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We extend our previous analysis of the motion of vortex lines in wave mechanics to the case of more elaborate vortex patterns and to a rotating harmonic trap.
Vol. 100 ( 2001) A CT A P HY SIC A POLON IC A A Suppl ement
V or tex L ines in Motion
I. Bia¤ynicki-Bi r u la Ê
Center for Theoret ical Physics and Insti tute of Theoreti cal Physi cs
Warsaw Uni versity, Warsaw, Poland
T. M¤ oduc ho wsk i
S.I. Witki ewicz Hi gh School, Elbl ¨ska 51, 01-737 Warsaw, Poland
T. Rado ây ck i
Departm ent of Physi cs, Warsaw Uni versity , Hoâa 69, 00-681 Warsaw, Poland
and C. Ïl iw a
Center for Theoreti cal Physi cs, al. Lo tni k§w 32/ 46, 02-668 Warsaw, Poland
(Received November 15, 2001)
We extend our previou s analysis of the motion of vortex lines in wave
mechanics to the case of more elaborate vortex patterns and to a rotatin g
harmonic trap .
PACS numb ers: 03.65.{w , 67.40.V s
1. I nt rod uct io n
The hydro dyna mic form ulatio n of wa ve mechanics discovered by Madelung
[1] o˜ers an opportuni ty to di rectly connect the moti on of quantum parti cleswi th
the m otion of a Ûuid | the probabi lity Ûuid. Thi s formulatio n not only o˜ers a
di ˜erent way to visualize the quantum mechanical evoluti on but also intro duces
new objects: qua nti zed vo rti ces. Vorti city Ùeld i sa very interesti ng physi cal object
already in the study of classical Ûuids but in the framework of wave mechanics
it acqui res a new f eature. In the three- dim ensional conÙgura tion space vorti city
can generically exist only on vortex lines and, in addi tion, the circul ation around
each vortex line must be quanti zed in uni ts of 2 ¤ ñh =m . The f ascinati on with vortex
Êcorr espon din g author; e-m ail : biru la@cft .ed
30 I . Bi a¤yni cki-Bi rula et al .
lines goes back m any centuri es when Empedocles, Ari stotl e [2], and D escartes [3]
tri ed to explain the form ati on of the Earth, its gravi ty, and the dyna mics of the
who le solar system as due to pri mordi al cosmic vorti ces. Later Lord Kelvi n [4] has
attem pted to descri be ato m sas vortex rings. The prop er m athem atica l descripti on
of hydro dynamic vorti ces has been started by Helmholtz [5]. The best summ ary
of the present day theory of vorti city in classical hydro dyna mics may be found in
a recent monograph by Sa˜m an [6].
There are two reasons why we are interested in the study of vorti cesembed-
ded i n soluti ons of wave equati ons. Fi rst, vorti ces are by them selves very interest-
ing structures tha t deepen our understa nding of wave phenomena. Second, recent
advances in exp eriments on the Bose{ Einstein condensati on made it possible to
create vorti ces in the laborato ry provi ding a testi ng ground for the theoreti cal
analysis of the vortex moti on. Surpri singly, despite the fact tha t one can Ùnd with
relati ve ease the behavi or of even fairly complicated vortex structures, there has
been very littl e acti vity in tha t area of research. The notable example is the re-
search carri ed out for a long ti me by Berry, Nye, and their collaborators [7{ 9]. We
wo uld like, ho wever, to emphasize the di ˜erences between thei r appro ach and our
appro ach. They stress the generic features of vo rtex lines and apply thei r analysis
mainly to m onochromatic waves in opti cs witho ut any reference to ti me evoluti on.
We restri ct ourselves to quantum mechanics and we study in detail speciÙc ex-
amples exhi biti ng the ti m eevoluti on of vorti ces. Thus, we believe tha t these two
compl ementary appro aches suppl ement each other.
In our Ùrst publ icati on [10] we have intro duced a general metho d of gen-
erating soluti ons of the Schrodinger equati on with embedded vorti ces of almost
arbi tra ry compl exity. In the second paper [11] we have extended the analysis to
the nonlinear wave equati on wi th harm onic interpa rti cle forces. In the present pa-
per we restri ct ourselves to the linear Schrodi nger equati on but in order to see a
more intri cate behavi or, not seen in the examples studi ed before, we extend the
scope of our anal ysis to cover m ore elab orate vortex structures . We also study the
case of a rotating harm onic trap tha t shows some new features and might also be
of experimenta l interest.
