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ПИСЬМА В ЭЧАЯ

PARTICLES AND NUCLEI, LETTERS

1[116]-2003

Proceedings of the International Workshop «Quantum Physics and Communication»

Издательский отдел ОИЯИ ДУБНА

JINR Publishing Department DUBNA РЕДАКЦИОННЫЙ СОВЕТ

А. Н. Сисакян — председатель А. И. Малахов — зам. председателя Д. В. Ширков — зам. председателя

Члены совета:

А. В. Белушкин, А. Е. Дорохов (ученый секретарь), П. И. Зарубин (ученый секретарь), С. П. Иванова, М. Г. Иткис, В. Д. Кекелидзе, Е. А. Красавин, И. В. Пузынин, Н. А. Русакович, А. Т. Филиппов

EDITORIAL BOARD

А. N. Sissakian — Chairman A. I. Malakhov — Vice-Chairman D. V. Shirkov — Vice-Chairman

Members of the Board:

A. V. Belushkin, A. E. Dorokhov (Scientific Secretary), P. I. Zarubin (Scientific Secretary), S. P. Ivanovo, M. G. Itkis, V. D. Kekelidze, E. A. Krasavin, I. V. Puzynin, N.'A. Russakovich, A. T. Filippov

1[116]-2003

ПИСЬМА О ФИЗИКЕ ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА

PHYSICS OF PARTICLES AND NUCLEI, LETTERS

Дубна 2003 В журнале «Письма о физике элементарных частиц и атомного ядра», кратко — «Письма в ЭЧАЯ», публикуются статьи, содержащие результаты ориги- нальных теоретических, экспериментальных, научно-технических, методических и прикладных исследований. Содержание публикуемых статей определяется тема- тикой научных исследований в ОИЯИ: теоретическая физика, физика элементар- ных частиц, релятивистская ядерная физика, физика атомного ядра и связанные вопросы общей физики, нейтронная физика, физика конденсированных сред, физика низких температур и криогенная техника, физика и техника ускорителей, методика физического эксперимента, компьютерные технологии в физике, прикладные работы по всем перечисленным разделам физики, включая радиобио- логию, экологию и ядерную медицину. Журнал зарегистрирован в Министерстве Российской Федерации по делам печати, телерадиовещания и средств массовых коммуникаций. Издателем журна- ла является Объединенный институт ядерных исследований. Журнал выходит шесть раз в год.

The journal PHYSICS of PARTICLES and NUCLEI, LETTERS, brief name PARTICLES and NUCLEI, LETTERS, publishes the articles with results of the original theoretical, experimental, scientific-technical, methodical and applied research. Sub- ject-matter of articles covers the principal fields of research at JINR: theoretical physics, elementary particle physics, relativistic nuclear physics, nuclear physics and related problems in other branches of physics, neutron physics, condensed matter physics, physics and technique at low temperature, physics and technique of accelera- tors, physical experimental instruments and methods, physical computer experiments, applied research in these branches of physics and radiology, ecology and nuclear medi- cine. The journal is registered in the Ministry of the Russian Federation for Press and is published bimonthly by the Joint Institute for Nuclear Research.

References to the articles of the PHYSICS of PARTICLES and NUCLEI, LETTERS should contain: — names and initials of authors, — title of journal, — year of publication, — publication index, — page number.

For example: Alexeev G. D. II Part. Nucl., Lett. 2000. No. 5[102]. P. 5. PROCEEDINGS OF THE INTERNATIONAL WORKSHOP

«QUANTUM PHYSICS AND COMMUNICATION»

Dubna, May] 6-17, 2002

Edited by V. V. Ivanov

QPC-2002

ТРУДЫ МЕЖДУНАРОДНОЙ КОНФЕРЕНЦИИ

«КВАНТОВАЯ ФИЗИКА И ИНФОРМАЦИЯ»

Дубна, 16-17 мая 2002 г.

Научный редактор В. В. Иванов УДК [530.145 : 004.9] (063) ББК [22.314+ 32.97] я 431

Organizing Committee: V. Kadyshevsky (JINR), chairman A. Sissakian (JINR), co-chairman

Members: I. Antoniou (Brussels, Belgium), V. Belokurov (MSU), V. Buzek (Bratislava, Slovakia), S. Ivanova (JINR), V. Ivanov (JINR), A. Korol (JINR), T. Strizh (JINR), A. Filippov (JINR), A. Chizhov (JINR), S. Lukyanov (JINR), E. Fedorova (JINR)

QPC-2002 is supported by the Joint Institute for Nuclear Research; Lomonosov Moscow State University; Solvay Institutes for Physics and Chemistry; Research Centre for Quantum Informa- tion, Institute of Physics, Slovak Academy of Sciences.

QPC-2002

Организационный комитет: В. Кадышевский (ОИЯИ) — председатель, А. Сисакян (ОИЯИ) — сопредседатель

Члены комитета: Я. Антониоу (Брюссель, Бельгия), В. Белокуров (МГУ), В. Бужек (Братислава, Словакия), С. Иванова (ОИЯИ), B. Иванов (ОИЯИ), А. Король (ОИЯИ), Т. Стриж (ОИЯИ), А. Филиппов (ОИЯИ), А. Чижов (ОИЯИ), C. Лукьянов (ОИЯИ), Е. Федорова (ОИЯИ)

Конференция проведена при поддержке Объединенного института ядерных исследований, Московского государственного университета, Международного института физики и химии (Брюссель), Исследовательского центра квантовой информации Института физики (Ака- демия наук Словакии). PREFACE

Quantum information processing is a new research area at the intersection of quantum physics and computer science. The main goal of this field is to improve on classical computers and classical complexity bounds as well as cryptographic communication protocols by making use of quantum mechanical phenomena. Simultane- ously, investigation of resources for quantum computation (such as quantum entanglement) enables us to understand from a different perspective the fundamental properties of quantum mechanics. Recent advances in the realization of elementary quantum logic gates (e. g., using ions traps, cavity QED and NMR technology), and the experimental implementation of phenomena such as quantum teleportation, entanglement swapping or quantum cloning, and, obviously, experimental implementation of quantum key distribution have led to a massive exploration of this field by sci- entists from varied disciplines. The purpose of the Workshop is to provide a compact and comprehensive overview of basic ideas of quantum information theory and to es- tablish perspective contacts for future collaboration. Main topics of the Workshop should cover entanglement as a resource for quantum computing and communication, multipartite quantum correlations, decoherence, physics of information processing, and others.

V. V. Ivanov СОДЕРЖАНИЕ CONTENTS

Е. Karpov, G. Ordonez, Т. Petrosky, [I. Prigogine| Microscopic Entropy and Nonlocality Е. Карпов, Г. Ордонец, Т. Петроский, | И. Пригожий | Микроскопическая энтропия и нелокальность .

V. V. Belokurov, О. A. Khrustalev, V. A. Sadovnichy, О. D. Timofeevskaya Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics В. В. Белокуров, О. А. Хрусталев, В. А. Садовничий, О. Д. Тимофеевская Условная матрица плотности: системы и подсистемы в квантовой механике 16

V. V. Belokurov, О. A. Khrustalev, V. A. Sadovnichy, О. D. Timofeevskaya Physics of Quantum Computation В. В. Белокуров, О. А. Хрусталев, В. А. Садовничий, О. Д. Тимофеевская Физика квантовых вычислений 30

F. De Martini, V. Buzek, F. Sciarrino, С. Sias Flipping Qubits Ф. де Мартини, В. Бужек, Ф. Шаррино, К. Сиас Обращаемые кубиты 39

I. Antoniou, Е. Karpov, G. Pronko, E. Yarevsky Quantum Zeno Effect for ./V-Level Friedrichs Model Я. Антониоу, Е. Карпов, Г. Пронко, Е. Яревский Квантовый эффект Зенона для TV-уровневой модели Фридрикса 53

A. Yu. Vlasov Universal Hybrid Quantum Processors А. Ю. Власов Универсальные гибридные квантовые процессоры 60 A. P. Alodjants, A. V. Prokhorov, S. M. Arakelian Formation of the 5'(/(3)-Polarization States in Atom-Quantum Electromagnetic Field System under Condition of the Bose-Einstein Condensate Existence A. П. Алоджанц, А. В. Прохоров, С. М. Аракелян Образование 5£/(3)-поляризованных состояний в атомной системе, взаимодействующей с квантовым электромагнитным полем при условии существования бозе-эйнштейновского конденсата . . . 66

G. P. Miroshnichenko Periodical Sequences (Trajectories) of Outcomes of Atomic State Measurement on Exit from the Micromaser Cavity Г. П. Мирошниченко Периодические последовательности (траектории) результатов измерений атомных состояний на выходе микромазерного резонатора . 74

I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, J. Timonen An Algebraic Method to Solve the Tavis-Cummings Problem И. П. Вадейко, Г. П. Мирошниченко, А. В. Рыбин, Д. Тимонен Алгебраический метод решения задачи Тэвиса-Каммингса 83

V. V. Ivanov, В. F. Kostenko, V. D. Kuznetsov, М. В. Miller, A. V. Sermyagin Quantum Teleportation of Nuclear Matter and Its Investigation B. В. Иванов, Б. Ф. Костенко, В. Д. Кузнецов, М. Б. Миллер, А. В. Сермягин Исследование квантовой телепортации ядерной материи 96

Указатель статей к журналу «Письма в ЭЧАЯ» за 2002 г.. . 108 PARTICLES and NUCLEI, LETTERS. 2002. Paper index 113

Именной указатель к журналу «Письма в ЭЧАЯ» за 2002 г. 118 PARTICLES and NUCLEI, LETTERS. 2002. Author Index 122 XJ0300178 Письма в ЭЧАЯ. 2003. № 1[116] Particles апитчшлст, деиы». ±wj. IN

MICROSCOPIC ENTROPY AND NONLOCALITY E. Karpov a, G. Ordonez a'b, Г. Petrosky a'b, /. Prigogine 0 International Solvay Institutes for Physics and Chemistry, Brussels 6 Centre for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas, USA

We have obtained a microscopic expression for entropy in terms of H function based on nonunitary Л transformation which leads from the time evolution as a unitary group to a Markovian dynamics and unifies the reversible and irreversible aspects of quantum mechanics. This requires a new representation outside the Hilbert space. In terms of H, we show the entropy production and the entropy flow during the emission and absorption of radiation by an atom. Analyzing the time inversion experiment, we emphasize the importance of pre- and postcollisional correlations, which break the symmetry between incoming and outgoing waves. We consider the angle dependence of the H function in three-dimensional situation. A model including virtual transitions is discussed in a subsequent paper.

INTRODUCTION

Irreversibility in macroscopic physics is associated with entropy increase. The usual formulation of dynamics (classical or quantum) does not include irreversible processes. We have shown that this contradiction comes from the limitation to integrable systems. For a class of nonintegrable systems, dynamics includes irreversibility. For integrable systems there exists a unitary transformation U, which is distributive:

U(ab) = (Ua}(Ub). (1)

For a class of nonintegrable systems (due to Poincare resonances), we have introduced a «star unitary» invertible operator Л which includes irreversibility and stochasticity [1-3]. This leads to a deep change in the formulation of dynamics, as it forces us to go outside the Hilbert space. Our Л is not distributive:

A(ab) £ (Ло)(ЛЬ). (2)

We have shown that there exists then an H function (or Lyapounov variable) [4-8] represented by the operator H = AfA. (3) Using Л we go from the time evolution as a unitary group to a Markovian dynamics. Note also that Л is a nonlocal transformation replacing points by ensembles. In our early work on the interaction of an atom with a field (e. g., the Friedrichs model) [9] we have shown that (3) reduces to the operator outside the Hilbert space introduced in [2,7,10]

W = l0i)<0i|, (4) Microscopic Entropy and Nonlocality 9 where \ф\) is the eigenstate of the Hamiltonian H outside the Hilbert space [10,11] with complex eigenvalue [10,12]

c Я|&) = zf- |&), г!=Ш1-г7, wb 7 > 0, (5) where c.c. denotes complex conjugate. This eigenstate is called the Gamow vector [11,12] and for the Friedrichs model, which we use in our paper, \ф\) is given by (9). Hence, in agreement with [1,2,7,13], Ti. function is an operator outside the Hilbert space with the Heisenberg evolution given by

H(t) = eimH *-"** = e-2*H. (6) The physical meaning of H is very simple. The expectation value of H decreases (and entropy increases) as the energy of the excited state is transferred to the field modes. We make a remark here that there have always been two points of view on entropy: the point of view of Planck, relating entropy to dynamics, and the point of view of Boltzmann, relating entropy to probabilities (ignorance) [14]. We understand now that Planck could not realize his program as he worked in the usual representation of dynamics, equivalent to the Hilbert space representation. We have now a «microscopic» formulation of thermodynamics which includes the decay and excitation of quantum states. The entropy creation is due to a resonance in the time evolution of \4>i). A one-dimensional situation was considered [3,9] where an initially localized wave packet of the field interacts with an excitable particle. An essential element is the consideration of correlations. Due to the nonlocal nature of Л function we can introduce pre- and postcollisional correlations between the particle and the field. They exist even if the wave packet corresponding to the field is at large distance from the particle. We show that the time inversion after the collision requires instant extraction of large amount of entropy from the system because the entropy after the collision is much higher than before. It means that, in order for the collision to take place, we have to create a state with a high order as we have to target our wave packet to the atom. Analyzing the scattering process we show that the second law of thermodynamics is valid as the excitation of the particle is associated with the flow of entropy that is provided by the incoming wave packet [9]. The amount of entropy provided decides the amount of excitation of the ground state. This will be further discussed in subsequent papers. The interest of these considerations is to introduce space dependence in the thermodynamic evolution. These effects were extended to the three-dimensional problem where the direction of the initial momentum of the wave packet and the possibility for the wave packet to be scattered in all direction of the three-dimensional space results in more complicated picture.

1. H FUNCTION IN THE FRIEDRICHS MODEL The Friedrichs model for the interaction of matter with radiation in the rotating wave approximation is given by the Hamiltonian (with units ft = с — I)1

Я = W1|1)<1| + fc|fc)(fc| + A Vk (\l)(k\ + |fc){l|), (7)

'We shall extend this consideration including virtual transitions in subsequent papers. 10 Karpov E. et al. where |1) represents a bare particle or atom in its excited state with no photons present, while the state |fc) represents a bare field mode of the momentum k and the particle in its ground state. The momentum in the volume L is quantized k — (2Tr)dn/L; n is integer vector with the space with the dimension d. The energy of the ground state is chosen to be zero; u>i is the bare energy of the excited level, and wjt = jfc| is the photon energy. The coupling constant A is dimensionless. For a, ft = l,k we have orthonormality and completeness relations a a = me (a|/3) = <$a,/3, Z)a=i k l )( l 1- I" infinite volume limit L —> со, the momentum k becomes continuous, i. e.,

v L°, (8) (2тг)« k where the integration is taken over «d» dimensional k space. When ш\ > / dk\2vl/uk the excited state is unstable due to resonance between the particle and the field. In this case, the Gamow vector \ф\) is [10, 12]

where N\ is a normalization constant and the distribution l/[uJk - z\ -c]_ is defined with the help of suitable test function

W c = I wk - -sf ]- J Note that the transition from the bare state to the Gamow vector is «star unitary» (see [10]). We shall consider the ~H^(t} function defined as the expectation value of the Ti. operator:

H«(t) = (t\eimHe-im\t), (И) where the initial state |£) corresponds to the particle located at x = 6 in its ground state and a localized wave packet formed by the field.

2. MOMENTUM INVERSION EXPERIMENT

We consider the momentum inversion experiment [9, 10] with the wave packet initially localized in a finite interval of size a: I - *- -0 • The wave packet is centered at XQ, and an initial momentum fco > 0 is directed to the particle. We observe the time evolution of |£) determined by the Hamiltonian H and calculate T~i^(t) (11) as a function of t. At some moment t\, we perform the momentum inversion by the antilinear time-inversion operator Tj [7] that inverts the sign of time t and the momentum of the field modes. Then we observe the time evolution of the transformed state further. The results of these observations are presented in Fig. 1. We see that H^(t) function decays Microscopic Entropy and Nonlocality 11 exponentially with time. This corresponds to the increase of the entropy in the system in accordance with the second law of thermodynamics. The time inversion made at time £» intro- duces the instant effect of the outside world which leads to a jump of the Ti.^(t) function. The jump depends on the time the momentum 0.2 - inversion is performed: 1) The time inversion made at £2, that is, after the collision leads to a drastic in- 250 t crease of Ht(t) function. The time inversion replaces the scattered outgoing field by an in- Fig. 1. The momentum inversion experiments per- coming wave targeted to the particle. This formed at time t\, that is, before the collision (de- creates correlations between the particle and stroys precollisional correlations) and at time tz, the field which are precollisonal correlations. that is, after the collision (replaces postcollisional This lowers the entropy. As a consequence, correlations by the precollisional ones) the expectation value of К$ (t) «jumps» to a higher value. 2) The inversion performed at time *i, that is, before the collision and replaces the incoming wave by an outgoing one, which will not collide with the particle. This destroys precollisional correlations and, therefore, increases the entropy. The expectation value of Л «jumps» to a lower value corresponding to postcollisional correlations. The basic distinction between precollisional and postcollisional correlations takes into account the change of the entropy. As we noticed, precollisional correlations increase the Ti.^(t) function. This observation has already been made many years ago [1]. The emission of photons corresponds to a postcollisional correlation decreasing the H^(t) function as photons escape from the particle. The details of the collision showing the difference between pre- and postcollisional correlations are given in the next section.

3. ENTROPY PRODUCTION AND ENTROPY FLOW DURING THE COLLISION

Using the completeness of the |1>, |fc) basis we can express the Щ (11) as a sum:

W«(t) = Wu(«) + Wi/(«) + «//(*), (13) where

, (14)

""|0 + c.c., (15)

(16) 12 Karpov E. el al. where Hn(t), T~i/f(t), and H\f(t) correspond to the particle, field, and field-particle correlation. Knowing \ф\) (see (9)) it is elementary to evaluate the time dependence of these components. In the continuous limit using the pole approximation, we have

(17)

Hif(t) « 2(7(1 - - 1) + с. с., (18)

(19) where С is a constant determined by the interaction AT4 and by the initial wave packet |£). In Fig. 2, we see that the total entropy production in the system expressed by the exponentially decreasing curve H^(t) may be divided into three different time intervals:

1) 0 < t < -(z0 - a/2), before the collision. Initially, the entropy of the system is low CH^(t) function is high) because the wave 0.8 packet is «targeted» to the particle; i.e., a higher order is created in the system by «anom- 0.6 alous» precollisional correlations. Here, only 0.4 the field part, H/f, contributes to the total entropy H^(t) of the system as the particle is 0.2 in its ground state, therefore, Hn = 0. As the wave packet approaches the particle H de- 20 40 60 80 100 t creases. -(x -o/2) 0 2) -(ZQ+ o/2) < t < -(zo-o/2), during the collision. The particle (curve 2, HU) goes Fig. 2. Entropy production and the entropy flow to its excited state with lower entropy. during the collision: curve 1 — Ji^(t); curve 2 — Hn(t); curve 3 — fti/(t); curve 4 — T-i (t) 3) -(zo + a/2) < t, after the collision. The ff excited particle contributes to Н$ (t) as entropy is produced during the irreversible decay of the particle. Note the analogy between excited states and «dissipative structures» introduced in thermodynamics. The excited particle is a nonequilibrium state due to the environment (here the wave packet) and decays when the constraint is released.

4. PRECOLLISIONAL CORRELATIONS AS A FUNCTION OF SPATIAL CONFIGURATION

As follows from the previous discussion, the initial correlations between the particle and the field determine the degree of the order in the system. It is natural to expect that the initial correlations must be larger for higher initial distance between the particle and the wave packet because for the larger distance «targeting» becomes more difficult problem. Microscopic Entropy and Nonlocality 13

ID П '~: .*,'-: I . I : > к- П П. .

