From quantum to classical: Photons, entanglement, and decoherenceLuiz Davidovich Instituto de Física Universidade Federal do Rio de Janeiro

Outline of the lecturesLECTURE 1 n Decoherence and the classical limit of the quantum

world n Cavities, fields, and Schrödinger cats LECTURE 2 n Introduction to entanglement n Entanglement and decoherence: new experimental

results n Multiparticle systems and decoherence 2

FIRST LECTURE

3

Schrödinger on the classical limit

n 1926: “At first sight it appears very strange to try to describe a process, which we previously regarded as belonging to particle mechanics, by a system of such proper vibrations.'' Demonstrates that “a group of proper vibrations” of high quantum number $n$ and of relatively small quantum number differences may represent a particle executing the motion expected from usual mechanics, i. e. oscillating with a constant frequency. 4

Schrödinger on the classical limit

5

n 1935: “An uncertainty originally restricted to the atomic domain has become transformed into a macroscopic uncertainty, which can be resolved through direct observation... This inhibits us from accepting in a naive way a `blurred model' as an image of reality...There is a difference between a shaky or not sharply focused photograph and a photograph of clouds and fogbanks.”

Quantum physics and localization

n “Let Ψ1 and Ψ2 be two solutions of the same Schrödinger equation. Then Ψ = Ψ1+Ψ2 also represents a solution of the Schrödinger equation, with equal claim to describe a possible real state. When the system is a macrosystem, and when Ψ1 and Ψ2 are `narrow’ with respect to the macro-coordinates, then in by far the greater number of cases, this is no longer true for Ψ. Narrowness in regard to macro-coordinates is a requirement which is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.”

Letter from Einstein to Born, January 1, 1954

Quantum measurement

Linear evolution:

Why interference cannot be seen?

n Decoherence: entanglement with the environment - same process by which quantum computers become classical computers!

n Dynamics of decoherence: related to elusive boundary between quantum and classical world

n Important reference: work by Caldeira and Leggett

End of XX century: Advent of Quantum Technology

9

Ion traps (Blatt, Wineland)

R. Blatt

Chapman, Haroche, Kimble, Rempe, Walther

Loss, DiVincenzo, Imamoglu, Awschalom

Decoherence and the quest for quantum computers

Josephson junctionsQuantum dotsIon traps

Nuclear Magnetic Resonance

Silicon

•

Wineland, Blatt

I. Chuang, D. Cory, R. Laflamme, E. Knill

B. Kane R. Clark

Van der Wal, Nakamura

DECOHERENCE

11

Cavity QED

Q = !⌧ ⇡ 108 � 1010

Microwave domain: Haroche, Walther Optical domain: Kimble, Rempe

up to a fraction of a second⌧

Field manipulation and measurement: Rydberg atoms (n=50 - 60), , lifetime 30 ms. Transition frequencies: 20 - 100 GHz. Transit time: 20 - 100 Transition dipoles: 1000 atomic units

` = n� 1µs

Quantized electromagnetic field

†

ˆ 1

ˆ 1 1

a n n n

a n n n

= −

= + +

( ) ( )

( )

( )

†

†

2 3

†

ˆˆ ˆ, 1

,

Electric field:

effective volume of mode,

defined

1ˆ ˆ ˆ ˆ ˆ

, , 0,

2

ˆ

1,2,

so that 1/ 4

ˆ ˆ

/

FH N N a a

E E au r

number operator

field per phot

a a N n n n n

E h V

V u r d

a u

r

r

onω

ω

π

ω

ε ε

ω

∗ ∗

⎛

⎡ ⎤ = = =⎣ ⎦

=

=

⎞= + =⎜ ⎟⎝ ⎠

⎡ ⎤= +⎣ ⎦

→

→

→∫

!

" " ""

…

"

"

( )2 3ˆ0 0 / 2E r d r ω=∫" "

!

Spatial dependence of the field mode

Polarization vector

( ) ( ) ( )i ru r u r e φ=!! !

