Collge de France abroad Lectures Quantum information with real or - - PowerPoint PPT Presentation

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

A series of six lectures at NUS in the Course «!QT5201E Special Topics in quantum information: advanced quantum optics!»

www.college-de-france.fr

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

• scillators 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 (Circuit- QED). In microwave CQED, the qubits are Rydberg atoms and the

• scillator 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

• scillator is a mode of an LC radiofrequency resonator.

The goal of these lectures is to analyze various ways to synthesise non- classical 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

• pportunity 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

• scillators.
• 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)

!," = cos! 2 e#i" /2 0 + sin! 2 ei" /2 1

! "

Any pure state of a qubit (0/1) is parametrized by two polar angles !," and is represented by a point on the Bloch sphere :

P

A qubit quantum state (pure or mixture) is fully determined by its Bloch vector

! P =1: pure state ; ! P <1:mixture; ! P >1:non physical (negative eigenvalue)

|0> |1>

A statistical mixture is represented by a density operator:

! = 1 2 I + !

• P. !

"

( )

; ! P #1

! x = 0 1 1 " # $ % & '; ! y = 0 (i i " # $ % & '; ! z = 1 (1 " # $ % & ' ! i

2 = I

(i = x,y,z)

which can be expanded on Pauli matrices:

! hermitian, with unit trace and " 0 eigenvalues

! = "qubit "qubit (pure state) ! = pi "qubit

(i) i

#

"qubit

(i)

( pi =1) (mixture)

i

#

The Bloch vector components are the expectation values of the Pauli operators: The qubit state is determined by performing averages on an ensemble of realizations: the concept of quantum state is statistical.

Description of a qubit (cont’d)

Calling pi+ et pi- the probabilities to find the qubit in the eigenstates of "i with eigenvalues ±1, we have also:

P

i = pi+ ! pi!

; " = 1 2 I + pi+ ! pi!

( )# i

i

$

% & ' ( ) *

A useful formula: overlap of two qubit states described by their Bloch vectors: S12 = Tr !1!2

{ } = 1

2 1+ ! P

1.

! P

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.

P

i = Tr!" i = " i

(i = x,y,z) ; ! = 1 2 I + " i " i

i

#

$ % & ' ( ) Tr! i! j = 2"ij

( )

Qubit rotation induced by microwave pulse

Coupling of atomic qubit with a classical field: H = H at + Hint(t) ; Hat = !!eg 2 " z ; Hint(t) = #D "# .E " #

mw(t)

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)

Microwave electric field linearly polarized with controlled phase #0: Emw(t) = E0 cos(!mwt "#0) = E0 2 expi(!mwt "#O)+ exp" i(!mwt "#O)

[ ]

Hence the Hamiltonian: and in frame rotating at frequency $mw around Oz: i! d ! dt = H ! ; " ! = exp i " z#mw 2 t $ % & ' ( ) ! * i! d " ! dt = " H " ! ; " H = ! #eg +#mw

( )

2 " z + ! ,mw 2 e

i" z#mwt 2 " xe +i" z#mwt 2

ei(#mwt+-O )+ e+i(#mwt+-O ) . / 1

H = !!eg 2 " z # ! $mw 2 " x expi(!mwt #%O)+ exp# i(!mwt #%O)

[ ]

; $mw = d.E0 / !

Bloch vector rotation (ctn’d)

e

i! z"mwt 2 ! xe #i! z"mwt 2

= ! x cos"mwt # ! y sin"mwt = ! x + i! y 2 $ % & ' ( )ei"mwt + ! x # i! y 2 $ % & ' ( )e#i"mwt

! H = " !eg "!mw

( )

2 # z " " $mw 2 e

i# z!mwt 2 # xe "i# z!mwt 2

ei(!mwt"%O )+ e"i(!mwt"%O ) & ' ( ) 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. The rotating wave approximation (rwa) neglects terms evolving at frequency ±2$mw:

! Hrwa = " !eg "!mw

( )

2 # z " " $mw 2 # +ei%0 +# "e"i%0

( )

