Time Evolution, Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I

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1 Commun. Theor. Phys. (Beijing China) 54 (200) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 54 No. 5 November Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I LIANG Guo-Dong ( Á ) YU Xiao-Min (ß ) 2 and YU Chao-Fan (ß ) 3 Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Education Institutes Department of Optoelectronic Engineering Jinan University Guangzhou China 2 Department of Physics Jinan University Guangzhou China 3 Department of Physics Guangdong Education College Guangzhou China (Received May ; revised manuscript received August 2 200) Abstract We have set up a new reduced model Hamiltonian for the polariton system in which the nonlinear interaction contains the rotating term k (a + b + ab + ) and the attractive two-mode squeezed coupling k 2(a + b + + ab). The dynamical evolution of this system has been solved and the nonclassical features relevant to the second-order and high-order squeezing have been obtained in an analytical form. For the first time in contrast to the existing result we have confirmed for the phonon field that the attractive two-mode squeezed interaction will not only result in the second-order and high-order squeezing in X-component the time evolution but also in time average. Furthermore the phenomena of collapse and revival of inversion will occur as well in the time evolution of the average number of photon and phonon as also in the second-order and high-order squeezing of photon field particularly in the high-order squeezing of phonon field. PACS numbers: c.30.Qc Key words: dynamic evolution of polariton system quantum fluctuation high-order squeezing Introduction For a long time the nonclassical properties of light have become the focus of widespread investigation in quantum optics. Some nonclassical states such as squeezed states pair-coherent states photon number states and sub-poissonian states have been extensively studied both theoretically and experimentally. [7 Particularly the new ideas of the quantum optics have inspired activities in the condensed matter physics [8 and that has been proved to be a great success. [8 4 For the same reason the research of the nonclassical phonon states is of importance subject not only in quantum optics but also in condensed matter physics. As is well expected to obtain the squeezed state of phonon we take the polariton system as a better candidate. In past several years some researches for the squeezing behavior of the phonon were devoted to the squeezing magnitudes and the non-poissonian statistics by the assumption of the two-mode squeezed state vector for the phonon or the squeezed-state Boqoliubov transformation but not the initial vacuum state or coherent state vector. In fact the squeezed state of phonon has not been really solved yet. Recently efforts have also been made to study the nonclassical behavior of phonons in model solid system. [3 9 For the first time Grovorkov and Shumovsky have studied the statistical distribution of phonons in the polariton system in terms of the model Hamiltonian [5 H = ω a a + a + ω b b + b + k (a + b + ab + ) + k 2 (a + b + + ab) (k = k 2 > 0) (where a + (a) b + (b) represent the phonon and polarization quanta creation operator respectively) which involves a one-mode photon field interacting a singlet optical phonon (GC-model). They have solved this model by a canonical transformation for this Hamiltonian. [6 8 Based on the calculation of the thermal Q-parameter and the thermal GSP function their results show that the phonon distribution of the polariton system is always super-poissonian. In spite of a number of works in physics of the nonclassical behavior the model Hamiltonian for the polariton system is not perfect up to now. In addition the profound cognition of physical substance for the nonclassical states and the quantum fluctuation effects in the polariton system are not clear yet. Especially the time evolution behavior for the dynamic squeezing of nonclassical field the dynamic quantum fluctuation and the nonclassical statistical distribution call in question still. We believe it is necessary to go into a matter seriously. It is a very important step for us to understand the nonclassical nature for the polariton system and study the more complicated systems further. In our recognition GC-model is not a corrective model: (i) As is well known for the atomic Bose Einstein condensate (BEC) the reduced non-attractive two-mode Supported by the Foundation of Scientific Research Education and Innovations under Grant No Jinan University yu-xm@26.com

