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

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

C. Bouchiat

To cite this version:

C. Bouchiat. Erratum : Cyclic geometrical quantum phases : group theory dérivation and manifestations in atomic physics. Journal de Physique, 1987, 48 (10), pp.1627-1631.

�10.1051/jphys:0198700480100162700�. �jpa-00210601�

Erratum : Cet article remplace ^l’article

"Cyclic

geometrical quantum phases : group

theory

dérivation and manifestations in atomic

physics"

^J.

Physique

⁴⁸

(1987)

1401-1406, qui ^{a été}

publié

^sans que les corrections nécessaires aient été faites, ^ce dont la rédaction prie ^{l’auteur et les} lecteurs de bien vouloir l’excuser.

Cyclic geometrical quantum phases : group theory dérivation 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 juin 1987, accepti le ler juillet}

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 _par Aharonov et 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 Aharonov and Anandan.

Classification

Physics

^Abstracts

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, cue)

^{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), a(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:0198700480100162700

1628

With the help ^of

(4)

^one finds that

I(t))

^obeys

the following Schr6dinger

equation :

with

The unitarity ^relation

U(R)t

⁼

U(R)-1

^implies

that

H1 (t)

^{is indeed a Hermitian operator. In order}

to evaluate explicitly

Hl _(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 :

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

O(T)

^{= Oo and}

cp(T )

^{= 2r.}

We are now in position ^to compute

H1 (t)

^{in term}

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

Using (10)

^and simple properties of the rotation _group

one gets :

where we have set

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

tions one gets ^the following expression

forfil _{(t) :}

It is convenient ^to introduce the projections ^of

the angular ^momentum Ju, J, Jw upon the orthonor- mal set of unit vectors

(u, v, w)

associated with Bo :

u ⁼

Bo/-B,

^{w = y, v = wAu. In} particular ^{we have :}

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

One readily obtains the following ^final

expression

for H1 (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} eigenstates

of

H(Bo) lot 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

H1 (t) :

Using

(7), (13)

^and

(6)

^{we obtain the} state vector at time t = T :

The Berry quantum ^adiabatic phase ^is readily

obtained from the two above formulae :

with

where

fl(C)

^is the solid angle ^enclosed by ^the

curve

(C).

A _very remarkable feature of the Berry phase ^{is the}

fact that it can be written as the circulation of a

classical _gauge field

along

^{a closed} loop. ^As Berry ^first

noted the _gauge field relevant for the case considered here is that of magnetic monopole ^of charge ^{m. This}

result is

easily

^recovered by noting ^{that :}

where Am ^{is the} monopole ^vector potential

(3).

Since it seems

difficult,

^{at least for} atoms, to per- form a true interference experiment involving angular

momentum eigenstates ^we suggest ^to study ^{the evo-}

lution of a coherent mixture of state vectors having

different magnetic quantum numbers. This can be done by looking ^{at the time} dependence ^{of the aver-}

age value of J. Rare _{gas atoms} with ^{a non zero} nuclear

spin appear to be convenient for our purpose. The in- teraction of a rare gas atom in its ground ^{state with a}

static magnetic ^{field can} be described

by

the effective Hamiltonian :

where I is the nuclear spin ^and -IN is given ^{in term of}

the nuclear magnetic ^moment by --IN Ih = UN - We associate with the spin

density

^matrix

p(t)

^a

rotated frame density ^matrix

p(t) :

We have the set of relations :

It

is

convenient to

study

the time variation of it ^is

given by

^the

equations :

Keeping ^{for the moment} only ^the diagonal part ^of

Hl (t),

^we obtain after

simple algebra

involving angu- lar momentum commutation relations :

with

One gets immediately using ^{the same}

manipula-

tions as in equations

(14) :

The operator I± having ^{non zero} matrix element

only ^{between states with Am = ±1 the time evo-}

lution of

(I:t:.)

involves, ^as expected, ^{the difference}

1m (C) - 1m:t:.l (C) = ±fl (C).

^For practical purposes

we give the values of

( I" y

^and

(Iw)

^{at time t =} T, assuming

(Iw (0))

^{= 0 :}

As a way to arrive at an

empirical

^determina-

tion of the difference of the Berry phases ^associated

with states differing by ^{one unit of} angular ^momen-

tum ym

(C)-

^ym,

(C) = - (m - m’)fZ(C),

^{we propose}

to measure the phase shift of the nuclear spin pre- cession of atoms interacting ^{with a} magnetic ^field

B which varies slowly following ^{a closed} path ^{on a} sphere ^{of radius} B, ^with respect ^{to that of atoms for} which the field stays ^fixed during ^{the same time in-}

terval T. Such an experiment would be the quantum spin counterpart ^{of the} geometrical light polarization phase ^shift measurement performed recently ^with op- tical fibres

[4].

It is, ⁱⁿ principle, possible ^to get ^{the difference} of Berry phases ^{for states} differing by ^{more than one}

unit of h when I >

1/2.

