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”Entanglement in Time and Space”

J.H. Eberly, Ting Yu, K.W. Chan, and M.V. FedorovUniversity of Rochester / Prokhorov Institute

We consider entanglement as a dynamic property of quantum states and examine its behavior in time and space. Some interesting findings: (1) adding more noise helps fight phase-noise disentanglement, and (2) high entanglement induces spatial localization, equivalent to a quantum memory force.

• Ting Yu & JHE, Phys. Rev. Lett. 93, 140404 (2004).• K.W. Chan, C.K. Law and JHE, Phys. Rev. Lett. 88, 100402 (2002)• JHE, K.W. Chan and C.K. Law, Phil. Trans. Roy. Soc. London A 361, 1519 (2003).• M.V. Fedorov, et al., Phys. Rev. A 69, 052117 (2004).

Quantum Optics II Cozumel, Mexico, December 5-8, 2004

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Entanglement means a superposition of conflicting information about two objects.

Can you handle the conflicting informatio

n here? Which face

is in the back?

Superposition of conflicting information,

but only one object.

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Try to see both at the same time.

Do they “flip” together?

A pair of conflicts can be “entangled”

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Measurement cancels contradiction

A pair of boxes, but only one view of them

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Schrödinger-cat “Bell State”:

|-> =|C*>|N*> |C>|N>

excited cat = C*, dead cat = C, excited nucleus = N*, ground state = N

Bell States provide a simple example

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Schrödinger-cat “Bell State”:

|-> =|C*>|N*> |C>|N>

excited cat = C*, dead cat = C, excited nucleus = N*, ground state = N

<N|-> = |C> (sorry, Cat)

Bell States provide a simple example

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Issue -- entanglement in time and space.

Illustration #1 -- two atoms are excited and entangled but not communicating with each other. Result #1 -- both atoms decay, diag e-2t and off-diag e-t , just as expected; but entanglement of the atoms behaves qualitatively differently.

Illustration #2 -- two atoms fly apart in molecular dissociation. Result #2 -- entanglement means localization in space (a “quantum memory force”).

Overview

For detailed treatments: Ting Yu & JHE, PRL 93 140404 (2004), and M.V. Fedorov, et al., PRA 69, 052117 (2004).

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Illustration #1:

HAT = (1/2)AZA + (1/2) BZ

B , HCAV = kkak†ak + kkbk

†bk

HINT = k(gk*-Aak† + gk+Aak) + k(fk*-Bbk

† + fk+Bbk)

At t=0 the joint initial state AB is entangled and mixed (not pure). The atoms only decay (no cavity feedback). After t=0, what happens to entanglement?

A B

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mixed initial states, C = 2/3

=

++ +- -+ --

++ a 0 0 0+- 0 1 1 0 -+ 0 1 1 0 -- 0 0 0 d

Initial state, entangled and mixed, where d = 1-a.

C = concurrenceEOF concurrence

1 ≥ C ≥ 0

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=

a 0 0 00 1 1 0 0 1 1 0 0 0 0 d

a(t) 0 0 00 b(t) z(t) 0 0 z*(t) c(t) 0 0 0 0 d(t)

Obvious point: the atoms go to their ground states, so d 3, and the other elements decay to zero.

No surprise: the decay of (t) is smooth, and exponential, measured by usual natural lifetime:

Time Evolution Result:

±A(t) = ±

A(0) exp[-At/2 ± iAt]

Reminder: the atoms decay independently.

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Kraus operators:

K1 A 0

0 1

B 0

0 1

K2 A 0

0 1

0 0

B 0

K3 0 0

A 0

B 0

0 1

K4 0 0

A 0

0 0

B 0

A = exp[-t / 2 ]A2 = 1- exp[-t ] , same for B.

˜ (t) = K(t) (0) K†(t)

Sol’n. in Kraus representation:

for details, see Ting Yu and JHE, PRB 68, 165322 (2003)

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Kraus matrix evolution:

a 0 0 0

0 b z 0

0 z* c 0 0 0 0

dDecays all depend on A(t) = exp(-At/2) and B(t) = exp(-

Bt/2).

a(t) = AB, b(t) = B2 +A

2 B2, c(t) = A

2 +B2 A

2,

d(t) = A2 +A

2 +A2A

2, z(t) = AB , where A2 = 1 - A

2, etc.