2. G ener at i ng fun ct i on
In contra st to the situa ti on in classical hydro dyna mics, vorti city in quantum
m echani cs canno t b e present i n a three- di m ensional volume. Due to the requi re-
ment that the wave functi on be single valued, vorti city in quantum mechani csis
generically concentrated on lines. Vortex lines are deÙnedas an intersecti on of the
two surf aces deÙned by the vanishing of the real and imaginary part of the wa ve
functi on.
The constructi on of soluti ons of wave equati ons, studi ed in [10], tha t exhibit
vari ous vo rtex structures, was ba sed on the idea of a generati ng functi on. As a
Vortex Li nes in Mot ion 31
generati ng functi on we may use any soluti on êkk (r;t) of the Schrodinger wa ve
equati on under study tha t sati sÙesthe initi al condi ti on
êkk (r; t = 0) = exp(ir Âk) ¢ 0(r) ; (1)
where k is an auxi liary wave vector and ¢ (r)is a smooth, non-v anishing functi on.
By ta king a linear combi natio n of deri vati ves with respect to the components of
the vector k of the generati ng functi on (1), we can produce an arbi tra ry polynomial
in the vari abl esx ; y , and zin front of thi s functi on,
[W R(r) + iWI(r) ] ¢ 0(r) ; (2)
where WRand WIare tw o real p olynom ials. Vortex lines are deÙned by the equa-
ti ons
WR(r) = 0 ; W I(r) = 0:(3)
In thi sm anner we obtain an initi al wave functi on that may conta in all typ es
of i ntri cate vortex structure s. The moti on of the vorti ces i s determ i ned by the
Schrodinger equati on. Since the Schrodinger equation is linear and it does not
inv olve k, all deri vati ves of êkk (r;t) wi th respect to the components of the vector
k also satisfy thi s equati on. These derivati ves at k =0 in all known cases, when
the soluti on of the Schrodinger equati on can be wri tten down explicitl y, have the
[W R(r; t) + iWI(r; t )] ¢ 0(r; t ) ; (4)
where now WRand WIhave coe£ cients tha t in general depend on ti me. The ti me
evoluti on of vorti cesis determined by the ti meevoluti on of the zeros of the wa ve
functi on. Thi s requi res solvi ng two simulta neous real algebraic equati ons
WR(r; t) = 0 ; W I(r; t ) = 0 : (5)
It is worth stressing that the m oti on of vorti cesdepends on the shape of the
\ envelope" wave functi on ¢0(r). The same vorti ces \ sitti ng" on di ˜erent envelop e
functi ons will m ove in a di˜erent way.
3. Si m ple co nÙgu r at io n s o f vo r tex l i nes an d vo r tex r i n gs
We have used in [10] an exampl eof two vortex lines moving accordi ng to the
free-parti cl e Schrodinger equati on to exhibit the phenom enon of vortex reconnec-
ti on. The dyna mics of three vortex lines in free moti on has already been described
in our prelim inary report [12] where we have shown tha t it exhibits novel features.
Namely, the three vortex lines go thro ugh the reconnecti on pro cessin a di˜erent
way; the reconnecti on occurs thro ugh the creati on of a closed ring tha t shrinks
and di sappears. Since the ful l soluti on of the Schrodi nger equati on is ti me reversal
sym metri c, f or negati ve ti mes the who le pro cess occurs in a reversed order: the
vortex ri ng suddenl y app ears, then grows, and Ùnally it is swallowed by the three
32 I. Bi a¤ynicki -Bi r ula et al .
vortex lines. A similar creation of an addi ti onal vortex ri ng accompaniesthe vortex
reconnecti on also for other conÙgurati ons studi ed in thi s section: a vortex circle
and a vo rtex line. W eshall make the simplest choiceof the generati ng functi on for
the free-parti cle Schrodinger equati on, viz., a pl ane wave
êkk (r; t) = exp(ir Âk)exp(Àik Âkt= 2 ) : (6)
W e use the uni ts ñh = 1 ; m = 1 thro ughout thi s paper with the only excepti on
of Ùnal form ulas (27){ (34) for the generati ng functi on in the rotati ng tra p. The
ini ti al wave f uncti on describi ng three ortho g onal vortex l ines has the form (2 ),
with the followi ng choice of the ini tial polynomial
WR(r; t = 0) + iWI(r; t = 0 ) = (x Àd + iy )( y Àd + iz )( z Àd + ix ) : (7)
The ti me-dependent polyno mials obtained by di˜erenti ati ng the functi on (6) and
setti ng k =0 are
WR(r; t) = Àd3+ t (3 d ÀxÀyÀz )
+ d2( x + y + z ) Àx y 2Àyz2Àz x 2+ x y z ; (8a)
WI(r; t) = Àt (x + y + z) + d 2( x + y + z )
Àd ( x 2+ y 2+ z 2+ x y + y z + z x ) + x z 2+ y x 2+ z y 2Àx y z (8b)
and the moti on is depicted in Fi g. 1.
