Formation of the 255-Kev Annihilation Line Near the Magnetic Poles of Pulsars V

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JETP Letters, Vol. 79, No. 6, 2004, pp. 241Ð244. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 299Ð302. Original Russian Text Copyright © 2004 by Kontorovich, Flanchik.

Formation of the 255-keV Annihilation Line near the Magnetic Poles of Pulsars V. M. Kontorovich1, 2 and A. B. Flanchik2 1 Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002 Ukraine 2 Kharkov State University, Kharkov, 61077 Ukraine Received December 22, 2003; in ﬁnal form, February 17, 2004

The possibility of existance is discussed for pulsars emitting soft γ radiation near their magnetic poles upon the annihilation of ultrarelativistic positrons from the magnetosphere and electrons from the surface of a star. With an increase in the energy of incident positrons, the photon energy of this backward radiation tends to a constant 2 value mec /2 = 255 keV. This radiation is shown to be directed opposite to the positron ﬂux direction. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 97.60.Gb; 97.10.Ld; 98.70.Rz

1. A pulsar is a source of electromagnetic radiation and, with allowance for the general-relativity effects, over a wide range from radio waves to hard γ radiation the coefficient A is given by the formula [5, 6] 8 γ up to a photon energy of 10 MeV. As is known, rays 3ΩB ω can be emitted due to both ultrarelativistic electrons A = ------0 4---- cosαε+ cosφ sinα . (2) 2 cR Ω 0 m and positrons moving in a strong magnetic field and various collision processes between charged particles. Here, h < H is the altitude over the pulsar surface, H is In the former and latter cases, this radiation is called the gap altitude determined by the condition for gener- bend radiation and collisional bremsstrahlung, respec- ating the secondary component of eÐe+ plasma, Ω is the tively. In this work, we analyze the possibility of gener- angular velocity of the pulsar, R is the pulsar radius, B0 ating soft γ radiation caused by the annihilation line ε Ω is the magnetic field at the pulsar surface, 0 = R/c formed due to the annihilation of ultrarelativistic is the pulsar geometric factor, α is the angle between positrons from the magnetosphere and electrons from 3 the magnetic and rotation axes, ω = ΩR /r3, and φ is the pulsar surface. Since the electron velocities in the g m pulsar surface are much lower than the speed of light, the azimuth angle measured from the direction of the northern magnetic semiaxis of the pulsar. The mecha- we consider electrons to be at rest. As is known [1], nism considered for generating γ radiation can be real- charged particles in the pulsar magnetosphere move ized only if E|| < 0. According to Eq. (2), there are mag- along magnetic lines of force, because the motion in the netic lines of force on which the inequality E|| < 0 is sat- plane perpendicular to the magnetic field rapidly isfied. Such lines were previously called preferable. becomes impossible due to synchrotron radiation. Since the second term in Eq. (2) is small, preferable Positrons can be accelerated in the pulsar magneto- lines are most magnetic lines of force on which sphere by an electric field. However, this acceleration is positrons moving along them are accelerated by field not possible over the entire magnetosphere, because, in (1) toward the pulsar surface. plasma-filled regions rotating with the pulsar, the con- The mechanism of forming the 255-keV annihila- dition E á B = 0 for electric E and magnetic B fields is tion line is shown in the figure. When moving along the fulfield [2]. Positrons are accelerated toward the pulsar magnetic lines of force, ultrarelativistic positrons from the magnetosphere are incident on the pulsar surface, surface in a gap, i.e., in the region over the magnetic γ pole of the pulsar, where E á B ≠ 0 and an electric field and radiation arises due to their annihilation with electrons. Radiation in the direction of positron motion, directed along the magnetic field exists. In this work, i.e., forward radiation, carries almost the entire positron the electric field of the pulsar is treated in the Arons energy to the pulsar interior, and a soft backward γ radi- model [3, 4]. In this model, the longitudinal electric ation with an energy of 255 keV is emitted opposite to field in the gap is represented in the form the positron beam. The spectrum and angular distribution of this radiation will be discussed below. 1 2 E||()h = ---Ah()Ð Hh , (1) 2. The formation of the annihilation line is primarily 2 determined by the two- and three-photon electronÐ

242 KONTOROVICH, FLANCHIK

From these relations, the desired energies of annihilation photons are expressed as ε χ ω ()ε + Ð p+ cos 1 = + + m ------ε χ, + + m Ð p+ cos (5) ()ε ω m + + m 2 = ------ε χ. + + m Ð p+ cos 3. Photon energies (5) are functions of two parameters: the energy of an incident positron and the emission angle χ. To determine the angular distribution of annihilation photons, we use the following general formula for the differential cross section for two-photon annihilation [8]:

π 2 2  2 2 2 dσ 8 re m m m ------= ------------+ ------ds tt()Ð 4m 2 smÐ 2 umÐ 2 Mechanism of the formation of annihilation line: I is the gap (6) 2 2 2 2 region, where E|| ≠ 0; II is the region of open ﬁeld lines, m m 1smÐ umÐ  + ------+ ------Ð ------+ ------, where E|| = 0; and III is the corotation region. The annihila- 2 2 42 2 tion of a positron from the pulsar magnetosphere and a sur- smÐ umÐ umÐ smÐ  face electron is accompanied by the backward emission of the 255-keV line. where s, t, and u are the standard kinematic invariants given by Eqs. (3) and re is the classical electron radius. Let us calculate expression (6) in the laboratory frame, positron annihilation events. In this work, three-photon where the electron is at rest. Using the expressions t = ε 2 ω annihilation affecting the width and wings of the anni- 2m(m + +) and u Ð m = Ð2m 2 and differentiating the 2 ω ε | | χ hilation line [7] is disregarded. expression s Ð m = Ð2 2( + Ð p+ cos ), we obtain ω ()ε χ ω χ Let us consider the two-photon annihilation kine- ds = Ð22d 2 + Ð p+ cos + 2 p+ d cos matics in the laboratory frame, where an electron is at ∂ω rest. We use the standard kinematic invariants [8] 2 = 2 ω p Ð ------()ε Ð p cosχ d cosχ 2 + ∂χcos + + sp==()Ð k 2 ()k Ð p 2, Ð 1 2 + ()ε  m + + m = Ð p+ ------ε χ (7) ()2 ()2 + + m Ð p+ cos tp==Ð + p+ k1 + k2 , (3) ()εε ()χ m + + m + Ð p+ cos  do ()2 ()2 Ð ------ ------, up==Ð Ð k2 k1 Ð p+ , ()ε χ 2 π + + m Ð p+ cos where pÐ, p+, k1, and k2 are the 4-momenta of the elec- do = 2πχsin dχ. tron, positron, and annihilation photons, respectively. Substitution of Eq. (7) into general expression (6) gives Let us express the frequencies of annihilation photons through the angle χ between the direction of positron 2 ()ε , χ re G + motion and the photon 3-momentum k (ﬁgure). In the dσ = ------F()ε , χ do, (8) 2 2 p + laboratory frame, the electron and positron have + ε 4-momenta pÐ = (m, 0) and p+ = ( +, p+), respectively. where Therefore, from Eqs. (3) and the 4-momentum conser- ()ε ()ε , χ m + + m vation law pÐ + p+ = k1 + k2, we obtain G + = ------ε χ + + m Ð p+ cos 2 2 sp==()Ð k ()k Ð p , m()εε + m ()Ð p cosχ , Ð 1 2 + Ð ------+ + + - []()ε χ 2 + + m Ð p+ cos m2 Ð 2mω = m2 Ð 2 p k , (4) 1 + 2 ε χ ()ε , + Ð p+ cos m ω ε ω ω χ ω ()ε χ F + m = ------+ ------ε χ- (9) m 1 = + 2 Ð p+ 2 cos = 2 + Ð p+ cos . m + Ð p+ cos

JETP LETTERS Vol. 79 No. 6 2004 FORMATION OF THE 255-keV ANNIHILATION LINE 243

2m21 1 with an energy of 255 keV are emitted opposite to the + ------ω ---- + ε------χ- positron beam incident on the surface. Indeed, as was 2 m + Ð p+ cos mentioned above, a photon with momentum k2 is emitted at a small angle to the direction of positron motion. m41 1 2 Ð ------+ ------. Let us calculate the angle φ between the momenta of 2 ε χ ε ω m + Ð p+ cos annihilation photons. From the relation t = 2m(m + +) = 2 ω ω φ ε 2k1k2 = 2 1 2(1 Ð cos ), we obtain Since the denominators in expressions (9) contain + + | | χ ε | | χ mm()+ ε m Ð p+ cos and + Ð p+ cos , cross section (8) is non- φ + χ cos= 1 Ð ------ω ω -. (15) zero only for small angles . This property is a manifes- 1 2 tation of the general property (relativistic aberration) of radiation from an ultrarelativistic particle [9]. Almost In view of limiting expressions (12)Ð(14), formula (15) the entire positron energy is emitted along its direction yields of motion. It follows from Eqs. (9) that there are two χ ≤ ε small-angle regions: the region χ < m/ε , where m/(ε Ð Ð1, m/ +, + +  |p |cosχ) ӷ 1, and the region χ < m/ε , where cosφ =  m2 2m χ2 m m (16) + + 12Ð ------Ð ------Ð -----, ----- < χ ≤ -----. m/(ε + m Ð |p |cosχ) ӷ 1. ε2 χ2 ε 2 ε ε + +  + + + + (i) In the region χ < m/ε , the product + φ χ Ð1 The limiting value of cos for 0 is equal to Ð1; m2 χ2 i.e., photons move in the opposite directions. Therefore, G()ε , χ F()ε , χ = m------+ ----- (10) + + ε2 2 the annihilation line is emitted backward, i.e., opposite 2 + to the positron beam. increases with an increase in the positron energy. 5. The intensity of the 255-keV annihilation line is estimated by expressing it through the quantities refer- ε χ ε (ii) In the region m/ + < < m/ + , Eqs. (9) take the ring to the positron component of plasma. To this end, form we consider a monoenergetic positron beam incident with energy ε+ on the pulsar surface. The probability of ()ε , χ m ()ε , χ 2m ε ε σ G + ==------, F + ------. (11) positron capture per unit time is dw( +, t) = j( +, t)d γγ, ε χ2 ε χ2 ε 1 + + /2m + where j( +, t) is the ﬂux density of positrons with ε σ An increase in the product GF stops due to a decrease energy + and d γγ is the cross section for two-photon in the quantity m/(ε Ð |p |cosχ) for these angles. annihilation given by Eq. (8). Multiplying the probabil- + + ity dw(ε , t) by the photon energy ω (ε , χ) and the 4. Taking into account the expression obtained for + 1 + the angular distribution, we now analyze expressions (5) number of electrons Ne, we obtain the intensity dI of for the energies of annihilation photons in the high- 255-keV line radiation in the form χ ≤ ε χ energy limit. For m/ +, one can set = 0 in expres- dσγγ ε ӷ dI= N ω ()ε , χ j()ε , t ------do. (17) sions (5). For + m, the energy in this case takes the e 1 + + do limiting value Integrating this expression with respect to the angles, ω m one obtains the total intensity of annihilation line. We 1 ==---- 255 keV (12) 2 take into account only the contribution from the region χ ε ω ε χ and the asymptotic value is < m/ +. According to Eq. (12), 1( +, ) = m/2. Then, the integration of Eq. (17) with respect to the angles ω ≈ ε 2 +. (13) yields ε χ ≤ ε m In the angular region m/ + < m/ + , we retain ()ε , ()σε , ()ε I + t = ----Ne j + t γγ + , (18) contributions ~χ2 in the expansions of expressions (5) 2 and obtain the photons energies in the form where σγγ is the total cross section for two-photon anni- ε2 χ2 ε hilation. The total intensity of annihilation line is ω ==------+ , ω ------+ -. (14) obtained by summing contributions (18) over the entire 1 ε χ2 2 ε χ2 2m + + 1 + + /2m energy spectrum of positrons. Therefore, According to Eqs. (12) and (13), photons with energies ∞ () N m ω form the 255-keV annihilation line and γ-ray pho- tot () e ()ε , ()σε , ()εε 1 I t = ------∫ f + t j + t γγ + d +, (19) ω 2 tons with energy 2 have continuous energy spectrum ε reﬂecting the positron energy spectrum. The existence min ε of a limiting energy of 255 keV in the process under where min is the lower limit of the positron energy ε consideration is the basic result of this work. Photons spectrum and f( +, t) is the positron energy distribution

JETP LETTERS Vol. 79 No. 6 2004 244 KONTOROVICH, FLANCHIK function. We treat Ne in Eqs. (18) and (19) as the num- which can provide valuable information about the rela- ber of electrons in a layer from which γ radiation can be tion between the masses and radii of neutron stars. The emitted outside. Therefore, Ne ~ neSL, where ne is the effect of suppressing the annihilation line by a strong electron density in the pulsar surface, S is the area of magnetic field [11] weakens significantly in this case, annihilation region, and L ~ 1/µ is the thickness of this because the suppression effect is absent for photons µ σ γ layer, where ~ Cne is the scattering coefficient of propagating along the magnetic field and is very weak σ for photons propagating at small angles to the field. radiation per unit length ( C is the Compton scattering σ cross section). Therefore, Ne ~ S/ C. The intensity of We are grateful to V.S. Beskin, B.M. Morozhenko, the 255-keV annihilation line is preliminarily estimated and the referee for stimulating discussions. as follows. Let us take the positron flux density in the ε ξ Ω π form j( +, t) ~ JGJ, where JGJ = B/2 e is the Gold- REFERENCES reichÐJulian density and ξ is a certain dimensionless σ 1. V. S. Beskin, Usp. Fiz. Nauk 169, 1169 (1999) [Phys. parameter. The cross section C is estimated as the σ Ð26 2 Usp. 42, 1071 (1999)]. Thompson cross section T ~ 10 cm . In this case, the intensity of 255-keV radiation arising upon the annihi- 2. P. Goldreich and W. H. Julian, Astrophys. J. 157, 869 ε (1969). lation of a positron beam with energy + for a pulsar Ω ε 3. J. Arons and E. T. Scharlemann, Astrophys. J. 231, 854 with rotation frequency ~ 1 Hz is equal to I( +) ~ (1979). ξ m × 20 ε----- 10 erg/s. This estimate is obtained without 4. E. T. Scharlemann, W. M. Fawley, and J. Arons, Astro- + phys. J. 222, 297 (1978). γ regard for the absorption and scattering of radiation in 5. V. S. Beskin, Pis’ma Astron. Zh. 16, 665 (1990) [Sov. the pulsar magnetosphere. Astron. Lett. 16, 286 (1990)]. 6. In this work, we predict the possibility of forming 6. A. G. Muslimov and A. I. Tsygan, Astron. Zh. 67, 263 the 255-keV annihilation line in the γ spectrum of a pul- (1990) [Sov. Astron. 34, 133 (1990)]. sar. As was shown above, this line arises due to the 7. Yung-Su Tsai, Phys. Rev. 137, B730 (1965). annihilation of ultrarelativistic positrons and surface γ 8. V. B. Berestetskiœ, E. M. Lifshitz, and L. P. Pitaevskiœ, electrons. Annihilation-induced radiation must be Quantum Electrodynamics, 2nd ed. (Nauka, Moscow, characterized by a sharply anisotropic angular distribu- 1979; Pergamon Press, Oxford, 1982). γ tion. High-energy -ray photons carry almost the entire 9. L. D. Landau and E. M. Lifshitz, The Classical Theory energy of positrons to the pulsar interior. However, soft of Fields, 6th ed. (Nauka, Moscow, 1979; Pergamon γ radiation from the surface must be observed. For high Press, Oxford, 1975). energies of incident positrons, the photon energy of this 10. M. P. Ulmer, S. M. Matz, J. E. Grove, et al., Astrophys. radiation approaches 255 keV. Analysis of this line pro- J. 551, 244 (2001). vides important information about the gap structure and 11. M. G. Baring, Mon. Not. R. Astron. Soc. 262, 20 (1993). processes of particle acceleration in the gap. The annihilation line under consideration must undergo a large (about 20%) gravitational redshift (see, e.g., [10]), Translated by R. Tyapaev

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 245Ð248. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 303Ð306. Original English Text Copyright © 2004 by Belavin, Chernodub, Polikarpov.

On Projection (In)dependence of the Dual Superconductor Mechanism of Confinement¦ V. A. Belavin, M. N. Chernodub*, and M. I. Polikarpov Institute of Theoretical and Experimental Physics, Moscow, 117259 Russia *e-mail: [email protected] Received January 29, 2004; in ﬁnal form, February 19, 2004

We study the temperature dependence of the monopole condensate in different Abelian projections of the SU(2) gauge theory on the lattice. Using the Fröhlich–Marchetti monopole creation operator, we show numerically that the monopole condensate depends on the choice of the Abelian projection. © 2004 MAIK “Nauka/Inter- periodica”. PACS numbers: 12.38.Gc; 11.15.Ha

The confinement of color in QCD is one of the most Since the qualitative features of the confinement interesting issues in the modern quantum field theory. mechanism in the real QCD with the SU(3) gauge Numerical simulations of non-Abelian gauge theories group and in the SU(2) gauge theory are the same, in on the lattice show [1] that the confinement of quarks this letter, we restrict ourselves to the simplest case of happens due to the formation of the chromoelectric the SU(2) gluodynamics. The most convincing results strings spanning between quarks and antiquarks. supporting the dual superconductor scenario were Although an analytical derivation of the color confine- obtained in the so-called maximal Abelian (MA) pro- ment is not available, the physical reason of the emer- jection [6]. This gauge is defined by the maximization σ gence of the string seems to be known. According to the of the lattice functional ( i are the Pauli matrices), dual superconductor model [2], the vacuum of a non- []Ω Abelian gauge model may be regarded as media of con- maxΩ RMA U , densed Abelian monopoles. The monopole condensate (1) squeezes the chromoelectric flux (coming from the [] ()σ ()σ, µ †(), µ quarks) into a flux tube due to the dual Meissner effect. RMA U = ∑Tr 3Us 3U s , This flux tube is an analogue of the Abrikosov vortex in s, µ˜ an ordinary superconductor. with respect to the gauge transformations, U(s, µ) The basic element of the dual superconductor is the UΩ(s, µ) = Ω(s)U(s, µ)Ω†(s + µˆ ). In the continuum Abelian monopole. This object does not exist on the limit, condition (1) corresponds to the minimization of classical level in QCD, but it can be identified with a 4 1 2 2 2 particular configuration of the gluon fields with the help the functional ∫d x((Aµ ) + (Aµ ) ). A local condition of the so-called Abelian projection [3]. The Abelian of the MA gauge can be written in the form of a differ- projection uses a partial gauge fixing of the SU(N) 3 1 2 ential equation, (∂µ + ig Aµ )(Aµ Ð i Aµ ) = 0, which is gauge symmetry up to an Abelian subgroup. In the invariant under the residual U(1) gauge transforma- Abelian projection, the Abelian monopoles appear nat- ω ω urally due to compactness of the residual Abelian tions, ΩAbel(ω) = diag(ei , eÐi ), where ω is an arbitrary group. scalar function. Many numerical simulations show that the Abelian The MA gauge is a good candidate for a realization degrees of freedom in an Abelian projection are likely of the dual superconductor scenario because the MA to be responsible for the confinement of quarks (for a gauge makes the off-diagonal gluon fields as small as review, see, e.g., [4]). For example [5], the Abelian possible, reducing their role. Thus, the Abelian domi- gauge fields provide a dominant contribution to the ten- nance [5] is a natural effect in the MA gauge. Accord- sion of the fundamental chromoelectric string (“Abe- ing to numerical simulations [7Ð9], the monopole con- lian dominance”). Moreover, the internal structure of densate in the MA gauge is formed in the low-temper- the string energy, such as energy profile and the field ature (confinement) phase and it disappears in the high- distribution, are described very well by the dual super- temperature (deconfinement) phase, in perfect agree- conductor model [1]. ment with the dual superconductor scenario. Besides the MA gauge, there are also various gauges ¦ This article was submitted by the authors in English. which are defined by a diagonalization of certain SU(2)

246 BELAVIN et al. functionals X[U] with respect to the gauge transforma- mations, the diagonal field θ transforms into an Abelian tions gauge field, the off-diagonal field χ changes into a dou- ()Ω ble charged matter field, and the field φ remains intact: X []U X []ΩU ≡ † X []ΩU = diag()λ , λ . (2) x x x 1 2 θ θ ω ω xµ xµ + x Ð x + µˆ , To define an Abelian gauge, the functional X[U] must χ χ ω ω transform into an adjoint representation of the SU(2) xµ xµ ++x x + µˆ , gauge group [3]. After the Abelian projection is fixed, φ φ the matrix X[U] becomes diagonal and the theory pos- xµ xµ. sesses the (residual) U(1) gauge symmetry. The most The SU(2) plaquette action contains [11] various inter- popular examples of the diagonalization gauges are the actions between these fields as well as the action for the Abelian Polyakov (AP) gauge and the Abelian field Abelian gauge field θ: strength gauge (F12 gauge). The Polyakov gauge corre- [] β[]φ θ … sponds to the diagonalization of the Polyakov line, SU = Ð∑ P cos P + . (3) []≡ P Xx U Ux,,x 0Ux, x + 10, …Ux, x Ð 1.0, 4 4 4 θ θ θ θ θ Here, P = 1 + 2 Ð 3 Ð 4 is a lattice analogue of the where x = {x, x4}. The F12 gauge is defined by the diag- β φ ≡ Abelian field strength tensor and P[ ] is an effective onalization of the 12-plaquette, Xx[U] Ux, 12, where coupling constant dependent of the fields φ [11]. † † Ux, µν = Ux, µUx + µ˜, νUx + ν˜, µUx, ν . Following [7], we apply the monopole creation There are conflicting reports on the gauge indepen- operator of Fröhlich and Marchetti [12] to the Abelian part of non-Abelian action (3). Effectively, this operator dence of the dual superconductor mechanism of the θ color confinement (a review of the current literature on shifts the Abelian plaquette variable P as follows: this subject can be found in [10]). Needless to say, this Φ ()x question is important for understanding of the proper- mon ties of the QCD vacuum. It is natural to think that the (4) β []φ []()θ θ () () confinement—as a gauge-invariant phenomenon—can- = exp∑ P cos P + W P x Ð cos P ,  not be described by a gauge-dependent model. On the P other hand, the Abelian projection by itself can be considered as just a gauge-dependent tool to associate the where the tensor WP can be written in a compact way confining gluon configurations with the Abelian mono- with the help of the differential form formalism on the poles. This tool may work well in one gauge and may lattice [4]. In this formalism, the δÐ (dÐ) operator is the not work in another gauge. backward (forward) derivative on the lattice which Analytical considerations of [10] show that, in the decreases (increases) by one the rank of the form on AP gauge (contrary to the MA gauge), the dual super- which it is acting. The rank of the form is determined conductor mechanism cannot be realized. It was con- by the dimensionality of the lattice cell on which this cluded that the Abelian projection mechanism is pro- form is defined. For example, a scalar function is a jection-dependent. However, the projection (in)depen- 0-form, the vector function is a 1-form, etc. Suppose that A is a lattice vector, then δA is a scalar (a lattice dence of the monopole condensate could not be proven ∂ within an analytical approach. To check this issue, we analogue of the divergence, µAµ), while dA is an anti- study below the condensate of the Abelian monopoles symmetric tensor (a lattice analogue of the curl, ∂ ∆ δ δ ∆Ð1 in the finite temperature SU(2) gauge theory in the AP [µ, Aν]). The lattice Laplacian is = d + d , and and the F projections and compare it with the conden- denotes the inverse Laplacian. The lattice Kronecker 12 δ sate in the MA gauge. symbol is denoted as x: it is a scalar function, which is We study numerically the SU(2) gauge theory with equal to unity at the site x and zero otherwise. The *-operator relates the forms on the dual and original lat- β the standard Wilson action, S[U] = Ð ∑Tr UP, where tice. For example, if B is a scalar function (0-form) on P the original lattice, then *B is a 4-form on the dual the sum goes over the plaquettes P and β is the lattice lattice. bare gauge coupling related to the continuum gauge The plaquette function W in Eq. (4) is defined as fol- coupling g, β = 1/4g2. The SU(2) link field is parame- 1 πδ∆Ð1 ω lows: W = 2 (Dx Ð x). Here, the lattice 1-form trized in the standard way: ω * x represents the Dirac string attached to the mono- ω θ χ pole on the dual lattice (thus, x is a 3-form on the orig- φ i xµ φ i xµ ω cos xµe sin xµe inal lattice). The form * x is zero outside the string U µ = . x χ θ position, and it is equal to plus or minus unity (depend- φ Ði xµ φ Ði xµ Ðsin xµe cos xµe ing on the orientation) on the string. The three-dimen-

In an Abelian projection, the allowed gauge trans- 1 We omit a lengthy derivation of the tensor W and refer an inter- formations have a diagonal form. Under these transfor- ested reader to [12].

JETP LETTERS Vol. 79 No. 6 2004 ON PROJECTION (IN)DEPENDENCE OF THE DUAL SUPERCONDUCTOR MECHANISM 247 sional form Dx is called the “Dirac cloud” because it represents a lattice analogue of the radially symmetric (Coulomb) magnetic field of a monopole. This form δ δ satisfies the equation *Dx = * x, which is a lattice analogue of the Maxwell equation in continuum, divH(x) = ρ(x). Here, ρ(x) is the density of the monopoles. In our case, we introduce one monopole at the origin of the lattice; therefore, the lattice monopole ρ δ density is just a Kronecker symbol, = x. A partition function of any compact U(1) model can be rewritten as a sum over closed monopole trajectories Φ [4, 7]. The quantum average of the operator mon(x) can be represented as a sum over all closed monopole trajectories plus one open trajectory, which begins at point x. Thus, this operator does create a monopole at the point x. The operator is invariant under local gauge transformations. Another property of operator (4) is that it is iαΦ defined up to a complex phase: the operator e mon(x) also creates a monopole at the point x. For the sake of definiteness, we study the positively defined operator (4) Φ Φ ≥ with Im mon and Re mon 0. The object of our interest in this paper is the effective potential on the monopole field. According to [7], this potential is defined as follows: ()Φ 〈〉 δΦΦ()() V = Ðln Ð mon x . (5) The minimum of V(Φ) corresponds to the monopole condensate. We simulate the SU(2) on the lattice 163 × 4 with C-periodic boundary conditions in space directions [13]. The C periodicity corresponds to the antiperiodic- ity for the Abelian gauge fields, which is required by the Gauss law [7]. In the case of SU(2) gauge group, the C-periodic boundary conditions are almost trivial: on Ω+ Ω Ω σ the boundary, we have Ux, µ Ux, µ , = i 2. To get the effective potential, we use 400 configurations of the SU(2) gauge fields. On each configuration, the distribution of the monopole creation operator is evaluated at 20 points. The logarithm of the distribution provides us with effective potential (5). To evaluate the errors of the potential, we use the bootstrap method. In Fig. 1, we show the effective potential V(Φ) (Eq. (5)) at various values of the gauge coupling β in the MA, AP, and F gauges. The potential is shown for 12 Fig. 1. The potential on the monopole field in the MA, AP, positive real values of the field Φ: ReΦ ≥ 0, ImΦ = 0. β and F12 gauges at various values of the gauge coupling . The minimum of the effective potential corresponds to The data for the MA gauge are taken from [7]. the value of the monopole condensate (in lattice units). The results for the MA gauge (quoted below) are taken from [7]. The critical gauge coupling corresponding to from AP and F12 potentials. According to Fig. 1a, in the the temperature phase transition at chosen lattice geom- strong coupling limit (β = 0.1), the minima of the β ≈ etry is c 2.3. Thus, Figs. 1aÐ1c correspond to the monopole potential in all three gauges are located at the Φ ≈ confinement phase, while Fig. 1d is plotted for the same point, min 1. However, as one can see from deconfinement phase. other figures, this coincidence is lifted as we increase β. First of all, we note that, for all considered values of β, As we increase the value of β, the difference in the (i) the minima of the potentials in the AP and F12 monopole condensates in the MA gauge and in the AP gauges coincide with each other within numerical and F12 gauges appears evident (Figs. 1b, 1c). More- errors and (ii) the potential in the MA gauge is different over, in the deep deconfinement phase (Fig. 1d), the

JETP LETTERS Vol. 79 No. 6 2004 248 BELAVIN et al. monopole condensate vanishes in the MA gauge, while 2. G.’t Hooft, in High Energy Physics: EPS International in the AP and F12 gauges, the condensate is nonzero. Conference, Ed. by A. Zichichi (Palermo, 1975); S. Mandelstam, Phys. Rep. 23, 245 (1976). Summarizing, we have presented evidence that the 3. G.‘t Hooft, Nucl. Phys. B 190, 455 (1981). monopole condensates in different Abelian projections 4. M. N. Chernodub and M. I. Polikarpov, in Conﬁnement, coincide with each other only in the (unphysical) strong Duality, and Nonperturbative Aspects of QCD, Ed. by coupling region. Generally, the condensate depends on P. van Baal (Plenum, New York, 1998), p. 387; hep- the choice of the Abelian projection. Our results are in th/9710205. contradiction with conclusions of [14], where conden- 5. T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42, 4257 sate was found to be projection-independent. Since our (1990); H. Shiba and T. Suzuki, Phys. Lett. B 333, 461 results are based on the well-justiﬁed theory of the (1994); G. S. Bali, V. Bornyakov, M. Müller-Preussker, Fröhlich–Marchetti operator, we suggest that the differ- and K. Schilling, Phys. Rev. D 54, 2863 (1996). ence between the results of our paper and [14] origi- 6. A. S. Kronfeld, M. L. Laursen, G. Schierholz, and nates in the improper choice of the monopole creation U. J. Wiese, Phys. Lett. B 198, 516 (1987); A. S. Kron- operator in [14]. feld, G. Schierholz, and U. J. Wiese, Nucl. Phys. B 293, 461 (1987). This work was supported by the Russian Foundation 7. M. N. Chernodub, M. I. Polikarpov, and A. I. Veselov, for Basic Research (grant nos. 02-02-17308 and 01-02- Phys. Lett. B 399, 267 (1997); Nucl. Phys. (Proc. Suppl.) 17456), by RFBRÐDFG (grant no. 03-02-04016), by 49, 307 (1996). INTAS (grant no. 00-00111), by CRDF award RPI 8. V. A. Belavin, M. N. Chernodub, and M. I. Polikarpov, (grant no. 2364-MO-02), and by MK (grant Pis’ma Zh. Éksp. Teor. Fiz. 75, 263 (2002) [JETP Lett. no. 4019.2004.2). 75, 217 (2002)]; Phys. Lett. B 554, 146 (2003). 9. A. Di Giacomo and G. Paffuti, Phys. Rev. D 56, 6816 (1997). REFERENCES 10. M. N. Chernodub, hep-lat/0308031. 11. M. N. Chernodub, M. I. Polikarpov, and A. I. Veselov, 1. V. Singh, D. A. Browne, and R. W. Haymaker, Phys. Phys. Lett. B 342, 303 (1995). Lett. B 306, 115 (1993); G. S. Bali, K. Schilling, and C. Schlichter, Phys. Rev. D 51, 5165 (1995); G. S. Bali, 12. J. Fröhlich and P. A. Marchetti, Commun. Math. Phys. C. Schlichter, and K. Schilling, Prog. Theor. Phys. 112, 343 (1987). Suppl. 131, 645 (1998); F. V. Gubarev, E. M. Ilgenfritz, 13. U. J. Wiese, Nucl. Phys. B 375, 45 (1992). M. I. Polikarpov, and T. Suzuki, Phys. Lett. B 468, 134 14. A. Di Giacomo, B. Lucini, L. Montesi, and G. Paffuti, (1999); Y. Koma, M. Koma, E. M. Ilgenfritz, and Phys. Rev. D 61, 034503 (2000); Phys. Rev. D 61, T. Suzuki, hep-lat/0308008. 034504 (2000).

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 249Ð252. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 307Ð310. Original English Text Copyright © 2004 by Machulin, Tolokonnikov.