2 94 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 (k k) interaction takes the form H = N 0 2V g (b + k b+ k + b kb k ). k Since the stable coupled-mode (b + k b+ k ) can violate the atomic boson (b k ) condensation we must require g > 0. As this result the coupled-mode (b + k b+ k ) does not form an elementary excitation. On the contrary the interaction responsible for superconducting electron gas the reduced interaction is H = (c + k c+ k + c k c k ). k In this case the two-mode coupled boson (c + k c+ k ) is a new type of elementary excitation. To assure the composite boson (c + k c+ k ) to form a stable coupled state the interaction between the two-mode c + k and c+ k must be attractive i.e. < 0. (ii) Now the polariton (a + b + ) is a composite boson for which the photon (a + ) and the phonon (b + ) rather couple together to form a new coupled-mode elementary excitation. In this paper we have obtained the nonlinear reduced interaction ±k 2 (a + b + +ab). To obtain the stable coupled-mode state we should choose k 2 (a + b + + ab) as the physical solution and abandon the non-physical solution +k 2 (a + b + + ab). Furthermore k (a + b + ab + ) represents the energy conservation process (rotating wave effect) while k 2 (a + b + +ab) the virtual state process (nonrotating wave effect). In particular k 2 (a + b + +ab) is not only related closely to the restrict condition to form the composite boson (a + b + ) but also to the generation of the ground state the broken symmetry as a result k k 2. So far based on the framework of the G-S model peoples have attempted to obtain the knowledge of the nonclassical properties of the polariton system by virtue of the calculation of thermal counterpart of the Mandel Q-parameter and the GSP function but their results are valueless to any real physics. From the viewpoint in the quantum mechanics the nonclassical effects (such as the squeezed state the dynamic quantum fluctuation the sub-poissonian distribution et al.) of the interaction system of the photon field (a(t)) the polarization wave of the solid (the associated elementary excitations are the TO-phonon b(t) for the polarization of the ion lattice) are decided by ψ(t) = e iht 0 due to the time-evolution the a(t) and b(t) fields but not by the thermodynamic evolution. Particularly these nonclassical effects are related closely to the nonclassical vacuum state of the interaction system. Recently we have found that the ground state of the polariton system is a nonclassical one i.e. the twomode rotation squeezed-vacuum state. As a result the nonclassical properties for the phonon subsystem can be explained as the evolution from the nonclassical ground state of the polariton system. The present work for the first time is to solve the time evolution of the polariton system in Sec. 2. Based on this framework in Sec. 3 we study the dynamic quantum fluctuation for this system. In addition in Sec. 4 by the investigation of the high-order dynamic squeezing for both the photon and phonon field we obtain the nonclassical behavior of the polariton system from another aspect. As the conclusion we will summarize the principle results of this study in Sec Dynamical Evolution of the Modeled Polariton System As is known when light interacts the lattice vibration the lattice polarization mode can couple the photon to form a new set of normal modes it is called a polariton. To express the model effective Hamiltonian clearly we introduce along the vector potential A the polarization of the dielectric P Â = kα ˆP = κα A 0 (a kα + a + kα )eik r P 0 (b κα + b + κα )eiκ r () where a kα and b kα denote the photon annihilation operator and the polarization quanta annihilation operator. The photon-polarization mode interaction depends on the product of the two fields E P = A P. According to Madelung we obtain the index k for the combination of summation indices in both case [9 then for the Hamiltonian of the photon-polarization quantum interaction we put Ĥ = ( [E k a + k a k + ) ( + E 2k b + k 2 b k + ) 2 k + E 3k (a + k b k a k b + k a kb k + a + k b+ k ). (2) It is seen from Eq. (2) that the terms (a + k b+ k a k b k ) and (a + k b k a k b + k ) describe different excitation processes and have different time-reversal symmetry. For this reason both the respective physical subspaces are different. If Π denotes the projection operator which projects out from the configuration space the states related to the physical subspace n a n b then E 3k (a + k b k a k b + k ) Π ±k (a + b + ab + ) and k E 3k (a + k b+ k a Π kb k ) ±k 2 (a + b + +ab) where a and k b are the reduced operators for photon and phonon respectively i.e. a = Tr {Π [ψna k } b = Tr {Π [ψnb k } ψ n = n a n b. Therefore for the composite boson (a + b + ) is to form the stable coupled-mode state the effective interaction of the one-mode photon the single (TO) phonon for the modeling polariton system can now

3 No. 5 Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I 95 be recast in the form (k k 2 ) Ĥ = ω a a + a + ω b b + b + k (a + b + ab + ) k 2 (a + b + + ab) (k 2 > 0). (3) As is well known a similar effective interaction Hamiltonian for the N- and V -particles systems was obtained by Ruijgrok et al. in quantum field theory (λ λ 2 ). [20 To study the dynamical evolution for the polariton system we firstly solve the time evolution behavior for a(t) and b(t) in the Heisenberg picture. By the Heisenberg equation of motion for a(t) and b(t) the model Hamiltonian (3) we can derive ( = ) ȧ(t) = i [a Ĥ = i[ω aa(t) + k b(t) k 2 b + (t) ḃ(t) = i [b Ĥ = i[ω bb(t) + k a(t) k 2 a + (t). (4) In terms of the state superposition principle we seek for the solutions a(t) = f (t)a + f 2 (t)a + + g (t)b + g 2 (t)b + (5) b(t) = g 3 (t)a + g 4 (t)a + + f 3 (t)b + f 4 (t)b +. (6) By virtue of symmetry