3

3 3

Microwaves: 0.7 cm , 1.5 mV/m

Optical fields: V 10 m , 150 V/cm

V EEω

ωµ

≈ =

≈ =

Note that ˆ 0n E n =!

Coherent states

( ) ( )

( )( )

2

2†

† *

†

Require that

Classical energy

Require also that , so th

ˆ / .

ˆat

coincides with the classical en

"Quasi-classical

ergy wh

ˆ ˆ ˆ

1

ˆ ˆ

ˆ

states."

en

Then

cl

F

E r h Vu r c c

n a a H

a a

E

a

α α ω εα

α α α α α

α

α α

ω α

α α

α

⇒

= +

= =

− −

=

=

≫

"

# ## #

2* †ˆ ˆ ˆ

ˆ Eigensta

0

tes o ˆ f

a a a

a a

α α α α α α α α

α α α

− − + =

⇒ = →

† 1ˆ ˆ2FH a aω⎛ ⎞= +⎜ ⎟

⎝ ⎠!

[

2

†

/ 2

0

ˆ ˆ ˆExpansion in terms of eigenstates o

ˆ ˆ1 1

0

f

!

:

!

n n

n

a n a n n n

N a

n e nn

a

nα

α α α α α α α

α αα α α

∞−

=

= ⇒ − = = −

⇒ = ⇒ =

=

∑

��q = 1/

p2

↵ = |↵| exp(i�)Phase quadrature

Amplitude quadrature

q

p

( )

( )

0 03

†0

†

†

0

1 0ˆ , 0 1

1 0ˆ ˆ0 12

ˆ

, ,

( real, rotating-wave and dipole

approximati

ˆ

1ˆ ˆ ˆ

o

22

ˆ ˆ

Jaynes-Cum

ˆ ˆ

0ming

1

ˆ

sn - Hamiltonian)

ˆ

ˆ

,

2,

ˆ

F A

AF

I

I

H

H a a

a

H

a

e gH H

H

H

ω ωσ

σ σ

σ σ

ω

+ −

+ −

⎛ ⎞= +

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝

⎜ ⎟

= + +

Ω

⎠ ⎝ ⎠

⎛ ⎞

Ω= +

= = ⎜ ⎟−⎝ ⎠

=

⎝

=

⎠

!

!

!!

( )

( )

1 2

0

1 ˆ ˆ0 0 2

/

, ,

, atomic center-of-mass2 /eg

i g e e g

d h Vu R R

σ σ σ σ

ε ω

+ −

⎛ ⎞= + = =⎜ ⎟

⎝ ⎠

Ω = →− ⋅" """

!

Two-level atom interacting with one mode of the electromagnetic field

0δ ω ω= −

0

Two-level :

,

s

δ ω ω≪

0 0,ω ωΩ ≪

Rabi frequency

Semiclassical approximation – Bloch equations

( )

†0 3

0 *

1

3 , ,

Go to interaction picture corresponding to

, and set

(slowly-varying envelope of classical field) :

w

ˆ(rotating frame)

here

1 1ˆ ˆ ˆ ˆ2

,

,

2

ˆ2 2

ˆ2

ˆ ˆ ˆ

R

R

I

I I I

H a

H

aa

V

δσ

α

α α

σ

σ σ

ω

σ+ −

→⎛ ⎞= + +⎜ ⎟⎝ ⎠

Ω

Ω

Ω+

=

⋅= + =!"

!

"

" " !

( )

1

0 12

1, 2 2, 3 3

2

,

, with .

Atomic dynamics may be described in terms of the precession

of a pseudo-spin around a pseudo-magnetic field .

Se ˆ , then ˆ ˆt ,

,

,I I Ir r r

V V iVV

dr rdt

σ σ

αδ

σ

Ω ≡ ≡

Ω

−

≡ Ω≡ ≡ = ×

!

! ! !

Spin-precession equation

Bloch vector

SHOW THAT!