Neutrino Flavor Relaxation or Neutrino Oscillations?¦ I. N. Machulin and S. V. Tolokonnikov Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia e-mail: [email protected], [email protected] Received December 22, 2003; in ﬁnal form, January 17, 2004

We propose the new mechanism of neutrino flavor relaxation to explain the experimentally observed changes of initial neutrino flavor fluxes. The test of neutrino relaxation hypothesis is presented using the data of modern reactor, solar, and accelerator experiments. The final choice between the standard neutrino oscillations and the proposed neutrino flavor relaxation model can be done in future experiments. © 2004 MAIK “Nauka/Interpe- riodica”. PACS numbers: 14.60.Pq; 13.15.+q

Now, the phenomena of changes in initial neutrino The time evolution of the neutrino states are gov- flavor flux are observed in different neutrino experi- erned by the Schrödinger equation: ments. The SNO experiment [1] evidently detects only d one-third of the initial electron neutrino flux from the i-----|〉ν()t = Hˆ ()t |〉ν()t , (1) Sun and two-thirds of muon and (or) tau neutrino dt fluxes. The KamLAND reactor experiment [2] detected only 61% of expected electron antineutrino events from where |ν(t)〉 is the neutrino vector of state and H(t) is the different reactors at a mean distance of 180 km. Con- time-dependent Hamiltonian of the system, whose vincing evidence of initial neutrino flavor flux changes form depends on in what basis it is given. is also observed in Super-K [3] and MACRO [4] atmo- In flavor basis, the total Hamiltonian in the random spheric neutrino data and the K2K [5] accelerator fluctuating vacuum field for rest reference frame can be experiment with muon neutrinos. written as The standard way of interpreting these results lies in ˆ () ˆ ˆ () the neutrino oscillation hypothesis, first proposed by H f t = Hm + V f t , (2) Pontecorvo [6] and developed in further works [7]. where Here, we discuss the alternative mechanism of neu-  trino flavor relaxation, which can also describe the m 00 observed changes of initial neutrino flavor fluxes with 1 Hˆ m = 0 m 0 , distance. 2 00m The proposed model is similar to the mechanism [8] 3 of spin relaxation in random fluctuating magnetic field (3) 〈 〉 () () () B with zero average B(t) = 0 and mean square fluctu- V ee t V eµ t V eτ t 〈 2 〉 ≠ ˆ () ating field value B (t) 0. The spin relaxation process V f t = V µ ()t V µµ()t V µτ()t . e is described by the Pauli master equation [9]. () () () V τe t V τµ t V ττ t Let us assume the existence of some small random

fluctuating vacuum field Vˆ , causing the transitions Here, Hm is the free Hamiltonian in mass basis. Note ˆ that, in this model, the neutrino flavor states are between different lepton flavors. The field V can have assumed to be the same as their massive states (or, in 2 mean zero value, with 〈〉Vˆ ()t different from 0 and Γ = other words, the mixing mass matrix is diagonal). 2 1//2 α µ τ 〈〉Vˆ ()t > mν , mν , mν . Interactions of neutrino V αα' ( = e, , ) is the vacuum field potential with e µ τ 〈〉() 〈〉()()2 ≠ flavors with such vacuum field should lead to flavor mean value V αα' t = 0 and V αα' t 0. relaxation process. The neutrino flavor evolution in time can be ¦ This article was submitted by the authors in English. obtained from Eqs. (1)Ð(3), using the density matrix

250 MACHULIN, TOLOKONNIKOV approach [10], and is given by the Pauli master equa- For γ = Eν/mν ӷ 1, the probability P of observing the tion: neutrino flavor with energy Eν at distance r from the source is given by  nν e d  ----- Pν ()r nνµ e 1/3 0 dt Ðt/Λ E () = + be 0 ν nν Pνµ r 1/3 1 τ  Pν ()r 1/3 Ð1 τ Ð()Wµ + Wτ W µ W τ (8) e e e e  = () (4) Wµe Ð Weµ + Wτµ Wµτ Ð1 Λ Ðr/ 1Eν () + c1/2 e , Wτe Wτµ Ð Weτ + Wµτ  1/2  nν e Ð1  where we use the new notation Λ = (2Wµτmν) and × , 0 nνµ Λ Ð1  1 = 2/3[(Weµ + Weτ)mν] . nν τ Equation (8) can be solved for different initial α ν ν ν experimental conditions of neutrino flavor fluxes. where nα(t) ( = e, µ, τ) are the probability of observ- For initial pure electron neutrino or antineutrino flux ing a neutrino with electron, muon, or tau flavor, Wαα' (the case of solar or reactor experiments), Pν ()0 = 1, corresponds to the neutrino transition rates from flavor e α α () () ' to flavor , and Pνµ 0 = Pντ 0 = 0, the solution is

() Λ ∑ nα t = 1. Ðr/ 1Eν Pν ()r =,1/3[] 1+ 2e αν,,ν ν e = e µ τ Λ The general solution of Eq. (4) is given by the sum Ðr/ 1Eν Pν ()r = 1/3[] 1 Ð e , (9) of two exponents and a constant: µ

Λ   Ðr/ 1Eν nν ()t Pν ()r = 1/3[] 1 Ð e . e a1 b1 c1 τ Ðt/T1 Ðt/T2 nν ()t = a ++b e c e , (5) µ 2 2 2 From KamLAND experiment data [2], it is possible to estimate the value of parameter Λ . Taking the effec- nν ()t a3 b3 c3 1 τ tive energy of reactor antineutrino ﬂux to be equal to with 4.8 MeV (here, we take into account the threshold of the KamLAND detector Eν > 3.4 MeV), mean reactor 3 3 3 ˜ e

∑ a ===1, ∑ b 0, ∑ c 0. (6) distance 180 km, and R = Pν˜ () /Pν˜ () = i i i e measured e expected i = 1 i = 1 i = 1 0.61, we obtain To illustrate the possibilities of the relaxation Λ = 43 km/MeV. model, the particular “simple” solution can be found 1 under the following assumptions: Figure 1 illustrates the distance dependence of reac- (a) Weµ = Wµe, Weτ = Wτe, Wµτ = Wτµ, tor antineutrino fluxes (for Eν˜ = 4.8 MeV). (b) Wµτ ӷ W µ, W τ. e e The final flavor survival probability 1/3 is consistent Now, Eq. (5) appears as with the SNO [1] experimental data with the ratio of measured neutrino flavor fluxes  nν ()t  e 1/3 0 Ð2Wµτt CC ± ()± () nν ()t = 1/3 + b1 e ------= 0.306 0.026 stat 0.024 syst , (10) µ  NC () 1/3 Ð1 nντ t (7) and with the absence of distortion in measured neutrino  spectrum at low energies, while such distortion is pre- Ð1 dicted by LMA solution [11]. Ð3/2()W µ + W τ t + ce e e . It also agrees with the Homestake results [12] for the 1/2 1/2 ratio R of observed and predicted by standard solar model (SSM) [13] neutrino rates: R = 0.34 ± 0.03.

JETP LETTERS Vol. 79 No. 6 2004 NEUTRINO FLAVOR RELAXATION OR NEUTRINO OSCILLATIONS? 251

Fig. 1. Probability of observing different ﬂavors from a Fig. 2. Probability of observing different ﬂavors from an reactor for Eν˜ = 4.8 MeV. accelerator muon neutrino beam for Eν = 1.3 GeV.

The deviation from the “simple” relaxation model neutrino flavor fluxes on the distance is described by the exists for Gallium experiments [14, 15] with R = sum of a constant and two relaxation exponents, instead 0.553 ± 0.034. But it is worth mentioning that the R of the oscillation case. Figure 3 illustrates this differ- value depends on the accuracy of SSM theoretical pre- ence for reactor antineutrino experiments. dictions. For the case of pure initial muon neutrino or Forthcoming reactor and accelerator neutrino exper- antineutrino flux (like in accelerator experiments), iments can provide the data necessary to choose between the neutrino oscillations and the proposed fla- Pν ()0 = 1, Pν ()0 = Pν ()0 = 0, the solution of Eq. (8) µ e τ vor relaxation model. Of course, the possibility of hav- is ing a mixture of neutrino oscillation and relaxation pro- Λ cesses also exists. Ðr/ 1Eν Pν ()r = 1/3[] 1 Ð e , e Λ Λ () Ðr/ 0Eν Ðr/ 1Eν Pνµ r = 1/3++ 1/2e 1/6e , (11)

Λ Λ () Ðr/ 0Eν Ðr/ 1Eν Pντ r = 1/3Ð 1/2e + 1/6e . Λ Preliminary estimation of parameter 0 can be done using K2K accelerator experiment data [5]. Taking the mean energy of muon neutrino flux Eν = 1.3 GeV, distance r = 250 km, CHOOZ RP==νµ()measured /Pνµ()expected 0.70 KamLAND we obtain Λ 0 = 0.21 km/MeV. Figure 2 illustrates the distance dependence of neutrino fluxes from muon neutrino beam (Eν = 1.3 GeV). It is necessary to note that the estimations of neu- Λ Λ trino relaxation parameters 0 and 1 are very preliminary and were made for illustration of the relaxation ν˜ Fig. 3. The ratio of measured to expected e flux from model. More precise calculations can be done also tak- KamLAND [2] and CHOOZ [16] reactor experiments: (dot- ing into account atmospheric neutrino data. ted curve) predictions of the oscillation model with 2 Finally, we would like to emphasize the main differ- sin 2θ = 0.91 and δm2 = 6.9 × 10Ð5 (eV)2 best-fit parame- ence between the standard oscillation theory and the ters from [2]; (solid curve) predictions of the “simple” relaxation model proposed here: the dependence of relaxation model.

JETP LETTERS Vol. 79 No. 6 2004 252 MACHULIN, TOLOKONNIKOV

The authors would like to thank Academician 8. C. P. Slichter, Principles of Magnetic Resonance, 3rd ed. S.T. Belyaev and Prof. M.I. Katsnelson for fruitful dis- (Springer, Berlin, 1989; Mir, Moscow, 1967); M. H. Le- cussions. vitt, Spin Dynamics: Basics of Nuclear Magnetic Reso- nance (Wiley, Chichester, 2001). 9. W. Pauli, Collected Scientiﬁc Papers, Ed. by R. Kronig REFERENCES and V. F. Weisskopf (Interscience, New York, 1964), 1. Q. R. Ahmad, R. C. Allen, T. C. Andersen, et al. (SNO Vol. 1. Collab.), Phys. Rev. Lett. 89, 011301 (2002); S. N. Ah- 10. K. Blum, Density Matrix Theory and Its Applications med, A. E. Anthony, E. W. Beier, et al. (SNO Collab.), (Plenum, New York, 1981; Mir, Moscow, 1983). nucl-ex/0309004 (2003). 11. A. Y. Smirnov, hep-ph/0311259 (2003). 2. K. Eguchi, S. Enomoto, K. Furuno, et al. (KamLAND 12. B. T. Cleveland, T. Daily, R. Davis, et al., Astrophys. J. Collab.), Phys. Rev. Lett. 90, 021802 (2003). 496, 505 (1998). 3. Y. Fukida, T. Hayakawa, E. Ichihara, et al. (SK Collab.), 13. J. N. Bahcal, M. H. Pinsonneault, S. Basu, et al., Astro- Phys. Rev. Lett. 81, 1562 (1998). phys. J. 555, 990 (2001). 4. M. Ambrosio, R. Antolini, G. Auriemma, et al. (MACRO 14. J. N. Abdurashitov, V. N. Gavrin, S. V. Girin, et al. Collab.), Phys. Lett. B 517, 59 (2001). (SAGE Collab.), Phys. Rev. C 60, 055801 (1999); 5. M. H. Ahn, S. Aoki, H. Bhang, et al. (K2K Collab.), J. N. Abdurashitov, V. N. Gavrin, S. V. Girin, et al., Nucl. Phys. Rev. Lett. 90, 041801 (2003). Phys. (Proc. Suppl.) 110, 315 (2002). 6. B. Pontecorvo, Zh. Éksp. Teor. Fiz. 34, 247 (1957) [Sov. 15. M. Altmann, M. Balata, P. Belli, et al. (GNO Collab.), Phys. JETP 7, 172 (1958)]. Phys. Lett. B 490, 16 (2000); M. Altmann, M. Balata, 7. Z. Maki, M. Nakagava, and S. Sakata, Prog. Theor. Phys. P. Belli, et al., Nucl. Phys. (Proc. Suppl.) 91, 44 (2001). 28, 870 (1962); B. Pontecorvo, Zh. Éksp. Teor. Fiz. 53, 16. M. Apollonio, A. Baldini, C. Bemporad, et al. (CHOOZ 1717 (1967) [Sov. Phys. JETP 26, 984 (1968)]. Collab.), Eur. Phys. J. C 27, 331 (2003).

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 253Ð256. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 311Ð314. Original Russian Text Copyright © 2004 by Rodionov, Chirkin.

Entangled Photon States in Consecutive Nonlinear Optical Interactions A. V. Rodionov and A. S. Chirkin* Faculty of Physics, Moscow State University, Vorob’evy gory, Moscow, 119992 Russia *e-mail: [email protected]; [email protected] Received January 27, 2004

Quantum theory of two consecutive light-wave parametric interactions with aliquant frequencies produced by a common pump wave in a crystal is developed. Using the differentiation method, the unitary evolution operator of the system is reduced to an ordered form that allows the calculation of the ﬁeld state and the statistical characteristics of interacting waves. It is shown that, for the initial vacuum ﬁeld state, the created photons obey the super-Poisson statistics at the interacting frequencies and are in a multiparticle entangled state. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 42.50.Dv; 42.65.Ky; 03.67.Mn; 03.65.Ud

The object of this letter is to develop the quantum parametric process in the field of low-frequency pump- theory of consecutive nonlinear optical interactions of ing waves with aliquant frequencies and analyze the states ω ω ω of the generated photons. It will be shown that the type p = 1 + 2 (1) of interaction considered in this work can be the source of multiparticle (multiphoton) entangled states. and the parametric frequency summation In quantum optics, one distinguishes between the ω ω ω entanglement of individual photons [1], quadrature 1 + p = 3. (2) field components [2], and Stokes parameters [3] char- acterizing the polarization state. Entangled quantum Processes (1) and (2) can simultaneously proceed states are finding use both in the experiments on check- during the collinear wave interaction in periodically ing the quantum mechanical principles and in various poled nonlinear crystals, e.g., in an LiNbO3 crystal applications of quantum information science [4]. [13]. In such crystals, the condition for the efficient energy exchange between the interacting waves (quasi- At present, two-photon entangled states arising in phase-matching condition) can simultaneously be met the course of spontaneous parametric down-conversion for processes (1) and (2) by choosing the poling period, (SPDC) in homogeneous nonlinear crystals lie at the i.e., the modulation period for the nonlinear wave-cou- basis of many applications of entangled states [5]. At pling coefficient [14]. the same time, for a number of reasons, the multiphoton entangled states are of special interest for applications. In the approximation of an undepleted pump field, For example, when checking the Bell’s inequalities [6] the simultaneous occurrence of processes (1) and (2) with two-particle entangled states, the contradiction can be described by the interaction Hamiltonian of the with local realism takes place only for statistical predic- form tions [7], whereas, in the case of multiparticle entangle- ប{}β ()β+ + ()+ + ment, it arises for determinate predictions [8]. By now, Hint = i 1 a1 a2 Ð a1a2 + 2 a3 a1 Ð a2a1 , (3) the methods of creating three- [9] and four-photon [10] entangled states using two independent pairs of polar- + ization-entangled photon states and four-photon entan- where a j (aj) is the creation (annihilation) operator for ω ប gled states arising as a result of the action of a high- a photon with frequency j, is the Planck’s constant, β intensity short pump pulse in the course of SPDC have and j is the effective nonlinear coefficient proportional been suggested [11], together with the method of creat- to the crystal quadratic susceptibility and pump-wave ing multiphoton entanglement in optical solitons [12]. + amplitude. The boson operators ak and aj satisfy the In this work, a new source of multiphoton entangled + δ δ states is considered based on two consecutive three-fre- standard commutation rules [aj, ak ] = jk, where jk is quency interactions of waves with aliquant frequencies Kronecker delta. The terms containing the coefficient ω ω ω β 1, 2, and 3 in the field of intense classical pumping 1 in Eq. (3) account for process (1), and the terms con- ω β with frequency p. These interactions consist of the taining the coefficient 2 account for process (2).

254 RODIONOV, CHIRKIN

The field state |Ψ(z)〉 at the output of a nonlinear θθζ αζ() sinh crystal is determined by the unitary evolution operator = arctanh------2, ≡ ប coshθζÐ ε U(z) exp[ÐiHintz/ ]: Ð1 Ð2 |〉Ψ()z = Uz()Ψ|〉()0 , (4) ηζ()= ε arcsinhεθ ()1 Ð coshθζ , (9) where |Ψ(0)〉 is the initial field state. ζ cosh[]α()x ξζ()= ------dx, However, the general form of a direct action of the ∫ cosh[]εη()x unitary operator on the initial state cannot be calculated 0 without the preliminary transformation of this operator. For this reason, we introduce the anti-Hermitian opera- where θ ≡ 1 Ð ε2 . The particular form of the function tors ξ(ζ) is immaterial to our analysis. σ = ()a+a+ Ð a a , Next, we apply the ordering procedure to each expo- d 1 2 1 2 nential operator in Eq. (7). The ordered form of the α ξ σ σ ε()+ + operator exp[ ( ) d] is presented, e.g., in [17]. As a u = a3 a1 Ð a3a1 , (5) result, we obtain the following final ordered expression σ ε()+ + for the unitary evolution operator: a = Ð a2 a3 Ð a2a3 , ()ζ []() α + + ε β β U = exp tanh a1 a2 where the coefficient = 2/ 1 is chosen to be real with- σ out loss of generality. The auxiliary operator a for- ×αexp[]Ðln()cosh()()1 ++a+a a+a mally describes the generation process for frequencies 1 1 2 2 ω ω ω ω 2 and 3 in a pump field with frequency 2 + 3. The × []α()εη + + []() σ σ σ exp Ðtanh a2 a3 exp Ðtanh a1a2 operators d, u, and a obey the SH(3) algebra [15]; i.e., they form a closed set with respect to the commu- ×εηexp[]tanh()tanh() αa+a tation operations: 3 1 []σσ , σ ≡σσ σ σ ×εηexp[]Ðln()cosh()()1 ++a+a a+a (10) d u d u Ð u d = a, 2 2 3 3 (6) []σσ , σ []σ , σ ε2σ ×εξ[]()+ a d ==Ð u, u a d. exp tanh a3 a1 ×εη[]εη() []()() εξ In the new notation, the unitary operator can be rep- exp tanh a2a3 exp tanh tan a1a2 resented in two equivalent forms: ×εξ[]()()()+ + ζσ()+ σ exp Ðln cos a1 a1 Ð a3 a3 U()ζ = e d u (7) []ηζαζ()σ []ξζ()σ []()σ ×εξexp[]Ðtan()a+a , = exp d exp a exp u , 1 3 ζ ≡ β α ζ η ζ ξ ζ where α ≡ α(ζ), η ≡ η(ζ), and ξ ≡ ξ(ζ). where 1z and ( ), ( ), and ( ) are the unknown functions to be determined by the differentiation Assume that only the intense pump wave is fed into ω ω method [16]. In this method, the differentiation of both a nonlinear crystal, while the initial states of the 1, 2, ω sides of the equality with respect to ζ is followed by the and 3 fields are the vacuum states; i.e., the initial state ζ |Ψ 〉 | 〉 | 〉 | 〉 ≡ | 〉 operator transformations of U( ) to set it off on the vector is (0) = 0 1 0 2 0 3 0, 0, 0 . According to right-hand side of the equality and by equating the coef- Eq. (10), the state at the crystal output is σ σ σ ficients of operators d, a, and u. Note that the ∞ sequence of exponential operators on the right-hand 1 1 m |〉Ψζ() = ------∑ ()tanhαζ() side of Eq. (7) is dictated by the convenience of solving coshαζ()cosh()εη() ζ the problem. mn, = 0 (11) To determine the functions α(ζ), η(ζ), and ξ(ζ), we tanh()εη() ζ n ()mn+ ! × Ð------|〉mm, + nn, . obtain the following system of nonlinear differential coshαζ() m!n! equations: ω Here, m is the number of photons with frequency 1, α'()ζ Ð1, εξ'()ζ sinh() ηζ()ε = ω m + n is the number of photons with frequency 2, and ω η'()ζ cosh()ξαζ()+ '()ζ cosh()αζηζ()ε sinh()() = 0, (8) n is the number of photons with frequency 3. The states |m, m + n, n〉 form the orthogonal basis in the η'()ζ sinh()ξαζ() + '()ζ cosh()αζηζ()ε cosh()() = 1 space of Fock’s states. In fact, Eq. (11) fully determines α η ξ the solution to our problem. Using state vector (11), one with the initial conditions (0) = (0) = (0) = 0. can write the expression for the density matrix ρ = One can verify that the solution to Eq. (8) for ε < 1 |Ψ(ζ)〉〈Ψ(ζ)| and calculate the statistical characteristics is given by the functions of the generated radiation at the crystal output.

JETP LETTERS Vol. 79 No. 6 2004 ENTANGLED PHOTON STATES 255

The photon-number distribution functions Pj(N) = ρ ˆ ˆ Sp( j N ), where N is the photon-number operator and ρ ω j is the density matrix of the field j, are given by the expressions 2N ()αζ() () tanh P1 N = ------, cosh2 ()εη() ζ C () 1 jk P2 N = ------cosh2 ()αζ()cosh2 () εηζ() (12) 2 ()εη() ζ N × tanh 2N ()αζ() 1 + ------tanh , sinh2 ()αζ() 2N ()εη() ζ () tanh P3 N = ------. cosh2 ()εη() ζ It follows that the statistics of generated photons has, in the general case, a super-Poisson character (the photon-number variance is much greater than the mean Fig. 1. Probability amplitudes Cjk of two-photon generation ω ω photon number). However, for small interaction lengths at frequency 2 and j and k photons at frequencies 1 and β ω ( 1z < 1), it is close to the Poisson statistics. 3, respectively, as functions of the reduced interaction ζ β ε β β One has for the mean photon numbers length = 1z for = 2/ 1 = (a) 0.5 and (b) 0.75. 〈〉()ζ 2 αζ()2 () εηζ() n1 = sinh cosh , The coefficients qij are determined from Eq. (11), and 〈〉()ζ 2 () εηζ() ᏺ2 ≡ 2 ()ζ 2 ()ζ 2 ()ζ Ð1 n3 = sinh , (13) (q20 + q11 + q02 ) is the normalization factor. For the frequencies ω and ω , we have an ana- 〈〉()ζ 〈〉()ζ 〈〉()ζ 1 3 n2 = n1 + n3 . logue of a three-level system for this set of photon num- Taking into account Eq. (9), one obtains bers. Such field states can probably be used as qutrits. An alternative method of obtaining qutrits is considered 〈〉θn ()ζ = Ð2 sinh2 ()θζ , in [19]. 1 ζ ᏺ ζ (14) The probability amplitudes Cjk( ) = qjk( ) are 〈〉ε()ζ 2θÐ2()()θζ 2 n3 = coshÐ 1 . shown in Fig. 1 as functions of the interaction length. One can see that the probability amplitudes for the field According to Eq. (13), the mean number of photons configurations considered differ only slightly at some increases exponentially at ε < 1 (θ is real). In this case, ζ β β interaction lengths. Estimates show that the value = 1 > 2; i.e., the pump-photon decay into the photons β z = 1 in an LiNbO crystal of length 1 cm is achieved ω ω 1 3 with frequencies 1 and 2 is more efficient than the 6 2 ω for the pump intensity I ~ 10 W/cm (e ee interac- photon creation at frequency 3. In the opposite case tion type). β β ε ( 1 < 2, > 1), the hyperbolic functions in Eqs. (9) ω ω ω To characterize the coupling of the 1, 2, and 3 should be replaced by the trigonometric functions photons, the three-photon correlation coefficient (sinh sin, cosh cos, and θ2 (ε2 Ð 1)), 〈〉 while the statistical parameters depend on the interac- ()3 (),, ≡ n1n2n3 g n1 n2 n3 ------〈〉〈〉〈〉 tion length in an oscillatory manner. n1 n2 n3 According to Eq. (11), the field at the crystal output 2 2 (16) 22[]+ 3sinh αζ()+ tanh () εηζ() is in the entangled state [18], because it cannot be rep- = ------|Ψ〉 ≠ |Ψ 〉 ⊗ |Ψ 〉 ⊗ 2 2 resented in the factorized form 1 2 sinh αζ()+ tanh () εηζ() |Ψ 〉 |Ψ 〉 ω 3 , where j is the field state at frequency j. was calculated, which can be measured in a triple-coin- Assume that one can set off, e.g., a field with two pho- (3) ω cidence scheme. The typical dependence of g on the tons with frequency 2. Then, the normalized vector of interaction length is shown in Fig. 2. It follows from the corresponding state is this figure that there is a strong nonclassical correlation ω ω ω |〉Ψζ() = ᏺ{q ()ζ |〉220,, between the 1, 2, and 3 photons at the initial inter- cond 20 (15) action stage, and this correlation weakens as the inter- ()ζ |〉,, ()ζ |〉,, } (3) + q11 121+ q02 022 . action length increases. Indeed, g = 1 for the coherent

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2. M. D. Reid and P. D. Drummond, Phys. Rev. Lett. 60, 2731 (1988); M. D. Reid, Phys. Rev. A 40, 913 (1989). 40 3. N. Korolkova, G. Leuchs, R. Loudon, et al., Phys. Rev. A 65, 052306 (2002); R. Schnabel, W. P. Bowen, N. Treps, et al., Phys. Rev. A 67, 012316 (2003). 4. The Physics of Quantum Information: Quantum Cryp- tography, Quantum Teleportation, Quantum Computa- tion, Ed. by D. Bouwmeester, A. K. Ekert, and A. Zeilinger (Springer, Berlin, 2000; Postmarket, Mos- cow, 2002). 5. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980). 6. J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).

(3) 7. S. J. Freedman and J. S. Clauser, Phys. Rev. Lett. 28, 938 Fig. 2. Three-photon correlation coefficient g (n1, n2, n3) (1972). ζ β as a function of the reduced interaction length = 1z for ε β β 8. D. M. Greenberger, M. A. Horne, A. Shimony, et al., Am. = 2/ 1 = 0.5. J. Phys. 58, 1131 (1990); D. M. Greenberger, M. A. Horne, and A. Zeilinger, Phys. Today 46 (8), 22 (1993). and statistically independent fields. At the same time, 9. D. Bouwmeester, J.-W. Pan, M. Daniell, et al., Phys. Rev. Lett. 82, 1345 (1999). g(3) = 6 for the fully statistically dependent fields obeing Gaussian statistics [20]. With an increase in the interac- 10. J.-W. Pan, M. Daniell, S. Gasparoni, et al., Phys. Rev. tion length, the coefficient g(3) approaches this value Lett. 86, 4435 (2001). from its larger values. 11. M. Eibl, S. Gaertner, M. Bourennane, et al., Phys. Rev. Lett. 90, 200403 (2003). In conclusion, it should be emphasized that the main result of our theory consists in the derivation of the 12. N. Korolkova and G. Leuchs, in Coherence and Statis- ordered expression for unitary evolution operator (10) tics of Photons and Atoms, Ed. by J. Perina (Wiley, New York, 2001), p. 111. describing consecutive nonlinear optical interactions with aliquant frequencies and in the derivation of 13. E. Yu. Morozov and A. S. Chirkin, J. Opt. A: Pure Appl. Eq. (11) for the field-state vector at the output of a non- Opt. 5, 233 (2003). linear crystal with the vacuum initial state. Equation (11) 14. A. S. Chirkin, V. V. Volkov, G. D. Laptev, et al., Kvan- clearly demonstrates that the created photons are in the tovaya Élektron. (Moscow) 30, 847 (2000). entangled state. For the photon numbers m and M at, 15. N. Ya. Vilenkin, Special Functions and the Theory of ω ω respectively, frequencies 1 and 2, n photons are gen- Group Representations (Nauka, Moscow, 1965; Ameri- ω can Mathematical Society, Providence, R.I., 1968). erated at the frequency 3, with n = M Ð m, and the numbers m and n can take values from 0 to M. Thus, the 16. J. Wei and E. Norman, J. Math. Phys. 4, 575 (1963). nonlinear optical interactions considered in this work 17. M. J. Collett, Phys. Rev. A 38, 2233 (1988). can serve as a source of multiphoton entangled states. 18. I. V. Bargatin, B. A. Grishanin, and V. N. Zadkov, Usp. We are grateful to S.P. Kulik for helpful discussion Fiz. Nauk 171, 625 (2001) [Phys. Usp. 44, 597 (2001)]. of the results and to V.I. Man’ko for the fruitful discus- 19. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, sion on the group theory. This work was supported by et al., Phys. Rev. A 60, R4209 (1999); L. A. Krivitskiœ, the Russian Foundation for Basic Research (project S. P. Kulik, A. N. Penin, et al., Zh. Éksp. Teor. Fiz. 124, no. 04-02-17554) and INTAS (grant no. 01-2097). 943 (2003) [JETP 97, 846 (2003)]. 20. S. A. Akhmanov, Yu. E. D’yakov, and A. S. Chirkin, Introduction to Statistical Radio Physics and Optics REFERENCES (Nauka, Moscow, 1981). 1. L. Mandel and E. Wolf, Optical Coherence and Quan- tum Optics (Cambridge Univ. Press, Cambridge, 1995; Nauka, Moscow, 2000). Translated by V. Sakun

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JETP Letters, Vol. 79, No. 6, 2004, pp. 257Ð261. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 315Ð319. Original Russian Text Copyright © 2004 by Prischepa, Shabanov, Zyryanov.