Fig. 1. Full history of the time evolution of three vortex lines that at t = 0 are mutually
p erp endicul ar and nonintersecting .
Vortex Li nes i n Mot ion 33
In the case of a line and a ri ng, the ini tial polynomial wi ll be chosen as
WR(r; t = 0) + iWI(r; t = 0 ) = (x 2+ y 2ÀR2+iR z )( x ÀdÏiz ) : (9)
Depending on the sign in the second term the vorti citi esof the ring and the stra ight
line are the same or opposite. The ti me dependent polynomials in thi s case are
WR(r; t) = ´2 tz + ( x Àd)( x 2+ y 2ÀR2)´R z 2;(10a)
WI(r; t ) = t ( 4x ´RÀ2d) Ï( x 2+ y 2ÀR2) z ÀdR z + R x z : (10b)
Finally, for the two rings we have chosen the initi al polynomial in the form
WR(r; t = 0) + iWI(r; t = 0 ) = (x 2+ y 2ÀR2ÀiR z )
È[ ( x Àx0)2+ y 2ÀR2+iR ( z Àz0)] ; (11)
that leads to the following tim e-dependent real and imaginary parts:
WR(r; t ) = 2 R z 0tÀ8t 2+ ( x 2+ y 2ÀR2)
È[ ( x Àx0)2+ y 2ÀR2] + z ( z Àz0) ; (12a)
WI(r; t ) = t [8 ( x 2+ y 2Àx0x ) À3 R 2À2 x 2
+ R z0( x 2+ y 2ÀR2)ÀR z ( x 2
0À2x 0x ) : (12b)
In Figs. 2, 3, and 4 we show the most characteri stic m ovie frames for three vortex
lines, for a vortex line and a vortex ri ng (for upp er signs in Eqs. (10a)), and for
two vortex ri ngs. In order to get a better vi ew, we have ti lted the axes.
Fig. 2. The creation of a vortex ring by three vortex lines.
34 I. Bi a¤ynicki -Bi r ula et al .
Fig. 3. A vortex ring and a vortex line create a second ring.
Fig. 4. Tw o vortex rings create a third ring.
4. R ot at i ng h ar m on i c t r ap
A general stati onary harm onic trap centered at the ori gin of the coordi nate
system is described by a quadratic functi on of the coordi nates V ( r) = (1 =2 ) v i j xixj.
A rotati ng tra p in the laboratory frame is described by the potenti al
V ( r; t ) = (1 = 2) v i j ( t ) x ixj= V ( r( t )) ; (13)
where the ti me dependence of the matri x vi j (t ) results from the rotation. In what
follows we shall treat only a special case when the axis of rotati on coincides with
one of the pri ncipal directi ons of the trap. In this case, the time dependence of the
potenti al has the form (we choosethe zaxi s as the axi s of rota tion)
Vortex Li nes i n Mot ion 35
[ v i j ( t )] =
xcos2( ¨ t ) + ! 2
ysin2( ¨ t ) ( ! 2
y)cos( ¨ t ) sin( ¨ t )
( ! 2
y)cos( ¨ t ) sin( ¨ t ) ! 2
xsin2( ¨ t ) + ! 2
ycos2( ¨ t ) :(14)
Conto ur lines of thi s potenti al are the ellipses rotati ng in the positi ve (anti clock-
wise) directi on in the x y plane. The ti me dependence of the parameters of the
potenti al may be replaced by the ti me dependence of the coordinates
V ( ; t) = V ( ( t )) : (15)
In our case the time dependence of ( t ) i s given by the form ul ae
x ( t ) = x cos( ¨ t ) + y sin( ¨ t ) ; (16a)
x ( t ) = Àxsin( ¨ t ) + y cos( ¨ t ) ; (16b)
z ( t ) = z : (16c)
The f uncti ons (t ) are the coordinates in the rotati ng fram e where the tra p is
stati onary . The Schrodi nger equati on for a rota ti ng ha rmonic tra p in the l abora tory
fra me has the form
i@ ê ( ; t ) = À1
2Â + V ( ( t )) ê ( ; t ): (17)
The ti me dependence of the Ha milto nian in thi sequati on may be eliminated by the
tra nsformati on to the rota ti ng fram e. The wa ve functi on ê ( ; t ) in the laboratory
