Cyclic geometrical quantum phases : group theory derivation and manifestations in atomic physics

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Cyclic geometrical quantum phases : group theory derivation and manifestations in atomic physics

C. Bouchiat

To cite this version:

C. Bouchiat. Cyclic geometrical quantum phases : group theory derivation and man- ifestations in atomic physics. Journal de Physique, 1987, 48 (9), pp.1401-1406.

�10.1051/jphys:019870048090140100�. �jpa-00210568�

Cyclic geometrical quantum phases : ^group theory derivation and manifestations in atomic

physics

C. Bouchiat

Laboratoire de Physique Théorique ^E.N.S. Paris*, 24, ^rue Lhomond, ⁷⁵²³¹ Paris Cedex 05,

France

(Reçu ^{le 5} juirc 1987, i accepti ^{le ler} juildet 1987)

Résumé.-Nous présentons ^une démonstration de la phase quantique adiabatique ^de Berry ^{basée sur} la théorie des groupes. Notre formalisme est ensuite utilisé _pour analyser des manifestations possibles ^en Physique Atomique ^et

pour étudier la nouvelle phase quantique cyclique introduite récemment by Aharonov and Anandan.

Abstract.-We present ^a derivation of the Berry quantum ^adiabatic phase using ^group theory. ^Our formalism is used to discuss possible manifestations in Atomic Physics ^{and to} investigate ^{the new} cyclic quantum phase recently introduced by Aliaronov and Anandan.

Classification

’

Physics ^Abstract

03.65 - 42.50

In this note we shall first derive the adiabatic quantum phase discovered by Berry [1,2] ^{by a group}

theoretical method. Our derivation applies ^{to Hamil-}

tonians having covariance properties ^with respect ^to

group transformations acting ^upon the external _pa- rameters space. Our method which allows a system- atic evaluation of the non adiabatic corrections will be used to discuss a possible ^{way to measure the} Berry phase ⁱⁿ optical pumping experiments. Finally ^{we in-} vestigate ^{within our} formalism the new cyclic quan- tum phase introduced by Aharonov and Anandan [9].

Let us consider a class of Hamiltonians H(B)

which describe quantum systems interacting ^{with an}

external uniform magnetic field B. The Hamiltonian

H(B) is assumed to be invariant _upon a simultaneous rotation of the quantum system ^{and the} external field B. More precisely ^let R(n, a) ^{be a} rotation of axis n

and angle ^a and J the total angular ^{momentum of the} system, ^the following operator identity is assumed to hold :

where

A time dependent Hamiltonian H(t) ^is generated by applying ^{to a} given initial field configuration Bo, ^a

time dependent ^rotation R(t) ⁼ R(n(t), c,(t)) :

Using (1) ^{we can rewrite} H(t) ^{as :}

In order to solve the time dependent Schrodinger equation :

it is convenient ^to introduce the rotated frame ^state

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090140100

1402

With the help ^of (4) ^{one finds} that )I(t)) ^obeys

the following Schr6dinger equation :

with

The unitarity ^relation U(R)l ⁼ U(R)-l ^implies

that fI,(t) ^{is indeed a Hermitian operator. In order}

to evaluate explicitly #i _(t) ^{we introduce a} system ^of spherical coordinates to describe the motion of B(t)

as given ⁱⁿ figure ¹ (X, ^{y, z are unit vectors along the}

x, y, z axis respectively). It is convenient to take Bo

in the z x plane :

Fig.l.-Spherical coordinates for the closed curve (C) ^drawn by B(t) during the adiabatic cycle.

