Microwave absorption saturation and decay heating of surface electrons on liquid helium

The microwave (MW) resonance absorption and decay heating of surface electrons (SEs) on liquid ⁴He are theoretically studied for the vapor atom scattering regime. The decay heating is shown to be an essential occurrence of a MW resonance experiment appearing even at low excitation rates. It stro...

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Datum:	2007
Hauptverfasser:	Monarkha, Yu., Konstantinov, D., Kono, K.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Schriftenreihe:	Физика низких температур
Schlagworte:	Кpаткие сообщения
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/120922
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Zitieren:	Microwave absorption saturation and decay heating of surface electrons on liquid helium / Yu. Monarkha, D. Konstantinov, K. Kono // Физика низких температур. — 2007. — Т. 33, № 08. — С. 942–945. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-1209222017-06-14T03:06:31Z Microwave absorption saturation and decay heating of surface electrons on liquid helium Monarkha, Yu. Konstantinov, D. Kono, K. Кpаткие сообщения The microwave (MW) resonance absorption and decay heating of surface electrons (SEs) on liquid ⁴He are theoretically studied for the vapor atom scattering regime. The decay heating is shown to be an essential occurrence of a MW resonance experiment appearing even at low excitation rates. It strongly affects the occupancies of surface levels and the broadening of resonance lines long before the absorption suturation condition is reached. Contrary to the model of cold SEs usually used for description of the MW resonance, the new theory leads to MW absorption saturation when only a very small fraction of electrons (less than 10%) is left on the ground and the first excited levels. 2007 Article Microwave absorption saturation and decay heating of surface electrons on liquid helium / Yu. Monarkha, D. Konstantinov, K. Kono // Физика низких температур. — 2007. — Т. 33, № 08. — С. 942–945. — Бібліогр.: 7 назв. — англ. 0132-6414 PACS: 67.90.+z, 73.20.–r, 73.25.+i, 78.70.Gq http://dspace.nbuv.gov.ua/handle/123456789/120922 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Кpаткие сообщения Кpаткие сообщения
spellingShingle	Кpаткие сообщения Кpаткие сообщения Monarkha, Yu. Konstantinov, D. Kono, K. Microwave absorption saturation and decay heating of surface electrons on liquid helium Физика низких температур
description	The microwave (MW) resonance absorption and decay heating of surface electrons (SEs) on liquid ⁴He are theoretically studied for the vapor atom scattering regime. The decay heating is shown to be an essential occurrence of a MW resonance experiment appearing even at low excitation rates. It strongly affects the occupancies of surface levels and the broadening of resonance lines long before the absorption suturation condition is reached. Contrary to the model of cold SEs usually used for description of the MW resonance, the new theory leads to MW absorption saturation when only a very small fraction of electrons (less than 10%) is left on the ground and the first excited levels.
format	Article
author	Monarkha, Yu. Konstantinov, D. Kono, K.
author_facet	Monarkha, Yu. Konstantinov, D. Kono, K.
author_sort	Monarkha, Yu.
title	Microwave absorption saturation and decay heating of surface electrons on liquid helium
title_short	Microwave absorption saturation and decay heating of surface electrons on liquid helium
title_full	Microwave absorption saturation and decay heating of surface electrons on liquid helium
title_fullStr	Microwave absorption saturation and decay heating of surface electrons on liquid helium
title_full_unstemmed	Microwave absorption saturation and decay heating of surface electrons on liquid helium
title_sort	microwave absorption saturation and decay heating of surface electrons on liquid helium
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2007
topic_facet	Кpаткие сообщения
url	http://dspace.nbuv.gov.ua/handle/123456789/120922
citation_txt	Microwave absorption saturation and decay heating of surface electrons on liquid helium / Yu. Monarkha, D. Konstantinov, K. Kono // Физика низких температур. — 2007. — Т. 33, № 08. — С. 942–945. — Бібліогр.: 7 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT monarkhayu microwaveabsorptionsaturationanddecayheatingofsurfaceelectronsonliquidhelium AT konstantinovd microwaveabsorptionsaturationanddecayheatingofsurfaceelectronsonliquidhelium AT konok microwaveabsorptionsaturationanddecayheatingofsurfaceelectronsonliquidhelium
first_indexed	2025-07-08T18:52:24Z
last_indexed	2025-07-08T18:52:24Z