^{One may} study ^{the time evo-}

lution of the spin alignment ^{normal to the} precession

axis :

We would like now to evaluate the non adiabatic corrections to formula

(18),

associated with the non

diagonal part ^of

Hl _{(t) :}

The equation giving ^{the time} evolution oaf gets ^an extra term involving

To lowest order in

(ill (t))

^n.d.

we

^{can neglect the}

time variation of

(Iu).

^The correcting ^{term A}

(I± (T))

to formula

(17)

^{is then obtained by a} simple quadra-

ture :

1630

with

To formulate the adiabatic approximation ^we

write

0(t)

⁼

Ðo/(t/T)

^and

p(t)

^{= 27r}

g(t/T)

^with

f (0) -

^1,

g(o) =

^{0 and}

f (1) = g(l) =

^{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

wNT --+

^oo.

Using

^an

integra-

tion

by

part ^one proves that the absolute value of the

correcting

^term given by

(20)

^{behaves like}

1 /WN T

when

IWNTI

^{--+ oo. If the} system ^is prepared ^{in such as}

way that

(Iu (0))

^{= 0, it can} be shown then that the

correcting

^{term is of} the order of

(WNT) 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

’He,

^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

been observed

by

^C.

Cohen-Tannoudji

^{et al.}

[5]

using ^an

optically

pumped 17 Rb cell as magnetometer. ^This very sensitive tech-

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

permanent electric dipole ^{of l2gXe}

[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} _mag-

netic field its modulus remains constant. In practice

such a condition _may be rather difficult to

satisfy.

^To

account for possible deviations one writes :

B(t) = A(t)U(R(t))Bo

^where

A(t)

^{is a}

slowly varying

positive

scale factor with

Ã(O)

⁼

A(T).

^The

Berry

phase ^re-

mains

unchanged

^{but the} dynamical phase ^is replaced

by

wN

fo ^A(t)dt.

^{There is, in}

principle

^a way ^{to can-} cel 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

⁺

6t) = -B(T) =

-Bo, ^{where 6t} > 0 is such that 6twN ^« 1, in order to insure the

validity

^of the sudden _approx- imation. 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

f3 (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

fT, ^{A (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

experimen-

tal proof ^{of the} topological invariance properties ^of

the Berry phase :

"1m (C)

^{is invariant upon the set}

of continuous 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 _prop- erty ^{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 11j1(t))

performs ^a cyclic ^evolution generated by ^a

Hamiltonian

H(t)

during ^{the time interval 0 t T}

i.e.

11j1(T))

coincides with

10(0))

^{up to a} phase ^fac-

tor :

11j1(T))

⁼

exp(io) ltk(O)). (The

^states

10(t))

^are

assumed to be of unit

norm).

^{Let us consider the set}

of vectors obtained from

11j1(t))

by multiplication by

an

arbitrary

^time dependent phase ^{factor :}

10 (t)) ^!4 Ip (t))’

⁼

exp(if(t)) 10 (t)) .

^{The A.A.}

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

The density ^matrix

p(t) = 10(t)) (,P(t)

^{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

H(t) _ 1i/2ü.b(t)

such that the

dynamical phase

^van-

ishes identically :

One verifies easily ^that

b(t)

⁼

p(t)

ⁿ

d/dt p(t)

^is

the correct choice by showing ^that

p(t)

^{obeys the}

evolution equation ^for

(a(t)).

^{In order to solve the}

Schrodinger equation ^for

H(t)

^{one writes}

p(t) = R(t)

.u. As before one introduces the rotated state

l"b(t))

=

U-1(R(t)) l,p(t)) ;

^{its evolution is} governed by ^the

Hamiltonian

H(t)

⁼

Ho (t)± H1 (t)

^where

H1 (t)

^is

given by equation

(11)

^and

Ho (t)

by :

Writing ^{the action of the} infinitesimal rotation

R-I(t)

^d

R(t)

^on any vector ^{x as}

w(t)dt 1B

^{x, the}

Hamiltonians

Ho (t)

^and

fl, (t)

^{read as follows :}

The total Hamiltonian

Íl(t)

⁼

Ho(t)

⁺

H1 (t)

^reduces

to :

We recognize ^the operator ^called previously

(H1

(t))d.

^{For t = 0 we have}

p(t)

^{= u so that}

1"’(0))

^is

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 in-}

stead 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

IF, M)

^with

F = 1, 0 stand for the hydrogen

hyperfine

^{states. Let}

us assume that the system ^is prepared ^{in a} coherent.

superposition ^of

)0,0) and 11, 1)

^states by ^{a 21 cm ra-}

diofrequency

pulse.

(One

^{can keep track of the}

(0,0) p 1,1)

^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

0, 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

is then detected by ^{its effects} upon the phase ^{of the}

(0, 0 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 like to acknowledge very

stimulating

discussions with Prof. M. Berry ^{and Prof. A. 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.

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. eds}

(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} STObOLSKY, L., Phys. ^Reu.

D35

(1987)

^2597.

[11]

TYCKO, R., Phys. ^{Rev. Lett. 58}

(1987)

^2281.