Entanglement in Time and Space

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

• Entanglement in Time and Space
• J.H. Eberly, Ting Yu, K.W. Chan, and M.V.
  Fedorov
• University 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).

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Entanglement means a superposition of conflicting
information about two objects.
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A pair of conflicts can be entangled
Try to see both at the same time.
Do they flip together?
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Measurement cancels contradiction
A pair of boxes, but only one view of them
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Bell States provide a simple example
Schrödinger-cat Bell State ??-?gt
CgtNgt CgtNgt excited cat C, dead cat
C, excited nucleus N, ground state N
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Bell States provide a simple example
Schrödinger-cat Bell State ??-?gt
CgtNgt CgtNgt excited cat C, dead cat
C, excited nucleus N, ground state N
ltN??-?gt Cgt (sorry, Cat)
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Overview

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-2gt and
off-diag ? e-gt , 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).
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
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mixed initial states, C 2/3
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Time Evolution Result
Obvious point the atoms go to their ground
states, so d ? 3, and the other elements decay to
zero. No surprise the decay of r(t) is smooth,
and exponential, measured by usual natural
lifetime
?sA(t)? ?sA(0)? exp-GAt/2 iwAt
Reminder the atoms decay independently.
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Soln. in Kraus representation
Kraus operators
gA exp-G?t / 2 ???????wA2 1- exp-Gt ,
same for B.
for details, see Ting Yu and JHE, PRB 68, 165322
(2003)
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Kraus matrix evolution
For detailed treatment Ting Yu JHE, PRL 93,
140404 (2004).
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Entanglement evolution
Entanglement has its own rules, and follows the
atom decay law only exceptionally. Entanglement
can be completely lost in a finite time!
art by Curtis Broadbent
Ting Yu JHE, Phys. Rev. Lett. 93, 140404 (2004).
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Noise Entanglement
Werner state density matrix
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.
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Two Noises Entanglement
Pure off-diagonal relaxation of qubits
Add some diagonal relaxation, for example via
vacuum fluctuations.
Add diag. to off-diag. relaxation of qubits
Werner entanglement gets some protection from
added noise!
Ting Yu JHE (in preparation).
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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.
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Physical examples of breakup
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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
How much correlation is realistic? How to measure
it?
Original paper Einstein, Podolsky and Rosen,
Phys. Rev. 47, 777 (1935).
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Localization - Entanglement
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Time-dependent EPR example
Given a dissociation rate gd, a post-breakup
diatomic Y is
M.V. Fedorov, et al., PRA 69, 052117 (2004) /
quant-ph/0312119.
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Joint Probability Density Ytotal2
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Dynamics of localization What do we know and when
do we know it?

• Massive particles ? spreading wavepackets
• Dx ? Dx(t) and DX ? DX(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

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Dynamics of localization - F ratios
Plots of Y(t)2 vs. x1 and x2
Experiments track localization via packet
spreading (i.e.,spatial variances). The
two-particle ratio h(t) ?x/2?X is a convenient
parameter connected with dynamical evolution.
Chan, Law and Eberly, PRL 88, 100402 (2002).
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Universal man-in-street theory
Model the breakup state as double-Gaussian .
K.W. Chan and JHE, quant-ph 0404093
This makes it easy to calculate the Fedorov
ratios (F1 F2 ) at t0 and for later times.
Question can we guess what happens to
localization?
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Entanglement migration to phase
Therefore P(x, X t) ??(x, X t)?2 has two
similar real exponents.
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Quantum memory force (QMF)
Atom Photon
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 gt 2.
M.V.Fedorov, et al. (in preparation).
Chan-Law-Eberly, PRL 88, 100402 (2002)
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Summary / dynamics of entanglement

• 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 h
• Man-in-street theory and phase entanglement
• Memory effects enforce spatial configurations
  (QMF)

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Acknowledgement Research 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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More sophisticated Schmidt analysis
Any bipartite pure state can be written as a
single discrete sum

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

e.g.,
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Discretization of continuum information, the
Schmidt advantage
Unique mode pairs
Continuous-mode basis
Schmidt-mode basis
Pure-state non-entropic measure of entanglement
Schmidt number counts experimental modes,
provides practical metric
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Interpreting K, the Schmidt number
K 1, no entanglement. K 2, perfect Bell
states. K 5, beyond Bell, more information. K
10, still more info. Quantum info is
always discrete and countable.
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Estimation of K for photodissociation
Comparing the photodissociation process with the
double-Gaussian model, we identify Dx0 v / gd
and DX0 DR0. If we take DR0 10 nm,

, and define td gd-1, then with td in sec,
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Retreat of entanglement into phase
The Fedorov ratios for double-Gaussian

Position
Momentum
where
, so .
Note non-equivalence of k-space and x-space for
these experimentally measurable quantities.
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Double-Gaussian Schmidt analysis
For the man-in-street double-Gaussian model (with
m1 m2)
The Schmidt modes are the number states and one
finds
while from the actual wave function we had
inferred
with h(t) ?x(t)/2?X(t). These are the same,
except for spreading!