Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter

Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter

For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the speci...

Повний опис

Збережено в:
Бібліографічні деталі
Опубліковано в: :Физика низких температур
Дата:2013
Автор: Rovenchak, A.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2013
Теми:
Поиск новых сверхтекучих систем
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/118821
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter / A. Rovenchak // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1141–1145. — Бібліогр.: 26 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-118821
record_format dspace
spelling Rovenchak, A.
2017-05-31T10:23:28Z
2017-05-31T10:23:28Z
2013
Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter / A. Rovenchak // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1141–1145. — Бібліогр.: 26 назв. — англ.
0132-6414
PACS: 05.30.Pr, 67.10.Fj,64.60.–i
https://nasplib.isofts.kiev.ua/handle/123456789/118821
For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the specific heat are made.
I am grateful to Yuri Krynytskyi and Prof. Volodymyr Tkachuk for discussion and useful comments. Remarks from the anonymous referees are highly appreciated. This work was partly supported by Project ФФ-110Ф (registration No. 0112U001275) from the Ministry of Edu-cation and Sciences, Youth and Sports of Ukraine.
en
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Физика низких температур
Поиск новых сверхтекучих систем
Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
spellingShingle Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
Rovenchak, A.
Поиск новых сверхтекучих систем
title_short Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
title_full Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
title_fullStr Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
title_full_unstemmed Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter
title_sort phase transition in a system of 1d harmonic oscillators obeying polychronakos statistics with a complex parameter
author Rovenchak, A.
author_facet Rovenchak, A.
topic Поиск новых сверхтекучих систем
topic_facet Поиск новых сверхтекучих систем
publishDate 2013
language English
container_title Физика низких температур
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format Article
description For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the specific heat are made.
issn 0132-6414
url https://nasplib.isofts.kiev.ua/handle/123456789/118821
citation_txt Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter / A. Rovenchak // Физика низких температур. — 2013. — Т. 39, № 10. — С. 1141–1145. — Бібліогр.: 26 назв. — англ.
work_keys_str_mv AT rovenchaka phasetransitioninasystemof1dharmonicoscillatorsobeyingpolychronakosstatisticswithacomplexparameter
first_indexed 2025-11-26T08:16:49Z
last_indexed 2025-11-26T08:16:49Z
_version_ 1850615306314055680
fulltext Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10, pp. 1141–1145 Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter Andrij Rovenchak Department for Theoretical Physics, Ivan Franko National University of Lviv 12 Drahomanov Str., Lviv, UA-79005, Ukraine E-mail: andrij.rovenchak@gmail.com Received February 25, 2013, revised March 28, 2013 For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the specific heat are made. PACS: 05.30.Pr Fractional statistics systems (anyons, etc.); 67.10.Fj Quantum statistical theory; 64.60.–i General studies of phase transitions. Keywords: Polychronakos statistics, phase transitions. 1. Introduction In recent decades, quantum systems obeying fractional statistics have been a subject of studies based on different approaches [1]. In real physical systems a description based on an intermediate statistics between the Fermi and Bose cases appears in a number of possibilities: the so- called exclusion statistics can be linked to the interaction in 1D systems [2]; ultracold gases can be described by means of anyon–fermion mapping [3]; thermodynamics of finite Bose-systems can be studied using the Gentile statistics [4]; and this list can be expanded further. In this work, a generalization of the Polychronakos statistics [5–7] is studied. The parameter of statistics is considered a complex number [8]. Complex-valued physical quantities can occur effectively in some physical systems: complex energy is connected to a dissipative process [9]; complex chemical potential is also used, e.g., in quantum chromodynamics [10] or physics of semicon- ductors [11]; complex external potential is applied to describe the interaction between the light field and atoms moving in crystals [12,13] and also can be used to study -symmetry breaking [14]. The one-dimensional problem is considered here for the sake of simplicity. A system of 1D harmonic oscillators effectively corresponds to a 2D homogenous system, which opens a possibility to apply the obtained results for studies of, e.g., critical phenomena in liquid-helium films [15–17] or surface-electron systems [18]. The paper is organized as follows. In Sec. 2 statistics is explained and calculation procedure is described. In Sec. 3 analytical expressions for the thermodynamic functions of a 1D harmonic oscillator system are derived. Main results are obtained in Sec. 4, where the critical temperature is calculated and temperature dependence of energy and heat capacity is analyzed. A discussion on possible experimen- tal observation of the phase transition is given in Sec. 5. 2. Initial expressions In the Polychronakos statistics [5,6] the occupation of the jth level with energy jε is given by /1 1= , e j Tj n z ε− − α (1) where z is fugacity and T is temperature. Here jn is a formal characteristic and can take a complex values. Note that this form for the occupation numbers was also suggested a priori by Acharya and Narayana Swamy to describe anyons [19]. To make a generalization, the parameter = e =iπνα = i′ ′′α + α is put a complex number; = 0–1ν provides a smooth transition between the bosonic and fermionic © Andrij Rovenchak, 2013 Andrij Rovenchak limits, while the value = 0α corresponding to the clas- sical Boltzmann statistics is avoided, see Fig. 1 [8]. It can be shown that in the limit 0ν → a simple correspondence can be established between a system with the real excitation spectrum jε obeying the Polychronakos statistics generalized as above and a system with a small dissipative term in the excitation spectrum obeying the Bose statistics. If the elementary excitation spectrum is j jiε + γ , where jγ corresponds to small energy dissipation, for high levels with 1j the link is =j jTγ πν ε [8]. Such systems are seen as most prospective candidates for experimental observations of the effects discussed in the present work. Calculations are made using a simple scheme as follows. The number of particles N is given by /1 = = , e j j j Tjj j g N g n z ε− − α ∑ ∑ (2) where jg is the degeneration of the jth level. Elementary excitation spectrum jε is assumed real. Since N is a real number, fugacity =z z iz′ ′′+ must be a complex number. Energy is then given by /1 = = = e j j j j j Tjj j g E g n i z ε− ε ε + Γ − α ∑ ∑  (3) with = ( , )z z N T defined from Eq. (2). Note that only such values of the statistics parameter ν and other quantities are physically reasonable which correspond to Γ  ; otherwise the system becomes essentially nonequilibrium making the whole subsequent consideration incorrect. Real and imaginary parts of the heat capacity equal = , = .d dC dT dT Γ Θ  (4) The peculiarities of the temperature behavior are better seen on the specific heat curves, and these will be given a closer look further. 3. 1D oscillators Let us consider a system of 1D harmonic oscillators with frequency ω. The degeneracy of the jth level equals unity, = 1jg . The excitation spectrum is = , = 0, 1, 2, 3,j j jε ω  (5) The quantity ω is also used as a unit of temperature. One-dimensional systems of oscillators can be realized in highly anisotropic harmonic traps [20] or in optical lattices [21,22]. To obtain results in an analytic form, the summation in Eqs. (2), (3) is substituted with integration using the densi- ty of state function ( )g ε : ( )j d g→ ∫ ε ε∑ . The accuracy of this approximation increases for the number of particles being large and if temperature is large comparing to the level separation, T ω . Numerical comparison of the results obtained using exact summation and integration shows a very good agreement already for the number of particles = 1000N . For a system of 1D harmonic oscillators ( ) = constg ε , which makes it equivalent to, e.g., a system of free particles trapped to a 1D harmonic potential (in the semi- classical approach [23]) or a 2D system of free particles in a box. In the case under consideration, ( ) = 1/g ε ω . The num- ber of particles is then 1 / 0 1= = ln (1 ) e T d TN z z ∞ − ε ε − − α ω ωα− α∫   (6) giving fugacity as ( )/1= 1 e .N Tz − ω α− α  (7) The energy equals 2 21 / 0 1= = ( ) =Li e T d TE z z ∞ − ε ε ε α ω ωα− α∫   ( ) 2 / 2= 1 e ,Li N TT − ω α− ωα   (8) where ( )Lis x is the polylogarithm: =1 ( ) = .Li k s s k xx k ∞ ∑ (9) This series definition is valid for all complex s and complex x with | | < 1x ; for other values of x an analytic continuation is used. Using the obvious property 1 ( ) 1Li = ( )Lis s d x x dx x − Fig. 1. Statistics explanation. +1 Bose –1 Fermi (Boltzmann) ′α ′′α α plane e iπν 1142 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics and taking into account that 1( ) = ln(1 )Li x x− − , for the heat capacity one obtains ( )/ 2/ / 2= = 1 e .Li 1 e N T N T dE N T TC i N N dT N − ω α ω α ω α + Θ + − ω α−     (10) As one should expect for extensive quantities, E N∝ and C N∝ , which is not immediately seen from the above expressions (8) and (10). To ensure extensive nature of energy and heat capacity, thermodynamic limit must be taken into consideration, which in the case of 1D harmonic oscillators reads = constNω . Such a result can be obtained in a number of different ways [4,20], cf. also [24]. The easiest interpretation of such a thermodynamic limit can be given as follows. If a D-dimensional system is contained in a harmonic potential with frequency ω, classical turning points confine the particles to some characteristic length 1/r ∝ ω . To keep an average density / DN r constant, the requirement = constDNω must be satisfied. Thus, in the case under consideration const= = and = , const T NT NT N N T T ω ω ω    therefore, the conditions E N∝ and C N∝ hold. 4. Critical temperature and behavior of thermodynamic functions Discontinuities are observed in both energy and heat capacity as functions of temperature. These discontinuities are due to 2( )Li zα , where the argument is changing the sign of the imaginary part. Indeed, in the calculation of the integral one obtains singularity “1/0” when / / /e = 0 = e 1 e =T T N Tzε ε − ω α− α − +  /= e 1 e cos sin N T T N Ni T T ω ′− αε ω ω ′′ ′′− + α − α       (11) if = and cos = ( 1) < 0.nN n n T ω ′′α π π −  A set of critical temperature values can thus be defined as follows: ( ) = , = 0, 1, 2, 3, , (2 1) k cT N k k ′′α ω + π   (12) cf. Fig. 2. For brevity, the index “(k)” will be dropped in further derivations. To estimate the contribution from the singularity into thermodynamic function, the following procedure can be applied. Let the singularity corresponds to 0ε such that /0e =T zε α . The energy E can be expressed as follows: 0 / / 0 0 = = e eT T z d z dE z z ε −∆∞ ε ε ε ε ε ε + ω ω− α − α∫ ∫   0 / // 0 0 0 , e e eT TT z d z d z ε +∆∞ ε εε ε +∆ ε −∆ ε ε ε ε + + ω ω− α − ∫ ∫   (13) where ∆ is a small positive number. In the limit of 0∆ → first two terms give a smooth function of temperature smoothE , and the last term can be written only to the first order of 0( )ε − ε : 0 /0smooth 0 0 = ( ) e .TzT dE E T ε +∆ −ε ε −∆ ε ε + ω ε − ε∫  (14) While 0ε is real for = cT T , it becomes complex as soon as T deviates from the critical point. The sign of the imagi- nary part is defined by sin ( / )N T′′ω α , see (11). There- fore, the denominator 0( )ε − ε must be substituted with 0( )iε − ε η for = 0cT T ± , where 0η → + . Using Sochocki's formulas [25], one can write 0 0 0 1 1= v.p. ( ),i i ± πδ ε − ε ε − ε η ε − ε (15) where “v.p.” denotes principal value and 0( )δ ε − ε is Dirac's delta-function. Thus, in the vicinity of critical points energy equals /0smooth 0( 0) = ( ) ( ) e .Tc cc c c T E T E T i z T −ε± ± π ε ω (16) For the real part one obtains /00( ) = ( 0) ( 0) = 2 ( ) e .Tc cc c c c T T T T z T −ε′′∆ + − − − π ε ω    (17) Taking into account the definition of critical tempera- ture (12) and writing the statistics parameter in the form Fig. 2. Real part of the specific heat C/N at = 0.25ν . Vertical lines show critical temperatures. The right-most one corresponds to = 0k . 