1000 \ \V«2V /

-5 S-r*\

Fig. 3. In (Hj(0)) as function of |xo| for Gaussian Fig. 4. The origin of initial fall of "H^ function wave packets centered in XQ. The size of the wave with |xo|: the further is the wave packet, the packet is о = 1, and the initial momentum is less portion of it will hit the atom fco = 1. Dashed line — ID; solid line — 3D. The position of the pole is z\ = 1 — O.Olt

In order to analyze this, we use the Gaussian wave packet of the size a centered at with the initial momentum ko:

1 x 2 etkotlcoxx exp / ( -—5-!- Xo)

(20) 1/2

«(k-k0)xo

Here, we use bold notations x and k in order to distinguish ID and 3D vectors from their absolute values x and k. For ID and 3D situations, we have in the continuous limit in the pole approximation

(21)

The same result for ID case was also obtained in [9]. In Fig. 3, we draw In (7^(0)) as function of XQ for ID (dashed line) and 3D (solid line). The normalization constants are chosen such that for (XQJ = 0 both curves coincide. In both cases, for large |XQ|, due to the factor e2^*0', the entropy exponentially decreases with the distance between the initial position of the wave packet and the atom. The larger the distance |xo|, the more order (a lower entropy) we must introduce into the system in order to target the wave packet to the particle. However, in 3D, in addition to this effect, we see an initial decrease of H% function due to the factor (a4fcg + я2»)"1 which is absent in ID. The origin of this can be explained by simple geometrical reasons (see Fig. 4). In the 3D case, initially localized wave packet spreads in all directions. However the major contribution to %£ is given by those components of the wave packet that have the momenta with the directions in some critical solid angle directed to the atom. This angle becomes nar- 14 Karpov E. el al.

rower with the distance and Ti^ decreases with

|XQ|. This effect dominates at small ]x0| but for large distances the exponential factor e"27'"0' that is due to interaction starts to overcome the geometrical effect and the 7^(0) curve starts to increase with |XQ|. 0.4 - Another important question is how precollisional correlations depend on whether the 0.2 - wave packet itself is targeted to the atom or not. In Fig. 5, we draw the dependence of W{(0) on 175 the angle 00 between the vector ko and the direction from the centre of the wave packet to the atom for different sizes of the wave packet.

Fig. 5. tt(00) 5= Щ(0)/(ЩШ»0=о) as a We see that the Ti.^ function is greater when the function of во for xo = 10, fco = Ш1 = 1, wave packet is directed to the atom and becomes 7 = 0.01 in the pole approximation. 1 — smaller with the deviation of ko from the di- a = 1; 2 — a = 1.5; 3 — a = 4; 4 — a = 6; rection to the atom. This is in agreement with 5 —a = 8 our statement that when the wave packet is targeted to the atom the order increases.

CONCLUSION

We may summarize the main interest in our Ti function as follows: 1) Irreversibility and entropy are not limited to macroscopic physics (contrary to the opinion of Boltzmann). Elementary processes like the decay of an excited state contribute to the entropy balance. 2) The symmetry between incoming waves and outgoing waves as used in 5 matrix theory is broken (contrary to the opinion of Pauli). Incoming waves have lower entropy. 3) Relation between dynamics (here time inversion) and entropy. On the one hand, velocity inversion leads to the change of entropy. On the other hand, it requires acceleration. Therefore, it is interesting to investigate the relation between acceleration and entropy. 4) Dual aspect of reversible and irreversible processes (according to J. von Neumann) is unified in our formalism in terms of Л transformation, as it combines both. Acknowledgements. We acknowledge the International Solvay Institutes for Physics and Chemistry, the European Commission Grant HPHA-CT-2001-40002, the Engineering Research Programme of the Office of Basic Energy Sciences at the U.S. Department of Energy, Grant No.DE-FG03-94ER14465, the Robert A. Welch Foundation Grant No. F-0365, the National Lottery of Belgium, and the Communaute Francaise de la Belgique.

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12. Antoniou L, Prigogine I. II Physica A. 1993. V. 192. P. 443-464.

13. Petrosky Т., Prigogine I. II Adv. Chem. Phys. 1997. V.99. P. 1-120.

14. Mehra J., Rechenberg H. The Historical Development of Quantum Theory. N. Y.: Springer Verlag, 1982. V.I. Parti. XJ0300179 Письма в ЭЧАЯ. 2003. № 1[116] Particles and Nuclei, Letters. 2003. No. 1[116]

CONDITIONAL DENSITY MATRIX: SYSTEMS AND SUBSYSTEMS IN QUANTUM MECHANICS V. V. Belokurov, O. A. Khrustalev1, V. A. Sadovnichy, O. D. Timofeevskaya Lomonosov Moscow State University, Moscow A new quantum mechanical notion — Conditional Density Matrix — proposed by the authors [5,6] is discussed and is applied to describe some physical processes. This notion is a natural generalization of von Neumann density matrix for such processes as divisions of quantum systems into subsystems and reunifications of subsystems into new joint systems. Conditional Density Matrix assigns a quantum state to a subsystem of a composite system on condition that another part of the composite system is in some pure state.

INTRODUCTION

A problem of a correct quantum mechanical description of divisions of quantum systems into subsystems and reunifications of subsystems into new joint systems attracts a great interest due to the present development of quantum communication. Although the theory of such processes finds room in the general scheme of quantum mechanics proposed by von Neumann in 1927 [1], even now these processes are often described in a fictitious manner. For example, the authors of classical photon teleportation experiment [2] write: The entangled state contains no information on the individual particles; it only indicates that two particles will be in the opposite states. The important property of an entangled pair is that as soon as a measurement on one particle projects it, say, onto |<-+) the state of the other one is determined to be ||), and vice versa. How could a measurement on one of the panicles instantaneously influence the state of the other particle, which can be arbitrary far away? Einstein, among many other distinguished physicists, could simply not accept this «spooky action at a distance». But this property of entangled states has been demonstrated by numerous experiments.

1. THE GENERAL SCHEME OF QUANTUM MECHANICS

It was W. Heisenberg who in 1925 formulated a kinematic postulate of quantum mechanics [3]. He proposed that there exists a connection between matrices and physical variables:

variable Т «=>• matrix (F)mn.

'e-mail: [email protected] Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 17

In the modern language the kinematic postulate looks like: Each dynamical variable Т of a system S corresponds to a linear operator F in Hilbert space H: dynamical variable F <$=>• linear operator F.

The dynamics is given by the famous Heisenberg's equations formulated in terms of commutators,

To compare predictions of the theory with experimental data, it was necessary to understand how one can determine the values of dynamical variables in the given state. W. Heisenberg gave a partial answer to this problem: If matrix that corresponds to the dynamical variable is diagonal, then its diagonal elements define possible values for the dynamical variable, i. e., its spectrum,

(F)mn = fmSmn «=» {/m} w spectrum T.

The general solution of the problem was given by von Neumann in 1927. He proposed the following procedure for calculation of average values of physical variables:

Here, operator /3 satisfies three conditions:

1) P+ = P, 2)1rp=l, 3) W> e ft (ф\рф) > 0.

By the formula for average values, von Neumann found out the correspondence between linear operators p and states of quantum systems:

state of a system p <=» linear operator p.

In this way, the formula for average values becomes quantum mechanical definition of the notion «a state of a system». The operator p is called Density Matrix. From the relation

one can conclude that Hennitian-conjugate operators correspond to complex-conjugate variables, and Hermitian operators correspond to real variables,

The real variables are called observables. 18 Belokurov V.V. et al.

From the properties of density matrix and the definition of positively definite operators

F+ = F, W> e К, (Ф\Рф) > 0, it follows that the average value of nonnegative variable is nonnegative. Moreover, the average value of nonnegative variable is equal to zero if and only if this variable equals zero. Now it is easy to give the following definition: variable Т has a definite value in the state p if and only if its dispersion in the state p is equal to zero. In accordance to general definition of the dispersion of an arbitrary variable

the expression for dispersion of a quantum variable f in the state p has the form

where Q is an operator: Q = F- If f is observable then Q2 is a positive-definite variable. It follows that the dispersion of Т is nonnegative. And all this makes clear the above-given definition. Since density matrix is a positive definite operator and its trace equals 1, we see that its spectrum is pure discrete, and it can be written in the form

where Pn is a complete set of self-conjugate projective operators:

••n = "ni *m*n = Omn*mi / _, •* n= n

Numbers {pn} satisfy the condition

Pn=Pn, 0

It follows that p acts according to the formula

/ЗФ =

The vectors фпа form an orthonormal basis in the space Ti. Sets An = {!,..., fcn} are defined by degeneration multiplicities kn of eigenvalues pn. Now the dispersion of the observable Т in the state p is given by the equation Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 19

All terms in this sum are nonnegative. Hence, if the dispersion is equal to zero, then

if pn ± 0, then Q<$>na = 0.

Using the definition of the operator Q, we obtain

if р„ ^ 0, then Рфпа = фпа(Р).

In other words, if an observable Т has a definite value in the given state p, then this value is equal to one of the eigenvalues of the operator F. In this case, we have

which proves the commutativity of operators F and p. It is well known that, if A and В are commutative self-conjugate operators, then there exists self-conjugate operator f with nondegenerate spectrum such that A and В are functions of f:

Ф =

С* = *па, tna ф tn-a,, if (n, а) ^ (n', a').

Suppose that F is an operator with nondegenerate spectrum; then if the observable Т with nondegenerate spectrum has a definite value in the state p, then it is possible to represent the density matrix of this state as a function of the operator F. The operator F can be written in the form

The numbers {/„} satisfy the conditions

/n=/«. /n^/n', if П^П'.

We obviously have 20 Belokurov V. V. el al.

From

we get Pn — 8nN- In this case, density matrix is a projective operator satisfying the condition

It acts as рФ where |Ф) is a vector in Hilbert space. The average value of an arbitrary variable in this state is equal to

(A) = (ФлИФлг). It is so-called pure state. If the state is not pure, it is known as mixed. Suppose that every vector in Л is a square integrable function Ф(х), where x is a set of continuous and discrete variables. Scalar product is defined by the formula

= / <£гФ*(г)Ф(х).

For simplicity, we assume that every operator F in H acts as follows:

(FV)(x) = f F(x,x')dx'V(x').

That is, for any operator F there is an integral kernel F(x,x') associated with this operator:

Certainly, we may use 5 function if necessary. Now the average value of the variable f in the state p is given by equation

(F}p = I F(x,x')dx'p(x',x)dx.

Here, the kernel p(x,x') satisfies the conditions

p*(x,x') = p(x',x),

I p(x,x)dx = 1,

УФ 6 U /Ф(х)йр(х,х')^х'Ф(х') > 0. Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 21

2. COMPOSITE SYSTEM AND REDUCED DENSITY MATRIX

Suppose the variables x are divided into two parts: x = {y, z}. Suppose also that the space W is a direct product of two spaces H\,Hi'-

So, there is a basis in the space that can be written in the form

The kernel of operator F in this basis looks like

In quantum mechanics it means that the system S is a unification of two subsystems Si and

S2:

S = Si U S2. The Hilbert space H corresponds to the system 5 and the spaces Hi and Hz correspond to the subsystems Si and S2. Now suppose that a physical variable f\ depends on variables у only. The operator that corresponds to FI has a kernel

Fi(y,z;y',z')=Fi(y,y')5(z-z').

The average value of FI in the state p is equal to

f (Fi)p = I F(y,y )dy'pi(y',y)dy, where the kernel pi is defined by the formula

Pi(y,y')= I p(y,z;y',z)dz.

The operator p\ satisfies all the properties of Density Matrix in Si. Indeed, we have

J Pi(y,

') > 0.

The operator Pi = Тг2 Pi+2 is called Reduced Density Matrix. Thus, the state of the subsystem Si is defined by reduced density matrix. 22 Belokurov V. V. et al.

The reduced density matrix for the subsystem 82 is denned analogously:

Pi = Tri /5i+2-

Quantum states p\ and p2 of subsystems are defined uniquely by the state p1+2 of the composite system. Suppose the system 5 is in a pure state, then a quantum state of the subsystem S\ is defined by the kernel

If the function Ф(у, z) is the product

«(»,*) = /(y)40, J\w(z)\2dz = l, then subsystem Si is a pure state, too:

As was proved by von Neumann, it is the only case when purity of composite system is inherited by its subsystems. Let us consider an example of a system in a pure state having subsystems in mixed states. Let the wave function of composite system be

where (f\w) = 0 and (f\f) = (гиги) = 1. The density matrix of the subsystem S\ has the kernel Pi (У, У') = \(!(У)Г(У') + w(y)w*(y'}).

The kernel of the operator p\ has the form

pi(w.y') = \(f(y)f*(y') + w(»K (!/'))•

Therefore, the subsystem Si is in the mixed state. Moreover, its density matrix is proportional to unity operator. The previous property resolves the perplexities connected with Einstein- Podolsky-Rosen paradox.

3. EPR PARADOX

Anyway, it was Schrodinger who introduced a term «EPR paradox». The authors of EPR themselves always considered their article as a demonstration of inconsistency of present quantum mechanics rather than a particular curiosity. Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 23

The main conclusion of the paper «Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?» [4] published in 1935 (eight years after von Neumann's book) is the statement: ...We proved that (1) the quantum-mechanical description of reality given by wave functions is not complete or (2) when the operators corresponding to two physical quantities do not commute, the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with noncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to negation of only other alternative (2). We can thus focus to conclude that the quantum-mechanical description of physical reality given by wave function is not complete. After von Neumann's works this statement appears obvious. However, in order to clarify this point of view completely, we must understand what is «the physical reality» in EPR. In EPR paper the physical reality is defined as: If, without in any way disturbing a system, we can predict with certainty (i. e., with probability equal to unity) the value of physical quantity, then there exists an element of physical reality corresponding to this physical quantity. Such a definition of physical reality is a step back as compared to von Neumann's definition. By EPR definition, the state is actual only when at least one observable has an exact value. This point of view is incomplete and leads to inconsistency. When a subsystem is separated, «the loss of observables» results directly from the definition of density matrix for the subsystem. «The occurrence» of observables in the chosen subsystem when the quantities are measured in another «subsidiary» subsystem can be naturally explained in the terms of conditional density matrix.

4. CONDITIONAL DENSITY MATRIX

The average value of a variable with the kernel

Fc(x,x') = JidMrttiWtiV), j \u(z)\2dz = is equal to where

РС{У,У') = ~ fu'(z)dzp(y,z;y',z')u(z')dz', г J p = u*(z)dz p(y, z; y, z1} u(z')dz'dy.

Since we can represent p in the form

p = Jp(z,z')dz'p2(z';z)dz, 24 Belokurov V. V. el al. we see that p is an average value of a variable P of the subsystem 83. Operator P is a projector (P2 = P). Therefore, it is possible to consider the value p as a probability. It is easy to demonstrate that the operator pc satisfies all the properties of density matrix. So the kernel pc(y,y') defines some state of the subsystem Si. What is this state? According to the decomposition of 6 function

6(z - z') =

{n(z}} being a basis in the space #2, the reduced density matrix is represented in the form of the sum

Here,

c p n(y,y') = - f*n(z)dzp(y,z;y',z')n(z')dz', Pn j and

Pn= I n(z)dzP(y,z',y'z')n(z')dz'dy,

= j Pn(z,z')dz'pi(z',z)dz.

The numbers pn satisfy the conditions

Pn=Pn, Pn>0, and are connected with a probability distribution.

The basis {фп} in the space H? corresponds to some observable G^ of the subsystem 62 with discrete nondegenerate spectrum. It is determined by the kernel

G2(z,z') =

The average value of GZ in the state p2 is equal to

fdyP2(z,z')dz'G(z',z) = ^gn fdyP2(z,z')dz'n(z')*n(z') = 5>„5„. J n J n

Thus, number pn defines the chance that the observable 62 has the value gn in the state p^. Obviously, the kernel p^(y,y') in this case defines the state of system Si on condition that the value of variable GI is equal to

Tr2 Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 25

It is (conditional) density matrix for the subsystem Si on condition that the subsystem S? is selected in a pure state fa = P^. It is the most important case for quantum communication. Conditional density matrix satisfies all the properties of density matrix. Conditional density matrix helps to clarify a sense of operations in some finest experiments.

5. EXAMPLES: SYSTEM AND SUBSYSTEMS

5.1. Parapositronium. As an example, we consider parapositronium, i.e., the system consisting of an electron and a positron. The total spin of the system is equal to zero. In this case the nonrelativistic approximation is valid, and the state vector of the system is represented in the form of the product

Ф(ге,о-е;гр,сгр) = Ф(ге,Гр)х(0-е,стр). The spin wave function is equal to

Here, Xn(cr) and X(-n)(^) are the eigenvectors of the operator that projects spin onto the vector n:

The spin density matrix of the system is determined by the operator with the kernel

p(a;a') = X^e.^phVe.^)- The spin density matrix of the electron is

= \(xn(v)X(-»)(°'} + X* (*)Х(-„)(

In this state the electron is completely unpolarized. If an electron passes through polarization filter, then the pass probability is independent of the filter orientation. The same fact is valid for the positron if its spin state is measured independently of the electron. Now let us consider quite a different experiment. Namely, the positron passes through the polarization filter, and the electron polarization is simultaneously measured. The operator that projects the positron spin onto the vector m (determined by the filter) is given by the kernel

Now the conditional density matrix of the electron is equal to

,. £(^)Хт(<г)хЖ*( pe/p(a,a ) 26 Belokurov V. V. et al.

The result of the summation is

Thus, if the polarization of the positron is selected with the help of polarizer in the state with well-defined spin, then the electron appears to be polarized in the opposite direction. Of course, this result is in an agreement with the fact that total spin of composite system is equal to zero. Nevertheless, this natural result can be obtained if positron and electron spins are measured simultaneously. In the opposite case, the more simple experiment shows that the direction of electron and positron spins are absolutely indefinite. A. Einstein said, «Raffiniert ist der Herr Gott, aber boschaft ist Er nicht.» 5.2. Quantum Photon Teleportation. In the Innsbruck experiment [2] on a photon state teleportation, the initial state of the system is the result of the unification of the pair of photons 1 and 2 being in the antisymmetric state x(crii<72) with summary angular momentum equal to zero and the photon 3 being in the state Хт(^з) (that is, being polarized along the vector m). The joint system state is given by the density matrix

where the wave function of the joint system is the product

Considering then the photon 2 only (without fixing the states of the photons 1 and 3), we find the photon 2 to be completely unpolarized with the density matrix

However, if the photon 2 is registered when the state of the photons 1 and 3 has been determined to be хС^ъстз). then the state of the photon 2 is given by the conditional density matrix 1^(1,3) (Pl,3 Pl,2,s) P2/{1,3} — rrv /p- ч — • ТГ (Fi,3 Pl,2,3) Here, Pi,3 is the projection operator

To evaluate the conditional density matrix, it is convenient to first find the vectors

and Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 27

The vector в equals to flfo) = -^Xmto) and the conditional density matrix of the photon 2 appears to be equal to

CT P2/{1,3} = Xm( °з) (with total angular momentum equal to zero), then the photon 2 appears to be polarized along the vector m. 5.3. Entanglement Swapping. In the recent experiment [7] in installation two pairs of correlated photons emerged simultaneously. The state of the system is described by the wave function

The photons 2 and 3 are selected into antisymmetric state What is the state of pair of photons 1 and 4? Conditional density matrix of the pair (1 and 4) is

ТГ23 PU/23 = ТГ (.Г23/Э1234) where operator that selects pair (2 and 3) is defined by

and density matrix of four-photon system is determined by kernel

-, a') = Ф(с7Ь a2) сг3, сг4)Ф*(ег1, <г2, <г'3, <т4)- Direct calculation shows that the pair of the photons (1 and 4) has to be in pure state with the wave function

The experiment confirms this prediction. 5.4. Pairs of Polarized Photons. Now consider a modification of the Innsbruck experiment. Let there be two pairs of photons (1, 2) and (3, 4). Suppose that each pair is in the pure antisymmetric state x- The spin part of the density matrix of the total system is given by the equation р(<г,<г') = where Ф(сг)=х(<71, If the photons 2 and 4 pass through polarizers, they are polarized along Xm(vi} and Xa( then the wave function of the system is transformed into

Here, n, m and r, s are pairs of mutually orthogonal vectors. 28 Belokurov V. V. et al.