fram e is related to the wa ve functi on ¢ ( ; t) in the rotati ng fram e thro ugh the
form ula
ê ( ; t ) = ¢ ( ( t ) ; t ) = exp (ÀiÂ^t ) ¢ ( ; t ) ; (18)
where ^isthe angul ar momentum operator. Thi s tra nsform ati on cancels the ti me
dependence of the potenti al since
exp (iÂ^t )x ( t ) exp(ÀiÂ^t ) = x ; (19a)
exp (iÂ^t )y ( t ) exp(ÀiÂ^t ) = y ; (19b)
but it intro duces an extra term into the Ham ilto nian. Upon substituti ng the rela-
ti on (18) into Eq. (17), we obtain
i@ ¢ ( ; t ) = À1
2Â + V ( ) À Â ^¢ ( ; t ) : (20)
The addi tional term À Â ^is responsible for the centri fugal force and the Coriolis
force tha t act in the rota ti ng frame. The generating functi on ¢ ( ; t ) for all poly-
nomial vo rtex structure s bui lt on the funda menta l, Gaussian state wave functi on
in a tra p in the rotati ng frame is the soluti on of the Schrodinger equation (17)
sati sfying the initi al condi ti on
¢ ( ; t = 0) = exp(iÂ) ¢ ( ) : (21)
36 I. Bi a¤ynicki -Bi r ula et al .
In the general case, for an arbi tra ry rotati on, the wa ve functi on ¢0(r)and
hence also ¢kk (r;t) cannot be explicitl y wri tten down since the set of algebraic
equati ons for the co cients of the Gaussian wave functi on cannot be solved in a
closed form. However, in a special caseconsidered in thi s paper, when the angular
velocity is aligned with one of the symm etry axes of the tra p, the general soluti on
is relati vely simpl e. In thi s case the moti on in the zdi recti on unco upl esfro m the
moti on in the x y pl ane and the generating functi on factori zesinto a product. The
part of the generati ng functi on describing the m otion in the zdirecti on does no t
involve the rota ti on and it has been already wri tten down in Ref. [10]. Thus, the
three- di mensional probl em reduces to the soluti on of the Schrodi nger equati on in
a ro tati ng ani sotropi c ha rm onic tra p in tw o di mensions. In wha t follows we shall
choose the coordi nate axes xand yalong the main axes of the potenti al and for
deÙnitness we assume tha t !x> ! y.
The soluti on of the equati ons of motion for the harm onic oscillator in a rota t-
ing fram ein classical and in quantum theory invol vestw ocharacteri stic frequencies
!+and !Àgiven by the form ula s (cf., for exampl e, [13])
2 ¨ 2+ ! 2
x+ ! 2
y+q( ! 2
xÀ!2)2+ 8¨ 2( ! 2+ ! 2)
! =
2¨ 2+ ! 2+ ! 2À( ! 2À!2)2+ 8¨ 2( ! 2+ ! 2)
The requirement tha t the expression under the outer square root for !b e positi ve
leads to the following condi ti on:
( ¨ 2À!2)( ¨ 2À!2) > 0 : (23)
We inf er from thi s that there are two regions of stable oscillati ons separated by
a gap: slow rota tions ¨ < ! < ! and f ast rotations ! < ! < ¨ . In the
slow rotati on regim e the trapping forces are stronger tha n the centri fugal force.
In the fast rota ti on regim e the centri fugal force overwhel ms the tra pping forces
but neverthel ess stabi lity holds due to the acti on of the Cori olis force. Thi s is
the same stabilizati on mechanism as in the Paul tra p or in the Trojan states of
electrons (cf. [13, 14]). It turns out tha t in the investigati on of the soluti ons of the
Schrodinger equati on it is convenient to work not with !+and !but with thei r
linear combinatio ns ¨1and ¨2
2¨ 2+ ! 2+ ! 2+ 2 ¯ ( ! 2À¨2)( ! 2À¨2)
2=!++ ¯ !
