7/28/2019 PhysRevLett.106.150403

1/4

7/28/2019 PhysRevLett.106.150403

2/4

0 mm (no coupling) s 9 mm (full coupling). The lefthalf of the cavity in Fig. 1 contains a movable semicircular

Teflon piece with diameter 60 mm and height 5 mm con-

nected via a thin snell to another stepper motor. Its dis-

placement with respect to the center of the cavity defines

the other parameter .A pointlike dipole antenna intrudes into each half of

the cavity. A vectorial network analyzer Agilent PNA

5230 A couples microwaves into the cavity through oneantenna a and measures amplitude and phase of the signalreceived at the same or the other antenna b relative to theinput signal. In this way the complex element Sba of thescattering matrix S is determined. To induce Tviolation,a cylindrical ferrite of diameter 4 mm and height 5 mm

[17] is put in the right part of the cavity and magnetized

by two permanent magnets, which are mounted above

and below the cavity. Magnetic field strengths 0 mT B 90 mT are obtained by varying their distance. Toscan the parameter space spanned by (s, ), the twostepper motors and the vectorial network analyzer are

controlled by a PC. The four S-matrix elements Sbaf

,

fa; bg 2 f1; 2g are measured in the parameter plane withthe resolution s 0:01 mm and a frequency stepf 10 kHz. The frequency range of 40 MHz is deter-mined by the spread of the resonance doublet. Lack of

reciprocity, i.e., S12 S21, is the signature for T viola-tion [17]. Figure 1 shows two typical spectra for jSaaj2with a 1, 2. Additional measurements include neigh-boring resonances, situated about 250 MHz away from

the region of interest, to account for their residual influ-

ence. In the considered frequency range the electric field

vector is perpendicular to the top and bottom plates.

Then, the Helmholtz equation is identical to the

Schrodinger equation for the corresponding quantum bil-liard [14,21]. Consequently, the results provide insight

into the associated quantum problem. In previous experi-

ments [11,12] an EP was located by determining for each

parameter setting the real and imaginary parts of the

eigenvalues as the frequencies f1;2 and the widths 1;2of the resonances from a fit of a Breit-Wigner function.

This procedure, however, fails at the EP because there the

line shape is not a first order pole of the S matrix butrather the sum of a first and a second order pole [4,13].

Therefore, the coalescence of the eigenvectors should also

be incorporated [12,13]. We determine the two-state

Hamiltonian H and its eigenvalues and eigenvectors ex-plicitly for every setting of s, , B from the measuredS-matrix elements via the method presented in [17].There, we showed that for two pointlike antennas a

resonance doublet is well described by the two-channel

S matrix Sf 1 2iWyf1 H1W. The matrixW Wa couples the resonant states 1, 2 to theantennas a 1, 2. It is real since that coupling conservesT. The Hamiltonian Hcomprises dissipation in the reso-nator walls and the ferrite, and T violation and thus is

neither Hermitian nor symmetric. Its general form reads

H e1 HS12 iHA12

HS12 iHA12 e2

!: (1)

The quantities e1 e2, HS12, and HA12 are complex expan-sion coefficients with respect to the unit and the Pauli

matrices. The ansatz for S was tested thoroughly by fittingit to S-matrix elements measured with and without mag-netization of the ferrite. In the latter case the antisym-

metric part HA12 vanishes [11,12]. Fitting the S matrixto the four measured excitation functions Sbaf yieldsthe matrices H and W up to common real orthogonalbasis transformations. We define the basis such that

HS12 iHA12=HS12 iHA12 exp2i is a phase factor.Thus, T violation is expressed by a real phase . This isthe usual practice in physics, e.g., for nuclear reactions

[22], and for weak and electromagnetic decay [23].

