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Atomic quantum phase studies with a longitudinal Stern-Gerlach interferometer
J. Robert, Ch. Miniatura, O. Gorceix, S. Le Boiteux, V. Lorent, J. Reinhardt, J. Baudon
To cite this version:
J. Robert, Ch. Miniatura, O. Gorceix, S. Le Boiteux, V. Lorent, et al.. Atomic quantum phase studies with a longitudinal Stern-Gerlach interferometer. Journal de Physique II, EDP Sciences, 1992, 2 (4), pp.601-614. �10.1051/jp2:1992155�. �jpa-00247659�
Classification Physics Abstracts
03.65W 32.60V 07.60L
Atomic quantum phase studies with a longitudinal Stern-Gerlach interferometer
J. Robert, Ch. Miniatura, O. Gorceix, S. Le Boiteux, V. Lorent, J. Reinhardt and J. Baudon
Laboratoire de Physique des Lasers(*), Institut Gali16e, Universitd Paris-Nord, Av. J-B- C16ment, 93430 Villetaneuse, France
(Received 31 January1992, accepted 5 February 1992)
R4sumd On donne une description g6n6rale des interf4romktres atomiques en termes d'opd-
rateur de diffusion. On montre que l'action d'un interf6rom+tre longitudinal sur un atome reprd-
sentd par un paquet d'onde polar1s6 se ramkne £ une transformation de r4fdrentiel. On examine le cas particulier d'un champ magn4tique I gradient longitudinal et celui d'un champ pr4cessant
spatialement. Une r4alisation exp4dmentale utilisant des atomes m6tastables d'hydrog+ne est ensuite prdsent6e et les r6sultats obtenus dans les deux situations pr6c6dentes sont discut6s. On dtudie enfin les firopr16t6s des atomes "en chalelet" produits par des champs de quelques 10 G,
£ partir du rayonnement induit par un champ 61ectrique.
Abstract A general description ofatomic interferometers in terms of the scattering operator is given. The action of a longitudinal interferometer
on an atom described by an incident polarized wavepacket is shown to be equivalent to a frame transformation, leading to a "beaded"
atom. The special case of a pure longitudinal gradient of magnetic field and that of a spatially precessing field are examined. An experiment «sing metastable hydrogen atoms is presented and the results obtained in both situations mentioned above are discussed. Properties of beaded atoms produced by relatively strong fields (od 10 G) are investigated by means of the intensity diagram of their electrically induced radiative decay.
1 Introduction.
In the representation, based upon the superposition principle, of any quantum system by
states arises the problem of the phase associated to a given state (cf. Dirac [ii) and more
generally the problem of bra and ket conjugation. This question is of importance when a definite system of kets is used as a representation of states insofar as it leads a priori to a
(*) assoc16 au CNRS, URA 282.
phase-dependent representation of the observables. However by chance, the usual way we
use
to perform measurements cancels all these difficulties, at least if a permanent and coherent choice of phases is adopted.
In this context, one may wonder about the necessity of a renewal of phase studies (for a general review see [2]). One possible answer is that, with systems containing several degrees of freedom, it is possible in a real or "gedanken" experiment to exhibit relative phases. The idea is to lock with respect to each other some of the degrees of freedom (by some filter) whereas
one observes, by means of an interferometric measurement, the evolution of the other degrees
of freedom, in an elTective sub-space depending on the preparation that has been made on the locked degrees of freedom. It may be noticed that this type of methods is commonly used in
scattering experiments on atoms or molecules, as well as in light-matter interaction studies.
What is really new in the recent atomic interferometry experiments is not the evidence for the de Broglie wavelength but rather the variety of methods to "freeze" or "lock" some degrees
of freedom made feasible by the use of atomic beam control techniques. AH these experiments
are of the one-particle type in the sense that each particle (atom) interferes with itself. So far
no collective coherent elTect has been observed or even, to our knowledge, proposed.
Atomic systems are particularly very well fitted for fine studies of quantum phases »ince inner and external degrees of freedom can be manipulated rather easily by means of external fields. The occurence of a rich internal structure in atoms is an advantage, compared to photons
or neutrons, which can be exploited to act on the external degrees of freedom via the internal ones, and vice versa. This provides a great flexibility in the methods to control or specify phaseshifts and consequently to build an interferometer.