Transformation of Director Configuration upon Changing Boundary Conditions in Droplets of Nematic Liquid Crystal O. O. Prischepa, A. V. Shabanov, and V. Ya. Zyryanov* Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia *e-mail: [email protected] Received December 31, 2003; in ﬁnal form, February 12, 2004

The transformation of the director conﬁguration upon changing boundary conditions from planar to homeotropic for bipolar nematic droplets dispersed in a polymeric matrix was studied. The characteristic textural patterns are presented for droplets with different concentrations of homeotropic surfactants, and their orientational structure is identiﬁed. The scenario predicted earlier by G.E. Volovik and O.D. Lavrentovich (ZhETF 85, 1997 (1983)) for the transformation of orientational structure of nematic droplets from bipolar to radial, without the formation of additional disclinations, is revealed. It is shown that, by using the computational method of minimization of the director elastic-strain energy in the droplet bulk and by introducing the inhomogeneous boundary conditions, one can obtain orientational structures that are analogous to the observed ones.© 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 61.30.Gd; 61.30.Eb

INTRODUCTION could be varied from homeotropic to planar by varying temperature in the range of nematic phase. In this case, Topological analysis is a rather efficient tool for the second structure-transformation scenario was real- studying spatially inhomogeneous structures in orienta- ized with the formation of additional disclinations. tionally ordered systems, including various director and However, considering that this composite consists of disclination configurations in liquid crystals (LCs). The spherical LC droplets dispersed in an isotropic liquid classification of the topologically stable defects in nem- matrix [2], one cannot assert that the observed scenario atic LC droplets has shown [1, 2] that the planar (tan- for the interconversion of orientational structures is gential) orientation of nematic molecules at the inter- universal for other objects as well, e.g., for the films of face is characterized by the bipolar direction configura- polymer-dispersed liquid crystal (PDLC) films that tion, with two point defects (boojums) arranged at the have been intensively studied in the last years. At the opposite sides of the droplet surface. In the case of a same time, a change in the boundary conditions by normal (homeotropic) nematic anchoring, the radial varying temperature is not the only approach to the director ordering with a point defect (hedgehog) at the study of this problem; a direct method is also possible. droplet center is the equilibrium structure, in accor- Namely, one can fabricate a set of samples with differ- dance with the experimental observations [3, 4]. ent concentrations of the required surface-active mate- The theoretical analysis [2] of nematic droplets with rial and perform comparative analysis of their structural varying boundary conditions suggests two possible sce- organization. This work was aimed at the detailed study narios for the interconversion of the bipolar and radial of the director configurations in nematic droplets dis- configurations. In the first case, one of the boojums in persed in a polymeric matrix, with the boundary condi- the bipolar structure gradually disappears and another tion varying as a result of varying concentration of the transforms to a hedgehog, which subsequently breaks corresponding surfactant. away from the surface and moves to the droplet center. In the second case, both boojums undergo changes in a similar manner, but the structure interconversion is EXPERIMENTAL accompanied by the formation of additional, including linear, disclinations. The well-known 4-n-pentyl-4'-cyanobiphenyl (5CB) nematic liquid crystal with the transition temper- In [2], a nematic with a lecithin impurity was dis- ° ° persed in glycerol and taken for the experimental study. atures “crystal 22 C nematic 35 C isotropic liquid” The surface anchoring of LC molecules in this system was chosen for the study. At T = 22°C and λ =

258 PRISCHEPA et al. ethanol evaporation rate was controlled so as to provide the same morphological parameters for the samples of composite film studied. The LC droplets formed a monolayer with a size dispersion of 4Ð16 µm. The textural patterns of PDLC films were studied using a polarizing microscope both in the geometry of crossed polarizers and in the linearly polarized light with a switched-off analyzer. Observations showed that the textures of all droplets in a composite film without lecithin are typical for the bipolar director configuration (Fig. 1a). In the crossed polarizers, two symmetri- cally arranged hyperbola-shaped extinction bands are seen (Fig. 2a) that emanate from the droplet poles (point defects) and are gradually expanded. In the geometry with one polarizer, point defects are clearly seen as dark spots because of a strong optical inhomo- geneity and, hence, intense local light scattering near the defects for any light polarization. For the same reason, the sections of droplet boundaries, where the light polarization coincides with the local director orientation and the gradient of refractive index n|| Ð np is large, are also clearly seen. By contrast, the boundary sections with the orthogonal arrangement of director and light polarization are seen least distinctly, because the gradient of refractive index n⊥ Ð np is minimal in this region. The sequence of director configurations arising in the nematic droplets upon an increase in the fraction of Fig. 1. The sequence of director configurations in nematic lecithin in the composite is schematically illustrated in droplets with different lecithin content. The arrows are Fig. 1. In the PDLC films with 0.08% lecithin, one of directed toward the increase in the surfactant content. (a) Bipolar droplet; (b) droplet with a destructed left boo- the boojums disappears in most (about 70%) of droplet jum; (c) monopolar droplet; (d) surface hedgehog structure; ensemble. This is clearly seen in the bottom region of and (e) radial structure. the droplet in Fig. 2b. The extinction band near the decaying boojum expands rather than narrows. In the remaining region of the droplet, the textural pattern has µ a form similar to the bipolar structure. The director- 0.589 m, the refractive indices of 5CB are n|| = 1.725 field distribution in these droplets can be represented by and n⊥ = 1.534 [5]. Poly(vinyl butyral) (PVB) of the the configuration shown in Fig. 1b. It should be noted 1PP type was used as a polymeric matrix. This polymer that the boundary conditions in this case are inhomoge- is transparent in the visible region and provides planar neous. Here, the director in the bottom region of the anchoring to the molecules of mesomorphic alkylcy- droplet is oriented homeotropically. When moving anobiphenyl derivatives [6]. The refractive index of along the surface to the remaining point defect, the ° λ µ PVB is np = 1.492 at Tc = 22 C and = 0.589 m. director orientation becomes tilted and, finally, planar. The homeotropic LC orientation at the interface was In samples with a higher lecithin content, the region produced using lecithin—surface-active substance of homeotropic and tilted director orientations relating to the class of phospholopids. In the LC drop- increases and the director lines further straighten out lets with lecithin impurity, the long axes of surfactant (Fig. 1c), so that the structure, in essence, becomes molecules are arranged perpendicular to the surface in monopolar. In [2], such a structure transformation was such a way that their polar groups are directed toward treated as “a continuous defect destruction accompa- the interface, while the nonpolar fragments (flexible nied by turning the size of its nucleus to infinity.” In the alkyl chains) are directed toward the bulk of the LC. As crossed polarizers with the geometry presented in a result of such structural ordering, the lecithin mole- Fig. 2c, only one extinction band is seen in the droplet cules provide the homeotropic orientation of LC mole- with such a director configuration. It emanates at the cules at the interface. top of the point defect, strongly expands, and fills Using a common solvent (ethyl alcohol) for all com- almost the entire lower half of the droplet. The texture ponents, a set of samples with an LC content of 55% of the upper half of the droplet is also analogous to the were prepared by the solvent-induced phase-separation bipolar structure. method with the PVB and lecithin concentrations vary- As the lecithin concentration increases, droplets ing within 41.5Ð45.0 and 0Ð3.5 wt %, respectively. The with the surface hedgehog structure (Fig. 1d) and radial

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TRANSFORMATION OF DIRECTOR CONFIGURATION 259

(a) (b) (c) (d) (e)

3 mm 3 mm 3 mm 4 mm 2 mm

Fig. 2. Textural patterns of the 5CB LC nematic droplets dispersed in poly(vinyl butyral) with different lecithin concentrations Clec. (bottom) Microphotographs in the geometry of crossed polarizers (shown by white arrows) and (top) photographs in the polarized light (the polarizer direction is shown by black arrows). (a) Bipolar droplet with symmetry axis directed at α = 11° with the polarizer, α ° α ° Clec = 0%; (b) droplet with a destructed bottom boojum, = 11 and Clec = 0.08%; (c) monopolar droplet, = 0 and Clec = 0.1%; α ° (d) droplet with the surface hedgehog structure, = 0 and Clec 2.0%; and (e) droplet with the radial structure, Clec = 2.6%. structure (Fig. 1e) appear. Droplets with the new struc- hedgehog are seen most distinctly for the light polar- ture appear and gradually increase in number. In this ized perpendicular to the symmetry axis (Fig. 2d). case, droplets with different director configurations can be observed simultaneously in the same PDLC film sample (Figs. 1aÐ1e), likely because of the inhomoge- CALCULATION OF DIRECTOR neous lecithin distribution over the film volume and due CONFIGURATION to some other factors, such as droplet shape, structure The problem of determining the director configura- of the transition layer at the droplet surface, etc. tion through the minimization of the orientational part In the sequence of orientational structures from of free energy monopolar (Fig. 1c) to the surface hedgehog (Fig. 1d), 1 2 2 the director field transforms in a smooth manner. One F = ---∫K[]()∇ ⋅ n + ()∇ × n dV (1) should identify these structures with care, because their 2 textural patterns are, on the whole, similar. The distinc- of the LC volume with a given boundary conditions can tions in these textures are most clearly seen in the be solved analytically for some simple geometries, e.g., observation geometry shown in Figs. 2c and 2d. Here, for the plane-parallel cells. Here, n is the nematic direc- only one extinction band is seen in the monopolar drop- tor and K is the elastic constant. For the droplet struc- let. Three extinction bands are observed in the droplet tures, the problem becomes more complicated, so the with a surface hedgehog (Fig. 2d). The central band director field in this case is calculated numerically [8]. goes along the droplet symmetry axis, but it is much narrower than in the monopolar structure. Two side The transition structures described above (Figs. 1bÐ bands emanate from the point defect at an angle of 1d) were calculated within the framework of a three- approximately 50° to the symmetry axis on both its dimensional model using the method developed in [9, sides. As a result, such droplets are cone-shaped in the 10] for the study of Friedericksz transitions in bipolar crossed polarizers (photographs of such textures were nematic droplets with rigidly fixed poles. The problem presented earlier in [7]), although they are, in fact, cir- was solved in a one-constant approximation with the cular. The appearance of the three aforementioned elastic modulus K = (K11 + K22 + K33)/3. The Kii values extinction bands can be understood from the consider- were taken from [11]. The minimum of F in Eq. (1) was ation of the corresponding director configuration found by the variational method in the Cartesian coor- shown in Fig. 1d. For the switched-off analyzer, the dinate system using the experimentally observed droplet boundaries adjacent to the defect are clearly boundary conditions. seen for the monopolar structure if the light polariza- For example, to determine the director configuration tion is parallel to the symmetry axis (Fig. 2c). By con- for the texture shown Fig. 2b, the anchoring was trast, the boundaries in the droplet with a surface assumed to be rigid and planar for all surface points

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260 PRISCHEPA et al. as in the droplet volume, from the condition that the free energy be minimal. The resulting distribution of director field (Fig. 3) was similar to the configuration presented in Fig. 1b. In this distribution, the surface director orientation is also homeotropic near the point −R. When moving away from the symmetry axis, the orientation becomes tilted and then smoothly changes to planar. Next, this information was used to calculate the corresponding textural patterns in crossed polarizers by applying the theoretical approach described in [12]. One can see (Fig. 4) that the theoretically calculated texture agrees well with the microphotograph of nematic droplet (Fig. 2b), taken for the same angle of incli- nation α of the symmetry axis to the polarizer. Fig. 3. The director configuration in the diametrical cross section passing through the symmetry axis of a spherical nematic droplet, as obtained by theoretical calculation CONCLUSIONS within the framework of a 3D model with inhomogeneous Thus, a set of orientational structures intermediate boundary conditions. The dashed line near the surface shows the region with a specified planar director orienta- between the bipolar and radial director configurations tion. have been revealed for nematic droplets in PDLC film samples with various lecithin content. The data obtained are evidence that the change of surface anchoring from planar to homeotropic results in a grad- ual transformation of the bipolar structure of nematic droplets into the radial structure through a sequence of equilibrium director configurations without the formation of additional disclinations, as was predicted previously by the topological analysis in [1, 2]. Clearly, the approach described above allows the observation of only the equilibrium director configurations in different samples, whereas the temperature-induced variations of boundaries [2] allow one to trace the droplet restruc- turization dynamics. However, it should be remem- bered that not only the boundary conditions change in the second case but so do the elastic moduli that strongly influence the distribution of director field, thereby hampering the interpretation of the experimental results. It should be emphasized that the surface anchoring in nematic droplets with intermediate configurations is inhomogeneous. Due to the use of the realistic inhomo- 4 mm geneous boundary conditions in our numerical calculation, the obtained director configurations and textural patterns of nematic droplets proved to be analogous to the experimentally observed ones. Fig. 4. Theoretically calculated texture of a spherical nem- This work was supported in part by the Presidium of atic droplet in crossed polarizers for the director configura- the Russian Academy of Sciences (project no. 8.1), the tion shown in Fig. 3 and an angle of 11° between the droplet symmetry axis and polarizer. Section of Physical Sciences of the Russian Academy of Sciences (project no. 2.10.2), and the Siberian Divi- sion of the Russian Academy of Sciences (integration project no. 18 and youth project no. 14). whose X coordinates fell within the interval Ð0.8R ≤ X ≤ +R. The azimuthal director direction in this surface region corresponded to the bipolar configuration with REFERENCES defects at the points ÐR and +R on the X axis. The 1. G. E. Volovik, Pis’ma Zh. Éksp. Teor. Fiz. 28, 65 (1978) boundary conditions were not specified for the rest of [JETP Lett. 28, 59 (1978)]. the surface with the coordinates ÐR ≤ X < Ð0.8R; i.e., 2. G. E. Volovik and O. D. Lavrentovich, Zh. Éksp. Teor. the director orientation in this region was determined, Fiz. 85, 1997 (1983) [Sov. Phys. JETP 58, 1159 (1983)].

JETP LETTERS Vol. 79 No. 6 2004 TRANSFORMATION OF DIRECTOR CONFIGURATION 261

3. R. B. Meyer, Mol. Cryst. Liq. Cryst. 16, 355 (1972). 9. A. V. Shabanov, V. V. Presnyakov, V. Ya. Zyryanov, et al., 4. S. Candau, P. LeRoy, and F. Debeauvais, Mol. Cryst. Pis’ma Zh. Éksp. Teor. Fiz. 67, 696 (1998) [JETP Lett. Liq. Cryst. 23, 283 (1973). 67, 733 (1998)]. 10. A. V. Shabanov, V. V. Presnyakov, V. Ya. Zyryanov, et al., 5. V. Ya. Zyryanov and V. Sh. Épshteœn, Prib. Tekh. Éksp. 2, 164 (1987). Mol. Cryst. Liq. Cryst. 321, 245 (1998). 11. J. D. Bunning, T. E. Faber, and P. L. Sherrell, J. Phys. 6. J. Cognard, Alignment of Nematic Liquid Crystals and (Paris) 42, 1175 (1981). Their Mixtures (Gordon and Breach, Paris, 1982; Uni- versitetskoe, Minsk, 1986). 12. R. Ondris-Crawford, E. P. Boyko, B. G. Wagner, et al., J. Appl. Phys. 69, 6380 (1991). 7. O. D. Lavrentovich, Pis’ma Zh. Tekh. Fiz. 14, 166 (1988) [Sov. Tech. Phys. Lett. 14, 73 (1988)]. 8. S. Zumer and J. W. Doane, Phys. Rev. A 34, 3373 (1986). Translated by V. Sakun

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 262Ð267. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 320Ð325. Original Russian Text Copyright © 2004 by Popov, Karnakov, Mur.

On the Relativistic Theory of Tunneling V. S. Popov1, *, B. M. Karnakov2, **, and V. D. Mur2 1 Institute of Theoretical and Experimental Physics, Moscow, 117218 Russia *e-mail: [email protected] 2 Moscow Institute of Engineering Physics (State University), Moscow, 115409 Russia **e-mail: [email protected] Received January 15, 2004

A relativistic generalization of the semiclassical theory of tunneling and multiphoton ionization of atoms and ions in the ﬁeld of a high-intensity electromagnetic wave with linear, circular, and elliptic polarization is con- structed. The exponential factor in the ionization probability is calculated for arbitrary values of adiabaticity parameter γ. In the case of low-frequency laser radiation, an asymptotically exact formula is derived for the ionization rate of the s atomic level, including the Coulomb, spin, and adiabatic corrections and the preexponential factor. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 32.80.Rm; 03.65.Xp; 03.65.Sq

1. Rapid progress in laser physics and technology example, a(η) = sinη corresponds to the monochro- has made it possible to attain the highest intensities J ~ matic laser light, a(η) = η corresponds to a constant 1021 W/cm2 [1, 2]; it is planned to increase these values crossed field, a(η) = tanhη corresponds to a soliton- by one to two orders of magnitude [3]. In such strong Ᏹ Ᏹ 2 η fields, atomic ions with charge Z ~ 40Ð60 will form; the like pulse (t, x) = 0/cosh , and so on. The equa- 2 tions of motion for the electron 4-momentum pi = (p, E) binding energy Eb = mec Ð E0 of electron energy levels in such ions becomes comparable to the rest energy have the form 2 mec . In this case, the tunneling electron motion leading Ᏹ Ᏹ() púx ==e v y, púy e 1 Ð v x , to the ionization can no longer be regarded as nonrela- (2) tivistic, and the generalization of the Keldysh ioniza- pú ===0, Eú e()%v eᏱv , tion theory [4Ð8] is required (this is the subject of this z y communication). Below, the equations defining the where the dot denotes the derivative with respect to the underbarrier trajectory in a wave field and the ioniza- laboratory time t. For any dependence Ᏹ(η), there tion rate of the s level are written for all values of the exists an integral of motion [5] adiabaticity parameter γ. The case of ionization by a low-frequency laser field (γ Ӷ 1), important for appli- () 2 η ωτ cations, is considered and some critical remarks con- JEp==Ð1x Ð v x /1Ð v =/ , (3) cerning the recently published article [9] are also made. In our calculations, we will use the imaginary-time where τ = ∫t 1 Ð v 2 dt is the particle intrinsic time. method (ITM) providing a pictorial description of the tunneling process, especially in the case of passage The second equation in Eqs. (2) gives through a time-varying barrier [6Ð8]. As a rule, hence- Ᏹ Ᏹ ប dpy e 0 e 0 forth = me = c = 1. ------= ------a'()η , p ()η = ------a()η = ÐeA ()η (4) dη ω y ω y 2. A plane electromagnetic wave with linear polarization is defined by the potentials (while choosing the integration constant, we took into Ᏹ account that the light-front variable η and the momen- ,,0 ()η ϕ ≡ tum p become purely imaginary quantities after the A =,0 Ð0------ω a 0, (1) y transition to the imaginary time, t it; cf. [10Ð12]). Ᏹ Ᏹ Ᏼ Ᏹ η Next, we find that where 0 is the wave-field amplitude, = = 0 a'( ), η = ω(t Ð x), the x axis is chosen along the wave-prop- η agation direction, the electric field is directed along the dy 1 dy p ()η eᏱ ------= ------= ------y -, y()η = ------0 a()ηη' d ', (5) y axis, and the magnetic field is directed along the z dη Jωdτ Jω ω2 ∫ J η axis. The function a(η) defines the pulse shape. For 0

0021-3640/04/7906-0262 $26.00 © 2004 MAIK “Nauka/Interperiodica” ON THE RELATIVISTIC THEORY OF TUNNELING 263 η η and the quantities px( ) and x( ) are defined analogously. Note that the solution can be obtained in the explicit form for any wave-field dependence on η. The explicit form of the tunnel trajectory for monochromatic laser radiation in the case a(η) = sinη is 1 sinh2η p ()η = ------20 Ð cosh η , x 2 η 4β J 2 0 ()η βÐ1 η ⑀ py = i sinh , = 1 iη sinh2η sinh2η (6) ⑀ ⑀ x = ------0 Ð ------, = 0 = 0.5 2 2η η 4ωβ J 2 0 2 1 y = ------()coshη Ð coshη , z ≡ 0, ωβJ 0 β ω Ᏹ η η γ ⑀ where = /e 0 and the substitution i corre- Fig. 1. Functions g( , 0) from formula (11) for monochro- sponding to the ITM is made. Here, η = Ðiωt , where matic light (linear polarization). The curves correspond to 0 0 the initial energy ⑀ 1 (nonrelativistic case), 0.9, 0.75, t0 is the initial (imaginary) instant of time for the tun- 0.5, 0.25, and 0 (from top to bottom). neling motion. η The quantities 0 and J appearing in Eqs. (6) can be determined from the initial condition [8, 10] along the tunnel trajectory, we obtain the ionization rate E()η ===p2 ++p2 1 ⑀, p ()η ⑀ Ð J (7) of the relativistic bound state with an exponential accu- 0 x y x 0 racy, ⑀ 2 ⑀ (here, = E0/mec , 0 < < 1, and E0 is the initial energy 2 of the energy level, including the electron rest energy); ∝ ()បÐ1 2mec η ()⑀ wR exp Ð2 ImW = expÐ------0 J Ð (10) this gives បω ⑀ 2 2 η γ 212Ð J + J sinh 0 = ------, or 1 Ð ⑀2 (8) η 2  sinh2 0 21 Ð J ∝ 2 ()γ, ⑀ ------12= + γ ------, wR expÐ------g , (11) η 2 3F 2 0 1 Ð ⑀  where γ is the adiabaticity parameter, which is a relativistic generalization of the well-known Keldysh param- 2ξ2 1ξ4 ξ2γ η ⑀ Ᏹ where g = 1 +/--- Ð --- 0(J Ð ), S = eter [4] 3 3 2 3 ប × 16 ω me c /e = 1.32 10 V/cm is the so-called Schwinger γω==T ------1 Ð,⑀2 (9) Ᏹ Ᏹ t eᏱ field, characteristic of QED [13, 14], F = 0/ ch, and 0 the characteristic field defined by the initial energy of and Tt is the characteristic tunneling time in electric the level is introduced: Ᏹ η field 0. Equations (7) and (8) are used to determine 0 ()ξ 3 1/2 and J as functions of the parameters γ and ⑀. Ᏹ 3 Ᏹ ξ 1⑀⑀()2 ⑀ ch = ------2- S, = 1 Ð --- + 8 Ð . (12) By calculating the “truncated action” function [8] 1 + ξ 2

0 Ᏹ The value of ch increases monotonically with decreas- W = ∫{}Ð 1 Ð v 2 ++e()Av ⑀ dt ing ⑀, i.e., with increasing level depth. In the nonrelativ- Ᏹ 3/2Ᏹ Ᏹ α3Ᏹ α t istic limit, ch = (2I) a, where a = S, = 1/137, 0 and I is the ionization potential (in atomic units). In this τ 0 case, relation (11) transforms into the well-known ⑀ = 1Ð e()Ap Ð ------dτ Keldysh formula [4, 15]. Equations (8)Ð(11) generalize ∫ this formula to the case of deep levels and can easily be v 2 0 1 Ð solved by computer methods (Fig. 1).

JETP LETTERS Vol. 79 No. 6 2004 264 POPOV et al. After rather cumbersome calculations, Eqs. (10) and (13) lead to the expansion

2 1 Ð ρ /3 2 4 g()γ,,⑀ ρ = 1 Ð ------γ + O()γ , (15) 10() 1 Ð ξ2/3 which is valid in the adiabatic region γ Ӷ 1 (for γ Շ 0.5, this expression has an accuracy of a few percent). In the nonrelativistic limit, when ξ ~ α I Ӷ 1, this formula is in accord with [6] for an arbitrary ellipticity ρ, while in the case of circular polarization, it fits to the result obtained in [5, 6]. Equation (15) directly shows that the increase in light ellipticity reduces the ionization prob- ⑀ ability wR; conversely, a decrease in , i.e., increase in the depth of a bound level, increases this probability Fig. 2. Function g(γ) in the case of linear and circular polarization (curves l and c, respectively) for a hydrogen-like (for a fixed value of reduced field F, which is itself a atom with Z = 60. function of the level energy). 3. The exponential factor in Eq. (11) is independent of the particle spin. In the framework of the ITM, the Analogously, we can calculate the level ionization spin correction to the action function is given by [17] rate by a wave with elliptic polarization (general case of 1 ie αβ λ µ monochromatic radiation). Instead of system (8), we δS = ------⑀αβλµ F u s dτ obtain the equations spin 2mc ∫ (16) 2 e sinh η Ð ρ2()coshη Ð sinη /η 2 = ------{}()sH Ð ()vs ()vH + []vs Ᏹ dt. 0 0 0 0 mc∫ = γ 2[]1 + ()J Ð ⑀ 2/1()Ð ⑀2 , Considering that the tunnel trajectory (6) lies in the (x, (13) y) plane and Ᏹ = Ᏼ, we obtain ()ρ2 η η ρ2[]()η η 2 1 Ð sinh2 0/2 0 + 2 sinh 0/ 0 Ð 1 0 eᏱ γ 2()2 ()⑀2 δS = ------0 s a'()η ()1 Ð v dt = 1+ 2 1 Ð J /1Ð , spin mc ∫ x x where ρ is the light ellipticity (Ð1 ≤ ρ ≤ 1, ρ = 0 corre- t0 η (17) ρ ± 0 sponds to linear polarization and = 1 corresponds to eᏱ J circular polarization). In this case, formula (10) = Ð ------0 s a'()τη d . η mc ∫ x remains valid for ImW; however, the quantities 0 and J should now be determined from Eqs. (13). These 0 equations are simplified in the case of linear (see Spin rotation in a uniform external field is described by Eq. (8)) and circular polarizations, when ρ2 = 1 and the BargmannÐMichelÐTelegdi equation [18]; for crossed fields, this equation implies that sz(t) = const 2 and ()η η 2 γ 21 Ð J sinh 0/ 0 = 1 + ------, 1 Ð ⑀2 Ᏹ (14) δ µ 0 ()η Sspin = Ð B------ω szai 0 . (17') 21 Ð ⑀J sinh2η /2η = 1 + γ ------. 0 0 2 η 1 Ð ⑀ In the case of a constant field, we have a(i 0) = Numerical calculation gives for function g the curves ω 3ξ2 i------; in addition, we must also take into depicted in Fig. 2. With increasing ellipticity, function Ᏹ 2 e 0 ()1 + ξ g = g(γ, ⑀, ρ) monotonically increases and the ioniza- account the change in the exponential factor in Eq. (11) tion probability accordingly decreases; in other words, ⑀ owing to the splitting of level 0 in a magnetic field, we have qualitatively the same situation as in the non- µ relativistic case [6]. because this factor contains magnetic moment of a bound electron, which differs from the Bohr magneton 1 In this case, the equations of motion are used in the ITM. A more if Zα ~ 1 [19, 20]. As a result, we obtain the spin factor elegant method for solving this problem is associated with the in the level ionization rate in a constant crossed field: application of the intrinsic-time method developed in Fock’s remarkable paper [16]. The application of Fock’s method makes ξ it possible to consider the general case of a wave with elliptic ITM ± 3 ()µ µ S± = exp------1 Ð / B , (18) polarization and leads to Eqs. (13); however, this requires a spe- ξ2 cial consideration. 1 +

JETP LETTERS Vol. 79 No. 6 2004 ON THE RELATIVISTIC THEORY OF TUNNELING 265 where the signs ±ប/2 correspond to the spin projections lomb factor Q [11] arising when the Coulomb interac- onto the direction of Ᏼ; for this reason, the states with tion between an electron and the atomic core (with different values of sz are characterized by different ion- charge Z) is taken into account is calculated using the ization rates. semiclassical perturbation theory [7]. For the elliptic An alternative method for calculating spin correc- radiation polarization, the exponential factor in Eq. (19) tion to the action is based on squaring the Dirac equa- should be replaced in accordance with Eq. (15), while the field index F (in the preexponential factor) in the case tion. This approach is similar to that used in [21, 22] for ν the relativistic Coulomb problem with Z > 137. In this of circular polarization should be replaced by 1 Ð 2 . case, instead of Eq. (17), we obtain In the nonrelativistic limit, Eq. (19) assumes the form σ ξ µ ξ 1 + 3 σ 3 S+ = ------expÐ------, = ------,(18')  1 Ð σ 2 µ 2 κ2 2 3F 2n* 12Ð n* 2 1 γ 2 1 + ξ B 11+ + ξ w = Cκ ------2 F expÐ------1 Ð ------, (21) π 3F 10 and S = 1/S . Expressions (18) and (18') are close Ð + Ᏹ κ3 γ ωκ Ᏹ numerically. According to Breit [19], for example, we where we now have F = 0/ , = / 0 is the Keldysh µ µ have = 0.933 B for the 1s1/2 ground level in a hydro- parameter, κ = 2I , and atomic units are used (in the gen-like atom with charge Z = 60, whence S+ = 1.046 case of circular polarization, we must omit factor ITM and S+ differs from it by only 1.5%; for Z = 92, we 3F/π and replace coefficient 1/10 in the exponent by have S+ = 1.121, and so on. Thus, the results of these 1/15). The generalization of formula (21) to the states two (independent) calculations virtually coincide. with an arbitrary orbital angular momentum l is given in [6]. α Ӷ ≈ ITM = + α 3 For Z 1, we have S± S± 1 O((Z ) ); i.e., Let us assess the range of application of the nonrel- the tunneling probability is independent of the electron ativistic theory of ionization. With an exponential accu- spin projection. This is not surprising since operator s racy, we obtain from Eqs. (11) and (21) commutes with the Schrödinger Hamiltonian and the spin component splits off. ∝ {}()ξ 3Ᏹ ()ξ2 Ᏹ w exp Ð23 S/3 1 + 0 , (22) 4. Maximal intensities J are attained for IR lasers; R for this reason, the case with γ Ӷ 1 deserves special 3 consideration. For the tunneling ionization rate of the s 2κ Ᏹ 2 3/2Ᏹ w ∝ expÐ------a = expÐ---[]21()Ð ⑀ ------S . level, we can derive an asymptotically exact formula in NR 3Ᏹ 3 Ᏹ Ᏹ Ӷ Ᏹ Ӷ 0 0 (22') the limit of a weak ( 0 ch, or F 1) field:

1 2 2 2 Eb 2 Ð()2ν Ð 3/2 Setting here ⑀ = 1 Ð --- α κ = 1 Ð ()Zα and taking wR = -----ប Cλ DF 2 into account the expansions (19) 2 γ 2 × expÐ------1 Ð ------, ρ = 0, 1 2 3F2 κ = Z 1 ++,---()Zα … 10() 1 Ð ξ /3 8

2 where E = mc (1 Ð ⑀) is the binding energy of the level, 7 3 2 3 b 3ξακ==++------()ακ … Zα ++---()Zα …, 2 72 9 ν = Zα⑀/1 Ð ⑀ is the relativistic analogue of the effective principal quantum number n* = Z/2I , and we obtain Cλ is an asymptotic (at infinity) coefficient of the free- 1 5Ᏹ atom wave function (in the absence of the wave field). ≈ ------()α ------S α Ӷ wNR/wR exp Ð Z Ᏹ , Z 1 . (23) Generally speaking, this coefficient can be obtained 36 0 only numerically from the HartreeÐFockÐDirac equations; however, for a hydrogen-like atom, there exists It can be seen that the range of applicability of the an analytic solution (see, e.g., [23]). For example, for Keldysh nonrelativistic theory is “extended” to quite the 1s1/2 ground level and any Z, we have large values of Z. For example, for Z = 40, 60, and 80 and for the radiation intensity J = 1023 W/cm2, the val- ⑀ ν ()α 2 2 2⑀ Ð 1 Γ()⑀ ==1 Ð,Z C1s =2 / 2 + 1 . (20) ues of wNR and wR differ by a factor of 1.15, 3, and 65, respectively. Using the formulas derived above, we can Finally, the factor D = D(⑀, Z) is independent of the easily refine this estimate and make it more exact wave amplitude and has a rather cumbersome form, and (Fig. 3). For want of space, we postpone the discussion it will not be given here. It should be noted that the Cou- of this question to a more detailed publication.

JETP LETTERS Vol. 79 No. 6 2004 266 POPOV et al. if the atom has only one electron in the K shell, while the remaining electrons are stripped (hydrogen-like atom with nuclear charge Z). In this particular case, η ≡ ⑀ and Q ≡ Q, while the asymptotic coefficient is defined by formula (20) and corresponds to [9]. How- ever, our formula for Q is considerably more general; it applies to multicharged ions with an arbitrary degree of 2 ionization if the atomic-level parameters ⑀ and Cλ are taken from independent calculations for a free (Ᏹ = Ᏼ = 0) atom or directly from the experiment, as in the nonrelativistic theory of ionization [5Ð8]. In [9], neither spin factor (18) in the tunneling probability nor correction (15) on the order of γ2 in the exponent were calculated and the adiabatic correction [6] Fig. 3. The ratio R = wNR/wR for the 1s1/2 ground level as a function of Z for various intensities of laser radiation. The changing the power of field in the preexponential factor figures on the curves indicate the values of logJ [W/cm2]. was not taken into account. For this reason, formula (8) from [9] could refer only to constant fields. However, in this case the authors of [9] assume, in fact, that the The ionization of a relativistic bound state by a con- Dirac bispinor Sˆ defined in the vicinity of atomic stant crossed field was considered by Nikishov and nucleus does not change in the course of electron tunnel Ritus [5]. Using the KleinÐGordon equation, they cal- motion (this obviously follows from the comparison of culated the ionization probability wR for the s level formulas (1), (2), and (5) in [9]), which is not correct. bound by short-range forces (Z = 0) in the case of a par- The word combination “Dirac theory” appearing in the ticle with zero spin. For Z = v = 0, expressions (19), title of [9] is also surprising since it is only the normal- (22), and (25) coincide with those obtained in [5], but 2 they are written in a more compact form. The coinci- ization factor Cλ mentioned in all textbooks on quan- dence of the results obtained by two independent meth- tum mechanics [23] (which, in addition, differs from ods is very important for the ITM. Although this 2 unity by only a few percent; e.g., Cλ = 1 for small Z and method possesses a heuristic potential and physical 2 clearness, it cannot be regarded as rigorously substanti- Cλ = 1.039 for Z = 60) that is related to the Dirac the- ated, in spite of some attempts made in this direction ory, while the factors P, Q, and Exp are independent of [17]. the particle spin and can be calculated on the basis of 5. In conclusion, we must consider the article by the KleinÐGordon equation, as in [5, 10Ð12]. Milosevic, Krainov, and Brabec “Semiclassical Dirac The formulas given in [9] literally reproduce the Theory of Tunnel Ionization” [9], which contains for- corresponding formulas2 from [10Ð12] with the same mula (8) for the ground-state ionization rate wr in con- notation (including the transition from the energy ⑀ to stant crossed fields. This formula (which is treated in the convenient variable ξ introduced in [10] and natu- [9] as the “main result” of that study) can easily be writ- rally arising in the ITM). The original contribution of ten in the form the authors of [9] lies in the multiplication of factors 2 2 ប Exp, Q, P, and Cλ , which were previously known. wr ====CλPQExp, mc1, (24) Thus, article [9] is a compilation of the studies [10Ð12] where the factors Exp and P identically coincide with carried out (and published) several years earlier (which, the corresponding expressions from [5, 10, 11], however, is not mentioned in [9]). At the same time, it is stated in [9] that “for the first time a quantitative 3 Ᏹ 2 23ξ S 1 1 Ð ξ /3 Ᏹ description of tunnel ionization of atomic ions” or “the Exp ==exp Ð------, P ------, (25) 2 Ᏹ ξ 2 Ᏹ first quantitative determination of tunneling in atomic 1 + ξ 3 + ξ S ions in relativistic regime” is given in that work (itali- while the Coulomb factor Q differs from our expres- cized by us); see abstract and the text preceding formula (9) in [9]. sion (formula (35) in [11]) only in that the exponent Reference [11] in [9] shows that our publications 2 η ≡ ν = Zα⑀/1 Ð ⑀ is replaced by the level energy ⑀. were familiar to the authors. However, our studies are The difference between Q and Q is due to the fact that, treated as the “analytic solution of the Klein–Gordon in [9], a special case of ionization of the 1s1/2 level with 2 See, for example, Eqs. (17), (35), and (50) in [11]. While compar- 2 ing these formulas with formula (8) from [9], it should be borne energy ⑀ = 1 Ð ()Zα was considered and the wave ≡ Ᏹ ≡ Ᏹ µ α ប in mind that eF /Fcr, Fcr S = 1/e, and = Z (for = m = function had the simplest form [23]. This can be done c = 1).