The eigenvalues ofHin Eq. (1) coalesce to an EP when

HS2

12 HA2

12 e1 e22=4 0 but not all three termsvanish. Figure 2 shows one example for a search of the

EP. Keeping s 1:66 mm and B 53 mT fixed, the realand imaginary parts of the eigenvalues,f

j

and j

, areplotted

vs . At sEP;EP 1:660:01;41:250:01 mm theencounter of the eigenvalues is closest. To find the valuefor the coalescence of the eigenvectors jrki, k 1, 2, theratios k rk1=rk2 jkjeik of their components aredetermined [6]. The right part of Fig. 2 demonstrates that

it also equals EP 41:25 mm. Indeed, as will be evi-denced in further tests below, both the eigenvalues and theeigenvectors cross at (sEP, EP). At the EP the only eigen-vector ofHis [3]

jrEPi iei

1

!: (2)

The ratio of its components is a phase factor. ForT-conserving systems the phase is EP =2 [12] asconfirmed by Fig. 3 at B 0. With T violation EP =2 and thus provides a measure for . Figure 3 showsthat goes through a maximum with increasing B.We identify this with the ferromagnetic resonance resulting

FIG. 2. Left panels: Real and imaginary parts of the eigenval-

ues, f1;2 and 1;2, as functions of at s sEP 1:66 mm andB 53 mT. Around EP 41:25 mm they are closest.Right panels: Modulus and phase of the ratios 1;2 j1;2jei1;2 of the components of the associated eigenvectors.They are also closest at 41:25 mm. For EP the uppercurves correspond to, respectively, f1; 1; j2j; 2.

PRL 106, 150403 (2011) P H Y S I C A L R E V I E W L E T T E R Sweek ending

15 APRIL 2011

150403-2

7/28/2019 PhysRevLett.106.150403

3/4

from the coupling of the rf magnetic field to the spins in the

ferrite. In Ref. [17]theT-violating matrix element iHA12 hasbeen expressed by a resonance formula which yields the

solid curve for EPB.Figure 4(a) shows at B 53 mT around an EP the

differences of the complex eigenvalues, f1;2 (blue, leftof EP) and 1;2 (orange, right of EP); 4(b) those of thephases 1;2 (orange, left of EP) and of the moduli

j1;2

j(green, right of EP). The darker the color the smaller is therespective difference. The darkest color visualizes the

branch cut along which it vanishes. The panels of

Fig. 2 are cuts through the respective ones of Fig. 4 at

s 1:66 mm. We observe that jf1 f2j and j1 2jare small and thus visible only to the left of the EP,j1 2j and jj1j j2jj to its right. That the branchcuts all extend from one common point into opposite

directions proves that this is an EP [2,6,16].

For systems with T violation the geometric phase gathered by the eigenvectors around an EP is predicted to

be complex, yielding a geometric amplitudeeIm 1 [1].To check this we choose contours around the EP for the six

values ofB considered in Fig. 3. One example is shown inFig. 4. It consists of two different loops [20]. The path isparametrized by a real variable t with initial value t 0 attheir intersection. After the first encircling t t1; after thesecond t t2. In [11,24] the geometric phase gatheredaround a diabolical point, respectively, an EP, was

obtained for just a few parameter settings, because the

procedurethe measurement of the electric field intensity

distributionis very time consuming. We now have the

possibility to determine the left and right eigenvectors,

hljtj and jrjti, j 1, 2 of H in Eq. (1) on a muchnarrower grid of the parameter plane. In distinction to

previous numerical works they are biorthonormalized ateach point t of the contour such that hljtjrjti 1.Defining in analogy to the T-conserving case [11]

B e1e22

=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHS

2

12 HA2

12

q, tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 B2

pB yields

for the right eigenvectors [1]

jr1ti ei=2 cos

ei=2 sin

!; jr2ti

ei=2 sinei=2 cos

!:

(3)

As in [11] the EPs are located at 1 B2 0 and encir-cling the EP once changes to =2. The T-violatingparameter is not constant along the contour, even though

the magnetic field is fixed. In fact, it varies with the open-ing s between both parts of the resonator, because theferrite is positioned in one of them (see Fig. 1). The

position of the EP does not depend on . Thus, the spacecurve st; t; t winds around the line (sEP, EP, )(see Ref. [19]). Since has no singular points in theconsidered parameter plane it returns to its initial value

after each encircling of the EP and the eigenvectors jr1;2tifollow the same transformation scheme as in the

T-conserving case [11], that is, jr1t1i jr20i,jr2t1i jr10i, and jr1;2t2i jr1;20i.