Now the conditional density matrix of the pair of photons 1 and 3 is

<7i, 0-3).

The wave function of the pair is the product of wave functions of each photon with definite polarization

We note that initial correlation properties of the system appear only when the photons pass through polarizers. Although the wave function of the system seems to be a wave function of independent particles, the initial correlation exhibits in correlations of polarizations for each pair. Pairs of polarized photons appear to be very useful in quantum communication. 5.5. Quantum Realization of Vernam Communication Scheme. Let us recall the main idea of Vernam communication scheme [8]. In this scheme, Alice encrypts her message (a string of bits denoted by the binary number mi) using a randomly generated key k. She simply adds each bit of the message with the corresponding bit of the key to obtain the scrambled text (s — mi® k, where ф denotes the binary addition modulo 2 without carry). It is then sent to Bob, who decrypts the message by subtracting the key (s9fc = mi©fc©fc = mi). Because the bits of the scrambled text are as random as those of the key, they do not contain any information. This cry ptosy stem is, thus, proved secure in sense of information theory. Actually, today this is the only probably secure cryptosystem! Although perfectly secure, the problem with this security is that it is essential for Alice and Bob to possess a common secret key, which must be at least as long as the message itself. They can only use the key for a single encryption. If they used the key more than once, Eve could record all of the scrambled messages and start to build up a picture of the plain texts and, thus, also of the key. (If Eve recorded two different messages encrypted with the same key, she could add the scrambled text to obtain the sum of the plain texts: si ф «2 = mi ф k ф Ш2 Ф k = mi ф m^ @ k ф k — m\ © mi, where we used the fact that ф is commutative.) Furthermore, the key has to be transmitted by some trusted means, such as a courier, or through a personal meeting between Alice and Bob. This procedure may be complex and expensive, and even may lead to a loophole in the system. With the help of pairs of polarized photons we can overcome the shortcomings of the classical realization of Vernam scheme. Suppose Alice sends Bob pairs of polarized photons obtained according to the rules described in the previous section. Note that the concrete photons' polarizations are set up in Alice's laboratory and Eve does not know them. If the polarization of the photon 1 is set up by a random binary number pi and the polarization of the

photon 3 is set up by a number mj фр4, then each photon (when considered separately) does not carry any information. However, Bob after obtaining these photons can add corresponding binary numbers and get the number m» containing the information (m, © pi @ Pi — ттц). In this scheme, a secret code is created during the process of sending and is transferred to Bob together with the information. It makes the usage of the scheme completely secure.

CONCLUSION

Provided that the subsystem £2 of composite quantum system 5 = Si + 82 is selected (or will be selected) in a pure state Pn, the quantum state of subsystem Si is conditional density Conditional Density Matrix: Systems and Subsystems in Quantum Mechanics 29

matrix pic/2n- Reduced density matrix pi is connected with conditional density matrices by an expansion: Pi = here

The coefficients pn are probabilities to find subsystem 52 in pure states Pn.

REFERENCES

1. Von Neumann J. II Gott. Nach. 1927. P. 1-57; 245-272. See also von Neumann J. Mathematische Grundlagen der Quantenmechanik. Berlin, 1932.

2. Bouwmeester D. et al. Experimental Quantum Teleportation // Nature. 1997. V. 390. P. 577-579.

3. Heisenberg W. Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen // Z. Physik. 1925. V.43. P. 172.

4. Einstein A., Podolsky В., Rosen N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? // Phys. Rev. 1935. V.47. P. 777-780.

5. Belokurov V. V., Khrustalev O. A., Timofeevskaya O. D. Regular and Chaotic Dynamics. Izhevsk, 2000.

6. Belokurov V. V. et al. System and Subsystems in Quantum Communication // Proc. of the 23th Solvay Congress, Greece, 2001.

7. Jennewein T. et al. Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping // Phys. Rev. Lett. 2002. V. 88, No. 1.

8. Vernam G. Cipher Printing Telegraph Systems for Secret Wire and Radio Telegraph Communica- tions // J. Am. Inst. Electr. Eng. 1926. V.45. P. 109-115. XJ0300180 Письма в ЭЧАЯ. 2003. № 1 [116] Particles and Nuclei, Letters. 2003. No. 1[116]

PHYSICS OF QUANTUM COMPUTATION V. V. Belokurov, O. A. Khrustalev, V. A. Sadovnichy, O. D. Timofeevskaya Lomonosov Moscow State University, Moscow

INTRODUCTION

As many other significant discoveries in physics, the invention of quantum computer was a result of the battle against thermodynamic laws. It was the middle of the 19th century when Maxwell proposed his demon — a creature with unique perceptivity able to detect every molecule and control its motion. Separating fast and slow molecules of a gas in different parts of a box, Maxwell's demon could have changed the temperature in these parts without producing any work. Thus, the intelligent machine could prevent thermal death. Later Smoluchowski proved that measuring molecular velocities and storing the information demon would increase entropy. Therefore, the perpetuum mobile of the second type could not be created. It became a common opinion that any physical measurement was irreversible. In the 1950s von Neumann applied these arguments to his computer science. He supposed that the penalty for computer operation was energy scattering with the rate of energy loss kT In 2 per one step of computation. This estimation was considered as convincing. However at the beginning of the 1960s Landauer proved that energy dissipation in computers was closely related to logical irreversibility of computation. There were no rigorous arguments for correctness of von Neumann's conclusion in the case of reversible computation. A hope appeared that if Maxwell's demon learned to perform reversible computations, the construction of reversible computers would become real. To formulate the latter thesis more precisely, let us recall some definitions. A function Т is Л4-computable if a computing machine M. can compute the function Т according to some program. For each computing machine M there is a set C(M) of Л4-computable functions. Among a great number of machines one can ever imagine there exists the universal Turing machine (T) which is able to replace any computer. It follows from the statement (Turing, 1936) that any M -computable function that maps the set of integers Z to itself belongs to the set C(T). It is well known that C(T) is a numerable set of all recursive functions and it is essentially smaller than the set of all functions that map Z to Z. In 1936, Church and Turing independently formulated a hypothesis, the so-called Church- Turing principle: Every function that is considered computable can be computed by the universal Turing machine. In 1985, Deutsch [1] proposed a physical version of Church-Turing principle: «Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.» Physics of Quantum Computation 31

Here, the term «finitely realizable physical system» means any physical object upon which experiment is possible. The «universal computing machine» is an idealized (but theoretically realizable) model. «Finite means» can be defined axiomatically, without restrictive assump- tions about the form of physical laws. If we think of a computing machine as proceeding in a sequence of steps whose duration has a nonzero lower bound, then it operates «by finite means» if (1) only a finite subsystem (though not always the same one) is changed during any step, (2) a change of a subsystem at any step depends only on a state of a finite number of subsystems, and (3) a rule of a change of a subsystem is finite in mathematical sense. The new version of Church-Turing principle is stronger than the original one. Indeed, the demands are so strong that they cannot be satisfied by Turing machine acting according to the laws of classical physics. Owing to continuity of spectra of variables in classical physics, possible states of a classical system form a continuum. But only a numerable set of states can be taken at the input of Turing machine. Consequently, Turing machine cannot perfectly simulate any classical dynamic system. Contrarily to classical systems, quantum systems are compatible with the new version of Church-Turing principle. Quantum Turing machines are so attractive because their action is controlled by unitary reversible transformations of quantum mechanics. A possible irrevesibility could be introduced only by an input data or by an inappropriate choice of a material of the machine's construction. Configurations of spins seem to be a particularly suitable construction material because these machines would not depend on wave packet spreading. Among the most important early works on the subject, the papers of Benioff [2], Bennett [3] and especially famous Feynman's articles [4] should be mentioned. However, contemporary science on quantum computation began in 1985 after Deutsch's paper [1].

1. QUANTUM TURING MACHINE

Deutch's quantum computer as well as Turing machine consists of two components — a finite processor and an infinite memory. During calculations always only a finite part is used. The processor is a system, all states of which are eigenstates of a set of observables

acting in two-dimensional Hilbert spaces. The memory consists of an infinite sequence of observables of the same kind:

М = {т4, г = 0,1,...}.

This system corresponds to the infinite memory's tape of Turing machine. The observable x corresponds to the head's (cursor's) position at the tape of Turing machine. Its spectrum is a set of integer numbers. One of the bases in the space of computer's states is a set of eigenvectors of these operators: \x\ n; m) = \x\ no, ...,njv-i; . ..m_i,mo,mi,...). 32 Belokurov V. V. et al.

They are called «computational basis states». The variables n, rh are chosen in such a way that all of them have spectrum {0, 1}. The evolution operator U for Deutsch computer is defined by its matrix elements

(x';n';m'\U\x-n;m) =

= {Sx'x+iU+(n',rn'x\n,mx) + 5x'x^iU-(n',rn'x\n,mx)}UyjtxSmymv- Quantum computer is equivalent to Turing machine if

U±(n',m'\n,m) = -t>A(n,m),n'bB(n,m),m{l±C(n,'™}}, where А, В, С are some functions with ranges of values (Zz)N ', Z% and {—1, 1}, correspondingly. For operator U to be unitary, it is necessary and sufficient to have a bijective mapping:

{n,m} & {A(n,m),B(n,m),C(n,m)}.

In the rest the functions, А, В, С are arbitrary. They can be chosen such that the constructed computer represents the universal quantum Turing machine T.

2. EVALUATION OF FUNCTIONS BY QUANTUM COMPUTER

Suppose the function /(г) = j transforms the set

into the set

Zn = {j=0,l,...,n-I}. Is it possible to associate with it a unitary transformation in some Hilbert space? The answer is positive and the constructions of the space and the transformation are rather transparent. Let Hmn be a Hilbert space with the dimension mn and

es = i,j}, ieZm, j e Zn, (i,j\i',j') = 5n'5jj> be a basis in this space. The transformation

is a unitary one. Symbol © means addition modulo n. A remarkable property of quantum computation is a possibility to define states in which all the values of the function / are considered simultaneously. If

m-l Physics of Quantum Computation 33

then m-l

\fm

In some cases the unitary operator 5 defined by the equation

appears to be useful. It converts the vector ip into the vector

i=0

In its turn, operator U/ transforms the vector в into the vector

t=0

If n = 2, i.e., the function / has only two values 0 and 1, then /(г) ф /(г) = 0 and

»=о

The scalar product of the initial and the final vectors in this chain is

2m- 1

An operation of a quantum computer is described by a d-dimensional unitary matrix Т that realizes the evolution operator in a computational basis. It is very essential that the matrix Т can be represented as a product of d(2d - 1) unitary matrices. Each of them corresponds to an operator that acts in a two-dimensional space formed by a pair of vectors of the mentioned basis. Any vector V with the components (vi,V2,...,Ud) in the computational basis can be transformed by d — 1 transformations of indicated form to the vector W that has the components (1, 0, . . . , 0) in the computational basis:

The reverse transformation looks like 34 Belokurov V. V. el al.

3. QUANTUM FOURIER TRANSFORM

Consider functions defined in ZN • Suppose

If numbers a, 6 £ ZN, then the numbers шаЬ form a unitary matrix F:

•)<* = £

The Fourier transform of a function /(a) is given by the equation

We can define the Fourier operator in the Hilbert space HN- If a vector Ф 6 HN equals

N-l Ф = £ И/(а), a=0 then we say that Fourier transform of the vector Ф is

It is clear that Fourier transform of a basis vector |a) is

Further, we suppose that N = 2'. Let a be represented in binary as ai • • • a/ 6 {0, 1}'

(and similarly c). It is possible to represent the computational basis in the form of direct product of the bases in two-dimensional Hilbert spaces

The product of the numbers a and с can be written in the form

1 1 l ac = aid + 2(oj_ic/ + a;c;_i) + . . . 2 " (a;Ci + . . . + ощ) + O(2 ). Physics of Quantum Computation 35

Since

oc we obtain 27ri 0 a (2тгг— ) = e ( - l)ci here,

-, a ' The Fourier transform of the vector |a) is

a Cl F\a) = - '> |c2)

It follows that

The Walsh-Hadamard operator Rj acting in a subspace Hj is given by the formula

The controlled phase shift operator Sjk acts in the join of subspaces "Hj U Jik as:

The angle ^^ is

It is easy to prove that the quantum Fourier transform [S] is realized by the following sequence of the operators. First, the operator R\ acts. Then the operators £2,1, 83,1, ..., §it\ act. Then there is a turn for the operators RZ, 833 and 54,2 and so on. In the interval between Rj and Rj+i all operators §k,j (k > j) act. The explicit formula is

. нч 2iri(0.a,h \i~/ • i"/ ~ /\i~/ • i~/ ~ / \i~/ • I*/*' /• This vector differs from the Fourier transform of the vector |oi,..., aj) in order of variables Oj only. We use unitary operator 5 that converts basis |oi,...,oj) into |aj,...,oi). Thus, we get SRr--Ri=F. In order to understand how important the factorization of the Fourier transform operator is, we recall some general concepts of quantum computation theory. 36 Belokurov V. V. el al.

The minimal quantity of information in the classical theory, a bit, is a basic notion of the theory. In the theory of quantum computations the minimal quantity of information (a quantum bit, or a qubit) is given by a unit vector in a two-dimensional Hilbert space H-2- In the classical theory a logic gate is a computing machine which has fixed numbers of input and output bits and can produce a fixed operation in a fixed period of time. A quantum gate is a device which performs a fixed unitary operation on the selected qubits in a fixed period of time. The quantum gate is an operator that transforms Hilbert space Tij into Ho- The operators R and SV,, defined above, are gates. The swapping operation is the sequence of gates. If one quantum gate is one step of computation, then Fourier transform in Hilbert space of N = 2l dimension requires about

2 ATqstep = 0((lg2 (AT)) ) steps. The best known classical algorithm for fast discrete Fourier transform is of size

4. EXPERIMENTAL REALIZATION OF QUANTUM FOURIER TRANSFORMATION

In the paper [6] QFT in two-qubit system is realized by NMR methods. Two qubits were nuclears A and В with spin 1/2 in constant magnetic field. The Hamiltonian for this system can be approximated as

Я = ш05за + шьЗзъ + 2тгша(,5з05зь + Henv,

where Henv is interaction with environment. In the first approximation it can be ignored. The eigenstates of Я are the eigenstates of operators &за and §зь- 31 J In the experiment, they chose Н2РОз as a sample, labeled P as A-qubit and H as B-qubit. The observed J-coupling between 31P and JH was 647.451 Hz. First, they produced effective pure state |00) by using «temporal averaging» [7]. —7Г \) —1-j — ( 2/y 2 7Г \ -J on :H to obtain the state |01). ( L ' X Second, we perform Fourier transformation on |01). The list of pulses and spin-spin interaction are as follows: p y' (-!L\P (_1L\H - —

1)Y\1)Y V4/xV4^ V 2/yV 2/x 4J' Physics of Quantum Computation 37

The first and the third lines in this formula are Walsh-Hadamard transformation and the second is phase-shift transformation. After these operations, the initial state |01) is transformed into the state

The third step is the reversion of qubits. It is well known that this operation is composed of three C-Not gates.

5. GEOMETRIC QUANTUM COMPUTATIONS

At present, the geometric quantum computation with NMR on the base of Berry phase [8] is considered as one of the most promising methods. This new approach to quantum gates may be important to the future, as it is naturally resilient to certain types of errors connected with interactions with environment. If Hamiltonian varies adiabatically through a circuit С in the space of parameters R(£), then the adiabatic evolution of the system is described by time-dependent Schrodinger equation. According to Berry, solutions have a form

exp -1 jf

here, |R(t)) is eigenvector of Hamiltonian at the moment t and 7n(*) is geometric phase which depends only on the path. For system of two spins 1/2 Berry's phase in the state IU>is 7ТД = TT(! - cos 9); here в is a solid angle that vector R(t) sweeps out in the space of parameters. Thus, we get the evolution of state's vector in time Ф = |a) e^s+^; S is dynamical phase which depends on the Hamiltonian. It is well known in NMR that it is possible to eliminate the dynamic phase by the «phase refocusing» procedure. As a result of suitable circuit, the spin state vectors are changed. One gets

The dynamic phases are eliminated and we are left with an exclusively geometric phase difference of 47 = 4тгсоз0. The conditional Berry phase gates depend only on the geometry of the path. They are completely independent of how the motion is performed, as long as it is adiabatic. Hence, the kind of quantum computation with the help of Berry's phase may be called geometric quantum computation. These techniques are readily implemented with current technology in quantum optics and have already been demonstrated by some of the authors using NMR [9]. 38 Belokurov V. V. el al.

REFERENCES

1. Deutsch D. Quantum theory, the Church-Turing Principle and the Universal Quantum Computer // Proc. Royal. Soc. Lond. A. 1985. V.400. P. 97-117.

2. Benioff P. Quantum Mechanical Hamiltonian Models of Turing Machines // J. Stat. Phys. 1982. V. 29. P. 515-545.

3. Bennett C.H. Thermodynamics of Computation // Intern. J. Theor. Phys. 1982. V.21. P. 219-253.

4. Feynman R. Quantum Mechanical Computers // Ibid. Nos. 6/7.

5. Shor P. W. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer// SIAM J. Comput. 1997. V.26, No.5. P. 1484-1509.

6. Fu L. et al. Experimental Realization of Discrete Fourier Transformation on NMR Quantum Com- puter. quant-ph/9905083. 1999.

7. Knill A., Chuang E. II Phys. Rev. A. 1998. V. 57. P. 3357.

8. Berry M. V. Quantum Phase Factor Accompanying Adiabatic Changes // Proc. Royal. Soc. Lond. A. 1984. V. 392. P. 45-57.

9. Jones J.A., Hansen R. H., Mosca M. II J. Magn. Reson. 1998. V. 135. P. 353. XJ0300181 Письма в ЭЧАЯ. 2003. № 1[116] Particles and Nuclei, Letters. 2003. No. 1[116]

FLIPPING QUBITS F. DeMartinia, V.Buzekb

On a classical level the information can be represented by bits, each of which can be either 0 or 1. Quantum information, on the other hand, consists of qubits which can be represented as two-level quantum systems with one level labeled |0) and the other |1). Unlike bits, qubits cannot only be in one of the two levels, but in any superposition of them as well. This superposition principle makes quantum information fundamentally different from its classical counterpart. One of the most striking difference between the classical and quantum information is as follows: it is not a problem to flip a classical bit, i. e., to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Flipping a qubit, however, is another matter: there exists the fundamental bound which prohibits to flip a qubit prepared in an arbitrary state |Ф> = a|0) + /9|0) and to obtain the state 1Ф-1) = /?*|0) - a*|l) which is orthogonal to it, i.e., (ФХ|Ф) = 0. We experimentally realize the best possible approximation of the qubit flipping that achieves bounds imposed by complete positivity of quantum mechanics.

