Transcript Dynamical Realization of Coherent Structures in Condensed Matter

Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Coherence 2006
Roma - April 21th, 2006
Dynamical Realization of
Coherent Structures in
Condensed Matter
Luca Gamberale
Pirelli Labs – Materials Innovation Advanced Research (Milano, Italy)
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Summary
•Overview of the ideas on Quantum Coherence and Coherence
Domain
•Coherent States: a simplified approach
•The effect of temperature
•Coherent Interactions
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Condensed Matter
• How can a system of weakly interacting atoms organize itself
and form highly ordered structures over large scales?
Examples:
Superfluidity/Superconductivity
Gas/Liquid/Solid Phase
Crystals
Biological Systems
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
“Orthodox” description
Paradigm of electrostatic hooks
Question: Is the interaction between neighbors sufficient to
guarantee long-range order?
An interesting example: like-charge attraction
Observation of long-ranged many-body attractive forces among
sub-micron latex sheres suspended in water, that cannot be
explained by means of short-range electrostatic interactions
A.E.Larsen, D.G.Grier, Nature 385, 230 (1997)
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
QED: a critical analysis
• Matter has been traditionally considered as a collection of
PARTICLES. What happens if we treat matter as a WAVE?
Matter states are
eigenvectors of number
operator
Quantum Phase completely
undefined
the radiation field is screened and may be neglected
Matter states are
eigenvectors of
destruction operator
Quantum Phase completely
fixed
the radiation field cannot be neglected in some circumstances
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
What happens if we take into account the
radiation field?
Contribution of order N
NEGATIVE!
(0)
(2)
H total  H matter
 H SR  H ra(1)d  H rad
 H em
Matter Field
Short-Range
Free e.m. field
Matter-Field Interaction
(usually neglected)
A2-term (required by gauge-invariance)
A careful analysis shows that COHERENT CONFIGURATIONS exist
whose energy is LOWER than those at zero-field
Symmetry Breaking
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Coherent condensation of a system of N
identical particles
Not equivalent to
Long range!!
Definition of quantum coherence
Ex: Bose condensate
0  x ,  ; t      x ,  ; t    0
Slow (classical) evolution
Locked phase
Quantum fluctuation

Random phase

Complex plane
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

1

N 
N
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Two-level system
The simplest of the many-body systems
But...
of enormous physical importance
Atomic Energy Levels
Many-level system
Two-level approximation
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Coherence equations for the Two-level system
(Preparata Equations)
i 1 ( x , )  g  2 ( x , )A ( x , )
i  2 ( x , )  g 1 ( x , )A * ( x , )
1
 A ( x , )  iA ( x , )  A ( x , )  g  d 3 yG ( x  y )  2* ( y, ) 1 ( y, )
2
g  eJ
Not present in laser eqs.
Responsible for runaway
4
302
e 2 N
 2
0 V
N
V
Interaction term
Electromagnetic field
A ( x , )
1 ( x , ),  2 ( x , ) Matter Field
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Electromagnetic structure of a single
Coherence Domain
The e.m. field has the
same phase for each point
of the CD
E ( x, t )  4
Quantum phase
 s
RA 0 j0 (r )sin Rt  3ˆ
30
B  x  cos  RP
t
 s 6
xˆ  3ˆ
B( x , t )  4
A 0 j1  r  cos Rt 
3
30 5
xˆ  3ˆ
r  0
6 2
x32 
2
 x1  x2  
5
2
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J
E  x  sin Rt 
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Total Reflection of trapped em field
1. CD does not radiate because
Poynting vector has zero mean value
2. Due to frequency renormalization,
photons cannot radiate because offshell
R  k
Total reflection: a natural
‘trapped’ laser
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Structure of bulk matter
Condensed Matter is
viewed as a collection of
COHERENCE DOMAINS
2-fluid model
Domain
(Coherent Matter)
Fluctuations
(Incoherent Matter)
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Coherent states: a simplified approach
H  HF  HA
H F    cs†cs
s 1,2
HA 
1
2M
2
2
Ze
1
e




P

A
(
R
)

p

A
(
r
)


i 
ij   V Ri , rij
 i
 ij
c
c
 2m i 1 j 1 

i 1 
N
N
Z


2
N


 Z  piz2

  Pi

H   
 Vij    
 v  i1 , i 2,..., iZ    i  E Ri   H F ,
i 1 


  2M j  i  z 1  2m

 
N
S. Sivasubramanian, A. Widom, and Y. N. Srivastava, Physica A 301, 241 (2001).
Z
i  e iz
z 1
 2  
E ( R)  i 

