Lorentz-group Berry phases in squeezed light

Volume 132, nurhber 2,3

LORENTZ-GROUP

26 September 1988

PHYSICS LETTERS A

BERRY PHASES IN SQUEEZED

LIGHT

Raymond Y. CHIAO Department of Physics, University of California, Berkeley, CA 94720. USA

and Thomas F. JORDAN Physics Department, University of Minnesota, Duluth, MN 55812, USA

Received 3 May 1988; revised manuscript received 12 July 1988; accepted for publication 13 July 1988 Communicated by J.P. Vigier

Berry phases for the Lorentz group are obtained from transformations that produce squeezed states of the electromagnetic field. They involve phases that could be observed in interference experiments with degenerate parametric amplifiers oflight or microwaves.

The Berry phases [ 1] most often observed in experiments [2] are produced by rotations. Berry phases produced by Lorentz transformations are almost as simple mathematically. They have been discussed at length [ 3-91, but there has been no suggestion for observing them in an experiment. Indeed, it has been shown [ 7,8] that in one context they can be removed by a canonical transformation so they appear to be a property of the mathematical description that could not be observed physically. Here we show that the essential element of these Berry phases could be observed as a change of phase of the electromagnetic field in squeezed states of light #’ or microwaves [ 141. It could be seen as a fringe shift in interference experiments. In this way, a physical effect of the squeezing could be measured without measuring anything like photon statistics or noise. Instead of the three spin matrices that are generators for rotations, we use three generators that satisfy the commutation relations of the Lorentz group for two space dimensions. The difference in the com” For a review that provides an introduction to squeezed states see ref. [ 101. For a survey of the state of the art see ref. [ 111, a special issue on squeezed states beginning with ref. [ 121. See alsoref. [13].

mutation relations is just one minus sign, but the result is that the sign of the Berry phase is opposite to what it is for rotations. This should produce a fringe shift in the opposite direction and thus be immediately observed. From a geometric point of view, this sign change is not surprising. Instead of a vector making a loop ‘on the unit sphere, which has positive curvature, we will have a vector making a loop on a unit hyperboloid, which has negative curvature. Consider a single mode of the electromagnetic field with creation and annihilation operators ut and a, where [a, at] = 1. Let K, = -i(au-utut)/4,

K2 = (uu+utut)/4,

J3 = (uu++u+u)/4.

(1)

Squeezed fields [1O-131 are produced by the unitary operators S=exp[ir(K,

cos O+K, sin e)] ,

(2)

made from the hermitian operators K, and K2. From S[ufexp(

-i@u+]S+

=exp(*r/2)[ufexp(-i@u+],

(3)

we can see that for a field, for which we can write the identity

03759601/88/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

77

a exp( -iiwt)+a+ =exp(i0/2) -iexp(if3/2)

exp(iot)

a’=exp(i@/2)a.

[a+exp( -i@z+]cos(wt+0/2) [a-exp(

-i@a+] sin(ot+8/2),

(4) the effect of the transformation by S is to squeeze the part of the field that oscillates in time as sin(ot+ 8/2) and, in the same proportion, stretch the part that oscillates as cos (wt + 8/2 ). The squeezing operations for 8 and 0+x are inverses of each other. What we have described as transformations of quantum field operators can be described equally well as transformations of classical field functions; they are not the origin of quantum effects. The operators K1, K2 and J3 satisfy the commutation relations [K,&l=-iJ,,

[&,J31=%,

[J~,KII=%,

+k2K2+k3J3,

(6)

where k,, k2, k3 are real numbers such that -k:-k:+k:=l.