INTRODUCTION

Let us assume the Poincare sphere, which represents a state space of a qubit. The points corresponding to |Ф) and 1Ф1} are antipodes of each other. The desired spin-flip operation is therefore the inversion of the Poincare sphere (see Fig. 1). It is well known that this inversion preserves angles (which is related to the scalar product |{Ф,Ф}| of rays). Therefore, by the arguments of the Wigner theorem, the ideal spin-Sip operation must be implemented either by a unitary or by an anti-unitary operation. Unitary operations correspond to proper rotations of the Poincare sphere, whereas anti-unitary operations correspond to orthogonal transformations with determinant —1. The spin-Шр is an anti-unitary operation; i. e., it is not completely positive. Due to the fact that the tensor product of an antilinear and a linear operator is not correctly defined, the spin-flip operation cannot be applied to a qubit, while the rest of the world is governed by unitary evolution1. On the other hand, if we consider a spin-flip operation, we have in mind a Universal NOT gate flipping an input qubit to its orthogonal state. The gate itself is an operation applied to the qubit, that is just a subsystem of a «whole universe». Therefore, a completely positive operation must be represented. It is well known that any

'in fact, exactly this property makes the spin-flip operation so important in all criteria of inseparability for two-qubit systems [1,2]. 40 De Martini F. el al

completely positive operation on a qubit can be realized by a unitary operation performed on the qubit and the ancillary system. Following -0 this arguments, we see that the ideal Univer- -1 sal NOT gate which would flip a qubit in an arbitrary state does not exist. Obviously, if the state of the qubit is known, then we can always perform a flip operation. In this situation the classical and quantum operations share many similar features, -0.5 since the knowledge of the state is a classical information, which can be manipulated accord- -1 -1 ing to the rules of classical information processing (e. g., known states can be copied, flipped, etc.). But the universality of the operation is lost. That is, the gate which would flip the state |0) —> |1) is not able to perform a flip Fig. 1. The state space of a qubit is a Poincare Ku/" sphere. Pure states are represented by points on since il is not possible to realize a perfect the sphere, while statistical mixtures are points Universal NOT gate [3] which would flip an inside the sphere. The Universal NOT operation arbitrary qubit state, it is of interest to study corresponds to the inversion of the sphere, since what is Ле best approximation to the perfect the states |Ф) and |ФХ) are antipodes Universal NOT gate. Here, one can consider two possible scenarios. The first one is based on the measurement of input qubit(s) — using the results of an optimal measurement, one can manufacture an orthogonal qubit, or any desired number of them. Obviously, the fidelity of the NOT operation in this case is equal to the fidelity of estimation of the state of the input qubit(s). The second scenario would be to approximate an anti-unitary transformation on a Hilbert space of the input qubit(s) by a unitary transformation on a larger Hilbert space which describes the input qubit(s) and ancillas. It has been shown recently that the best achievable fidelity of both flipping scenarios is the same [4-6]. That is, the fidelity of the optimal Universal NOT gate is equal to the fidelity of the best state-estimation performed on input qubits [7-9] (one might say that in order to flip a qubit we have to transform it into a bit). In what follows we briefly describe the unitary transformation realizing the quantum scenario for the spin-flip operation; that is, we present the optimal Universal NOT gate. Then we describe our experiment (see also Ref. [10]) in which we «flip» qubits encoded in polarization states of photons.

1. THEORETICAL DESCRIPTION OF SPIN FLIPPING

Let H = C2 denote the two-dimensional Hilbert space of a single qubit. Then the input state of TV systems prepared in the pure state |Ф) is the /V-fold tensor product \$)®N e H®N'. The corresponding density matrix is a = p®N, where p = |Ф){Ф| is the one-particle density matrix. An important observation is that the vectors |Ф)®-'¥ are invariant under permutations Flipping Qubits 41 of all N sites; i. e., they belong to the symmetric, or «Bose»-subspace 7ifN С H®N. Thus, as long as we consider only pure input states, we can assume all the input states of the device under consideration to be density operators on H®N. We will denote by S(Ji) the density operators over a Hilbert space H. Then the U-NOT gate must be a completely positive trace preserving map Г: S (Ti.®N) —>• S(H). Our aim is to design Т in such a way that for any pure one-particle state p € S(H) the output T(p®N) is as close as possible to the orthogonal qubit state /r1 = 1 — p. In other words, we are trying to make the fidelity f := Tr [p*-T(p®N)} = 1 - Д of the optimal complement with the result of the transformation Т as close as possible to unity for an arbitrary input state. This corresponds to the problem of finding the minimal value of the error measure Д(Т) defined as

Д(Г)=тахТг[рГ(р®")]. (1) p.pure

Note that this functional Д is completely unbiased with respect to the choice of input state. More formally, it is invariant with respect to unitary rotations (basis changes) in Л. When N N Т is any admissible map, and U is a unitary on U, the map Tv(a) = U*T(U® vU*® )U is also admissible and satisfies A(T[/) = A(T). We will show later on that one may look for optimal gates T, minimizing A(T), among the universal ones, i. e., the gates satisfying TV — Т for all U. For such U-NOT gates, the maximization can be omitted from the definition (1), because the fidelity Tr [pT(p®N)] is independent of p. 1.1. Measurement-Based Scenario. An estimation device by definition takes an input state a e S(H®N) and produces, on every single experiment, an «estimated pure state» p e S(H). As in any quantum measurement this will not always be the same p, even with the same input state p, but a random quantity. The estimation device is, therefore, described completely by the probability distribution of pure states it produces for every given input. Still simpler, we will characterize it by the corresponding probability density with respect to the unique normalized measure on the pure states (denoted «ЙФ» in integrals), which is also invariant under unitary rotations. For an input state а б S(H+N), the value of this probability density at the pure state |Ф) is р(Ф,<г) = (#+ !)<**", ***">• (2) To check the normalization, note that /йФр(Ф,<т) = ТгрСст] for a suitable operator X, because the integral depends linearly on a. By unitary invariance of the measure «ЙФ» this operator commutes with all unitaries of the form U®N, and since these operators restricted to H+N form an irreducible representation of the unitary group of H (for d = 2, it is just the spin N/2 irreducible representation of SU(2)), the operator X is a multiple of the identity. To determine the factor, one inserts a = 1, and Uses the normalization of <«2Ф» to verify that X = l. 2 лг лг Note that the density (2) is proportional to КФ.Ф)! ^, when

p(est) = г(ст)

The fidelity required for the computation of Д from Eq. (1) is then equal to (see also [8,9])

2 ФКФ,Ф)Г(1-|{Ф)Ф)| ) = ! (4)

where we have used the fact that the two integrals have exactly the same form (differing only in the choice of N), and that the first integral is just the normalization integral. Since this expression does not depend on p, we can drop the maximization in the definition (1) of Д and find Д(Т) = l/(JV + 2), from which we find that the fidelity of creation of a complement to the original state p is

Finally, we note that the result of the operation (3) can be expressed in the form

p(°ut) = s^p1 H ^Г^1' ^

with the «scaling» parameter SN = N/(N + 2). From here it is seen that in the limit TV —> oo, perfect estimation of the input state can be performed, and, consequently, the perfect complement can be generated. For finite N, the mean fidelity is always smaller than unity. The advantage of the measurement-based scenario is that once the input qubit(s) is measured and its state is estimated, an arbitrary number M of identical (approximately) complemented qubits can be produced with the same fidelity, simply, by replacing the output М function /(Ф) = (1 - |Ф)(Ф|) by /М(Ф) = (1 - |ФХФ|)® 1.2. Quantum Scenario. Let us now present a transformation which produces complements whose fidelity is the same as those produced by the measurement-based method. Assume we have N input qubits in an unknown state (Ф), and we are looking for a transformation which generates M qubits at the output in a state as close as possible to the orthogonal state |ФХ). The universality of the proposed transformation has to guarantee that all input states are complemented with the same fidelity. If we want to generate M approximately complemented qubits at the output, the U-NOT gate has to be represented by 2M qubits (irrespective of the number, N, of input qubits), M of which serve as ancilla, and M of which become the output complements. We will indicate these subsystems by subscripts «a» = input, «6» = ancilla, and «c» = (prospective) output. The U-NOT gate transformation, UNM, acts on the tensor product of all three systems. The gate is always prepared in some state \X}bc, independently of the input state |Ф). The transformation is determined by the following explicit expression, valid for every unit vector |Ф) 6 Ti: M М} ®\X)bc - ^2^' \Ху(^)}аЬ®\{(М -j^^-J^^c (7) j=o Flipping Qubits 43 with

N ) \ M

влг where |./У~Ф)а = |Ф) is the input state consisting of N qubits in the same state |Ф). Х In the right-hand side of Eq. (7), |{(M — ЯФ ;^Ф})С denotes symmetric and normalized states with (M — j) qubits in the complemented (orthogonal) state (Ф1) and j qubits in the original state |Ф). Similarly, the vectors |.У,-(Ф))вь consist of N + M qubits and are given explicitly by

Note that with this choice of the coefficients 7- 'N', the scalar product of the right-hand side with a similar vector, with Ф replaced by Ф, becomes (Ф, Ф)^. This is consistent with the unitarity of the operator UNM • Each of the M qubits at the output of the U-NOT gate is described by the density operator

(6) with SN — N/(N + 2), irrespective of the number of complements produced. The fidelity of the U-NOT gate depends only on the number of inputs. This means that this U-NOT gate can be thought of as producing an approximate complement and then cloning it, with the quality of the cloning independent of the number of clones produced. The universality of the transformation is directly seen from the «scaled» form of the output operator (6). We stress that the fidelity of the U-NOT gate (7) is exactly the same as in the measurement- based scenario. Moreover, it also behaves as a classical (measurement-based) gate in a sense that it can generate an arbitrary number of complements with the same fidelity. We have also checked that these cloned complements are pairwise separable. The N + M qubits at the output of the gate, which do not represent the complements, are individually in the state described by the density operator

(10)

N 2N with the scaling factor s = — — - + тг — .,.,.. — —r\ i.e., these qubits are the clones N + 2 (N г+ M)(N + 2) of the original state with a fidelity of cloning larger than the fidelity of estimation. This fidelity depends on the number, M, of clones produced out of the N originals, and in the limit M — > со the fidelity of cloning becomes equal to the fidelity of estimation. These qubits represent the output of the optimal N — > N + M cloner introduced by Gisin and Massar [11-13]. This means that the U-NOT gate, as presented by the transformation in Eq. (7), serves also as a universal cloning machine. 13. Optimality of U-NOT Gate. At this point, the question arises whether the transformation (7) represents the optimal U-NOT gate via quantum scenario. If this is so, then it would mean that the measurement-based and the quantum scenarios realize the U-NOT gate with the same fidelity. In what follows we present a proof due to Werner in Refs. [4, 5]. Theorem. Let H be a Hilbert space of dimension d = 2. Then among all completely positive trace preserving maps Т : S (H®N) — » S(T~t), the measurement-based U-NOT scenario (3) attains the smallest possible value of the error measure defined by Eq. (1), namely, A(T) = l/(N + 2). 44 De Martini F. et al.

We have already shown (see Eq. (4)) that for the measurement-based strategy the error Д attains the value l/(N + 2). The more difficult part, however, is to show that no other scheme (i. e., quantum scenario) can do better. Here, we will largely follow the arguments in [14]. Note first that the functional A is invariant with respect to unitary rotations (basis changes) in H. When Т is any admissible map, and U is a unitary on H, the map TU(

A(T)< dUb(Tu) = b(T), (11) where Т = f dUTu is the average of the rotated operators Тц with respect to the Haar measure 'on the unitary group. Thus, Т is at least as good as Т and has the additional «covariance property» Тц = Т. Without loss we can therefore assume from now on that Tu = Т for all U. An advantage of this assumption is that a very explicit general form for such covariant operations is known by a variant of the Stinespring Dilation Theorem (see [14] for a version adapted to our needs). The form of T is further simplified in our case by the fact that both representations involved are irreducible: the defining representation of SU(2) on H, and the representation by the operators U®N restricted to the symmetric subspace H® . Then Т can be represented as a discrete convex combination Т = £^ • XjTj, with Aj > 0, ^ Aj = 1, and Tj admissible and covariant maps in their own right, but of an even simpler form. Covariance of Т already implies that the maximum can be omitted from the definition (1) of A, because the fidelity no longer depends on the pure state chosen. In a convex combination of covariant operators we therefore get . (12)

Minimizing this expression is obviously equivalent to minimizing with respect to the discrete parameter j. We write the general form of the extremal instruments Tj in terms of expectation values of the output state for an observable X on H:

Tr (T(a)X) = TV [aV*(X 1)V] , (13)

where V : H^.N — » H <8> C2j'+1 is an isometry intertwining of the respective representations of SU(2), namely, the restriction of the operators U®N to H®N (which has spin TV/2), on the one hand, and the tensor product of the defining representation (spin-1/2) with the irreducible spin-j representation, on the other. By the triangle inequality for Clebsch-Gordan reduction, this implies j = (TV/2) ± (1/2), so only two terms appear in the decomposition of T. It remains to compute A(l}) for these two values. The basic idea is to use the intertwining property of the isometry V for the generators 2j+1 N Sa, JQ, and La, a = 1, 2, 3 of the 5C/(2)-representations on H, C and Hf , respectively. Flipping Qubits 45

We will show that

V*(Sa®lj)V = »jLa, (14) where /j.j is some constant depending on the choice of j. That such a constant exists is clear from the fact that the left-hand side of this equation is a vector operator (with components labeled by a = 1,2, 3), and the only vector operators in an irreducible representation of SU(2) are multiples of angular momentum (in this case La). The constant fj,j can be expressed in terms of a 6j symbol, but can also be calculated in an elementary way using the intertwining property, VLa = (Sa ® 1 + 1 <8> Ja)V, and the fact that the angular 2 2 2 momentum squares J = £Q J* = j(j + 1), S = 3/4, and L = N/2(N/2 + 1) are multiples of the identity in the irreducible representations involved, and can be treated as scalars:

The sum in the right-hand side can be obtained as the mixed term of a square, namely, as \ (53 V*(S<* ® X + ! ® J«)2^ - S2 - a Combining these equations, we find 1 , . N 1 JV f°r J = Y + 2 -1 . ЛГ 1 ' (17)

We combine equations (13) and (14) to get the error quantity A from equation (1), with the pure one-particle density matrix p = 1/2 1 + 83:

(18)

With equation (17) we find

ЛГТ2 T~2 The first value is the largest possible fidelity for getting the state p from a set of N copies of p. The fidelity 1 is expected for this trivial task, because taking any one of the copies will do perfectly. On the other hand, the second value is the minimal fidelity, which we were looking for. This clearly coincides with the value (4), so the Theorem is proved. The Theorem as it stands concerns the task of producing just one particle in the U-NOT state of the input. From the results of the previous section we see that it is valid also in the case of many outputs. We see that the maximum fidelity is achieved by the classical process via estimation: in equation (3) we just have to replace the output state (1 — |Ф){Ф|) by the 46 De Martini F. et al. desired tensor power. Hence, once again the optimum is achieved by the scheme based on classical estimation. Incidentally, this shows that the multiple outputs from such a device are completely unentangled, although they may be correlated. We can conclude this section by saying that in the quantum world governed by unitary operations anti-unitary operations can be performed with the fidelity which is bounded by the amount of classical information potentially available about states of quantum systems.

2. EXPERIMENT

In our experiment we will consider a flipping of a single qubit. In this case the flipping transformation reads

/2" _,_ /Т х V з ° V з To be specific, Eq. (20) describes a process when the original qubit is encoded in the system a, while the flipped qubit is in the system c. The density operator describing the state of the system с at the output is

The fidelity of the spin flipping is Т — 2/3. A natural way to encode a qubit into a physical system is to utilize polarization states of a single photon. In this case, the Universal NOT gate can be realized via the stimulated emission. The key idea of our experiment is based on the proposal that universal quantum machines [15] such as quantum cloner can be realized with the help of stimulated emission in parametric down conversion [16,17]. In particular, let us consider a qubit to be encoded in a polarization state of a photon. This photon is injected as the input state into an optical parametric amplifier (OPA) physically consisting of a nonlinear (NL) BBO (/3-barium-borate) crystal cut for Type II phase matching and excited by a pulsed mode-locked ultraviolet laser UV, having pulse duration т к 140/ s and wavelength (wl) \p = 397.5 nm, associated to pulse duration [18]. The relevant modes of the NL three-wave interaction were the spatial modes with wave-vector

(wv) fci and fc2 each supporting the two horizontal (Я) and vertical (V) linear polarizations

(П) of the interacting photons, e. g., П1Я is the horizontal polarization unit vector associated with ki. The OPA was frequency degenerate; i.e., the interacting photons had the same wl's Л = 795 nm. The action of OPA under suitable conditions can be described by a simplified Hamiltonian

Яш1 = к(а^х-в^Ь^) + Ь.с. (22) A property of the device, of key importance in the context of the present work, is its amplifying behaviour with respect to the polarization П of the interacting photons. It has been shown by theory [16,19] and in a recent experiment on «universal quantum cloning» [17] that the amplification efficiency of this type of OPA under injection by any externally injected quantum field, e. g., consisting of a single photon or of a classical «coherent» field, can be made independent of the polarization state of the field. In other words, the OPA «gain» is Flipping Qubits 47

independent of any (unknown) polarization state of the injected field. This precisely represents the necessary universality (U) property of the U-NOT gate. For this reason in Eq. (22) we

have denoted the creation a\, (£Ф) and annihilation йф (6ф) operators of a photon in mode ki (fo) with subscripts Ф (or Фх) indication of the invariance of the process with respect to polarization states of the input photon. Let us consider the input photon in the mode ki to have a polarization Ф. We will describe

this polarization state as a4|0,0)fc1 = |1,0}*,, where we have used notation introduced by Simon et al. [16]; i.e., the state |m, n)^ represents a state with m photons of the mode fci having the polarization Ф, while n photons have the polarization Ф-1-. Initially, there are no excitations in the mode k%. The initial polarization state of these two modes reads

|l,0)fc1 <8> |0, 0)fc2, and it evolves according the Hamiltonian (22):

exp(-»#intt)|l,0)fcl ® 10,0)*, ^ |l,0)fcl ® |0,0)*,-

-iKt (\/2|2,0)fcl ® 10,1)*, - 11,1)*, ® 11,0}*,). (23)

This approximation for the state vector describing the two modes at times t > 0 is sufficient since the values Kt are usually very small (see below). The zero-order term corresponds to the process when the input photon in the mode ki does not interact in the nonlinear medium, while the second term describes the first-order process in the OPA. This second term is formally equal (up to a normalization factor) to the right-hand side of Eq. (20). Here, the state 12,0})^ describing two photons of the mode k\ in the polarization state Ф corresponds to the state |ФФ}. This state-vector describes the cloning of the original photon [16, 17]. The vector |0, l)fc_2 describes the state of the mode k2 with a single photon with the polarization Ф1. That is, this state vector represents the flipped version of the input. To see that the stimulated emission is indeed responsible for creation of the Sipped qubit, let us compare the state (23) with the output of the OPA when the vacuum is injected into the nonlinear crystal. In this case, to the same order of approximation as above, we obtain

|0,0)*, к 10,0)*,

)fc/®|0,l)fc2HO,l)fc1®|l,0>fc2). (24)

We see that the flipped qubit described by the state vector |0, 1}*2 in the right-hand sides of Eqs. (23) and (24) does appear with different amplitudes corresponding to the ratio of probabilities to be equal to 1 : 2. This ratio has been measured in our experiment. 2.1. Universality. On the «microscopic» quantum level the justification of this {/-property of the OPA amplifier resides in the SU(2) rotational invariance of the NL interaction Hamil- tonian when the spatial orientation of the OPA NL Type II crystal makes it available for the generation of two-photon entangled «singlet» states by Spontaneous Parametric Down Conversion (SPDC), i.e., by injection of the «vacuum field» [16,19]. However we should note that in the present context the universality property, i. e., the Ti-insensitivity of the parametric amplification «gain» g, is a «macroscopic» classical feature of the OPA device. As a consequence, it can be tested equally well either by injection of «classical», e. g., coherent (Glauber) fields or of a «quantum» states of radiation, e. g., a single-photon Fock state. We have carried out successfully both tests, leading to identical results. We only report here the ones corresponding to the injection by attenuated «coherent» laser field (see Fig. 2). 48 De Martini F. et at.

