damping time in the
100 millisecond range.
Atoms cross it one at
a time.
Cavity half-mounted…
atom
...and fully-mounted
Orders of magnitude
w~+=6mm
V ~ *w2L/4 ~ 4+3
~ 700 mm3
L=9+/2=2,7 cm
2d 2
q2
!
&
!
!
% ra2 % !
V
"
# 0 "V"
$# 0 "c
! ! 10 "6 # = 2 $ % 50kHz
1/2
q2
ra2 ' 1 *
%
=)
,
4 $# 0 "c & 2 ( 137 +
ra
! 10 -6
&
Parameter determined by
geometric arguments
Number of vacuum Rabi flops during cavity damping time (best cavity):
N RF =
!TC
! 5000
2"
Atom-cavity interaction time (depends on atom velocity):
t int !
Number of vacuum Rabi flops during atom-cavity interaction time:
Strong coupling in
CQED
1
!
3.10 "6 s
< t int
3.10 "5 s
! Tc
10 "1 s
w
! 10 to 50µ s
vat
!t int / 2 " ! 1 to 3
Many atoms
cross one by one
during Tc
Cavity Quantum Electrodynamics:
a stage to witness the interaction between light and matter at
the most fundamental level
The best
mirrors in the
world: more
than one billion
bounces and a
folded journey
of 40.000km
(the earth
circumference)
for the light!
One atom interacts with
one (or a few) photon(s)
in a box
A sequence of atoms crosses the
cavity,couples with its field and
carries away information about the
trapped light
Photons bouncing on mirrors
pass many many times on the
atom: the cavity enhances
tremendously the
light-matter coupling
Photons are
trapped for
more than a
tenth of a
second!
6 cm
I-D
Entanglement and quantum gate experiments
in Cavity QED
Resonant Rabi flopping
e
0 photon
g
e
R2
R1
0 photon
0
1 photon
R2
R1
e
cos ( !t / 2 ) e, 0
+
R1
R2
g
+ sin ( !t / 2 ) g,1
Spontaneous emission and absorption involving atom-field
entanglement
When n photons are present, the oscillation occurs faster (stimulated
emission):
(
)
cos ! n + 1t / 2 e, n
(
)
+ sin ! n + 1t / 2 g, n + 1
Simple dynamics of a two-level system (|e,n>, |g,n+1>)
Rabi flopping in vacuum or in small coherent field:
a direct test of field quantization
p(n)
n
" ! n + 1t %
Pe (t) = ( p(n)cos $
;
'
2
#
&
n
2
n
p(n) = e
n!
)n
0
n=0
1 2 3
(n th = 0.06)
n = 0.40 (±0.02)
n = 0.85 (±0.04)
n = 1.77 (±0.15)
Pe(t) signal
Fourier transform
Inferred p(n)
Brune et al,
PRL,76,1800,1996.
