DECOHERENCE, ENTANGLEMENTAND ALL THAT

Chen Chia-Chu

NCHU, Physics Department

April/17/2009

Richard FeynmanI think it is safe to say that no one understands

Quantum Mechanics

The Quantum Principles

• State Vector of Hilbert space defined by the Hamiltonian H

• Physical Observables are Self-Adjoint (Hermitian) Operators

• Schroedinger Equation

(1)

• Fundamental Commutator Relation

• Probabilistic Interpretation

( )t

di Hdt

[ , ]q p i

2.x prob density

The success of QM

• Atomic spectra.• Conductivity of metal.• Magnetism.• Prediction of Semiconductor.• Explanation of Superconductivity.• The existence of Antimatter.• Fundamental interactions and elementary

particles

Consequences of QM

Gravitational Wave

• Quadruple Wave instead of vector wave as EM Radiation.

• Frequency ~

• Amplitude ~ Hulse-Taylor

• LIGO~

• Speed of light c

• Source: Binary Star, Supernova Explosion

7 1110 ~10 Hz

2010h 2610h

225 10h

Laser Interferometer Gravitational Wave Observatory (LIGO)

The LIGO detector

• Adapted from LIGO website

Dominant Sources of Noise

• seismic noise – mechanical vibrations carried through the earth – transmitted into interferometer via infrastructure – caused by plate tectonics, ocean tides, logging, cars, ....

• thermal noise – molecular motion associated with non-zero temperature – most problematic in the suspension system.

• photon shot noise – statistical fluctuations in number of photons measuring mirror

position – can be reduced by increasing laser power

                                               

             

A quantum-enhanced prototype gravitational-wave detector K. Goda1, O. Miyakawa2, E. E. Mikhailov3, S. Saraf4, R. Adhikari2, K. McKenzie5, R. Ward2, S. Vass2, A. J. Weinstein & Mavalvala1

• The quantum nature of the electromagnetic field imposes a fundamental limit on the sensitivity of optical precision measurements such as spectroscopy, microscopy and interferometry.

• The so-called quantum limit is set by the zero-point fluctuations of the electromagn

etic field, which constrain the precision with which optical signals can be measured. • The sensitivity of currently operational and future gravitational-wave detectors is li

mited by quantum optical noise. • They demonstrate a 44% improvement in displacement sensitivity of a prototype gr

avitational-wave detector with suspended quasi-free mirrors at frequencies where the sensitivity is shot-noise-limited, by injecting a squeezed state of light.

Nature Physics 4, 472 - 476 (2008)

Superposition and Coherence

Linearity of (1) Superposition

e.g. or the famous Schrodinger’s Cat

Such Coherence superposition of states does not occur in Classical Mechanics!!

In daily life, a cat is either alive or dead, but not a superposition ofboth!!

n nn

a

1( )

2

COPENHAGEN SCHOOL• Probabilistic Interpretation

• The need of Classical mechanics for measurement

From Landau’s QM: “The fact that the electron has no definite path means that it has also , in itself, no other dynamical characteristics. Hence it is clear that, for a system composed only of quantum objects, it would be entirely impossible to construct any logically independent mechanics. The possibilitiy of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy.”

Double Slits Interference HITACHI 1989

• Double-slit experiment is in the heart of in QM. In Reality, it contains the only mystery Feynman’s Lectures in physics

• This is the superposition of single particle which results with interference effect.

• What would happen if one consider the two-particle interference?

• That corresponding to the problem of Entanglement

“which is not one but rather the characteristic trait of QM” ( According to Schroedinger).

The detection of quantum object

• Physical Review, Vol.43, p. 491,Carl D. Anderson• The first Antiparticle: The Positron

Where is the boundary?

Adapted from Zurek’s Physics Today Article(199

1)

The Universe as a whole

• If QM needs Classical system for interpretation, then there is a problem for Quantum Mechanical description of our Universe.