X = 795 nm PBS,

0.9 8< ft- SHG J 1 A. - 397 5 nm kD ^^^-*. 1 P BBO iНПь Type II Ti:Sa laser Coherent MIRA

Fig. 2. Schematic description of the experimental verification of the universality of the flip operation

A coherent state of attenuated laser field with wl Л = 795 nm is used. The source is Ti:Sa Coherent MIRA pulsed laser providing by Second Harmonic Generation (SHG) the

OPA «pump» field associated with the spatial mode with wv kp and wl Ap. A small portion of the laser radiation at wl A was directed along the OPA injection mode k\. The parametric amplification, with calculated «gain» g = 0.31, was detected at the OPA output mode k\

-40 -20 0 20 40 60 -40 -20 0 20 40 60 Mirrors position Z, цт Mirrors position Z, цт

Fig. 3. Experimental verification of the universality of OPA Flipping Qubits 49 by D{, a Si linear photodiode SGD100. The time superposition in the NL crystal of the «pump» and of the «injection» pulses was assured by micrometric displacements (Z) of a two-mirror optical «trombone». Various П-states of the injected pulse were prepared by the set (WPi + <2) consisting of a Wave-plate (either A/2 or A/4) and of a 4.3-mm X-cut Quartz plate. These states were then analyzed after amplification and before detection on mode £2 by an analogous optical set (WPg + Q + Tl-analyzer), the last device being provided by the Polarizing Beam Splitter PBS2. In the two experiments reported in the present work, all the 4.5-mm thick X-cut quartz plates (Q) provided the compensation of the unwanted beam walk-off effects due to the birefringence of the NL crystal. The universality condition is demonstrated by the plots of Fig. 3, showing the amplification pulses detected by D'2 on the OPA output mode, fo. Each plot corresponds to a definite П- state, |Ф) = (соз(1?/2)|Я) + exp(i0)sin(i?/2)|y)], either linear — П, i.e., i?.= О, тг/2, тг; ф = 0, or circular — П, i.e., i? = тг/2; ф = ±тг/2, or elliptical — П, in the very general case: t? = 5тг/18; ф = —тг/2. We may check that the corresponding amplification curves, each corresponding to a standard injection pulsei with an average photon number N w 5 • 103, are almost identical. For more generality, the universality condition as well as the insensitivity of this condition to the value of N is also demonstrated by the single experimental data reported, with different scales, at the top of each amplification plot and corresponding to injection pulses with N w 5 • 102. Single-photon tests of the same conditions were also carried out with a different experimental setup, as said. 2.2. Optimality. Let us move to the main subject of the present work, i.e., the quantum U-NOT gate. In virtue of the tested universality of the OPA amplification, it is of course sufficient to consider here the OPA injection by a single-photon in just one П-state, for instance in the vertical П-state. Accordingly, Fig. 4 shows a layout of the single-photon, N = 1, quantum-injection experiment with input state |Ф) = \V).

Consider the kp pump mode, i. e., the «towards R» excitation. An SPDC process created single photon-pairs with wl A = 795 nm in entangled singlet Л-states, i. e., rotationally invariant, as said. One photon of each pair, emitted over k\ was reflected by a spherical mirror M onto the NL crystal where it provided the N = 1 quantum injection into the OPA amplifier M excited by the UV «pump» beam associated to the back reflected mode

—kp. We consider flipping of a single photon in a state |Ф) = \V). In this experiment, owing to a spherical Fig. 4. Experimental realization of the quantum U-NOT mirror Mp with 100% reflectivity and gate micrometrically adjustable position Z, the UV pump beam excited the same NL OPA crystal amplifier in both directions kp and

—kp, i. e., correspondingly oriented towards the right (R) and left (L) sides of the figure. Because of the low intensity of the UV beam, the two-photon injection probability N = 2 50 De Martini F. el al. has been evaluated to be и 3.5 • 10~4 smaller that for the N = 1 condition. The twin photon emitted over k-z was Ii-selected by the devices (W?2+ PBS2) and then detected by D%, thus providing the «trigger» of the overall conditional experiment. All detectors in the present experiment were equal active SPCM-AQR14 with quantum efficiency: QE « 55%. Because of the EPR nonlocality implied by the singlet state, the H-selection on channel k-z provided the realization on fci of the state |Ф) = \V) of the injected photon. As for the previous experiment, all the X-cut quartz plates Q provided the compensation of the unwanted beam walk-off effects due to the birefringence of the NL crystal. Consider the «towards L» amplification, i.e., the amplification process excited by the mode — kp, and do account, in particular, for the OPA output mode £2- The П-state of the field on that mode was analyzed by the device combination (WP£+ PBS^) and measured by the detector D\. The detectors Da, Db were coupled to the field associated with the mode fci. The experiment was carried out by detecting the rate of the four-coincidences involving all detectors [D^D^DaDb]. From, the analysis presented by [16,17], it follows that the state of the field emitted by the OPA indeed realizes the U-NOT gate operation, i. e., the «optimal» realization of the «anticloning» of the authentic qubit originally encoded in the mode k\. The flipped qubit at the output is in the mode k%. As has been shown earlier, the state created by the U-NOT gate is not pure. There is a minimal amount of noise induced by the process of flipping which is inevitable in order to preserve complete-positiveness of the Universal NOT gate. This mixed state is described by the density operator (21). The polarization state of the output photon in the mode kz in our experiment is indeed described by this density operator. The plot of Fig. 5 reports our experimental F=0.623 ±0.025 four-coincidence data as function of the time 150 - 4*4ч, -г superposition of the UV pump and of the injected single-photon pulses. That superposition was expressed as function of the micrometric displacement Z of the back-reflecting mirror а u TJ Mp. The height of the central peak expresses .3 so the rate measured with the Tl-analyzer of mode k-z set to measure the «correct» horizontal (H) polarization, i. e., the one orthogonal to the (V) I i i i i i i i i i i i i i i polarization of the П-state, |Ф) = |V) of the

-80 -60 -40 -20 0 20 40 60 80 injected, input single photon, TV = 1. On turn-

UV mirror displacement, um ing by 90 оm e П-anafyzer, the amount of the «noise» contribution is represented by a «flat»

Fig. 5. Experimental verification of the optimality curve In our systeiri( the «noise» was provided of the U-NOT gate by me opA ampiification of the unavoidable «vacuum» state associated with the mode k\. Our main result consists of the determination of the ratio R* between the height of the central peak and the one of the flat «noise» contribution. To understand this ratio, we firstly note that the most efficient stimulation process in the OPA is achieved when a perfect match (overlap) between the input photon and the photon produced by the source is achieved. This situation corresponds to the value of the mirror position Z equal to zero (see Eq. (23)). As soon as the mirror is displaced from the position, the two photons do not overlap properly, and the stimulation is less efficient. Correspondingly, the spin-flip operation is more noisy. In the limit of large displacements Z the spin flipping is totally random due to the fact that the Flipping Qubits 51 process corresponds to injecting the vacuum into the crystal (see Eq. (24)). The theoretical ratio between the corresponding probabilities is 2. In our experiment, we have found the ratio to be R* = (1.66 ± 0.20). This corresponds to a measured value of the fidelity of the U-NOT apparatus: F* = (0.623 ± 0.025) to be compared with the theoretical value: F — 2/3 = 0.666. Note that the height of the central peak does not decrease towards zero for large Z's. This effect is due to the finite time-resolution of our four-coincidence electronic apparatus, which is in the nanosecond range. It would totally disappear if the resolution could be pushed into the subpicosepond orange, Le4 of the order of the .time duration of the OPA pump and injection pulses. By taking a, little time to think, it can be easily found that the spurious out of resonance plateau of the central peak should indeed reproduce the size of the «noise» condition measured on the mode k?. As we can see, this is indeed verified by the experiment.

CONCLUSIONS

In summary, we have realized and investigated experimentally the first quantum machine which performs the best-possible approximation to an anti-unitary quantum operation. In particular, we have realized the universal spin flipping of a qubit (U-NOT gate). The universality of the performed operation is of the paramount importance since in general the fidelity of the flipping should not depend on the state of the input qubit. We have achieved almost optimal fidelity of the gate which is determined by the complete positivity of quantum mechanics. Acknowledgements. We thank Mark Hillery and Reinhard Werner for many discussions and collaboration on theoretical aspect of the U-NOT gate. This work was supported in part by the European Union projects ATESIST, EQUIP and QUBITS. V. B. would like to thank the SFI for the E. T. S. Walton award.

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20. De Martini F., Di Giuseppe G., Padua S. II Phys. Rev. Lett. 2001. V. 87. P. 150401. i XJ0300182 Письма в ЭЧАЯ. 2003. № Ц116] Particles and Nuclei, Letters. 2003. No. 1 [ТIS]

QUANTUM ZENO EFFECT FOR W-LEVEL FRIEDRICHS MODEL I.Antonioua, E.Karpov", G.Pronkoa'b, E. Yarevskya>°

" International Solvay Institutes for Physics and Chemistry, Brussels 6 Institute for High Energy Physics, Protvino, Russia c Laboratory of Complex Systems Theory, Institute for Physics, St. Petersburg State University, St. Petersburg, Russia

We study the short-time behaviour of the survival probability in the framework of the TV-level Friedrichs model. We show that depending on initial conditions the decay can be considerably slowed down or even stopped. By choosing proper parameters of the system, the Zeno time can also be considerably extended.

INTRODUCTION

Since the very beginning of the quantum mechanics, the measurement process has been a most fundamental issue. The main characteristic feature of the quantum measurement is that the measurement changes the dynamical evolution. This is the main difference of the quantum measurement compared to its classical analogue. On this framework, Misra and Sudarshan pointed out [1] that repeated measurements can prevent an unstable system from decaying (the quantum Zeno effect, QZE). The QZE has been discussed for many physical systems including atomic physics [2,3] and mesoscopic physics [4, 5], and has been even proposed as a way to control decoherence for effective quantum computations [6]. Recently, however, it has been found [7,8] that under some conditions the repeated observations could speed up the decay of the quantum system (the quantum anti-Zeno effect). The anti-Zeno effect has been further analyzed in [9-11]. While there exist experiments [12,13] demonstrating the perturbed evolution of a coherent dynamics, the demonstration of the QZE for an unstable system with exponential decay, as originally proposed in [1], has long been an open question. Only recently, both Zeno and anti-Zeno effects have been observed in the experiment {14]. In order to analyze the short-time behaviour of an unstable system, we use here the Friedrichs model [15], which is very appropriate for the discussion of the particle decay and for the description of dressed unstable states [16,17]. The analytical structure of the ЛГ-level Friedrichs model has been widely discussed, see, e.g., [18,19] and references therein. 54 Antoniou I. et al.

1. JV-LEVEL FRIEDRICHS MODEL

The Hamiltonian of the Friedrichs model [15] generalized to N level is

Я = Яо + XV, (1) where N I fc=i

fe=l w

Here, |fc) represent states of the discrete spectrum with the energy uk, wk > 0. The vectors |w) represent states of the continuous spectrum with the energy w, fk(u) are the form factors for the transitions between the discrete and the continuous spectrum, and A is the coupling parameter. The vacuum energy is chosen to be zero. The states \k} and \u>) form a complete orthonormal basis:

(k\k') = 5kk>, (ш\ш')=6(ш-ш'), H*) = 0, k,k' = l,...,N, (3)

^f4 kk+Jduw Jdwu ш -I,

where 5kk' is the Kronecker symbol; 6(u> — ш') is Dirac's delta function, and / is the unity operator. The Hamiltonian Я0 has the continuous spectrum on the interval [0, oo) and the discrete spectrum wi,..., u>k embedded in the continuous spectrum. As the interaction XV is switched on, the eigenstates \k) become resonances of Я as in the case of the one-level Friedrichs model [15]. The total evolution leads to the decay of an initial unstable state

<Ф|Ф) = 1. (5)

Decay is described by the survival probability p(t) to find,, after time t, the initial state evolving according to the evolution exp (—iHt) in the same state [3]:

p(t) = |{Ф|е~гН*|Ф)|2 = |A(t)|2, (6) where A(t) is the survival amplitude. The survival amplitude can be explicitly expressed in terms of the form factors /fc(o>) [19]. In order to calculate the short-time behaviour for the system (2), we will use the Taylor expansion of the survival probability. We shall assume here the existence of all necessary matrix elements, and denote {•} = (Ф| • |Ф). Then we find

0(t6) = 1 - + + 0(t6). (7) Quantum Zeno Effect for N -Level Friedrichs Model 55

The expressions for the times ta and tb can be deduced using the special structure of the potential V (2): / \ 2 - E

7 + ^ E i"*i2*4 - E м2"* Е 6 \ fc / k k k

+ А2Л2 (^ E Ы2"2 + 1 Дз - I Й2 E M2**) + Л"Л4 + R ' (9) where

ifc (ID

ifc

ifc Here, where the dimensionless form factor Д(х) is expressed as

fk(x) = -=Д(Лх), and the parameter Л has the dimension of energy.

2. ZENO EFFECT AND ZENO TIME

The probability that the state Ф after N equally spaced measurements during the time interval [0,T] has not decayed, is given by [1]: pjv(T) = pN(T/N). We are interested in the behaviour of pjv(T) as N —> oo or, equally, when the time interval between the measurements т = T/N goes to zero:

0, when p'(0) = -oo, e~cT, when p'(0) =-c, (14) { I,1 whe1_ n p'(0f/f\\) = 0f\. The results (14) are found in case of continuously ongoing measurements during the entire time interval [О, Т]. Obviously, this is an idealization. In practice, we have a manifestation of the 56 Antoniou I. et al.

Zeno effect, if the probability рн(Т) increases as the time interval т between measurements decreases. Formula (14) may be accepted as an approximation for a short time interval т < tj,. For longer times we cannot use the Taylor expansion, therefore, Eq. (14) is not valid. As one refers in discussions about the Zeno effect to the Taylor expansion (7) of survival probability for small times, and specifically to the second term, we shall define the Zeno time tz as corresponding to the region where the second term dominates. Hence, as in paper [11], we introduce the Zeno time tz as a natural boundary where the second and third terms have the same amplitude:

§ = ^f, so tz = t%/ta. (15) 'о Ч Expressions (8), (9) include many different parameters and can hardly be analyzed in general case. We consider here few specific representative cases I-III for the physically motivated weak coupling model [11] with

I. The decay of one level. In this case, the only level I is initially occupied: a; = 1, ak = 0 for fc ^ I. Expressions (8), (9) become

1 _ \2A2 0 -J - А Л /-;,F , La

1 1 л2Л2 ( 1 , ,2ciO 1 д, , F _L •*• A2zp2 \ _i_ \4л4 I И*н) , tf= V12 ' " ~ 6i " 12 " J (~^~ 1

It is not surprising that the expressions for ta and if, practically coincide with those for the one-level Friedrichs model [11]. Therefore, the Zeno time tz is also the same:

2 The only difference in the last term (X)m ^hn^mi instead of (Рц) for one-level model) does not influence the results for the moderate number of levels N for A2

= la

I = A'A> (i^(^) - IMF')+ where О, (,РР) = О Quantum Zeno Effect for N-Level Friedrichs Model 57

for some a iff the form factors Д(х) are linearly dependent:

(17)

This case resembles case I when all averages {•} are separated from zero, e.g., the form factors Д(х) are linearly independent. The Zeno time is of the same order of magnitude:

However, the situation may change for special initial conditions Ф. Assuming for the sake of simplicity that the form factors are identical, we have in this approximation

1 Л2Л4 _,. .о 1 /12F° ' v-- p « -jg-^H. *2«ду-р2-, where a = ^ak. (18)

We can now see that the Zeno time tz is independent of the initial conditions. However, both ta and tb increase to the infinity when a goes to zero. This means that the state Ф with a = 0 does not decay, while the relation between ta and tb is unchanged. The same is true when the form factors are not identical but linearly dependent: there also exists a nondecaying state Ф (17). In this sense, the Zeno time definition (15) may not work for the JV-level model and should be modified. We would like to notice that this problem does not exist for the one-level model analyzed in [11]. Ш. TV-level model with one different level. In this case, energy of one level differs from the others: u>k = ш, k = 1, . . . , N — 1, WN = ш + Д, and the form factors are identical: fk(x) = /(я). Then we have for the time fa

^ = А2(|алг|2 - |ajv|4) + A2A2F°|a|2.

We first analyze the time tb on condition (16):

A i\ ., ,o i ,A\ А Л

Both these times and the Zeno time depend on the initial vector. One can easily see that the Zeno time always has a maximum as the function of energy difference A. We can easily estimate the position and the value of this maximum:

АЛ/Г

One can see that the Zeno time is increased by the factor ~ 1/\/Л with respect to the one-level model. When conditions (16) are not satisfied, the prolongation of the Zeno time still takes place. We illustrate this prolongation in the figure using exact expression (9) for the time tb- 58 Antoniou I. et al.

tz, units of \IL 600 г

400 -

200 The Zeno time tz as the function of the energy difference Д for two-level Friedrichs model. The parameters of the model are: Л = 8.498 • 1018 s"1, ш = 1.55 • 1016 s-1, A2 = 6.43 • 1(T9. From above, the curves correspond to the initial condition 0.01 0.02 0.03 Ф: (ai.aa) = (1.-0.6), (1,1), (1,0.1) and (1,0), D, units of L respectively

We use the parameters of the model associated with the hydrogen atom [2]. One can see that for this system the maximum (19) cannot be reached. However, for small Д we can find

(20) therefore, a prolongation always takes place.

CONCLUSION

We have analyzed the short-time behaviour for the ./V-level Friedrichs model. Compared to the one-level model, there exists one important difference: the speed of the decay depends on initial conditions. As a result, there exist situations when the decay is considerably slowed down or even stopped. By choosing proper parameters of the system, the Zeno fine can also be considerably extended. Acknowledgements. We would like to thank Prof. Ну a Prigogine for helpful discussions. This work was supported by the European Commission Project No. IST-1999-11311 (SQID).

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Письма в ЭЧАЯ. 2003. № 1[116] Particles and Nuclei, Letters. 2003. No. 1[116]

UNIVERSAL HYBRID QUANTUM PROCESSORS A. Yu. Vlasov1 FRC7IRH, St. Petersburg, Russia

A quantum processor (the programmable gate array) is a quantum network with a fixed structure. A space of states is represented as tensor product of data and program registers. Different unitary operations with the data register correspond to «loaded» programs without any changing or «tuning» of network itself. Due to such property and undesirability of entanglement between program and data registers, universality of quantum processors is subject of rather strong restrictions. Universal «stochastic» quantum gate arrays were developed by different authors. It was also proved, that «deterministic» quantum processors with finite-dimensional space of states may be universal only in approximate sense. In the present paper it is shown that, using hybrid system with continuous and discrete quantum variables, it is possible to suggest a design of strictly universal quantum processors. It is also shown that «deterministic» limit of specific programmable «stochastic» U(l) gates (probability of success becomes unit for infinite program register), discussed by other authors, may be essentially the same kind of hybrid quantum systems used here.

INTRODUCTION

The quantum programmable gate array [1—4] or quantum processor [5,6] is a quantum circuit with fixed structure. Similarly with usual processor here are data register \D) and program register \P). Different operations u with data are governed by a state of the program; i.e., it may be described as

U:(\P}®\D))^\P')®(uP\D)). (1)

Each register is a quantum system2 and may be represented for particular task using qubits [1-4], qudits [5,6], etc. It can be simply found [1] that Eq. (1) is compatible with unitary quantum evolution, if different states of program register are orthogonal, due to such a requirement number of accessible programs coincides with dimension of Hilbert space, and it produces some challenge for construction of universal quantum processors. Few ways were suggested to avoid such a problem: to use specific «stochastic» design of universal quantum processor [1-3,6], to construct (nonstochastic) quantum processor with possibility to approximate any gate with given precision [2-5] (it is also traditional approach to universality [7-9], sometimes called «universality in approximate sense» [10]). Here is discussed an alternative approach for strictly universal quantum processor — use of continuous quantum variables in program register and discrete ones for data, i.e.,

1 e-mail: [email protected] 2Usually finite-dimensional. Universal Hybrid Quantum Processors 61

hybrid quantum computer [12]. In such a case, number of different programs is infinite, and it provides possibility to construct strictly universal hybrid quantum processor for initial («deterministic») design described by Eq. (1). It is enough to provide procedures for one-qubit rotations with three real parameters together with some finite number of two-gates [10,11]. It is also shown that hybrid quantum gates used in this article can be considered not only as limit of deterministic design [4,5], but also coincide with «deterministic limit» [3] of a special case of programmable (7(1) «stochastic» gates with probability of failure, which tends to zero for infinite program register.