Bloch vector and density matrix

ˆ ee egA

ge gg

ρ ρρ

ρ ρ

⎛ ⎞= ⎜ ⎟⎝ ⎠

Atomic density matrix: Pure state:2 *

2*ˆ e e gA

e g g

c c c

c c cρ

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

e gc e c gψ = +

2 2 21 2 3Show that , the

equality holding if and only if t1

h:

e state is purer rExe ise rrc + + ≤

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )

1 1

2 2

3 3

ˆˆTr 0 2Re

ˆˆTr 0 2 Im

ˆˆ

coherences

populatir 0 onT

IA eg ge ge

IA eg ge ge

IA ee gg

r t

r t i

r t

σ ρ ρ ρ ρ

σ ρ ρ ρ ρ

σ ρ ρ ρ

⎡ ⎤= = + =⎣ ⎦

⎡ ⎤= = − =⎣ ⎦

⎡ ⎤= = −⎣ ⎦

⎫⎪⎬⎪⎭

→

SHOW THAT!

Bloch sphere

( )( )

/ 2

/ 2

cos / 2

sin / 2

i

i

e e

e g

ϕ

ϕ

ψ θ

θ

−=

+

e gc e c gψ = +

or

( )( )

*1

*2

223

2Re sin cos

2 Im sin sin

cos

e g

e g

e g

r c c

r c c

r c c

θ ϕ

θ ϕ

θ

= =

= =

= − =

1

2

3

r!Bloch vector

Bloch sphere

SHOW THAT!

2Im( )ρ

2Re( )ρ

ρ − ρ

e

e

e e

g

g

gg

Geometrical interpretation of atom dynamics

( ) 2 21 2, , , , dr r V V V

dtδ δ=Ω× Ω = Ω = +

! ! ! !!

Dispersive case: Vδ ≫

Bloch sphere

Bloch vector

0δ ω ω= −

2Im( )ρ

2Re( )ρ

ρ − ρ

e

e

e e

g

g

gg

φ

Resonant dynamics/ 2 rotationπ

( )12

ie e e gφ→ +

( )12

i ie e g e gφ φ+ →

( )12

ig e e gφ−→ − +

( )2

iie e e g e

φφ−− + →−

π Phase shift: 2π rotation of a spin 1/2

,e n

, 1g n +

δ0

Atoms and photons: The dressed atom

Uncoupled system

( )†0

2ˆ ˆˆ ˆI a aH σ σ+ −

Ω= +!

0δ ω ω= −

Couples only states within each doublet!

E

Effect of interaction

[ ]( )

( )

0

0

1 1 / 22

1 / 2 12

n

n nH

n n

δω

δω

⎡ ⎤+ − Ω +⎢ ⎥= ⎢ ⎥

⎢ ⎥Ω + + +⎢ ⎥⎣ ⎦

!

The dressed atom and the dispersive limit

0 1nΩ +

, 1g n +

,e n

,n+

,n−

δ0

Exercise: Show that, for AC Stark shift ⇒

0 ,1nδ Ω +≫

E( ) ( )2 2

, 01 12nE n nω δ± = + ± Ω + +!

!

Possible level scheme

|ei

|gi

|ii

!C

!0

Dispersive interaction between atom and

cavity mode

|↵ei�i

Manipulate through resonant classical fields

Dispersive interaction in classical physics

Transparent material (dispersive interaction): frequency change ⇒ phase change

ʹ ʹ

Application: Optical lattices

Periodic potential ⇔ Stationary light waves Electrons ⇔ Atoms

Insulator-superfluid Mott transition (quantum phase transitions: Greiner et al, Nature 415, 39 (2002)

( )†

Hubbard model:1ˆ ˆ ˆ 12i j i i i i

i j i i

H J a a n U n nε≠

= + + −∑ ∑ ∑

27

Decoherence dynamics

MEASURING DECOHERENCE IN CAVITY QED

e ge ge g

Atomic oven

Microwave

Microwave

Ionization detectors

e; g;

e; g; e;

g;

( )( )

/ 2

/ 2

e e g

g e g

→ +

→ − +

iF

eψα αΨ ∝ + − 0ge ψ π

ψ

=

⇒ =

⇒M. Brune, J.M. Raimond, S. Haroche, L.D. et N. Zagury, PRA 45, 5193 (1992)

HOW TO DETECT THE COHERENCE?