“ Given almost any initial condition the universe described by

evolves into state that simultaneously contains many alternatives never seen to coexist in our world. Moreover, while the ultimate evidence for the choice of one such option resides in our elusive “consciousness” there is every indication that the choice occurs long before consciousness ever gets evolved.” Zurek Physics Today 1991

Density Matrix

Equivalent Formulation

More generally,

Pure State

otherwise is Mixed State

1[ , ]

dH

dt i

i i ii

( ) 1Tr

2

STERN-GERLACH EXPERIMENT

Pure Sate Mixed State

0 0

0 1

10

21

02

The Meaning of

Diagonal elements is the probability density of state

The off-diagonal elements provides quantum coherence

When all for then the system behaves classically, such as a class compose of 50% male students and with remaining 50% being female

ii

i

ij

0ij i j

ij

Assuming

The time evolution is given by Instantaneous disturbance

Denoting the nth column elements of U asThen

If the disturbance are randomly distributed in time and average overresults with zero off-diagonal elements

1

0, ,nn

lm l m n

A simple example for vanishing

10 0( ) '( )U t U t

1 2,,a a

0 0' ( ) exp[ ( )] i jij i j ij ij

E Et a a i t t

0t

QM for Closed SYSTEM

• In general QM is a good description for closed system.• The total system: System Environment is closed,

thus the system is described by a total wave function

where and represented the system and the environment respectively.

The density matrix of the total system is a pure state

,ij i j

i j

Another Look at Mixed state: Open System

• Any system that couples to environment is an open system

• The density of an open system can be obtained by tracing out the environment, namely

e.g. For two spins system

and the Reduced Density Matrix

Mixed State

( )s EnvTr 1

( )2

1( )

2

10

21

02

s

Quantum Measurement (1)• Instead of using a classical apparatus ,von Neumann proposed to measure with quantum device.• The Q device : begins with which clicks when the system

is

• The system state is

• The composite state

d

d d click

1( )

2

1( )

2i d

Quantum Measurement (2) Time evolution If the detector is seen in the state then the system is in Sounds Good, but there is something wrong!!

This state is the famous Bell State which is the same written in any basis!

where and are the eigenstates of

1( )

2c d d

d

1( )

2c d d

x

Quantum Measurement (3)

• The results of the outcome are undefined!!• In contrast to a classical result where the possible

results are well known. Such as the envelops containing split coin if one has the head then the other must have the tail

• Classically,

• Thus Q Meas. Provides Coherent Records!!

1 2 1 21/ 2( )c H H T T T T H H

Quantum measurement (4)

• In order to make sense for Q measurement, von Neumann proposed a further step which is not contained in the principle of QM, namely,

• By including the environment, such ad hoc step can be avoided and the problem of measurement can be resolved

(details is provided in Zurek’s article)

1/ 2( )c c

c d d d d

The ways to decoherence

• Macroscopic objects are constantly subjected to interaction with the environment which is unavoidable.

• Decoherence can also occur with the Fluctuating Vacuum, once again this is also inevitable.

EPR Paradox1935 Einstein, Podolsky and Rosen raised the question of completeness of Q

M.

The criteria of completeness contain the following 2 requirements

(1) Physical reality If without disturbing the system, the value of a physical quantity can b

e predicted with certainty, then there is an element of reality corresponding to this physical quantity

(2) Separability (or Locality) A good discussion can be found in Bohm’s Quantum Theory

At the top of Einstein's list of complaints was what he called "spooky actions at a distance". Einstein's "spookiness" is now called nonlocality

EEinstein-instein-PPodolsky-odolsky-RRosen pairosen pair

A B

101 10

2

Separable state

Non-separable state

(Entangled state)

(Maximally entangled state)

Erwin Schrödinger said, “If two bodies, each by itself

known maximally, enter a situation in which they

influence each other, and separateseparate again again, then there

occurs regularly that which I have [just] called

entanglement of our knowledge of the two bodiesentanglement of our knowledge of the two bodies”

The Bohm Experiment

• In Bohm’s proposal, the state of a system can be represented by

• The measurement of one particle’s implies the information of the other one.

• If Reality as they proposed is true, then the properties of the second particle pre-exist before the measurement of the first one.

• The measurement can be performed in any direction all three components exist

• Resulting QM is incomplete.