±A(t) = ±

A(0) exp[-At/2 ± iAt]

Usual Born-Markov solutions for the separate atoms:

For detailed treatment: Ting Yu & JHE, PRL 93, 140404 (2004).

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art by Curtis Broadbent

Entanglement has its own rules, and follows the atom decay law only exceptionally.

Entanglement can be completely lost in a finite time!

Entanglement evolution:

Ting Yu & JHE, Phys. Rev. Lett. 93, 140404 (2004).

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Noise + Entanglement

W 1

3(1 F) I

1

3(4F 1) |( )( ) |

Werner qubits undergo only off-diagonal relaxation under the influence of phase noise.

All entanglement of a Werner state is destroyed in a finite time by pure phase noise.

Werner state density matrix

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Two Noises + Entanglement

Add some diagonal relaxation, for example via vacuum fluctuations.

Ting Yu & JHE (in preparation).

Werner entanglement gets some protection from added noise!

Pure off-diagonal relaxation of qubits

Add diag. to off-diag. relaxation of qubits

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Spatial localization and entanglement

With just two objects, high entanglement can be reached by allowing each object a wide

variety of different states. The idealized two-particle

wave function used by Einstein, Podolsky and Rosen in their

famous 1935 EPR paper used continuous variables (infinite number of states) to get the

maximum degree of entanglement.

Experiments following the original EPR “breakup”

scenario have not been done yet.

Physical examples of breakup

Spontaneous emission (K ≈ 1)Chan, Law and Eberly, PRL 88, 100402 (2002)Fedorov, et al. (in preparation, 2004)

Raman scattering (K > 100)Chan, Law, and Eberly, PRA 68, 022110 (2003)Chan, et al., JMO 51, 1779 (2004).

Down conversion (K ≈ 4.5 — 1000’s)Huang and Eberly, JMO 40, 915 (1993) Law, Walmsley and Eberly, PRL 84, 5304 (2000)Law and Eberly, PRL 92, 127903 (2004)

Ionization/Dissociation (K > 10)Fedorov, et al., PRA 69, 052117 (2004).Chan and Eberly, quant-ph 0404093

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EPR system is created by break-up

The variables entangled are positions (x1 and x2), or momenta (k1 and k2). Perfect correlation is implied in the EPR wavefunction:

Original paper: Einstein, Podolsky and Rosen, Phys. Rev. 47, 777 (1935).

How much correlation is realistic? How to measure it?

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Localization - Entanglement

• All information is in . We can plot the two-particle density ||2 vs. x1 and x2.

• Knowledge of one particle gives information about the other particle.

• Joint localization information is packet entanglement.

||2

Fedorov ratios for particle localization:

F1

x1

x1

condF

2x

2

x2

cond

We can calculate these for a simple dissociation model.

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relative part CM part

Time-dependent EPR example

Given a dissociation rate d, a post-breakup diatomic is:

(r1,r2;t) ~rr2 (vt r)exp d t

r

v

exp

R2

4R02

rel

2

(r vt)d v

CM

2

R R0

M.V. Fedorov, et al., PRA 69, 052117 (2004) / quant-ph/0312119.

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Joint Probability Density |total|2

rel

2

CM

2

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• Massive particles spreading wavepackets: x x(t) and X X(t)

• EPR pairs: [x, P] = 0 and [X, p] = 0 nonlocality

• Spreading is governed by the free-particle Hamiltonian.

• Time evolution is merely via phase in the momentum picture:

Dynamics of localizationWhat do we know and when do we know it?

˜ (k1;t)1

exp

k12

4k 2 iht

4m1

k12

˜ (k1;0)1

exp

k12

4k 2

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Dynamics of localization - F ratios

Experiments track localization via packet spreading (i.e.,spatial variances). The two-particle ratio (t) = ∆x/2∆X is a convenient parameter [*] connected with dynamical evolution.

Plots of |(t)|2 vs. x1 and x2 :

F(t) ~ A( t)B( t)

Calculation

Inferred dependence

F1

x1

x1

cond

* Chan, Law and Eberly, PRL 88, 100402 (2002).

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Universal man-in-street theory

This makes it easy to calculate the Fedorov ratios (F1 ~ F2 ) at t=0 and for later times.

Question: can we guess what happens to localization?