JETP LETTERS Vol. 79 No. 6 2004 ON THE RELATIVISTIC THEORY OF TUNNELING 267 equation for πÐ atoms in static electric and magnetic 23, 924 (1966)]; Zh. Éksp. Teor. Fiz. 51, 309 (1966) fields,” although we never mentioned πÐ atoms.3 On the [Sov. Phys. JETP 24, 207 (1966)]. contrary, it is repeatedly emphasized in [10Ð12] that our 7. A. M. Perelomov and V. S. Popov, Zh. Éksp. Teor. Fiz. goal is the generalization of the imaginary time method 52, 514 (1967) [Sov. Phys. JETP 25, 336 (1967)]. to the case of relativistic tunnel electron motion and its 8. V. S. Popov, V. P. Kuznetsov, and A. M. Perelomov, application to the theory of ionization of deep levels Zh. Éksp. Teor. Fiz. 53, 331 (1967) [Sov. Phys. JETP 26, (including the K shell) in heavy atoms. 222 (1968)]. One of the authors of this paper (V.P.) was obliged 9. N. Milosevic, V. P. Krainov, and T. Brabec, Phys. Rev. to note (in connection with the so-called ADK theory Lett. 89, 193001 (2002). [24]; see [25] for details) a certain tendency to use the 10. V. S. Popov, B. M. Karnakov, and V. D. Mur, Pis’ma Zh. results obtained by other authors without the appropri- Éksp. Teor. Fiz. 66, 213 (1997) [JETP Lett. 66, 229 ate citation. The “main result” in [9] was obtained by (1997)]; V. S. Popov, V. D. Mur, and B. M. Karnakov, simply rewriting the formulas from original publica- Phys. Lett. A 250, 20 (1998). tions [5, 10Ð12]. 11. V. D. Mur, B. M. Karnakov, and V. S. Popov, Zh. Éksp. Teor. Fiz. 114, 798 (1998) [JETP 87, 433 (1998)]. We are grateful to M.I. Vysotskii, S.P. Goreslavskii, V.R. Zoller, N.B. Narozhnoi, L.B. Okun’, S.V. Popru- 12. B. M. Karnakov, V. D. Mur, and V. S. Popov, Yad. Fiz. 62, zhenko, Yu.A. Simonov, and also to participants of the 1444 (1999) [Phys. At. Nucl. 62, 1363 (1999)]. 17th Conference on Fundamental Atomic Spectroscopy 13. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). (Zvenigorod, December 2003) and Scientific Confer- 14. J. Schwinger, Phys. Rev. 82, 664 (1951). ence MIFI-2004 (Moscow, January 2004) for fruitful 15. L. D. Landau and E. M. Lifshitz, Course of Theoretical discussions, valuable remarks, and moral support. We Physics, Vol. 3: Quantum Mechanics: Non-Relativistic thank S.G. Pozdnyakov, who carried out numerical cal- Theory, 3rd ed. (Nauka, Moscow, 1974; Pergamon, New culations, and N.S. Libova and M.N. Markina for the York, 1977). help in preparing the manuscript. 16. V. A. Fok, Izv. Akad. Nauk SSSR, Ser. Fiz. 4Ð5, 551 This study was supported in part by the Russian (1937). Foundation for Basic Research (project no. 01-02- 17. M. S. Marinov and V. S. Popov, Yad. Fiz. 15, 1271 (1972) 16850). [Sov. J. Nucl. Phys. 15, 702 (1972)]. 18. V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Lett. 2, 435 (1959). REFERENCES 19. G. Breit, Nature 122, 649 (1928). 1. G. Mourou, C. P. J. Barty, and M. D. Perry, Phys. Today 20. A. M. Perelomov and V. S. Popov, Yad. Fiz. 14, 661 51 (1), 22 (1998). (1971) [Sov. J. Nucl. Phys. 14, 370 (1971)]. 2. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 21. V. S. Popov, Yad. Fiz. 12, 429 (1970) [Sov. J. Nucl. Phys. (2000). 12, 235 (1970)]; Yad. Fiz. 14, 458 (1971) [Sov. J. Nucl. 3. T. Tajima and G. Mourou, Phys. Rev. ST Accel. Beams Phys. 14, 257 (1971)]; Yad. Fiz. 64, 421 (2001) [Phys. 5, 031301 (2002). At. Nucl. 64, 367 (2001)]. 4. L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1945 (1964) [Sov. 22. Ya. B. Zel’dovich and V. S. Popov, Usp. Fiz. Nauk 105, Phys. JETP 20, 1307 (1964)]. 403 (1971) [Sov. Phys. Usp. 14, 673 (1971)]. 5. A. I. Nikishov and V. I. Ritus, Zh. Éksp. Teor. Fiz. 50, 23. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, 255 (1966) [Sov. Phys. JETP 23, 168 (1966)]; Zh. Éksp. New York, 1968; Inostrannaya Literatura, Moscow, Teor. Fiz. 52, 223 (1967) [Sov. Phys. JETP 25, 145 1957). (1967)]. 24. M. V. Ammosov, N. B. Delone, and V. P. Kraœnov, 6. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. Éksp. Teor. Fiz. 91, 2008 (1986) [Sov. Phys. JETP Zh. Éksp. Teor. Fiz. 50, 1393 (1966) [Sov. Phys. JETP 64, 1191 (1986)]; N. B. Delone and V. P. Kraœnov, Usp. Fiz. Nauk 168, 531 (1998) [Phys. Usp. 41, 487 (1998)]. 3 To ionize πÐ atoms, the electric field strength should be increased Ᏹ ∝ 2 25. V. S. Popov, Usp. Fiz. Nauk 169, 819 (1999) [Phys. Usp. by five orders of magnitude (since S m ) to attain intensities J տ 1023 W/cm2, which is far beyond the limits of the wildest 42, 733 (1999)]. fantasies and can hardly be realized in principle due to the pro- duction of e+eÐ pairs from vacuum by an external field (Schwinger effect [14]). Translated by N. Wadhwa

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 268Ð271. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 326Ð329. Original Russian Text Copyright © 2004 by Akhmedzhanov, Zelensky.

Experimental Study of the Resonance Radiation Group Delay in Degenerate Systems R. A. Akhmedzhanov and I. V. Zelensky* Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia *e-mail: [email protected] Received February 12, 2004

The dependence of the polarization- and intensity-modulation group delay on the polarization of electromag- 87 netic wave was studied experimentally for different transitions between the hfs components of the Rb D1 absorption line. It was found that the polarization-modulation delay strongly depends on the degeneracy structure of resonant transition and, in the general case, on the ellipticity of light-wave polarization. It is demonstrated that the polarization-modulation delay does not occur for the transitions not involving dark states. The polarization delay was studied as a function of the polarization ellipticity angle. The intensity-modulation delay 87 was measured for the resonance radiation to show that it is observed for all Rb D1-line transitions and is independent of polarization. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 42.25.Ja; 42.50.Gy

The discovery of coherent population trapping magnetic quantum number degeneracy is suggested for (CPT) and electromagnetically induced transparency use in magnetic-field measurements [6–9]. (EIT) has stimulated the study of resonance nonlinear effects in media with induced atomic coherence [1Ð3]. The CPT in degenerate systems is caused by the The appearance of a transparency window upon the res- polarization-induced separation of excitation channels. onant wave-field interaction in multilevel systems is In such a situation, the behavior of the system depends accompanied by the appreciable lengthening of laser on the mutual orientation of the polarizations of optical radiation path. The strong medium dispersion under the fields. The population trapping occurs in the so-called EIT conditions brings about anomalously strong delay dark state representing a sublevel superposition nonin- of a signal pulse in the pump-wave field. For the special teracting with field. The occurrence of a dark state and radiation-controlled regime, “light stop,” i.e., in fact, the population trapping in this state are the common implementation of optical memory, is possible [4, 5]. property of degenerate systems and are well under- The combination of a low velocity of wave-packet stood. In particular, the CPT in a magnetically degener- propagation and increase in the wave-interaction path ate two-level system interacting with a polarized radia- under resonance conditions renders the EIT regime tion was studied in detail in the early works devoted to promising for the observation of various nonlinear the theory of this phenomenon [10, 11]. effects. For the EIT in degenerate systems, the traditional Degenerate schemes have gained the widest accep- scheme of studying the behavior of a weak signal wave tance in the implementation of the CPT and EIT effects. in the field of an intense pump wave becomes conven- This is caused by both the widespread occurrence of the tional, because the corresponding wave polarizations degenerate systems and their obvious advantages. In are chosen arbitrarily. In theoretical work [12], a gen- particular, due to the fact that the frequencies of the eral approach was suggested to the study of the evolu- fields used in these schemes are equal in the case of tion of a wave with slowly varying polarization inter- exact resonance, the Doppler broadening of the EIT- acting resonantly with a medium of degenerate atoms inducing two-photon transition can be reduced, in under CPT conditions. It has been shown that a change effect, to zero, and, in addition, the possibility of using in the light-wave polarization propagates through such a common generation source for the interacting waves a medium with a group velocity that is much lower than allows the coherence of excitation events to be substan- the velocity of light and dependent on the wave polar- tially improved. For this reason, degenerate systems prove to be suitable for the observation of EIT and, ization and the transition type. therefore, are widely used in experiment. In particular, Another effect inherent in multilevel systems con- 87 the so-called light stop was observed for the Rb 5s1/2, sists in the delay of a change in the intensity of a reso- F = 2 5p1/2, F = 1 transition [4]. It should also be nant light wave as a result of population pumping out to noted that the EIT phenomenon in the systems with the long-lived levels noninteracting with field [13]. Evi-

EXPERIMENTAL STUDY OF THE RESONANCE RADIATION GROUP 269 dently, a dark state can play the role of such level in degenerate systems. In this work, the group delay of a change in polarization and intensity was studied as a function of the electromagnetic-wave polarization for different transi- 87 tions between the hfs components of the Rb D1 absorption line. To reduce the influence of disturbing factors such as spurious magnetic fields, measurements were performed in a pure rubidium vapor without a buffer gas and with a relatively high laser intensity (100 mW/cm2) [8]. The delay times under these conditions were relatively short (0.5Ð2 µs, which corresponds to a mean group velocity of 25Ð100 km/s) but sufficient for the measurements of their dependence on polarization. The scheme of the experimental setup for studying the group delay of a change in polarization is presented Fig. 1. (a) Scheme of the setup for measuring the polariza- in Fig. 1a. An external cavity diode laser 1 was used as tion-modulation delay: 1 laser, 2 polarizer, 3 Pockels cell, 4 a source of monochromatic radiation. The radiation 87Rb vapor cell, 5 quarter-wave plate, 6 polarizer, 7 photo- was linearly polarized at the laser output. The polariza- diode, 8 magnetic screen, and 9 heater. (b) Setup for study- tion was additionally controlled by polarizer 2. The ing the intensity-modulation delay: 1 laser, 2 polarizer, light-wave polarization was varied from linear to circu- 3 Pockels cell, 4 polarizer, 5 quarter-wave plate, 6 87Rb lar using a Pockels cell 3 and modulated about its mean vapor cell, 7 photodiode, 8 magnetic screen, and 9 heater. value with a frequency of ~60 kHz and a modulation depth of ~10%. A laser beam with a cross section of 2 × 5 mm passed through a cylindrical cell 4, 56 mm in diameter and 55 mm in length, filled with rubidium vapor. To prevent the influence of the laboratory magnetic field, magnetic screen 8 was used. The rubidium concentration in the cell was varied using heater 9. The signal was detected using the scheme consisting of quarter-wave plate 5, polarizer 6, and photodiode 7. Quarter-wave plate 5 and polarizer 6 could rotate about the axis, allowing the modulation to be detected near the arbitrary elliptic radiation polarization.

Among the hfs transitions of the D1 absorption line (Fig. 2), the F = 1 F' = 2 transition does not involve a dark state, the F = 1 F' = 1 and F = 2 F' = 2 transitions involve a simple one-dimensional dark state, and the F = 2 F' = 1 transition has a double (two- dimensional) dark state, which is to say that, for every light-wave polarization, the lower-level sublevels have 87 Fig. 2. (a) Hyperﬁne structure scheme of the Rb D1 line. a two-dimensional space noninteracting with the reso- 87 nant electromagnetic ﬁeld [10, 11]. Measurements (b) Transitions between the hfs levels of the Rb D1 line. The magnetic sublevels of the corresponding levels are showed that the delay in the propagation of polarization shown. The magnetic quantum numbers are numbered and structure is absent for the F = 1 F' = 2 transition. the arrows indicate the allowed transitions for the right σ+ The group delay of a change in polarization was and left σÐ circular polarizations. observed for the F = 1 F' = 1 and F = 2 F' = 2 transitions. This delay did not depend on the azimuthal rotation angle of polarization because of the axial sym- The situation with the F = 2 F' = 1 transition metry of the system. The delay for the F = 1 F' = 1 proved to be more complicated; in addition to the delay transition did not depend on the ellipticity. For the F = of polarization structure, it changed and the radiation 2 F' = 2 transition, the delay of the polarization- modulation propagation was found to be dependent on was depolarized. Such a behavior is apparently caused the mean ellipticity of light-wave polarization; the by the complex structure of the dark state of this transi- delay was maximal for the linear polarization and min- tion and calls for further analysis. It is worth noting that imal for the circular polarization. These data are in the polarization modulation with a frequency of 60 kHz agreement with the theoretical ideas developed in [12]. and a depth of 10% did not change the absorption at this

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270 AKHMEDZHANOV, ZELENSKY

wave and different transitions between the hfs compo- 87 nents of the Rb D1 line. Our studies showed that the intensity-modulation delays in the polarization-delay experiments (laser intensity ~100 mW/cm2 and rubidium concentration × 12 Ð3 µ 87 ~2 10 cm ) did not exceed 0.5 s for all Rb D1 transitions. Measurements with a higher concentration (~5 × 1012 cmÐ3) and lower laser intensity (50 mW/cm2) gave the following results: 1.2 ± 0.2, 0.4 ± 0.2, 1.2 ± 0.2, and 2.0 ± 0.2 µs for the F = 1 F' = 2, F = 1 F' = 1, F = 2 F' = 2, and F = 2 F' = 1 transition, respectively. The small delay for the F = 1 F' = 1 transition is caused by its low oscillator strength (one- fifth of the other transitions). It was found that the intensity-modulation delay is independent of the radiation polarization within the measurement accuracy. τ Note that the delay of a change in intensity occurs for Fig. 3. Dependence of the group delay of a change in the F = 1 F' = 2 transition, for which no delay of polarization on the mean ellipticity angle ⑀ of the electromagnetic-wave polarization resonant with the F = 2 polarization structure was observed. This is probably 87 due to the population pumping to the F = 2 level. F' = 2 transition of the Rb D1 absorption line. In summary, the polarization- and intensity-modulation group delay have been studied for a light wave transition, indicating the adiabatic character of the interacting resonantly with a medium consisting of dark-state rearrangement in our experiment [3]. degenerate atoms. It has been shown that the evolution of resonance radiation depends on the degeneracy The maximal delay in the propagation of polariza- structure of transition. For the transitions not involving tion structure at the F = 2 F' = 2 transition was a dark state, the polarization modulation was not observed for a rubidium concentration of ~2 × 1012 cmÐ3. delayed. For the transitions involving a one-dimen- The corresponding dependence of the group delay τ of sional dark state, the polarization-modulation group a change in polarization on the mean ellipticity angle ⑀ delay was observed, much as the electromagnetic-wave is shown in Fig. 3. One can see that the delay in the propagation is retarded under EIT conditions. In the propagation of polarization structure for the linear simplest configuration (F = 1 F' = 1 transition), the polarization ⑀ = 0° is more than three times larger than polarization delay does not depend on the radiation the delay for the circular polarization ⑀ = 45°. With an polarization. For the transitions with a more complex increase in the concentration, the delay decreased, degeneracy structure, the polarization-modulation likely due to the radiation trapping, i.e., breaking of decay depends on the polarization of a resonant wave. atomic coherence in the reemitted light [14Ð16]. The Due to this dependence, the shape of the polarization group delay measured under the same conditions for pulse can be distorted upon its propagation in medium. the polarization structure at the F = 1 F' = 1 transi- These results are in close agreement with the conclu- tion was independent of the radiation polarization and sions of theoretical work [12]. For the transitions equal to 0.75 ± 0.10 µs. involving a two-dimensional dark state, the delay of polarization structure was accompanied by its change Along with the delay of a change in polarization, the and the radiation depolarization. Evidently, such a delay of a change in the intensity of a resonant light behavior is caused by the complex structure of a dark wave, analogous to that observed previously in ruby state and calls for further investigation. The delay of a crystal [13], was also studied in our experiment. The change in the laser-radiation intensity was observed for scheme of the setup for studying the intensity-modula- all the transitions studied. The delay in the propagation tion delay is shown in Fig. 1b. The radiation from laser of a change in intensity is caused by the population 1 passed through linear polarizer 2 and fell on Pockels pumping out to the long-lived levels noninteracting cell 3, to which an ac voltage with a frequency of with the field. Both the dark state and the long-lived 60 kHz and a bias voltage were applied. Upon passing nonresonance levels can play the role of such levels. It through polarizer 4, the laser beam acquired linear should be noted that the delay of a change in intensity polarization and amplitude modulation. A quarter-wave does not depend on the polarization and, under analo- plate allowed the radiation polarization to be changed gous conditions, is several times shorter than for the from linear to circular. The radiation passing through a polarization delays. cell with 87Rb vapor inside magnetic screen 8 with These results can prove to be important for the solu- heater 9 was detected using photodiode 7. The inten- tion of various fundamental and applied problems in sity-modulation delay was studied for the resonant light nonlinear optics and, in particular, the problem of data

JETP LETTERS Vol. 79 No. 6 2004 EXPERIMENTAL STUDY OF THE RESONANCE RADIATION GROUP 271 storage and processing of optical and, in the future, 8. P. N. Anisimov, R. A. Akhmedzhanov, I. V. Zelenskiœ, quantum information. and E. A. Kuznetsova, Zh. Éksp. Teor. Fiz. 124, 973 (2003) [JETP 97, 868 (2003)]. We are grateful to L.A. Gushchin, A.G. Litvak, and V.A. Mironov for helpful discussions. This work was 9. P. M. Anisimov, R. A. Akhmedzhanov, I. V. Zelenskiœ, supported by the Russian Foundation for Basic et al., Zh. Éksp. Teor. Fiz. 124, 973 (2003) [JETP 97, 868 (2003)]. Research (project nos. 04-02-17042 and 03-02-17176). 10. V. S. Smirnov, A. M. Tumaœkin, and V. I. Yudin, Zh. Éksp. Teor. Fiz. 96, 1613 (1989) [Sov. Phys. JETP REFERENCES 69, 913 (1989)]. 11. A. M. Tumaœkin and V. I. Yudin, Zh. Éksp. Teor. Fiz. 98, 1. B. D. Agap’ev, M. B. Gornyœ, B. G. Matisov, and 81 (1990) [Sov. Phys. JETP 71, 43 (1990)]. Yu. V. Rozhdestvenskiœ, Usp. Fiz. Nauk 163 (9), 35 (1993) [Phys. Usp. 36, 763 (1993)]. 12. I. V. Zelenskiœ and V. A. Mironov, Zh. Éksp. Teor. Fiz. 121, 1068 (2002) [JETP 94, 916 (2002)]. 2. E. Arimondo, Prog. Opt. 35, 257 (1996). 13. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Phys. 3. A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, et al., Rev. Lett. 90, 133903 (2003). Adv. At. Mol. Opt. Phys. 46, 191 (2001). 14. I. Novikova, A. B. Matsko, and G. R. Welch, Opt. Lett. 4. D. F. Phillips, A. Fleischhauer, A. Mair, et al., Phys. Rev. 26, 1016 (2001). Lett. 86, 783 (2001). 15. I. Novikova and G. R. Welch, J. Mod. Opt. 49, 349 5. O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, (2002). Phys. Rev. Lett. 86, 628 (2001). 16. A. B. Matsko, I. Novikova, M. O. Scully, and 6. R. Wynands and A. Nagel, Appl. Phys. B 68, 1 (1999). G. R. Welch, Phys. Rev. Lett. 87, 133601 (2001). 7. R. A. Akhmedzhanov and I. V. Zelenskiœ, Pis’ma Zh. Éksp. Teor. Fiz. 76, 493 (2002) [JETP Lett. 76, 419 (2002)]. Translated by V. Sakun

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 272Ð276. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 330Ð334. Original Russian Text Copyright © 2004 by Zharova, Litvak, Mironov.

Self-Action of Laser Radiation under the Conditions of Electromagnetically Induced Transparency N. A. Zharova*, A. G. Litvak, and V. A. Mironov Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhni Novgorod, 603950 Russia *e-mail: [email protected] Received January 28, 2004; in ﬁnal form, February 17, 2004

The laser-radiation self-action dynamics in the electromagnetically induced transparency band were studied analytically and numerically for an atomic gas with the Λ-type energy-level scheme. The self-consistent system of equations describing the spatiotemporal evolution of a finite-amplitude wave packet in the field of a uniform pump wave was derived. The self-action in the presence of competing diffraction, nonlinear dispersion, and radiation absorption in the system was qualitatively analyzed; in particular, the conditions for self-focusing of a probe beam were determined. The results were confirmed by the numerical simulation of wave-packet spatiotemporal evolution. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 42.50.Gy; 42.65.Jx

The studies of interference phenomena [1Ð3] of the tions under which the radiation self-focusing is possi- type of coherent population trapping (CPT) and elec- ble are determined. The results of qualitative consider- tromagnetically induced transparency (EIT) in multi- ation are confirmed by the numerical simulation of self- level atomic ensembles have culminated in the inten- action dynamics. sive development of a new direction in nonlinear reso- 1. Let us consider the spatiotemporal evolution of nance optics. It is characterized by a negligible the bichromatic laser radiation absorption of resonance radiation upon its parametric interaction with a medium. As a result, the traditional (),, ()ω EE= 1 r⊥ ztexp i 1tikÐ 1z nonlinear effects can be observed for exceedingly low (1) (),, ()ω intensities of interacting waves (at a level of a few pho- + E2 r⊥ ztexp i 2tikÐ 2z tons) and with less stringent phase-matching require- ments. interacting resonantly with a three-level system (Fig. 1) The quantum interference effects are responsible for containing a metastable state |2〉. We assume that the ω ω some unusual features in the propagation of a bichro- wave-packet frequencies ( 1, 2) coincide with the fre- ω ω ω ω matic resonance radiation in a system of three-level quencies of atomic transitions, 1 = 13 and 2 = 23, atoms. A broad class of self-consistent two-pulse struc- so that a two-photon resonance takes place. For a rar- tures were found in a one-dimensional case. In the efied gas, the relation between the wavenumbers and absence of phase-coherence relaxation in a medium frequencies can be taken in the form of the dispersion Λ ω with the -type energy-level scheme, they propagate relation in vacuum k1, 2 = 1, 2/c. without any noticeable change in shape at a distance The evolution of wave packets E (r⊥, z, t) is much greater than the absorption length of an individ- 1, 2 ual wave packet (see [4Ð6] and references cited described by the reduced equations therein). Of particular interest is the study of stationary ∂ ∂ self-action (self-focusing) of the bichromatic radiation E j 1 E j 2 2ik ------+ ------+4∆⊥E = k πP , (2) [7Ð10] in a medium with EIT. j∂z c ∂t j j j In this work, the self-action dynamics are considered for spatially restricted laser pulses under the EIT where Pj is the complex polarization amplitude induced conditions in a medium of Λ-type atoms. The density by the field Ej (j = 1, 2). It is determined by the nondi- ρ matrix formalism is used to obtain a self-consistent sys- agonal elements of density matrix j3: tem of equations describing the self-action dynamics of ρ a bichromatic radiation in a system with long-lived P j = Nb3 j j3, (3) phase coherence. The spatiotemporal evolution of a finite-amplitude wave packet in the field of a strong where N is the atomic density and d3j are the electric pump wave is qualitatively examined and the condi- dipole moments of allowed transitions.

SELF-ACTION OF LASER RADIATION 273

The speciﬁc features of wave-packet interaction in the case considered are caused by the fact that the relaxation rate Γ of low-frequency coherence is much lower than the upper-level decay rate γ, Γ Ӷ γ. The relaxation processes in this system are conditioned by the strongly different characteristic times. Clearly, the population n3 | 〉 ρ ρ of the upper level 3 and the coherences 13 and 23 reach their stationary values most rapidly. For wave ﬁelds smoothly varying at times τ ӷ γ Ð1 E , (4) the corresponding density-matrix elements are assumed to be quasi-stationary. The resulting dynamic Λ equations for the density-matrix components of a Fig. 1. Energy-level scheme of a -type atom. The wave E1 | 〉 | 〉 Λ-type atom [1, 2] take the form is resonant with the 1 3 transition, the E2 wave is resonant with the |2〉 |3〉 transition, and the low-fre- ()ρ γ inú1 = Ð g1 31 Ð c.c. + i n3, (5) quency |1〉 |2〉 transition is forbidden. ()ρ γ inú2 = Ð g2 32 Ð c.c. + i n3, (6) iρú = Ð g ρ + g*ρ Ð iΓρ . (7) of the time derivatives in Eqs. (5)Ð(7) by the perturba- 12 1 32 2 13 12 tion method brings about equation The quasi-stationary components n , ρ , and ρ are 3 13 23 ∂ ∂ determined from the following relationships: 1 g1 i ∆ ------∂ - + ∂----- + ------⊥ g1 ()ρ γρ c t z 2k1 Ð g1 n3 Ð n1 + g2 12 = i 13, (8) ()ρ γρ ∂ µ 2 ∂ Ð g2 n3 Ð n2 + g1 21 = i 23, (9) µ g1 µΓ 2 g1 g1 = Ð ------Ð ------g1 + ------(13) g2 ∂t g2 g4 ∂t n1 ++n2 n3 = 1. (10)

2 2 3 Here, n1, n2, and n3 are the populations of the respective µγ∂ g µγ ∂ g + ------1 Ð ------1, ρ ρ* 4 2 6 3 levels, ij = ji are the nondiagonal density-matrix ele- g ∂t g ∂t ប ments, and gi = diEi/ are the Rabi frequencies. Expres- sion (10) ensures the particle-number conservation in a which describes the wave-field evolution under the con- three-level system. ditions of electromagnetically induced transparency 2 Γγ One can readily find from Eqs. (5)–(10) that the time (g > ). Here, the field E1 is expressed through its of establishing stationary population is µ π ω 2 ប Rabi frequency g1 and the parameter = 4 N d13 / c 2 2 Ð1 is introduced. The first term on the right-hand side of g + g τ ≈ 1 2 Eq. (13) accounts for the well-known decrease in the N ------γ , (11) propagation velocity of a probe pulse in the Λ-type sys- while the relaxation time of the low-frequency coher- tem. The second and fourth terms are responsible for ρ the pulse dissipation on the propagation path; a “uni- ence 12 is given by the expression form” (along the pulse length) wave-packet decay (sec- 2 2 Ð1 ond term) is determined by the relaxation rate Γ of low- τ ≈ Γ g1 + g2 12 + ------γ . (12) frequency coherence, and the decay in the case of Γ = 0 is due to the diffusional wave-packet spreading (fourth It then follows that characteristic times (11) and (12) term). The nonlinear third term appears because the shorten with strengthening field and become equal group velocity of a wave packet in the Λ-type system | |2 | |2 ӷ Γγ under the CPT conditions ( g1 + g2 ). depends on the weak amplitude g1. Finally, the last term Let us consider the long-time evolution of a “weak” on the right-hand side of Eq. (13) accounts for the wave | |2 Ӷ | |2 dispersion in the electromagnetic transparency win- wave packet ( g1 g2 ) in more detail. We assume 1 that the radiation at frequency ω is continuous and uni- dow. One can see from Eq. (13) that the dissipative 2 effects in the wave packet with a small (|g | Ӷ |g|), form; i.e., g = const = g. The field amplitude g is much 1 2 though finite amplitude, predominate over the disper- smaller than the saturation amplitude for the optical sion effects. Of all the dissipative effects, we will con- transitions in the Λ-type scheme (g Ӷ γ). In the case of sider only the dissipation associated with the finite a slowly varying (on the relaxation time scale (11), (12)) wave field g1, we will determine the polarization 1 Note that the dispersion relation obtained and analyzed in [11] P1 using the adiabatic approximation. The calculation follows from Eq. (13) in the linear approximation.

JETP LETTERS Vol. 79 No. 6 2004 274 ZHAROVA et al.

quency. By substituting the ﬁeld ψ = G(z, r⊥)exp(iΩτ) into Eq. (14), we obtain the equation

∂G 2 2 Ð i------+0.∆⊥G Ð Ω G G Ð iΩ G = (15) ∂z

It accounts for the radiation self-action in a medium with cubic (Kerr) nonlinearity and dissipation. The nonlinear term is proportional to the frequency shift, and, for the negative detuning from the resonance (Ω < 0), the nonlinearity in the Λ-type scheme is focusing. The absorption coefﬁcient of a monochromatic radiation in the transparency window increases as Ω2, so that the nonlinear effects predominate over the dissipation at small values of detuning |Ω| Ӷ 1. The condition for the predominance of the nonlinear wave-beam refrac-

tion over the diffraction in the axially symmetric case implies that the power is higher than its self-focusing Fig. 2. |ψ|-Field isolines for the initial distribution ψ(z = 0) = critical value. In the dimensional variables, one obtains 1.5exp(Ð0.5(r/5)2 Ð 0.5(τ/20)2 Ð 0.15iτ) and various dis- for the self- focusing critical power tances from the entrance into a medium with electromagnetically induced transparency. cg4 P = 5.58------. (16) cr 8πបµΩ width of the EIT window (i.e., Γ = 0 in Eq. (13)) and introduce the dimensionless variables Evidently, the critical power decreases with increasing the detuning Ω. However, the frequency shift Ω must ψ τ 2 ()γ not exceed the width of EIT window; i.e., ==g1/gm, 2gm tzÐ /v gr / , Ω < g2/γ . (17) ()µ 2 Ð1 µ 4 4γ v gr ==1/c + /g , znew 4 gmz/g , Accordingly, the critical power cannot be smaller than 2 បµ ()2 γ its minimal value rnew = 2gm 2 / g ,

2 min cg γ where gm is a free parameter on the order of the maxi- P = 5.85------. (18) ξ cr πបµ mal field value gm ~ max(g1( , z)) in the wave packet. 8 As a result, we arrive at the equation Critical power (18) decreases also with decreasing 2 amplitude |g| of the controlling field, but it is bounded ∂ψ 2∂ψ ∂ ψ Ð i------++∆⊥ψ i ψ ------+ i------= 0, (14) 2 γΓ ∂ ∂τ 2 from below by the EIT condition g > . In reality, the z ∂τ critical power of a probe wave may become greater than (18) due to the finite pulse duration. which describes the wave-packet self-action dynamics in the coordinate system associated with the pulse 3. Below, we present the results of numerical simulation of the self-action dynamics performed for a finite group velocity vgr. In the experiments, the group velocity, as a rule, is much lower than the velocity of light EIT band using Eq. (14). The evolution of an axially ≈ 2 µ symmetric distribution of the form [3], and the deceleration vgr/c g /c is determined by the medium parameters and the pump-wave amplitude. ψ(),,τ ()τ2 2 2 2 Ωτ r z = 0 = A0 exp Ð /2aτ Ð r /2ar + i (19) In the one-dimensional case (∆⊥ = 0), the self-action dynamics are governed by the competition of the non- was studied, where A0 is the field amplitude, aτ and ar linearity introduced by the amplitude dependence of are the characteristic scales of the distributions in the group velocity with the dissipation and are manifested respective coordinates, and Ω is the detuning from the in the steepening of the pulse leading edge. resonance in the Λ-type scheme. 2. To illustrate the salient features of self-action, we First of all, it should be noted that, in the case of ini- consider the spatial evolution of a continuous radiation tial Gaussian distribution with zero frequency shift at the frequency detuned by Ω from the resonance fre- (Ω = 0), field strengthening (focusing) was attained for

JETP LETTERS Vol. 79 No. 6 2004 SELF-ACTION OF LASER RADIATION 275 none of the amplitudes A0. The calculations with a positive shift (Ω > 0) point to additional defocusing and a more rapid decrease in the field amplitude. Of greater interest is the behavior of the distribution with a negative frequency detuning. The evolution of the initial distribution with parameters A0 = 1.5, ar = 5, and aτ = 20 and negative frequency shift Ω = Ð0.15 is demonstrated in Fig. 2. At the first stage, the field self- focusing occurs against the background of leading- edge steepening, typical of nonlinear dispersion. The rise in the maximal field amplitude is stabilized as the spectrum broadens during the processes of leading- edge modification and spectral harmonic generation outside the transparency band. A dissipative decrease in the wave-field power to a value smaller than critical subsequently brings about wave-packet spreading Fig. 3. Dynamics of the maximal field amplitude A0 and the mainly in the transverse direction. Inasmuch as the total energy E for the same initial conditions as in Fig. 2. All strong decay occurs on the paths of lengths on the order quantities are normalized to their initial values. Inset: the η of the breaking length, the radiation self-focusing dis- dependence of the maximum field strengthening max = |ψ | Ω tance can be estimated using one-dimensional Eq. (14). max /A0 on the initial frequency shift . The solid curve is It is well known that the profile of the shock-wave type for the initial field amplitude A0 = 1.5, and the dashed curve is for a slightly larger amplitude A0 = 1.7. Ӎ τ 2 τ appears in this case on the paths z 0/A0 , where 0 is the pulse duration and A0 is the maximal field amplitude. For the parameters used in our numerical calcula- ditions of increasing (with increasing frequency shift) tion, this estimate gives zf = 9. A somewhat smaller absorption. ≈ value, zf 8, obtained in the numerical simulation (Fig. 3) is likely caused by the increase in the maximal Appreciable radiation self-focusing is possible in field on the axis of the system upon self-focusing. As the case of negative wave-field detuning from the reso- the initial amplitude A increases, the self-focusing pro- nance frequency. For Gaussian beams, it occurs only in 0 Ω ceeds more intensely; i.e., field strengthening with the frequency range < 0, i.e., in half of the transpar- respect to the initial level becomes more and more pro- ency window. In the other half of the EIT band, the radi- nounced (inset in Fig. 3). It should also be noted that the ation self-focusing takes place. The competition of the detuning interval where the radiation self-focusing nonlinearity and absorption in the transparency band occurs in the Λ-type scheme becomes noticeably wider gives rise, in particular, to the optimal frequency shift with an increase in amplitude; the increase in the ampli- for which the self-focusing critical power is minimal. Expression (18) for the self-focusing critical power can tude A0 of the initial distribution from 1.5 to 1.7 (solid and dashed curves in the inset) doubles this interval. conveniently be represented in the form γω 4. Our study of the laser-radiation self-action P = ------λ2 p, dynamics under the EIT conditions has revealed a num- cr Ω2 ber of special features of this process. It should prima- c rily be noted that the self-action in the case of resonant where p = P /L2 is the pump power density, P is wave-field interaction is determined by the amplitude pump pump the pump power, L is the transverse size of the pump dependence of the wave-packet group velocity. In this λ Ω π ω 2 ប 1/2 case, the pulse leading edge steepens upon pulse prop- field, is the wavelength, and c = (8 N d / ) is the agation. For a spatially nonuniform (with respect to the cooperative frequency of the “probe” transition. With transverse coordinate) initial field distribution, the the parameters of the EIT experiment carried out in [3] γ Ӎ 7 Ð1 ω Ӎ 15 Ð1 Ω Ӎ 9 10 Ð1 group-velocity dispersion brings about horseshoe dis- ( 10 s , 10 s , c 10 Ð 10 s ), the tortion of the leading edge of wave packet. self-focusing critical power for beams of size L equal to several thousand wavelengths is on the order of (10−5Ð If the radiation frequency is detuned from reso- 10Ð2)P . nance, the nonlinear effects in the Λ-type scheme pump appear as the Kerr-type nonlinearity, resulting in the We are grateful to M.D. Tokman for helpful discus- radiation self-focusing (self-defocusing). Due to the sions. This work was supported by the Russian Founda- finiteness of the transparency band (determined by the tion for Basic Research (project nos. 01-02-17388 and pump-wave field), these processes proceed under con- 04-02-17147).