1. CONSTRUCTION OF HYBRID QUANTUM PROCESSORS

In finite-dimensional case, unitary operator U satisfying Eq. (1) can be simply found [4,5]. Let us consider the case with |P') = |P) in Eq. (1). It was already mentioned that states \P) of program register corresponding to different operators UP are orthogonal and, thus, may be chosen as basis. In such a basis, UP is simply a set of matrices numbered by integer index P, and operator U (1) can be written as block-diagonal JVM x NM matrix: \

u2 0 U = (2) 0

with -N x N matrices up, if dimensions of program and data registers are M and N, respectively; M up. (3)

It is conditional quantum dynamics [13]. For quantum computations with qubits M = 2m, ЛГ = 2П. Generalization of hybrid system with program register described by one continuous quantum variable and qubit data register is straightforward. The states of program register may be described as Hubert space of functions on line ф(х). In coordinate representation, a basis is

\q) = S(x-q), <<7 1 V(*)> = tf'te)- (4)

To represent some continuous family of gates U(9) acting on data state, say, phase rotations

0(g) = exp(2iriq{r3), (5) it is possible to write continuous analog of Eq. (3):

"(,)), (6)

= S(x - q)Tl>(q)\u(q}s)dq = V(x)|u(x)s>, (7) 62 VlasovA. Yu. where <8> is omitted because \ф]|s) can be considered as product of scalar function ^(x) on complex vector |s). Finally, (8) It is also convenient to use momentum basis, i. e.,

iPx^(x)dx = ф(р), (9)

(where is Fourier transform of -ф) and operator :

и{р)), (10)

™ф(х - q)\u(p)S)dqdp. (11)

Here is not rewriting U in momentum basis, it is the other operator with property:

U(|p>|e» = |p)|u(p)a>. (12)

Using such an approach with hybrid program register (few continuous variables for different qubit rotations and discrete ones for two-gates like CNOT), it is possible to suggest a design of universal quantum processor with qubits data register. Hilbert space of hybrid system with k continuous and M = 2m discrete quantum variables can be considered as space of CM-valued functions with k variables

-Af F(xi,...,xfc):

For construction of universal processor, it is possible to use three continuous variables1 for each qubit together with discrete variables for control of two-qubit gates. It should be mentioned that it is a rather simplified model. More rigorous consideration for different physical examples may include different functional spaces, distributions, functions localized on discrete set of points, and symbol «/» or scalar product used in formulas above. In such a case it should be defined with necessary care. Due to such a problem, in many works about quantum computations with continuous variables Heisenberg approach and expressions with operators like coordinate Q and momentum P are used [14]. Heisenberg approach may simplify description, but hides some subtleties. For example, in many models variables could hardly be called «continuous», because they may be described as set of natural numbers; i. e., terms «infinite», «nonfinite» perhaps fits better for such quantum variables. Let us consider an example with qubit controlled by above-described continuous variable

= 0(g) Eq. (5). In such a case, it is enough to use operator U in Eq. (8) only at the

'Angular parametrization of 311(2). Universal Hybrid Quantum Processors 63 such as phases. But in such a case in dual space momenta one has only discrete set of values p € Z and both spaces are connected by Fourier transform, it is an example of relation between periodical functions of continuous variable and functions defined on infinite, but discontinuous set Z of integer numbers. Here is an important issue: the commutation relations like t[P, Q] = 1 are not compatible with linear algebra of any finite matrix1, but may be simply satisfied by infinite-dimensional operator algebras, like Schrodinger representation Q = x, P = —id/dx. But here is yet another problem: integer and real numbers are used for representation of infinite quantum variables, but cardinality of the sets are different, cardN = NO, cardR = N. To avoid discussion, related with the cardinality issues, Russell paradox, etc., some formal cardinality KQP of «quantum infinite variables» is used here; i. e., any model of infinite numbers appropriate for introduction of Heisenberg relations. It should be mentioned that the term «hybrid» is also used with other meaning [15]. Formally, it is a different thing, but for discussed strategy for hybrid quantum processors, these two topics are closely connected. Let us discuss it briefly. For realistic design of quantum computers, it is useful to have some language for joint description with more convenient classical microdevices, which could be used as some base for development of quantum processors. Generally, such a task is very difficult (if possible at all) and has variety of different approaches. But there is an especially simple idea that could be applied for the model under consideration. The quantum gates and «wires» may be roughly treated as (pseudo)classical, if only elements of computational basis are accepted in a model as states of system, and also gates may not cause any superposition and directly correspond to set of invertible classical logical gates [9]. Really, such a model is still quantum, but has closer relation with usual classical circuits and, thus, may reduce some difficulties in description of hybrid classical-quantum processor design. It was already discussed in [4,5] that from such point of view program register can be treated as pseudoclassical2. It was designed with finite number of state in program register. Similar procedure without difficulties may be extended for continuous case, but now it corresponds to continuous classical variables; i. e., it is similar either to analogous classical control or to more detailed description of usual microprocessor, when inputs and outputs are not described as abstract zeros and ones, but as real dynamically changed classical continuous signals (fields, currents, laser beams, etc.).

2. COMPARISON WITH LIMIT OF «STOCHASTIC» MODELS

In this paper, design of universal hybrid quantum processor was used that could be considered as some limit of approximately universal «deterministic» quantum processors [4,5], when the size of program register formally becomes unlimited. On the other hand, in [2,3] a design of programmed «stochastic» [7(1) gates is considered with probability of success

'it is simple to show, taking trace of the commutator for D x D matrices: iTr[P, Q] = iTr(PQ) - r(QP) = O^Trl = £>. 2In fact, there was used even more specific design with intermediate register (bus) between program and data. 64 VlasovA. Yu. comming arbitrary close to unit with extension of program register and, thus, such design formally also becomes deterministic for infinite size of program register. Conceptually, the «probabilistic», «stochastic» design of quantum processors [1-3,6, 16] is a rather tricky question, but it is not discussed here in details. For our purposes it is enough to use «stochastic» quantum circuit [2, 3] for applica- m tion of gate 0(g) (5) with probability of success p = 1 - 1/M for size M = 2 of (m-qubits) program register with existance of «deterministic» limit p — 1 for M — > oo [3]. Let us, without plunging into discussion about specific problems of «stochastic» model, consider the limit and show that it is essentially the same programmable phase gates discussed in Sec. 1.

The construction is straightforward. For «encoding transformations» Qa to state of m-qubits program register in [2, 3] a family of states is used

ia/2 ia 2 l*e,m> = ®|^O, ^ere |-M = -4(e |0)+e- / |l)). (13) fc=0 V^ It can be rewritten as1

ia(M-l)/2 M^- ... e гКа т Фа.т) = ^ е~ \К) (М = 2 ) (14) and for а = —1-кр/М with integer p states (14) coincide with usual momentum basis |p) =

|Ф-2тгР/лг,т} (p € Z, 0 < p < M) of M-dimensional Hilbert space. Such elements \p) may be used as M orthogonal basic states of program register in «deterministic» quantum processor [c.f. Eq. (3)],

M-l 0= 5] \P}(P\®ed«P/M)- (15) p=0

The «deterministic» approach uses only the computational basis p, and it prevents the entanglement between program and data registers. Stochastic C/(l) approach [2,3] uses |Фа,т) with arbitrary a and for finite-dimensional case such states are not always orthogonal, but here it is possible not to discuss the issues related with interpretation of quantum measurements used for «probabilistic» calculations of 0(a) for entangled case а ^ 2тгр/М, because for infinite-dimensional case all states |Фа,оо) are orthogonal. So, continuous (infinite) limit of «stochastic» f/(l) programmable gates suggested in [2,3] is essentially2 the same as deterministic hybrid gate like Eq. (12) discussed in Sec. 1. Acknowledgements. Author is grateful to Prof. V. Buzek for interesting discussion and also to all organizers of the Workshop «Quantum physics and communication».

'Here is used «inverted» binary notation for \K), 0 ^ К < M, i.e., |6obib2---) corresponds to К = bo + 2bi + 22b2 + ... Another choice is to save standard binary notation, and to change order of terms in initial tensor product Eq. (13) to opposite one. 2There is some difference, if state of program register changes in [2,3], even if there is no entanglement in continuous limit under consideration. Universal Hybrid Quantum Processors 65

REFERENCES

1. Nielsen M.A., Chuang l.L. Programmable quantum gate arrays // Phys. Rev. Lett. 1997. V.79. P. 321-324.

2. Vidal G., Cirac J. I. Storage of quantum dynamics on quantum states: A quasi-perfect programmable quantum gate, quant-ph/0012067. 2000.

3. Vidal G., Masanes L., Cirac J. I. Storing quantum dynamics in quantum states: stochastic programmable gate for U(\) operations, quant-ph/0102037. 2001.

4. Vlasov A. Yu. Classical programmability is enough for quantum circuits universality in approximate sense, quant-ph/0103119. 2001.

5. Vlasov A.Yu. Universal quantum processors with arbitrary radix n ^ 2. // ICQI. 2001; quant- ph/0103127. 2001.

6. Hillery M, Buzek V., Ziman M. Probabilistic implementation of universal quantum processors. quant-ph/0106088. 2001.

7. Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer // Proc. Royal. Soc. Lond., ser. A. 1985. V. 400. P. 97-117.

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9. Deutsch D., Barenco A., Ekert A. Universality in quantum computation // Proc. Royal. Soc. Lond., ser. A. 1995. V.449. P. 669-677.

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11. Barenco et al. Elementary gates for quantum computation // Phys. Rev. A. 1995. V.52. P. 3457- 3467.

12. Lloyd S. Hybrid quantum computing, quant-ph/0008057. 2000.

13. Barenco A. et al. Conditional quantum dynamics and logic gates // Phys. Rev. Lett. 1995. V.74. P. 4083^086.

14. Lloyd S., Braunstein S. L Quantum computation over continuous variables // Phys. Rev. Lett. 1999. V. 82. P. 1784-1787; quant-ph/981082. 1998.

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16. Hillery M.. Ziman M., Buzek V. Implementation of quantum maps by programmable quantum processors; quant-ph/0205050. 2002. XJ0300184 Письмав ЭЧАЯ. 2003. №1[116] Particles ana Nuclei, Eetters; 2003. No. Ц116]

FORMATION OF THE £[/(3)-POLARIZATION STATES IN ATOM-QUANTUM ELECTROMAGNETIC FIELD SYSTEM UNDER CONDITION OF THE BOSE-EINSTEIN CONDENSATE EXISTENCE A. P. Alodjants1, A. V. Prokhorov1, S.M.Arakelian2 Vladimir State University, Vladimir, Russia

We consider the problem of multipartite entanglement of two-level atom system in Bose-Einstein condensate (ВЕС) and quantized electromagnetic field. The main accent is made on polarization properties of such a system. The St/(3)-polarization states are introduced for three-mode quantum system satisfying both the Gell-Mann symmetry and the isopolarization property for atoms in condensate state. Superstructure of revivals and collapses of quantum state in the system is described by degree of polarization changing in time.

INTRODUCTION

Quantum states of ultracold atoms in Bose-Einstein condensate (ВЕС) are under intensive study in modern quantum and atomic optics [1]. Such a mesoscopic system is of great interest for both basic and applied research. In the latter case, the problem of quantum computing is considered in many papers (e. g., [2]). Entangled multipartite states of condensate atoms and external electromagnetic (EM) field become principal for the case [3]. The effect is very well known for classical optical field when two hyperfine levels of atoms are coupled by the field and the Rabi oscillations for population imbalance and phase difference of two-level system take place and demonstrate the nonlinear behaviour [4]. Another case is the Josephson effect resulting in nonclassical statistics and squeezing in atomic variables for junctions of two conductors at low temperature [5]. In our previous paper [6], we considered the problem of interaction of two-level atoms with quantized EM field under the Jaynes-Cummings (or Dicke) model. In the frameswork of the model, nonclassical effects of collapse and revivals for quantum atomic system and also the squeezing effects in terms of the SU(9) observables (i. e., the spin operators) have been predicted. For some conditions a quantum steady-state excitation can appear. In experiments [7] with ultracold sodium atoms, the light velocity reduction has been demonstrated for passing EM radiation, and such a behaviour can be explained by the effect of electromagnetic field induced transparency for «bright» and «dark» polaritons (i. e., the spin-wave excitations) arising in three-level atomic system [7, 8]. Mathematically, the excitations represent an

'e-mail: [email protected] 2e-mail: [email protected] Formation of the SU (^-Polarization States in Atom-Quantum Electromagnetic Field System 67 example of phenomena under formalism of so-called 5C/(2)-algebra deformation (see, e. g., [9])- On the other hand, complete description of the three-component bosonic system should be presented by using the 5J7(3)-algebra symmetry, being the mathematical apparatus in both quantum chromodynamics and physics of elementary particles [10]. Although such an approach is well known for describing the atom-field interaction [11, 12], the SU(3)- polarization problem is still open now. In the present paper we develop the S£/(3)-polarization approach for the two-level atomic and EM field system interaction.

1. QUANTUM DESCRIPTION OF THE 5C/(3)-POLARIZATION PROBLEM

Let us describe the SU(3) symmetry of Bose system in the Schwinger representation by Hermitian Gell-Mann operators A., (j; = 0, 1, . . . , 8) (cf. [10]):

АО = afai -f a^a-i + 0,30,3, (1)

AI = Oj"a2 -j-ajai, A2 = i (ajai — a]1" 02) , АЗ = afai — a^az, (2)

A4 = a+аз + ajai, A5 = г (ajai - a+аз) , (3)

A6 = ajo3 + aja2, A7 = г (ajo2 - а%а3) , (4)

AS = -7= (a+ai + <4a - 2<4a ) , (5) V3 2 3 where aj(at), j = l; 2, 3 are the bosonic annihilation (creation) operators, respectively, under commutation relation:

[о4; at] = 6i:i , [ai; uj] = 0. (6) Using Eqs. (1), (6), we have the 5(7(3)-algebra commutation relations for A-operators:

[Aj ; Afc] = i£jfcmAm, j, k, т = 1, 2, 3, (7)

[A4; A5] = г (Аз + \/3A8) , [A6; A7] = г (\/ЗА8 - A3) , (8)

[A0;Ai]=0, t = l,...,8, (9) where structural coefficients ejkm are the completely antisymmetric with the values (cf. [12])

£123 = 2, £584 = £678 = Л/3, £l47 — £246 = £257 = £345 = ^516 = £637 = 1. (10)

According to definition (1), the operator АО describes a total number of particles in Bose system. The operators AI^.S can be considered as an isospin of SU(6) subgroup in SU(3) algebra under Gell-Mann symmetry [10]. The operators AI^.S characterize a two-level atomic system for Bose condensate. Another set of the operators \j (j > = 4, . . . , 8) determines the interaction of atoms and mode of quantum field аз. Now consider the 5f/(3)-polarization problem for three-mode system by analogy with the case discussed above (see, e.g., [13]). 68 Alodjants A. P., Prokhorov A. V., Arakelian S. M.

Let us introduce a unit «vector of polarization» e: ea = ejai + GZU^ + 6303, (11) where a is the annihilation operator for a total three-mode system; BJ(J = 1,2,3) are the orthogonal basic vectors under the condition

(12)

Expression (11) can be represented in the form

a = elai + e^at + (13) where e = e*ej. According to the SU(3) symmetry (see, e.g., [12]), we can rewrite Eq. (13) in terms of the four phase parameters в, ф, fa, fa:

e\ = e1 siinn G sin ф, e% = e**2 sin 0 cos ф, 7Г (14) e3 = cos в, 0 < 8, ф < -, 0 < fa>2 < 2тг.

A simple geometric interpretation of introduced quantities is presented in Fig. 1. In fact, the phase parameters в, ф, fa, fa as well as the operators Oj determine completely the

Fig. 1. 3D geometric presentation of the SU(S) polarization for three-level bosonic system. The angles ф, в and relative phase V>i,2 determine the parameterization of polarization state of the system

«polarization» states of the Sf7(3)-symmetry system in a Hilbert space. The polarization states of atomic system can be associated with isopolarization states of the 517(6) subgroup. Thus, the degree of isopolarization looks like (cf. [13])

,!/2 (15) Formation of the SU (^-Polarization States in Atom-Quantum Electromagnetic Field System 69 and we can introduce the parameter for total system

1/2

being independent of the P\p parameter. The numerical factor \/3/2 in Eq. (16) arises due to normalization condition for the P quantity. For absolutely polarized state of the system, we have

(17)

2. QUANTUM POLARIZATION IN THE ВЕС ENTANGLED STATE

For the system including the two-level ultracold atoms and the quantum EM field (described by the operators a3, aj), we have the Hamiltonian [6]

H = Hf + ЯВЕС + ЯОРТ, (18) where

Hf = fiw/ajoa, (19)

2 ЯВЕС = fftt ( wiawiaf ai + w2ajo2 + -7ia?" af + -£1ъа%*а\ J , (20)

ЯОРТ = Uk (a3a2ai + a* 0,20,3) (21) and rotating wave approximation and lossless cavity taken into account. The operators oi(of ), 03(03") describe an annihilation (creation) of the atoms in condensate states being induced by two hyperfine levels «1» and «2», respectively. For «small» number of particles (N и 200, see, e.g., [5]) that is exactly the case, we can neglect the motion of the atoms. The parameters 71,2 are proportional to the scattering length, and characterize the atomic collisions in the Born approximation. The Hamiltonian ЯОРТ (see (21)) describes a direct coupling between the atom levels arising due to quantum EM modes. The Hamiltonian Я/ (see (19)) corresponds to the energy of the free EM field. Let us characterize the entangled state determined by a single-photon state and atoms and described by the following ansatz of the state vector of the system in Schrodinger representation:

№(t))=Y/{PP(t)\p)l\N-p),\l}3 + Dp(t)\p + l)1\N-p-l)2\())3}, (22) p=0

where Pp (t) and Dp (t) are the unknown coefficients being subject of calculating by variational method. Physically, the Pp (t) amplitudes determine the probability to find the number (N — p) and p atoms in the lower and upper level of the condensate, correspondingly. 70 Alodjants A. P., Prokhorov A. V., Arakelian S. M.

The coefficient Dp (t) characterizes the probability of a single-photon state absorption in the condensate. The following normalization condition takes place:

2 2 (Ф (t) | Ф (t)) = £ (\PP (t)| + \DP (t)\ ) = 1. (23) p=0

For in-+itial state, we have

+ N I* (0)) = == К + b ) |0)12 |1>3 (24) in the limit of a large number of atoms (N » 1) (cf. [3]).

Using the variational method to obtain the coefficients Pp (t) and Dp (t), we have the equations

ДЦ.) _ <25)

XD = "2 (N - p - 1) + ал (p +1) + -72 (ЛГ - p - 1) (JV - p - 2) + -7lp (p + 1), (26)

Stationary solutions of the equations (25) reduce to the form

iwat st PP(t) = QPe- Up, Dp(t) = Qpe-^ UD, (27) •//vT where 015 is the frequency for the evolution behaviour in time; O0 = — VP ,/2"(N-p)\pl Coefficients Up, UD do not depend on time t and characterize the initial distribution of the atoms in condensate. In particular, we have

<28) where 5 = %£> — xo = A^ — 72 (•'V — P ~ 1) + 7ip is the phase difference describing the «self-interaction» effects for the atom/field modes; Дш = uj\ —w-z—Uf is the phase retardation. Formation of the SU(3)-Polarization States in Atom-Quantum Electromagnetic Field System 71

The dispersion relation associated with solutions (27) determines the two frequency branches fli,2 for quantum excitations (i. e., spin waves) in such a system:

Oi,a = (29)

In optical region, the coupling for two atomic levels «1» and «2» by frequencies fli,2 results in a «bright» and a «dark» condensate polaritons, correspondingly. The frequencies fii,2 depend on the quantum number p characterizing the population of atomic levels in condensate state (see Fig. 2).

40 i i i i 30 i...--"' S 20 nio 2,---'' a

-• --•"" -8 0 20 40 60 80 100 P Fig. 2. The frequency for «bright» (1) and «dark» (2) excitations as a function of particle number p for s -1 9 1 N = 100, Aw = 0, 71 = 72 = 2.1 • 10 s , fc = 1.7 • 10 s' . The magnitude wi2/27r = 5.5 MHz determines an initial and final energy gap between the two types of excitations

Existence of two excitations, i. e., positive (flj > 0) and negative (fij < 0), associated with two quasiparticles propagating with supersonic velocities results in instabilities for the condensate, and macroscopic effect of formation of shock waves takes place [7]. The parameter wi2 = VS2 + 402 determines an energy gap for the two branches of excitations which have a minimum for p = 0 and p = N — 1. 2 For «dark» polariton in the case when П2 = 0 (Fig. 2), the condition xoXo = 0 (see Eq. (29)) determines the frequency of the EM field driving the atomic system (p is fixed). For the state (22), the expressions (15), (12) reduce to the form

(30)

p j _ 3«aj-ai>(4oa)+#<4os)-; 2= (31)

In the case described by Eqs. (24) and (21), we have „, . „o . ЗЛГ (32) (ЛГ+1)

The first expression in (32) corresponds to polarization of own coherent state for atomic system, but depolarization of total system is determined by a single-photon state. The time evolution behaviour for P, PIP under resonance condition Да; = О is shown in Fig. 3. 72 Alodjants A. P., Prokhorov A. V., Arakelian S. M.