Vortex Li nes i n Mot ion 37
2¨ 2+ ! 2
x+ ! 2
yÀ2 ¯ q( ! 2
xˬ2)( ! 2
2=!+À¯ ! À
where ¯is the sign parameter equal to 1 for slow rota ti ons and equal to { 1 for
fast rota ti ons. It follows from these deÙniti ons that for slow rotati ons ¨1> ¨ 2
and for fast rota ti ons ¨2> ¨ 1. By inverti ng the form ulas(24) we obtain the tra p
frequenci es expressed i n term s of ¨2and ¨1
1+ ¨ 2
2À¨2+ 2q( ¨ 2
1À¨2)( ¨ 2
2ˬ2) ; (25a)
! = ¨ 2
1+ ¨ 2
2À¨2À2 ( ¨ 2
1À¨2)( ¨ 2
2ˬ2) : (25b)
We shall bui ld the generati ng functi on accordi ng to the formula (21), cho os-
ing the ground state as the envelope functi on. Owi ng to our special choice of
the rota ti on axis, the generating functi on in three dimensions is a pro duct of the
generati ng functi on in the xand yvariables and the generating functi on for the
one-dimensional oscillator in the zvariable
¢ ( ; t ) = ¢ ( x ; y ; t ) ¢ ( z ; t ) : (26)
The last part has been already obtained in Ref. [10] and it has the form (we leave
out norm alizatio n constants for they play no role in our considerations)
¢ ( z ; t ) = exp Ài! t
2Àie ñh k 2
2m! sin( ! t ) exp Àm!
2ñh z2
Èexp (ie k z ) : (27)
The generati ng functi on in the xand yvari abl eshas the same general structure.
It is also a product of the ti me-dependent phase factor, the ti me-independent
Gaussian, and the phase facto r that is linear in the coordi nates
¢ ( x ; y ; t ) = exp (i' ( t )) exp Àm
2ñh Â^
MÂexp (i(t ) Â) ; (28)
where ^
Mis a 2È2symmetri c constant matri x, ( t) is a two-dimensional vector,
' ( t ) is a phase, and is the pro jection of the vector into the x y pl ane. Up on
substi tuti ng thi s ansatz into the Schrodinger equati on (20), one obta ins a set
of algebraic equati ons for the components of the matri x ^
M, a Ùrst-order linear
di ˜erenti al equati on for the vecto r ( t) and an expression for the deri vati ve of the
phase ' ( t ) ,
¨ ; ^
M ] À^
V = 0; (29a)
d( t )
dt= ( Ài^
M + ^
¨ ) ( t ) ; (29b)
d' ( t )
2Àñh (t)2
2 m ;(29c)
where ^
¨ = ff0 ; ¨ g;f À ¨ ; 0gg is the matri x representi ng the rotati on and ^
V =
ff!2; 0g;f0; ! 2g g is the matri x of the potenti al energy divided by m. The i ni tial
values are (0 ) = ; ' (0 ) = 0 . Equati ons (29) m ay besuccessively solved starti ng
38 I. Bi a¤ynicki -Bi r ula et al .
with the equati on for ^
M. The soluti on for the matri x ^
M =
"¨1(1 + ç ) i¨ ç
i¨ç ¨1(1 Àç )
ç = ¯
1À¨2= ¯
t2¨ 2À!2
y+ 2 ¯ q(! 2
xˬ2)( ! 2
2¨ 2À!2
xÀ!2À2¯ (! 2À¨2)( ! 2À¨2):(31)
Note that since ( ¨ 2
1À¨2)( ¨ 2
2À¨2) = ( ! 2À!2)2; ç is always real. In addi ti on,
the values of çat the boundari es of the stability region !and !are equal to
1 and { 1, respectively and the derivati ve of çwi th respect to ¨in the stability
regions is always positi ve
d ¨ =2 ¨ ¯ ç
(! 2ˬ2)( ! 2ˬ2)> 0 : (32)
Typi cal behavi or of ças a functi on of ¨is shown in Fig. 5. Thus, the absolute value
of çinside the stability region is lesstha n 1. Theref ore, the matri x ^
Malwayshas a
positi ve deÙnite real part and hence it deÙnes a bounded Gaussian wave functi on.