; # ± = (# x ± i# y) / 2

The rwa hamiltonian is t-independant. At resonance ($eg=$mw), it simplifies as: ! Hrwa = !" "mw 2 # +ei$0 +# !e!i$0

( ) = !" "mw

2 # x cos$0 !# y sin$0

( )

Rotation angle control by pulse length Rotation axis control by pulse phase

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: ! U(!) = exp("i ! Hrwa! / ") = exp("i #mw! 2 $ u) $ u = $ x cos%0 "$ y sin%0

Description of harmonic oscillator (phonons or photons)

!"#$%&#

Phase space (with conjugate coordinates x,p or E1,E2) p (E2) x (E1) x Particle in a parabolic potential or field mode in a cavity

P(n) n

Photon (or phonon) number distribution in a coherent state: Poisson law

Mechanical or electromagnetic

• scillation.

Coupling qubits to

• scillators is an

important ingredient in quantum information.

n

Gaussian Gaussian x Hermite Pol.

X = a + a† 2 ; P = a ! a† 2i X,P

[ ] = i

2 I

Basic formulae with photon annihilation and creation operators

a,a† ! " # $ = I ; a n = n n %1 ; a† n = n +1 n +1 ; n = a†n n! N = a†a ; H field = !&N ; eiN&tae%iN&t = e%i&ta Displacement operator : D(') = e'a† %'*a Coherent state : ' = D(') 0 = e

% ' 2 /2

'n n! n

n

(

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:

HQ = !!a†a +V = !!a†a + " ei!t + e#i!t

( ) a + a† ( )

(" :constant proportional to current amplitude in antenna)

Interaction representation:

! ! field = exp(i"a†at) ! field # i" d ! ! field dt = ! HQ ! ! field with ! HQ = $ ei"t + e%i"t

( )ei"a†at(a + a†)e%i"a†at

Rotating wave approximation (keep only time independent terms):

! HQ(rwa) = ! a + a†

( )

Field evolution in cavity starting from vacuum at t=0:

! ! field(t) = exp "i # " a + a† $ % & 't ( ) * + ,

• 0 = exp .a† ".*a

( ) 0 = e"..* /2e.a†e".*a 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)

! = Cn n , Cn = e

" ! 2 /2 ! n

n!

n

#

, n = !

2

, P(n) = Cn

2 = e"n n n

n!

A Poissonian superposition of photon number states: n=0

n

Produced by coupling the cavity to a source (classical

• scillating

current)

Im(# ) " = " 0$ % c t Re(#) Amplitude |# | Complex plane representation

Uncertainty circle («!width!» of Wigner function- see next page) The larger n, the more classical the field is !n = " = n

Representations of a field state

Density operator for a field mode:

! = " field " field (pure state) ; ! = pi " field

(i) i

#

" field

(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

• scillator energy eigenstates in the x basis (Hermite polynomial multiplied by

gaussian functions).

Representation in phase space: the Wigner function:

W (x, p) = 1 ! duexp("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. Ground state

Gaussian function

Coherent state

Gaussian function translated

Fock state

Positive and negative rings

'

W<0 : non-classical

I-B Coupling a qubit to a quantized field mode: the Jaynes-Cummings Hamiltonian

Classical current Quantum field mode Quantum (qubit) Classical field Quantum field mode Matter Radiation

Coherent states

Cavity QED Qubit rotations

Semi- classical

Quantum (qubit)

Fully quantum

Qubit-quantum field mode coupling

V = ! ! D. ! EQ(0) ; EQ(0) = iE0 a ! a†

( )

!0V E0

2 = !"

2 # E0 = !" 2!0V Coupling qubit dipole to quantum electric field operator: with:

! + = ! x + i! y 2 = 0 1 " # $ % & ' = e g ; ! ( = ! x ( i! y 2 = 0 1 " # $ % & ' = g e Absorption and emission

• f photons while qubit

jumps between its energy states.

0 EQ

2(0) 0 = E0 2 0 aa †+a†a 0 = E0 2

Vacuum Rabi frequency in cavity mode of volume V:

V = !idE0" x(a ! a†) = !idE0 " + +" !

