Vida y Muerte del Entrelazamiento Juan Pablo Paz

Departamento de Física “Juan José Giambiagi”FCEyN, UBA

Instituto de Física de Buenos Aires (IFIBA, UBA CONICET)

http://www.qufiba.df.uba.ar

Coloquio CNEAAgosto 13, 2010

HOY: ENTRELAZAMIENTO

ENTRELAZAMIENTOUna breve revisión. Por qué es interesante?

Entrelazamiento en sistemas cuánticos abiertos.

UN EXPERIMENTO PROPUESTO (una simulación cuántica en una trampa de iones).

UN MODELO SIMPLE EN EL QUE LA EVOLUCION DEL ENTRELAZAMIENTO NO ES TRIVIAL

Dos osciladores acoplados a un único reservorio. Dinámica para tiempos largos. Fases dinámicas.

CARACTERIZAR LA DECOHERENCIA. Tomografía de procesos cuánticos: Teoría y experimentos.

EL ENTRELAZAMIENTO ES UNA PROPIEDAD BASICA DE LA MECANICA CUANTICA

Schrödinger (1935) "Discussion of probability relations between separated systems" en Proceedings of the

Cambridge Philosophical Society

“When two systems, of which we know the states … enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then

they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own.”

“As a consequence of the interaction the systems became entangled”“Vershrankung”

“I would not call that one but rather the characteristic trait of quantum

mechanics, the one that enforces its entire departure from classical lines of

thought.”

Propiedades extrañas de los estados entrelazados…

1 2 Ψ 12 =12

↑ z 1↓ z 2

− ↓ z 1↑ z 2( )

Pr ob dA = ROJO( ) = Pr ob dA = VERDE( ) =12

Pr ob dB = ROJO( ) = Pr ob dB = VERDE( ) =12

ALEATORIEDAD COMPLETA EN LABOS A Y B

Propiedades:dA

Labo AdB

Labo B

Pr ob dA = ROJO ; dB = ROJO( ) = Pr ob dA = VERDE ; dB = VERDE( ) =14

1 − ˆ d A • ˆ d B( )Pr ob dA = ROJO ; dB = VERDE( ) = Pr ob dA = VERDE ; dB = ROJO( ) =

14

1 + ˆ d A • ˆ d B( )

CORRELACIONES FUERTES ENTRE A Y BLas correlaciones admitidas por los estados entrelazados son DEMASIADO

FUERTES: incompatibles con cualquier modelo ‘realista y local’.

EINSTEIN (EPR): LOS ESTADOS ENTRELAZADOS NOS MUESTRAN QUE LA MECANICA CUANTICA LLEVA LA

SEMILLA DE SU PROPIA DESTRUCCION!“Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

A. Einstein, B. Podolsky, N. Rosen.Physical Review 47, 1935, 777-780.

Two particles (A and B) in an entangled state: Eigenstate of total momentum and relative coordinate (two commuting observables).

Measuring position in Lab A we predict position in Lab B (no disturbance in B).Measuring momentum in Lab A we predict momentum in Lab B (no disturbance in B).

Therefore: Position and momentum must be “written” in the stuff that is in Lab B (nothing in Lab B can depend on what I decide to do in Lab A!!).

Therefore: Quantum Mechanics is incomplete

CUALQUIER teoría que acepte el “realismo local” da lugar a predicciones cuantitativas (desigualdades de Bell)

que pueden ser testeadas en experimentos.

La mecánica cuántica predice violaciones a las desigualdades de Bell (para estados entrelazados).