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fulltext	Fizika Nizkikh Temperatur, 2007, v. 33, No. 8, p. 942–945 Microwave absorption saturation and decay heating of surface electrons on liquid helium Yuriy Monarkha B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: monarkha@ilt.kharkov.ua Denis Konstantinov and Kimitoshi Kono Low Temperature Physics Laboratory, RIKEN, Hirosawa 2-1, Wako 351-0198, Japan Received March 28, 2007 The microwave (MW) resonance absorption and decay heating of surface electrons (SEs) on liquid 4He are theoretically studied for the vapor atom scattering regime. The decay heating is shown to be an essential occurrence of a MW resonance experiment appearing even at low excitation rates. It strongly affects the oc- cupancies of surface levels and the broadening of resonance lines long before the absorption suturation con- dition is reached. Contrary to the model of cold SEs usually used for description of the MW resonance, the new theory leads to MW absorption saturation when only a very small fraction of electrons (less than 10%) is left on the ground and the first excited levels. PACS: 67.90.+z Other topics in quantum fluids and solids; liquid and solid helium; 73.20.–r Electron states at surfaces and interfaces; 73.25.+i Surface conductivity and carrier phenomena; 78.70.Gq Microwave and radio-frequency interactions. Keywords: surface state electrons on liquid helium, Rydberg states, microwave absorption, power broadening. The microwave (MW) resonance absorption experi- ment [1] had given the first direct observation of im- age-potential-induced Rydberg levels outside liquid he- lium. For weak holding fields E� , positions of surface levels were shown to be well described by the simple for- mula � �l R l� � / 2, where l �1 2, , ... , and � R is the corre- sponding Rydberg energy which is about 7.6 K for liquid 4 He. Because the dielectric constant of liquid helium � is close to unity (� �1 0 057� . ), the effective Bohr radius of these Rydberg levels, b me� � �4 1 12 2( / )( ) / ( )� � � , is large (about 76 �). Therefore, surface electrons (SEs) can move freely along the interface, forming a remarkable two-dimensional (2D) electron liquid, useful for studying many-body effects in a highly correlated 2D electron gas (for a review see [2]). The recent interest in the MW resonance is evoked by experimental development of the idea of using SEs as electronic qubits (see [3] and references there in). These qubits would be controlled by the MW field whose fre- quency � is close to �0 2 1� �( ) /� � � (usually, the sys- tem is tuned to resonance by varying E� ). Since the elec- tron potential is anharmonic, it is assumed that coupling to outer levels (l � 2) is negligible, and a simple two-level model is an excellent approximation. As an important achievement in this field, observation of MW absorption saturation of Rydberg states of SEs was reported [3]. The electron decay from the excited surface level to the ground level due to scattering by vapor atoms and ripplons is a negative factor for SE qubits. MW absorp- tion saturation appears when the stimulated absorption (emission) rate r is larger than the decay rate of the first excited level 2 1 1� � / . The dimensionless parameter r could be considered as a measure of electron excitation by MW. In the simple model of cold SEs [3], for low excita- tion r 1, the fractional occupancy of the first excited level n2 is much smaller than that of the ground level n1. In this case, power absorbed by electrons from the filed PA increases linear with r, or with the in-put power Pin . In the oppos i te l imi t o f h igh exc i ta t ion r �� 1, n n2 1 1 2� � / and P n n rA � �( )1 2 0�� saturates, while © Yuriy Monarkha, Denis Konstantinov, and Kimitoshi Kono, 2007 absorption lines become substantially affected by power broadening. The above mentioned results were based on the key as- sumption that electron temperature Te remains to be close to the ambient temperature T as the excitation parameter r increases. This was not proven for experimental condi- tions of Ref. 3. Moreover, already in 1980, there was an evidence supported by a simple estimate that MW can substantially heat SEs [4] at the resonance condition. In the important temperature range, electron scattering by vapor atoms and ripplons is accompanied by a very small energy exchange. When decaying back to the ground level an electron acquires huge kinetic energy of the in-plane motion, � k k m� �� 2 2 2/ � R T�� , and decaying becomes a sort of shooting electrons along the surface. High electron-electron collisions quickly redistribute this kinetic energy among other electrons, which should lead to decay heating of SEs. Therefore, an accurate analysis of MW resonance absorption of decay-heated SEs is highly desirable. In this work we report the results of