( )(a ! a†)

Rotating wave approximation:

Vrwa = !i !"0 2 # +a !# !a†

( )

; "0 = 2d.E0 / !

Requires large dipole d and small cavity volume V

!0 = 2d 2" #0!V

Qubit-oscillateur coupling: the Jaynes-Cummings Hamiltonian

! ="qb #"oh

&

g,0

g,1

g,2 g,3 e,2

e,1

e,0

e,n

g,n+1

±,0

±,1

±,2

±,n

g,0

uncoupled states Coupled states (dressed qubit)

g,n +1

g,n +1 e,n

e,n

! n +1 +,n

!,n

(

E±,n = n+1/ 2

( )!!oh ± !

2 "2 +#2 n+1

( )

Anticrossing of dressed qubit

! ! 2 ! 3 ! n+1 Light shift

(()0)

H = !!qb " z 2 + !!oha†a # i !$ 2 " +a #" #a† % & ' (

±,n = 1 2 e,n ± i g,n +1

( )

at resonance ((=0)

e,n ! " ! cos # n +1t 2 e,n + sin # n +1t 2 g,n +1 g,n +1 ! " ! $sin # n +1t 2 e,n + cos # n +1t 2 g,n +1

Non-Resonant coupling: light shifts in CQED

g,n+1

g,n+1 e,n

e,n

! n+1

+,n

!,n

(

E±,n = n+1/ 2

( )!!oh ± !

2 "

2 +# 2 n+1

( )

Anticrossing of dressed qubit

Energy shift due to off-resonant coupling (distance between energy level and asymptot):

E+,n ! Ee,n " ! #2(n +1) 4$ +... ; E!,n ! Eg,n+1 " !! #2(n +1) 4$ +...

Second order perturbation theory: E±,n ! n +1/ 2

( )!"C ± ! #

2 + $2(n +1) 4# % & ' ( ) *

Light shift induced on eg transition by n-photons:

Vacuum shift (Lamb-shift):

!g,0 = 0 ; !e,0 = E+,0 " Ee,0 # ! $2 4% E+,n ! E!,n!1 = !"eg + ! #2 2$ n +...%&("eg) = #2 2$ n ; '0 = #2t 2$

#0=Phase shift per photon accumulated during time t

Measuring qubit phase shift in CQED: the Ramsey interferometer

Qubit initially in e R1 pulse rotates Bloch vector by */2 around Ox Qubit phase shift #c during C crossing amounts to Bloch vector rotation around Oz R2 pulse realizes */2 rotation around Ou whose direction depends on R2-R1 relative phase #r

exp !i " 4 # x $ % & ' ( )

exp !i "C 2 # z $ % & ' ( )

R1 R2 C z x z

exp !i " 4 # u $ % & ' ( ) = exp !i " 4 # x cos*r !# y sin*r + ,

• .

$ % & ' ( )

exp(!i"# u) = cos" I ! isin" # u for any Pauli operator

( )

Ramsey interferometer (ctn’d)

A useful formula:

Rotation induced by R1 Cavity phase-shift Rotation induced by R2

Hence the rotation induced by Ramsey interferometer:

R = exp !i " 4 # u $ % & ' ( )exp !i *C 2 # z $ % & ' ( )exp !i " 4 # x $ % & ' ( ) = 1 2 1 !iei*r !ie!i*r 1 $ % & ' ( ) e !i*C /2 ei*C /2 $ % & ' ( ) 1 !i !i 1 $ % & ' ( ) = !i ei*r /2 sin *c +*r 2 $ % & ' ( ) ei*r /2 cos *c +*r 2 $ % & ' ( ) e!i*r /2 cos *c +*r 2 $ % & ' ( ) e!i*r /2 sin *c +*r 2 $ % & ' ( ) $ % & & & & ' ( ) ) ) )

and the evolution of |e> and |g> states:

R e = !iei"r /2 sin "c +"r 2 # $ % & ' ( e ! ie!i"r /2 cos "c +"r 2 # $ % & ' ( g R g = !iei"r /2 cos "c +"r 2 # $ % & ' ( e ! ie!i"r /2 sin "c +"r 2 # $ % & ' ( g

Ramsey fringes detected by sweeping #r. Fringe-phase used to measure cavity shift #C.