contained in the a- and b-operators after some algebric operation we find the coupled system of equations as follows (k 2 > 0) where ẋ = i[ω a y + (k + k 2 )y 2 ẏ = i[ω a x + (k k 2 )x 2 ẋ 2 = i[ω b y 2 + (k + k 2 )y ẏ 2 = i[ω b x 2 + (k k 2 )x (7) ẋ 3 = i[ω b y 3 + (k + k 2 )y 4 ẏ 3 = i[ω b x 3 + (k k 2 )x 4 ẋ 4 = i[ω a y 4 + (k + k 2 )y 3 ẏ 4 = i[ω a x 4 + (k k 2 )x 3 (8) x = f + f 2 y = f f 2 x 2 = g 3 + g 4 y 2 = g 3 g 4 (9) x 3 = f 3 + f 4 y 3 = f 3 f 4 x 4 = g + g 2 y 4 = g g 2. (0) Let us denote these basic solutions to be x = A e λt x 2 = A 2 e λt y = B e λt y 2 = B 2 e λt () x 3 = C e λt x 4 = C 2 e λt y 3 = D e λt y 4 = D 2 e λt (2) where the initial conditions are x (0) = x 2 (0) = 0 y (0) = y 2 (0) = 0 x 3 (0) = x 4 (0) = 0 y 3 (0) = y 4 (0) = 0. With the use of Eqs. (7) and () we can obtain the set of linear equations for (A A 2 B B 2 ) λ 0 iω a i(k + k 2 ) A 0 λ i(k + k 2 ) iω b A ω 2 kba 2 B = 0 (3) 0 0 kab 2 ω 2 B 2 where ω 2 = λ 2 i + ω 2 a + (k 2 k 2 2) ω 2 = λ 2 i + ω 2 b + (k 2 k 2 2) (4) k 2 ab = (k k 2 )ω a + (k + k 2 )ω b k 2 ba = (k k 2 )ω b + (k + k 2 )ω a (5) and the eigen-values λ i are λ = ix + λ 2 = ix λ 3 = ix + λ 4 = ix (6) x ± = ξ ± ξ 2 (7) ξ = 2 [(ω2 a + ωb) 2 + 2(k 2 k2) 2 ξ 2 = [(ω a +ω b ) 2 2 4k2 2[(ω a ω b ) 2 +4k 2+6k2 k2 2. (8) By solving Eq. (3) we have (i = 2 3 4) A (i) = (k + k 2 )ω 2 i + ω ak 2 ba A (i) k 2 ba x(i) 2 = (k +k 2 )k 2 ba +ω bω 2 i k 2 ba x(i) B (i) B (i) B(i) 2 = ω2 i k 2 ba B (i) (x () = x + x (2) = x x (3) = x + x (4) = x ) (9) and B (i) satisfy the following set of equations 0 0 B () c () 0 0 B (2) 0 0 α λ α 2λ B (3) = c (2) c (3) 0 0 α 3λ α 4λ B (4) (20) c (4) where 2a λ 2a λ2 α λ = α 2λ = a λ a λ2 a λ a λ2 2a 2λ 2a 2λ2 α 3λ = α 4λ = (2) a 2λ a 2λ2 a 2λ a 2λ2 c () = ω2 2 ω2 x 2 + c (2) x2 = x 2 + x2 c (3) = a λ k 2 ba a λ a λ2 + c (4) = a 2λ a 2λ a 2λ2 + ω2 x 2 + x 2 ω2 x 2 + (22) x2 a λi = (k + k 2 )ω 2 i ω ak 2 ba x (i) a 2λi = (k + k 2 )k 2 ba + ω bω 2 i x (i). (23)

4 96 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 From Eq. (20) we have B () = c () α 4λc (3) α 2λc (4) α λ α 4λ α 2λ α 3λ B (2) = c (2) + α 3λc (3) α λc (4) α λ α 4λ α 2λ α 3λ = α 4λc (3) α 2λc (4) α λ α 4λ α 2λ α 3λ B (3) = α 3λc (3) α λc (4). (24) α λ α 4λ α 2λ α 3λ B (4) On the other hand the set of equations for (C C 2 D D 2 ) corresponding to Eq. (8) take the form λ 0 iω b i(k + k 2 ) C 0 λ i(k + k 2 ) iω a C ω 2 kab 2 D = 0. (25) 0 0 kba 2 ω 2 D 2 In the same manner the solutions for C (i) C(i) 2 D(i) and D (i) 2 respect to Eq. (25) are found as follows C (i) = (k + k 2 )ω 2 i ω bk 2 ab C (i) k 2 ab x(i) 2 = (k + k 2 )k 2 ab + ω aω 2 i D (i) 2 = ω2 i k 2 ab k 2 ab x(i) where D (i) are determined by β λ β 2λ 0 0 β 3λ β 4λ in which 2b λ D (i) D (i) D (i) (26) D () D (3) D (4) = c () c (2) c (3) c (4) 2b λ2 (27) β λ = β 2λ = b λ b λ2 b λ b λ2 2b 2λ 2b 2λ2 β 3λ = β 4λ = (28) b 2λ b 2λ2 b 2λ b 2λ2 c () = ω2 2 ω2 x 2 + x 2 c (2) = x 2 + x 2 c (3) = b λ k 2 ab b λ b λ2 + ω2 x 2 + x2 c (4) = b 2λ + ω2 b 2λ b 2λ2 x 2 + x 2 (29) b λi = (k + k 2 )ω 2 i ω bk 2 ab x (i) b 2λi = (k + k 2 )k 2 ab + ω aω 2 i x (i). (30) In this way the solutions D (i) for Eq. (27) read D () = c () β 4λc (3) β 2λc (4) β λ β 4λ β 2λ β 3λ = c (2) + β 3λc (3) β λc (4) β λ β 4λ β 2λ β 3λ = β 4λc (3) β 2λc (4) β λ β 4λ β 2λ β 3λ D (3) D (4) = β 3λc (3) β λc (4). (3) β λ β 4λ β 2λ β 3λ Obviously even though the Hamiltonion (3) is symmetric under the gauge transformation U = e a+ a+b +b all these coefficients (f f 2 g g 2 ) and (f 3 f 4 g 3 g 4 ) do not have the corresponding symmetry. This reason is attributed to the broken symmetry caused by the nonlinear coupling terms k (a + b + ab + ) and k 2 (a + b + + ab). We now assume that the light field is initially in a pure coherent state D(α) 0 a and the polarization quantum system is in the vacuum state 0 b then the time evolution of the interacting system can be expressed as ψ(t) = e iht D(α) 0 0 ab (32) where 0 0 ab = 0 a 0 b and D(α) = e αa+ α a is the unitary displacement operator of the photon field the coherent parameter α. We now denote by N a (t) = ψ(t) a + a ψ(t) (33) the average photon number and by N b (t) = ψ(t) b + b ψ(t) (34) the average phonon number of the polarization quanta field respectively. Using we can find D + (α)a(t)d(α) = a(t) + z D + (α)b(t)d(α) = b(t) + z 2 (35) z = f α + f 2 α z 2 = g 3 α + g 4 α (36) N a (t) = f g z 2 N b (t) = f g z 2 2. (37) Neglecting the contribution from the high-order correction due to n 0 and choosing α to be real we arrive at N a (t) [ f 2 + f Re (f f 2 ) n 0 = [(A () + A (3) )2 cos 2 (x + t) + (A (2) + A (4) )2 cos 2 (x t) + (B () B (3) )2 sin 2 (x + t) + (B (2) B (4) )2 sin 2 (x t) + 2(A () + A (3) )(A(2) + A (4) )cos(x +t)cos(x t) + 2(B () B (3) )(B(2) B (4) )sin(x +t)sin(x t) n 0 (38)