1( )

2

.

z

Elements of Reality

Taking the electron as an example

Resulting no precise values for measurement, except the defining values without violating the uncertainty principle.

Furthermore, their values depend on how we measure as such the separate reality belonging to the electron is in doubt.

Nonlocality or Nonseparability is the essence of QM

[ , ]x p i

佛 曰 佛 曰 : : 不 可 說不 可 說

Bell States

• There are 4 Bell States which are Linear Independent• Bell Singlet 1

( )2

1

2

3

1( )

21

( )2

1( )

2

Application of Entanglement

• All above analysis rely on the fact that the system is an Entangled State.

• Entanglement is the cornerstone for Quantum Information Science(QIS) including Quantum Cryptography, Teleportaion, Quantum Algorithm, Quantum Error Correction…

• More recently Entanglement has been introduced to analyze Many-body system and also Quantum Phase transition.

Life is not so easy --- decoherenceLife is not so easy --- decoherence

• The information is encoded in Quantum bit(qubit), such as a ½ spin state: and

State flipping

State phase changing

Such processes can lead to

“disentanglement”

ie

Decoherence by Vacuum Fluctuation

• Entangled state

• Disentangled by photon emission

Product state!!

11 0

21 0

1 01 1

0 0 02

1 02

Definition of Degree of EntanglementDefinition of Degree of Entanglement(restrict on bipartite system)

• Pure state Von Neumann entropyVon Neumann entropy maximally entangled

For example for with

Maximally Entangled

( ) - logS Tr ( ) - logS Tr

( ) ( ) - logtotal A A AE S Tr A B totalTr

0 1

( ) 0, ( ) 0non seperable seperable

E

E E

1:

2

maximally entangled

1 01

0 12A B

1S

Mixed State

• Mixed state ConcurrenceConcurrence, C, which is proposed by W. K. Wootters in 1998

1 2 3 4( ) [0, ]C Max

*eigenvalues of matrix and ( ) ( )A B A Bi y y y yR R

0 1C

Stability of Entanglement

• Physical systems are not isolated and always coupled to a environment.

• Decoherence is a big problem for entanglement whose stability is essential for all QIS.

• The time scale of decoherence depends on the system, it can varies from millisecond to nanosecond.

• Understand the mechanism of Dec. can help to control and manipulating of Q system.

Naïve Arguments

• Spin system is a representation of Bell states.• The basic interaction among spins is the Heisenber

g hamiltonian,

• This H can be viewed as the lowest order scattering result of electrons.

• The bell states are stable against this interaction. Pachos and Solano

1 2H J S S ����������������������������

More Careful Analysis

• Electrons are Identical Fermions.• Magnetic interactions are basically relativistic.• A consistent treatment requires a full analysis by

keeping relativistic correction to the same order.• To order of , this is equivalent to include

two Photons exchange diagrams

2( / )v c

Interactions Including Relativistic Corrections

NRQED to order of 2( / )v c

2 Electrons Wave Function

21 13 2

2

2

1( , )

2

with

1 Bell singlet state

2

can not be written down.

1 1 2 2 2 1 1p x p x p x p x1 2p p

ES

ES

EA

EA

e e

D CGC D G

D CG

1 2 1 2(total system) ( , ) ( , )x x s s

Consequences of scattering on Entanglement

• The spin singlet state is the only stable Bell state.• Other Bell states will evolve to no spin entangle stat

es by including all magnetic interactions in contrast to the stability from Heisenberg Hamiltonian.

• Bell singlet state can not be generated from other spin states regardless its state of entanglement.

• Bell singlet state is the only useful state for QI processes.

Ru-Fen Liu and CC Chen 2003 PRA, N. L. Harshman and S. Wickramasekara Physical Review Letters 98 080406 (2007) , Michael Bußhardt and Matthias Freyberger Physical Review A 75 052101 (2007)

Disentanglement and Spontaneous Emission

• Entangled state

• Disentangled by photon emission

Product state!!