P

e!g = cos2 "c +"r

2 # $ % & ' (

General scheme of cavity QED experiments

Atom preparation Rydberg atoms High-Q-cavity storing the trapped field Detector of states e and g Microwave pulses preparing (left) the atomic qubit and rotating it (right) in the detection direction

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

• f e and g. After C, a second pulse, applied in R2, maps the measurement direction
• f 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

R1 R2 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.

De Dg #e #g e g

B2 B1 M M' a b D

In Mach-Zehnder, the splitting and recombination

• ccurs on particle
• trajectories. Beam splitters

replace Ramsey pulses. Fringes inform about differential phase shifts induced on the pathes

!r

P

g,P e

2!

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

Complex preparation with lasers and radiofrequency

Atom in ground state: electron on 10-10 m diametre

• rbit

Atom in circular Rydberg state: electron on giant orbit (tenth of a micron diameter)

Electron is localised on orbit by a microwave pulse preparing superposition of two adjacent Rydberg states: |e> & |e> + |g>

Delocalized electron wave

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 in the equatorial plane of the Bloch sphere. e (n=51) g (n=50)

n!dB = nh mv = 2"rR # rR = n! mv ; E = mv2 2 = q2 8"$0rR = q2 8"$0 mv n! # v = q2 4"$0!c c n = % fs c n & 1 137 c n # E = 1 2n2 mc2% fs

2

Rydberg formulae !n,n"1 ! n"3 ; P

rad = "!n,n"1

TR ! !n,n"1

2

rR

( )

2 ! n"8 # TR ! n5

Very long radiative lifetime TR

rR = n2 ! ! fsmc = n2a0

The path towards circular states: an adiabatic process involving 53 photons

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

Controling the atom-cavity interaction time by selecting the atom’s velocity via Doppler effect sensitive optical pumping

Detecting circular Rydberg states by selective field ionization

Adjusting the ionizing field allows us to discriminate two adjacent circular

• states. The global

detection efficiency can be > 80%.

n=51 n=50

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… ...and fully-mounted atom

! " ! 2d 2 #0"V" ! q2 $#0"c % r

a 2 % &

V ! q2 4$#0"c % r

a 2

&2 = 1 137 ' ( ) * + ,

1/2 r a

& ! 10-6

L=9+/2=2,7 cm w~+=6mm V ~ *w2L/4 ~ 4+3 ~ 700 mm3

Orders of magnitude

! ! 10"6# = 2$ % 50kHz

Parameter determined by geometric arguments Atom-cavity interaction time (depends on atom velocity): tint ! w vat ! 10 to 50µs Number of vacuum Rabi flops during cavity damping time (best cavity):

NRF = !TC 2" ! 5000

Number of vacuum Rabi flops during atom-cavity interaction time:

!tint / 2" ! 1to 3

1 ! < tint ! Tc 3.10"6s 3.10"5s 10"1s

Strong coupling in CQED

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 6 cm One atom interacts with

• ne (or a few) photon(s)

in a box Photons bouncing on mirrors pass many many times on the atom: the cavity enhances tremendously the light-matter coupling A sequence of atoms crosses the cavity,couples with its field and carries away information about the trapped light

The best mirrors in the world: more than one billion bounces and a folded journey

• f 40.000km

(the earth circumference) for the light! Photons are trapped for more than a tenth of a second!

I-D Entanglement and quantum gate experiments in Cavity QED

Resonant Rabi flopping

R1 R2 0 photon

e

R1 R2 0 photon

e

R1 R2 0 photon

g

1

+ e g

cos !t / 2

( ) e,0

+ 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>)

n p(n)

0 1 2 3

Rabi flopping in vacuum or in small coherent field: a direct test of field quantization

P

e(t) =

p(n)cos2 ! n +1t 2 " # $ % & '

n

(

; p(n) = e)n n

n

n!

n = 0

(n th = 0.06)

n = 0.40 (±0.02) n = 0.85 (±0.04)

n = 1.77 (±0.15)

Brune et al, PRL,76,1800,1996.