The phenomena described above determine the principal dynamics behaviour of the polarization states for the system, and the collapse and revival effects (see Fig. 3 and cf. [6]) arise. At the same time the depolarization occurs in the system.

0 0.06

Fig. 3. Time dependence of the degree of polarization P (a) and isopolarization PIP (b) for nonstationary regimes. The parameters of the atom-field interaction: N = 15", До> = 0, 71 = 72 = 2.1 • 10s s"1, fc= 1.7-109 s"1

The difference between two parameters, i. e., the degree of isopolarization for atoms in condensate state and the degree of polarization for total system, can be explained by the influence of EM field. In fact, the P behaviour demonstrates the superstructure of the revivals as a result of the atom-field interaction [6].

CONCLUSION

Nonclassical effects of collapse and revival for degree of polarization are obtained in the atom-EM-field system in condensate state. For experimental observation of predicted effects, the interferometric methods are necessary [14], but special analysis should be carried out for the SU(3) interferometer to measure simultaneously all the Gell-Mann observables with some ultimate accuracy. Finally, the problem of the Elliott 5f/(3) symmetry for atomic system [10, 11], which we do not consider in the present paper, demands that the angular momentum be taken into account as well (cf. [15]). This work has been partly supported by the Russian Foundation for Basic Research (Grant No. 01-02-17478) and also carried out in the framework of the Scientific Programme of the Ministry of Industry, Science and Technologies, and of the Ministry of Education of the Russian Federation.

REFERENCES 1. Anderson M.H. et al. II Science. 1995. V.269. P. 189; Bradley C. C. et al. II Phys. Rev. Lett. 1995. V. 75. P. 1687; Davis K. B. et al. II Ibid. P. 3969. Formation of the SU(S)-Polarization States in Atom-Quantum Electromagnetic Field System 73

2. J. Mod. Opt. 2000. V.47, special issue.

3. Cirac J. I. et al. II Phys. Rev. A. 1998. V. 57. P. 1208; Naraschewski M. et al. II Phys. Rev. A. 1996. V.54. P. 3185.

4. Marino I. et al. II Phys. Rev. A. 1997. V.60. P. 487; Raghavan S. et al. II Phys. Rev. A. 1999. V.59. P. 620.

5. Milburn G. et al. II Phys. Rev. A. 1997. V.55. P. 4318.

6. Prokhorov A. V., Alodjants A. P., Arakelian S. M. II Rus. J. Opt. Spectrosc. 2002 (in press).

7. Hau L. N. et al. II Nature, Lett. 1999. V. 397. P. 594.

8. Fleischhauer M., Lukin M. D. quant-ph/0106066. 2001.

9. Delgado J. et al. II Phys. Rev. A. 2001. V.63. P. 063801.

10. Huang K. Quarks, Leptons and Gauge Fields. Singapore: World Scientific Publishing, 1992.

11. Hioe F. T. II Phys. Rev. A. 1983. V.28. P. 879; Ни G., Aravind P. K. II J. Opt. Soc. Am. B. 1989. V.6. P. 1757.

12. Khanna G. et al. II Annals of Phys. 1997. V. 253. P. 55.

13. Gantsog Ts., Tanas R. II J. Mod. Opt. 1991. V.38. P. 1537.

14. Rauschenbeutel A. et al. II Phys. Rev. Lett. 1999. V. 83. P. 5166.

15. Law C.K., Pu H., Bigelow N.P. II Phys. Rev. Lett. 1998. V.81. P. 5258. XJ0300185 Письма в ЭЧАЯ. 2003. №1[116] Particles ana iNuciei, Letters, гиш. No. HII6J

PERIODICAL SEQUENCES (TRAJECTORIES) OF OUTCOMES OF ATOMIC STATE MEASUREMENT ON EXIT FROM THE MICROMASER CAVITY

G. P. Miroshnichenko1 Institute of Fine Mechanics and Optics, St. Petersburg, Russia

A method of recurrence relations, developed in the paper [1], is applied for simulation of a random sequence (trajectory) of detector clicks that controls a state of atoms leaving the one-atom micromaser. The random sequence of relative frequencies of detector clicks for the fixed time interval was calculated. The change of the frequencies along the trajectory is chaotical and close to the mean level, which is independent of time. The character peculiarities of such a sequence were observed experimentally in the paper [2] where it was shown that the mean levels may alternate each other stepwise, but their value was reproduced. Hence, the mean levels is the observable quantity. A method of calculation of a stationary reduced density matrix of the field pst of a subensemble, which is determined by the given mean relative frequencies of the detector clicks, is proposed here. The main idea is that the random sequence could be approximated by the simpler periodical one. The micromaser would approach pst along the preset (imposed) periodical trajectory. A problem to find pst is reduced to the eigenvalue problem for the evolution operator during the period. The latter is solved by means of the linearization procedure applied to the evolution operator of one period. Then the matrix pst is obtained as the solution of the inverse problem that is a reconstruction of the field statistics from the statistics of the random trajectory. We notice an important property of the periodical trajectories: probabilities to observe leaving atom in the definite state at the end of every period are close to the relative frequencies calculated with help of statistical processing of the detector clicks.

1. A ONE-ATOM MICROMASER MODEL

At the present time, a lot of fundamental aspects of quantum mechanics are tested directly and applied in practice due to a high level of experimental achievements in atomic physics and quantum optics. Fast developing fields like quantum information processing, quantum computation and communication connected very close to a concept of entangled states of quantum-mechanical systems. An operation of quantum informational and communicational schemes is based on a realization of quantum measurement in informational elements which results in state reduction. A role of qubits is played by individual trapped atoms or ions and by single photons. Such a kind of experiments when a measurement is carried out under specially prepared individual quantum system solves general theoretic questions of interpretation of basis concepts in quantum theory [3]. In this connection, a so-called micromaser possesses interesting potentials [4]. Here, individual quantum system — a chosen mode of the microwave resonator interacts (and as a

'e-mail: [email protected], [email protected] Periodical Sequences (Trajectories) of Outcomes of Atomic State Measurement 75

result entangles) during each period of operation with another quantum system — Rydberg atom which is initially prepared in the excited (maser) energy level. When the atom exits the cavity (and the systems stop interacting but stay in an entangled state), the atom undergoes quantum measurement of its energy. The measurement allows concluding indirectly about state of the quantum mode at the moment of detection. The measurement process is repeated for each atom flyby, resulting in peculiar dynamics of the field mode due to so-called back- action of the detector on a detected quantum object. The one-atom micromaser model is based on the Hamiltonian of Jaynes-Cummings H [5] and developed in the paper [4]. Here,

Я = ша!а + w0S3 + g(a*S- + aS+).

In the expression, operators S+,S-,Ss belong to the group 517(2), Bose operators а*,а, of creation and annihilation of quanta of the field mode. Now let us consider the dynamics of the main diagonal — distribution law of the photons number probability in Fock's basis — in basis of the photon number operator. Denote by a symbol p(l) a vector of the main diagonal of the reduced density matrix of the field (MDRDM) in the beginning of Ith period satisfying the normalization condition (1) n=0

Here, n is Fock's number; Tr/ is the field trace. Let us denote by S(£,t) an operator of the evolution p(l) over a micromaser operation cycle of time duration t. We introduced a notation of the random variable £ taking a zero value £ = 0, if the detector measured an atom in the lower state, and the value £ = 1, if the atom was detected in the upper state, and £ = 2, if the detector did not click because of nonideal efficiency. According to paper [4], this operator could be represented by the evolution operator £>(£) of the MDRDM, describing the atom-filed interaction in interval т, and by an operator W(T) of the MDRDM evolution in the relaxation process (interval T). Here,

, T) = W(T)D($, W(T) = exp (T7 L), (2)

D(2) = (1 - eo)QO + (1 - ei)Ql, (3) QO = a* (sin2 (gr^tfa + 1 ) / \fa\a + 1 ) ,

(4)

f t / L = -(2nb + 1) a a - nb + (nb + 1) aa& + nb a v aal. (5) In the relations, 7 is decay constant; пь — average Plank's number of photons in the resonator; £i> £o — efficiencies of detection of atoms in the upper (lower) states. We keep a notation of the photons creation and annihilation operators acting on the main diagonal in the case. The MDRDM in the end of /th period (in the beginning of (I + l)th) is equal to 76 Miroshnichenko G. P.

100 80 CT1 60 40 20 80 CT2 60 40 20 50 100 150 t,s

Fig. 1. The results of O.Benson, G.Raithel, and H. Walther experiment [2]

Here, £i denotes the value of the random variable £ found on lui cycle. Below the dependence of the random variable on the number of the micromaser, £j, would be called the trajectory. Let us introduce a notation of a probability to detect the atom in the lower state immediately before measurement on (/ + l)th cycle: (7) correspondingly, for the upper state ai = Tr/£>(l) p(l).

The relative frequencies of detector clicks for the fixed time interval Ata are defined as Pi (for atoms in the upper state) and po (in the lower): po= R). (8) Here, R is injection rate; k — number of atoms detected in the lower state; т — number of atoms detected in the upper state from the whole number of atoms Atav.R passed during averaging time Atav. In the experiment [6] (Fig. 1), these probabilities were determined from the sample of about 300 passed atoms (Atav w 0.1 s) by direct accounting of favourable outcomes. The experiment was carried out with the following parameters: gr w 0.92, A/ex = Д/7 яз 200, R и 3300 s-1. The outcomes are obtained by detector with efficiency of order e\ = EQ и 0.35. Our results of simulation are presented in Fig. 2. Here, the abscissa axis corresponds to the moment of time following one by one with interval At taken as equal to 1. We assumed that the interaction time т fulfills a condition т <^ At. Number of intervals At between successive atoms is random and is generated in our model according to Poisson distribution

Here, P(j) is the waiting probability of j elementary intervals between successive atoms or of the time Т = jAt. Particularly, Fig. 2 is plotted for 7? At = 1/4. The axis of ordinates corresponds to the relative frequency (8) to detect unexcited atom on the time interval Atav = Ne*/R (Fig. 2). The solid line in Fig. 2 represents average steady (quasistationary) level of the relative frequency pQ, obtained by averaging of the histogram over observation time. Specifically for plotting Fig. 2, we used the following parameters:

Л/ex = 71, Atav = 284, nb = 0.1, gr = 0.92, £0 = £1 = 1/3- (10) Periodical Sequences (Trajectories) of Outcomes of Atomic State Measurement 77

0.30

g 0.25

0.20 Fig. 2. 1 — the relative frequency to detect the atom exiting the micromaser in the ground • 0.15 state po, waiting times are random and distrib- 0.10 uted according to Poisson law; 2 (solid horizon- 0.05 tal line) — time averaged relative frequency p~0. The abscissa axis corresponds to time measured 0.00 in units of elementary interval At. Parameters 10000 20000 30000 40000 50000 of the simulation are given in the present paper

As we mentioned above, experimentally measured observable is an average over the time

Atav, relative frequencies p\ and po- It is natural to average over the number of passed atoms

which is larger (or equal) than number Nex(Atg.vR > Nex). In such a case, the analysis could be simplified replacing operators (2) by their average over the distribution (9). Average formulas have a form

ДА* ехр (RAtL/N ) W = ex £ = 0,1,2, (11)

I-(l- ДД<) ехр (RbtL/Nm)'

(12)

(13) Here, / is identical operator. The equation (13) describes the evolution of MDRDM without measurement. The steady state of this equation denotes />(*•). Comparing recurrence relations (12), (13), one can conclude that the evolution process of MDRDM in the nonmeasurement case is linear (13), whereas the atomic state measurement result is unpredictable and the stochastic recurrence relation (12) is nonlinear. In Fig. 3 results of simulation with the same parameters (10) are presented, but the calculations are based on (!!)-(13). As follows from

Fig. 3. The relative frequencies po and p0 to detect the atom exiting the micromaser in the ground state. For simulation we have used the evolution operator W. The average time interval 0.05 between successive atoms is 4Д<. The notations 0.00 and simulation parameters are the same as for 10000 20000 30000 40000 50000 Fig. 2 t= t/&t 78 Miroshnichenko G. P. the comparison of the figures, the statistical characteristics of both figures po are close to each other. The presented comparison gives a possibility to study the model with the help of formulas (11)-(13), without generation of the Poisson variable T, which significantly simplifies our analysis.

2. PECULIAMTIES OF THE EVOLUTION OPERATORS AND THEIR PRODUCTS. PERIODICAL TRAJECTORIES

Let us consider some features of the system dynamics evolving to the stationary state by choosing the period operator as an evolution operator

SL=

Here, Д is a symbol denoting product of operators. The vector of MDRDM is calculated in the end of each period; the periods are enumerated by integer number p. The equation describing nonlinear dynamics has an analogous form to equation (12), but in the present case this is not a stochastic relation:

In contrast to (6), (12), equation (14) describes the micromaser evolution according to preset (imposed) trajectory. All operators SL have eigenvalues less than 1, and as a consequence, they do not preserve the trace (in contrast to the operator in recurrence relation (13)). Nev- ertheless, due to nonlinearity the relation (14) has a stationary solution. We write the general solution of (14) by applying Fourier method. For this purpose, we expand the initial vector p(0) by eigenvectors of SL:

(15)

Here, CA are the coefficients of the expansion, p\ and Л are eigenvector and eigenvalue of SL, respectively:

= \Px. (16) We denote the maximal eigenvalue in the expansion (15) by A and get the solution of (14) on period p in the. form

(17)

It follows from (17) that the stationary state p of the equation (14) could be any eigenvector of the initial state expansion if its eigenvalue is the largest. So the relation (14) has infinitely

many stationary states. The vector with the maximum possible eigenvalue Amax is the main

stationary state pst:

Pst=PAmax- (18) Periodical Sequences (Trajectories) of Outcomes of Atomic State Measurement 79

As follows from (16), the eigenvector of SL is normalized according to (1) and corresponds to eigenvalue A = Tr/ SL p\. But a value Tr/ SL p\ is equal to conditioned probability that L atoms are found by the detector in states described by the random variable £ taking successive values from a set {£,- , 1 < j; < L} (on condition that the initial state of the field was p\). So the eigenvalue problem (16) for a product of the operators is of special importance. The main stationary state (18) determines (average) probability levels to detect the atom in the lower (upper) state in the end of each period — OQ, a\:

ao = Tr/ D(Q) pst, Si = Tr/ D(l) РЙ. (19)

These average a priori probabilities satisfy an exact relation

00/60 + 01/6! = !. (20)

The parameters of a periodical trajectory are: a period L = k + m + r; a number of atoms in the upper and in the lower states during the period k and m, correspondingly; r atoms are left undetected. Then the average relative frequencies of detector clicks pl and p0 can be evaluated: p\ = m/L, po = k/L. (21) Let us find the main stationary state (18) solving an eigenvalue problem (16) for the operator of the period SL = S(0)*S(l)mS(2)r. (22)

Here, S(£) is defined in (11). Now we want to study a connection between a relative frequencies (21) and their a priori analog (19). In further analysis, we assume the following connection between parameters of the operator (22):

Po/£o + Pi/£i = k/(Le0) + m/(Lei) = 1. (23)

For this purpose, we perform a shift of operators &, a in formulas (3)-(5):

where /Д fj, are numbers, and linearization of the operator (22), keeping its linear and quadratic term with respect to fit, a. After that we find /^,ц from a condition that the linear terms with respect to a*, a become zero in SL. The method of linearization is restricted by the condition gr

but Fillipowicz parameter 0 = g r^/Nex can have arbitrary value. Let us consider another simpler case of equal detector efficiencies eo = £o = 6. We have

m + r), x = Po/£ = */(* + "»), v = ^n/Nm, (25)

-x)(^-X), (26) 80 Miroshnichenko G. P.

(27)

We give simplified expressions of functions F(v) and G(v) that are correct for small пь 1. Zeros of function F(v) determine values of parameter v for which linear terms of а^, a in SL become zero. Function F(v) has many zeros. Each one is connected to a series of close to equidistant eigenvalues, and distances between those eigenvalues depend on the series number. In each series there is an eigenvector of the ground state. The vector is very localized in Fock's space, its maximum lies close to the corresponding zero of function F(v], and the eigenvalue is determined by value of the function G(v] taken in the chosen zero. Functions F(v) and G(y) are connected by important relation: locations of maxima of the function G(v) are determined by zeros of the function F(v). The analysis of quadratic terms shows that the stable states are localized in the so-called trapping regions [6], whose location is defined by inequality

(7г(0.5 + Я/9)2^1/<(7г(1+Я/9)2, .7 = 1,2... (28) Here, integer j is a number of the trapping region. The trapping regions have an important property: their location (28) does not depend on the measurements result (of numbers k,m,r) and on the detector efficiencies. We want to point out that functions F(v) and G(v) do not depend on the order of efficients in the period operator (22), and therefore they determine properties of the beam (m + n + r)!/(m!n!r!) of periodical trajectories. There is dependence of coefficients of the quadratic form of operators at, a on permutation of the efficients. To justify formulas (23), (25), we note that for such a chosen connection the equation F(i>) = 0 has a root v = v satisfying two approximate conditions: sin2(9-\/^) ~ X> v K X-

It is a root defining the main steady state pst corresponding to the maximal eigenvalue of SL that we were looking for. Hence the ground state is a very localized vector, according to 2 formulas (19), (3) in the approximation of linearization OQ = Tr/ £>(0)pst ~ £sin (9л/?) , then we have a connection of (average) probability levels to detect the atom in the lower state in the end of each period and relative frequencies of detector clicks pQ и а0.

3. DISCUSSION

The method developed in our paper is based on assumption that the random trajectory £i could be approximated in quasistationary conditions by the most simple periodical trajectory whose parameters k,m,r should be taken from the experiment. Then our search of the distribution law of photon numbers in the resonator MDRDM takes a form of calculating eigenvector of the period operator (22) corresponding to the maximal eigenvalue. Our estimate under condition (24) shows that equation of motion (14) in definite periodical trajectory ф is self-consistent that is, it leads the quantum system to evolve into the state when, in order to detect the atom in the lower (upper) state in the end of each period oo and 01, (average) probability levels become close to mean relative frequencies of Periodical Sequences (Trajectories) of Outcomes of Atomic State Measurement 81

Probability G(v) 0.35 0.25

0.20

0.15

0.10

0.05

0.0 0.2 0.4 0.6 0.8 1.0 v 10 20 30 40 Proton number л Fig. 5. Function G(i/) for parameters k = 1, m = 1, x — 0-5. 1 — function G(v) for Fig. 4. Eigenvectors for the largest eigenvalue of r = 1, e = 1; 2 — function G(v) • 10 for the period operator SL. 1 — p t for k = 1, т = 4, s r = 2, e = 0.5; 3 — function G(v) • 100 for г = 0; 2 — pst for fc = 1, m = 1, r = 0; 3 — 2 ( г = 20, s = 0.091; 4 — cos (Q^/W. Pet for fc = 1, m = 1, r = 20; 4 — p ">; 5 — 2 The simulation parameters are the same as со8 (рт-у/п)/8. The simulation parameters are the for Fig. 2 same as for Fig. 2

detector clicks po and p\ which determine the definite trajectory. A statistical treatment of

the random sequence shown in Figs. 2,3 gives the following results: pQ = k/L и 0.162, pi = m/L « 0.163. The result agrees with (23): p0 + Pi « e = 1/3. To get the minimal period, one can take k = 1, m = 1, % = k/(k + m) = 1/2. Then the period L = 1/po ~ 6. We determine parameters of the periodical trajectory found by detector with efficiency e = 1/3: k'= 1, m = 1, r = 4. In Fig. 4 we plotted MDRDM />(es) (13) and pst (18) for k = 1, m = 1,4; r = 0,20. As one can see from Fig. 4, p^ almost coincide with pst that have been predicted since for chosen parameters of simulation //") have only one peak. As follows from the obtained results, a problem of experimenter is to determine parameter of the trajectory: its minimal period and number of clicks for the upper and lower states of atoms. From Fig. 5 we see that it is not important what type of detector — with small or large efficiency — is used to find the parameters. Indeed, location of the maxima of function G(v) in Fig. 5 depends weakly on efficiency. Obviously, high-efficiency detectors are more preferable, since they allow observation of trajectories with small time of stay in quasistationary state. In conclusion we want to note once more that the results presented in the paper are correct in the approximation of the linearization method. The main states of eigenvalue problem (16) localized in the regions do not overlap and weakly interact. It is the case when a probability of any part of the trajectory weakly depends on order of events and is determined by number of happened events. The analysis becomes significantly complicated when condition (24) breaks down. The system has unstable behaviour; mean relative frequencies may have two (or more) competitive values. In case the trajectory also becomes unstable, and abrupt jumps from one state to another are possible. The jumps happen in the time interval much smaller than the time of living in quasistationary state. Such frustration results from events with very small prob- 82 Miroshnichenko G. P. ability — critical fluctuation in the sequence of clicks. The peculiarities of the critical fluctuation and the statistics of cavity mode photons are planned to be studied in succeeding works.