theoretical de- scription of MW resonance of SEs on liquid helium for the conditions of the experiment [3]. Electron tempera- ture is found from the energy balance equation. We show that the decay heating cannot be disregarded even for rather low excitations r � �10 3, and, under the saturation condition r � 1, electron temperature more than 40 times (!) exceeds the ambient temperature. As a result, the limit- ing fraction of electrons left on the ground and the first excited levels which provides the decay heating of the rest electrons is very small (less than 10%). For vapor atom scattering regime, the excitation rate r has its usual Lorentzian form [5]: r R� � 1 2 2 2 2 � � � � , (1) where � � �� 0, � is the linewidth, � R eE z� � �0 1 2\| \| / � is the Rabi frequency, E0 is the MW field amplitude, and � �1 2\| \|z is the electric dipole length for the transition. The line shape of power absorption PA ( )� coincides with that of r( )� , when n n1 21� �� . If n1 is substantially reduced by excitation, the line-shape of PA ( )� will be changed. Fractional occupancies nl should satisfy the rate equa- tions for all surface levels. In the simple model of cold SEs [3], coupling to outer levels is disregarded, n r r2 1 2� � / ( ), and, therefore, P r A � �� 21 1 1 2 ( ) , (2) where � � �l l l l� �� . At r � 1, PA saturates, while the line-shape PA ( )� acquires a substantial broadening pro- portional to � R 2 (power broadening). This simple picture changes drastically, if we do not fix electron temperature to T . The energy absorbed from the MW field is eventually transferred to vapor atoms. Therefore, electron temperature Te as a function of r is determined by the energy balance equation ( ) ( )~( )n n r T T Te e1 2 21� � �� , (3) where ~( ) Te is the energy relaxation rate. Because the electron mass (m) is much smaller than the vapor atom mass (M), ~( ) T is estimated to be about 10 3� . Equation (3) makes decay heating obvious even by a simple esti- mate: if ~ 1and n n1 2 1� � , at r �1, electron tempera- ture Te should be much larger than � 21. In other words, Te should be high enough to compensate the lack of electron energy relaxation. For electron scattering by helium vapor atoms, the en- ergy relaxation rate is found as [7] ~ (\| \| ) /( ) , � � � � � � ��m M n Ta l l l l l l l e 0 2exp [ ]� � � � �� [( / ) (\| \|/ ) ]� �R e l l l l e l lT u T s2 , (4) where a ( )0 is the momentum relaxation rate for electron scattering within the ground level [6], s B B B dz z zl l l l l l l l� � � � � �� 11 1 0 2, [ ( ) ( )]� � , u b B C C dz d dz z zl l l l l l l l� � � � � �� 2 11 1 0 2 , [ ( ) ( )]� � , � R mb� � 2 22/ , and � l z( ) is the SE wave function. Usu- ally, it is assumed that fractional occupancies nl are given by the simple Boltzmann distribution with an effective Te . In this case, Eq. (4) gives the same energy relaxation rate as that found previously in the theory of nonlinear conductivity of hot SEs [6]. As we shall see below, this approximation is not sufficient for description of MW ab- sorption saturation. At least for two lowest levels, the the- ory should be extended beyond the Boltzmann approxi- mation. The rate equations dn dtm / � 0 (here m �1 and 2) con- sist of the terms for transitions between the two lowest levels induced by the MW field ( )'n n rm m� , and terms for transitions between all levels due to scattering by vapor atoms. The rates for latter transitions are given by w s n Tl l a ll l l l l l e� � � � �� ( ) exp [ (\| \| ) / ]0 2� � (5) [Eq. (5) defines also the life-time �1 0 21/ ( ) a s which does not depend on Te]. A simple and quite accurate solu- tion of the rate equations could be found, if we assume that the distribution of outer electrons (l ! 3) is close to the Boltzmann distribution (this assumption was verified by Microwave absorption saturation and decay heating of surface electrons on liquid helium Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 943 the exact numerical solution of 400 rate equations). In this case, the problem is reduced to the effective 3-level model which can be solved even analytically, though the final expressions for fractional occupancies n T re1( , ) , n T re2( , ) and n T r ne l l out ( , ) � � � 2 are cumbersome. Equation (3) establishes the relation between Te and r . Its solution is shown in Fig. 1,a for six distinctive val- ues of Rabi frequency (the in-put power is proportional to � R 2 ). Even at low excitations, Te substantially exceeds T � 0 9. K. At high excitations, Te is limited by MW ab- sorption saturation: T Te e ( ) .max K� 41 5 . The typical line Te ( )� has a resonance form. It is important that the line-width at the half-height of Te ( )� increases strongly with � R even at r 1. The same is valid for power ab- sorption