Pe(t) signal Fourier transform Inferred p(n)

30 60 90 0.0 0.2 0.4 0.6 0.8

P e(t)

time ( ? s)

51 (level e) 50 (level g) 51.1 GHz

Brune et al, PRL 76, 1800 (96)

Initial state |e,0> * / 2 pulse creates atom-cavity entanglement |e,0> , |e,0> + |g,1> µ

Useful Rabi pulses (quantum « knitting ») Microscopic entanglement

e,0 ! cos "t 2 # $ % & ' ( e,0 + sin "t 2 # $ % & ' ( g,1

30 60 90 0.0 0.2 0.4 0.6 0.8

P e(t)

time ( ? s)

51 (level e) 50 (level g) 51.1 GHz

Brune et al, PRL 76, 1800 (96)

|e,0> , |g,1> |g,1> , |e,0> |g,0> , |g,0> (|e> +|g>)|0> , |g> (|1> +|0>) µ

Useful Rabi pulses (quantum knitting)

e,0 ! cos "t 2 # $ % & ' ( e,0 + sin "t 2 # $ % & ' ( g,1

* pulse copying atomʼs state on field and back

30 60 90 0.0 0.2 0.4 0.6 0.8

P e(t)

time ( ? s)

51 (level e) 50 (level g) 51.1 GHz

Brune et al, PRL 76, 1800 (96)

2* pulse : conditional dynamics and quantum gate |e,0> , - |e,0> |g,1> , - |g,1> |g,0> , |g,0> µ

Useful Rabi pulses (quantum knitting)

e,0 ! cos "t 2 # $ % & ' ( e,0 + sin "t 2 # $ % & ' ( g,1

V(t) e1

g2

Atom pair entangled by photon exchange

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

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 Resonant classical ' /2

pulses in auxiliary cavities R1-R2 (with adjustable phase offset ( between the two) prepare and analyse atom state superpositions

Probabilities Pe ( or Pg=1-Pe) for finding atom in e or g oscillate versus ( .

g,1

!t =2"

# $ ## ei" g,1 i,1

!t =2"

# $ ## i,1

2' Rabi flopping on transition e-g in 1 photon field induces a ' phase shift between the g and i amplitudes

i + g

( ) 1 !

i + ei" g

( ) 1

i + g

( ) 0 !

i + g

( ) 0

g-i fringes are inverted when photon number in C increases from 0 to 1

g e i

R1-R2

C Cavity C resonant with e-g (51-50) transition. Ramsey R1-R2 interferometer resonant with g-i (50-49) transition.

Effect of 2' Rabi flopping on Ramsey signal

Ramsey fringes conditioned to one photon in C

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

R1 R2 0 photon R1 R2 1 photon

0 photon 1 photon

g i

R1 R2 0 photon

e (source) g (probe)

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

The quantum gate with photon as control bit realises a quantum non-demolition measurement (QND) of field

| a > | a > | b > | a ) b >

*x

Control bit (photon): a = 0/1 Target bit (atom): b = 0 (g) / 1 (i)

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) C.Guerlin et al, Nature, 448, 889 (2007) The atom carries away information about field energy without altering the photon number (2' Rabi pulse).This is very different from usual photon detection, which is destructive

Repetitive QND measurement of photons stored in super high Q cavity. Second lecture (Wednesday)

+ + +

' /2 Rabi pulse 2 ' Rabi pulse (QND) ' Rabi pulse

First atom prepares 1 photon with 50% probability (pulse ' / 2) and second atom reads photon by QND (pulse 2' ) Third atom absorbs field (pulse ' ), producing a three atom correlation. Three particle engineered entanglement (GHZ) (Rauschenbeutel et al, Science, 288, 2024 (2000))

Combining Rabi pulses for entanglement knitting