EXPERIMENTOS DE DETECCION DE VIOLACIONES A DESIGUALDADES DE BELL (Aspect 1982,…)

•Fuente de pares de objetos entrelazados•Separar los objetos de modo tal que no se des-entrelacen

FUENTES• LUZ

• MATERIAX

Kwiat et al. Phys. Rev. Lett. 75, 4337 (1995)

BBO-Kristall

UV-Pump

+| ⟩ | ⟩H VA A| ⟩ | ⟩V HB B

AB

V

H

kpump = kphoton1 + kphoton2

wpump = wphoton1 + wphoton2

The evolution of SPDC polarization-entanglement

01

10100

1,00010,000

100,0001,000,000

10,000,000

1985 1990 1995 2000 2005

Rep

orte

d ra

te (s

-1)

BBOBBO

AliceAlice

LaserLaser

BobBob

PICT 2004 ANPCyT objetivos:•Armar una fuente de fotones entrelazados,

un sistema de detección y manipulación•Caracterizar la fuente detectando violación

de desigualdades de Bell•Realizar experimentos originales de

manipulación de información cuántica (algoritmos cuánticos)

Primera evidencia de violaciones a desigualdades de Bell detectada en Argentina. Febrero de 2009.

QM: es válida aquí también!- 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0- 1 0 0

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

Coi

ncid

enci

as

β ( g r a d o s )

α = 0 º α = 4 5 º α = 9 0 º a = 1 3 5 º

S=2.72 (0.06)

FOTONES ENTRELAZADOS EN BUENOS AIRES!!Colaboración QUFIBA-CEILAP (Citefa)

Christian Schmiegelow, Miguel Larotonda, Alejandro Hnilo

ES UN RECURSO FISICO!SIGLO XXI: GENERALCION,

MANIPULACION Y CONTROL DEL ENTRELAZAMIENTO EN ESCALA

MACROSCOPICA

(2010)TELEPORTACION

DISTRIBUCION CUANTICA DE CLAVESCOMPUTACION CUANTICA

PERO HOY EL ENTRELAZAMIENTO NO SOLAMENTE ES UNA FUENTE DE SORPRESAS PARA DISCUSIONES FUNDAMENTALES…

TELEPORTACION

Entrelzamiento

Teleportación in an ion trap: Wineland et al; Blatt et al

(Nature, 2004)

Environment

System

…

Environment

…

System

HOY: Evolución del entrelazamiento en un sistema cuántico abierto?

• Entrelazamiento entre grados de libertad de traslación de dos partículas. Un sistema sencillo con evolución compleja

• Tres “fases dinámicas” para la evolución del entrelazamiento (y su detección en experimentos de trampas de iones).

POR QUE NO EXISTEN HOY TECNOLOGIAS CUANTICAS VIABLES ?(BASADAS EN EL

USO DEL ENTRELAZAMIENTO)ES CULPA DE LA DECOHERENCIA

(que induce des-entrelazamiento y la transición cuántico-clásica)

Environment

System

… …

HOW TO FIGHT AGAINST DECOHERENCEANTES: NUESTRO PRIMER EXPERIMENTO ORIGINAL

Cómo caracterizar eficientemente el proceso de decoherencia que sufre un sistema cuántico?

PRIMERA IMPLEMENTACION EXPERIMENTAL DE UN NUEVO METODO PARA “TOMOGRAFIA DE PROCESOS CUANTICOS”

Para ver esta película, debedisponer de QuickTime™ y de

un descompresor .

“Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?A. Einstein, B. Podolsky, N. Rosen.

Physical Review 47, 1935, 777-780.

Entangled states of two particles

r r =

r r 1 −

r r 2 ;

r P =

r P 1 +

r P 2

Ψ 12 =r r =

r r 0 ;

r P =

r P 0

Ψr r ,

r P ( ) = δ

r r −

r r 0( )δ

r P −

r P 0( )

1 2

These are non-physical states. A set of physical (yet entangled) states:Two mode squeezed states.

a 12

= x 12

+ i p 12

; Ψ 12 = exp r a 1+ a 2

+ − a 1 a 2( )( ) 0 12

m Ω δ x −

δ p −

=δ p +

m Ω δ x +

= exp 2 r( )

δ x −

δ p −δ x +

δ p +Gaussian states with finite entanglement

Measure of entanglement E=2r

Can be transfered to spins by means of local operations (in two distant labs) and then use

it for teleportation, etc…

PREVIEW: ANIMALS IN THE ZOO OF SOLUTIONS (NUMERICS)

Environment

System

…

Zero temperature, ohmic environment

It is possible to have entanglement in the

final state! Induced by the environment.