Once ^
Mhas been determined, the remaining two equati ons may be solved. W e
shall wri te down the soluti ons with the physi cal constants ñh and minserted
( t ) = exp[ ( Ài^
¨) t ] Â=e[cos2t ) Ài(Â)sin( ¨ 2t)] Â;(33)
' ( t ) = ñh
2m Â[ f ( t ) + g ( t) ¥ + h ( t ) ¥ ] Â À ¨1t; (34)
where ¥;¥, and ¥are the standa rd Pauli m atri ces (i ntro duced here o nly for
conveni ence; there is no spin in our problem), is the following complex uni t
Vortex Li nes i n Mot ion 39
¨2fiç¨ ; À¨ ; ç¨ 1g;(35)
and f ; g, and hare three complex functi ons of ti me
f ( t ) = Ài( ¨ 2
2¨ 1( ¨ 2
2)+ieÀ2i ¨1t
Ȩ1( ¨ 2
2À¨2)[ ¨ 1cos(2 ¨ 2t ) + i¨2sin(2 ¨ 2t)] + ¨ 2( ¨ 2
2 ¨ 1¨2
2( ¨ 2
g ( t ) = ç eÀ2i ¨1t¨sin2( ¨ 2t )
h ( t ) = iç ( ¨ 2
2¨ 1( ¨ 2
Ȩ1[¨ 1( ¨ 2
2À¨2)cos(2 ¨ 2t ) + i¨2( ¨ 2
1À¨2)sin(2 ¨ 2t )] + ¨ 2( ¨ 2
2¨ 1¨2
2( ¨ 2
W e il lustra te our results wi th two simpl e conÙgura ti ons of vo rtex lines. The
Ùrst example inv olves tw o parallel vortex lines with the same circul ati on. These
vortex lines remain straight and parallel all the ti me (cf. Fig. 6). In the second
exam ple, two vortex lines have opposite circul ati on. In thi s casethe ti me evoluti on
is compl etely di ˜erent; b oth vortex lines become curved and m ove around i n a
compl icated fashion (cf. Fi g. 7).
40 I. Bi a¤ynicki -Bi r ula et al .
Fig. 7. Tw o vortex lines with opp osite circulatio n in a rotating trap.
5. Co n cl usion s
The m ain result of thi s paper is the extension of our m ethod of generati ng
functi ons to the case of a rota ting harm onic tra p. Such tra ps are now often used
in the exp eriments on the Bose{Ei nstein condensates. We have deri ved a com-
pl ete expression for the generati ng functi on in the case when the rotati on axis
coincides with one of the pri ncipal axes of the trap. In order to show its pra ctical
value, we have used thi s expression in a very simple case to Ùnd the motion of
two vortex lines. Our generati ng functi on may be used to study systemati cally
the m otion of more compl icated vortex structures. The serious limita ti on of thi s
metho d in applications to realisti c Bose{Ei nstein condensates is our use of the
linear appro ximati on to the wave equati on. However, as it wasargued in Ref. [11],
thi s appro ximatio n might be adequate to describe the moti on of vorti ces.
R efer en ces
[1] O. Madelung, Z. Phys. 40, 342 (1926).
[2] A ristotle, D e Caelo ( On t he H eavens ), Bo ok II , C h. X I I I .
[3] R. Descartes, P ri n cipia Ph ilo sophi ae, A msterdam 1644.
[4] W. Thomson ( Lord K elvin), Phil os. Mag., 34, 15 (1867).
[5] G. H elmholt z, Cr elles J., 55, 25 (1858) (English translation P.G. Tait, Phi l os.
Ma g., 33 , 485 (1867).
[6] P.G. Sa˜man, Vort ex Dy nami cs, Cambridge U niversity , Cambridge 1992.
[7] J.F. N ye, M. V . Berry , 165 (1974).
[8] M. V . Berry , in: , Eds. R. Balian,
M. K eman, J.-P. Poirier, North- H olland , A msterdam 1981, p. 453.
[9] M. V . Berry , M. R. Dennis, 2251 (2001).
Vortex Li nes i n Mot ion 41
[10] I. Bia ¤ynicki- Birul a, Z. Bia¤ynick a-Birula, C. Ïliw a, Ph ys. Rev. A 61, 032110
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... To find a complete analysis of these structures, it is necessary to turn to, for example, the following articles by Białynicki-Birula et al. [9,10]. Furthermore, a review of their citations might serve to put into context the impact that this initially marginal topic is having on the study of various few-body processes such as, for example, the simple ionization of atoms, as we will see in this article. ...
... In particular, the two zeros observed at an angle of 45 degrees had opposite vorticities. This was compatible with one of the mechanisms for vortex appearance identified by Białynicki-Birula et al. [9,10]. ...
... Here, it is worth highlighting that reaching the kinematic limit in momentum space is equivalent to reaching infinity in coordinate space, because the distance between an electron escaping with the maximum allowed kinetic energy and any other with a smaller speed grows linearly with time. As the deflection angle of the positron increases, the ends of both vortices get closer together, until they join to form a single line as shown in the Figure 9, in one of the forms of vortex evolution studied by Białynicki-Birula [9,10]. If the deflection angle of the positron continues to increase, this vortex line is deformed until, in an inverse process to that observed in the previous figure, it separates into a line and a ring, as shown in Figure 10 for = 120 . ...