Send a second atom! [L.D., A. Maali, M. Brune, J.M. Raimond, and S. Haroche, PRL 71, 2360 (1993); L.D., M. Brune, J.M. Raimond, and S. Haroche, PRA 53, 1295 (1996)].

Results for phase difference equal to π:•Coherent superposition: preparation and probing atoms detected in the same state→Pee=1•Statistical mixture: second atom detected in |e〉 or |g〉 with 50 % chance→ Pee=1/2

EFFECT OF DISSIPATION

Φ = π/2

tcav/2〈n〉 Decoherence time: tcav/D

〈n〉 = average number of photons in cavity

L. D., M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A 53, 1295 (1996).

Exponential decay!

EXPERIMENTAL RESULTS

[Brune et al., PRL 77, 4887 (1996)]

Plot of Pee − Peg

QUIZZ

Where has the coherence gone? Can you describe what happens with the environment?

Why are coherent states more stable than superpositions?

Does this answer Einstein’s question?

1935

ENTANGLED STATES

Individual state of each part is not known: only global state is known!

State of the system cannot be written as a product of the states of the two systems

Classical X quantum situation

35

Global properties: Center-of-mass momentum

= total momentum Relative position

Individual properties: Momentum of each particle Position of each particle

36

Schrödinger on Entanglement

“This is the reason that knowledge of the individual systems can decline to the scantiest, even zero, while that of t h e c o m b i n e d s y s t e m r e m a i n s continually maximal. Best possible knowledge of a whole does not include best possible knowledge of its parts – and that is what keeps coming back to haunt us.”

Naturwissenschaften 23, 807 (1935)

ATOM-PHOTON ENTANGLEMENT

Blinov et al, C. Nature 428, 153 (2004)

MULTI-PARTITE ENTANGLEMENT

Multiparticle entanglement

Multiphoton entanglement

Magnetic salt LiHoxY1-xF4

Entanglement andEntanglement andmagnetic susceptibilitymagnetic susceptibility

Magnetic saltLiHoxY1-xF4

MACROSCOPIC SIGNATURE OF ENTANGLEMENT

ENTANGLEMENT AS A RESOURCE

• Entanglement is useful for communications and quantum computation

Dynamics of entanglement

n Multiparticle system, initially entangled, with individual couplings of particles to independent environments: each particle undergoes decay, dephasing, diffusion.

n How is local dynamics related to nonlocal loss of entanglement?

n How does loss of entanglement scale with number of particles?

n Need measure of entanglement!

Reminder of density operatorIf one measures a complete set of commuting operators, one determines the state of a system

Suppose however one does not measure a complete set, or that the measurements are not precise.Then one can only say that there is a probability pi that the state of the system is

Then, the average of any operator A is

ψi

A = pi ψi A ψi = Tr ρA( )i∑

where ρ = pi ψi ψii∑ Density operator

Entangled and separable states

n Separable states: – Pure states:

– Mixed states (R. F. Werner, PRA, 1989):

n Entangled state: non-separableBell states - Maximally entangled states: ρA,B =

121 00 1

⎛⎝⎜

⎞⎠⎟

Reminder of partial traceIf one has an entangled state of two parties, and one is interested in calculating the averages of operators acting only on one of the parties, it is useful to consider the density operator corresponding to it, which is obtained by calculating the partial trace of the total density operator with respect to the other party.Example:

⇒ ρA = TrB ψABAB

ψ( ) = 121 00 1

⎛

⎝⎜

⎞

⎠⎟

SHOW THAT!