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REFERENCES 6. V. G. Arkhipkin and I. V. Timofeev, Kvantovaya Élek- tron. (Moscow) 30, 180 (2000). 1. B. D. Agap’ev, M. B. Gornyœ, B. G. Matisov, and 7. B. G. Matisov and I. E. Mazets, Pis’ma Zh. Tekh. Fiz. 20 Yu. V. Rozhdestvenskiœ, Usp. Fiz. Nauk 163, 1 (1993) (4), 16 (1994) [Tech. Phys. Lett. 20, 140 (1994)]; [Phys. Usp. 36, 763 (1993)]. S. F. Huelga, C. Maochiavello, T. Pellizzari, et al., Phys. 2. M. O. Scully and M. S. Zubairy, Quantum Optics (Cam- Rev. Lett. 79, 3865 (1997). bridge Univ. Press, Cambridge, 1997). 8. J. T. Manassah and V. Gross, Opt. Commun. 124, 418 3. M. D. Lukin, P. Hemmer, and M. O. Scully, Adv. At. (1996). Mol. Opt. Phys. 42, 347 (2000); A. B. Matsko, O. Koch- 9. M. Jain, A. J. Merriam, A. Kasapi, et al., Phys. Rev. Lett. harovskaya, Y. Rostovtsev, et al., Adv. At. Mol. Opt. 75, 4385 (1995). Phys. 46, 191 (2001). 10. Tao Hong, Phys. Rev. Lett. 90, 183901 (2003). 4. J. H. Eberly, M. L. Pons, and H. R. Haq, Phys. Rev. Lett. 11. S. E. Harris, J. E. Field, and A. Kasapi, Phys. Rev. A 46, 72, 56 (1994). R29 (1992). 5. A. V. Andreev, Zh. Éksp. Teor. Fiz. 113, 747 (1998) [JETP 86, 412 (1998)]. Translated by V. Sakun

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JETP Letters, Vol. 79, No. 6, 2004, pp. 277Ð281. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 335Ð339. Original English Text Copyright © 2004 by Artemenko.

Impurity-Induced Stabilization of Luttinger Liquid in Quasi-One-Dimensional Conductors¦ S. N. Artemenko Institute for Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009 Russia e-mail: [email protected] Received December 9, 2003

It is shown theoretically that the Luttinger liquid can exist in quasi-one-dimensional conductors in the presence of impurities in a form of a collection of bounded Luttinger liquids. The conclusion is based on the observation by Kane and Fisher that a local impurity potential in Luttinger liquid acts, at low energies, as an inﬁnite barrier. This leads to a discrete spectrum of collective charge and spin density ﬂuctuations, so that interchain hopping can be considered as a small parameter at temperatures below the minimum excitation energy of the collective modes. The results are compared with recent experimental observation of a Luttinger-liquid-like behavior in thin NbSe3 and TaS3 wires. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 71.10.Pm; 71.27.+a; 71.45.Lr; 73.21.Hb

Electronic properties of one-dimensional (1D) met- undergo the Peierls transition from a metallic state als are known to be very different from those of ordi- either to a semiconducting CDW state (e.g., blue nary three-dimensional (3D) metals (for a review, see bronze K0.3MoO3, TaS3, (TaSe4)2I etc.) or to semimetal- [1Ð3]). 3D electron gas is well described by Landau’s lic CDW state (NbSe3) [5]. Typically, these transitions Fermi-liquid picture, in which interaction modifies free occur in the temperature range 50Ð250 K. From the the- electrons, making them quasiparticles that behave in oretical point of view, the formation of Luttinger liquid many respects like noninteracting electrons. In contrast in quasi-1D conductors at low enough temperatures is to the 3D case, in 1D electronic systems, the Fermi-liq- also problematic because of the instability towards 3D uid picture breaks down even in the case of the arbi- coupling in the presence of arbitrarily small interchain trarily weak interaction. In 1D metals, single-electron hopping [6Ð10]. So, the interchain hopping induces a quasiparticles do not exist, and the only low energy crossover to 3D behavior at low energies, while Lut- excitations turn out to be charge and spin collective tinger liquid behavior can survive only at a high enough modes with the sound-like spectrum. These modes are energy scale where it is not affected by 3D coupling. dynamically independent, giving rise to a spin-charge In contrast to the interchain hopping, the Coulomb separation in 1D systems. Furthermore, correlation interaction between the electrons at different chains functions at large distances and times decay as a power does not destroy the Luttinger liquid state, the main dif- law with interaction-dependent exponents. Such a ference from the 1D case being the absence of simple behavior has been given the generic name Luttinger liq- scaling relations between the exponents of the various uid [4]. correlation functions [11Ð14]. The concept of Luttinger liquid is of great interest in However, in recent experimental studies of temper- view of its application to real physical systems, such as ature and field dependence of conductivity of TaS3 and carbon nanotubes and semiconductor heterostructures NbSe3 in nanosized crystals, a transition from room- with a confining potential (quantum wires and quantum temperature metallic behavior to nonmetallic one Hall effect edge states). The case of special interest is accompanied by the disappearance of the CDW state at quasi-1D conductors, i.e., highly anisotropic 3D con- temperatures below 50Ð100 K was observed [15Ð17]. ductors with chainlike structure. Numerous experimen- The low temperature nonmetallic state was character- tal studies of both organic and inorganic quasi-1D con- ized by power law dependences of the conductivity on ductors at low temperatures typically demonstrate bro- voltage and temperature like that expected in Luttinger ken-symmetry states, like superconductivity, spin- or liquid, or by a stronger temperature dependence, corre- charge-density wave (CDW) states, and a metallic sponding to the variable-range hopping. Similar depen- behavior above the transition temperature with nonzero dences of conductivity were also reported in focused- single-particle density of states at Fermi energy. For ion beam processed or doped relatively thick NbSe3 instance, the most studied inorganic quasi-1D metals crystals [17]. In order to account for such behavior, we study the ¦ This article was submitted by the author in English. possibility of impurity-induced stabilization of a gap-

278 ARTEMENKO less Luttinger liquid state in quasi-1D metals. Impuri- Kane and Fischer [18] found that the backscattering ties in Luttinger liquid are known to act as infinite bar- impurity potential for repulsive potential (Kρ < 1) flows riers, forming the effective boundaries for low energy to infinity under scaling. Their arguments were gener- excitations [18Ð20]. This leads to a dimensional quan- alized by Fabrizio and Gogolin [21] to the case of many tization and, consequently, to a minimal excitation impurities. It was shown that the impurity potential can ω energy 1. As a result, the interchain hopping does not be considered as effectively infinite, provided that the destroy the Luttinger liquid phase at temperatures T Ӷ mean distance, l, between impurities satisfies the con- ω 1, producing only small perturbations of the 1D pic- dition ture. To show this, we consider first the gapless 1D 2/() 1 Ð Kρ ӷ 1 D TomonagaÐLuttinger (TL) model with impurity poten- l -----------, (3) tial included and make certain that the system with kF V 0 impurities breaks up into a set of independent segments where D is the bandwidth. We assume that the impurity described as bounded Luttinger liquid with discrete Շ spectrum. Then, we calculate corrections caused by the potential is of atomic scale, V0 D, and the interaction interchain hopping to thermodynamic potential and to between electrons is not too weak (i.e., Kρ is not too ӷ the one-particle Green’s function and find that such cor- close to 1). Then, condition (3) is satisfied for l 1/kF ~ rections are small at low temperatures. Finally, we dis- α, which is of the order of interatomic distance. So, the cuss modifications introduced by generalization of the limit of strong impurity potential should be a good TL model to the more realistic case of Coulomb poten- approximation in a wide range of impurity concentra- tial and compare our results with experimental data. tions. First of all, we start with the TL model, ignoring Further, Πν, Θν, and Φν must obey the commutation interchain hopping integral t⊥ in order to formulate the relations (see [1Ð3]) ensuring anticommutation of elec- problem in the zeroth approximation in t⊥. Electronic tronic operators (1). Using then the analogy of Eq. (2) operators for right (r = +1) and left (r = Ð1) moving with the Hamiltonian of an elastic string strongly electrons with spin s are given in terms of phase fields pinned at impurity sites, we can write solutions for the as (see [1, 3]) phase operators Φν and Θν in the region between impurity positions at x = xi and xi + 1 as irkF x ψ () e η iAr ∞ rs, x = lim ------rs, e , α → 0 πα Kν + 2 Φν()x = ∑ ------()b + b sinq x˜ (1) n n n n 1 []Θ Φ ()Θ Φ n = 1 Ar = ------ρ Ð r ρ + s σ Ð r σ . 2 Φ ()Φ x˜ i + 1 Ð x˜ Ð li i x˜ + ------δνρ Ð ∑π∆Nν Ð π∆Nν -- , (4) Φν ν ρ j i Here, phase fields (x) are related to charge ( = ) li 2li ji< and spin (ν = σ) densities, while fields Θν(x) are related Π π ∂ Θ ∞ to the momentum operators ν = (1/ ) x ν canonically Φ η Θ () 1 ()+ θ conjugate to ν. Further, r, s are Majorana (“real”) Fer- ν x = ∑ ------bn Ð bn cosqn x˜ + ν, Kνn mionic operators that assure proper anticommutation n = 1 relations between electronic operators with different π Φ spin s and chirality r, and the cut off length α is where x˜ = x Ð xi, li = xi + 1 Ð x, qn = n/li, and i is the π ∆ ∆ assumed to be of the order of the interatomic distance. modulo 2 residue of 2kFxi. Further, Nρi = ( N↑i + ∆ ∆ ∆ ∆ We describe the intrachain properties of the system N↓i)/2 , Nσi = ( N↑i Ð N↓i)/2 , and N↑i ( N↓i) is the by the standard TL Hamiltonian [1, 3] with added 2kF number of extra electrons with spin up (down) in the impurity backscattering term [2]. In the bozonized region between ith and (i + l)th impurities; and finally, θ ∆ θ form, it reads νi is the phase canonically conjugate to Nνi ([ νi, ∆ Nνi] = i). πv νKν 2 v ν 2 Π ()∂ Φ Excitation spectra of the eigenmodes are ων = H = ∑ ∫dx------ν + ------x ν 2 2πKν ω ≡ ω π νρσ= , n 1, ν vνqn, where 1, ν = vν/li is the minimum exci- (2) tation frequency for mode ν. Note that, if we consider the open boundary condi- + ∑V dδ()xxÐ cos()2Φρ + 2k x cos()2Φσ()x , 0 i F tions at the sample ends, x = 0, and x = L (instead of i periodic boundary conditions that are commonly used) ν ρ η where vν are velocities of the charge ( = ) and spin then operators s in Eq. (1) are the same for electrons (ν = σ) modes; Kν = v /vν is the standard Luttinger liq- going right and left. In this case, the electron field oper- F ψ ψ ψ uid parameter, describing the strength of the interac- ator, s(x) = s+(x) + sÐ(x), vanishes at impurity posi- α tion; and V0 and d ~ are amplitude and radius of the tions, x = xi, and expressions for the phase fields scattering potential, respectively. between the impurity sites turn out to be similar to

JETP LETTERS Vol. 79 No. 6 2004 IMPURITY-INDUCED STABILIZATION 279 those found for bounded 1D Luttinger liquid in [21Ð Then, we use Eq. (4) in (1) and calculate the average 23], the main difference being the summation over j < i in (7) using the relation that ensures proper commutation relations between the 〈〉{}() {}() electron operators related to different segments T τ exp iArm, 1 exp ÐiArm, 2 between impurities. Thus, the system breaks up into a  set of independent segments described as bounded Lut- 1〈 2 () 2 () = expÐ --- Arm, 1 + Arm, 2 (8) tinger liquid with discrete spectrum.  2 Now, we consider the role of interchain hopping  adding to (2) the hopping Hamiltonian []〉() () Ð2T τ Arm, 1 Arm, 2 . ψ+ ()ψ ()  H⊥ = t⊥ ∑ ∫dx rsm,, x rsn,, x + HC mnrs,,, ∝ ω Neglecting small corrections exp[Ð 1, ν/T] due to η η Planck’s distribution functions, we find for the average it⊥ rsn,, rsm,, [ () = ∑ ∫dx------πα - sin Arm, Ð Arn, (5) in the exponent mnrs,,, 〈〉() () T τ Ar 1 Ar 2 + sin()A , Ð2A , + irk x ], rm Ðrn F Ð1 ()Kν Ð Kν 1 ()coshzyÐ cos where indices n and m, denoting the chain numbers = --- ln ------+ - (9) ()Ð1 related to the nearest neighbors, are added. 8 ()Kν + Kν coshzyÐ cos Ð Arguments by Schulz [8] on instability of the Lut- tinger liquid in the presence of the interchain hopping Ð1 sinyÐ were based on calculations of temperature dependence + itan ------z - , of the thermodynamic potential at low temperatures. e Ð cosyÐ So, we calculate the contribution of the interchain hop- π ˜ ± ˜ π α v |τ τ | ping to the thermodynamic potential per unit volume where y± = (x1 x2 )/li, z = ( + ν 1 Ð 2 )/li (chain given by the standard expression [24] indices are dropped for brevity here). In the integration over 1 and 2, the leading contribu- 1/T ≈ | | Ӷ Ӷ tions comes from the region 1 2, i.e., yÐ 1, z 1 ∆Ω = Ð TSln 〈〉/V, S = Tτ expÐ H⊥()ττ d , (6) ∫ where expression (7) reduces to 0 2 []()() τ t⊥mL cos rrÐ ' kF xÐ dxÐd Ð where V is the volume, Tτ stands for imaginary time ------∫------δ , (10) 〈 〉 π2α2 2 2 12+ ν ordering, and … means thermodynamic averaging T ∏ []()1 + ⑀ντ + ()x /α over the unperturbed state. Ð Ð νρσ= , ӷ ω At temperatures T 1, ν, the discreteness of the 1 excitation spectrum can be neglected; hence, according where ⑀ν = vν/πα, δν = --- (Kν + 1/Kν Ð 2), τ = |τ Ð τ |, to [6Ð10], interchain hopping is expected to make sig- 4 Ð 1 2 nificant contributions, destroying the Luttinger liquid. x = x Ð x , and m is the number of the nearest-neighbor Ӷ Ð 1 2 We examine the opposite limit, T w1, ν, which does chains. Contribution to expression (7) from integration | | տ տ α 2δ δ not exist in pure infinite Luttinger liquid. over the region yÐ 1, z 1, is small, ~( /li) , = δ δ δ Consider first the second order correction in t⊥. The ρ + σ, because is not too small in the assumed case leading contribution to 〈S〉 in Eq. (6) is given by items of the not too small interaction (cf. Eq. (3)). in which the term related to a given chain contains con- Additional items in 〈S〉 in Eq. (6), in which the terms tributions from the electrons with the same chirality, r, related to the same chain contain contributions from only, electrons moving both left and right, is smaller than that K + K 2 given by Eq. (10) by a factor of ~(α/l ) ρ σ . For rea- t⊥ 〈 {}[]() () i ∑ ------∫d1d2 T τ exp iArm, 1 Ð Ar', n 1 sonable values of Kν, this contribution is small and can 8π2 Ð α2 mrr,,' (7) be neglected. ()() irÐ r' kF x1 Ð x2 ∆Ω × exp{}〉ÐiA[], ()2 Ð A , ()2 e , Similarly, the leading contribution to from rm r' n higher-order terms in series expansion of the exponen- τ τ where 1 = {x1, 1} and 2 = {x2, 2}. Other items in which tial in Eq. (6) was found to come from even powers 2n the terms related to the same chain contain contribu- in t⊥ that can be represented as a sum of (2n Ð 1)!! items tions from electrons moving both left and right make like (7) with almost coinciding times and coordinates. small a contribution, and we do not discuss them in Therefore, summing up the leading contributions and detail. inserting them into Eq. (6), we can calculate the varia-

JETP LETTERS Vol. 79 No. 6 2004 280 ARTEMENKO tion of the thermodynamic potential per single chain realistic Coulomb potential. It is reasonable to assume and per unit length that the long-range part of the interaction is dominated 2 by the Coulomb potential, while the backscattering is t⊥m described by relatively small coupling constant g . This ∆Ω = Ð a------, 1 π2v enables us to concentrate on the spin isotropic case and F to ignore the possibility of the spin gap. The problem of ()α τ (11) 12+ cos kF x dxd the long-range Coulomb potential on an array of chains a = ------δ-. ∫ 12+ ν was solved in [11Ð13]. It was found that interaction of []()τ 2 2 ∏ 1 + /Kν + x electrons on a given chain is screened by the electrons νρσ= , on other chains, and the Coulomb interaction can be For moderate repulsion, δ ~ 1, a ~ 1. In the limit of described by the TL Hamiltonian with coupling con- 2 stants dependent on wave vector, strong repulsion, Kρ Ӷ Kσ ~ 1, a is small, a ~ Kρ . 4πe2 Thus, ∆Ω is much smaller than the thermodynamic g ==g ------, (15) Ω 2 4 2 2 2 potential of purely 1D Luttinger liquid, 0 ~ (1/Kρ + s ()q|| + ⑀⊥q⊥ 1/Kσ)ε k , F F where s is the lattice period in the direction perpendic- 2 ular to the chains and ⑀⊥ is a background dielectric con- ∆Ω Ω ∼ t⊥ / 0 ε----- , stant for the transverse direction. Coupling constants in F spin channel remain unaffected. In principle, the cou- and temperature-dependent corrections to Eq. (11) are pling constants must be determined by matrix elements determined by small thermally activated contributions of Coulomb potential that depend on the details of wave ∝ ω exp[Ð 1, ν/T]. functions and on chain arrangement and must contain Now, we calculate modification of the one-particle an infinite sum over transverse reciprocal lattice vec- Green’s function due to the interchain hopping tors. So, expression (15) is not universal and depends on material. (), 〈〉ψ()ψ() 〈〉 G 11' = Ð T τ 1 1' S / S . (12) Equation (15) leads to q-dependent velocities Again, we consider the low-temperature limit, T Ӷ v ω ω F 1, ν, not existing in a pure infinite system. Consider ρ = ------q||, Kρ first the second order correction in t⊥ to the Green’s 2 (16) function of a pure 1D system, G0(1, 1'), 1 ℵ 2 2 8e ------= 1 +,------ℵκ= s = ------, (), 〈〉ψ()ψ() 2 2 2 បv G2 11' = Ð T τ 1 1' S2 Kρ s ()q|| + ⑀⊥q⊥ F (13) 〈〉ψ()ψ()〈〉 + T τ 1 1' S2 . where κ is the inverse ThomasÐFermi screening length. Calculation is similar to that considered above (cf. (7)Ð We do not perform explicit calculations, restricting (9)). However, in contrast to the case of the thermody- ourselves to estimations. For the case of q-dependent namic potential, where the leading contribution was coupling, expressions for the thermodynamic potential made by regions of almost coinciding values of times and Green’s functions contain various integrals of cor- and coordinates, such contributions from two terms in relation functions over q⊥. One can show that the (9) cancel each other. So, the second-order correction is results obtained above can be generalized qualitatively estimated as to the case of long-range Coulomb interaction if we substitute q⊥ in Eq. (16) for its characteristic value, 2 2δ t⊥l α (), Շ  (), q⊥ ~ π/s. For example, integrals for corrections to the G2 11' --------- G0 11' . v F l thermodynamic potential are dominated by close values of coordinates and times, similar to Eq. (10), and Estimation of the fourth-order correction in t⊥ gives coupling parameters should be substituted by their 2 α 2δ G4 ~ (t⊥l/vF) ( /l) G2. Therefore, we conclude that at averages over q⊥, Ӷ ω T 1, ν the interchain hopping gives small corrections 1 to the one-particle Green’s function, provided that δ = ---()Kρ +++1/Kρ kσ 1/Kσ Ð 4 2 2δ 2 2δ Ð 2 4 t⊥l α t⊥ α ------∼ ------Ӷ 1, (14) v  ε  2 2 F l F l 1 s d q⊥1 ∼ ------Kρ + ------Ð 2 . α ∫ 2  where we estimated the cut-off parameter as ~ 1/kF. 4 ()2π Kρ So far we considered the TL model in which inter- ℵ α α action is described by coupling constants related to for- Note that = 8 (c/vF), where is the fine structure ward- and backscattering. In order to make a compari- constant. Since vF is much smaller than the velocity of ℵ ≈ × 7 son with experimental data, we must consider a more light, the factor is large. For vF 2 10 cm/s, which

JETP LETTERS Vol. 79 No. 6 2004 IMPURITY-INDUCED STABILIZATION 281 is a typical value for transition metal trihalcogenides, Foundation for Basic Research, by INTAS, CRDF, and ℵ ~ 90. This corresponds to the case of strong interac- by collaborative program with the Netherlands Organi- tion and leads to quite large values of coupling param- zation for Scientific Research. eters, 1/Kρ ~ ℵ/⑀⊥ ~ 3Ð8, δ ~ 1/2Ð2. Now, we discuss conditions for observation of the REFERENCES Luttinger liquid in quasi-1D conductors stabilized by 1. J. Voit, Rep. Prog. Phys. 58, 977 (1995). impurities. First, we discuss the condition for the tem- 2. M. P. A. Fisher and L. I. Glazman, in Mesoscopic Elec- perature limiting from above the region where the Lut- tron Transport, Ed. by L. L. Sohn, L. Kouwenhoven, and Ӷ ω tinger liquid can exist. This condition reads T 1, ν. G. Schön (Kluwer Academic, Dordrecht, 1997). ω The minimal excitation energies, 1, ν, can be estimated 3. H. J. Schulz, G. Cuniberti, and P. Pieri, Field Theories as for Low-Dimensional Condensed Matter Systems (Springer, Berlin, 2000); cond-mat/9807366. បv ω F ∼ ε 1 ∼ ε 4. F. D. M. Haldane, J. Phys. C 14, 2585 (1981). 1, ν = ------1/Kν F------1/Kν Fci1/Kν, l kFl 5. G. Grüner, Density Waves in Solids (Addison-Wesley, Reading, 1994). where ci stands for dimensionless impurity concentra- 6. V. N. Prigodin and Yu. A. Firsov, Zh. Éksp. Teor. Fiz. 76, tion corresponding to number of impurities per one 736 (1979) [Sov. Phys. JETP 49, 369 (1979)]; Zh. Éksp. electron. As Fermi energy in NbSe3 and TaS3 is about Teor. Fiz. 76, 1602 (1979) [Sov. Phys. JETP 49, 813 1 eV, we find that ω ρ is about 100 K for an impurity (1979)]; Yu. A. Firsov, V. N. Prigodin, and Ch. Seidel, 1, Phys. Rep. 126, 245 (1985). concentration of c ~ 10Ð2Ð10Ð3. i 7. S. A. Brazovskiœ and V. M. Yakovenko, Zh. Éksp. Teor. Another condition to be fulfilled is the smallness of Fiz. 89, 2318 (1985) [Sov. Phys. JETP 62, 1340 (1985)]. corrections to the Green’s function due to interchain 8. H. J. Schulz, Int. J. Mod. Phys. B 5, 57 (1991). hopping. According to Eq. (14), the corrections are 9. D. Boies, C. Bourbonnais, and A.-M. S. Tremblay, Phys. small, provided that Rev. Lett. 74, 968 (1995); cond-mat/9604122. 2 10. E. Arrigoni, Phys. Rev. Lett. 83, 128 (1999). t⊥ δ ∨ 2 Ð 2 Ӷ ∨ ----- ci 1. 11. S. Baris ic , J. Phys. (Paris) 44, 185 (1983). ε ∨ ∨ ∨ F 12. S. Botric and S. Baris ic , J. Phys. (Paris) 44, 185 (1984). If the interaction is strong enough, δ տ 1, this condition 13. H. J. Schulz, J. Phys. C: Solid State Phys. 16, 6769 is no stricter than the condition for the limiting temper- (1983). ature discussed above. For lower strength of interac- 14. R. Mukhopadhyay, C. L. Kane, and T. C. Lubensky, tion, δ < 1, this condition reads Phys. Rev. B 64, 045120 (2001). 15. S. V. Zaœtsev-Zotov, V. Ya. Pokrovskiœ, and P. Monceau, 1/() 2Ð 2δ ӷ t⊥ Pis’ma Zh. Éksp. Teor. Fiz. 73, 29 (2001) [JETP Lett. 73, ci ε----- . 25 (2001)]. F 16. S. V. Zaitsev-Zotov, Microelectron. Eng. 69, 549 (2003). Estimating t⊥ as being of the order of the Peierls transi- 17. S. V. Zaitsev-Zotov, M. S. H. Go, E. Slot, and H. S. J. van ε tion temperature, TP ~ 100Ð200 K ~ 0.01 F, we find that der Zant, Phys. Low-Dimens. Semicond. Struct. 1Ð2, 79 this condition can be fulfilled easily even at δ = 1/2 for (2002); cond-mat/0110629. relatively small impurity concentration, c ӷ 10Ð2. 18. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68, 1220 i (1992). Thus, we find that Luttinger liquid can be stabilized 19. K. A. Matveev and L. I. Glazman, Phys. Rev. Lett. 70, by impurities in relatively pure linear-chain compounds 990 (1993). at rather high temperatures corresponding to experi- 20. A. Furusaki and N. Nagaosa, Phys. Rev. B 47, 4631 mental observation [15Ð17] of the transition from (1993). metallic to nonmetallic conduction characterized by 21. M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51, 17827 power law dependences of conductivity and by conduc- (1995). tivity resembling the variable-range hopping. However, 22. S. Eggert, H. Johannesson, and A. Mattsson, Phys. Rev. in order to make a detailed comparison with the exper- Lett. 76, 1505 (1996). imental data, calculation of the conductivity in a ran- 23. A. E. Mattsson, S. Eggert, and H. Johannesson, Phys. dom network made of weakly coupled bounded Lut- Rev. B 56, 15615 (1997). tinger liquids is needed. 24. A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskiœ, The author is grateful to S.V. Zaitsev-Zotov for use- Methods of Quantum Field Theory in Statistical Physics ful discussions. This work was supported by Russian (Fizmatgiz, Moscow, 1962; Dover, New York, 1963).

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 282Ð285. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 340Ð343. Original Russian Text Copyright © 2004 by Zavaritskaya, Karavanskiœ, Mel’nik, Pudonin.

Diamond-Like and Graphite-Like Carbon States in Short-Period Superlattices T. N. Zavaritskaya*, V. A. Karavanskiœ**, N. N. Mel’nik*, and F. A. Pudonin* * Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia ** Natural Sciences Center, Institute of General Physics, Russian Academy of Sciences, Moscow, 119991 Russia Received January 29, 2004; in ﬁnal form, February 9, 2004

The Raman and luminescence spectra are studied in superlattices consisting of carbon layers separated by thin SiC barrier layers. It is shown experimentally that, upon the avalanche annealing of an initially amorphous superlattice, the carbon layers can crystallize into either a diamond-like or graphite-like structure, depending on the geometrical parameters of the superlattice. A method is proposed for obtaining carbon films with a specified crystal modification within a unified technology. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 78.67.Pt; 68.65.Cd

To date, a great many investigations have been firmed that the annealing of the grown samples resulted devoted to studying the physical properties of carbon in the formation of a graphite-like or diamond-like structures and technological processes providing the structure in the samples of superlattices with the corre- basis for their formation. This is connected with the sponding thickness of carbon layers. wide variety and unique properties of carbon com- Thus, we proposed and implemented a method for pounds, which have determined their widespread prac- obtaining carbon films with a desired crystal modifica- tical use in areas from nanoelectronics to spacecrafts. tion. It is presumed that this method will allow speci- However, the preparation of various carbon modifica- fied carbon modifications to be obtained in a controlled tions within a unified technology is a complicated prob- way within a unified technology. lem, which still remains unsolved [1]. Radio-frequency sputtering was used to obtain car- Previously [2], the authors have shown that the peri- bon superlattices. The C and SiC deposition rates were odic potential arising as a result of the alternation of 0.7 and 3.3 nm/min, respectively. Graphite and poly- layers differing in the band gap can affect the modifica- crystalline silicon carbide served as the targets for tion in which the substance crystallizes. The grown obtaining the carbon and SiC layers. The substrate was superlattices are also characterized by the existence of KEF-4.5 (100) silicon. A Co layer ~50-nm thick, which a periodic potential, which can be favorable to the for- was necessary for the subsequent annealing of the as- mation of a specific carbon modification. grown superlattice, was sputtered on the substrate. The In this work, it is proposed that the annealing of layer thicknesses were controlled by the sputtering amorphous superlattices based on carbon and a broad- time, because the deposition rate in this method band semiconductor (here, SiC) with varying layer remained constant during the entire technological pro- thicknesses be used to obtain various carbon modifica- cess and did not depend on the thickness of the depos- tions. It is assumed that, upon pulse annealing, carbon ited film. The details of the process and the growth fea- in these structures will transform to a certain modifica- tures for such superlattices are given in [2]. tion determined by the superlattice parameters. The conventional thermal annealing is unsuitable Multilayer periodic structures C/SiC with specified for short-period superlattices, because the mutual diffu- thicknesses of the C and SiC layers were manufactured sion of layer atoms in the course of annealing leads to for these experiments. The thickness of carbon layers the mixing of layers and the degradation of the super- was chosen to be a multiple of the lattice constant of a lattice. Therefore, as in [2], a Co underlayer was used known carbon structural modification. Thus, the thick- in annealing. Upon short-term low-temperature heat- nesses of carbon layers for the graphite- and diamond- ing, this underlayer transformed from an amorphous like structures were 1.34 and 0.8 nm, respectively. It state to a crystalline state in a time of ~10Ð11Ð10Ð10 s was assumed that carbon in the structures with the with heat evolution. This heat is necessary for the graphite-like thickness will crystallize in the course of superlattice annealing. The new type of annealing pro- annealing into a graphite-like crystalline configuration, posed in [2] was based on the fact that the internal whereas carbon in the structures with the diamond-like energy of a layer in an amorphous (disordered) state is thickness will transform into a diamond-like structure. higher than the energy of the same layer in a crystalline Our Raman scattering and luminescence studies con- or polycrystalline modification. This difference in ener-

DIAMOND-LIKE AND GRAPHITE-LIKE CARBON STATES 283 gies is evolved upon layer transition from an amor- Table phous to crystalline state. However, for an amorphous C thickness SiC thickness Number phase to transform to the crystalline phase, it is neces- Sample no. sary to overcome an activation barrier. For a number of (Å) (Å) of periods materials, for example, Co, this barrier is not very high 1143220 and can be overcome by the short-term (~1Ð2 min) low- 2141620 temperature (300Ð400°C) heating of this material. The transition rate of a material from an amorphous state to 3 13.4 16 20 the crystalline state is on the order of the sound veloc- 4 8 16 20 ity, and the amount of evolved heat is proportional to the mass of this material. As was shown in [2], the mutual diffusion of superlattice layer atoms does not ing from both Si and SiO2. The parameters of the peri- proceed under this type of annealing. This type of ava- odic SL potential in the case of carbon SLs are deter- lanche annealing is in many ways similar to the pulse mined not only by the C and SiC band gaps but also by laser annealing. the SL thicknesses. To provide a stronger effect of this It should be noted that, according to [2], two pro- potential, the parameters of superlattice potential cesses can take place during avalanche annealing. The should be brought closer to the parameters of the KP first one is a simple annealing of the C and SiC layers. potential. For this purpose, we manufactured SLs with Depending on the thickness of the C layers, carbon narrow barriers (1.6 nm), whose periodic potential was transforms to either a diamond-like (diamond-like similar to the KP potential, and with wide SiC barrier superlattice, DSL) or graphite-like (graphite-like super- layers (3.2 nm), for which the difference of the superlattice, GSL) modification if the number of carbon lattice potential from the KP potential was more pro- atoms corresponds to only an integer number of mono- nounced. layers of these two modifications. Note that, because of Porous structures were prepared from all obtained the difference in the densities of the deposited carbon, SLs by anodic etching. It is known that anodic etching diamond, and graphite, it is diamond-like and graphite- starts (and predominantly continues) at defects. Previ- like modifications, rather than pure crystalline diamond ously, we showed that porous films could form with a or graphite structures, that can be obtained in the course lower concentration of crystal defects and doping of annealing. By varying the thickness of the carbon impurity [3] than in the starting crystal. Moreover, in layers, it is possible to achieve the synthesis of predom- some cases [3], anodic etching is accompanied by the inantly diamond or graphite structures. It is not incon- formation of nanocrystalline objects, whose properties ceivable that certain intermediate crystalline carbon are also of great interest. states can exist that have not been synthesized previously. It should be noted here that the thicknesses of The parameters of the synthesized samples are given carbon layers in an SR must be small, because the dif- in the table. ference between the DSL and GSL will disappear as the The Raman spectra of the synthesized samples are thickness increases. On this basis, carbon SLs were shown in Fig. 1. The Raman spectra of the sputtered grown with carbon layer thicknesses of 0.8 nm (the dia- C/SiC SLs are almost identical in shape (Fig. 1a), and mond-like layer with d ~ 4a, where a = 0.205 nm is the their shape corresponds to the Raman spectrum of a interplanar spacing in diamond in the (111) direction) strongly disordered carbon film (Fig. 1c). After anneal- and 1.34 nm (the graphite-like layer with d ~ 4a, where ing, the spectra of the samples under study became a = 0.335 nm is the interplanar spacing in graphite in noticeably different (Fig. 1b). The Raman spectrum of the (002) direction). sample 1 with the largest thickness of the SiC layers The second process that can proceed during the changed insignificantly (Fig. 1b, curve 1). The spectra course of SL annealing is associated with the one- of the other samples exhibit shoulders in the region of dimensional periodic SL potential arising as a result of D (~1350 cmÐ1) and G (~1580 cmÐ1) bands (see lower the alternation of materials differing in the band gap spectrum in Fig. 1c) differing in the intensity ratios. and its action on the crystallization process. As was These bands correspond to the presence of sp3 and sp2 shown in [2], in the course of annealing, when superlat- bonds, respectively, in the carbon material [4]. The tice atoms are in a nonequilibrium state, the superlattice Raman spectra of the annealed samples subjected to potential is the only strong potential in the structure anodic etching are shown in Fig. 1c. As was already being annealed. This potential, by analogy with the noted, atoms with a lower binding energy (defects, KronigÐPenny (KP) potential, will strongly influence impurities, etc.) are removed first upon anodic etching. the formation of electronic and crystal structures of the Therefore, the remaining part of the material can have sample. In the case of Si/SiO2 superlattices with a small a more perfect crystal structure [3]. We observe a simi- period (of order 1.0Ð1.5 nm), it was found that the one- lar picture for the structures under study as well. The D dimensional periodic potential formed, in effect, the and G bands are clearly seen in the Raman spectrum of superlattice crystal structure, resulting in a material the etched annealed samples (Fig. 1c); however, these with hexagonal symmetry and lattice parameters differ- bands have different halfwidths and intensities. We note

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284 ZAVARITSKAYA et al.

