Question

I need help with 1.b)

1a. Assume the unitary operator U-exp(1?/4lâ??? +â?â01. show that the matrix relation for the beam splitter, is identical with the unitary transformation Hint: use the Baker-Hausdorf lemma on page 13 of Gerry/Knight. 1b. operator U-exp(??/2la??, + âtaol). Obviously, for ?-?/2 the result in 1a is retrieved. Calculate the beam splitter matrix corresponding to the more general unitary

Answer 1

U^bar = exp(i product/4 [a_0^+ a_1^+ a_1^+ a_0^]) given matrix relation for beam splitter, [a_2^a_3^] = 1/ squareroot 3 [1 i I 1] [a_0^a_4^] i.e a_2^= 1/squareroot (a_0^+ I a_1^) a_3^= 1/squareroot (i a_0^+ I a_1^) we have to show it is identical with the unitary transformation. [a_2^a_3^] = u_^+ (a_0/a_1)u [a} a_2^= u_^+ a_0 u & a_3^= u^a_1 u a_3^= u_^+ a_0 u = exp(-i product/4 (a-1^+ a_0 + a_0^+ a_1^]).a_0 exp i product/4 (a-1^+ a_0 + a_0^+ a_1^]) rightarrow using Baker-Hausdorf lemma e^ial A^e^-ial = A + i lambda [a, A] +(i lambda)^2/2! [a_2(a, A)] + ... \r\n so a2^= exp i product/4 [-(a_1 + a_0 + a_0^+ a_1^)] a_0 exp (- i product/4[- (a_0^+ a_1 + a_1^+ a_0^)]) = a_0 + i product/4[- (a_0^+ a_1 + a_1^+ . a_0)] + (i product/4)^2/2! [-(a_0^+ + a_1^+ a_0), - (a_0^+ a_1 + a_1^+