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Vortices are structures known in our daily lives and observed in a wide variety of systems, from cosmic to microscopic scales. Relatively recent studies showed that vortices could also appear in simple quantum systems. For instance, they were observed experimentally and theoretically as isolated zeros in the differential cross section in atomic ionization processes by the impact of charged particles. In this work, we show that the appearance of these quantum vortices as point structures was not due to any intrinsic property of them, but to the use of restrictive geometries in their visualization. In particular, we show that by studying the fully differential cross section for hydrogen ionization by positron impact, these vortex points are actually a manifestation of a more complex and hitherto unexplored structure, a 3D “vortex surface”.
... Electrons as the prototypical example of a de Broglie wave came to the fore as the first candidate for a matter vortex beam. Although suggestions for particle vortex beams were made first by Bialynicki-Birula et al. [17][18][19], the main thrust of research specifically on electron vortex beams is due to the work by Bliokh et al. [20] and electron vortices were then experimentally realized in earlier studies [21][22][23] inside electron microscopes from 2010 onwards. The main technique for the realization of electron vortex beams includes computer-generated holographic masks applied in similar ways to those routinely adopted in the creation of optical vortex beams [24,25]. ...
... Although much of electron vortex research is inspired by concepts and techniques in optical vortex beams, due to the equivalence of scalar Maxwell equations in free space and the Schrödinger equation for a free particle [17][18][19][20], electron vortex beams are characterized by a number of different properties [20]. These are attributable to the fact that electrons have finite mass, carry electric charge and have half-integer intrinsic angular momentum (spin). ...
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Electron vortex beams constitute the first class of matter vortex beams which are currently routinely produced in the laboratory. Here, we briefly review the progress of this nascent field and put forward a natural quantum basis set which we show is suitable for the description of electron vortex beams. The normal modes are truncated Bessel beams (TBBs) defined in the aperture plane or the Fourier transform of the transverse structure of the TBBs (FT-TBBs) in the focal plane of a lens with the said aperture. As these modes are eigenfunctions of the axial orbital angular momentum operator, they can provide a complete description of the two-dimensional transverse distribution of the wave function of any electron vortex beam in such a system, in analogy with the prominent role Laguerre–Gaussian (LG) beams played in the description of optical vortex beams. The characteristics of the normal modes of TBBs and FT-TBBs are described, including the quantized orbital angular momentum (in terms of the winding number l ) and the radial index p >0. We present the experimental realization of such beams using computer-generated holograms. The mode analysis can be carried out using astigmatic transformation optics, demonstrating close analogy with the astigmatic mode transformation between LG and Hermite–Gaussian beams. This article is part of the themed issue ‘Optical orbital angular momentum’.
... Knotted structures appear in physical fields in a wide range of areas of theoretical physics; in liquid crystals [24,29,30], optical fields [14], Bose-Einstein condensates [32], fluid flows [17,18], the Skyrme-Faddeev model [39], quantum mechanics [4,8,9] and several others. ...
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Persistent topological structures in physical systems have become increasingly important over the last years. Electromagnetic fields with knotted field lines play a special role among these, since they can be used to transfer their knottedness to other systems like plasmas and quantum fluids. In null electromagnetic fields the electric and the magnetic field lines evolve like unbreakable elastic filaments in a fluid flow. In particular, their topology is preserved for all time, so that all knotted closed field lines maintain their knot type. We use an approach due to Bateman to prove that for every link L there is such an electromagnetic field that satisfies Maxwell’s equations in free space and that has closed electric and magnetic field lines in the shape of L for all time. The knotted and linked field lines turn out to be projections of real analytic Legendrian links with respect to the standard contact structure on the 3-sphere.
Formation of quantum vortices in laser-induced photodetachment from negative ions is analyzed. The driving laser field consists of a single ultrashort pulse of circular polarization and the unperturbed ground-state wave function of the anion is found in either the s or p state. In particular, numerical illustrations for the photodetachment from H−, O−, K−, and a model A− anion are presented. Special attention is paid to the symmetry of the ground-state wave function and ionization potential over the final vortex pattern. It is shown that the two-dimensional spectra of photoelectrons in momentum space comprise three well-defined regions: The low-energy (central) region, multiphotonlike zone, and supercontinuum. While the supercontinuum does not contribute to vorticity and the multiphoton zone depends only on the laser field characteristics, vortices in the low-energy region strongly depend on the bound-state wave function and its ionization potential.