PA ( )� shown in Fig. 1,b in units of its limiting value Psat obtained for r � � (Psat � 0 049 221. /� ). For the model of cold SEs, the area under the resonance line increases, while the line-width at the half-height remains approximately the same. Solid and dashed lines represent P PA sat/ for decay heated electrons (solid line takes into account the increase of � with Te because of scattering to outer levels [5]). The line-width at the half-height of these lines increases strongly with � R even at r 1 (P PA sat ). Power absorbed by SEs as a function of the excitation parameter r is shown in Fig. 2, a. The dependence P rA ( ) expected for cold SEs (dotted line) holds only at r " �7 10 4 . At larger values of r , the solid curve bends due to the decay heating. Up to about r � 01. , this effect can be described also by pure Boltzmann distribution of SEs (dashed line). Still, pure Boltzmann distribution can- not give power saturation up to r �100, in spite the fact that n n1 2� decreases with Te . This can be understood if we consider the left and right sides of Eq. (3) as functions of Te and r . For pure Boltzmann distribution, n1 and n2 depend only on Te , and the left side, representing PA , saturates if Te increases with r . If Te � const, PA will increase linear with r . At the same time, the right side of Eq. (3) do not depend explicitly on r, and increases with Te in the actual range of parameters. Therefore, saturation 944 Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 Yuriy Monarkha, Denis Konstantinov, and Kimitoshi Kono 0.5 1.0 1.5 2.00 0.05 0.10 0.15 1 2 3 b 2 30 10 20 30 40 T , K e a 1 � , 10 s 9 –1 � , 10 s 9 –1 P /P A sa t Fig. 1. Te (a) and PA (b) versus � ��0 for T � 09. K, E� � 106 V / cm, and different values of �R. For Te( )� , lines correspond (from bottom to top) to �R equal to 1.7, 5.5, 17.3, 54.8, 172, and 103 MHz. For PA( )� , model of cold SEs (dotted lines), decay heating theory with �( )Te � const (dashed lines), and decay heating theory with �( )Te given in Ref. 5 (solid lines) correspond (from bottom to top) to 1 (1), 1.73 (2), and 4.47 (3) MHz of �R. 0.01 0.1 1 10 –3 10 –3 10 –2 10 –2 10 –1 10 –1 10 0 10 0 10 1 10 1 10 2 10 2 0.01 0.1 1 10 a cold hot F ra ct io n al o cc u p an ci es , n l r r b n1 n2 cold hot P /P A sa t Fig. 2. Power absorption (a) and fractional occupancies (b) versus the excitation parameter r : model of cold SEs (dotted line), decay heating theory (solid line), Boltzmann approxima- tion (dashed line). T and E� are the same as in Fig. 1. requires Te � const. This contradiction is eliminated for real occupancies n1 and n2 which depend explicitly on both Te and r . For T re ( ) obtained from Eq. (3), n1 and n2 are shown in Fig. 2,b. As we can see, strong deviations from the model of cold SEs occur at r �10 3. For example, n2 in- creases with r much faster than it is for cold electrons. At higher excitations (r � " �7 10 3), n2 reaches its maximum, and then decreases with r . At r � 01. , both n1 and n2 devi- ate strongly from those given by the Boltzmann approxi- mation (dashed lines), and approach each other much faster when r � �. For decay-heated SEs, the limiting value of n1 and n2 is approximately an order of magnitude smaller than that obtained previously for T Te � . In conclusion, our study indicates that absorption of MW radiation by electrons on liquid helium is strongly af- fected by the decay heating, and the simple two-level model fails long before the saturation condition is reached. The decay heating leads to an additional power broadening which can be strong even at low excitations. The important point is that for decay-heated electrons, the absorption saturation appears when the theory is extended beyond the conventional Boltzmann approximation. This work is partly supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promo- tion of Science (JSPS); D.K. thanks JSPS for a postdoc- toral fellowship. 1. C.C. Grimes and T.R. Brown, Phys. Rev. Lett. 32, 280 (1974). 2. Yu.P. Monarkha and K. Kono, Two-Dimensional Coulomb Liquids and Solids, Springer-Verlag (2004). 3. E. Collin, W. Bailey, P. Fozooni, P.G. Frayne, P. Glasson, K. Harrabi, M.J. Lea, and G. Papageorgiou, Phys. Rev. Lett. 89, 245301-1 (2002). 4. V.S. Edel’man, Usp. Fiz. Nauk 130, 675 (1980) [Sov. Phys. Usp. 23, 227 (1980)]. 5. T. Ando, J. Phys. Soc. Jpn. 44, 765 (1978). 6. M. Saitoh and T. Aoki, J. Phys. Soc. Jpn. 44, 71 (1978). 7. The detailed theoretical description will be given elsewhere. Microwave absorption saturation and decay heating of surface electrons on liquid helium Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 945

Microwave absorption saturation and decay heating of surface electrons on liquid helium

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