5 No. 5 Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I 97 N b (t) [ g g Re (g 3 g 4 ) n 0 = [(A () 2 + A (3) 2 )2 cos 2 (x + t) + (A (2) 2 + A (4) 2 )2 cos 2 (x t) + (B () 2 B (3) 2 )2 sin 2 (x + t) + (B (2) 2 B (4) 2 )2 sin 2 (x t) + 2(A () 2 + A (3) 2 )(A(2) 2 + A (4) 2 )cos(x +t)cos(x t) + 2(B () 2 B (3) 2 )(B(2) 2 B (4) 2 )sin(x +t)sin(x t) n 0 (39) the average photon number at t = 0 n 0 = ψ(0) a + a ψ(0) = α 2. The time evolution for both the photon and phonon average numbers obtained above are shown in Figs. and 2. then we have N a (t) = n! nn 0 e n0 [ f 2 + f Re (f f2 ) n N b (t) = n=0 = n=0 P n F a (t) n n=0 n! nn 0 e n0 [ g g Re (g 3 g 4 ) n = n=0 P n G b (t) n. As the result of the nonlinear interaction k (a + b + ab + ) and k 2 (a + b + +ab) the time evolution of F a (t) and G b (t) will be appeared in a coherent quantum oscillations form and overlapped the relative weight P(n). When t = 0 the polariton system is prepared in a definite state and therefore all these coherent quantum oscillations are correlated. As time increases the quantum oscillations involved in F a (t) and G b (t) associated different excitation have different frequencies and therefore become uncorrelated leading to the envelope of the quantum oscillation of F a (t) and G b (t) collapse to zero i.e. a collapse of inversion (t = τ c ) occurs. As time is further increased the correlation is restored due to the presence of the correlated behavior caused by the nonlinear interaction (t = τ R ). We encounter a revival of the collapsed inversion. This behavior of collapse and revival of inversion is repeated increasing time as shown in Fig.. The results also show that the τ c is dependent on (k k 2 ) but not n 0 but τ R is related to both (k k 2 ) and n 0 obviously. Fig. Time evolution of the average photon number N a(t) and phonon number N b (t) the initial state D(α) 0 for the polariton system where the light field is initially in a coherent state. (a) k =.0 k 2 = 0.5 ω a =.0 ω b = 0.0 n 0 = 20 (b) k 2 =.0 ω a =.0 ω b = 0.0 n 0 = 20. In the present case due to the absorption of photon the creation of elementary excitation in the ionic crystals it will result in the a + b ab + a + b + and abprocesses. Notice that the initial photon field is in the coherent state α = F n n n=0 F n = e n0/2 n0 n n! Fig. 2 The time average for the photon and phonon average numbers respectively. k =.0 k 2 = ω a =.0 ω b = 0.0 n 0 = 20.

6 98 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 It is also shown that the N a (t) and N b (t) are not only dependent on (k k 2 ) but their time average N a (t) T and N b (t) T are related to k /k 2 or k 2 /k as show in Fig. 2. It should be noted that for the GC-model Hamiltonian N a (t) = (/2)[ + cos(2kt)n 0 N b (t) = (/2)[ cos(2kt)n 0 obviously here the phenomena of collapse and revival for N a (t) or N b (t) will not occur. 3 Nature of Squeezed States the Time Evolution in Polariton System To study the nature of quantum fluctuation in the polariton system we now introduce the quadrature operators for both the photon field and the polarization quantum field X = 2 (a + a+ ) Y = 2i (a a+ ) X 2 = 2 (b + b+ ) Y 2 = 2i (b b+ ). (40) It is convenient as usual to define a second-order squeezing index S X = X2 ψ X 2 ab (4) where X 2 ψ = X 2 ψ X 2 ψ. If S Xi < (i = 2) (or S < ) a squeezed state of the photon field (or phonon field) is obtained. Remembering 0 0 Xi = 0 0 Yi = /4 so that S X = 0 0 [a +2 (t) + a 2 (t) (z + z ) 0 0 a + (t)a(t) + a(t)a + (t) 0 0. Using Eq. (5) responsible for the time evolution operator a(t) we obtain S X = f + f g + g 2 2 = x 2 + x 4 2. (42) In such a manner the corresponding squeezing index for the other components are S Y = f f g g 2 2 = y 2 + y 4 2 (43) S X2 = f 3 + f g 3 + g 4 2 = x x 3 2 (44) S Y 2 = f 3 f g 3 g 4 2 = y y 3 2. (45) By use of the Eq. () and Eq. (2) we have obtained for S Xi and S in an analytical form S X = 4(A () A(3) + C () 2 C(3) 2 )sin2 (x + t) 4(A (2) A(4) + C (2) 2 [ C(4) 2 )sin2 (x t) 4[(A () A(4) + A (2) A(3) ) + (C() 2 C(4) 2 + C (2) 2 C(3) 2 ) sin2 2 (x + + x )t [ 4[(A () A(2) + A (3) A(4) ) + (C() 2 C(2) 2 + C (3) 2 C(4) 2 ) sin2 2 (x + x )t (46) S Y = 4(B () B(3) + D () 2 D(3) 2 )sin2 (x + t) 4(B (2) B(4) + 2 [ D(4) 2 )sin2 (x t) 4[(B () B(4) + B (2) B(3) ) + (D() 2 D(4) D(3) 2 ) sin2 2 (x + + x )t [ 4[(B () B(2) + B (3) B(4) ) + (D() 2 D(2) 2 + D (3) 2 D(4) 2 ) sin2 2 (x + x )t S X2 = 4(C () C(3) + A () 2 A(3) 2 )sin2 (x + t) 4(C (2) C(4) + A (2) 2 [ A(4) 2 )sin2 (x t) 4[(C () C(4) + C (2) C(3) ) + (A() 2 A(4) 2 + A (2) 2 A(3) 2 ) sin2 2 (x + + x )t (47) [ 4[(C () C(2) + C (3) C(4) ) + (A() 2 A(2) 2 + A (3) 2 A(4) 2 ) sin2 2 (x + x )t (48) S Y 2 = 4(D () D(3) + B () 2 B(3) 2 )sin2 (x + t) 4( D(4) + B (2) 2 [ B(4) 2 )sin2 (x t) 4[(D () D(4) + D(3) ) + (B() 2 B(4) 2 + B (2) 2 B(3) 2 ) sin2 2 (x + + x )t [ 4[(D () D(2) + D (3) D(4) ) + (B() 2 B(2) 2 + B (3) 2 B(4) 2 ) sin2 2 (x + x )t. (49) With the time evolution the second-order squeezing index S Xi and S Yi are shown in Figs Here we summarize our new findings: (i) The squeezed photon states can be produced by the polariton-one-photon interaction to evolve into the twophoton coherent states due to the non-adiabatic phonon-two-photon correlation action. However as the photon energy (ω a a + a) increases it will destroy the stable state of polariton and weaken the development of the two-photon coherent states. Equally the squeezed phonon states can also be produced by the polariton-one-phonon interaction to evolve into the two-phonon coherent states due to the non-adiabatic photon-two phonon correlation. At the same time the effect of the phonon energy is also damaged to the evolution of the two-phonon coherent states.

