QUANTUM INTERFEROMETERS, EULER ANGLES ...

February 12, 2013

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QUANTUM INTERFEROMETERS, EULER ANGLES, UNITARY REPRESENTATIONS OF SU(2)

CLAUDIO PARMEGGIANI Università di Milano (Italy) E-mail: clparm@in…nito.it

The "one particle" and the "two particles" interferometers can be (quantum mechanically) described in terms of Hilbert spaces of states and scattering operators. In this case the scattering operators realize a unitary representation of SU(2) and the Euler angles of the SU(2) group are related to the interferometers parameters (transmission coe¢ cients, phase shifts)

1. Introduction. Quantum Interferometers A quantum interferometer can be described, as any physical system, by a Hilbert space of “states”and, if we are interested to transition probabilities, by a scattering operator (see1 , for example). For a (quantum) one particle interferometer, the describing Hilbert space, H, is the direct sum of two isometric closed subspaces, H1 and H2 ; the interferometer scattering operator is realized by mean of a (…xed) self-adjoint isometry from H1 to H2 , J; then, if I is the identity operator of H, J:J = I. In the next sections we shall examine the so called Mach-Zehnder interferometer and we shall see as the scattering operator realizes a unitary representation of the SU(2) group, on the Hilbert space H. In a similar way it is possible to describe a (quantum) two particles interferometer, for example the EPR (Einstein, Podolski and Rosen) one. In this case the Hilbert space of the physical system is the tensor product of two copies of the same Hilbert space; that is H = H H where H is the one particle space. Now, for a suitable choice of the “initial conditions”, we shall obtain the well know interference and correlation (entanglement) e¤ects. 1

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2. Beam splitters, phase shifter 2.1. The mathematical beam splitter The unitary operator (E =

I J 2 ):

S( ) = cos( 2 ):I + i:sin( 2 ):J

(1)

= exp(+i 2 ):E+ + exp( i 2 ):E is the scattering operator of a beam splitter (a semitransparent mirror, a perfect crystal1 ); cos( 2 ) is the transmission coe¢ cient of the splitter; then if cos( 2 ) = 0 (and sin( 2 ) = 1) we have a true, re‡ecting mirror. A splitter because, if the ongoing unit vector 1 belong to H1 , the outgoing vector: S( ):

1

= cos( 2 ):

1

+ i:sin( 2 ):

2

(2)

(where 2 = J: 1 ) is split between the closed orthogonal subspaces H1 and H2 . For a physical splitter cos( 2 ) is always > 0, but we can obtain a mathematical splitter combining a physical one and a mirror (as S( ):S( ) = S( + )). 2.2. The phase shifter If E1 and E2 are, respectively, the orthogonal projectors on the closed subspaces H1 and H2 , the unitary operator (K = E1 E2 ): T (#) = exp(+i #2 ):E1 + exp( i #2 ):E2 =

cos( #2 ):I

+

(3)

i:sin( #2 ):K

describes, quantum mechanically, a phase shifter (the shift is generally a consequence of strong, electromagnetic or gravitational interactions1 ); # is the phase di¤erence generated by the shifter. Observe that J:K = K:J and J:T (#):J = T ( #) so: E :K:E = 0

(4a)

E :K:E = E :K.

(4b)

3. The Mach-Zehnder interferometer 3.1. The scattering operator. The Euler angles An one particle, Mach-Zehnder interferometer is described by the unitary scattering operator : U ( ; ; #) = S( ):T (#):S( ),

(5)

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that is splitter, shifter, splitter (plus mirror). Now we can easily verify that U is a unitary representation of SU (2) (on the Hilbert space H); , , # are the corresponding Euler angles. In fact, let consider the SU (2) matrix: a b b a

(6)

where, as usual2 : a = cos( #2 ):exp(+i b=

+ 2

i:sin( #2 ):exp(+i

)

(7a) );

2

(7b)

hence we shall obtain, remembering the de…nitions of U , S and T (Section 2) and the relations (4a, 4b): U ( ; ; #) = V (a; b) = ei 2 E+ + e

i2

E

cos( #2 )I + i:sin( #2 )K

a:E+ + a :E + b:E+ :K

ei 2 E + + e

i2

E

= (8)

b :E :K

Then we can directly verify the multiplication formula of the SU (2) group: V (a; b):V (c; d) = (ac

bd ):E+ + (a c

+(ad + bc ):E+ :K + ( a d

b d):E +

(9)

cb ):E :K

3.2. Transition Probabilities. Interferences If the unit vectors 1 and 2 (= J: 1 ) belong, respectively, to H1 and H2 , the Mach-Zehnder interferometer transition probabilities are (remembering the de…nition of the transition probability, in a quantum contest): P rob( P rob(

1; 1;

1) 2)

= j(

1 jU (

; ; #):

2 1 )j

= j(

1 jU (

; ; #):

2

2 )j

= A( ; )

2:C( ; ):cos(#) (10a)

= B( ; ) + 2:C( ; ):cos(#) (10b)

Here A; B; C are polynomials of the sin and cos of and , A + B = 1. The cos(#) term, coming from the phase shifter, is clearly responsible of the interference e¤ects. 4. The EPR Interferometer 4.1. The scattering operator Now the Hilbert space of the physical system (the EPR, Einstein, Podolski and Rosen interferometer) is the tensor product (eventually symmetric or

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antisymmetric) H H of two copies of the same Hilbert space H (the one particle space); for every particle there is a phase shifter and a beam splitter. The splitters have two di¤erent transmission coe¢ cients, cos( 2 ) and cos( 2 ); the shifter have two di¤erent phase di¤erences, #A and #B . Then, if the S and T operators are de…ned as in the previous sections, the interferometer scattering operator is: (S( )

S( )):(T (#A )

T (#B )) = (S( ):T (#A ))

(S( ):T (#B ))

(11)

4.2. Transition Probabilities. Entanglement Now, for a suitable choice of the ongoing unit vector, , we shall obtain the well know interference and correlation (entanglement) e¤ects. In fact, let be: = c1 :(

1)

1

+ c2 :(

2)

2

(12)

where ( 2 = J: 1 ): 1 2 H1 , 2 2 H2 and (c1 )2 + (c2 )2 = 1, c1 and c2 are real numbers. Then we have, repeating the calculations of the previous section: P rob(

1

P rob(

1

) = (c1 )2 A1 + (c2 )2 B1

1;

2

2;

2c1 c2 :C:cos(#)

(13)

2

) = (c1 ) A2 + (c2 ) B2 + 2c1 c2 :C:cos(#)

Here A1 ; A2 ; B1 ; B2 are, again, polynomials of the sin and cos of and , C ( ; ) = 41 sin( ):sin( ), D ( ; ) = 41 cos( ): cos( ), # = #A + #B . If c1 = 0 or c2 = 0 (disentanglement) we lose the cos(#) interference term. But if c1 = c2 the entanglement is maximal and we have the “correlation” formulae: P rob(

1

1;

) = P rob(

2

2;

)=

P rob(

1

2;

) = P rob(

2

1;

)=

and, eventually, P rob(

1

1;

) = 0 or P rob(

1 4 1 4 1

+D

C:cos(#)

(14)

D

C:cos(#)

(15)

2;

) = 0.

References 1. H. Rauch, Reality in Neutron Interference Experiments, in J. Ellis, D. Amati (Ed.), Quantum Re‡ections, Cambridge UP, 2000. 2. A.R. Edmonds, Angular momentum in Quantum Mechanics, Princeton UP, 1996.