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Density Operator: the Coherent State and Squeezed State Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017)

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Density Operator: the Coherent State and Squeezed State Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017) Phase space representation of density operator: the coherent state and squeezed state Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 21, 2017) Here we discuss the phase space representation of the density operator, including Wigner function for the coherent state and squeezed state in quantum optics. The density operator of a given system includes classical as well as quantum mechanical properties. __________________________________________________________________ 1. P function representation for the Density operator We start with a density operator defined by ˆ d 2P() where P() is called the Glauber-Sudarshan P function (representation) and is an coherent state. We note that Tr[ˆ] Tr d 2P() d 2P()Tr[ ] d 2P() d 2P() 1 or Tr[ˆ] Tr d 2P() 1 d 2P() d 2 1 2 d 2P() d 2 1 2 2 d 2P() d 2 exp( * * ) 1 2 2 d 2P()exp( ) d 2 exp( 2Re[ * ])) 1 2 d 2P() exp( )exp( 2 ) d 2P() 1 where 1 d 2 1ˆ (formula) 2 exp{ 2 2 2Re[ * ]} Note that and are complex numbers, x iy , a ib Since Re[ *] Re[(a ib)(x iy)] ax by we get the integral as d 2 exp( 2 2Re[ *])) dxexp(x2 2ax)dx dyexp(y2 2by) exp(a2 b2 ) exp( 2 ) We also note that the density operator can be rewritten as 1 ˆ d 2 d 2 ˆ 2 1 d 2 d 2 ˆ 2 using the formula 1 d 2 1ˆ ((Example)) The average number of photons can be written as nˆ aˆ aˆ Tr[ˆaˆ aˆ] Tr[ d 2P() aˆ aˆ] d 2P() aˆ aˆ d 2P() 2 So P() is normalized as a classical probability distribution. ((Note)) Here we note that ˆ d 2P() 2 d 2P() d 2P()exp[ 2 ] since 2 exp{ 2 2 2Re[ *]} exp[ 2 ] Since exp[ 2 ] is not a delta function, the diagonal elements ˆ P( ) Although ˆ must be positive. However, the integral d 2P()exp[ 2 ] does not require to be positive. 2. Two-dimensional Dirac delta function of the complex variable Two-dimensional Dirac delta function of the complex variable, 'i". (2) () (') (") 1 dx dy exp[i('x " y)] (2 )2 1 d 2 exp( * * ) 2 1 d 2 exp( * * ) 2 or 1 (2) () d 2 exp( * * ) 2 where * * 2i Im( * ) i('x " y) and y x 1 i (y ix) 2 2 2 We also note that the 2D Dirac delta function is given by 1 (2) ( ) d 2 exp[( ) * ( * *) ] 2 3. Expression of P() How can we find the expression of P() ? We consider the density operator which is defined by ˆ d 2P() Using the formula for the scalar product for the coherent states, 1 2 1 2 exp[ * ] 2 2 we get the matrix element of the density operator as u ˆ u d 2P() u u exp[u 2 ] d 2P()exp[ 2 ]exp[ *u u*] or exp[u 2 ] u ˆ u d 2P()exp[ 2 ]exp[ *u u* ] Here we use the 2D Dirac delta function. d 2u exp[u 2 ] u ˆ u exp[ *u u* ] d 2 P( )exp[ 2 ] d 2u exp[ *u u* ]exp[ *u u* ] d 2 P( )exp[ 2 ] d 2u exp[( )u* ( )*u] 2 d 2 P( )exp[ 2 ] ( ) 2 P()exp[ 2 ] or 1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2 4. Another expression of P() )) The expression of P() can be also obtained as follows. * eaˆ eaˆ Tr[ˆeaˆ e aˆ ] * Tr[ d 2 P( ) eaˆ e aˆ ] * d 2P( ) eaˆ e aˆ * * d 2P( )e Here we use the expression of the 2D Dirac function as 1 (2) ( ) d 2 exp[( ) * ( * *) ] 2 Thus we get 1 * * * 1 * * * * d 2 eaˆ e aˆ e d 2 ( d 2 P( )e )(e ) 2 2 1 * * * d 2 P( ) d 2e ( ) ( ) 2 d 2 P( ) (2) ( ) P() or 1 * * * P() d 2 eaˆ e aˆ e 2 1 * * d 2Tr[ˆe (aˆ )e (aˆ ) ] 2 Here we note that Tr under the integral is characteristic function of the normally ordered aˆ , aˆ operators, aˆ *aˆ aˆ *aˆ N ( ) e e Tr[ˆe e ] Therefore we have 1 * * * P() d 2 eaˆ e aˆ e 2 1 * * d 2 ( )e 2 N This relation represents a mapping from the density operator ˆ , which is a function of the two operators aˆ and aˆ , to the scalar function N ( ) , which is function of the complex variables . P() is the Fourier transform of the normally ordered characteristic function. 5. P function for the coherent state (I) We consider the density operator for the pure coherent state ; ˆ . We note that u ˆ u u u 1 2 1 2 1 2 1 2 exp[ u u* ]exp[ u *u] 2 2 2 2 exp(u 2 2 u* *u) where 1 2 1 2 u exp[ u *u] 2 2 1 2 1 2 u exp[ u u* ] 2 2 Then we get 1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2 1 2 2 2 2 * * * * e e u e( u u u)e(u u )d 2u 2 1 2 2 * * * * e e(u u)e(u u )d 2u 2 1 2 2 * * * e e(u ( )u( )d 2u 2 2 2 e (2) ( ) (2) ( ) or P() (2) ( ) for the density operator (pure state ˆ ). 6. P function for the coherent state (II) We consider the density operator for the pure coherent state ; ˆ . Thus we get aˆ *aˆ N ( ) Tr[ˆe e ] * Tr[ eaˆ e aˆ ] * eaˆ e aˆ * * e 1 * * * P() d 2 eaˆ e aˆ e 2 1 * * d 2 ( )e 2 N 1 * * * * P() d 2e e 2 1 * * * d 2 e ( ) ( ) 2 (2) ( ) 7. P function for the number state n (I) We consider the density operator for the pure coherent state n ; ˆ n n . We note that * n 2 (u u) u ˆ u u n n u e u n! Thus we get 1 2 2 * * P() e e u u ˆ u e(u u )d 2u 2 * n 1 2 2 2 (u u) * * e e u e u e(u u )d 2u 2 n! 2 e 1 * * (u*u)n e(u u )d 2u n! 2 Formally, we can write 2 2n e 1 * * P() e(u u )d 2u n! n * 2 2 e 2n n () n! n * 8. P function for the number state n (II) We consider the density operator for the pure number state n ; ˆ n n . Thus we get aˆ *aˆ N ( ) Tr[ˆe e ] * Tr[ n n eaˆ e aˆ ] * n eaˆ e aˆ n 1 * * P() d 2 ( )e 2 N 1 * * * d 2 n eaˆ e aˆ n e 2 1 * * * d 2 n eaˆ e aˆ n e 3 1 1 * * * d 2 n n d 2e ( ) ( ) 2 1 d 2 n n (2) ( ) 1 2 n or 2n 1 2 2 P() n e n! which is a Poisson function. where * * e aˆ n * n eaˆ e n * * e aˆ n e n 9. Q-representation There are other orderings possible for the two operators aˆ and aˆ . This time we use the antisymmetric ordering and define the antinormally ordered characteristic function *aˆ aˆ A ( ) Tr[ˆe e ] Its Fourier transform is called Q representation. This representation is sometimes called the Husimi function. 