7 No. 5 Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I 99 Fig. 3 Time evolution of the second-order squeezing index S Xi (t) and S Yi (t) the initial photon coherent states respectively. (a) k =.0 k 2 = 0.5 ω a =.0 ω b = 0.0 (b) k =.0 k 2 = 0.2 ω a =.0 ω b = 0.0 (c) k = 0.5 k 2 =.0 ω a =.0 ω b = 0.0 (d) k = 0.5 k 2 = 0.5 ω a =.0 ω b = 0.0. Fig. 4 The time average of the second-order squeezing index in X and Y -component S Xi (t) T and S Yi (t) T as function of k /k 2 and k 2/k where the initial light field is in the coherent states. (a) k =.0 k 2 = ω a =.0 ω b = 0.0 (b) k 2 = 0.4 k = 0.0 ω a =.0 ω b = 0.0. (ii) As time increases the evolution behavior for S Xi (t) inclines to S Xi (t) < in a regular form so as to the time average S Xi (t) T < ; while S Yi (t) inclines to S Yi (t) > and thus S Yi (t) T >. Above results are identical the quantum fluctuation of the nonclassical vacuum state of the polariton system in which 2 αβ < /4 and 2 αβ > /4. Furthermore since the initial photon field is in the coherent state α consider the effects of

8 920 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 the quantum fluctuation of the nonclassical vacuum state in the polariton system and the optical coherence action of the photon field at the initial time t = 0 the time evolution term ( f + f g + g 2 2 ) involved in S X (t) may be prepared in a coherent quantum oscillation and is correlated due to their initial phases are identical. Thus as the time evolves the quantum oscillations involved in ( f + f g + g 2 2 ) associated different excitation have different frequencies and become uncorrelated leading to the envelope of the quantum oscillations collapse to zero. By this way we also encounter a revival of the collapsed inversion for S X (t). The similar explanation also holds for S Y (t). As for the phonon field its initial state is vacuum state 0 b but not the coherent states correlation effects between the non-coherent quantum oscillations caused by the quantum fluctuation of the phonon vacuum state are too weak due to the different initial phase thus the phenomena of collapse and revival do not occur. (iii) S Xi (t) and S (t) (i = 2) are not only closely related to (k k 2 ) but also to (ω a ω b ). When k = 0 S Xi (t) T > and S Yi (t) T > when k 2 = 0 S Xi (t) T = and S Yi (t) T = it is because the interaction system is not a two-mode rotation squeezed vacuum state. In particular when ω a ω b the polariton system will incline to polariton-one-photon interaction but not to the polariton-one-phonon interaction. As this result no matter k > k 2 or k < k 2 S X (t) T < while for the phonon field under the condition k >.8k 2 but not k k 2 S X2 (t) T < and vice versa as shown in Figs. 4(a) and 4(b). It should be noted that for the GC-model using the perturbative approximation S X2 (t) = + (k/ω)sin(2ωt)sin(2kt) S Y2 (t) = (k/ω)sin(2ωt)sin(2kt) (k = k 2 ω a = ω b = ω) consequently the time average S X2 (t) T = and S Y2 (t) T =. (iv) In fact noting that the essence of the squeezed states is caused by the nonlinear interaction in physical process we can confirm above results to be correct: S X (t) T < S X2 (t) T <. As an example we firstly consider the squeezed state of phonon in the polariton system. Noting Eqs. (7) (8) relevant to S X2 (t) and S Y2 (t) we find ẋ 2 i(k + k 2 )(f f 2) ẋ 3 i(k + k 2 )(g g 2 ) ẏ 2 i(k k 2 )(f + f 2 ) ẏ 3 i(k k 2 )(g + g 2). When k 2 > 0 the effect of the nonlinear behavior for i(k + k 2 )(f f2 ) is more important than i(k k 2 )(f + f2 ) and i(k + k 2 )(g g2 ) is more important than i(k k 2 )(g + g2). As a result S X2 T < S Y 2 T the squeezed state of phonon will occur in X-component but not in Y -component. In contrast when k 2 < 0 the nonlinear effect for i(k k 2 )(f + f2 ) is approximated to i(k + k 2 )(f f2 ) and i(k k 2 )(g +g2 ) to i(k +k 2 )(g g2 ). Consequently S X2 T S Y 2 T the squeezed state of phonon will occur neither in X nor in Y -component. In particular when k = k 2 even though S X2 T < but S Y 2 T the squeezed state of phonon also does not occur in X or Y -component due to the restriction of the uncertainty equality S X2 T S Y 2 T. In a similar discussion we can also obtain the same conclusion for the squeezed states of photon in the polariton system. We particularly emphasized that for the nonlinear interaction k(a + b+ab + +a + b + +ab) (k = k 2 > 0) given by GC-model no matter k k 2 or k = k 2 even though the second-order squeezing for photon still occurs in the Y -component due to the initial photon coherent state the phonon subsystem have no squeezing i.e. S X2 (t) T > and S Y2 (t) T >. This reason should be attributed to the non-attractive two-mode squeezed coupling. 