The biorthonormality defines the eigenvectors hljj and

jrj

iup to a geometric factor, so that

hLj

tj h

ljtjeijt

and jRjti jrjtieijt. The geometric phases jt arefixed by the condition of parallel transport [1,5],

hLjtj ddtRjti 0. This yields d1tdt 12 cos2t dtdt d2tdt . The initial value of the phases is chosen asj0 0, such that 1t 2t. For theT-conserving case 1;2t 0. The phase 1t is deter-mined successively for increasing t from the product ofhl1tj and j ddt r1ti. In the lower (upper) panel of Fig. 5 isplotted the phase 1t accumulated by jR1ti when encir-cling twice the EP along the contour in Fig. 4 (along its

outer loop). The orientation is such that the EP is always tothe left. The cusps occur at d

dt 0. In each panel the

triangle marks the start point, the pentagon t1, and thediamond t2. In both examples 1t1 0 and, as predictedin Ref. [1], the geometric phase 1t1 is not real. If theEP is encircled twice along the same loop, we obtain

1t2 0 [19,20] within 107. Thus we measured the

90

110

130

0 20 40 60 80

EP(deg)

Magnetic field B(mT)

FIG. 3. Phase (dots) of the ratio EP of the eigenvector com-ponents at the EP as a function of the magnetization of the

ferrite. For the T-conserving case the known result [12] of 90

(dashed horizontal line) is recovered. The model for the

T-violating matrix element iHA12 in terms of the ferromagneticresonance yields the solid line. The vertical bar indicates the

range ofB where the ferromagnetic resonance is expected.

a) b)

| - |

| || | - | |1 2

|f -f |

1 2

1 2

| - |1 2

)mm(s)mm(s

)mm(

EP EP

FIG. 4 (color). Differences of the complex eigenvalues

f1;2 i1;2 (a) and of the ratios 1;2 j1;2jei1;2 of the eigen-vector components (b) in an area of the parameter plane (s, )around the EP. The darker the color the smaller is the respective

difference. The black dots in both panels indicate the chosen

contour for the encircling of the EP (see text).

PRL 106, 150403 (2011) P H Y S I C A L R E V I E W L E T T E R Sweek ending

15 APRIL 2011

150403-3

7/28/2019 PhysRevLett.106.150403

4/4

s up to an accuracy manifestly better than 102.Accordingly, from the value of1t2 given in the captionof Fig. 5, we may conclude that, when the loops are differ-

ent, 1t does not return to its initial value [20]. Whenencircling this double loop again and again we observe a

drift of 1;2 in the complex plane away from the origin.Thus, one geometric amplitude increases and the other one

decreases, yielding the experimental proof for the pre-

dicted reversible geometric pumping process [1]. The de-

pendence on the choice of the path signifies that the

complex phases 1;2t are geometric and not topological.In order to obtain the complete phase gathered by

jR1

t

iduring the encircling of the EP, one has to add the topo-logical phase accumulated by jr1ti. In summary, whenencircling the EP we obtain geometric factors different

from unity and the transformation scheme Eq. (3) for

j~r1;2ti. These results provide an unambiguous proof thatthe EP lies inside both loops of the contour shown in Fig. 4.

Their precision is illustrated by the dense sequence of

points in Fig. 5. It allows the determination of the size of

T violation and of the geometric factor along arbitrary

contours in the parameter plane. Predictions on geometric

amplitudes and phases could be confirmed.

We thank G. Ripka and H. A. Weidenmuller for illumi-

nating discussions. This work was supported by the DFG

through SFB 634 and by the Alexander von HumboldtFoundation.

*richter@ikp.tu-darmstadt.de

[1] J. C. Garrison and E. M. Wright, Phys. Lett. A 128, 177

(1988); M. V. Berry, Proc. R. Soc. A 430, 405 (1990);

K. Yu. Bliokh, J. Phys. A 32, 2551 (1999).

[2] N. Moiseyev and S. Friedland, Phys. Rev. A 22, 618

(1980); W. D. Heiss and A. L. Sannino, J. Phys. A 23,

1167 (1990); E. Hernandez, A. Jauregui, and A.

Mondragon, J. Phys. A 33, 4507 (2000); A. P. Seyranian,

O. N. Kirillov, and A. A. Mailybaev, J. Phys. A 38, 1723

(2005).