Environment

System

… …

THE ENTANGLEMENT MEASURE

We restrict to cinsider initial Gaussian states (to get analytic results). They remain Gaussiann.

REDUCED STATE OF THE SYSTEM IS CHARACTERIZED BY COVARIANCE MATRIX “V”

Entanglement measure: Logarithmic Negativity

: minimum symplectic eigenvalue of the partially transpose covariance matrix

Assumptions: 1. Linear system

2. Continuum spectral density

3. Gaussian states

Environment

System

…

THE MODEL: CALDEIRA-LEGGETT

THE SOLUTION

Resonant oscillatorsEnvironment

ALMOST IDENTICAL TO THE USUAL “QUANTUM BROWNIAN MOTION” MODEL

INTEGRATING OUT (TRACING OUT) THE ENVIRONMENT WE OBTAIN AN EXACT MASTER EQUATION

Damping term Normal diffusion Anomalous diffusionRenormalized Hamiltonian

Ohmic environment J ω( )= 2mγ ωπ

θ Λ −ω( )→ 2mγ ωπ

; γ t( )→ γ t >> Λ−1( )high TD → 2mγT

f → − 2γTπΩ

log Λ + ΩΛ − Ω

T=0

D→mγΩ+2mγ2

π2logΛ

Ω−1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

f →2γπ

logΛΩ

Asymptotic regime: ,

A PROPERTY OF THE SOLUTION

CONSIDER THE ASYMPTOTIC STATE FOR QUANTUM BROWNIAN MOTION

Δp =D2γ

; ΩRΔx =D

2m2γ−

fm

K =p2

2m=

D4mγ

V =12

mΩR2 x 2 =

D4mγ

−f2

rc =12

log mΩRΔxΔp

⎛

⎝ ⎜

⎞

⎠ ⎟ =

14

log 1−2mγ f

D⎛ ⎝ ⎜

⎞ ⎠ ⎟

ASYMPTOTIC STATE VIOLATES EQUIPARTITION!

ASYMPTOTIC STATE IS SQUEEZED!!

QUANTUM WEIRDNESS??

FOLLOWS FROM MASTER EQUATION… COEFFICIENT f IS THE ONE TO BLAME!

In the asymptotic regime:

The equilibrium state is squeezed due to f (low temperature regime)

THEN THE ASYMPTOTIC FORM OF THE COVARIANCE MATRIX IS

V =

Free oscillator of mass m and frequency

Correlations between the oscillators are zero in the asymptotic regime

Equilibrium moments of oscillator

COMPUTING ENTANGLEMENT FOR LONG TIMES

EVALUATE ANALYTICALLY THE LOGARITHMIC NEGATIVITY FOR THE OSCILLATORS

ENTANGLEMENT IN THE ASYMPTOTIC STATE

˜ E N → Maximal Sqeezing( )− Entropy( )ΔEN → Minimal Sqeezing( )

WE CONCLUDE THAT THERE ARE THREE QUALITATIVELY DIFFERENT ASYMPTOTIC BEHAVIORS!!

Mean value:

Amplitude of oscillations:

Where:

Squeezing of the oscillator

Squeezing of equilibrium for the oscillator

Analytical expressions can be obtained for the asymptotic values of the coefficients of the master equation for an ohmic environment. We can use them to construct a phase diagram

Phase diagram (complete description of the asymptotic behavior) :

Accurately describe all numerically simulations !

THREE DYNAMICAL PHASES FOR ENTANGLEMENT

Interesting points in the phase diagram :

T0 is such that Δx T =T0

=h

2mΩ

Δx

r1 is such that

r1 = log

h

2 m ΩΔ x T = 0( )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

r 2 = logΔ p T = 0( )

m Ω h

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

r1 =12

log π2

1 − γ 2 / Ω 2

arccos γ / Ω( )⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ r1 =

12

log 1 − 4 γ 2 / Ω 2

1 − γ 2 / Ω 2arccos γ / Ω( ) +

4π

γΩ

log ΛΩ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

THREE DYNAMICAL PHASES FOR ENTANGLEMENT

THREE DYNAMICAL PHASES FOR ENTANGLEMENT

SDR region (also non-Markovian) decreases width for ohmic environment as .