4 High-Order Dynamical Squeezing for Both the Photon and Polarization Quanta Field in the Polariton System In addition to the second-order squeezing effects from quite another aspect the high-order dynamical squeezing for both the photon and polarization quanta field describes the nonclassical nature of the polariton system. According to the criteria provided by Hong and Mandel for the 2N-order field squeezing the 2N-order quantum fluctuation for the dynamical variable X will be satisfied by the existence condition [20 ( X) 2N ψ < 0 ( X) 2N 0 (50) 0 ( X) 2N 0 = (2N)!/(N!2 2N ). (5) In terms of the field operators a(t) and b(t) we obtain X (t) = A X a + A Xa + + B X b + B Xb + Y (t) = A Y a + A Y a + + B Y b + B Y b + (52) X 2 (t) = A X2 a + A X2a + + B X2 b + B X2b + Y 2 (t) = A Y 2 a + A Y a + + B Y 2 b + B Y 2b + (53) A X = 2 (f + f 2 ) B X = 2 (g + g 2 ) A Y = 2i (f f 2) B Y = 2i (g g 2) (54) A X2 = 2 (g 3 + g 4) B X2 = 2 (f 3 + f 4 ) A Y 2 = 2i (g 3 g 4) B Y 2 = 2i (f 3 f 4 ). (55) We now consider the photon field is the initially in the squeezed vacuum state s(β) 0 a but the polarization

9 No. 5 Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I 92 quantum field is in the phonon vacuum state 0 b then the time evolution for the interaction system of the photon the polarization quantum is expressed as ψ(t) = e iht S(β) 0 0 (56) here S(β) is the squeezing operator of the photon field S(β) = exp[ 2 (βa+2 β a 2 ) (57) and β = β e iθ. As is well know the 2N-order squeezing index is defined by D (2N) X = ψ(t) ( X)2N ψ(t) 0 0 ( X) 2N ( X) 2N (58) 0 0 and the 2N-order squeezing condition is D (2N) X < 0 the maximum squeezing index D (2N) X =. By using the squeeze operator algebra we find S + (β)x (t)s(β) = F X a + FXa + + B X b + BXb + S + (β)y (t)s(β) = F Y a + FY a+ + B Y b + BY b+ (59) S + (β)x 2 (t)s(β) = F X2 a + FX2 a+ + B X2 b + BX2 b+ S + (β)y 2 (t)s(β) = F Y 2 a + FY 2 a+ + B Y 2 b + BY 2 b+ (60) where F X = A X cosh β + A X e iθ sinh β F Y = A Y cosh β + A Y e iθ sinh β (6) F X2 = A X2 cosh β + A X2e iθ sinh β F Y 2 = A Y 2 cosh β + A Y 2e iθ sinh β. (62) As a result we get ψ(t) ( ) 2N ψ(t) = ψ(t) ( ) 2N ψ(t) = 2N m=0 2N m=0 Noting that the normal product expansion (fa + f a + ) n = (2N)! m!(2n m)! 0 0 (F Xia + F Xia + ) 2N m (B Xi b + B Xib + ) m 0 0 (2N)! m!(2n m)! 0 0 (F a + F a+ ) 2N m (B b + B b+ ) m 0 0. (63) [n/2 k=0 n 2k s=0 n!(f f/2) 2k (2k)!s!(n 2k s)! (f a + ) s (fa) n 2k s (64) then the result becomes ψ(t) ( X ) 2N ψ(t) = (2N)! N!2 N ( F X 2 + B X 2 ) N. (65) In a similar manner we get ψ(t) ( Y ) 2N ψ(t) = (2N)! N!2 N ( F Y 2 + B Y 2 ) N (66) ψ(t) ( X 2 ) 2N ψ(t) = (2N)! N!2 N ( F X2 2 + B X2 2 ) N (67) ψ(t) ( Y 2 ) 2N ψ(t) = (2N)! N!2 N ( F Y B Y 2 2 ) N. (68) In this way finally we can write the analytical expression for the 2N-order squeezing index as D (2N) X = 2 2N [ A X 2 cosh2 β + Re(A 2 X eiθ ) sinh(2 β ) + B X 2 N = ( X + )N. (69) In a similar way we also have where D (2N) Y = ( Y + )N (70) D (2N) X2 = ( X2 + )N (7) D (2N) Y 2 = ( Y 2 + )N (72) X = 22 [ A X 2 cosh2 β + Re(A 2 Xe iθ )sinh(2 β ) + B X 2 (73) Y = 22 [ A Y 2 cosh2 β + Re (A 2 Y e iθ )sinh(2 β ) + B Y 2 (74) X2 = 22 [ A X2 2 cosh2 β + Re(A 2 X2e iθ )sinh(2 β ) + B X2 2 (75) Y 2 = 22 [ A Y 2 2 cosh2 β + Re (A 2 Y 2 eiθ )sinh(2 β ) + B Y 2 2. (76) A X 2 = 4 A() A(3) sin 2 x + t A (2) A(4) sin 2 x t (A () A(4) + A (2) A(3) )sin2 2 (x + + x )t (A () A(2) + A (3) A(4) )sin2 2 (x + x )t (77) B X 2 = C () 2 C(3) 2 sin 2 x + t C (2) 2 C(4) 2 sin 2 x t (C () 2 C(4) 2 + C (2) 2 C(3) 2 )sin2 2 (x + + x )t (C () 2 C(2) 2 + C (3) 2 C(4) 2 )sin2 2 (x + x )t (78)

10 922 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 A Y 2 = 4 B() B(3) sin 2 x + t B (2) B(4) sin 2 x t (B () B(4) + B (2) B(3) )sin2 2 (x + + x )t (B () B(2) + B (3) B(4) )sin2 2 (x + x )t (79) B Y 2 = D () 2 D(3) 2 sin 2 x + t 2 D(4) 2 sin 2 x t (D () 2 D(4) D(3) 2 )sin2 2 (x + + x )t (D () 2 D(2) 2 + D (3) 2 D(4) 2 )sin2 2 (x + x )t (80) A X2 2 = A () 2 A(3) 2 sin 2 x + t A (2) 2 A(4) 2 sin 2 x t (A () 2 A(4) 2 + A (2) 2 A(3) 2 )sin2 2 (x + + x )t (A () 2 A(2) 2 + A (3) 2 A(4) 2 )sin2 2 (x + x )t (8) B X2 2 = 4 C() C(3) sin 2 x + t C (2) C(4) sin 2 x t (C () C(4) + C (2) C(3) )sin2 2 (x + + x )t (C () C(2) + C (3) C(4) )sin2 2 (x + x )t (82) A Y 2 2 = B () 2 B(3) 2 sin 2 x + t B (2) 2 B(4) 2 sin 2 x t (B () 2 B(4) 2 + B (2) 2 B(3) 2 )sin2 2 (x + + x )t (B () 2 B(2) 2 + B (3) 2 B(4) 2 )sin2 2 (x + x )t (83) B Y 2 2 = 4 D() D(3) sin 2 x + t D(4) sin 2 x t (D () D(4) + D(3) )sin2 2 (x + + x )t (D () D(2) + D (3) D(4) )sin2 2 (x + x )t. (84) Xi Obviously from Eqs. (69) (72) we can see that: if < 0 then D(2N) Xi < 0; and if < 0 then D(2N) < 0. Therefore the second-order and 2N-order squeezed states appear simultaneously but the 2N-order squeezing is a new