0(x

1,x

2)

1

abexp

x 2

4x0

2

exp

X 2

4X0

2

(x,X;t) ~ exp x 2

4(x02 it /2)

exp

X 2

4(X02 it /2M)

* K.W. Chan and JHE, quant-ph 0404093

Model the breakup state as double-Gaussian [*].

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(x,X;t) ~ exp x 2

4(x02 it /2)

exp X 2

4(X02 it /2M)

Therefore P(x, X; t) = (x, X; t)2 has two similar real exponents.

1

2

x1 x

2 2x

0

2 (t /2)2 x0

2

1

2

m1

MX

1

m2

MX

2

2

X0

2 (t /2M)2 X0

2

Entanglement migration to phase

When these exponents are added, the

nonseparable x1x2 term 0 at a specific time t0:

t02 M x

0X

0

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Quantum memory force (QMF)

The dissociation example has a close analog in spontaneous emission. These atom-photon space functions show a “force” arising from shared quantum information, a “quantum memory force” (QMF). The first four bound states are shown for Schmidt number K = 3.5, which is slightly “beyond-Bell.,” i.e., K > 2. M.V.Fedorov, et al. (in preparation).

Chan-Law-Eberly, PRL 88, 100402 (2002)

Atom Photon

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• Entanglement dynamics are largely unknown (time or space)

• Noisy environment kills entanglement but not intuitively

• Individual atom decay is not a guide for entanglement

• Diag. + off-diag. noise has a cancelling effect

• EPR-type breakup is ubiquitous / creates two-party correlation

• Conditional localization vs. entanglement ?

• Packet dynamics, Fedorov ratio and control parameter • Man-in-street theory and phase entanglement

• Memory effects enforce spatial configurations (QMF)

Summary / dynamics of entanglement

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AcknowledgementResearch supported by NSF grant PHY-00-72359, MURI Grant DAAD19-99-1-0215, NEC Res. Inst. grant, and a Messersmith Fellowship to K.W. Chan.

References Ting Yu & J.H. Eberly, PRL 93, 140404 (2004) and in preparation.M.V. Fedorov, et al., PRA 69, 052117 (2004). C.K. Law and J.H. Eberly, PRL 92, 127903 (2004).M.V. Fedorov, et al., PRA (in preparation).K.W. Chan, C.K. Law and J.H. Eberly, PRL 88, 100402 (2002).K.W. Chan and J.H. Eberly, quant-ph/0404093.A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).

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Any bipartite pure state can be written as a single discrete sum:

• Continuous basis discrete basis• Unique association of system 1 to system 2

Size of ent.

e.g.,

More sophisticated Schmidt analysis

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Discretization of continuum information,

the Schmidt advantage

Continuous-mode basis Schmidt-mode basis

Pure-state non-entropic measure of entanglement:

Schmidt number counts experimental modes, provides practical metric

Uniquemode pairs

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K = 1, no entanglement.

K = 2, perfect Bell states.

K = 5, beyond Bell, more information.

K = 10, still more info.

Quantum info is alwaysdiscrete and countable.

Interpreting K, the Schmidt number

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x0

2X0

~v

dR0

~ 1011 d

Comparing the photodissociation process with the double-

Gaussian model, we identify x0 = v ⁄ d and X0 = R0.

If we take R0 = 10 nm, ,

and define d = d-1, then with d in sec,

v2E

~ c

1 eV

10 GeV~ 103 m s

K 12

1

100

K

d (s)

Estimation of K for photodissociation

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Retreat of entanglement into phase

The Fedorov ratios for double-Gaussian :

G1 k1

k1 co cosh rK

e2 r e2r (t / t0)2

1 e2r(t / t0)2where

F1 x1

x1 co cosh r KC( t)

C(t)1

t0 2mx0X0

, so .

Note non-equivalence of k-space and x-space for these experimentally measurable quantities.

0(x1,x2 )

Position:

Momentum:

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0(x1, x2) n n1 (x1) n

2(x2)n0

The Schmidt modes are the number states

and one finds:

ni (x i) x i n

Double-Gaussian Schmidt analysis

while from the actual wave function we had inferred

For the man-in-street double-Gaussian model (with m1 = m2)

with (t) = ∆x(t)/2∆X(t). These are the same, except for spreading!