n
Useful Rabi pulses
(quantum « knitting »)
Initial state
|e,0> , |e,0> + |g,1>
* / 2 pulse
creates atom-cavity
entanglement
P e (t)
|e,0>
0.8
51 (level e)
0.6
51.1 GHz
50 (level g)
0.4
# "t &
# "t &
e, 0 ! cos % ( e, 0 + sin % ( g,1
$ 2'
$ 2'
0.2
time ( ?µ s)
0.0
Brune et al, PRL 76, 1800 (96)
0
30
60
90
Microscopic entanglement
Useful Rabi pulses
(quantum knitting)
|e,0> , |g,1>
|g,1> , |e,0>
P e (t)
|g,0> , |g,0>
(|e> +|g>)|0> , |g> (|1> +|0>)
* pulse copying
atomʼs state on
field and back
0.8
51 (level e)
0.6
51.1 GHz
50 (level g)
0.4
# "t &
# "t &
e, 0 ! cos % ( e, 0 + sin % ( g,1
$ 2'
$ 2'
0.2
time ( ?µ s)
0.0
Brune et al, PRL 76, 1800 (96)
0
30
60
90
Useful Rabi pulses
(quantum knitting)
|e,0> , - |e,0>
|g,1> , - |g,1>
P e (t)
|g,0> , |g,0>
2* pulse :
conditional dynamics
and quantum gate
0.8
51 (level e)
0.6
51.1 GHz
50 (level g)
0.4
# "t &
# "t &
e, 0 ! cos % ( e, 0 + sin % ( g,1
$ 2'
$ 2'
0.2
time ( ?µ s)
0.0
Brune et al, PRL 76, 1800 (96)
0
30
60
90
Atom pair entangled by photon exchange
V(t)
g2
e1
Electric field F(t) used to
tune atoms 1 and 2 in
resonance with C for times t
corresponding to ' / 2 or '
Rabi pulses
Hagley et al, P.R.L. 79,1 (1997)
+
+
Ramsey interferometer with phase controled
by field in the cavity
Probabilities Pe ( or Pg=1-Pe) for finding
atom in e or g oscillate versus ( .
Resonant classical ' /2
pulses in auxiliary
cavities R1-R2 (with
adjustable phase offset
( between the two)
prepare and analyse
atom state
superpositions
The phase of the atomic fringes
and their amplitude depend
upon the state of field in C,
which affect in different ways
the probability amplitudes
associated to states e and g
Effect of 2' Rabi flopping on Ramsey signal
Cavity C resonant with e-g
(51-50) transition.
Ramsey R1-R2 interferometer
resonant with g-i (50-49)
transition.
= 2"
g,1 #!t##
$ ei" g,1
e
g
C
R1-R2
i
= 2"
i,1 #!t##
$
(i
(i
2' Rabi flopping on transition e-g in
1 photon field induces a ' phase shift
between the g and i amplitudes
i,1
(
)
+ g ) 1 ! i + e i" g 1
+ g
)0
!( i +
g
)0
g-i fringes are inverted when
photon number in C increases
from 0 to 1
Ramsey fringes conditioned to one photon in
C
g
photon
00 photon
R2
R1
1 photon
1 photon
R2
R1
i
0 photon
g (probe)
e R(source)
1
R2
Experiment with 1st atom acting as
source emitting 1 photon with
probability 0.5 ('/2 pulse on e-g
transition) and 2nd probe atom
undergoing Ramsey interference on g-i
transition
With proper phase
choice, atom is detected
in g if n = 0, in i if n = 1:
quantum gate with
photon (0/1) as control
qubit and atom (i/g) as
target qubit
The quantum gate with photon as control bit
realises a quantum non-demolition
measurement (QND) of field
The atom carries away
information about field
|a>
|a>
energy without altering the
photon number
(2' Rabi pulse).This is very
*x
different from usual
|b>
|a) b>
photon detection, which is
Target bit (atom): b = 0 (g) / 1 (i)
destructive
Control bit (photon): a = 0/1
M.Brune et al, Phys.Rev.Lett. 65, 976 (1990)
G.Nogues et al, Nature 400, 239 (1999)
A.Rauschenbeutel et al, Phys.Rev.Lett. 83, 5166 (1999)
S.Gleyzes et al, Nature, 446, 297 (2007) Repetitive QND measurement of
photons stored in super high Q cavity.
C.Guerlin et al, Nature, 448, 889 (2007)
Second lecture (Wednesday)
Combining Rabi pulses for entanglement
knitting
2 ' Rabi
' /2 Rabi
pulse
pulse (QND)
+
First atom prepares 1 photon with 50% probability (pulse ' / 2) and
second atom reads photon by QND (pulse 2' )
' Rabi pulse
+
Three particle engineered
entanglement (GHZ)
(Rauschenbeutel et al,
Science, 288, 2024 (2000))
+
Third atom absorbs field (pulse ' ), producing a three
atom correlation.