INSTITUTE of ATOMIC PHYSICS Magurele-Bucharest Gamma Laser Controlled by High External Fields M Apostol Institute of Physics and Nuclear Engineering, Magurele-Bucharest May 2010 1
Laser Dichotomy , usually : (two levels) Narrow width for coherence, broader width for pumping Optical Laser: third broader level, for pumping ( ∼ 1 eV ) Nuclear laser: large energy ( 10 MeV ), Doppler e�ect, loss of coher- ence Irrealizable! (yes or not?) A further di�culty : coupling constant To see the Di�erence Opt Laser vs Nuclear Laser we need a Theory 2
Laser Theory : Does not exist! Discovery of the maser and the laser: 1950-1960... by engineers, physicists... Townes, Maiman, Basov, Prokhorov, (Weber), ... As regards the Theory, Lamb : We know everything and there exist Three Schools of Thought: Lamb&Scully, 2)Lax&Louisell, 3)Haken&Risken Three Schools of Thought=No Theory! 3
The di�culty and the Failure of the Current "Theories" Non-linear equations Possible non-analyticity Perturbation theory: fails They "see" (predict) many things which do not exist and do not see what does exist 4
What I mean by a Theory? A simple problem: Given : two quantum levels, interacting external and polarization �elds (everything ideal) Find : the population of the two levels, the population (intensity) of the �elds as functions of time, preferrably stationary, coherence 5
A new concept : Coherent coupling , all the atoms "excited" ("disexcited") in phase (stationary regime) Direct coupling ( 3 rd level not necessary), simple model Su�cient condition: high external (pumping) �eld Results: In principle, realizable, extremely low e��ciency Indeed non-analyticity 6
Practical idea (M Ganciu) : Relativistic electrons accelerated by intense laser pulses, Bremsstrahlung radiation, many photons, coupling with a 2 -level nuclear system Usual problems with cross-section and Doppler scattering: in the co- herent interaction context we may have surprises here (not discussed) Still, another di�culty : coupling constant 7
Coherent interaction Two levels � ω 0 = ε 1 − ε 0 (dipoles), mean inter-particle distance a , J 01 matrix el particle current, interacting with a classical electromagnetic �eld A coupling constant � λ = 2 g 2 π J 01 = 3 a 3 � ω 0 � ω 0 ω 0 Critical condition λ > 1 8
(at �nite temperature T < T c ) Second-order phase transition (super-radiance): macroscopic occu- pation of the two levels, macroscopic occup photon state, long-range order (of the quantum phases) Typical atomic matter: λ ∼ 0 . 17 Typical nuclear matter: λ ∼ 10 − 9 (this disparity makes the di�erence for the two lasers) No chance for this transition 9
Mathematical Machinery: Fields Vector potential (usual notations, transverse) � � 2 π � c 2 � e α ( k ) a α k e i kr + e ∗ � α ( k ) a ∗ α k e − i kr � � � A ( r ) = V ω k α k Fields E = − (1 /c ) ∂ A /∂t , H = curl A curl E = − 1 Three Maxwell's equations satis�ed: c ∂ H /∂t , div H = 0 , div E = 0 Similar expression for the external vector potential A 0 ( r ) , the corre- sponding Fourier coe�cients being denoted by a 0 α k , a 0 ∗ α k 10
Classical lagrangian of radiation � L f = 1 E 2 − H 2 � � d r 8 π Interaction lagrangian L int = 1 � d r · j ( A + A 0 ) = c � 2 π � � � � � �� e α ( k ) j ∗ ( k ) a α k + a 0 + e ∗ a ∗ α k + a 0 ∗ = � α ( k ) j ( k ) α k ω k α k α k Current density 1 j ( k ) e i kr � j ( r ) = √ V k (with div j = 0 , continuity equation) 11
Euler-Lagrange equations for the lagrangian L f + L int lead to the wave equation with sources � 8 πω k − α − k + ω 2 a ∗ � a α k + a ∗ � e ∗ ¨ a α k + ¨ = α ( k ) j ( k ) k − α − k � which is the fourth Maxwell's equation curl H = (1 /c ) ∂ E /∂t + 4 π j /c 12
Mathematical Machinery: Particles N independent, non-relativistic, identical particles i = 1 , ...N Hamiltonian (internal degrees of freedom) � H s = H s ( i ) i Orthonomal eigenfunctions ϕ n ( i ) � d r ϕ ∗ H s ( i ) ϕ n ( j ) = ε n δ ij , n ( i ) ϕ m ( j ) = δ ij δ nm Normalized eigenfunctions (for the whole ensemble) 1 e iθ ni ϕ n ( i ) � � √ ψ n = c ni ϕ n ( i ) = N i i 13