[3] H. L. Harney and W. D. Heiss, Eur. Phys. J. D 29, 429

(2004); W. D. Heiss, J. Phys. A 39, 10 077 (2006).

[4] I. Rotter, J. Phys. A 42, 153 001 (2009).

[5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984).

[6] M. V. Berry and M. R. Dennis, Proc. R. Soc. A 459, 1261

(2003).[7] T. Kato, Perturbation Theory for Linear Operators

(Springer, Berlin, 1966).

[8] O. Latinne et al., Phys. Rev. Lett. 74, 46 (1995); E.

Narevicius and N. Moiseyev, Phys. Rev. Lett. 84, 1681

(2000); H. A. Weidenmuller, Phys. Rev. B 68, 125326

(2003); H. Cartarius, J. Main, and G. Wunner, Phys.

Rev. Lett. 99, 173003 (2007); P. Cejnar, S. Heinze, and

M. Macek, Phys. Rev. Lett. 99, 100601 (2007); R.

Lefebvre, O. Atabek, M. Sindelka, and N. Moiseyev,

Phys. Rev. Lett. 103, 123003 (2009); R. Lefebvre and

O. Atabek, Eur. Phys. J. D 56, 317 (2009).

[9] J. Wiersig, S. W. Kim, and M. Hentschel, Phys. Rev. A 78,

053809 (2008); J.-W. Ryu, S.-Y. Lee, and S. W. Kim,

Phys. Rev. A 79, 053858 (2009); A. C. Or, Q. J. Mech.Appl. Math. 44, 559 (1991); M. Kammerer, F. Merz, and

F. Jenko, Phys. Plasmas 15, 052 102 (2008); O. N. Kirillov,

Proc. R. Soc. A 464, 2321 (2008); S. Klaiman, U. Gunther,

and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008);

Ch. E. Ruter et al., Nature Phys. 6, 192 (2010).

[10] M. Philipp, P. von Brentano, G. Pascovici, and A. Richter,

Phys. Rev. E 62, 1922 (2000).

[11] C. Dembowski et al., Phys. Rev. Lett. 86, 787 (2001);

Phys. Rev. E 69, 056216 (2004).

[12] C. Dembowski et al., Phys. Rev. Lett. 90, 034101

(2003).

[13] B. Dietz et al., Phys. Rev. E 75, 027201 (2007).

[14] H.-J. Stockmann and J. Stein, Phys. Rev. Lett. 64, 2215

(1990); H.-D. Grafet al., Phys. Rev. Lett. 69, 1296 (1992).[15] T. Stehmann, W. D. Heiss, and F. G. Scholtz, J. Phys. A 37,

7813 (2004).

[16] S.-B. Lee et al., Phys. Rev. Lett. 103, 134101 (2009); Y.

Choi et al., Phys. Rev. Lett. 104, 153601 (2010).

[17] B. Dietz et al., Phys. Rev. Lett. 98, 074103 (2007).

[18] F. Haake, Quantum Signatures of Chaos (Springer, New

York, 2001).

[19] A. A. Mailybaev, O. N. Kirillov, and A. P. Seyranian, Phys.

Rev. A 72, 014104 (2005); Dokl. Math. 73, 129 (2006).

[20] H. Mehri-Dehnavi and A. Mostafazadeh, J. Math. Phys.

(N.Y.) 49, 082 105 (2008).

[21] S. Sridhar, Phys. Rev. Lett. 67, 785 (1991); A. Richter, in

Emerging Applications of Number Theory, edited by D. A.

Hejhal et al., IMA Vol. 109 (Springer, New York, 1999),

p. 479; H.-J. Stockmann, Quantum Chaos: An

Introduction (Cambridge University Press, Cambridge,

2000).

[22] H. Driller et al., Nucl. Phys. A317, 300 (1979); J.M.

Pearson and A. Richter, Phys. Lett. 56B, 112 (1975).

[23] A. Richter, in Interaction Studies in Nuclei, edited by H.

Jochim and B. Ziegler (North-Holland, Amsterdam 1975),

p. 191.

[24] H.-M. Lauber, P. Weidenhammer, and D. Dubbers, Phys.

Rev. Lett. 72, 1004 (1994).