(7)

Successive transformations move the tip of the vector k= (k,, k2, k,) on a unit hyperboloid. Suppose a sequence of transformations takes k around a closed loop from (0, 0, 1) back to (0, 0, 1); the operator (6 ) goes around a loop from J3 back to J3. Then the product of the unitary operators is So...S,S, =exp( -i@J,)

(10)

This could be observed as a change of phase of light waves or microwaves in an interference experiment. The change of phase would be observed for any state where the expectation value of the annihilation operator is not zero, particularly for a coherent state produced by a laser or a microwave source. For a more exact parallel with the usual presentation of Berry phases, we can consider a state represented by an eigenvector )n) of u+u with eigenvalue n. The sequence of transformations with S,, S,, ... So changes the state vector In) to

exp(i~J~)In)=exp[i(~/2)(n+l/2)llnl.

(8)

(11)

The state is not changed; the state vector is changed only by the Berry phase

exp[i(W)(n+1/2)1. (5)

which are characteristic of the Lorentz group for two space dimensions. Therefore a sequence of transformations with unitary operators S,, S,, .... So of the form (2 ) takes J3 to S,...S,S,J,S~St...St,=k,K,

26 September 1988

PHYSICS LETTERS A

Volume 132, number 2,3

(12)

The essential element in the Berry phase ( 12 ) is exp(i@/2). It can be observed as a change of phase of light waves or microwaves if they can be squeezed in a sequence of operations that corresponds to k going around a closed loop. We see no reason in principle why this cannot be done. The phase could be observed in interference between two parts of a split beam after each part is squeezed twice. Consider squeezing represented by unitary operators S,, S, for one part of the beam and S,, S, for the other part. We require that S,S, J&St

=S.,S,J,S:St

Then &S&S, S&&S,

.

(13)

commutes with J3 so

=exp( -i@J,)

(14)

for some angle @,or

for some angle @. When the sequence of transformations is complete, its effect on the field is just to change the annihilation operator a to

S,S, =S&

exp( -i@J,)uexp(i@J~)=exp(i@/2)u.

The transformations produce a phase difference between the two parts of the beam. It is the same as if one part were not changed and the other part were transformed by S~S&S,. There are two advantages: no part of the beam needs to be squeezed more than twice; and both parts can be treated identically

(9)

Shifting to the Schriidinger picture, we see that a coherent state represented by an eigenvector I a) of the operator a with eigenvalue (Yis changed to a coherent state represented by an eigenvector 1a’ ) with eigenvalue 78

exp( -i@J3> .

(15)

From (9 ) we see that S,S, aStS$ =exp(i@/2)S&uSfSd.

(16)

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PHYSICS LETTERS A

except for the angles 8 of the operators S defined by eq. (2). For example, let

cos($/2)={cosh(t)+

S, =exp(irK,),

S,=exp(i&)

sin(@/2)= -{cash(r)-

S,=exp(irK,),

S,=exp(i&).

,

(17)

The requirement ( 13) is sinh(r)=cosh(r)

sinh(s)

(18)

or tanh ( r) = sinh (s). The transformation from J3 to S, J,Sl to S,S, J,SlS’j takes R from (0, 0, 1) to (0, sinh(r), cash(r))

(19)

to (-cash(r)

sinh(s), sinh(r), cash(r) cash(s))

= (-sinh(r),

sinh(r), cash(r) cash(s))

(20)

and the transformation from J3 to S3J3Sf S&J3SfSf takes R from (0, 0, 1) to (-sinh(r),

0, cash(r))

to (21)

to (-sinh(r),

cash(r) sinh(s), cash(r) cash(s))

= (-sinh(r),

sinh(r), cash(r) cash(s))

.

(22)

(See fig. 1.) The angle @in eq. ( 14) is determined by

-sinh(r) t-7 B 53

”

"

s2

Sl

A.

c

sinh(r)

kl Fig. 1. The changes of k,, k2 made by the transformations S,, .Sz, S,, S, defined by eqs. ( 17) and ( 18). The other component ks is determined by the hyperboloid condition (7).