1 * * Q() d 2 ( )e 2 A and has a simple form 1 Q() ˆ 0 Thus Q() is the probability of finding a system in the density operator ˆ in the coherent state . This proof is given below. *aˆ aˆ A ( ) Tr[ˆe e ] 1 * d 2 Tr[ˆe aˆ eaˆ 1 * * 1 * Q() d 2 e d 2 Tr[ˆe aˆ eaˆ ] 2 1 * * 1 * * d 2 e d 2 Tr[ˆ eaˆ ]e 2 1 1 * * * d 2 Tr[ˆ ] d 2 e ( ) ( ) 2 1 d 2 [ ˆ (2) ( ) 1 ˆ ((Husimi function)) Kôdi Husimi (June 29, 1909 – May 8, 2008, Japanese:) was a Japanese theoretical physicist who served as the president of the Science Council of Japan. Husimi trees in graph theory, and the Husimi Q representation in quantum mechanics, are named after him. https://en.wikipedia.org/wiki/K%C3%B4di_Husimi Fig. Photo of Prof. Dodi Husimi. http://www.47news.jp/PN/200805/PN2008050901000355.-.-.CI0003.jpg 10. Q representation for a pure coherent state For the density operator ˆ we have 1 2 1 2 2 Q() exp[ * * ] since 1 2 1 2 exp[ * ] 2 2 11. Q representation for the pure number state For the density operator ˆ n n we have 2n 1 2 2 Q() n exp( ) n! which is the Poisson function 12. Wigner representation For symmetric order of two operators aˆ and aˆ . The characteristic function is defined as * ˆ ˆ S ( ) Tr[ˆ exp(aˆ aˆ)] Tr[ˆD( )] D( ) Its Fourier transform is called Wigner representation. 1 * * W () d 2 ( )e 2 S 1 * * d 2 e Tr[ˆDˆ ( )] 2 13. Another expression of Wigner function 1 1 Xˆ xˆ (aˆ aˆ ) (aˆ aˆ ) , 2 2 2 2 1 1 m 1 Yˆ pˆ 0 (aˆ aˆ ) (aˆ aˆ ) 2 2 2i 2i or aˆ Xˆ iYˆ , aˆ Xˆ iYˆ . Note that 1 1 1 1 [Xˆ ,Yˆ] [ xˆ, pˆ] [xˆ, pˆ] [xˆ, pˆ] i1ˆ , 2 2 2 2 2 2 Dˆ ( ) exp(aˆ *aˆ) exp[ (Xˆ iYˆ) * (Xˆ iYˆ)] exp[( * )Xˆ i( * )Yˆ] exp[2i " Xˆ 2i 'Yˆ] exp(i ' ")exp(2i " Xˆ )exp(2i 'Yˆ) This operator can be rewritten as 1 Dˆ ( ) exp(i ' ")exp(2i " xˆ)exp(2i ' pˆ) 2 2 2 ' pˆ exp(i ' ")exp(i 2 "xˆ)exp( i ) Then we get 2 ' pˆ Dˆ ( ) x' exp(i ' ")exp(i 2 "xˆ)exp( i ) x' 2 ' exp(i ' ")exp(i 2 "xˆ) x' 2 ' 2 ' exp(i ' ")exp(i 2 "(x' ) x' 2 ' exp(i ' ")exp(i 2 "x') x' Wigner function 1 * * W () d 2 ( )e 2 S 1 * * d 2 e Tr[ˆDˆ ( )] 2 1 * * d 2 e dx' x' ˆDˆ ( ) x' 2 1 * * 2 ' d 2 e dx'exp(i ' "i 2 "x') x' ˆ x' 2 Note that * * 2i '"2i "' * * i ' "i 2 "x' 2i '"2i "'i ' "i 2 "x' 2i '"i(2' ' 2x') " We use the Dirac delta function as d "exp[i(2' ' 2x') "] 2 (2' ' 2x') Then we have 2 2 ' W () dx' d 'exp(2i '") (2' ' 2x') x' ˆ x' 2 2 dx'exp[2i"(2' 2x')] x' ˆ x' (2' 2x') 2 2 2 dx'exp[4i"(' x')] x' ˆ 'x') 2 Here we define u 2 2 u x' x , 'x' x 2 2 2 ' x , " p 2 1 dx' du 2 4"(' x') 4"(' x') 2 2 2 4 "(x' ') 2 u 4 "(x x) 2 2 2 "u 2 2 pu 1 u u W () W (x, p) du e2ipu x ˆ x 2 2 which is the Fourier transform of non-diagonal density matrix element in the position eigenstates basis.
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