The recent prediction and subsequent creation of electron vortex beams in a number of laboratories occurred after almost 20 years had elapsed since the recognition of the physical significance and potential for applications of the orbital angular momentum carried by optical vortex beams. A rapid growth in interest in electron vortex beams followed, with swift theoretical and experimental developments. Much of the rapid progress can be attributed in part to the clear similarities between electron optics and photonics arising from the functional equivalence between the Helmholtz equations governing the free-space propagation of optical beams and the time-independent Schrödinger equation governing freely propagating electron vortex beams. There are, however, key differences in the properties of the two kinds of vortex beams. This review is primarily concerned with the electron type, with specific emphasis on the distinguishing vortex features: notably the spin, electric charge, current and magnetic moment, the spatial distribution, and the associated electric and magnetic fields. The physical consequences and potential applications of such properties are pointed out and analyzed, including nanoparticle manipulation and the mechanisms of orbital angular momentum transfer in the electron vortex interaction with matter.
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Relativistic definition of the phase of the electromagnetic field, involving two Lorentz invariants, based on the Riemann-Silberstein vector is adopted to extend our previous study [I. Bialynicki-Birula, Z. Bialynicka-Birula, and C. Śliwa, Phys. Rev. A 61, 032110 (2000)] of the motion of vortex lines embedded in the solutions of wave equations from Schrödinger wave mechanics to Maxwell theory. It is shown that time evolution of vortex lines has universal features; in Maxwell theory it is very similar to that in Schrödinger wave mechanics. Connection with some early work on geometrodynamics is established. Simple examples of solutions of the Maxwell equations with embedded vortex lines are given. Vortex lines in the Laguerre-Gaussian beams are treated in some detail.
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The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schrödinger equation in a harmonic potential proposed recently [J. J. García-Ripoll, V. M. Pérez-García, and V. Vekslerchik, Phys. Rev. E 64, 056602 (2001)] is extended to the many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics, and results in the separation of the center-of-mass motion from the internal dynamics of many-body systems. The center-of-mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid.
We describe the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of determinstic trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler–Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens’ principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie–Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie–Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.
The influence of the nonlinear, quantum terms in the Maxwell equations on the evolution of vortex lines was analyzed. The quantum corrections led to the deformation and disappearance of the vortex ring in the considered configuration of the constantly expanded vortex ring. The topological change was expected as a result of nonlinearity introduced by vacuum polarization. It was found that similar structures appeared in classical and linear fields by the appropriate perturbation of the vortex configurations.
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The notion of the rotational frequency shift, an analog of the Doppler shift, is introduced. This new frequency shift occurs for atomic systems that lack rotational invariance, but have stationary states in a rotating frame. The rotational frequency shift is given by the scalar product of the angular velocity and the angular momentum of the emitted photon in full analogy with the standard Doppler shift which is given by the scalar product of the linear velocity of the source and the linear momentum of the photon. The rotational frequency shift can be observed only in a Mössbauer-like regime when the angular recoil is negligible.
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When an ultrasonic pulse, containing, say, ten quasi-sinusoidal oscillations, is reflected in air from a rough surface, it is observed experimentally that the scattered wave train contains dislocations, which are closely analogous to those found in imperfect crystals. We show theoretically that such dislocations are to be expected whenever limited trains of waves, ultimately derived from the same oscillator, travel in different directions and interfere - for example in a scattering problem. Dispersion is not involved. Equations are given showing the detailed structure of edge, screw and mixed edge-screw dislocations, and also of parallel sets of such dislocations. Edge dislocations can glide relative to the wave train at any velocity; they can also climb, and screw dislocations can glide. Wavefront dislocations may be curved, and they may intersect; they may collide and rebound; they may annihilate each other or be created as loops or pairs. With dislocations in wave trains, unlike crystal dislocations, there is no breakdown of linearity near the centre. Mathematically they are lines along which the phase is indeterminate; this implies that the wave amplitude is zero.
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We exhibit analytically and confirm numerically the existence of stable, though nonstationary, quantum states of electrons moving on circular orbits that are trapped in an effective potential well made of the Coulomb potential and the rotating electric field produced by a strong circularly polarized electromagnetic wave. These states are direct counterparts of the Trojans-two clusters of asteroids moving around the Sun in the vicinity of the stable Lagrange points in the Sun-Jupiter two-body system.
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