-0.1

0.0

0.1

-0.4 0 0.4

Im(1

(t))

Re( 1(t))

b)

-0.1

0.0

0.1

Im(1

(t))

a)

FIG. 5. The complex phase 1t for B 53 mT when encir-cling twice the EP along the contour shown in Fig. 4 (lower

panel) or along its outer loop (upper panel). The triangle marks

the start point, the pentagon t1 after encircling the EP once, thediamond t2 after a second encircling. There, 1t2 equals3:123 i3:474 107; i.e., it is close to zero in the uppercase, 0:0931 i0:0152 in the lower one.

PRL 106, 150403 (2011) P H Y S I C A L R E V I E W L E T T E R Sweek ending

15 APRIL 2011

150403-4

http://dx.doi.org/10.1016/0375-9601(88)90905-Xhttp://dx.doi.org/10.1016/0375-9601(88)90905-Xhttp://dx.doi.org/10.1016/0375-9601(88)90905-Xhttp://dx.doi.org/10.1016/0375-9601(88)90905-Xhttp://dx.doi.org/10.1098/rspa.1990.0096http://dx.doi.org/10.1098/rspa.1990.0096http://dx.doi.org/10.1098/rspa.1990.0096http://dx.doi.org/10.1088/0305-4470/32/13/007http://dx.doi.org/10.1088/0305-4470/32/13/007http://dx.doi.org/10.1088/0305-4470/32/13/007http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1088/0305-4470/33/24/308http://dx.doi.org/10.1088/0305-4470/33/24/308http://dx.doi.org/10.1088/0305-4470/33/24/308http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1088/0305-4470/39/32/S09http://dx.doi.org/10.1088/0305-4470/39/32/S09http://dx.doi.org/10.1088/0305-4470/39/32/S09http://dx.doi.org/10.1088/1751-8113/42/15/153001http://dx.doi.org/10.1088/1751-8113/42/15/153001http://dx.doi.org/10.1088/1751-8113/42/15/153001http://dx.doi.org/10.1098/rspa.1984.0023http://dx.doi.org/10.1098/rspa.1984.0023http://dx.doi.org/10.1098/rspa.1984.0023http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1103/PhysRevLett.74.46http://dx.doi.org/10.1103/PhysRevLett.74.46http://dx.doi.org/10.1103/PhysRevLett.74.46http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevLett.99.100601http://dx.doi.org/10.1103/PhysRevLett.99.100601http://dx.doi.org/10.1103/PhysRevLett.99.100601http://dx.doi.org/10.1103/PhysRevLett.103.123003http://dx.doi.org/10.1103/PhysRevLett.103.123003http://dx.doi.org/10.1103/PhysRevLett.103.123003http://dx.doi.org/10.1140/epjd/e2009-00315-2http://dx.doi.org/10.1140/epjd/e2009-00315-2http://dx.doi.org/10.1140/epjd/e2009-00315-2http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1103/PhysRevA.79.053858http://dx.doi.org/10.1103/PhysRevA.79.053858http://dx.doi.org/10.1103/PhysRevA.79.053858http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1063/1.2909618http://dx.doi.org/10.1063/1.2909618http://dx.doi.org/10.1063/1.2909618http://dx.doi.org/10.1098/rspa.2008.0021http://dx.doi.org/10.1098/rspa.2008.0021http://dx.doi.org/10.1098/rspa.2008.0021http://dx.doi.org/10.1103/PhysRevLett.101.080402http://dx.doi.org/10.1103/PhysRevLett.101.080402http://dx.doi.org/10.1103/PhysRevLett.101.080402http://dx.doi.org/10.1038/nphys1515http://dx.doi.org/10.1038/nphys1515http://dx.doi.org/10.1038/nphys1515http://dx.doi.org/10.1103/PhysRevE.62.1922http://dx.doi.org/10.1103/PhysRevE.62.1922http://dx.doi.org/10.1103/PhysRevE.62.1922http://dx.doi.org/10.1103/PhysRevLett.86.787http://dx.doi.org/10.1103/PhysRevLett.86.787http://dx.doi.org/10.1103/PhysRevLett.86.787http://dx.doi.org/10.1103/PhysRevE.69.056216http://dx.doi.org/10.1103/PhysRevE.69.056216http://dx.doi.org/10.1103/PhysRevE.69.056216http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevE.75.027201http://dx.doi.org/10.1103/PhysRevE.75.027201http://dx.doi.org/10.1103/PhysRevE.75.027201http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevLett.69.1296http://dx.doi.org/10.1103/PhysRevLett.69.1296http://dx.doi.org/10