We write the field B(t) corresponding ^{to a} given position ^{on the closed curve} (C) ^{as :}

For t = t B (t) is assumed to coincide again ^with Bo ; ^it implies 0 (T) ^{= 8o and} p(T) ^{= 27r.}

We are now in position ^to compute ill (t) ^{in term}

of the angular ^momentum operator ^{J of the} system :

Using (10) ^and simple properties ^of the rotation _group

one gets :

By making ^{a first order} expansion ^{of the} unitary operator associated with the above product ^{of rota-}

tions one gets ^the following expression for f¡ 1 (t) :

It is convenient to introduce the projections ^of

the angular ^momentum Ju, Jv, Jw upon the orthonor- mal set of unit vectors (u, v, w) associated with Bo :

u ⁼ Bo IB, ^{w = y, v = way. In} particular ^{we have :}

One readily obtains the following ^final expression

for f¡ 1 ( t) :

We shall assume now that H(Bo) is the Hamil- tonian of a complex ^atom interacting ^with Bo, ^writ-

ten in the _gauge Ao ⁼ 1/2r A ^{Bo. The eigen states}

of H(Bo) la m) ^{are, in general, non} degenerate ^and eigenstates ^{of J.Bo = JuB :}

Let us take as initial state an eigenstate ^of H(Bo) :

Following ^the usual adiabatic approximation, ^{we shall}

first keep only ^the diagonal part ^of ill (t) :

Using (7), (13) ^and (6) ^we obtain the state vector

at time t = T :

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The equation giving the time evolution of gets ^an extra term involving

To lowest order in (jli (t)) we ^{can neglect the}

time variation of (Iu) . ^The correcting ^{term A} (I:f: (T))

to formula (17) ^{is then obtained} by ^a simple quadra-

tiire :

with

To formulate the adiabatic approximation ^we

write 0(t) ⁼ Oof (tlT) ^and V(t) ^{= 21r} g(tlt) ^with f (0) ^{= 1,} g(0) = ^{0 and} f (1) = g(1) ^{= 1. We as-}

sume that the first and second order derivatives of the functions f (s) ^and g(s) ^are finite in the closed interval 0 s 1. The adiabatic approximation ^cor- responds ^{to the limit} IWNTI -+ ^00. Using ^an integra-

tion by part ^one proves that the absolute value of the correcting ^term given by (20) behaves like 1 /WNT

when IWNTI , ^{oo. If the system is prepared in such a}

way that (1,, (0)) ^{= 0, it can} be shown then that the

correcting ^term is of the order of (WN T) 2

In the above considerations ^we have ignored ^the

relaxation _processes during the time interval T. They

do not really constitute a problem since, for instance in the case of 3He, the relaxation time T2 ^{which de-}

scribes the decay ^{of the} transverse polarization ^can

be as long ^as several hours. The Larmor precession

of 3He nuclear polarization has bPen observed by ^C.

Cohen-Tannoudji ^{et al.} [5] using ^an optically pumped 87 Rb cell as magnetometer. ^This very sensitive tech-

nique, which has been used recently ^to search for a

permanent ^electric dipole ^{of 129Xe} [6], ^{seems to be}

well adapted ^{to a} measurement of the Berry phase ⁱⁿ quantum spin systems.

To measure the phase shift in formula (18) ^one

has to know accurately ^the dynamical phase wNT.

We have assumed that in the cyclic evolution of the

magnetic field its modulus remains constant. In _prac- tice such a condition _may be rather difficult to sast-

isfy. ^{To account for} possible deviations ^one writes :

B(t) ⁼ A(t)U(R(t))Bo ^where A() ^{is a slowly vary-} ing positive scale factor with A(0) ⁼ A (T). ^{The Berry}

phase ^remains unchanged ^{but the} dynamical phase

is replaced by ^WN fo ^{A(t)dt. There is, in principle a}

way ^to cancel out the dynamical phase and, ^{at the}

same time, ^to double the Berry phase. ^{At t = T}

one performs ^a rapid reversal of the magnetic ^{field :} B (T + 8t) = -B(T) ^{= -Bo, where 8t} > 0 is such that 6twN 1, ^{in order to} insure the validity ^{of the}

sudden approximation. ^{Then a new adiabatic} cycle ^is

started for T t 2T with a magnetic ^field B(t) = -B(t - T) ⁼ A(t - T) R(t - T) (-Bo). ^{The Berry} phase depends upon B(t) only through the rotation

R(t) ⁼ R(t - T) which is identical to R (t) ^{up to a}

time translation. It follows then that :

On the contrary ^the dynamical phase mWN :T ^Ã(t-

T)dt changes ^its sign. ^The average value of I+ ^{is then} given ^at time t = 2T by :

In practice, ^the validity of the above result de-

pends upon the _accuracy with which the magnetic

field can be monitored during the time intervals 0 t T, ^{T t} 2T, and reversed at time t ⁼ T + 0.