Fig. 1. Raman spectra of (a) sputtered SL samples, (b) samples after avalanche annealing, and (c) porous annealed samples. The Raman spectra of a disordered graphite (two lines) and a carbon ﬁlm obtained by the decomposition of hydrocarbons (broad asymmetric band) are given at the bottom of graph (c) for comparison. In all graphs, the Raman spectra are shifted vertically with respect to each other for convenience. The numbers at the curves correspond to the sample number.

Fig. 2. PL spectra of (a) sputtered samples, (b) samples after avalanche annealing, and (c) annealed samples after anodic etching. two features of the observed Raman spectra. First, the curve 3) became similar to the Raman spectrum of a smallest change in the bandshape is observed for sam- strongly disordered graphite. ple 1, in which the thickness of the SiC layers is twice The photoluminescence (PL) spectra of the synthe- as large as in the other samples. This fact suggests that sized samples are shown in Fig. 2. Note that the PL the interaction between carbon layers influences the spectra of sample 1 remained virtually unchanged after process of carbon structure formation. Second, the annealing and etching and, hence, are not given in the bandshapes of the Raman spectra of samples 2 and 3 figure. The spectra of the starting samples were almost differ substantially after annealing and anodic etching identical (Fig. 2a). After annealing, the luminescence (Fig. 1c, curves 2, 3), although the thicknesses of the intensity of sample 4 increased significantly (Fig. 2b, carbon layers differ by only 5% (1.4 and 1.34 nm). We curve 4). The thickness of the carbon layers in this sam- add that the spectrum of graphite-like sample 3 (Fig. 1c, ple is close to the size of a diamond monolayer. Previ-

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DIAMOND-LIKE AND GRAPHITE-LIKE CARBON STATES 285 ously, we showed [5] that the simultaneous presence of This work was supported by the Russian Foundation the sp3 and sp2 bonds is necessary for the luminescence for Basic Research, the Federal Target Program “Inte- of carbon nanostructures. It can be assumed that the gration of Higher Educational Institutions and Basic number of sp3 bonds appeared in sample 4 upon anneal- Science,” Complex Scientific Problem of the Federal ing was greater than in the other samples, which led to Target Scientific and Technical Program “Physics of the increase in the PL intensity. After anodic etching, the Solid-State Nanostructures,” the Program of the Presid- PL intensity of sample 2 increased (Fig. 2c, curve 2). ium of the Russian Academy of Sciences “Low-Dimen- We believe that this sample must be intermediate in sional Quantum Structures,” and the Program of the relation to the number of sp3 bonds. In fact, the spectra State Support of Leading Scientific Schools (project demonstrate that graphite-like sample 3 has the lowest no. NSh-1923.2003.2). PL intensity, diamond-like sample 4 has the highest PL intensity, and sample 2 is intermediate in PL intensity. REFERENCES Hence, based on the data of Raman and PL spectra, it may be stated that the crystal structure of carbon lay- 1. Proceedings of 2nd International Conference on Car- bon: Fundamental Problems of the Science, Materials ers formed upon the avalanche annealing of superlat- Science, and Technology (Moscow, 2003). tices depends on the superlattice geometrical parameters. 2. A. F. Plotnikov, F. A. Pudonin, and V. B. Stopachinskiœ, Pis’ma Zh. Éksp. Teor. Fiz. 46, 443 (1987) [JETP Lett. In summary, a method has been proposed and 46, 560 (1987)]. implemented for obtaining carbon films with a pre- 3. N. N. Melnik, T. N. Zavaritskaya, M. V. Rzaev, et al., scribed crystal modification within a unified technol- Proc. SPIE 4069, 212 (2000). ogy. It has been shown experimentally that various modifications of the carbon crystal structure can be 4. M. Ramsteiner and J. Wagner, Appl. Phys. Lett. 51, 1355 obtained in the superlattice layers by choosing the (1987). appropriate thickness of these layers. On the basis of 5. V. A. Karavanskiœ, N. N. Mel’nik, and T. N. Zavar- this effect, the technology can be developed for obtain- itskaya, Pis’ma Zh. Éksp. Teor. Fiz. 74, 204 (2001) ing carbon nanostructures with prescribed properties. [JETP Lett. 74, 186 (2001)]. We are grateful to Prof. N.N. Sibel’din for useful discussions. Translated by A. Bagatur’yants

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 286Ð292. From Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 344Ð350. Original English Text Copyright © 2004 by Shaginyan.

Universal Behavior of Heavy-Fermion Metals Near a Quantum Critical Point¦ V. R. Shaginyan St. Petersburg Nuclear Physics Institute, Gatchina, 188300 Russia e-mail: [email protected] CTSPS, Clark Atlanta University, Atlanta, Georgia 30314, USA Received November 24, 2003; in ﬁnal form, February 9, 2004

The behavior of the electronic system of heavy-fermion metals is considered. We show that there exist at least two main types of the behavior when the system is near quantum critical point, which can be identified as the fermion condensation quantum phase transition (FCQPT). We show that the first type is represented by the behavior of a highly correlated Fermi liquid, while the second type is depicted by the behavior of a strongly correlated Fermi liquid. If the system approaches FCQPT from the disordered phase, it can be viewed as a highly correlated Fermi liquid which at low temperatures exhibits the behavior of Landau Fermi liquid (LFL). At higher temperatures T, it demonstrates the non-Fermi liquid (NFL) behavior which can be converted into the LFL behavior by the application of magnetic fields B. If the system has undergone FCQPT, it can be considered as a strongly correlated Fermi liquid which demonstrates the NFL behavior even at low temperatures. It can be turned into LFL by applying magnetic fields B. We show that the effective mass M* diverges at the very point that the Neél temperature goes to zero. The BÐT phase diagrams of both liquids are studied. We demonstrate that these BÐT phase diagrams have a strong impact on the main properties of heavy-fermion metals, such as the magnetoresistance, resistivity, specific heat, magnetization, and volume thermal expansion. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 71.27.+a; 71.10.Hf; 71.10.Ay

In heavy-fermion (HF) metals with strong electron It is the very nature of HF metals that suggests that correlations, quantum phase transitions at zero temper- their unusual properties are defined by a quantum phase ature may strongly influence the measurable quantities transition related to the unlimited growth of the effec- up to relatively high temperatures. These quantum tive mass at its QCP. Moreover, a divergence to infinity phase transitions have recently attracted much attention of the effective electron mass was observed at a mag- because the behavior of HF metals is expected to follow netic field-induced QCP [3–5]. We assume that such a universal patterns defined by the quantum mechanical quantum phase transition is the fermion condensation nature of the fluctuations taking place at quantum criti- quantum phase transition (FCQPT), an essential feature cal points (see, e.g., [1, 2]). It is widely believed that the of which is the divergence of the effective mass M* at proximity of the electronic system of an HF metal to its QCP [6, 7]. FCQPT takes place when the density x quantum critical points may lead to non-Fermi liquid of a system tends to the critical density xFC, so that ∝ | (NFL) behavior. The system can be driven to quantum M* 1/r, where r is the distance from the QCP, r = x Ð | critical points (QCPs) by tuning control parameters xFC . Such a behavior does not qualitatively depend on other than temperature, for example, by pressure, by the system’s dimensions and is valid in cases of both magnetic field, or by doping. When a system is close to two-dimensional (2D) and three-dimensional (3D) a QCP, we are dealing with the strong coupling limit Fermi systems [8, 9]. Beyond FCQPT, the system pos- where no absolutely reliable answer can be given on sesses fermion condensation (FC) and represents a new pure theoretical first principle grounds. Therefore, the state of electron liquid with FC [7, 10]. As soon as only way to verify what type of quantum phase transi- FCQPT occurs, the system becomes divided into two quasiparticle subsystems: the first is characterized by tion occurs is to consider experimental facts which describe the behavior of the system. Only recently have quasiparticles with the effective mass M*FC , while the experimental facts appeared which deliver experi- second one is occupied by quasiparticles with mass mental grounds to understand the nature of quantum M*L . The quasiparticle dispersion law in systems with phase transition producing the universal behavior of FC can be represented by two straight lines, character- HF metals. ized by the effective masses M*FC and M*L and inter- ¦ This article was submitted by the author in English. secting near the binding energy E0. Properties of these

UNIVERSAL BEHAVIOR OF HEAVY-FERMION METALS 287

∝ new quasiparticles with M*FC are closely related to the electron liquid and the effective mass behaves as M* state of the system, which is characterized by the tem- 1/T, leading to the NFL behavior even at low tempera- ≥ perature T, pressure P, or by the presence of supercon- tures. The application of a magnetic field (B Ð Bc) ∝ 2 ≥ ductivity. We may say that the quasiparticle system in B*(T) T restores the LFL behavior. At (B Ð Bc) ∝ Ð1/2 the range occupied by FC becomes very “soft” and is to B*(T), the effective mass M*(B) (B Ð Bc) , being be considered as a strongly correlated liquid. Nonethe- approximately independent of the temperature at T ≤ less, the basis of the Landau Fermi liquid theory [11] ∝ ≥ survives FCQPT: the low energy excitations of the T*(B) BBÐ c . At T T*(B), both the 1/T depen- strongly correlated liquid with FC are quasiparticles, dence and the NFL behavior are reestablished. We while this state can be considered as a quantum protec- show that the effective mass M* diverges at the very torate [6]. The only difference between the Landau point that the Neél temperature goes to zero. It is dem- Fermi liquid and Fermi liquid after FCQPT is that we onstrated that obtained BÐT phase diagrams have a have to expand the number of relevant low energy strong impact on the main properties of HF metals, degrees of freedom by introducing a new type of quasi- such as the magnetoresistance, resistivity, specific heat, magnetization, and volume thermal expansion. particles with the effective mass M*FC and the energy We start with the case of a highly correlated electron scale E0 [6]. It is possible to provide a consistent picture of high-T metals as the strongly correlated Fermi liquid liquid when the system approaches FCQPT from the c disordered phase. FCQPT manifests itself in the diver- [12]. gence of the quasiparticle effective mass M* as the den- When a Fermi system approaches FCQPT from the sity x tends to the critical density xFC or the distance disordered phase, its low energy excitations are Landau r 0 [8, 9] quasiparticles which can be characterized by the effec- 1 1 tive mass M*. This mass strongly depends on the dis- M* ∝∝------. (1) tance r, temperature, and magnetic fields B [8]. At low xxÐ FC r temperatures, it becomes a Landau Fermi liquid with Since the effective mass M* is finite, the system exhib- the effective mass M*, provided that r > 0. This state of its the LFL behavior at low temperatures T ~ T*(x) ∝ the system, with M* strongly depending on T, r, and B, |x Ð x |2 [8]. The quasiparticle distribution function resembles the strongly correlated liquid. In contrast to FC the strongly correlated liquid, there is no energy scale n(p, T) is given by the equation E0 and the system under consideration is the Landau δΩ 1 Ð n()p, T ------==ε()µp, T Ð ()T Ð0.T ln------(2) Fermi liquid at T 0. Therefore, this liquid can be δn()p, T n()p, T called a highly correlated liquid. Such a highly correlated Fermi liquid was observed in nonsuperconducting The function n(p, T) depends on the momentum p and Ω µ La1.7Sr0.3CuO4 [8, 13]. the temperature T. Here, = E Ð TS Ð N is the thermodynamic potential and µ is the chemical potential, In this Letter, we continue to show that, within the while ε(p, T), framework of FCQPT, it is possible to understand the δ []() NFL behavior observed in different strongly and highly ε(), Enp p T = ------δ (), , (3) correlated Fermi liquids such as high-Tc superconduc- n p T tors [12] and HF metals. We apply the theory of fermion is the quasiparticle energy. This energy is a functional condensation to describe the behavior of the electronic of n(p, T) just like the total energy E[n(p)], entropy system of HF metals and to show that there exist at least S[n(p)], and the other thermodynamic functions. The two main types of the behavior. If the system entropy S[n(p)] is given by the familiar expression approaches FCQPT from the disordered phase, it can be viewed as the highly correlated electron liquid and Snp[]() = Ð2 [n()p, T lnn()p, T the effective mass M* depends on temperature, M* ∝ ∫ TÐ1/2. Such a dependence of M* leads to the NFL behav- dp (4) +1()Ð n()p, T ln()1 Ð n()p, T ]------, ior of the electronic system. The application of a mag- ()π 3 ≥ ∝ 3/4 2 netic field (B Ð Bc) B*(T) T restores Landau Fermi liquid (LFL) behavior. Here, Bc is a critical mag- which results from purely combinatorial consider- ≥ netic field. At (B Ð Bc) B*(T), the effective mass ations. Equation (2) is usually presented as the FermiÐ ∝ Ð2/3 Dirac distribution depends on the magnetic field, M*(B) (B Ð Bc) , being approximately independent of the temperature at Ð1 ≤ ∝ 4/3 ≥ Ð1/2 ()ε()µp, T Ð T T*(B) (B Ð Bc) . At T T*(B), the T depen- n()p, T = 1 + exp ------. (5) dence of the effective mass and the NFL behavior are T reestablished. At T 0, the system becomes LFL with the effective mass M* ∝ 1/r. When the system has At T 0, one gets from Eqs. (2), (5) the standard θ ε Ӎ undergone FCQPT, it becomes a strongly correlated solution nF(p, T 0) (pF Ð p), with (p pF) Ð

JETP LETTERS Vol. 79 No. 6 2004 288 SHAGINYAN

µ FC = pF(p Ð pF)/M*, where pF is the Fermi momentum, where pF corresponds to xFC. Solving Eq. (9) with θ (pF Ð p) is the step function, and M* is the Landau respect to M*, we obtain [8] effective mass [11] 1 1 1 dε()pT, 0 M*()T ∝ ------. (10) ------= ------. (6) T M* p dp pp= F The behavior of the effective mass given by Eq. (10) It is implied that, in the case of LFL, M* is positive and can be verified by measuring the thermal expansion finite at the Fermi momentum pF. As a result, the coefficient, which is given by [15] T-dependent corrections to M*, to the quasiparticle energy ε(p), and other quantities start with T2 terms 1∂()logV x ∂()S/x α()T ==------Ð------. (11) being approximately temperature-independent. The 3∂T 3K∂x Landau equation relating the mass M of an electron to P T the effective mass of the quasiparticles is of the form Here, P is the pressure and V is the volume. Substitut- [11] ing Eq. (4) into Eq. (11), one finds that in the LFL theory coefficient is of the order of α(T) ~ M* T/ p2 K. By p p ()∇,, ()dp1 L F ------= ----- + ∫FL pp1 x p n p1 ------. (7) M* M 1 ()2π 3 employing Eq. (10), one finds that, at T ~ T0 [16], Applying Eq. (7) at T < T*(x), we obtain the common α()T ∝ aT+ bT, (12) result with a and b being constants. This result is in good M M* = ------. (8) agreement with experimental facts obtained in mea- 1 () surements on CeNi Ge [17]. 1 Ð N0FL x /3 2 2 The application of magnetic fields B leads to Zee- Here, N0 is the density of states of the free Fermi gas 1 man splitting of the Fermi level. As a result, two quasi- and F (x) is the p-wave component of the Landau 1 L particle distribution functions with Fermi momenta p interaction. At x x , the denominator in Eq. (8) F FC 2 1 2 ∆ 2 tends to zero and one obtains Eq. (1). The temperature and pF appear, so that pF < pF < pF and pF = (pF Ð smoothing out the step function θ(p Ð p) at p = 1 F F p ) ~ µ BM*/p . Here, µ is the electron magnetic (x/3π2)1/3 induces the variation of the Fermi momentum F 0 F 0 ∆ moment. In the same way Eq. (10) was derived, we can pF ~ TM*/pF. We assume that the amplitude FL has a Ӷ obtain the equation determining M*(B) [8]. The only short range q0 pF of interaction in the momentum difference is that there are no contributions coming space. It is a common condition leading to the existence ∆ from the terms proportional to pF, and we have to take of FC and nearly localized Fermi liquids [7, 14]. If the ∆ 2 radius is such that q ~ ∆p ~ T M*/p , corrections to into account terms proportional to ( pF) . Assuming 0 F 0 F that the system is close to the critical point, we obtain the effective mass are proportional to T at T ~ T0. Here, T ∝ |x Ð x | is a characteristic temperature at which the ε 2/3 0 FC ()∼ F system’s behavior is of the NFL type. On the other M* B M------. (13) Ӷ ӷ Bµ hand, at T*(x) T0, we have q0 T*(x)M*/pF and the system behaves as LFL at T ~ T*(x), so that the correc- At T ~ T*(x), Eq. (13) is valid as long as M*(B) ≤ tions to the effective mass start with T2 terms. We can M*(x); otherwise, we have to use Eq. (1). It follows also conclude that the transition region is rather large from Eq. (13) that the application of magnetic fields compared with T*(x), being proportional to T0. reduces the effective mass. If there exists a magnetic In the case of T ~ T0, we again can use Eq. (8) with order in the system which is suppressed by magnetic 1 ∆ 1 ∆ ∝ 1 field B = Bc0, then the quantity (B Ð Bc0) plays the role FL (pF Ð pF) ~ FL (pF) + A pF, where A dFL (x)/dx. of zero field, and Eq. (13) has to be replaced by the 1 ∆ equation Substituting this expansion of FL (pF + pF) into Eq. (8), we find that 2/3 ()∝ 1 M* B ------. (14) ∼ M ∝ M BBÐ c0 M* ------∆ ------. (9) A pF M*T At high magnetic fields, we expect Eq. (14) to be ∆ ∆ In deriving Eq. (9), we assumed that the system is close invalid, because pF becomes too large, so that pF > 1 Ӷ ∆ to FCQPT, so that (1 Ð N0 FL (pF)) N0A pF. We can q0. In that case, the effective mass still depends on the say that, at T ~ T , ∆p induced by T becomes larger magnetic field, but the proportionality given by Eq. (14) 0 F is not preserved, and the dependence on the magnetic ∆ | FC | than the distance r from FCQPT, pF > pF Ð pF , field becomes weaker.

JETP LETTERS Vol. 79 No. 6 2004 UNIVERSAL BEHAVIOR OF HEAVY-FERMION METALS 289

At elevated temperatures T ~ T0, the effective mass Consider the system when r 0. Then, its prop- starts to depend on both the temperature and the mag- erties are determined by the magnetic fields B and the netic field. A crossover from the B-dependent effective temperature T because there are no other parameters to mass M*(B) to the T-dependent effective mass M*(T) describe the state of the system. At the transition tem- takes place at a transition temperature T*(B) as soon as peratures T Ӎ T*(B), the effective mass depends on both M*(B) Ӎ M*(T). This requirement and Eqs. (10) and T and B, while at T Ӷ T*(B), the system is LFL with the (14) give effective mass being given by Eq. (14), and at T ӷ T*(B), the mass is defined by Eq. (10). Instead of solv- ()∝ ()4/3 T* B BBÐ c0 . (15) ing Eq. (8), it is possible to construct a simple interpo- lation formula to describe the behavior of the effective At T > T*(x), Eq. (15) determines the line on the BÐT mass over all the region, phase diagram which separates the region of the LFL behavior taking place at T < T*(B) from the NFL behav- 1 , ∝ , ------ior occurring at T > T*(B). At T < T*(B), the system M*()BT FB( T ) = 2/3 . (21) behaves like LFL with the effective mass M*(B) and c1()BBÐ c0 + c2 fy() T corrections to the effective mass start with T2 terms. In Here, f(y) is a universal monotonic function of y = Ӎ accordance with the LFL theory, the specific heat c 2/3 Ӷ γT, with T /(B Ð Bc0) such that f(y ~ 1) = 1 and f(y 1) = 0. It is seen from Eq. (21) that the behavior of the effective γ ()∝∝() ()Ð2/3 B M* B BBÐ c0 . (16) mass can be represented by a universal function FM of only one variable y if the temperature is measured in the ρ ρ ρ 2 The resistivity behaves as = 0 + A(B)T , where the units of the transition temperature T*(B), see Eq. (15), coefficient and the effective mass is measured in the units of M*(B) ()∝∝()()2 ()Ð4/3 given by Eq. (14), FM(y) = aF(B, T)/M*(B) with a being AB M* B BBÐ c0 . (17) a constant. This representation describes the scaling It follows from Eqs. (16) and (17) that the KadowakiÐ behavior of the effective mass. As is seen from Woods ratio K = A/γ2 [18] is conserved. All these results Eqs. (19) and (21), the scaling behavior of the magneti- obtained from Eqs. (14)Ð(17) are in good agreement zation can be represented in the same way, provided the with experimental facts observed in measurements on magnetization is normalized by the saturated value at the HF metal YbAgGe single crystal [19]. The critical each field given by Eq. (19), ∝ β behavior of the coefficient A(B) (B Ð Bc0) at B M ()BT, 1 β ------B ∝ ------, (22) Bc0 described by Eq. (14) with = Ð4/3 is in accordance M ()B 1 + c fy()y with experimental data obtained in measurements on B 3 β CeCoIn5, which displayed the critical behavior with = where c3 is a constant. It is seen from Eq. (22) that mag- Ð1.37 ± 0.1 [4]. netization is a monotonic function of y. Upon using the χ ∂ ∂ In the LFL theory, the magnetic susceptibility χ ∝ definition of susceptibility, = MB/ B, and differenti- M*/(1 Ð Fa ). Note, that there is no ferromagnetic insta- ating both sides of Eq. (22) with respect to B, we arrive 0 at the conclusion that the susceptibility also exhibits the bility in Fermi systems related to the growth of the scaling behavior and can be presented as a universal a effective mass, and the relevant Landau amplitude F0 > function of only one variable y, provided it is normal- Ð1 [14]. Therefore, at T < T*(B), the magnetic suscep- ized by the saturated value at each field given by tibility turns out to be proportional to the effective mass Eq. (18), Ð2/3 χ()BT, 1 fy()+ ydf() y /dy χ()B ∝∝M*()B ()BBÐ , (18) ------∝ ------+ 2c y------. (23) c0 χ()B 1 + c fy()y 3 ()() 2 3 1 + c3 fyy while the static magnetization MB(B) is given by It is of importance to note that the susceptibility is not ()∝∝()1/3 MB B Bχ BBÐ c0 . (19) a monotonic function of y because the derivative is the At T > T*(B), as follows from Eq. (10), Eq. (18) has to sum of two contributions. The second contribution on be rewritten as the right-hand side of Eq. (23) makes the susceptibility have a maximum. The above behaviors of the magneti- 1 zation and susceptibility are in accordance with the χ()T ∝∝M*()T ------. (20) T facts observed in measurements on CeRu2Si2 [20]. Note that the magnetic properties of CeRu Si do not χ 2 2 The behavior of (B) and MB(B) as a function of mag- show any indications of the magnetic ordering at the netic field B given by Eqs. (18) and (19) and the behav- smallest temperatures and in the smallest applied mag- χ ior of (T), see Eq. (20), are in accordance with facts netic fields [20], which is Bc0 0 in that case. As a observed in measurements on CeRu2Si2 with the criti- result, we can conclude that the QCP is driven by the cal field Bc0 0 [20]. divergence of the effective mass rather than by mag-

JETP LETTERS Vol. 79 No. 6 2004 290 SHAGINYAN ∆ netic fluctuations and FCQPT is the main cause of the At 1 0, the critical temperature Tc 0. We NFL behavior. We can also conclude that the Neél tem- see that the ordered phase can exist only at T = 0, and perature is zero in this case, because the magnetic sus- the state of electron liquid with FC disappears at T > 0 ceptibility diverges at T 0, as is seen from Eq. (20). [6]. Therefore, FCQPT is not the endpoint of a line of A more detailed analysis of this issue will be published finite-temperature phase transitions. This conclusion is elsewhere. in accordance with Eq. (2), which does not admit the Consider the case when the system has undergone existence of the flat part of spectrum at finite tempera- FCQPT. Then, there exist special solutions of Eq. (2) tures. As a result, the quantum-to-classical crossover associated with the so-called fermion condensation [7]. upon approaching a finite-temperature phase transition Being continuous and satisfying the inequality 0 < does not exist. In the considered case, one can expect to observe such a crossover at T ~ Tf. On the other hand, n0(p) < 1 within some region in p, such solutions n0(p) ∆ admit a finite limit for the logarithm in Eq. (2) at T 1 becomes finite if we assume that the pairing interac- 0 yielding [7] tion is finite and the corresponding x Ð T phase diagram becomes richer. Moving along this line, we can con- ε() µ <<() ≤≤ sider the high-T superconductivity as well (see, e.g., p Ð0,if0= n0 p 1; pi ppf , (24) c [6, 7]). ε Ӷ where (p) is given by Eq. (3). At T = 0, Eq. (24) defines At finite temperatures T Tf, the occupation num- a new state of electron liquid with FC [7, 10] which is bers in the region (pf Ð pi) are still determined by characterized by a flat spectrum in the (pf Ð pi) region Eq. (24) and the system becomes divided into two qua- and which can strongly influence measurable quantities siparticle subsystems: the first subsystem is occupied Ӷ up to temperatures T Tf. In this state, the order param- by normal quasiparticles with the finite effective mass eter of the superconducting state κ(p) = M*L independent of T at momenta p < pi, while the sec- ()() () 1 Ð n0 p n0 p has finite values in the (pf Ð pi) ond subsystem in the (pf Ð pi) range is characterized by region, whereas the superconducting gap ∆ 0 in 1 the quasiparticles with the effective mass M*FC (T) this region, provided that the pairing interaction tends [6, 24] to zero. Such a state can be considered as superconducting, with an infinitely small value of ∆ , so that the p Ð p 1 M* Ӎ p ------f -i. (25) entropy S(T = 0) of this state is equal to zero [6, 7]. FC F 4T

When pf pi pF, the flat part vanishes and There is an energy scale E0 separating the slow dispers- Eq. (24) determines the QCP at which the effective ing low energy part, related to the effective mass M* , mass M* diverges and FCQPT takes place. When the FC density approaches the QCP from the disordered phase, from the faster dispersing relatively high energy part, * Eq. (24) possesses nontrivial solutions at x = xFC as defined by the effective mass ML . It follows from soon as the effective interelectron interaction as a func- Eq. (25) that E0 is of the form [6] tion of the density, or the Landau amplitude, becomes Ӎ sufficiently strong to determine the occupation num- E0 4T. (26) bers n(p) which deliver the minimum value to the The described system can be viewed as a strongly energy E[n(p)], while the kinetic energy can be consid- correlated one, it has the second type of the behavior ered as frustrated [6]. As a result, the occupation num- and demonstrates the NFL behavior even at low tem- bers n(p) become variational parameters and Eq. (24) peratures. By applying magnetic fields, the system can has nontrivial solutions n0(p), because the energy be driven to LFL with the effective mass [25] E[n(p)] can be lowered by alteration of the occupation 1 numbers. Thus, within the region pi < p < pf, the solu- M*()B ∝ ------. (27) tion n0(p) deviates from the Fermi step function nF(p) BBÐ c0 in such a way that the energy ε(p) stays constant, while outside this region, n(p) coincides with n (p) [7]. Note In the same way as it was done above, we find from F Eqs. (25) and (27) that a crossover from the B-depen- that a formation of the flat part of the spectrum has been dent effective mass M*(B) to the T-dependent effective confirmed in [21–23]. mass M*(T) takes place at a transition temperature At r > 0 when the system is on the disordered side, T*(B) that is, κ(p) ≡ 0, and the density x moves away from T*()B ∝ ()BBÐ . (28) QCP located at xFC, the Landau amplitude FL(p = pF, c0 p1 = pF, x) as a function of x becomes smaller, the Equation (28) determines the line in the BÐT phase dia- kinetic energy comes into a play and makes the flat part gram which separates the region of the LFL behavior at vanish. Obviously, Eq. (24) has only the trivial solution T < T*(B) from the NFL behavior occurring at T > ε µ (p = pF) = , and the quasiparticle occupation numbers T*(B). The existence of the BÐT phase diagram given θ are given by the step function, nF(p) = (pF Ð p). by Eqs. (15) and (28) can be highlighted by calculating

JETP LETTERS Vol. 79 No. 6 2004 UNIVERSAL BEHAVIOR OF HEAVY-FERMION METALS 291 the resistivity and the magnetoresistance [8]. The resis- conclude that Eqs. (21)Ð(23) determining the scaling tivity, which at T > T*(B) demonstrates the NFL behav- behavior of the effective mass, static magnetization, ρ ρ ior, at T < T*(B), exhibits the LFL behavior, = 0 + and the susceptibility are also valid in the case of A(B)T2. The BÐT diagram of the dependence of the strongly correlated liquid, but the variable y is now effective mass on the magnetic field can be highlighted given by y = T/BBÐ c0 , while the function f(y) can be by calculating the magnetoresistance. At (B Ð Bc0) > dependent on (pf Ð pi)/pF. This dependence comes from B*(T), the magnetoresistance is negative, and at (B Ð Eq. (25). As a result, we can find that at T < T*(B), the Bc0) < B*(T), it becomes positive. This behavior of both ρ ∝ ∝ factor d /dT A(B)T behaves as A(B)T T/(B Ð Bc0), the magnetoresistance and the resistivity is in agree- and at T > T*(B), it behaves as A(B)T ∝ 1/T. These ment with measurements on YbRh2Si2 [3], when the observations are in good agreement with the data system exhibits the second type of the behavior, see obtained in measurements on YbRh2(Si0.95Ge0.05)2 [5]. Eq. (28), while CeCoIn5 and YbAgGe demonstrate the first type of behavior consistent with that given by It is worthy to note that, in zero magnetic fields, the Eq. (15) [4, 19]. Neél temperature is zero at FCQPT, because, as follows from Eq. (10), the effective mass tends to infinity at At T < T*(B), the coefficients γ ∝ M*(B), χ(B) ∝ FCQPT and makes the susceptibility divergent. On the M*(B), and A(B) ∝ (M*(B))2, and we find that the Kad- other hand, if there is the magnetic order and the Neél owaki ratio K arid the SommerfeldÐWilson ratio R ∝ temperature is not equal to zero, the effective mass is χ(B)/γ(B) are preserved due to Eq. (27). The obtained finite and there is no FCQPT. As soon as the magnetic BÐT phase diagram and the conservation of both the order is suppressed at B Bc0, that is, the Neél tem- Kadowaki and the SommerfeldÐWilson ratios are in perature tends to zero, the effective mass M*(B) ∞, full agreement with data obtained in measurements on as follows from Eq. (14). If the system has undergone YbRh2Si2 and YbRh2(Si0.95Ge0.05)2 [3, 5, 17]. Taking FCQPT, again at B Bc0, the Neél temperature goes ∞ into account Eqs. (11) and (25), we find that in the case to zero and M*(B) , see Eq. (27). If Bc0 = 0, the of the two quasiparticle subsystems the thermal expan- effective mass diverges at T 0, see Eq. (25), and the χ sion coefficient α(T) ∝ a + bT + c T , with a, b, and c susceptibility tends to infinity, being proportional to being constants. Here, the first term a is determined by the effective mass. In this case, the Neél temperature is the FC contribution, the second bT is given by normal equal to zero as well. Therefore, one may say that the effective mass M* diverges at the very point that the * quasiparticles with the effective mass ML , and the Neél temperature goes to zero. third c T comes from a specific contribution related to A few remarks are in order at this point. To describe ε the spectrum c(p) which ensures the connection the behavior of HF metals, we have introduced the sys- between the dispersionless region (pf Ð pi) occupied by tem of quasiparticles, as is done in the Landau theory of FC and normal quasiparticles [16, 24]. At finite temper- normal Fermi liquids, where the existence of fermionic atures, the contribution coming from the third term is quasiparticles is a generic property of normal Fermi expected to be relatively small because the spectrum systems independent of microscopic details. As we ε Ӷ c(p) occupies a relatively small area in the momentum have seen, at T Tf, these quasiparticles have universal space. Since at T 0, the main contribution to the properties which determine the universal behavior of ε specific heat c(T) comes from the spectrum c(p), the HF metals. One can use another approach, constructing ∝ the singular part of the free energy, introducing notions specific heat behaves as c(T) a1 T + b1T, with a1 and of the upper critical dimension, hyperscaling, etc. (see, b1 being constants. The second term b1T comes from e.g., [1, 2]). Moving along this way, one may expect the contribution given by FC and normal quasiparticles. difficulties. For example, having only the singular part, Measurements for YbRh2(Si0.95Ge0.05)2 show a power one has to describe at least the two types of the behav- low divergence of γ = c/T ∝ TÐα with α = 1/3 [17]. This ior. We reserve a consideration of these items for future result is in reasonable agreement with our calculations publications. α giving = 0.5. At lower temperatures, the relative con- In conclusion, we have shown that our simple model tribution of the first term a1 T becomes bigger and we based on FCQPT explains the critical behavior expect that the agreement will also become better. Now, observed in different HF metals. In the case of such HF we find that the Grüneisen ratio Γ(T) = α(T)/c(T) metals as CeNi2Ge2, CeCoIn5, YbAgGe, and CeRu2Si2, Γ ∝ the behavior can be explained by the proximity to diverges as (T) 1/T [16]. This result is in good FCQPT, where their electronic systems behave like agreement with measurements on YbRh2(Si0.95Ge0.05)2 highly correlated liquids. In the case of such HF metals [5, 17]. as YbRh2(Si0.95Ge0.05)2 and YbRh2Si2, the critical As follows from Eq. (27), the static magnetization behavior is different. This can be explained by the pres- ∝ ence of FC in the electronic systems of these metals, behaves as MB(B) BBÐ c0 , in accordance with i.e., by the fact that the electronic systems have under- measurements on YbRh2(Si0.95Ge0.05)2 [5]. We can also gone FCQPT and behave as strongly correlated liquids.