0.2 0.4 0.6 0.8 1.0 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 C N / /T Nω Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1143 Andrij Rovenchak = cos siniα πν + πν, after simple transformations the fol- lowing results can be obtained for jumps of the real parts of energy and heat capacity: ( ) 3 (2 1) cot 2 ( )1 2sin= ln 1 e (2 1) kcT N N k − + π πν∆ πν + ω + π  , (18) ( ) 2 (2 1) cot1 4sin( ) = ln 1 e (2 1) k cC T N k − + π πνπν ∆ + + . (19) In the bosonic limit 0ν → : 2 3 (2 1)/ 2 ( )1 2= e (2 1) kcT N N k − + ν∆ π ν ω +  , (20) 2 2 (2 1)/1 4( ) = e (2 1) k cC T N k − + νπ ν ∆ + . (21) The jumps are finite as soon as > 0ν , but the most pro- nounced one corresponds to = 0k and then they rapidly vanish for > 0k , see Table 1. Note that for values 0.35ν  the oscillations of ( )C T in the low-temperature domain increase and lead to some unphysical regions with negative specific heat. Table 1. Jumps of the specific heat /C N at the critical temperature ν = 0k = 1k = 2k 0.01 461.5 10−⋅ 1347.5 10−⋅ 2216.6 10−⋅ 0.05 102.4 10−⋅ 284.7 10−⋅ 451.7 10−⋅ 0.10 52.4 10−⋅ 143.2 10−⋅ 237.7 10−⋅ 0.15 31.7 10−⋅ 92.6 10−⋅ 156.7 10−⋅ 0.20 21.8 10−⋅ 61.1 10−⋅ 101.1 10−⋅ 0.25 28.5 10−⋅ 55.4 10−⋅ 86.0 10−⋅ 0.30 12.5 10−⋅ 49.3 10−⋅ 65.8 10−⋅ 0.34 15.0 10−⋅ 35.7 10−⋅ 41.1 10−⋅ Notes: While the jump values of 10–14 and smaller are expe- rimentally unobservable, they are preserved in the Table to show how rapidly C∆ vanish as 0ν → and/or k increases. Low- and high-temperature limits of the heat capacity are easily calculated using series expansions in Eqs. (8) or (10). A linear dependence C T∝ observed in Fig. 3 in the low-temperature domain is 2 2 2= (1) = (cos sin ),Li 3 C i T T i N N N + Θ π πν − πν ω α ω  (22) where the link between the polylogarithm and Riemann's zeta-function is taken into account: 2 2(1) = (2) = / 6Li ζ π . Different approaching to the asymptotic value / 1C N → (from below at < 1/ 4ν and from above at > 1/ 4ν ) is determined by the following limiting behavior of the specific heat: 2 2cos 2 sin 2= 1 , 36 36 C i N Ni N T T + Θ πν ω πν ω   − −          (23) where the temperature-dependent term in the real part changes sign at = 1/ 4ν , while the imaginary part remains negative for all ν , see Fig. 3. At = 1/ 4ν for the real part of the specific heat the leading deviation from the asymptotic value is given by 41= 1 . 1200 C N N T ω −      (24) 5. Discussion Since a discontinuity is found in the energy as a func- tion of temperature, the first-order phase transitions occur at ( )k cT . It is worth to estimate whether this effect can be tested experimentally. As it was mentioned in Secs. 1 and 2, complex para- meter in the statistics can be obtained as an effective influence of external laser field or dissipation in the ele- mentary excitation spectrum. It is expected, therefore, that the value of ν is small, 1ν . Hence, only the highest critical temperature corresponding to the value = 0k sin=cT N πν ω π  (25) Fig. 3. (Color online) The real (a) and imaginary (b) parts of the specific heat C/N for different values of the parameter = 0.0ν (black, ); 0.1 (red, – – –); 0.2 (green, ); 0.25 (blue, ); 0.3 (magenta, -- --); 0.34 (cyan, ). –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.70 0.2 0.4 0.6 0.8 1.0 1.2 C / N Θ / N /T Nω /T Nω (a) (b) 1144 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics is relevant for the analysis since the jumps of the specific heat become experimentally negligible at > 1k , see Table 1. Recovering the Boltzmann constant Bk in order to obtain the value of cT in Kelvins: sin= ,B ck T N πν ω π  (26) one can make simple estimations as follows. Assuming experimental precision in the specific heat measurements as high as 0.01%, which corresponds to = 0.10–0.15ν (see Table 1), trapping potential frequency = 1ω kHz, and the number of particles 4= 10N , critical temperature is ob- tained as 510cT −  K. The effect thus is available for experimental observations as soon as a proper system, which can be treated within the Polychronakos statistics with a complex parameter, is prepared. It is interesting to derive from the above results some estimation for a 2D homogenous system. Its density of states (for spin-0 particles of mass m on the area S) is easily calculated as 2( ) = / 2 = constg mSε π , which, comparing to the oscillator result ( ) = 1/g ε ω , leads to an effective correspondence: 2 