11 0

21 0

1 01 1

0 0 02

1 02

Decoherence phenomenon in cavityDecoherence phenomenon in cavity

*and ( ) ( )A B A By y y yR *and ( ) ( )A B A By y y yR

i nitial state set up

(0) 0 0AB

A Bz z z

A B

• Source(environment) Spontaneous emissionSpontaneous emission

0

1

21

2

*

a ph a ph

a z

ph k k kk

a ph k k k kk

H H H H

H

H a a

H g a g a

Equation of MotionEquation of Motion• Equation of motion of total system

A B totalTr A B totalTr

0

( )( ), ( )

( ) ( ), (0) ( ), ( ), ( )

( ) ( ( ))

( ), (0)

( ),

a ph

t

a ph a ph a ph

AB

a ph

a ph

b

b

b a

T

d ti H t t

dt

t i H t dt H t H t t

t t

i H t

r

Tr

Tdt H tr H

0

( ), ( )t

ph t t

: in Interacting pictureO

Born-Markov approximationBorn-Markov approximation• Interaction between system and bath is weakweak and

assume the bath is almost stationary Born approx.

• Due to weak interaction, memory effect is not important Markov approx.

• Equation of motion of system

A B totalTr A B totalTr

0

( ) ( ), ( ), ( ) 0 0t

AB a ph ab pht dt H t H t tTr

, 0 0 0b a phTr H

3(0) (0) 0 0 ( ) ( ) 0 0 ( )AB ABt t O g

3( ) ( ) 0 0 ( ) ( ) ( ) 0 0AB ABt t O g t t

Master EquationMaster Equation

( ) / ( ) is the real/imaginary part of ( )R If t f t f t

In Schrödinger PictureIn Schrödinger Picture

Master EquationMaster Equation

Lamb ShiftLamb Shift

Unitary evolutionUnitary evolution

Dipole interactionDipole interactionEnvironment effectEnvironment effect

Evolution solutions Evolution solutions initial state set up

(0) 0 0

11

22 23

32 33

44

( ) 0 0 0

0 ( ) ( ) 0( )

0 ( ) ( ) 0

0 0 0 ( )

t

t tt

t t

t

23(0) Im (0)I

Time evolve

0( )

2

t

Rdt f t t

0( )

2

t

Idt f t t

22 33 23 32

22 33 23 32

(0) (0) (0) (0)

(0) (0) (0) (0)

S

S

11( ) 4 (0)S t S t

Correlated statesCorrelated states

The classically correlated states separable state

• A general interesting state Werner stateWerner state

easy to characterize entangled mixed state

A B totalTr A B totalTr A B totalTr

A Bi i i

i

p

WERNER STATES

10 0 0

31 2 1 2

0 06 6

1 2 1 20 0

6 61

0 0 03

F

F

F F

WF F

F

1entangled state

2F

1separable state

2F

Dynamical Evolution

One can set initial state as Werner state , then see what happened after evolution

( , ) 0 entangled state

( , ) 0 no-entangled state

if t F

if t F

Entanglement preservedEntanglement preserved

With F<1/2, no-entangled With F<1/2, no-entangled state can evolve to state can evolve to entangled stateentangled state

Defined by PPT Defined by PPT

Fig. 2 The concurrence under evolution has been shown for the case of F=0.75 Fig. 2 The concurrence under evolution has been shown for the case of F=0.75 and F=0.25. One can see the entanglement is enhanced and become stable for and F=0.25. One can see the entanglement is enhanced and become stable for both case within finite time. both case within finite time. The quantities of stabilized entanglement are just The quantities of stabilized entanglement are just the ingredients of singlet of initial states and they are distillablethe ingredients of singlet of initial states and they are distillable..

No entangled initial stateNo entangled initial state

Evolve to a Evolve to a entangled state entangled state and stableand stable

entangled entangled initial stateinitial state

Entanglement is enhanced and Entanglement is enhanced and getting to stablegetting to stable

Summary

• By including interactions between atoms, it is shown that disentanglement is suppressed even with photon emission.

• For Werner State the evolution is controlled by non-unitary evolution.

• The existence of singlet state is crucial for persistence of entanglement.(R Tanas et al 2004 J. Opt. B: Quantum Semiclass. Opt. 6 S90-S97)

Ru-Fen Liu and Chia-Chu Chen Physical Review A 74 024102 (2006)

Thank You!!