In part 2 of this article, a general description of atomic interferometers is presented and the problem of dynamical and topological phases h discussed. In part 3, the principle of the
longitudinal Stern-Gerlach interferometer is given and the theoretical methods used to study
its operation are presented. Part 4 is devoted to the description of the experiJnent and to the results obtained either with a transverse magnetic field with a longitudinal gradient (pure dynamical phases) or with a conical field configuration (dynamical and topological phases). In part 5, experiments using a strong field are presented. They are devoted to the study of special optical properties exhibited by the atoms emerging from the interferometer l'~beaded" atoms).
2. Description ofatomic interferometers.
2.I GENERAL CONSIDERATIONS. Any atonfic interferometer consists of an arrangement
of external fields (among which one includes slits, grating, etc...) which perturbs the atomic evolution within a finite domain of space. From this view point, it is quite s1nlilar (except
for the size) to a scattering device. Just like a collision, the operation of the interferometer is contained in a "scattering operator" § which transforms
any asymptotic (free) incident state (~l~~) into an asymptotic outgoing state (~1°~~)
l~outj §j~inj (~)
Let us define now the '~principal path states" ofthe interferometer, [p(£)), labelled by a vectorial variable £ (discrete or continuous), as being the eigenstates of this operator:
=
/ dP(f)lw(f))e'~~~~(~2(t)1 (2)
A Stieltjes integral with a measure p(£) has been used in order to get a similar expression
in discrete and continuous cases. The quantities s(£) are real phases insofar as all scattering
channels are open. This is actually the case in most of the experiments using thermal atomic
beams (the case of ultra-cold atoms should be examined more carefully). Actually this descrip-
tion is that of the "core" of the interferometer (say the separatrices and the arms), in which the atomic evolution is in most cases adiabatic. It does not include the '~mixers" which are in charge to prepare or analyse the atomic beam before and after the interferometer, I.e. to
specify (~li~, °~~) which are definite linear combinations of the principal path states [~g(£)). In order to describe further the elTect of the interferometer on the atomic state, one can expand
the asymptotic states [~1°) (a = in, out) on a basis set realizing a complete description of the
asymptotic Hamiltonian #~. The interaction, in the interferometer being of the short-range
internal ariabletonian, then a ossiblechoije of the
basis set Is iven by states [k, E, #,
eigenstates of il, flint
and #~, with genvalues ~ ~~, e(x), = ~ ~~ + e(x)spectively.
m m
lt~°) = / dk dEP(x)a"(k, E,x)lk, suchEi # (3)
2 2 SELECTED ASYMPTOTIC WAVE-PACKETS. If asymptotic (incoming or outgoing) states are described, for each atom in the beam, by a selected (prepared or analysed) wave-packet, this implies a choice of a phase reference insofar as the wavepacket is obtained by projection
of [~l") onto the bra jr,t,y[, where y labels the internal state:
tt°(r,t, d = jr, i,y j~t ° (4)
We further restrict our choice to quasi-monochromatic wavepackets, the momentum distri- bution of which is peaked at some value kP, and such that the (y[ selection corresponds to a
given eigenstate of flint, say (x~[ With these asumptions and reminding that these asymptotic wavepackets are free, one gets:
a°(k, E,x) = a°(k,x)b E ~l~ ~ (x)j
jr,t k, E) = exp[I(k r Et/h)]
with k
=
kP + u, [u[ « (kfl
E = EP~
=
~~~~
+ ~ (x~)
2m
h h2 (kfl)2
Let vfl
=
-kfl the
group (particle) velocity, and Ef
= ~
In such conditions, the
m m
asymptotic wavepacket now labelled with kfl and x~ can be written in the form of a carrier
wave multiplying a slowly variable envelop:
1~" (kfl,x~; r, t) = exp ; kfl
r (~t)j £° (r vflt; x~) (5)
where the envelop is given by:
£" (r vflt; x~)
= exp [-ie (x~) t/h] / du a° (u, x~)
exp (I u (r vflt)) (6)
The special case of a plane wave is readily obtained by setting in a° (u,x~) = a° (x~) b(u).
2. 3 SPATIAL AND TEMPORAL SHIFTS IN A DEGENERATE LONGITUDINAL INTERFEROMETER.
A degenerate longitudinal interferometer h a device such that (I) the asymptotic channels
are an degenerated with respect to the internal variables, I.e. there exists a single value e of
e(x) (we shall take e = 0 in the following) (it) any incoming beam of atoms at a given direction results into an outgoing beam propagating in the same direction, regardless to the external motion (all trajectories if this concept is relevant) within the arms of the interferometer.