-sinh(r)

-sinh(r)

[ 1 +tanh2(r)]‘/’

tanh(r)}/2,

tanh(r)}/2,

[ l+tanhZ(t)]“2

(23)

which is easy to calculate with Pauli matrices. This is not the same as the angle determined by the area on the unit hyperboloid [ 5 1. The transformations in our example do not satisfy the condition in the definition of Berry phases [ 51 that leads to the area formula. It is not important in this physical setting. Once again, the idea of Berry phases has proved to be more general than any of the definitions that attempt to pin them down. The squeezing could be done with degenerate parametric amplifiers for either light [ 10-l 31 or microwaves [ 141. For microwaves, the squeezing transformation represented by the operator (2) could be implemented by a unidirectional parametric amplifier, for example, a circulator-isolated varactor paramp operated in a degenerate mode with the signal and idler frequencies exactly equal. The pump would operate at twice the signal frequency. The amplitude of the pump would determine the gain of the paramp and the parameter r in the operator (2). The phase of the pump would determine the parameter 19.We would cascade amplifiers in series inside a single-mode waveguide, so we would need to consider only a single mode. Phase shifters in the pump circuitry would determine the angles 0 for successive squeezing operations. For example, consider the microwave degenerate parametric amplifier circuit shown in fig. 2. The heavy solid lines denote single-mode waveguides for the signal (or idler) frequency. A signal is injected into the Y-junction at point A, where it is divided equally into the two arms of an interferometer. One arm implements the transformation represented by the operator S,S,, and the other S&. The interference signal is detected upon recombination at the Yjunction at point B of the interferometer. For op erators satisfying eqs. ( 17) and ( 18), the resulting phase difference is given by eq. (23 ) . The dynamical phases cancel if the optical path lengths of the two arms are the same. A Berry phase determined by an area on the unit 79

Volume 132, number 2,3

PHYSICS LETTERS A

SIGNAL

+A

PUMP -@

26 September 1988

t3 +DETECTOR 1r

Fig. 2. Microwave interferometer circuit for measuring Berry phases for the Lorentz group. The heavy solid lines denote single-mode waveguides for the signal (or idler), and the light solid lines denote pump waveguides at twice the signal frequency. The triangular symbols denote degenerate paramps, the square symbols phase shifters and attenuators, and A, B the Y-junctions.

hyperboloid can be made to appear or disappear by making a canonical transformation [ 7,8 1. It might also be observed physically in an experiment. There is no contradiction. The effect of the canonical transformation is physically observable. It is easy to see this explicitly. Let (Q+iP)/fi,

a=

(24)

with Q and P hermitian operators. Then

[Q,f'l=i

(25)

K2=(Q2-P2)/4,

Js=(Q*+P*)/4,

(26)

k,K, +k&+k,J, =

[aP2+8(QP+pQ)+yQ21/2,

(27)

where a=(k~

-k2)/2,

~=(k,+k2)/2.

P=k/&

(28)

Let

P=p’ - /?Q/a .

Then 80

(29)

(30)

and k,K, +k,K,+k,J,=[aP’*+yQ*]/2 =k,K;+k,J;,

(31)

where K; = (Q*-p’*)/4,

and K,=(QP+PQ)/4,

[Q,P’]=i

J; = (Q’+P”)/4.

(32)

From the formula (28) for (Yand the hyperboloid condition (7), we see cr is positive, so P’ is well defined. When we look at the operator (3 1) in terms of K2 and J; we see no Berry phase; the vector (0, k2, k3) moves only in the 2, 3 plane and circumscribes no area on the unit hyperboloid. But as k moves around the loop, a! and j? change, so P’ and K;, J; change. What we see when we look at things in terms of Q and P’ or K2 and J” is that no Berry phase is produced by change of electromagnetic fields relative to fields that are already changed by the changes from Q, P to Q, P’. The Berry phase would not be physically observable if the changes from Q, P to Q, P’ were not physically observable. If we were considering an oscillator using P only as a canonical variable to calculate the motion of Q, and not assuming a particular relation between P and the phys-

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ical momentum or velocity, we would conclude the Berry phase is not physically observable. Since we are considering electromagnetic fields, the changes from Q, P to Q, P’ are physically observable and so is the Berry phase. Indeed, we have exp[ir(K,+Jj)]Qexp[-ir(Kz+Js)]=Q, exp[iT(KZ+53)]Pexp[ =P-rQ.