2= .mNN Sm π ω  (27) The obtained expression for critical temperature appears to have a structure similar to that for the Kosterlitz–Thouless transition KTT [26]: 2 2 2 22 2 2sin= versus = , 2 2 B c B KT sk T k T m m π πν π ρ ρ π   (28) where the 2D density 2 = /mN Sρ and 2sρ is the superfluid density. With a 2D concentration 188 10⋅ m–2 correspon- ding to the bulk liquid helium density of 0.146 g/cm3 [17], the value of the critical temperature is 1cT  K. Note that this estimation is relevant to an ideal homogenous Bose- gas with parameters corresponding to liquid 4He. To apply this approach to a realistic planar system, one can proceed from studying phonons — excitations with a linear dis- persion — thus a 2D oscillator problem must be analyzed, which is yet to be considered in detail. Acknowledgments I am grateful to Yuri Krynytskyi and Prof. Volodymyr Tkachuk for discussion and useful comments. Remarks from the anonymous referees are highly appreciated. This work was partly supported by Project ФФ-110Ф (registration No. 0112U001275) from the Ministry of Edu- cation and Sciences, Youth and Sports of Ukraine. 1. A. Khare, Fractional Statistics and Quantum Theory, 2nd edition, Singapore, World Scientific (2005). 2. M.T. Batchelor, X.W. Guan, and N. Oelkers, Phys. Rev. Lett. 96, 210402 (2006). 3. M.D. Girardeau, Phys. Rev. Lett. 97, 100402 (2006). 4. A. Rovenchak, Fiz. Nizk. Temp. 35, 510 (2009) [Low Temp. Phys. 35, 400 (2009)]. 5. A.P. Polychronakos, Phys. Lett. B 365, 202 (1996). 6. B. Mirza and H. Mohammadzadeh, Phys. Rev. E 82, 031137 (2010). 7. S. Zare, Z. Raissi, H. Mohammadzadeh, and B. Mirza, Eur. Phys. J. C 72, 2152 (2012). 8. A. Rovenchak, J. Phys.: Conf. Ser. 400, 012064 (2012). 9. G.E. Cragg and A.K. Kerman, Phys. Rev. Lett. 94, 190402 (2005). 10. J.R. Ipsen and K. Splittorff, Phys. Rev. D 86, 014508 (2012). 11. P.K. Chakraborty, B. Nag, and K.P. Ghatak, J. Phys. Chem. Solids 64, 2191 (2003). 12. D.O. Chudesnikov and V.P. Yakovlev, Laser Phys. 1, 110 (1991). 13. M.K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, Phys. Rev. Lett. 77, 4980 (1996). 14. A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Vola- tier-Ravat, V. Aimez, G.A. Siviloglou, and D.N. Christodou- lides,, Phys. Rev. Lett. 103, 093902 (2009). 15. D.J. Bishop and J.D. Reppy, Phys. Rev. Lett. 40, 1727 (1978). 16. D. Finotello, K.A. Gillis, A. Wong, and M.H.W. Chan, Phys. Rev. Lett. 61, 1954 (1988). 17. V.E. Syvokon, Fiz. Nizk. Temp. 32, 65 (2006) [Low Temp. Phys. 32, 48 (2006)]. 18. T.I. Zueva, Yu.Z. Kovdrya, and S.S. Sokolov, Fiz. Nizk. Temp. 33, 3 (2007) [Low Temp. Phys. 33, 1 (2007)]. 19. R. Acharya and P. Narayana Swamy, J. Phys. A 27, 7247 (1994). 20. F. Dalfovo, S. Giorgini, and L.P. Pitaevskii, Rev. Mod. Phys. 71, 463 (1999). 21. O. Morsch, J.H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, Phys. Rev. Lett. 87, 140402 (2001). 22. L. Fallani, L. De Sarlo, J.E. Lye, M. Modugno, R. Saers, C. Fort, and M. Inguscio, Phys. Rev. Lett. 93, 140406 (2004). 23. V. Bagnato and D. Kleppner, Phys. Rev. A 44, 7439 (1991). 24. A. Posazhennikova, Rev. Mod. Phys. 78, 1111 (2006). 25. V.S. Vladimirov, Equations of Mathematical Physics, New York, Marcel Dekker (1971). 26. V.E. Syvokon and K.A. Nasyedkin, Fiz. Nizk. Temp. 38, 8 (2012) [Low Temp. Phys. 38, 6 (2012)]. Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 10 1145 http://publish.aps.org/search/field/author/G.%20J.%20Salamo http://publish.aps.org/search/field/author/D.%20Duchesne http://publish.aps.org/search/field/author/R.%20Morandotti http://publish.aps.org/search/field/author/M.%20Volatier-Ravat http://publish.aps.org/search/field/author/M.%20Volatier-Ravat http://publish.aps.org/search/field/author/V.%20Aimez http://publish.aps.org/search/field/author/G.%20A.%20Siviloglou http://publish.aps.org/search/field/author/D.%20N.%20Christodoulides http://publish.aps.org/search/field/author/D.%20N.%20Christodoulides http://publish.aps.org/search/field/author/L.%20Fallani http://publish.aps.org/search/field/author/L.%20De%20Sarlo http://publish.aps.org/search/field/author/J.%20E.%20Lye http://publish.aps.org/search/field/author/M.%20Modugno http://publish.aps.org/search/field/author/R.%20Saers http://publish.aps.org/search/field/author/C.%20Fort http://publish.aps.org/search/field/author/M.%20Inguscio

Схожі ресурси