1Λ

Entanglement in the NSD Island COMES FROM the squeezing of the equilibrium state! (from the environment)

NSD Island: Non Markovian, non perturbative

SD region: entanglement dies because entropy is too large.

NSD “continent”: entanglement persists because it is in a protected state.

˜ E N → Maximal Sqeezing( )− Entropy( )ΔEN → Minimal Sqeezing( )

HOW TO OBSERVE THESE PHASES IN AN ION TRAP\

•Three ions ina linear trap

• Central ion (dffierent species) continuously laser cooled. Central ion is the gate to a dissipative channel (sympathetic cooling)

• Central ion copules to TWO normal modes only (we use radial modes; breathing mode is decoupled from the environment)

Environment

…

System System

H =p i

2

2 m i

+12

m iω x2 x j

2 + m iω z2 z i

2( )⎛

⎝ ⎜

⎞

⎠ ⎟

i∑ +

e 2

4 π ε 0

1

x i − x j( )2+ z i − z j( )2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ i

∑

d eq =5

16 ε 0

e 2

m ω z

, H =p i

2

2 m ii∑ +

m2

γ ij x i x jij

∑ ; γ ij =e 2

2 π ε 0 z i − z j( )3

Ý ρ = − i H , ρ[ ] +Γ2

2 a ρ a + − a + a ρ − ρ a + a( )a → CM − mod e

Δ p +2 =

11

2 mc 1

2

ω 1

+c 3

2

ω 3

⎛

⎝ ⎜

⎞

⎠ ⎟

Δ x +2 =

m2

ω 1c 12 + ω 3 c 3

2( )

rcirt =12

ln m ν 2Δ x +

Δ p +

⎛

⎝ ⎜

⎞

⎠ ⎟

S cirt =12

ln 4 Δ x + Δ p + δ q − δ p −( )

ENTRANGLEMENT OF MOTIONAL DEGRESS OF FREEDOM

World recordTwo Be atoms separated about 1mm

January 2009 (D. Wineland, NIST Boulder)

Una hamaca de 7 átomos de Calcio

52 átomos de Calcio “fotografiados”

Una acordeón de 7 átomos de CalcioCa: R. Blatt (Innsbruck)

*** F. Schmidt-Kahler (Ulm)

Be: D. Wineland (NIST, Boulder)

Ba: C. Monroe (Maryland)

Etc, etc…

A POSSIBLE EXPERIMENT IN AN ION TRAP

A ROUGH GUIDE THROUGH THE EXPERIMENT

• Three ions are cooled, y-potential is tight. Radial (x) modes are used for the simulation.

• Then central ion is cooled continuously. This cools two normal modes. “Temperature”: free parameter

• Squeezing of radial mode is created by varying x-trap frequency. Initial squeezing is free parameter.

• Final state dispersions are measured (or vibrational x-state is transferred to the internal degree of freedom and

fluorescence experiment is performed)

System System

24 Mg 24 Mg9Beωx =1.68ωz ⇒ rcrit ≈0.75, Scrit ≈0.32ωx =2.2ωz ⇒ rcrit ≈0.47, Scrit ≈0.40

A POSSIBLE EXPERIMENT IN AN ION TRAP

Entanglement can be transferred to the internal degree of freedom (qubit). How? Using sequence of pulses (state dependent forces)?

How many pulses? Three or four is enough. How much entanglement transferred? (dotted lines)

System System

24 Mg 24 Mg9Beωx =1.68ωz ⇒rcrit ≈0.75, Scrit ≈0.32ωx =2.2ωz ⇒rcrit ≈0.47, Scrit ≈0.40

CONCLUSION: NON TRIVIAL ENTANGLEMENT DYNAMICS CAN BE OBSERVED IN AN ION TRAP

Environment

System

…

System System

24 Mg 24 Mg9Be