|�±⌅ =1⌃2

(| ⇥⇥⌅± | ⇤⇤⌅)Partial transposition:1A � TB : �⇥ �TB

12

�

⇧⇧⇤

1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⇥

⌃⌃⌅⇥12

�

⇧⇧⇤

1 0 0 00 0 ±1 00 ±1 0 00 0 0 1

⇥

⌃⌃⌅

|�±⌅ =1⌃2

(| ⇥⇥⌅± | ⇤⇤⌅)Partial transposition:1A � TB : �⇥ �TB

12

�

⇧⇧⇤

1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⇥

⌃⌃⌅⇥12

�

⇧⇧⇤

1 0 0 00 0 ±1 00 ±1 0 00 0 0 1

⇥

⌃⌃⌅ρ =

ρ = ψABAB

ψ

47

Measures of entanglement for pure states

reduced density matrix of A or B

Separable state (two qubits):

Maximally entangled state:

Von Neumann entropy

Linear entropy

S(ρr ) = 0

48

What about mixed states?

n Let n Could try to define n But this decomposition is not unique!

18

49

Entanglement of mixed states

n Define

n Difficult to calculate!

A mathematical interlude: partial transposition of a matrix

��00 �01

�10 �11

⇥�

��00 �10

�01 �11

⇥Transposition: a positive map T : �� �T

|�±⌅ =1⌃2

(| ⇥⇥⌅± | ⇤⇤⌅)Partial transposition:1A � TB : �⇥ �TB

12

�

⇧⇧⇤

1 0 0 ±10 0 0 00 0 0 0±1 0 0 1

⇥

⌃⌃⌅⇥12

�

⇧⇧⇤

1 0 0 00 0 ±1 00 ±1 0 00 0 0 1

⇥

⌃⌃⌅

Negative eigenvalue!

Mixed states: Separability criterium

n If ρ is separable, then the partially transposed matrix is positive (Asher Peres, PRL, 1996):

n For 2X2 and 2X3 systems, ρ is separable iff it remains a density operator under the operation of partial transposition (Horodecki family 1996) - that is, it has a partial positive transpose (PPT)

52

Negativity as a measure of entanglement

=1 for a Bell state

Negative eigenvalues of partially transposed matrix

Dimensions higher than 6: =0 does not imply separability!

N (⇥AB) � 2�

i

|�i�|

A paradigmatic example: Atomic decay•Qubit states:

•“Amplitude channel”:

Usual master equation for decay of two-level atom, upon tracing on environment (Markovian approximation)

Weisskopf and Wigner (1930)!

Our strategy: follow evolution as a function of p, not t

Apply evolution to two qubits, take trace with respect to environment degrees of freedom, find evolution of two-qubit reduced density matrix, calculate entanglement

Realization of amplitude map with photons

HV

Sagnac-like interferometer

|g⇤|0⇤ ⇥ |g⇤|0⇤|e⇤|0⇤ ⇥

�1� p|e⇤|0⇤+

⇧p|g⇤|1⇤

H

V0 E

1 E

p = sin2(2✓)

Investigating the dynamics of entanglement

“Sudden death” of entanglement

“Entanglement Sudden Death”(Yu and Eberly)

N (p = 0) = 2|↵�|N

egat

ivity

Decay of entanglement for N qubits, other environments?

n Independent individual environments

59

Does entanglement become more robust with increasing N?

60

Is ESD relevant for many particles?

Role of environment

n Usually one traces out environment, and one looks at irreversible evolution of system

n As entanglement decays and eventually disappears, what is its imprint onto the environment?

61

Measuring the environment?

63

64

Collaborators: entanglement dynamics

65

Leandro Aolita Fernando de Melo Rafel Chaves Malena Hor-Meyll Alejo Salles

Osvaldo Jiménez-Farías Gabriel Aguillar

Daniel Cavalcanti

Marcelo P. de Almeida

Antonio Acín

Andrea Valdés-Hernandéz

Paulo Souto Ribeiro Stephen Walborn Joe Eberly Xiao-Feng Qian

THANK YOU!