 N 
1/ 2
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  e
s 1,2
s
 ik  R
cs   s*eik  Rcs† 
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
The trial Quantum State
EM Glauber state
cs
   R, 1 ,  2 ,...,  Z  
N
em
Matter field Anzats
 i 1 , 2 ,...,  Z  p sin 
e
i
k R
2
V
  s1 N 
0 
N
em

2
 1 ,  2 ,...,  Z  s cos 
e
i
k R
2
V
01   p   s  0, 0  0
Full variational quantum state
  ,   
N
    Ri , i1, i 2 ,..., iZ  d Ri Ri
N
3
em
i 1
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Z
3
d
 iz iz
z 1
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Evaluation of the energy-per-particle
2 2
E ( ,  ) 1
k

  ,   H   ,   
 Es    sin 2    2   sin 2 
N
N
8M
  E p  Es
  0
2

Minimum exists when   1
2
 E
   0

 E  0
 
Emin  Es 
  crit

4
2  
  Mc 2
2
  2 


202
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Graphical representation of the energy
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Energy as a function of density (example)
Energy per particle
0.1
Energy gap only
Lennard-Jones plus Energy gap (Rc=R0, V0=Delta/40)
0.05
0
-0.05
-0.1
-0.15
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
/ crit
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1.9
2
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Important Remarks 1/2

  N sin 201 E (0),
4
and this happens only if the matter field has a modulation with period
k
Condensation occurs only if we have
N
i 1
i E ( Ri )
m
The ith atom is in a configuration TOTALLY delocalized in space, in contrast
with the particle-like character of its incoherent counterpart
The wave function of the single atoms in the coherent state is different from
that of the lowest energy state without em condensate, since it contains a
certain fraction of the excited state. This very important issue implies that,
when matter interacts with external fields, it may exhibit unexpected behavior,
a phenomenon referred to as violation of asymptotic freedom.
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Important remarks 2/2
Simplified approach not applicable in this form
to liquid water because of important dispersive
contributions arising from excited levels of the
water molecule (developed by G.Preparata and E.Del Giudice).
Complete revision of theory and numerical
calculations is presently under way (myself and E.Del
Giudice).
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Energy of a coherent domain. The effect of
temperature
Incoherent (thermal) Excitations
Boltzmann-distributed
Quasi-particle excitations
(Bogoliubov Spectrum)
Two-fluid
picture
Normal
Perturbative energy
Energy Gap (per particle)
Forbidden
Coherent
Zero-temperature coherent particles
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
The effect of temperature

Free
particles
massons
rotons
phonons
T=0

Free
particles
massons
rotons
phonons
k
T>0
k
Coherent phase
Coherent phase

1° sound
Free
particles
massons
Quasi-particle spectrum
phonons
rotons
T>>0
(Bogoliubov)
k
Coherent phase
(progressively depleted)
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Theoretical consequences of quantum
coherence in matter at finite temperature
•
The existence of coherent configurations in matter implies the
emergence of COHERENT SCATTERING
N
Incoherent scattering
Atot   A1e
j 1
i j
 tot  N1
Kinematics completely different!
N
Coherent scattering
Atot   A1
 tot  N  1
j 1
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2
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Coherent interactions: features
•
•
•
•
•
•
Increased probability of interaction by orders of magnitude
Different kinematics
Extremely low energy exchange
Virtually no entropy generation
Non-local interaction
Seems adequate to the description of BIOLOGICAL PROCESSES
When
•
Energy exchanged unable to overcome the energy gap
In most cases both coherent and incoherent interactions occur (e.g.
Moessbauer effect) with relative balance depending on temperature
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Conditions for coherent scattering
Incident particle must not be able to
overcome the coherent energy gap
Incident particle
 , k 
Phonon excitations
Perturbative energy
Energy Gap (per particle)
Forbidden
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Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Quantum Coherent Interactions
N scatterers
Quantum phase
Quantum incoherent
Quantum coherent
Cross section goes like N
Cross section goes like N2
N~1023 !!!
Pirelli Labs Materials Innovation – Advanced Research
Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter
Coherence 2006 – Roma - April 21th, 2006
Conclusions
•The generally accepted theory of condensed matter
misses the contribution of the radiative field, that in
particular circumstances cannot be neglected.
•Consideration of the radiative term brings to a
potentially rich and powerful theroretical tool.
•New kinds of many-body, non-local interactions are
possible even at high temperature (biology).
Although all that seems very promising, the theory is still in a
preliminary stage and a large effort must be made before these ideas
be universally accepted
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