-ir(&+.&)] (33)

The changes from Q, P to Q, P’ are made with squeezing operators generated by K2+ J3. They are active transformations that change the physical fields, not passive transformations that change only the mathematical framework. We conclud that the essential element of the Berry phase arising efrom a cycle of Lorentz transformations can be seen physically as a fringe shift in interference of squeezed electromagnetic fields. We propose to use microwave varactor degenerate paramps to do thle experiment. Two different kinds of Berry phase for electromagnetism have already been observed: Pancharatnam’s phase associated with cyclic changes~of polarization; and the phase associated with cy ‘c changes of direction of photon spin [ 2 1. In this p er we point out a third kind of Berry phase in elect$ magnetism, and we suggest a way to observe it. It is a physical effect of the squeezing that might be useful; it can be measured without measuring anything like photon statistics or fluctuations.

k

We would li e to thank J.C. Garrison, L. Mandel and E.H. Wichmann for helpful discussions. This in part by the U.S.A. National under grant ECS 86-13773 and Resech under contract NOOO14-88K-0126.

26 September 1988

References [ 1] M.V. Berry, Proc. R. Sot. A 392 ( 1984) 45; Nature 326 (1987) 277; C.A. Mead and D.G. Truhlar, J. Chem. Phys. 70 ( 1979) 2284; R.Y. Chiao and Y.-S. Wu, Phys. Rev. Lett. 57 (1986) 933; I. Bialynicki-Birula and 2. Bialynicka-Birula, Phys. Rev. D 35 (1987) 2383; J. Anandan and L. Stodolsky, Phys. Rev. D 35 ( 1987) 2597; T.F. Jordan, J. Math. Phys. 28 ( 1987) 1759; Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58 (1987) 1593. [2] A. Tomitaand R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937; R. Tycko, Phys. Rev. Lett. 58 (1987) 2281; T. Bitter and D. Dubbers, Phys. Rev. Lett. 59 (1987) 25 1; D. Suter, G. Chingas, R.A. Harris and A. Pines, Mol. Phys. 61 (1987) 1327; R. Bhandari and J. Samuel, Phys. Rev. Lett. 60 ( 1988) 1211; R.Y. Chiao, A. Antaramanian, K.M. Ganga, H. Jiao, S.R. Wilkinson and H. Nathel, Phys. Rev. Lett. 60 ( 1988) 12 14; D. Suter, K.T. Mueller and A. Pines, Phys. Rev. Lett. 60 (1988) 1218. [3]M.V.Berry,J.Phys.A18(1985) 15. [4]J.H.Hannay,J.Phys.A18(1985)221. [ 51 T.F. Jordan, Berry phases and unitary transformations, J. Math. Phys., to be published. [6] L. Vinet, Phys. Rev. D 37 (1988) 2369. R. Jackiw, Three elaborations of Berry’s connection, curvature and phase, in: Workshop on Nonintegrable phases in dynamical systems, University of Minnesota, 1987. Ph. de Sousa Gerbert, A systematic derivative expansion of the adiabatic phase, Massachusetts Institute of Technology preprint. R.G. Littlejohn, Phase anholonomy in the classical adiabatic motion of charged particles, Los Alamos National Laboratory preprint. [lo] D.F. Walls, Nature 306 (1983) 141; R.E. Slusher and B. Yurke, Sci. Am. 258 (1988) 50. [ll]J.Opt.Soc.Am.B4,No. 10(1987). [ 121 H.J. Kimble and D.F. Walls, J. Opt. Sot. Am. B 4 (1987) 1450. [13] H. Han, Y.S. Kim, and M.E. Noz, Phys. Rev. A 37 (1988) 807. [ 141 B. Yurke, P.G. Kaminsky, R.E. Miller, E.A. Whittaker, A.D. Smith, A.H. Silver and R.W. Simon, Phys. Rev. Lett. 60 (1988) 764.

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