nonclassical effect distinct from the second-order squeezing. The time evolution of the second-order squeeze index related to high-order squeezing for the photon field ( X D(2) Y ) and polarization quanta field (D(2) X2 D(2) Y 2 ) are displayed in Figs. 5 6 respectively. In view of these results we can get some important conclusion as follows: (i) (a) The high-order squeezing has quite new features different from the second-order squeezing. If C 0 and C 2N are constant and X = (b+b + )/2 Y = (b b + )/(2i) then we have X Y = ic 0 2 and X 2N Y 2N ic 2N 2 respectively. (b) The high-order squeezing of photon field was caused by the non-adiabatic coupling between the secondorder squeezed field of photon and the squeezed-vacuum state of photon field S(β) 0 α i.e. A X 2 cosh2 β + Re (A 2 X e iθ )sinh2 β A X = (/2)(f +f2 ) = (/2)x and S () On the other hand although X (t) = x 2. the phonon field is initially in the vacuum state the high-order squeezing can also be produced by the nonadiabatic coupling between the second-order squeezed field of phonon and the squeezed-vacuum photon field namely A X2 2 cosh2 β + Re (A 2 X 2 e iθ )sinh2 β A X2 = (/2)(g 3 + g 4 ) = (/2)x 3 and S () X 2 (t) = x 3 2. (ii) The phenomena of collapse and revival occur not only in the X (t) and Y (t) but also in the X 2 (t) and Y 2 (t). As a matter of fact since the initial photon field is in the nonclassical vacuum state s(β) 0 α the photon vacuum state 0 α will evolve into the coherent state α s of the squeezed-state space ( H a = ω a a + s a s + ) n s = a + s a s 2 α s = D s (α) 0 s = e αa+ s α a s 0 s where a s = a cosh β + a + e iθ sinh β. Considering the nonlinear kinetic interaction in the polariton system the time evolution terms [( A X 2 A Y 2 A X2 2 A Y2 2 ) cosh(2 β ) [ Re (A 2 X eiθ ) Re (A 2 Y eiθ ) Re (A 2 X2 eiθ ) Re (A 2 Y 2 eiθ ) sinh(2 β ) involved in (t) and (t) will appear in a coherent quantum oscillations. At the initial time t = 0 these coherent quantum oscillations are correlated since their oscillating phase are identical. As time increases the quantum oscillations associated different excitation have different frequencies and become uncorrelated leading to the envelope of the quantum oscillation collapse to zero (t = τ c ). When t = τ R the correlation is restored we can see a significant revival for (t) or (t). (iii) (a) The high-order squeezing occurs only in the X- component D (2N) (t) and this fact is essentially consistent the quantum fluctuation inequality ( Xi 2 αβ < /4 Yi 2 αβ > /4 i = 2) of the two-mode squeezed vacuum state. (b) Similarly (t) and (t) (i = 2) are not only related to (k k 2 ) but also to (ω a ω b ). When k = 0 or k 2 = 0 it shows that (t) T > 0 and (t) T > 0. Moreover when ω a < ω b the polariton

11 No. 5 Time Evolution Dynamical Quantum Fluctuation and High-Order Squeezing Feature in Polariton System I 923 system is favorable to the non-adiabatic coupling between the second-order squeezed photon field and the squeezed vacuum state of photon but not the non-adiabatic coupling between the second-order squeezed phonon field and the squeezed vacuum state of photon as this result no matter whether k > k 2 or k < k 2 we always k >.4k 2 but not k k 2 then we have D (2N) X 2 (t) T < 0 and vice versa. Fig. 6 The time average of high-order squeezing (t) T and (t) T versus k /k 2 and k 2/k where the initial light field is in the squeezed state. (a) k =.0 k 2 = 0.4 ω a =.0 ω b = 0.0 θ = π/7 β = 0.5 (b) k 2 = 0.5 k = 0.25 ω a =.0 ω b = 0.0 θ = π/7 β = 0.5. Fig. 5 Time evolution of the high-order squeezing (t) and (t) for the photon and phonon the initial photon squeezed state respectively. (a) k =.0 k 2 = 0.8 ω a =.0 ω b = 0.0 θ = π/30 β = 0.5 (b) k =.2 k 2 = 0.5 ω a =.0 ω b = 0.0 θ = π/7 β = 0.5 (c) k 2 = 0.8 k = 0.8 ω a =.0 ω b = 0.0 θ = π/7 β = 0.5. find D (2N) X (t) T < 0 while for the phonon field when (c) In particular when the two-mode squeezing coupling is non-attractive no matter k k 2 or k = k 2 the high-order squeezing of the phonon field is always inclined to X 2 (t) > 0 and Y 2 (t) > 0 to evolve so as to X 2 (t) T > 0 and Y 2 (t) T > 0. 5 Conclusion and Discussion In this paper we have set up a new reduced model Hamiltonian for the polariton system for which the nonlinear interaction involves the rotating term k (a + b + ab + ) and the attractive two-mode squeezed coupling k 2 (a + b + + ab). The dynamical evolution behavior for this interaction system and the nonclassical nature relevant to the second-order and high-order squeezing have been solved analytically. In summary some new results are found as follows: (i) As time increases as a result of the initial coherent state of photon field N a (t) N b (t) reveal the phenomena