Field operator � Ψ = b n ψ n n boson-like commutation relations [ b n , b ∗ m ] = δ nm , [ b n , b m ] = 0 Large, macroscopic values of the number of particles b ∗ � N = n b n n The lagrangian � � � − L s = 1 � Ψ ∗ · i � ∂ Ψ /∂t − i � ∂ Ψ ∗ /∂t · Ψ d r Ψ ∗ H s Ψ d r 2 or L s = 1 � b ∗ b ∗ � ε n b ∗ n ˙ b n − ˙ � � i � n b n − n b n 2 n n 14
The hamiltonian ε n b ∗ � H s = n b n n The corresponding equation of motion i � ˙ b n = ε n b n is Schrodinger's equation It is worth noting that the same equation is obtained for b n viewed as classical variables Current density associated with this ensemble of particles J ( i ) δ ( r − r i ) = 1 1 J ( i ) e − i kr i e i kr = j ( k ) e i kr � � � √ j ( r ) = V V i i k k 15
The interaction lagrangian � 2 π � a α k + a 0 α k + a 0 ∗ � e α ( k ) I ∗ � � + e ∗ � a ∗ �� b ∗ � L int = mn ( k ) α ( k ) I nm ( k ) n b m α k α k V ω k nmα k where I nm ( k ) = 1 J nm ( i ) e − i ( θ ni − θ mi ) e − i kr i � N i J nm ( i ) are the matrix elements of the i -th particle current 16
Mathematical Machinery: Coherence Interaction lagrangian re-written � 2 π � � a α k + a ∗ � b ∗ � L int = F nm ( α k ) n b m − α − k V ω k nmα k F nm ( α k ) = 1 e α ( k ) J nm ( i ) e i kr i − i ( θ ni − θ mi ) � N i First arrange a lattice of θ ni Reciprocal vectors k r , r = 1 , 2 , 3 , � ω k = ε n − ε m > 0 Arrange phases k r r pi − ( θ ni − θ mi ) = K 17
Then, L int non-vanishing Two levels: n = 0 , n = 1 Macroscopic occupation, use c -numbers β 0 , 1 for operators b 0 , 1 (co- � � herent states b 0 , 1 � � ) � β 0 , 1 = β 0 , 1 � β 0 , 1 � � Photon perators a α k r , k r = k 0 , � ω 0 = ck 0 , replaced by c -numbers α Interaction lagrangian � 2 π � α ∗ + α 0 ∗ �� � β ∗ α + α 0 � �� � 1 β 0 + β 1 β ∗ L int = J 01 + � 0 V ω 0 18
The "classical" lagrangian α 2 + ˙ α ∗ 2 + 2 | ˙ − � ω 0 α 2 + α ∗ 2 + 2 | α | 2 � � α | 2 � � � L f = ˙ 4 ω 0 4 ε 0 | β 0 | 2 + ε 1 | β 1 | 2 � L s = 1 � � � β ∗ β ∗ 0 β 0 + β ∗ β ∗ 0 ˙ β 0 − ˙ 1 ˙ β 1 − ˙ 2 i � 1 β 1 − α ∗ + α 0 ∗ �� � g �� α + α 0 � � � β 0 β ∗ 1 + β 1 β ∗ L int = √ + 0 N Coupling constant � π � / 6 a 3 ω 0 J 01 g = 19
Equations of motion 0 A = 2 ω 0 g � � A + ω 2 β 0 β ∗ 1 + β 1 β ∗ ¨ √ 0 � N g � A + A 0 � i � ˙ β 0 = ε 0 β 0 − √ β 1 N g � A + A 0 � i � ˙ β 1 = ε 1 β 1 − √ β 0 N A = α + α ∗ , A 0 = α 0 + α 0 ∗ 20
Total hamiltonian � 2 + � ω 0 A + A 0 � 2 A + ˙ � � H tot � ˙ A 0 = f 4 ω 0 4 H s = ε 0 | β 0 | 2 + ε 1 | β 1 | 2 H int = − g � A + A 0 � � � β 0 β ∗ 1 + β 1 β ∗ √ 0 N Conserved, energy E , H tot + H s + H int = E f Number of particles, conserved | β 0 | 2 + | β 1 | 2 = N 21
Stationary solutions β 0 , 1 = B 0 , 1 e iθ ; equations of motion become 0 A = 4 ω 0 g A + ω 2 ¨ √ N B 0 B 1 � g � A + A 0 � i � ˙ B 0 − � ˙ θB 0 = ε 0 B 0 − √ B 1 N g � A + A 0 � i � ˙ B 1 − � ˙ θB 1 = ε 1 B 1 − √ B 0 N The last two equations tell that B 0 , 1 and ˙ θ = Ω are constant Particular solution of the �rst equation 4 g √ A = B 0 B 1 � ω 0 N 22
In the absence of the external �eld ( A 0 = 0 ) the solutions are given by √ 1 − ( � ω 0 / 2 g ) 4 � 1 / 2 A = 2 g � N � ω 0 0 = 1 � 1 + ( � ω 0 / 2 g ) 2 � B 2 2 N 1 = 1 � 1 − ( � ω 0 / 2 g ) 2 � B 2 2 N and frequency 2 + 2 g 2 � � − 1 Ω = ω 0 � 2 ω 2 0 where ε 1 − ε 0 = � ω 0 has been used and ε 0 was put equal to zero. 23
We can see: the ensemble of particles and the associated electromag- netic �eld can be put into a coherent state, the occupation amplitudes oscillating with frequency Ω , providing the critical condition g > g cr = � ω 0 / 2 , λ = 2 g/ � ω 0 > 1 The total energy of the coherence domain is given by E = − g 2 1 − ( � ω 0 / 2 g ) 2 � 2 = − � Ω B 2 � N 1 � ω 0 It is lower than the non-interacting ground-state energy Nε 0 = 0 It may be viewed as the formation enthalpy of the coherence domains 24
This e�ect of seting up a coherence in matter is di�erent from the lasing e�ect, precisely by this formation enthalpy Rather, the picture emerging from the solution given here resembles to some extent a quantum phase transiton The coupled ensemble of matter and radiation is unstable for a macro- scopic occupation of the atomic quantum states and the associated photon states. 25
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