.1103/PhysRevLett.69.1296http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1103/PhysRevLett.103.134101http://dx.doi.org/10.1103/PhysRevLett.103.134101http://dx.doi.org/10.1103/PhysRevLett.103.134101http://dx.doi.org/10.1103/PhysRevLett.104.153601http://dx.doi.org/10.1103/PhysRevLett.104.153601http://dx.doi.org/10.1103/PhysRevLett.104.153601http://dx.doi.org/10.1103/PhysRevLett.98.074103http://dx.doi.org/10.1103/PhysRevLett.98.074103http://dx.doi.org/10.1103/PhysRevLett.98.074103http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1134/S1064562406010352http://dx.doi.org/10.1134/S1064562406010352http://dx.doi.org/10.1134/S1064562406010352http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1103/PhysRevLett.67.785http://dx.doi.org/10.1103/PhysRevLett.67.785http://dx.doi.org/10.1103/PhysRevLett.67.785http://dx.doi.org/10.1016/0375-9474(79)90484-6http://dx.doi.org/10.1016/0375-9474(79)90484-6http://dx.doi.org/10.1016/0375-9474(79)90484-6http://dx.doi.org/10.1016/0370-2693(75)90279-8http://dx.doi.org/10.1016/0370-2693(75)90279-8http://dx.doi.org/10.1016/0370-2693(75)90279-8http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1103/PhysRevLett.72.1004http://dx.doi.org/10.1016/0370-2693(75)90279-8http://dx.doi.org/10.1016/0375-9474(79)90484-6http://dx.doi.org/10.1103/PhysRevLett.67.785http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1063/1.2968344http://dx.doi.org/10.1134/S1064562406010352http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1103/PhysRevA.72.014104http://dx.doi.org/10.1103/PhysRevLett.98.074103http://dx.doi.org/10.1103/PhysRevLett.104.153601http://dx.doi.org/10.1103/PhysRevLett.103.134101http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1088/0305-4470/37/31/012http://dx.doi.org/10.1103/PhysRevLett.69.1296http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevLett.64.2215http://dx.doi.org/10.1103/PhysRevE.75.027201http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevLett.90.034101http://dx.doi.org/10.1103/PhysRevE.69.056216http://dx.doi.org/10.1103/PhysRevLett.86.787http://dx.doi.org/10.1103/PhysRevE.62.1922http://dx.doi.org/10.1038/nphys1515http://dx.doi.org/10.1103/PhysRevLett.101.080402http://dx.doi.org/10.1098/rspa.2008.0021http://dx.doi.org/10.1063/1.2909618http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1093/qjmam/44.4.559http://dx.doi.org/10.1103/PhysRevA.79.053858http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1103/PhysRevA.78.053809http://dx.doi.org/10.1140/epjd/e2009-00315-2http://dx.doi.org/10.1103/PhysRevLett.103.123003http://dx.doi.org/10.1103/PhysRevLett.99.100601http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevLett.99.173003http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevB.68.125326http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevLett.84.1681http://dx.doi.org/10.1103/PhysRevLett.74.46http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1098/rspa.2003.1155http://dx.doi.org/10.1098/rspa.1984.0023http://dx.doi.org/10.1088/1751-8113/42/15/153001http://dx.doi.org/10.1088/0305-4470/39/32/S09http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1140/epjd/e2004-00049-7http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1088/0305-4470/38/8/009http://dx.doi.org/10.1088/0305-4470/33/24/308http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1088/0305-4470/23/7/022http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1103/PhysRevA.22.618http://dx.doi.org/10.1088/0305-4470/32/13/007http://dx.doi.org/10.1098/rspa.1990.0096http://dx.doi.org/10.1016/0375-9601(88)90905-Xhttp://dx.doi.org/10.1016/0375-9601(88)90905-X