After the writing ^{of a} preliminary version of this note we become aware of a preprint of D. Suter et al.

[7] ^{where the} Berry phase ^{for a} quantum spin system is measured by ^a high field nuclear magnetic ^reson-

nance method following ^a suggestion ^{of F.} Moody ^et

al. [8]. Although the N.M.R. experiment gives ^{a clear}

evidence for the Berry phase associated with simple

conical circuits, ^{it does not} provide ^an experimental proof ^{of the} topological invariance properties ^{of the} Berry phase : ïm (C) is invariant _upon the set of con-

tinuous deformations of the circuit (C) ^which preserve the solid angle ^inclosed by (C). ^The optical pumping

method which involves low magnetic fields could be a

suitable _way to test this fundamental property ^{of the} Berry phase ^for quantum spin systems.

Another application of the formalism developed

in this note concerns the cyclic geometrical phase ^in-

troduced recently by Aharonov and Anandan called thereafter the A.A. phase [9]. ^{A quantum} physi-

cal system ^described by ^{a Hilbert} space ^{state vec-}

tor I1/; (t)) ^{performs a cyclic evolution} generated by ^a

Hamiltonian H(t) ^during the time interval 0 t T i.e. 11/;(T)) coincides with ltfi(O)) ^{up to a} phase ^fac-

tor : 11/;(T)) ⁼ exp(i4» 10(0)). (The ^states 10(t)) ^are

assumed to be of unit norm). ^{Let us} consider the set of vectors obtained from ]1b (t)) by multiplication by

an arbitrary ^time dependent phase ^{factor :}

phase ^is defined in such a way ^{as to} stay ^invariant under the above Abelian _gauge transformation :

The density ^matrix p(t) = 10(t)) (1/J(t) I gives ^a

gauge invariant description ^{of the} system ^{at time t.}

The A.A. phase ^{can be} then associated with a closed

curve in the _space E(p) ^{of the pure state} density ^ma-

trices i.e. such that p2 ⁼ p. In the case of a spin 1/2

system ^the general ^{form of} p is :

where p(t) = (6(t)) ^{is a unit vector. The quantum}

cycle is associated with a closed curve drawn on the unit sphere ^{S. It is} easily ^seen that different Hamil- tonians could lead to the same curve (C). ^{One takes} advantage ^{of this freedom to} search for a Hamiltonian

Hl(t) = n/2u.b(t) such that the dynamical phase ^van-

ishes identically :

One verifies easily ^that b(t) = p(t) A d/dt p(t) ^is

the correct choice by showing ^that p(t) ^{obeys the}

evolution equation ^for (Ô’{t)). ^{In order to solve the}

Schr6dinger equation ^for H(t) ^{one writes} p(t) ⁼ R(t)

xu. As before one introduces the rotated state l"b(t))

= U-l(R(t)) 11jJ(t)) ; its evolution is governed by ^the

Hamiltonian H(t) = Ho (t)+ ^{Hl (t) where jli(t) is}

given by equation (11) ^and Ho(t) ^{by :}

Writing the action of the infinitesimal rotation

R-l(t) ^d R(t) on ^{any vector x as w(t)dt 1B x, the}

Hamiltonians No (t) ^and ill (t) ^{read as follows :}

The total Hamiltonian to :

We recognize ^the operator ^called previously (H1 (t))d. ^{For t = 0 we have} p(t) ^{= u so that} 11/J(0)) ⁱⁿ

an eigenstate ^{of s.u with} eigenvalue h/2. ^{For now on,}

the calculation is identical to that of the Berry phase

for m ⁼ 1/2 ^{so that the A.A.} phase ^{reads :}

Unlike the case of the Berry phase ^{where one had}

to rely upon the adiabatic approximation (neglect ^of

the (H1 (t))n,d,), the result given ⁱⁿ equation (26) ^is

exact. It has to be noted it can also be obtained within the adiabatic approximation by taking, ^this time, ^as Hamiltonian : H(t) ⁼ w,,s.p(t).