In such a case, the matrix form of the § operator reduces to (k, E',x'(S(k, E, #
= (k,E,x'(S[k,E,x)b(k k')b(E E')
and the principal path states are simply:
1~2(t)ilk, E,z) (7)
where y stands for the internal state. The corresponding phaseshifts defined by equation (2)
are:
S(t) = S(k,z)
An incoming state prepared (polarized) along (x') leads to the outgoing state:
J~2~2
(l#°~~) # dk dp (X')dp(y) (X' y)e'~~~'~~ (y X')a~'~ (k,X') k, -j X' (8)
2m
The outgoing wave packet polarized along (x°) is then:
lV°~~ (r,t,x°)
= (r,t,x° fit°~~)
=
/ dk dp(y) (x° y)e'~(~,Y) (y x')a~" (k,x')e'~~ ~ ~~~~'~~ (9)
By expanding s(k, y) in powers of u
= k k' in the vicinity of k"
s(k, y) ci s° (k",fl + u s'(k", £
where s' = (Vk s)~,
,
one makes appear explicitely the incident wavepacket within the expression
of the outgoing one:
~1°~~ (r,t,x°) =
=
/ dv(y) (x° y) (y x")
tt~" [r + Ar (k',fl
,
t + At (k',fl
,x"] (lo)
The spatial and temporal shifts are given by:
Ar (k~,fl =2 ~
~
s° ~s' 2 ~'
~
(k' s') = Ar° + Ar'
(k') (k')
At (k",fl
=
~~
~ (s° k" s') = At° + At' (11)
h (k')
((°) refers to the s° dependent part of Ar and At)
It is worth noting that as Ar°/At°
= hk'/m
= v', shifts Ar° and At° do not alTect the
envelop of the packet.
In the one (spatial) dimension case, where Ar
= Az hi and k"
= k~6z, one has: Az'/At' = V", the phase velocity, which indicates that these shifts are absent in the carrier wave of the
packet.
To conclude this part, let us make a remark about the topological and dynamical parts of the phaseshift as regards their elTects on the wavepacket, in the adiabatic case. The topological part, as well as Ar°, does not involve k" and then does not aiect the envelop but only the carrier wave. In a real experiment one is dealing with atoms of dilTerent velocities, I.e. with
an incoherent distribution of k'. The adiabatic topological phaseshifts will not be involved in the averaging of the interference pattern. The result is similar to that observed in optics when
an extended monochromatic light source h used (cf the Young's holes experiment used as a spatial correlator),
3. The longitudinal Stern-Gerlach polarization interferometer.
3. I PRINCIPLE AND DESCRIPTION. This device is a longitudinal-degenerated interferom- eter of the type described in section 2.3. The core of this interferometer consists of a magnetic
field profile B(z) restricted t6
a finite range. The gradient of its modulus is longitudinal, with
opposite directions in the in and out-sides in such a way that the integral of the force exerted
on the atom is zero, which means that there is finally no net transfer of momentum neither of energy to the atom.
The core of the interferometer is preceeded by a preparation section consisting of a polar- izer, which selects a specific Zeeman state (x"), followed by a mixer which builds a linear
superposition of such states. Experimental details will be given further. The principle of this mixer is basically to turn the direction of a magnetic field rapidly (with respect to the Larmor
frequency). Similarly the core is followed by a second mixer and an analyzer selecting (x°).
Actually, (x'>°) are eigenstates of the component of the atomic spin J along a fixed axis u".
Here the asymptotic Hamiltonian fl~ reduces to the kinetic
energy operator. The principal path states [~g(y)), belonging to the [k E J) manifold, can be determined following the lines
developed by Majorana [3]. Within both mixers and core domains, the Hamiltonian can also be written in the form:
fl(r) ~
=
fi gvBJ B(r)
where g is the Land6 factor and pB the Bohr magneton. The field has the general form:
B(r) = B(r)fiB(r), with LimjrjcoB = 0, so that fl reduces to fl~ in the limit [r[oo.
