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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 Pirelli Labs Materials Innovation – Advanced Research 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) Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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 N Complex plane Pirelli Labs Materials Innovation – Advanced Research 1 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 Pirelli Labs Materials Innovation – Advanced Research 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 N 302 V e 2 N 2 0 V Interaction term Electromagnetic field A ( x , ) 1 ( x , ), 2 ( x , ) Matter Field Pirelli Labs Materials Innovation – Advanced Research 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 30 B x cos RP t s 6 xˆ 3ˆ B( x , t ) 4 A 0 j1 r cos Rt 3 30 5 xˆ 3ˆ r 0 6 2 x32 2 x1 x2 5 2 Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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 H A 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 Pirelli Labs Materials Innovation – Advanced Research e s 1,2 s ik R cs s*eik 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 01 p s 0, 0 0 Full variational quantum state , N Ri , i1, i 2 ,..., iZ d Ri Ri N 3 em i 1 Pirelli Labs Materials Innovation – Advanced Research 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 2 202 Pirelli Labs Materials Innovation – Advanced Research Luca Gamberale - Dynamical Realization of Coherent Structures in Condensed Matter Coherence 2006 – Roma - April 21th, 2006 Graphical representation of the energy Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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 201 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. Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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) Pirelli Labs Materials Innovation – Advanced Research 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 N1 Kinematics completely different! N Coherent scattering Atot A1 tot N 1 j 1 Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research 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 Pirelli Labs Materials Innovation – Advanced Research