JETP LETTERS Vol. 79 No. 6 2004 292 SHAGINYAN

We have shown that the basis of the Landau Fermi liq- 8. V. R. Shaginyan, Pis’ma Zh. Éksp. Teor. Fiz. 77, 104 uid theory survives in both cases: the low energy exci- (2003) [JETP Lett. 77, 99 (2003)]; Pis’ma Zh. Éksp. tations of both strongly correlated Fermi liquid with FC Teor. Fiz. 77, 208 (2003) [JETP Lett. 77, 178 (2003)]. and the highly correlated Fermi liquid are quasiparti- 9. V. M. Galitsky and V. A. Khodel, cond-mat/0308203; cles. It is also shown that the effective mass M* V. M. Yakovenko and V. A. Khodel, Pis’ma Zh. Éksp. diverges at the very point that the Neél temperature Teor. Fiz. 78, 850 (2003) [JETP Lett. 78, 398 (2003)]; goes to zero. The BÐT phase diagrams of both the cond-mat/0308380. highly correlated liquid and the strongly correlated one 10. G. E. Volovik, Pis’ma Zh. Éksp. Teor. Fiz. 53, 208 (1991) have been studied. We have shown that these BÐT phase [JETP Lett. 53, 222 (1991)]. diagrams strongly influence the effective mass and such 11. L. D. Landau, Zh. Éksp. Teor. Fiz. 30, 1058 (1956) [Sov. important properties of HF metals as magnetoresis- Phys. JETP 3, 920 (1956)]. tance, resistivity, specific heat, magnetization, suscepti- 12. M. Ya. Amusia and V. R. Shaginyan, Pis’ma Zh. Éksp. bility, and volume thermal expansion. Teor. Fiz. 76, 774 (2002) [JETP Lett. 76, 651 (2002)]. I thank P. Coleman for valuable comments. I am 13. S. Nakamae, K. Behnia, N. Mangkorntong, et al., Phys. grateful to the CTSPS for hospitality during my stay in Rev. B 68, 100502(R) (2003). Atlanta. This work was supported in part by the Russian 14. M. Pfitzner and P. Wölfe, Phys. Rev. B 33, 2003 (1986). Foundation for Basic Research, no. 04-02-16136. 15. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics, 3rd ed. (Nauka, Mos- cow, 1976; Addison-Wesley, Reading, MA, 1970), REFERENCES Part 1. 1. S. Sachdev, Quantum Phase Transitions (Cambridge 16. M. Ya. Amusia, A. Z. Msezane, and V. R. Shaginyan, Univ. Press, Cambridge, 1999). Phys. Lett. A 320, 459 (2004). 2. M. Vojta, Rep. Prog. Phys. 66, 2069 (2003). 17. R. Küchler, N. Oeschler, P. Gegenwart, et al., Phys. Rev. 3. P. Gegenwart, J. Custers, C. Geibel, et al., Phys. Rev. Lett. 91, 066405 (2003). Lett. 89, 056402 (2002). 18. K. Kadowaki and S. B. Woods, Solid State Commun. 58, 4. J. Paglione, M. A. Tanatar, D. G. Hawthorn, et al., Phys. 507 (1986). Rev. Lett. 91, 246405 (2003); J. Paglione et al., cond- 19. S. L. Bud’ko, E. Q. Morosan, and P. C. Canfield, Phys. mat/0304497. Rev. B (in press); cond-mat/0308517. 5. J. Custers, P. Gegenwart, H. Wilhelm, et al., Nature 424, 20. D. Takahashi, S. Abe, H. Mizuno, et al., Phys. Rev. B 67, 524 (2003). 180407(R) (2003). 6. M. Ya. Amusia and V. R. Shaginyan, Pis’ma Zh. Éksp. 21. I. E. Dzyaloshinskii, J. Phys. I 6, 119 (1996). Teor. Fiz. 73, 268 (2001) [JETP Lett. 73, 232 (2001)]; 22. D. Lidsky, J. Shiraishi, Y. Hatsugai, and M. Kohmoto, S. A. Artamonov and V. R. Shaginyan, Zh. Éksp. Teor. Phys. Rev. B 57, 1340 (1998). Fiz. 119, 331 (2001) [JETP 92, 287 (2001)]; M. Ya. Amusia and V. R. Shaginyan, Phys. Rev. B 63, 23. V. Yu. Irkhin, A. A. Katanin, and M. I. Katsnelson, Phys. 224507 (2001); V. R. Shaginyan, Physica B & C Rev. Lett. 89, 076401 (2002). (Amsterdam) 312Ð313, 413 (2002). 24. M. V. Zverev, V. A. Khodel’, V. R. Shaginyan, and 7. V. A. Khodel and V. R. Shaginyan, Pis’ma Zh. Éksp. M. Baldo, Pis’ma Zh. Éksp. Teor. Fiz. 65, 828 (1997) Teor. Fiz. 51, 488 (1990) [JETP Lett. 51, 553 (1990)]; [JETP Lett. 65, 863 (1997)]. V. A. Khodel, V. R. Shaginyan, and V. V. Khodel, Phys. 25. Yu. G. Pogorelov and V. R. Shaginyan, Pis’ma Zh. Éksp. Rep. 249, 1 (1994). Teor. Fiz. 76, 614 (2002) [JETP Lett. 76, 532 (2002)].

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JETP Letters, Vol. 79, No. 6, 2004, pp. 293Ð297. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 351Ð355. Original Russian Text Copyright © 2004 by O. Tkachenko, V. Tkachenko, Baksheev.

Ballistic Electron Wave Functions and Negative Magnetoresistance in a Small Ring Interferometer O. A. Tkachenko*, V. A. Tkachenko, and D. G. Baksheev Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia *e-mail: [email protected] Received December 11, 2003; in ﬁnal form, February 12, 2004

By two-dimensional ballistic magnetotransport calculations, it is demonstrated that large-scale resistance peaks, typical of small ring interferometers in zero magnetic ﬁeld, are suppressed at B ~ 1 T. This result is explained by the peculiarities of the interference pattern at the conﬂuence sites of quantum wires and is in qualitative agreement with experimental data. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 73.23.Ad; 73.63.Nm

An ideal ring interferometer is a device consisting of ~2e2/h as the magnetic field grows [9]. An explanation one-dimensional quantum wires. The first theoretical of all these facts requires numerical investigation of the publications [1, 2], in which the points of connection of magnetic-field effect on the behavior of a ballistic elec- quantum-wire leads with the ring were assumed to be tron in a realistic potential of the interferometer. Note structureless and the wires were considered as homoge- that the magnetotransport in an interferometer was neous, predicted a periodic 100% conductance modula- modeled earlier using simple modeling potentials that tion in a varying magnetic field, i.e., the AharonovÐ neglect the electrostatics of the structure [14, 15]. Bohm (AB) oscillations. The origin of the effect is in In this paper, we study the wave functions and the the passage through the ring resonance levels. How- magnetotransport in small interferometers, for which ever, in reality, the AB oscillations [3Ð5] have a much the effect of triangular dots is most pronounced. We smaller amplitude and, in some cases, are observed show that, in zero magnetic field, the Fermi-energy against the background of large-scale features in the dependence of the conductance of a single-mode inter- conductance [5Ð8]. Another effect that is often ferometer consists of the alternating portions with high, observed in experimental measurements is the presence G ≈ 2e2/h, and low, G Ӷ 2e2/h, transparencys. In the of a negative magnetoresistance [7Ð12]. region of a triangular dot, the wave functions corre- Obviously, a real interferometer is much more com- sponding to the low-conductance intervals have a sym- plicated than the one-dimensional model because of the metry that suppresses the transmission of a ballistic finite wire width and because of the formation of deep electron into the ring arms. Magnetic field violates this triangular potential wells, i.e., quantum dots, at the sites symmetry and increases the transmission probability of wire separation into the ring arms. According to the through the interferometer. self-consistent calculations of the three-dimensional To calculate the interferometer transparency, we electrostatic potential and electron density [6, 8, 13], used the method described in [16] and representing the these quantum dots have many single-particle levels, extension of the recursive Green’s-function technique which should affect the transmission through the inter- [17]. This method allows the Schrödinger equation in a ferometer. Calculations performed for a two-dimen- magnetic field to be solved for the realistic potentials of sional ballistic transport in zero magnetic field show a complex geometry. The advantage of the method [16] that the triangular dots cause additional reflection and is that the matrices of transmission and reflection even interferometer blocking. This conclusion follows amplitudes and the wave functions are calculated from the comparison of the arrangements of large-scale simultaneously for each point of a grid used for the dis- peaks and dips in the energy dependence of transpar- cretization of the computational region. The stability is ency for the whole device and for one triangular dot an important feature of this method. The conductance [6, 8]. of the device is determined from the multichannel Lan- A similar sequence of quasiperiodic resistance dauer–Büttiker formula [18]. The two-dimensional peaks in the gate-voltage dependence was experimen- effective potential necessary for calculating the con- tally obtained for small ring interferometers (r = ductance and wave functions was obtained by modeling 130 nm) [7, 8]. It was found that a large negative mag- three-dimensional electrostatics of an interferometer netoresistance is present in the region of conductance fabricated by etching on the basis of a GaAs/AlGaAs dips and that the conductance approaches the value heterostructure [6]. The lithographic dimensions were

294 TKACHENKO et al.

Fig. 1. Conductances calculated for a small symmetric interferometer Tring (solid line) and for a triangular quantum dot Tdot (dashed line) as functions of the Fermi energy at B = 0. taken smaller than in the cited publication (the average ring radius was r = 105 nm). For simplicity, we chose a symmetric etch profile and ignored the fluctuation impurity potential (uniform doping). In our case, the channel widths in the narrowest parts of the ring and between the ring and reservoirs with 2D electron gas were approximately the same and rather small. This Fig. 2. Probability density distribution in an electron wave made it possible to obtain a single-mode regime simul- incident in the first mode: (a) the interferometer transparency is Tring = 0.99 at EF = 2.34 meV and B = 0; (b) Tring = taneously for all constrictions over a sufficiently large × Ð4 energy interval from 1.3 to 8 meV. The lower boundary 1.5 10 at EF = 5.66 meV and B = 0 (the dashed lines of the interval corresponds to the first transverse-quan- show the boundaries of the classically allowed region EF = tization level in the narrow part of the interferometer Ueff); (c) Tring = 1 at EF = 5.62 meV and B = 1 T. The probability-density isolines correspond to the levels of 0.01, input channel, and the upper boundary corresponds to 0.0316, 0.1, 0.316, and 1 and to the maximal values (a) 1.2, the second level in the ring arms. The choice of this sit- (b) 1.4, and (c) 2.3. uation is motivated by the following reasons. If the channels connecting the ring with reservoirs become much narrower than the ring-arm channels and have relate with each other, and additional resonances on the tunnel barriers, a single-electron charging of a multi- curve Tring(EF) obtained for the interferometer appear as mode ring becomes possible [19]. The opposite condi- a result of a circular motion in the ring. The question tion leads to the case where two open triangular dots are arises of why the interferometer conductance has wide separated by the tunnel barriers formed in the ring and deep dips in the region 1.3Ð8 meV in zero magnetic arms. Evidently, both these tunneling situations make field while the transparency of the ring arms and the the device transparency rather low. However, these connecting channels is close to its total value. cases could hardly be realized in experiment [9], because, in a magnetic field of ~ 1 T, the interferometer Let us consider the plots of probability density in the conductance was close to 2e2/h in a considerable por- interferometer for the states corresponding to the total tion of the gate-voltage interval, which is impossible in transmission (Fig. 2a) and total reflection (Fig. 2b). We the tunneling regime. assume that electrons are incident from the left in the first transverse-motion mode. Since the ring and the tri- The calculated dependence Tring(EF) of the interfer- angular dots have a small size and the energy of the ometer transparency on the energy of ballistic electrons incident particles is low, the transmission is determined shows that, in zero magnetic field, the transmission by quantum interference. The state with the energy peaks alternate with deep dips (Fig. 1). For comparison, EF = 2.34 meV corresponds to the resonant transmis- we calculated the transparency Tdot(EF) of a half of the sion with Tring = 0.99 through the combined dotÐring interferometer (the outgoing wave is assumed to prop- system. One can see from Fig. 2a that the wave function agate through two parallel channels extending the ring is delocalized over the whole system, and the standing arms). One can see that two curves, on the whole, cor- wave along the ring contains 16 antinodes.

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BALLISTIC ELECTRON WAVE FUNCTIONS 295

Fig. 3. Interferometer conductance as a function of magnetic ﬁeld for Fermi energy EF = (a) 2.34 and (b) 5.6 meV. (c) Effects of negative magnetoresistance and suppression of AB oscillations observed for an asymmetric ring in [11]. The values of EF are indicated in millielectron volts near the calculated curves. The experimental curve taken from [11] is presented for comparison.

Now, we consider the state with EF = 5.66 meV and the symmetry in the positions of the probability-density × Ð4 maxima at the ring-input triangular dot, and the trans- Tring = 1.5 10 (Fig. 2b), which corresponds to a suppressed transmission. In zero magnetic field, the wave mission probability through the interferometer function represents a standing wave that arises between increases. Figure 2c represents the state of total trans- the antidot and the reservoir and has a symmetry forbid- mission in the magnetic field B = 1 T for EF = 5.62 meV. ding the transmission into the ring arms at the first tri- The magnetic-field effect manifests itself in the AB angular dot. Specifically, two maxima of the probability oscillations (Fig. 3a) and in the negative magnetoresis- density at the arm input correspond to the second trans- tance (Fig. 3b). Note that the one-dimensional interfer- verse-motion mode. This mode, being not mixed with ometer model gives the AB oscillations with a full the first one, is totally reflected, because its transmis- amplitude modulation and a period corresponding to sion through the narrow parts of the channels is not the flux quantum through the ring [1, 2]. Similar results allowed. Figure 2b shows that electron tunnels at a were obtained by a simplified two-dimensional model- small depth into the classically forbidden region of the ing of single-mode ring interferometers [14, 15]. How- two-dimensional effective potential E < U . Similar ever, the calculation for a realistic interferometer poten- F eff tial gives a qualitatively different result. For the half- pictures of probability density also occur for other integer magnetic flux quanta, the conductance of a sym- states of minimal transmission at B = 0 (EF = 3.83 meV metric interferometer with a finite channel width is not or EF = 4.9 meV). Note that, in symmetric single-mode suppressed to zero, as was predicted by the one-dimen- interferometers that have no quantum wells at the chan- sional model. The presence of levels in triangular dots nel-confluence sites, the reflection states in zero mag- and their displacement from the ring levels in a magnetic field appear only as a result of the destructive netic field is seen in the G(B) curve as a large-scale con- wave interference at the ring output [15]. In this case, ductance modulation and as Fano profiles of the peak– the electron distribution density in the ring is always dip type. Because of the presence of Fano resonances, nonzero [20]. By contrast, we investigate the reflection the form of the AB oscillations is not sinusoidal and the states corresponding to the case where an electron does period and phase vary. Calculations show that, for the not penetrate into the ring at all, although the first-mode states with Tring = 1 at B = 0, despite the large-scale con- motion is open. An increase in a magnetic field breaks ductance modulation, the transmission remains, on the

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296 TKACHENKO et al.

Fig. 4. Dependence of the conductance at the Fermi energy Fig. 5. Experimental dependence of the conductance on the for the ring, shown in Fig. 2, in magnetic fields B = 0, 0.5, gate voltage for a small ring at a temperature of 1.5 K. The and 2 T at a temperature of 1.5 K. dependence is plotted using the data reported in [9]. average, high with an increase in the magnetic field. the edge current states start to form (the asymptotic val- However, the states corresponding to the total reflection ues of 2e2/h and 4e2/h are approached). For compari- become more transparent in a magnetic field. One can son, Fig. 5 shows the experimental results reported in see from Fig. 3b that the conductance G(B) grows with [9]. The measured dependence of the conductance of a 2 increasing magnetic field and reaches the value 2e /h small interferometer on the gate voltage Vg at B = 0 even at B ≈ 1 T. Thus, at the given energy, we obtain the shows dips that can be related to the conductance dips giant negative magnetoresistance effect. For other in Fig. 4. One can see that, in the case of modeling and states of minimal transmission at B = 0, a similar behav- in the experiment, the dips decrease with increasing ior takes place for G(B). magnetic field in a similar way. The electrostatic calculation allows us to compare the lengths of the intervals As for the amplitude of AB oscillations, it depends of E and V , within which a qualitatively similar in a complicated way on the Fermi energy, magnetic F g field, and the degree of ring asymmetry. The effect of behavior of conductances is observed. In the calcula- AB-oscillation suppression upon passing to a tunneling tions, a 0.05-V variation of the upper gate voltage Vg regime in one of the ring arms was analyzed in our pre- causes an approximately 5.5-meV shift of one-dimen- vious papers [8, 11] (together with the corresponding sional subbands in the channels of symmetric interfer- experimental data). Figure 3c shows the conductance ometer. Hence, the values obtained for the lengths of G(B) calculated for different Fermi energies of a small the single-mode motion intervals (G ≤ 2e2/h) in the cal- interferometer (r = 110 nm) with different channel culations and measurements are in good agreement. widths in the ring. The first mode opens in one of the Thus, the two-dimensional modeling of the trans- ring arms at EF = Ð1 meV, while the other arm is open − mission through a small interferometer in the single- for the transmission of two modes. At EF = 2 meV, the mode regime predicts high-resistance peaks caused by ring is interrupted by a tunnel barrier and the AB oscil- the backscattering of a ballistic electron in the triangu- lations are suppressed. At EF = Ð1 meV, the AB oscilla- lar quantum dots. With an increase in a magnetic field, ≈ tions show a 5% modulation, and, at EF 0, a modula- the height of these peaks decreases because of the tion from 7 to 10% (in the experiment, a 5% modulation increase in the probability of electron penetration into is observed). It follows from the analysis of Fig. 3 that the ring arms. This effect may be the cause of the neg- the model of a symmetric single-mode interferometer ative magnetoresistance observed in the experiments exaggerates the effects of AB oscillations and negative [7Ð11]. We have also shown that the amplitude of AB magnetoresistance, compared to the experiment [9]. oscillations nonmonotonically depends on the Fermi However, when the fluctuation potential [8, 13] and the energy, magnetic field, and the degree of ring asymme- nanolithography errors [8, 11] are taken into account, try. When one of the ring arms switches to the tunneling the results of calculations become closer to the mea- regime, the AB oscillations become suppressed. surements [11]. We are grateful to M.V. Éntin, Z.D. Kvon, Figure 4 shows the dependences of conductance on A.A. Bykov, and J.C. Portal for discussions and to the Fermi energy, averaged for an effective temperature A.K. Bakarov for providing us with the experimental of T = 1.5 K. The dependences are obtained for differ- data from previous publications. This work was sup- ent magnetic fields and correspond to the conditions of ported by the Russian Foundation for Basic Research experiments with small rings [7Ð9]. One can see that (project no. 16516), the Federal Target Scientific and the deep dips typical for conductance G(EF) in low Technical Program “Physics of Solid-State Nanostruc- magnetic fields begin to blur at B = 0.5 T. At B = 2 T, tures” (project no. 01.40.01.09.04), and the program of

JETP LETTERS Vol. 79 No. 6 2004 BALLISTIC ELECTRON WAVE FUNCTIONS 297 the Russian Academy of Sciences “Low-Dimensional 9. A. A. Bykov, D. V. Nomokonov, A. K. Bakarov, et al., Quantum Structures.” Pis’ma Zh. Éksp. Teor. Fiz. 78, 36 (2003) [JETP Lett. 78, 30 (2003)]. 10. C. H. Yang, M. J. Yang, K. A. Cheng, and J. C. Culbert- REFERENCES son, Phys. Rev. B 66, 115306 (2002). 1. M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett. A 96A, 11. V. A. Tkachenko, Z. D. Kvon, D. V. Shcheglov, et al., 365 (1983); Y. Gefen, Y. Imry, and M. Ya. Azbel, Phys. Pis’ma Zh. Éksp. Teor. Fiz. 79, 168 (2004) [JETP Lett. Rev. Lett. 52, 129 (1984); M. Büttiker, Y. Imry, and 79, 136 (2004)]. M. Ya. Azbel, Phys. Rev. A 30, 1982 (1984). 2. Jian-Bai Xia, Phys. Rev. B 45, 3593 (1992). 12. O. Estibals, Z. D. Kvon, J. C. Portal, et al., Physica E (Amsterdam) 13, 1043 (2002). 3. G. Timp, A. M. Chang, J. E. Cunningham, et al., Phys. Rev. Lett. 58, 2814 (1987); C. J. B. Ford, A. B. Fowler, 13. V. A. Tkachenko, Z. D. Kvon, O. A. Tkachenko, et al., J. M. Hong, et al., Surf. Sci. 229, 307 (1990); K. Ismail, Pis’ma Zh. Éksp. Teor. Fiz. 76, 850 (2002) [JETP Lett. S. Washburn, and K. Y. Lee, Appl. Phys. Lett. 59, 1998 76, 720 (2002)]. (1991); A. A. Bykov, Z. D. Kvon, E. B. Ol’shanetskiœ, 14. T. Nakanishi and T. Ando, Phys. Rev. B 54, 8021 (1996). et al., Pis’ma Zh. Éksp. Teor. Fiz. 57, 596 (1993) [JETP Lett. 57, 613 (1993)]. 15. K. N. Pichugin and A. F. Sadreev, Phys. Rev. B 56, 9662 4. Z. D. Kvon, L. V. Litvin, V. A. Tkachenko, and A. L. Aseev, (1997). Usp. Fiz. Nauk 169, 471 (1999) [Phys. Usp. 42, 402 16. T. Usuki, M. Saito, M. Takatsu, et al., Phys. Rev. B 52, (1999)]. 8244 (1995). 5. A. A. Bykov, Z. D. Kvon, E. B. Olshanetsky, et al., Phys- 17. T. Ando, Phys. Rev. B 44, 8017 (1991). ica E (Amsterdam) 2, 519 (1998). 18. Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 6. O. A. Tkachenko, V. A. Tkachenko, D. G. Baksheev, (2000). et al., Pis’ma Zh. Éksp. Teor. Fiz. 71, 366 (2000) [JETP Lett. 71, 255 (2000)]. 19. A. Fuhrer, S. Lüsher, T. Ihn, et al., Nature 413, 822 7. A. A. Bykov, D. G. Baksheev, L. V. Litvin, et al., Pis’ma (2001). Zh. Éksp. Teor. Fiz. 71, 631 (2000) [JETP Lett. 71, 434 20. E. K. Heller and F. C. Jain, J. Appl. Phys. 87, 8080 (2000)]. (2000). 8. V. A. Tkachenko, A. A. Bykov, D. G. Baksheev, et al., Zh. Éksp. Teor. Fiz. 124, 351 (2003) [JETP 97, 317 (2003)]. Translated by E. Golyamina

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 298Ð303. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 356Ð361. Original Russian Text Copyright © 2004 by Tarasov, Kuz’min, Stepantsov, Agulo, Kalabukhov, Fominskii, Ivanov, Claeson.

Terahertz Spectroscopy with a Josephson Oscillator and a SINIS Bolometer M. Tarasov1, 4, L. Kuz’min2, 4, E. Stepantsov3, 4, I. Agulo4, A. Kalabukhov2, 4, M. Fominskii1, 4, Z. Ivanov4, and T. Claeson4 1 Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow region, 114112 Russia 2 Moscow State University, Moscow, 119899 Russia 3 Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow, 117333 Russia 4 Chalmers University of Technology, Gothenburg, Sweden Received January 26, 2004; in ﬁnal form, February 17, 2004

The voltage response of a thin-ﬁlm normal-metal hot-electron bolometer based on a SINIS (superconductor– insulatorÐnormal metalÐinsulatorÐsuperconductor) structure to the radiation of a high-temperature Josephson junction in the terahertz frequency region was measured. Bolometers were integrated with planar log-periodic and double-dipole antennas, and Josephson junctions were integrated with log-periodic antennas. Measure- ments showed that the Josephson junction at a temperature of 260 mK was overheated by the transport current, so that its electron temperature exceeded 3 K at a bias voltage of 1 mV. The maximum response of a bolometer with a double-dipole antenna was observed at a frequency of 300 GHz, which agreed well with the calculated value. The Josephson radiation was observed at frequencies up to 1.7 THz. The voltage response of a bolometer reached 4 × 108 V/W, and the total noise-equivalent power reached 1.5 × 10Ð17 W/Hz1/2. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 74.50.+r; 07.57.Kp; 85.25.Pb

1. SINIS bolometers. Normal-metal hot-electron evaporation. The next step consisted in the formation of bolometers with a capacitive decoupling of the supercon- the tunnel junctions and the absorber. The pattern was ductorÐinsulatorÐnormal metalÐinsulatorÐsupercon- made by direct electron lithography. The films were ductor (SINIS) structure were proposed in [1] and deposited by thermal evaporation at different angles experimentally tested in [2]. The response to an exter- through a suspended double-photoresist mask. This nal microwave signal and the noise-equivalent power of method made it possible to deposit films of different such a bolometer are determined by its electron temper- metals in a single process in vacuum and provide their ature. To improve the noise and signal characteristics, a overlap in the tunnel junction regions. A 65-nm-thick direct electron cooling of a normal-metal absorber by a aluminum film was deposited at an angle of 60° to the superconductorÐinsulatorÐnormal metal (SIN) tunnel substrate and oxidized for 2 min in oxygen at a pressure junction was proposed in [3]. The electron cooling of 0.1 mbar to obtain a tunnel barrier. A double-layer effect was demonstrated in [4] and further developed in absorber film consisting of chromium and copper with [5]. a total thickness of 75 nm was deposited perpendicular A general view of a substrate with bolometers is pre- to the substrate. The absorber volume was 0.18 µm3. sented in Fig. 1a. A broadband log-periodic antenna The outer cooling SIN junctions of the test structures with a frequency range of 0.2Ð2 THz is positioned at had a resistance of 0.86 kΩ each, and the inner junc- the center of the substrate; two double-dipole antennas tions had 5.3 kΩ each. The inner junctions had a simple with a central frequency of 300 GHz are on the right, crosslike geometry, where a segment of a normal-metal and one double-dipole antenna with a central frequency strip crossed the oxidized aluminum electrode. Their of 600 GHz is on the left. At the top and bottom, test overlap area was 0.2 × 0.3 µm. The structure of the structures are positioned with two pairs of SIN junc- outer junctions was such that the ends of the normal- tions for studying the electron cooling effect. The latter metal absorber covered the corner of each of the outer is described for such a structure in [6]. An atomic-force oxidized aluminum electrodes, and the junction area microscopic image of the central part of the bolometer was 0.55 × 0.82 µm. An increase in the size of alumi- is shown in Fig. 1b. num electrodes made it possible to improve the diffu- The first step in fabricating the samples was the for- sion of hot quasiparticles carried away from the mation of gold contact pads and traps for hot quasipar- absorber by the tunneling current and to avoid a reab- ticles. The pattern was made by standard photolithogra- sorption in the normal metal for the phonons emitted phy. Gold 60 nm in thickness was deposited by thermal upon quasiparticle recombination. Additionally, for the

TERAHERTZ SPECTROSCOPY WITH A JOSEPHSON OSCILLATOR 299 same purpose, one of the test structures had hot-particle traps in the form of normal-metal films located in the junction region under the aluminum film. In the experiments with electron cooling, the application of a bias voltage close to the energy gap of the outer SIN junctions caused a decrease in the electron temperature measured by the inner SIN thermometers from 260 to 100 mK. 2. High-temperature Josephson oscillators. According to [7, 8], the maximal Josephson-oscillation frequency and power are determined by the critical current Ic, the normal resistance Rn, and their product Vc = IcRn. The characteristic frequency of a Josephson junction is fc = (2e/h)Vc, where e is the electron charge and h is the Planck’s constant. The characteristic voltage of a Josephson junction does not exceed the energy gap, which corresponds to frequencies on the order of 700 GHz for niobium junctions, while, for high-Tc superconductors (HTSCs) with a critical temperature above 77 K, the corresponding frequency may reach 10 THz and higher. However, high characteristic voltages and oscillation frequencies are realized only at temperatures much lower than the critical temperature. In particular, values of Vc above 5 mV are observed for the HTSC junctions at liquid-helium temperatures and lower. For the HTSC junctions on bicrystal substrates, the choice of the substrate material is highly important. The best dc characteristics are obtained for junctions on the strontium titanate substrates, but the high dielectric constant and the substantial losses at high frequencies render them unsuitable for use in the submillimeter wave range. Sapphire substrates proved to possess the most suitable characteristics, and they were used to fab- Fig. 1. (a) General view of a substrate with bolometer and ricate the Josephson oscillators. Unlike standard bic- (b) atomic-force microscopic image of the bolometer cen- rystal substrates with a misorientation in the substrate tral part. plane, we used substrates with a crystallographic axis inclined to the substrate plane. Epitaxial YBaCuO films For structures with four SIN junctions, it was possi- were grown by laser ablation on the substrates whose c ble to apply power to the inner pair of junctions and axis in the 〈100〉 direction was inclined at an angle of measure the response of the outer pair. The measured 14° + 14°. Films 250 nm thick were deposited on a dependences of the response on the bias voltage are CeO2 buffer layer. The critical temperature of the films × was T = 89 K, and the transition width was ∆T = 1.5 K. shown in Fig. 2. The maximal voltage response was 4 c c 108 V/W for a pair of 70-kΩ junctions, and the maximal The bicrystal Josephson junctions were from 1.5 to Ω 6 µm wide and, at a temperature of 4.2 K, had the char- current response was 550 A/W for a pair of 10-k junc- ≥ tions. These values correspond to the noise-equivalent acteristic voltage Vc 4 mV. Under an external submillimeter radiation, their IÐV characteristics exhibited power (NEP) Shapiro steps at voltages up to 4 mV, which corresponds to frequencies above 2 THz. NEP ==In/Si or NEP V n/Sv , (1)