3.2 THE SCHWINGER'S TREATMENT. In his paper "Non-adiabatic processes in non-
homogeneous fields" [4], Schwinger has developed all the theoretical background needed in
a time-dependent description of the spin evolution in the spin-state subspace. However to describe properly the interferometer we cannot simply transpose this time-dependent theory
in assuming that the external motion is given by the classical expression z = vi. One has to solve the complete atomic motion, by adding the kinetic energy operator to the internal
Hamiltonian. The field configuration will be assumed to be as follows: In a first part the field has a longitudinal gradient but a fixed direction, its magnitude rhing from zero up to some
value B. In a second part of length L it keeps constant in magnitude but its direction pre-
cesses uniformly around z : B(z)
= BfiB(z), with a (spatial) angular velocity Q
= 2~/L and
making a constant angle with z-axis. In a third part, the field goes down to zero keeping
a constant direction. In the first and third parts no transition occurs among the spin states referred to the B ads and then these parts can be treated without any difficulty. We shall
concentrate our attention to the central part in which the evolution is not adiabatic. Let fi(z)
be a z-dependent unitary vector of the form:
6 = sin fl cos a(z)6~ + sin fl sin a(z)iiy + cos phi
where fl is a constant and a(z) = Qz. The eigenstates fltM of J u(z) are parametrized by 6,
with: fltM (6) = £ l8~,~,(0, -fl, -a(z))lV Mi (hi
Mi
As the total Hamiltonian is time-independent it is allowed to look for a state of definite energy E. The interaction being independent of z, y, these variables are separated. The problem will be restritted to the only z coordinate. The complete state can be expanded over the IVM (6)
basis: fit
=
£ FM(z)fltM (6(z)). Functions FM(z) describing the external motion
are solutions
M
of the coupled dilTerential equations:
E FM(z) = ~0)FM + £ (-gpB) (M(J B(M') ~2(M(0z(M') 0zFM/-
m
~, m
-~ (M
(0)(M')MJj
(12)
m
where all matrix elements are taken in the fltM (6) basis. The standard angular momentum
algebra [5] allows one to replace matrix elements of 0z and 0) by those of Jz and J) : (M (0z M')
= -iQ (M (3~ M') (M(0)(M')
=
-Q~ (M(J) M')
Using the JWKB approximation one is able to compare the dilTerent terms of equation (12):
0z FM is of the order of K(z)FM, where K is the local wavenumber very close to the asymp- totic value K = 2~/A because the Zeeman energies are very small compared to the kinetic
energy. On another hand Q is of the order of I/L. As L » A, one has Q~ « QK, then
the terms in (M (0)( if') are negligible with respect to those in (M (0z M'). Obviously these latter terms are small compared to E but not compared to the "static" interaction terms
(M [-gpBJ B[M') and they must be kept in the treatment. The eikonal form of the ampli- tudes is FM = AM(z)exp [iSM(z)], where AM is a slowly varying function and SM a rapidly varying real phase such that SM
= S + aM with aM < S. The approximate JWKB form of the coupled equations is now:
E AM(z) = )S'~AM £ (MgpBJ. B + ~~' Q6zj M')
~, gPBm
The oIT-diagonal coupling terms contain a static part (in J B) and a dynamical part (in QS')
due to the z dependence of the basis set. In the present case it is possible to make these two
parts cancel eachother by a convenient choice of 6(z) : if 6 is such that:
B +
~~ hQ6z = Be6(z) = Be(elTective magnetic field)
gfiBm
then the equations decouple in:
~2 E =
-S'~ (M(gpBJ Be M')
Finally the solution takes the form:
+CO
FM = Amexp ikz + ~co dz wM(z)
4. Reafisation of the interferometer, experimental results.
4. I GENERAL DESCRIPTION OF THE EXPERIMENT (see Fig. I). The experimental setup
has already been described in details elsewhere [6] and only its main features will be pre- sented here. The general principle developed in parts 2.3 and 3.I has been applied to the
case of a metastable Hydrogen atom H* (2si/2) beam. This beam is produced by electronic bombardment of a thermal molecular beam. In most of the experiments this bombardment h continuous, but it can be easily pulsed in view of an analysis or selection of the time of flight (TOF). The TOF dhtribution is well fitted by the following function:
KA Bp C '~ P C'B~ D la)
~j
~ @~~~ j ~ ~H~m~ [--,--)
)fj~j~i~~j~flj)~
--_zP
(b)
Fig. 1. (a) Scheme of the apparatus. K, A: electron gun, Bp,A Poladzing and analysing fields, CC' quasi-zero field chambers, p: magnetic shieldings, iC axisparallel intensities, D detector. (b) Principle of the experiment. P,A: polarizer, analyzer; M, M' : mixers; R: region where the longitudinal magnetic field gradient induces phaseshifts among Zeeman states (a J
= 1 value being assumed).
f(z) = C~~z~~exp(-z~)
where z = t/to,to being the most probable time of flight, corresponding to the velocity
vo =10 km/s.
The preparation of the beam at the entrance of the interferometer consists of the two fol-
lowing steps:
