Post on 23-Aug-2021
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)
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!