3. Bolometer response to changes in temperature where In is the current noise, Vn is the voltage noise, Si = and absorbed power. The main bolometer characteris- dI/dP is the current response, and Sv = dV/dP is the tics were measured at a temperature of 260 mK in a cry- bolometer voltage response. Setting the noise voltage ostat with a closed-cycle He-3 absorption refrigerator. 1/2 The maximal voltage response to the temperature vari- of our preampliﬁer equal to 3 nV/Hz , we obtain the ations was 1.6 mV/K for a 10-kΩ SIN junction, and the technical noise-equivalent power: maximal current response was equal to 55 nA/K for a Ð17 1/2 6-kΩ junction. TNEP = 1.25× 10 W/Hz . (2)

JETP LETTERS Vol. 79 No. 6 2004

300 TARASOV et al. Substituting the measured values of the response to changes in temperature and power, we obtain the bolometer heat conductivity

∂P ∂V/∂T Ð11 G ==------= 0 . 8× 1 0 W/K. (3) V ∂T ∂V/∂P The thermodynamic noise-equivalent power is determined by the expression 2 2 NEPTD = 4kT G. (4) Taking into account that G = 5ΣνT4 = 10Ð11 W/K, we × Ð18 1/2 obtain NEPTD = 1.4 10 W/Hz and, substituting the heat conductivity in a voltage-bias mode, we obtain × Ð18 1/2 Fig. 2. Current and voltage responses of bolometer with a NEPV = 1.3 10 W/Hz . (5) tunnel junction resistance of 10 kΩ vs. the bias voltage at a temperature of 260 mK. The use of a SQUID-based low-noise reading device with a current resolution of 50 fA/Hz1/2 will allow the noise-equivalent power to be improved by an order of magnitude as compared to the technical noise- equivalent power determined by the noise of a warm transistor amplifier. 4. Measurement of the Josephson junction radiation at 260 mK. In the first series of experiments, the backside of a substrate with the Josephson oscillator was directly attached to the backside of a substrate with the bolometer (Fig. 3), and this assembly was cooled to 260 mK. Since the planar antennas were deposited on the substrates with a dielectric constant higher than 10, the main lobes of their directivity patterns were oriented toward the dielectric and, when the radiating antenna was positioned opposite the receiving antenna, an efficient power transfer occurred from the Josephson junction to the bolometer. The log-periodic antennas of Fig. 3. Schematic diagram of the experiment at 260 mK the oscillator and the receiver were identical and rated with the backside of the Josephson oscillator substrate for a frequency range from 200 GHz to 2 THz. pressed to the backside of the bolometer substrate. The dependence of the bolometer response on the voltage applied to the Josephson junction is shown in Fig. 4. By applying a magnetic field, the critical current of the Josephson junction can be suppressed to zero, and then the junction will be a simple current-heated resistor. In this case, depending on the bias on the radiating junction, the bolometer receives thermal radiation from a cold or heated load. The dependence of the response on the bias is found to be parabolic, which corresponds to the Joule heating proportional to the square of applied voltage. This experiment allows one to separate the Josephson radiation component, whose frequency corresponds to the bias voltage, from the broadband thermal component, whose power is proportional to the square of bias voltage. It is significant that, above 1 mV, the Josephson junction is strongly overheated both in the absence and in the presence of magnetic field and its effective electron temperature considerably exceeds the refrigerator equilibrium temperature. Fig. 4. Bolometer response to the radiation of a Josephson The maximal Josephson radiation power can be esti- × Ð9 source. The solid parabola corresponds to Joule heating. mated as Posc = 0.1IcVc = 2 10 W [7]. In the case of

JETP LETTERS Vol. 79 No. 6 2004 TERAHERTZ SPECTROSCOPY WITH A JOSEPHSON OSCILLATOR 301 a log-periodic radiation antenna with a knife-edge pattern and double-dipole reception antenna with a pencil- beam pattern, the mismatch of the directivity patterns takes place and the losses increase by more than 10 dB. The oscillator substrate had five log-periodic antennas on its surface, and the bolometer substrate had two double-dipole antennas and one log-periodic antenna. The oscillator and receiver antennas were never directed toward each other, resulting in the losses of no less than 10 dB. No convergence lenses were placed between the Fig. 5. Schematic diagram of the quasi-optical experiment, oscillator and receiver, so that the received beams where the bolometer was positioned on the flat surface of a diverged, adding another 10-dB (or greater) loss. The hyperhemispherical sapphire lens at 260 mK and the substrate with the Josephson oscillator was placed on a similar mismatch with antenna, the mismatch between the radi- silicon lens at 1.8 K. The oscillator and receiver were 3 cm ating and receiving beams, the inaccurate alignment of apart, so that the tested object could be placed between the directions of different antennas, the reflections from them. the sapphireÐsilicon boundary, and the difference in polarizations—all this provides a total loss of no less than 30 dB at frequencies on the order of 1 THz. As a hf ≈ kT. Applying Eq. (7) to Eq. (1) and neglecting the result, the power received by the bolometer is less than phonon temperature, we arrive at the expression Ð12 10 W. Setting the response of the bolometer under 2 2 0.6 e V consideration to S = 1.1 × 108 V/W, we find that the Prad = ------2 ------, (8) maximal voltage response is equal to approximately 4π h 1.1 × 10Ð4 V. In our experiments, the response to the µ which fits the quadratic dependence observed in the Josephson radiation reached a level of 10 V. The dif- experiment. ference of one order of magnitude may be caused by such factors as nonideal characteristics of the Joseph- 5. Irradiation of bolometer by the Josephson son junction, excess current, and overheating, which junction at 1.8 K. To increase the output power of the reduce the output power. The response to the non- Josephson source and the oscillation frequency, it is Josephson radiation on the parabolic portion of the necessary to increase the characteristic voltage of the response curve may be ascribed to the receipt of the Josephson junction, i.e., its critical current. Placing the submillimeter and infrared radiation from the matched Josephson junction at the He-4 cooling step, we prevent load integrated in the broadband antenna circuit and the bolometer overheating by the dc bias-current power radiating into the quasi-optical channel. of the Josephson junction. For example, in a junction with a resistance of 10 Ω, a power of 0.2 µW is If we accept the overheating model for the Joseph- absorbed at an oscillation frequency of 300 GHz, and son junction as a bridge of variable thickness [4], we this power increases to 2.5 µW at a frequency of 1 THz. can estimate the temperature in the middle of the bridge In the quasi-optical configuration (Fig. 5), the as bolometer was placed on the flat surface of a hyperhemispherical sapphire lens at a temperature of 2 260 mK, and the substrate with the Josephson oscillator 2 eV T m = T b +.3------π (6) was placed on a similar silicon lens at a temperature of 2 k 1.8 K. The oscillator and the receiver were spaced 3 cm apart. From this relation, we obtain an equivalent electron The topology of the Josephson samples was the temperature of about 3 K for a Josephson junction bias same as in the experiments at 260 mK, but the critical of 1 mV. Considering that the thermal radiation is scat- π current exceeded 0.5Ð1 mA at 4 K. In the absence of an tered within a solid angle of 4 and that the bolometer external magnetic field, the value of I R exceeded is placed at a distance of about 1 mm in a dielectric, the c n value of 5 mK of the effective temperature measured by 5 mV. With increasing magnetic field, the critical cur- the bolometer becomes understandable. Since the rent oscillated. Under the irradiation by a backward- power is first radiated and then received, it is also nec- wave oscillator, the Shapiro steps were observed up to essary to take into account the quantum character of 4 mV. The experimental curves are shown in Fig. 6. radiation according to the Planck law: Figure 6a represents the response of a bolometer with a double-dipole antenna rated for a central frequency of 300 GHz and the response of a bolometer with a log- hf Pr = ------0.3 f . (7) periodic broadband antenna in the range 0.2Ð2 THz. ehf /kT Ð 1 Figure 6b shows the response of a bolometer with a double-dipole antenna for two values of magnetic field, According to this relation, the maximal radiation i.e., for the higher (upper curve) and lower (lower occurs at a frequency corresponding to the temperature: curve) values of critical current and characteristic volt-

JETP LETTERS Vol. 79 No. 6 2004 302 TARASOV et al. 6. Discussion. A simple analytic expression for the voltage response of bolometer [9] gives a rough estimate of the practically attainable power response at a temperature of 260 mK: 2k Smax ==------b -108 V/W. V Σν 4 e T e A more accurate calculation, according to [9], yields a value of 4 × 108 V/W, which coincides with the experimental data. With allowance made for the bolometer noise at the operating point and the ampliﬁer noise VN = 6 nV/Hz1/2 in a current bias mode, the noise-equivalent power is max × Ð17 1/2 NEPV ==V N/SV 1.5 10 W/Hz . Let us also estimate the characteristic values for the voltage bias mode with electron cooling. The main heat ﬂow occurs from hot phonons to electrons that are subjected to electron cooling: Σν 5 PPhÐ e ==T Ph 0.5 pW. To remove this power from the electron system, it is necessary to apply cooling current

ePPhÐ e × Ð8 Ic ==------2.2 10 A. kbT This current gives rise to a shot noise. Taking the theo- × retical value of the current response SI = e/2kbT = 6 103 A/W, we obtain the noise-equivalent power

Σν 5 × Ð17 1/2 NEPI ==4kbT e T Ph 1.3 10 W/Hz . (9) This value is smaller than the measured value of 5 × 10Ð17. This can be explained by the fact that the current response of high-resistance SIN junctions is weaker than the theoretical value obtained for an optimal resistance of 1 kΩ. The voltage response has been measured for a SINIS bolometer at 260 mK to give 4 × 108 V/W. The noise-equivalent power limited by the bolometer and amplifier noise is 1.5 × 10Ð17 W/Hz1/2. The measure- Fig. 6. (a) Responses measured by a bolometer with a double-dipole antenna (upper curve) and a bolometer with a ments of the radiation from an HTSC Josephson junc- log-periodic antenna (lower curve). (b) Response of a tion have shown that, at bias voltages on the order of bolometer with a double-dipole antenna for two values of 1 mV, the junction is overheated, and its effective tem- magnetic field and critical currents of 400 (upper curve) and perature exceeds 3 K for a substrate temperature of 150 µA (lower curve). (c) Response measured at high bias voltages without magnetic field. The last maximum corre- 260 mK. The use of HTSC junctions at low tempera- sponds to a frequency of 1.7 THz. tures is advantageous, because these junctions allow one to obtain high values of characteristic voltage IcRn and increase the radiation frequency at least to 1.7 THz. A combination of a Josephson terahertz oscillator with age. The response corresponding to the highest fre- a high-sensitivity SINIS bolometer makes it possible to quency (1.7 THz) is shown in Fig. 6c. The same curve realize a compact cryogenic terahertz network analyzer exhibits the third and fifth harmonics of the antenna with a frequency resolution on the order of several giga- fundamental mode. hertz.

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This work was supported by the VR and STINT 5. M. Leivo, J. Pecola, and D. Averin, Appl. Phys. Lett. 68, foundations (Sweden) and by the INTAS (project 1996 (1996). nos. 01-686 and 00-384). 6. M. Tarasov, L. Kuz’min, M. Fominskiœ, et al., Pis’ma Zh. Éksp. Teor. Fiz. 78, 1228 (2003) [JETP Lett. 78, 714 (2003)]. REFERENCES 7. K. K. Likharev and B. T. Ul’rikh, Systems with the 1. L. Kuzmin, Physica B (Amsterdam) 284Ð288, 2129 Josephson Contacts (Mosk. Gos. Univ., Moscow, 1978). (2000). 2. M. Tarasov, M. Fominskiœ, A. Kalabukhov, and 8. M. Tinkham, M. Octavio, and W. J. Skocpol, J. Appl. L. Kuz’min, Pis’ma Zh. Éksp. Teor. Fiz. 76, 588 (2002) Phys. 48, 1311 (1977). [JETP Lett. 76, 507 (2002)]. 9. D. Golubev and L. Kuzmin, J. Appl. Phys. 89, 6464 3. L. Kuzmin, I. Devyatov, and D. Golubev, Proc. SPIE (2001). 3465, 193 (1998). 4. M. Nahum, T. M. Eiles, and J. M. Martinis, Appl. Phys. Lett. 65, 3123 (1994). Translated by E. Golyamina

JETP LETTERS Vol. 79 No. 6 2004

JETP Letters, Vol. 79, No. 6, 2004, pp. 304Ð307. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 79, No. 6, 2004, pp. 362Ð366. Original Russian Text Copyright © 2004 by Safonov, Demukh, Kharitonov.

Dissipation of Ripplon Flow at the Surface of Superfluid 4He A. I. Safonov*, S. S. Demukh, and A. A. Kharitonov Russian Research Centre Kurchatov Institute, pl. Kurchatova 1, Moscow, 123182 Russia *e-mail: [email protected] Received February 18, 2004

Within the framework of quantum hydrodynamics of a superfluid helium surface, the momentum relaxation rate caused by the annihilation of two ripplons with phonon creation, inelastic phonon scattering with ripplon annihilation, and in the case of helium films one-particle ripplon scattering from the surface-level inhomogeneities introduced by the substrate roughness (new relaxation mechanism) was obtained for a ripplon gas at T Շ 0.25 K. The contribution from the inelastic phonon scattering is negligible at these temperatures. For a film at T ≤ 0.15 K, one-particle scattering dominates, leading to a temperature dependence of the form K ∝ T5/3 for the convective thermal conductance. At higher temperatures, phonon creation with annihilation of two ripplons is the dominant mechanism, giving K ∝ TÐ3. These results are in quantitative agreement with the available experimental data. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 67.40.Pm; 67.40.Hf; 68.03.Kn

The hydrodynamical flow drag of a gas of elemen- the phonon subsystem with the creation of a phonon tary excitations (ripplons) [1] at the surface of super- with wavevector (q, k), where q and k are the compo- fluid 4He determines the heat transfer in helium films at nents, respectively, along and perpendicular to the sur- T Շ 0.2 K [2, 3], the possibility of and the conditions face, and with the annihilation of two ripplons with for the observation of the second surface sound [4], and wavevectors q' and q". Denote the ripplon and phonon also plays an important role in the experiments with temperatures as TR and TP. The ripplon frequency and two-dimensional atomic hydrogen (2D H↓) at the wavevector are related to each other by the well-known helium surface [5, 6]. In this work, the phonon-induced expression ω2 = q'3α/ρ [1], where α = 0.378 dyne/cm ripplon-flow decay at the surface of a bulk liquid is cal- q' 3 culated and the flow dissipation mechanism is sug- and ρ = 0.145 g/cm are, respectively, the surface ten- gested for a helium film covering rough substrate. sion and the density of liquid 4He at T = 0. Similarly, ω 2 2 The ripplon gas flow along the helium surface sepa- one has for phonons qk = c q + k , where c = rately from bulk liquid was considered theoretically by 238 m/s is the sound velocity in helium. Due to the Andreev and Kompaneets in [7] and observed experi- energy and momentum conservation laws, q Ӷ q' and, mentally by Mantz, Edwards, and Nayak in [2, 3]. Orig- hence, q" ≈ Ðq'. According to [11], the matrix element inally, the main role in the ripplon-flow stagnation was for the R + R P transition is ascribed to the creation of a long-wavelength ripplon with annihilation of two short-wavelength ripplons [2, ប3c2 1/2 〈〉qkHq'', q' ≈ ------q'kδ , , (1) 8], followed by a rapid decay of the long-wavelength Vρω qq' + q'' ripplon, whose velocity field penetrates across the qk whole film width. Later on, it was recognized, however, where V is the liquid volume. The momentum transfer that both the theoretical probability of this process [9] rate to phonons is given by the expression presented in and the actual decay of acoustic ripplons [10] are quite [11] for the heat transfer rate: insignificant. It was established theoretically that the π Π 2 ប 〈〉, 2 inelastic phonon (P) scattering at the surface with the RR+ ↔ P= V------∫ qqkHq'' q' creation of a ripplon (R) [8, 9] or the annihilation of two ប2 ripplons with phonon creation [11] are the main mech- × []()()() anisms that are responsible for the interaction of rip- Nq' Nq'' 1 + nqk' Ð nqk' 1 + Nq' 1 + Nq'' (2) plon subsystem with bulk helium at temperatures con- 2 2 sidered. Next, Reynolds, Setija, and Shlyapnikov [11] ×δ() ω ω ω d q'd qdk q' + q'' Ð qk ------, showed that the energy transfer from the ripplon to the ()2π 5 phonon subsystem is mainly governed by the second process. where Nq' (TR), Nq'' (TR), and nqk' (TP) are the equilib- Following the reasoning exactly as in [11], we cal- rium occupation numbers for, respectively, ripplons culate the momentum-transfer rate from the ripplon to and phonons in the reference system associated with

0021-3640/04/7906-0304 $26.00 © 2004 MAIK “Nauka/Interperiodica” DISSIPATION OF RIPPLON FLOW 305 ω ripplons. The integration over q' goes over a half of the with the shifted ripplon distribution function N'( Q) = momentum space, lest the same pairs (q', q") of rip- ω ≈ បω2 plons be taken twice. N( Q Ð uQ) NQ + uQ(T/ Q ) and the integration going to Q ~ q = T /បc, where q is the wavevector of Let ripplons move relative to the bulk liquid with a T P T velocity u much lower than the sound velocity. Then, a thermal phonon. This gives the occupation numbers in the reference system associ- π ប 6 Π ∼ u cT P ' ≡ ω ω PPR↔ + ------T R, (7) ated with ripplons are nqk n'( qk) = n( qk Ð uq), 480 c α បc where n’s are the occupation numbers in a fixed coordinate system. By expanding n' in powers of uq and i.e., | | Ӷ ≤ ω retaining only the linear term (because uq cq qk), Ð5 g 6 Π ∼ × ------we obtain n' ≈ n Ð uq(dn /dω ). Due to the symme- PPR↔ + 7.1 10 3 7 uT PT R, (8) qk qk qk qk cm K try, all terms proportional to q in Eq. (2) turn to zero upon the integration, and q(uq) can be replaced by which is several orders of magnitude smaller than the 1 value obtained for the R + R P process at temper- uq2/2. In addition, ω ≈ ω ≈ --- ω and N ≈ N . Շ q' q'' 2 qk q' q'' atures T 0.2 K. Thus, the characteristic time of momentum transfer from the ripplon to the phonon sub- The subsequent calculation yields system is, evidently, 4/3 2 20/3 16 uρ ប T P ρ Ð7 5 Π ↔ = ------Ru 1.26× 10 sK RR+ P 2 α 4ប τ ≈≈ π c ρ R ------5 5/3 -, (9) 45 c P ↔ β ()β RR+ P T P F ∞ (3) 2x βx × 20/3 e e + 1 ρ × Ð10 5/3 2 Ð5/3 ∫ x dx ------------β - . where R = 1.67 10 T R (g/cm ) K is the ripplon ()e2x Ð 1 e x Ð 1 0 mass density. β The convective thermal conductance In this expression, = TP/TR. The substitution of numerical values gives τ 2 R KR = TSRρ------, (10) Π × Ð4 g 20/3 ()β R RR+ ↔ P= 1.33 10 ------3 20/3- uT P F , (4) cm K due to the heat transfer by the ripplon hydrodynamical × Ð2 4/3 Ð2 Ð7/3 where F(β) stands for the integral in Eq. (3). For small flow [2], where SR = 1.52 10 T erg cm K is the β Ӷ ripplon gas entropy, is one of the possible experimen- , i.e., for TP TR, the evaluation of this integral gives F(β) ≈ 7.91/β, and it becomes constant F ≈ 12.9 in the tally measured quantities. It follows that, if the ripplon opposite limit β ӷ 1. If the temperatures of both sub- stagnation is controlled by the R + R P process, Ð3 systems coincide, the integral is F(1) ≈ 14.3. the value of KR must decrease as T with increasing Let us now turn to the inelastic phonon scattering temperature. However, the experiments of Mantz, Edwards, and Nayak [2] showed a cardinally different P(qk) P(q'k') + R(Q). One can readily see from the kinematic considerations [8] that the ripplon frequency behavior (Fig. 1): the thermal conductance increased ω ω with temperature up to T Ӎ 0.15 K, whereupon it Q is small compared to the phonon frequencies qk ω became saturated and even slightly decreased. Such a and q'k' , whereas the vectors q, q', and Q are, gener- distinction can be explained by the occurrence of a dif- បω Ӷ ally speaking, comparable. Consequently, Q TP, ferent and a more efficient ripplon-stagnation mecha- because the main contribution to the momentum trans- nism in helium films. It follows from the experimen- Շ fer comes, clearly, from phonons with energies TP tally observed temperature dependence of thermal con- (the phonon and ripplon temperatures are assumed to ductance that the characteristic ripplon-momentum be comparable). According to the experimentally con- relaxation time at low temperatures depends weakly on firmed [10] calculations of Roche, Roder, and Williams temperature. This allows one to disregard multiparticle [9], the reciprocal decay time caused by the inelastic processes such as direct ripplon interaction with the phonon scattering is equal for acoustic ripplons to substrate phonons and restrict oneself to a search for the mechanisms determining the finite lifetime of an indi- π2 ប 4 1 ≈ QT P vidual ripplon. -----τ ------ρ-ប------. (5) Q 60 c We assume that the ripplon decay is due to the scattering from the film surface inhomogeneities intro- Similarly to Eq. (2), the momentum transfer rate can be duced by the substrate roughness (Fig. 2a). Since the estimated by the integral ripplon propagation is unrelated to the surface orienta- បQ d2Q tion, only the variable surface curvature K(r) can be the Π ↔ = ------N' ------(6) cause of scattering. The effective potential for ripplon PPR+ ∫ τ Q 2 Q ()2π scattering (Fig. 2b) can easily be found from the dimen-

JETP LETTERS Vol. 79 No. 6 2004 306 SAFONOV et al. where l is the roughness scale. Since the constant component of effective potential does not induce scattering ӷ and Kp Km, the effective potential V(r) can be replaced by the zero value everywhere except for the vicinities of peaks, where, due to the relation between ប α ρ 1/2 Ð3/2 s) the potential and the curvature, it is Vp ~ ( / ) d . The vicinity size a is determined by the surface slope φ φ Ӎ Ð7 = Kml/4 near the peak and is a = d 10 cm for l = 1 µm. Thus, the scattering centers are rather widely spaced, so that each of them can be considered sepa- rately.

Note that Kp is small compared to the wavevector ρ α 1/3 ប 2/3 × 6 Ð1 qT = ( / ) (T/ ) ~ 3 10 cm of the typical rip- Ӎ plon, whereas qT a 0.3. One can, therefore, use the Born approximation for slow particles, which, as is known, gives the isotropic scattering. The scattering cross section of a ripplon with wavevector q can be obtained from the standard expression [12] for the two- dimensional case by the substitution of the ripplon effective mass according to the definition 1/m = ∂ ∂ 2 3 ប α ρ 1/2 Ð1/2 Fig. 1. The convective thermal conductance of ripplons at Eq/ (q ) = --- ( / ) q . The result is independent the surface of 4He film: (solid curve) according to Eq. (16) 4 with H = 6 cm and l = 2.9 µm, and (᭿) experimental results of q: of Mantz et al. [2]. The calculated dependence (light dots) takes into account the experimentally measured thermal ρ π2 2 4 ρ σ 4  ()2 2 4 V pa  conductance of substrate. The dashed curve shows the rip- Ӎ --------- ∫V r d r Ӎ --------- , (11) plon thermal conductance restricted only to the interaction 9ប2 α 9ប2 α with helium phonons. or, using the expressions for Vp and a, 32φ4π2 σ ∼ ------d. (12) 9 The ripplon mean free path is, as usual, λ = l2/σ, and the τ reciprocal relaxation time q of the momentum projection onto the initial direction is found from the relation τ ω λ q q = q . The momentum transfer rate from ripplons to the substrate is calculated by analogy with Eq. (6): ∞ 1 d2q បuq2 Π = ------dNω . (13) V 2∫ 2 τ ------ω- ()2π q d q 0 The integration with respect to q and the substitution of Fig. 2. (a) The profile of a substrate surface and a coating φ give helium film. (b) Effective potential produced by the surface curvature and leading to the ripplon scattering. 2 ρ 5 Π ∼ T Ru()4 2 V 0.14------ប -α--- gH l d. (14) ប α ρ 1/2| |3/2 sional considerations: V(r) ~ ( / ) K(r) . Due to Accordingly, the characteristic time of momentum the capillary effect, a film covering a rough surface fills transfer from ripplons to the substrate is the deepenings between the substrate peaks to form ρ α 3 Ð1 Ð15 1/3 7 menisci with curvature Km = gH/ ~ 10 cm , where ρ u 810× K cm s τ ≈≈------R ------. (15) H ~ 1 cm is the height above the level of bulk liquid. At R Π 1/3 4 2 the peaks, the curvature coincides, in the order of mag- V T R H l d Ӎ Ӎ nitude, with the reciprocal film thickness, Kp 2/d Thus, the thermal conductance at low temperatures 6 × 105H1/3 cmÐ1 (the height H is in centimeters). The is proportional to T 5/3, in full agreement with the exper- number of peaks per unit surface area is equal to ~1/l2, imentally observed dependence.

JETP LETTERS Vol. 79 No. 6 2004 DISSIPATION OF RIPPLON FLOW 307

For several simultaneously acting mechanisms, the tension in pure 4He [13] and in the measurements of momentum relaxation rates add up. Hence, the final thermal conductance of helium films [2]. As to the expression for the film thermal conductance at TP = experiments with atomic hydrogen [5], the time of TR = T has the form establishing thermal equilibrium in a two-dimensional system is determined by the ripplon scattering from the 7 Ð1 3 910× 4 2 H atoms adsorbed at the helium surface [14]. For a K ≈ 83T + ------H l d []erg/s K . (16) 5/3 characteristic density of ~1012 cmÐ2 of these atoms and T Շ τ the temperature T 100 mK, this time is 2 ~ 60 ns, In Eq. (16), the temperature is in K and the roughness which is two orders of magnitude shorter than the heat scale l is in cm. The temperature dependence of KR for transfer time from ripplons to phonons. Thus, a two- H = 6 cm and l = 2.9 µm is shown in Fig. 1 by the solid dimensional system is also in thermal equilibrium in curve. The agreement with the experiment [2] (᭿) looks this case, but its temperature can be different from the rather conclusive, the more so as all calculations were phonon temperature. carried out using the ab initio values, while the only fit- We are grateful to L.A. Maksimov for helpful dis- ting parameter—substrate roughness scale l—seems to cussions and to I.I. Lukashevich for support. This work be quite reliable from the viewpoint of experimental was supported by the Russian Foundation for Basic conditions. In the calculations, the experimentally mea- Research (project no. 02-02-16652a) and INTAS (grant sured thermal conductance of the substrate (mylar film no. 2001-0552). coated with a layer of solid neon) was taken into account. For comparison, the ripplon convective thermal conductance restricted only to the interaction with REFERENCES liquid phonons (R + R P process), as is the case for 1. K. R. Atkins, Can. J. Phys. 31, 1165 (1953). the surface of bulk helium, is shown in Fig. 1 by the dashed curve. 2. I. B. Mantz, D. O. Edwards, and V. U. Nayak, Phys. Rev. Lett. 44, 663 (1980); 44, 1094(E) (1980). As is known, the equilibrium film thickness d is 3. I. B. Mantz, D. O. Edwards, and V. U. Nayak, J. Phys. related to the height above the level of bulk liquid by Colloq. 39, C6-300 (1978). ≈ × Ð6 Ð1/3 d 3 10 H , where both quantities are expressed 4. W. M. Saslow and A. A. Kumar, Phys. Rev. B 30, 6402 in cm. For this reason, the dependence of K on the film (1984). thickness observed by Mantz et al. in [3] can be 5. S. A. Vasilyev, J. Järvinen, A. I. Safonov, and explained qualitatively by formally replacing the height S. Jaakkola, Phys. Rev. A 69, 023610 (2004). × Ð6 3 H by (3 10 /d) in Eq. (16). It should also be noted 6. In recent experiments (with the participation of the that the fact that the momentum relaxation time authors of this work) on the thermal compression of 2D depends on the substrate roughness scale and on the H↓, a gas flow of H atoms along the surface of helium film height above the bulk liquid can be the reason why film was observed. The momentum transfer time from the times observed in experiments with two-dimen- the hydrogen subsystem to ripplons and from ripplons to sional atomic hydrogen [6] were much shorter than in the substrate are the key words in the description of this the experiments of Mantz et al. [2, 3]. flow. The material is prepared to publication. s12. Theo- retical Physics. III. Quantum Mechanics. In conclusion, let us discuss the validity of using the concept of ripplon temperature. The point is what is the 7. A. F. Andreev and D. A. Kompaneets, Zh. Éksp. Teor. ratio between the rate of establishing equilibrium in the Fiz. 61, 2459 (1972) [Sov. Phys. JETP 34, 1316 (1972)]. ripplon system and the rate of equilibrium breaking due 8. W. F. Saam, Phys. Rev. A 8, 1918 (1973). to the interaction with phonons (one-particle elastic rip- 9. P. Roche, M. Roger, and F. I. B. Williams, Phys. Rev. B plon scattering by the surface inhomogeneities does not 53, 2225 (1996). affect the distribution function). The characteristic heat 10. P. Roche, G. Deville, K. O. Keshishev, et al., Phys. Rev. transfer time from ripplons to phonons is mainly deter- Lett. 75, 3316 (1995). mined by the R + R P process and can be esti- 11. M. W. Reynolds, I. D. Setija, and G. V. Shlyapnikov, τ × Ð10 Ð13/3 13/3 mated as rp ~ RrpCR = 3 10 T K s, where Phys. Rev. B 46, 575 (1992). × Ð2 4/3 Ð7/3 Ð2 CR = 1.52 10 T erg K cm is the ripplon heat 12. L. D. Landau and E. M. Lifshitz, Course of Theoretical capacity (coinciding with the ripplon entropy) and Physics, Vol. 3: Quantum Mechanics: Non-Relativistic × Ð8 Ð17/3 20/3 2 Theory, 4th ed. (Nauka, Moscow, 1989; Pergamon, New Rrp = 1.8 10 T K cm /erg is the thermal resis- York, 1977). tance between the ripplon and phonon subsystems [11]. 13. D. O. Edwards and W. F. Saam, in Progress in Low Tem- One can readily see that, in the absence of an external perature Physics, Ed. by D. F. Brewer (North-Holland, heat inflow and efficient relaxation mechanisms in the Amsterdam, 1978), Vol. 7A, p. 283. ripplon system, the indicated process leads to the rip- 14. D. S. Zimmerman and A. J. Berlinsky, Can. J. Phys. 61, plon thermal distribution with temperature equal to the 508 (1983). phonon temperature (the phonon system is assumed to be in equilibrium). Such a situation is realized, e.g., when measuring the temperature dependence of surface Translated by V. Sakun

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JETP Letters, Vol. 79, No. 6, 2004, p. 308.

Erratum: “Structural Transformations in Liquid, Crystalline and Glassy B2O3 under High Pressure” [JETP Lett. 78 (6), 393Ð397 (2003)] V. V. Brazhkin, Y. Katayama, Y. Inamura, M. V. Kondrin, A. G. Lyapin, S. V. Popova, and R. N. Voloshin PACS numbers: 64.70.Kb; 62.50.+p; 61.43.-j; 61.50.Ks; 99.10.Cd

The study of the chemical composition of borate glasses obtained in graphite ampoules under pressures of 3−7 GPa and at temperatures above 1700 K have shown the presence of a considerable (5Ð10 at. %) amount of calcium that enters the glass, most likely, from the pressure-transmitting medium. As a result, the physical properties and structural characteristics that were initially assigned to boron oxide glasses with oxygen nonstoichiom- etry (6th section in the article) relate in reality to triple glasses with a high calcium content.