For spin higher ^than 1/2 ^{the Berry and A.A. phases,}

except ^{for those} quantum cyclic evolutions which can

be generated by Hamiltonians linear in the angular

momentum, ^are expected ^{to be} different, ^{the reason} being ^{that the} manifold associated with the _pure state

density matrices is no longer ^the sphere S2 but instead the complex projective space Pn (C) ^{with n = 2S.}

Ground state atomic hydrogen ^{could be a} good testing ground for the A.A. phase. ^Let [F, M) ^with

F ⁼ 1, 0 stand for the hydrogen hyperfine ^{states. Let}

us assume that the system ^is prepared ^{in a coherent} superposition ^of 10, 0) and 1,1) ^states by ^{a 21 cm ra-} diofrequency pulse. (One ^can keep track of the (0,01 I

p 11, 1) ^coherence by measuring its beat with a refer-

ence hydrogen maser). ^{Then one} applies ^{a variable} magnetic ^field B (t) ^{with an intensity low} enough ^so

that its action _upon the state 10, 0) ^{can be neglected.}

The field is monitored in such a way that the state

[1,1) ^{undergoes a cyclic} evolution. The A.A. phase

in then detected by its effects _upon the phase ^{of the} (0, 0 1 p 11, 1) ^coherence.

A discussion of the Berry phase involving group

theory considerations has appeared ⁱⁿ print recently

[10]. ^{The point} of view of the authors is rather differ- ent from the one adopted ^{here :} they stay ^{within the} adiabatic approximation while the method developed

here allows, ⁱⁿ principle, ^a systematic evaluation of the non adiabatic corrections.

We would to acknowledge very stimulating ^dis-

cussions with Prof. M. Berry ^{and Prof. Alexander}

Pines.

After completion of this work appeared ^a paper which reports ^{on the} observation of the Berry phase involving ^a sample ^{rotation in nuclear} quadrupole ^res-

onance [11]. ^{However, to} the best of our knowkedge,

an experimental investigation ^{of the} topological ^in-

variance properties ^{of the} Berry phase ^{is still} lacking.

1406

References

[1] BERRY, M.V., ^Proc. Roy. ^Soc. London, ^Ser

A392 (1984) ^45.

[2] SIMON, B., Phys. ^{Rev. Lett. 51} (1983) ^2176.

[3] COLEMAN, S., ^{Les Houches 1981 Session 37} Gauge Theories in H.E. Physics, GAILLARD,

M.K. and STORA, ^{R. edit.} (North Holland) ^1983,

464.

[4] CHIAO, ^{R.Y. and} WU, Y.S., Phys. Rev. Lett.

57, (1986) 933 ;

TOMITA, ^{A. and} CHIAO, R.Y., Phys. Rev. Lett.

57, (1986) ^937.

[5] COHEN-TANNOUDJI, ^{C. et} al., Phys. ^{Rev. Lett.}

22 (1969) ^758.

[6] ^{VOLD, T.G. et} al., Phys. ^{Rev. Lett. 52} (1984)

2229.

[7] SUTER, ^{D. et} al., University ^of California _pre-

print, Berkeley ^1987.

[8] ^{MOODY, J. et} al., Phys. Rev. Lett. 56 (1986)

893.

[9] ^{AHARONOV, Y. and} ANANDAN, J., Phys. ^Rev.

Lett. 58 (1987) ^1593.

[10] ^{ANANDAN, J. and} STODOLSKY, L., Phys. ^Rev.

D 35 (1987) ^2597.

[11] TYCKO, R., Phys. Rev. Lett. 58 (1987) ^2281.