12 924 LIANG Guo-Dong YU Xiao-Min and YU Chao-Fan Vol. 54 of collapse and revival of inversion in which the collapse time τ c depends only on (k k 2 ) but the revival time τ R is related to both (k k 2 ) and n 0. The dynamical behavior for the N a (t) and N b (t) are closely related to k 2 /k or k /k 2. (ii) Providing that the initial phonon state is in the vacuum state and the photon field is in the coherent state for the first time we have confirmed the existence of the second-order and high-order squeezing of the phonon field in the polariton system up to now. As the matter stands the squeezed photon states can be produced by the polariton-one-photon interaction to evolve into the two-photon coherent states due to the non-adiabatic phonon-two-photon correlation in which the photon energy is damage to the development of the two-photon coherent states while the squeezed phonon states can also be produced by the polariton-one-phonon interaction to evolve into the two-phonon coherent states due to the non-adiabatic photon-two phonon correlation in which equally the phonon energy will be weaken for the evolution of the two-phonon coherent states. As time increases due to the initial photon coherent state the evolution behavior of the second-order squeezed index for the photon field will encounter the phenomena of revival of the collapsed inversion rather than the phonon field. As a matter of fact the second-order squeezing S Xi (t) and S Yi (t) are related closely to k k 2. Furthermore when the second-order squeezed condition for the photon field relevant to k 2 /k or k /k 2 is satisfied S Xi (t) is inclined to S X (t) < thus the time average S X (t) T < but S Y (t) is inclined to S Y (t) > and S Y (t) T >. Similarly for the phonon field S X2 (t) and S Y2 (t) also tend to S X2 (t) < and S Y2 (t) > respectively so S X2 (t) T < and S Y2 (t) T >. These results are considered as the result of the quantum fluctuation of the new vacuum state of the polariton system in which the quantum fluctuation of the nonclassical vacuum state are Xi 2 αβ < /4 and Yi 2 αβ > /4 (i = 2) respectively. It should be noted that if the two-mode squeezed coupling is non-attractive no matter k k 2 or k = k 2 the second-order squeezing of phonon field will not appear i.e. S X2 (t) T > and S Y2 (t) T >. Particularly when k = 0 S Xi (t) T > and S Yi (t) T > when k 2 = 0 S Xi (t) T = and S Yi (t) T = it is because the interaction system is not a two-mode rotation squeezed vacuum state. (iii) The high-order squeezing D (2N) (t) and D (2N) (t) (i = 2) is another quantum fluctuation effect which is independent from the quantum fluctuation of the nonclassical vacuum state when the initial photon field is in the squeezed vacuum state. In particularly for D (2N) (t) and D (2N) (t) (i = 2) the phenomena of collapse and revival of inversion in the polariton system can not only occur in the photon field but also in the phonon field up to now. In the same manner no matter photon or phonon field the high-order squeezing occurs in the X-components but not in the Y -components. That is to say the time evolution behavior of (t) (i = 2) are always inclined to (t) < 0 so the time average (t) T < 0 but the Y -components are inclined to (t) > 0 and (t) T > 0. Equally when k = 0 or k 2 = 0 (t) > 0 and (t) T > 0. Particularly when the two-mode squeezed coupling is non-attractive since the initial photon field is in the squeezed state though the high-order squeezing appears in Y -component of photon field it will not occur in the phonon field and then X 2 (t) T > 0 and Y 2 (t) T > 0. References [ H.P. Yuen Phys. Rev. 3 (976) [2 H.P. Yuen and J.H. Schapiro Opt. Lett. 4 (979) 334. [3 G.S. Agawal Phys. Rev. Lett. 57 (986) 827. [4 R.L. Loudon and P.L. Knight J. Mod. Opt. 34 (987) 70. [5 G.S. Agawal J. Opt. Soc. Am. B 5 (988) 940. [6 G.S. Agqwal and J.K. Tara Phys. Rev. A 46 (992) 485. [7 L. Gilles and P.L. Knight Phys. Rev. A 48 (993) 582. [8 X. Hu and F. Nori Phys. Rev. Lett. 76 (996) [9 X. Hu and F. Nori Phys. Rev. Lett. 79 (997) [0 K.L. Wang J.L. Yang and T.Z. Li Phys. Lett. A 236 (997) 03. [ S.L. Wang S.G. Ma and K.L. Wang Chin. Phys. Lett. 20 (2003) 7. [2 C.F. Lo and R. Sollie Phys. Rev. B 48 (993) 083. [3 B.K. Chakraverty D. Feinberg Z. Hang and M. Avignon Solid State Commun. 64 (987) 47. [4 H. Zheng Phys. Lett. A 3 (988) 5. [5 B.B. Govorkov Jr and A.S. Shumovsky Preprint of the Jaint Institute for Nuclear Research Dubna (99). [6 S. Ghoshul and A. Chatterjee Phys. Rev. B 52 (995) 982. [7 S. Ghoshul and A. Chatterjee Phys. Lett. A 223 (996) 95. [8 S. Ghoshul and A. Chatterjee Phys. Rev. B 59 (999) 396. [9 O. Madelung Solid-State Theory Springer-Verlag Berlin Heidelberg (98). [20 Th. W. Ruijgrok and L. Van Hove Physica 22 (956) 880.