REFERENCES

1. Meystre P., Wright E.M. II Phys. Rev. A. 1988. V.37. P. 2524.

2. Benson O., Raithel G., Walther H. II Phys. Rev. Lett. 1994. V.72. P. 3506.

3. RaimondJ.M., Brune M., Haroche S. II Rev. Mod. Phys. 2001. V.73. P. 565.

4. Filipowicz P., Javanainen L, Meystre P. II Phys. Rev. A. 1986. V.34. P. 3077.

5. Jaynes E. Т., Cummings F. W. II Proc. IEEE. 1963. V. 51. P. 89.

6. Vadeiko 1. P. et al. II Optics and Spectroscopy. 2000. V. 89. P. 300; Vadeiko 1. P. et al. II JETP Lett. 2000. V. 72. P. 449. I XJ0300186

Письма в ЭЧАЯ. 2003. № 1[116] Particles and Nuclei, Letters. 20бЗ. No. 1[116]

AN ALGEBRAIC METHOD TO SOLVE THE TAVIS-CUMMINGS PROBLEM /. P. Vadeikoa>\ G. P. Miroshnichenko °'2, A. V. Rybin6'3, J. TimonenM " Department of Mathematics, Saint-Petersburg Institute of Fine Mechanics and Optics, St. Petersburg, Russia 6 Department of Physics, University of Jyvaskyla, Jyvaskyla, Finland

We study cooperative behaviour of the system of two-level atoms coupled to a single mode of the electromagnetic field in the resonator. We have developed a general procedure allowing one to rewrite a polynomial deformed SU(2) algebra in terms of another polynomial deformation. Using these methods, we have constructed a perturbation series for the Tavis-Cummings Hamiltonian and diagonalized it in the third order. Based on the zero-order Hamiltonian we calculate an intensity of spontaneous emission of Л/" two-level atoms inside a cavity, which are in thermal equilibrium with the reservoir. The atom-atom correlation determining superradiance in the system is analyzed.

INTRODUCTION

We consider here the collective behaviour of the system of Л/" two-level atoms coupled to a single mode of the electromagnetic field in a resonator. The useful form for the atom-field interaction was proposed in the rotating wave approximation (RWA) by Tavis and Cummings [1]. In their model jV identical two-level atoms interact via dipole coupling with a single-mode quantized radiation field at resonance, so that the Hamiltonian is given by

H = HQ + V, HQ = и>а*а + w0 ,

Here, w is a frequency of the electromagnetic field and WQ is the level splitting of the two- level atoms. The operators £3, S± are collective spin variables of jV two-level atoms. These operators are defined as

M JV s3 = £oi, s± = £4> (2) 3=1 3=1

'e-mail: [email protected] 2e-mail: [email protected] 3e-mail: [email protected] 4e-mail: [email protected] 84 Vadeiko I. P. et al. where CT'S are Pauli matrices. They satisfy the SC/(2) algebra, o, a^ are the annihilation and creation operators of the field. Due to historical reasons, the Tavis-Cummings (TC) model is often called the Dicke model [2]. We concentrate here on the case of exact resonance, i. e., w = WQ. In this case, the system exhibits a most interesting collective behaviour. For simplicity, time will be measured in units of the coupling constant g; i. e., we assume in the following that g = 1. The Hamiltonian (1) belongs to a class of operators that can be expressed in the form:

H = f(A0) + g(A++A-). (3) Here, f(x) is an analytic function of x with real coefficients, while the operators -A± satisfy the commutation relations

[A0,A±} = ±A±. (4) We also assume that the operators Ao,A± must satisfy the conditions

(Ao? = Ao, (A-? = A+. (5) We reformulate the Hamiltonian in terms of an algebra that allows one to diagonalize it in terms of perturbation series with some small parameter which should be introduced. This idea was already used by Holstein and Primakoff [3]. They expressed the generators 83, S± of the SU(2) algebra in terms of boson operators b, tf:

S_ = (S+)t. (6)

Here, r is an index that characterizes the irrep of 517(2). However, in [3] the square root in the transformation (6) was in the end replaced by unity, which amounts to applying the so-called «weak field» approximation ((tfb)

nomial deformation SUn(2) of the Lie algebra 517(2) [5-7]. Numerous physical applications exist for polynomially deformed algebras [3, 8-18]. A particularly interesting, in view of the present problem, application of deformed algebras was developed by Karassiov (see [8] and references therein). The method to be introduced below is an extension of Karassiov's method. We introduce here the notion of a Polynomial Algebra of Excitations (РАЕ). In this algebra, the coefficients of the structure polynomials are с numbers. We derive an exact mapping between isomorphic representations of two arbitrary РАЕ. We formulate then an analytical approach that allows us to expand the Hamiltonian, when expressed in terms of РАЕ, as a perturbation series. For completely symmetric states of atoms our results agree with those reported in [19, 20]. Our formalism provides however a solution of the problem for any value of r, which allows us to discuss new physical effects in the Dicke model. The article is organized as follows. In Section 1, we discuss the irreducible representations of РАЕ and apply the general approach to the Tavis-Cummings model in order to construct An Algebraic Method to Solve the Tavis-Cummings Problem 85

the perturbation theory. We solve eigenvalue problem of the Hamiltonian up to third order. The generalized ЛА-atom quantum Rabi frequency is defined for arbitrary quantum states of the system. In Section 2, we use the zero-order approximation for the TC Hamiltonian to calculate the intensity of spontaneous emission of atoms prepared in the state of thermal equilibrium with the resonator mode. We show that the correlation of the atoms due to interaction with the field gives rise to the enhancement of spontaneous emission as compared to the atoms in the absence of resonator. In conclusion, we discuss possible further applications of the methods developed here.

1. TAVIS-CUMMINGS MODEL

The coefficients of the structure polynomial of a polynomially deformed algebra are usually expressed through the Casimir operators of the algebra. We discuss representations of a special class (РАЕ) of polynomially deformed algebras when the coefficients of the structure polynomial are с numbers. We denote a PAE with a structure polynomial of order к

as UK. Formally UK is defined by three generators A±,Ao. These operators satisfy two basic commutation relations, Eq. (4). As can be readily seen from these commutation relations,

[A0, A+A-] = 0. We can thus assume that

A+A^ =Рк (А,) = со П (Ao - ft-). (7) i=l

Here, pK(x) is a structure polynomial of order к, whose coefficients are generally complex numbers. The terminology is chosen in analogy to the structure functions of quantum algebras ((/-deformed algebras) [21], and the structure constants of the linear Lie algebras. The set of к real roots of the structure polynomial is denoted as {ft}?-!- In physical applications the operators A± of Eq. (4) often play the role of creation and annihilation operators of collective

excitations. Therefore, hereafter the algebra UK will be referred to as the polynomial algebra of excitations (PAE) of order к. To begin, we assume, without loss of generality, that CQ = ±1. Indeed, in the case

ICQ | Ф 1, it is always possible to renormalize the generators of UK,

Л±^|соГЛ±, (8)

such that the commutation relations (4) remain intact. The most simple and important example of a PAE of first order, U\, is provided by the well-known Heisenberg-Weyl-Lie algebra, viz. t 6 -->Ab Ь->Л_, tfb->A0, CQ = 1,91=0. (9) Here, 6, &t are the usual boson operators. For the sake of simplicity, in what follows we will denote the generators of U\ by b, tf. The algebra U\ allows us to construct the irrep of any other PAE of higher order к > 1 as a multiple tensor product of U\.

An irrep of PAE is characterized by a set of parameters {k-,k+,d}, where d is a dimension of the invariant subspace and k- is an order of the left and k+ of the right roots defining the corresponding irreducible representation (see Ref. [22]). We will denote such 86 Vadeiko I. P. el al.

irrep by R(k-,k+,d). For instance, Л(1,0, oo) means the representation of U\, while the irrepof Sr is Д(1, l,2r + 1).

An isomorphism between irreps of UK. and IA'K, that belong to the same class jR(fc_, fc+, d) is given by

A П ( 'o (10) -A'+, A- = A'_ \ or by

П (9J' П W' + «J - Л - t=i (11)

\ t'=l

In Eqs. (10), (11) the operator argument of the square root function should be taken after identical multipliers in the nominator and denominator are cancelled. The roots qj and

a* a - S M_=o5+, M+ = ot5_, 3 (12)

It is plain that these generators satisfy the commutation relations (4). The generators of the

algebra M0,M± commute with the operators

а а з г, з 2 ( + +).

Hereafter we use the same notation M both for the Casimir operator and its eigenvalue, if no confusion arises. We show below that the eigenvalues M, r(r + 1) of the operators of Eq. (13) parameterize the РАЕ in question. We thus denote this РАЕ as Мм,г- The structure polynomial of Мм,г can be expressed in the form

2 p3 (Mo) = M+M_ = 0*0 (S - 5| - S3) = a*a (r - 53) (r + S3 + 1) = Л, М-Л (,, M-3rx = - Mo + —— An Algebraic Method to Solve the Tavis-Cummings Problem 87

The parameters of this structure polynomial are M — r M — r M — r co = -l, qi = —, 92—2 r, q3 = — l-r + 1. (15)

We turn next to the description of finite dimensional irrep of Мм,г- In physical applications the parameter r has the meaning of collective Dicke index. This index runs from 1 _ (-iyv tf e(M) — —-— to — with unit steps, while M can be any natural number including zero. Thus, 93 is the biggest positive root of first order. If M < 2r, then qi > q2; if M > 2r then 91 < 92; the case M = 2r => q\ = 92 corresponds to a root of second order. The different values of M and r define different algebras Мм,г. whose single physical finite dimensional representation we will call a zone. The case M <2r corresponds to nearby zones. The two largest roots are q\ and 93 and the irrep has the type R(l, 1, M + 1). Consequently, the well-known weak-field limit corresponds to nearby zones. The case M > 2r corresponds to remote zones. The two largest roots are 92 and 93, and the corresponding irrep is of the type Д(1,1,2г + 1). Notice that the region M » 2r is usually called the strong-field limit.

In the special case 2r = M, referred to intermediate zone, the algebra Мм,г possesses an irrep of the type R(2,1, IT + 1). It is the only irreducible representation that principally differs from all the others. The simplest РАЕ with irrep of the type R(l,l,d) is Sf (we use here f to distinguish it from the (physical) collective index r). It would be convenient to solve the eigenvalue problem in terms of the simplest algebra Sf.

To begin with, we consider the transformation of Мм,г to Sf for the case of remote zones. The dimension of a remote zone is 2r + 1, and the algebra Sf should be characterized by f = r. The finite dimensional irrep of Sf is isomorphic to the corresponding irreducible representation of the algebra of Eq. (2) of the atomic subsystem. For the pivotal root QJ, we choose the largest root that bounds the irrep of MM,г from the right (the root 93 in (15)), while as the root 9'-, we take the root q\ = —r of Sr. Applying the mapping (11), we obtain

(16)

M_ = J (м-г + 1-S3)S+.

The spectrum {m} of the operator 5з belongs to the region — r < rh < r, consequently, the argument of the square root function in Eq. (16) is positive in the remote zones (M — r > г). The relations (16) express the generators of algebra Мм,г as analytic function of the generators of the Sr algebra. Thus, they allow us to approximate the more complex algebra Мм,г of third order by a simpler algebra of second order. We turn now to the nearby zones M < 2r. For this region, the mapping of the algebra MM,T to the algebra Sf is realized through procedure similar to that described above for remote zones. Notice that the dimension of nearby zones is d = qs — q\ — M + 1, and therefore f = M/2. Applying Eq. (11), we obtain

r - //4r — M - ^ M0=--53, M+ = S-J —— + l-53 , M+ = (M_)t. (17) 88 Vadeiko I. P. et al.

Since all the eigenvalues of the operator 83 belong to the interval — f to f, the argument of the square root function does not have zero eigenvalues in the nearby zones. Let us introduce an A/"-atom generalization for arbitrary values r and M of the well-known quantum Rabi frequency such that

-, M >2r (18) /4Г-М + 1 „, 2л/ , M<2r

For r = jV/2, our definition agrees with that used in [19, 20]. Introducing a small parameter а = (1/2^д)\- 2 , we can rewrite the realizations of M± in nearby and remote zones (see Eqs.(16), (17)) in the form

M_=M|. (19)

The diagonalization problem for the operator in Eq. (1) can now be solved in each zone by means of perturbation theory with respect to the small parameter a. One can show that the eigenvalues of the argument of the square root function in Eq. (19) are less than unity. Hence, we can expand the square with respect to a and find thereby the interaction part of the Hamiltonian. In the interaction representation, the Hamiltonian coincides with V, we only need to diagonalize the latter. Up to third order in a, we find that

where the V^ are terms of nth order in a and are given in the Appendix 2. In the Appendix, we also show the unitary transformations Uk, k = 0,1,2,3, which bring the interaction operator into diagonal form

1 V = UVU- = Од5з/1 + (-V [55|-3f(f + l) + l] j, U = U3U2UiU0. (20)

The spectrum of the operator V as given by Eq. (20) agrees with the results of [19, 20] for the symmetric states of the atoms. We compared the third-order solution of Eq. (20) with the exact numerical diagonalization of V and found that the result (20) is very accurate, especially for increasing values of \M — 2r\. The results of this comparison are shown in Fig. 1. Figure 2 compares the energies calculated numerically and in accordance with the analytical solution of Eq. (20). In the intermediate region of M, the curves for nearby and remote zones overlap and coincide, thus providing still satisfactory correspondence to the exact solution. However evidently the expansion for the remote zone breaks down in the nearby zone and vice versa. This means that the classification of zones introduced in this paper is indeed adequate. An Algebraic Method to Solve the Tavis-Cummings Problem 89

0 10 20 30 40 SO 60 70 80 90 100 M Fig. 1. Deviation of the eigenvalues of V from their numerical values in zero (curve 1) and second (curve 2) order in a, for r = 6

20-, 18- 16-

12- » ' 8- 6-

0123456789 10 11 12 M Fig. 2. The dependence on M of the maximal eigenvalue of V for г = 3, as calculated numerically and from Eq. (20): ] — remote zones; 2 — neaby zones; о — numerical values

2. ENHANCEMENT OF SPONTANEOUS EMISSION IN THE RESONATOR DUE TO COLLECTIVE EFFECTS

In the previous section we developed an algebraic approach to the Tavis-Cummings model. We introduced the operators S± describing collective excitations in the atom-field system. In terms of these operators, we constructed a perturbation series for the Tavis- Cummings Hamiltonian (1). The derived perturbation series gives us a tool to distinguish and classify cooperative (multiparticle) effects of different orders that are involved in calculations of different physical observables characterizing the atom-field system. In this section, we study a contribution of cooperative effects into the rate of spontaneous emission generated by the atom-field system. The atom-field system is assumed to be prepared in the state of thermal equilibrium. This state is described by the canonical Gibbs ensemble with the thermostat temperature T. We demonstrate that nontrivial physical results for the intensity of spontaneous emission of Л/" two-level atoms placed inside the cavity can be already obtained 90 Vadeiko I. P. et al. for the zero-order approximation of the exact Hamiltonian (1). The thermal state is given by

1 Ptherm = -^ exp (21) kT

Here Z is a normalization factor. When calculating the rate of spontaneous emission (or the intensity proportional to this quantity) we merely follow the ideas of Dicke's paper (see, e. g., [2]). According to this theory, the rate of spontaneous emission in the system is proportional to the average of the square of the atomic dipole, viz.,

/ = /o{5+5_) = /o Tr {PthermS+S_}. (22)

It can be shown (Ref. [22]) that the intensity of spontaneous emission is given by

,. oo ЛГ/2

3 x exp r-(2r--r+-j+-mf Л !-'- (23) kT rh=—f

ЛЛ(2г — is the number of equivalent r)! representations with the same r.

2.0-

1.5-

1.0

0.5-

0.0 0 10 20 30 40 50 60 70 kT Fig. 3. The intensity of spontaneous emission per atom (in units of /o) versus cavity temperature. The curve / is the classical result given by Eq. (25), the curves 2, 3, 4 correspond to N = 10, Л/" = 50, N = 100, respectively, and w/g = 10

Let us consider the intensity per atom, i. e., /i = I/Af. This intensity consists of two terms; i.e., the first is given by a single-particle contribution /Smgie and the second one is An Algebraic Method to Solve the Tavis-Cummings Problem 91 proportional to the two-particle correlation function Cor, I\ = /single + /о(Л/" — l)Cor. They are found to be

/single - /О А/-(2^ 0+<Г-) - Io [ о + А/"( / (24)

It is plain that in the absence of the cavity, the correlation function vanishes and the only contribution to I\ is given by the first term,

/single = /cl = /0 (l + e<"/feT) ~* . (25)

Notice that if the number of atoms is big enough, the intensity of radiation exhibits a high maximum. In a cavity at low T, the cluster of N two-level atoms emits much more intensively than it does in the free space. It should be possible to drive the system to thermal equilibrium at the temperature where the spontaneous emission exhibits maximum. The marked amplification of spontaneous emission should be observed in cavity experiments. Concluding this section, we recapitulate our main results. We consider spontaneous emission of the system comprised of N two-level atoms strongly coupled to the cavity mode and prepared in the state of thermal equilibrium. In the absence of the cavity the atoms in the thermal equilibrium would be uncorrelated. In this case, the spontaneous emission would be described by the conventional formula (25). For high-Q resonators the strong coupling to the resonator mode should necessarily be taken into account. We demonstrate that in this case the intensity of spontaneous emission can be greatly enhanced. This phenomenon can be explained by additional correlation between atoms established by the cavity mode. Analytically, we have replaced the exact Tavis-Cummings Hamiltonian (1) by its zero-order approximation derived in the previous sections. This allowed us to represent the intensity of spontaneous emission in the simple analytical form Eq. (23). It is appropriate to emphasize once again that the zero-order approximation of the Hamiltonian contains strong coupling and, thus, describes cooperative effects in the atomic subsystem. This is the consequence of the fact that the operators 5± describe collective excitations in the atom-field system.

CONCLUSION

In this work, we solved the Tavis-Cummings problem by applying the technique of polynomially deformed algebras. We constructed the transformations that map one polynomial algebra of operators onto another. This allowed us to reformulate the problem in terms of a simpler algebra of second order, Sr, and develop a specific perturbation theory. We were able to find analytical expressions for all the eigenvalues of the Hamiltonian up to third order in the small parameter a. For the nearby zones, we showed explicitly how the collective quantum Rabi frequency depends on Dicke index r. Since this index characterizes the symmetry of atomic states, the result has significant physical implications. The dependence on atomic symmetry is revealed already in zeroth order in the perturbation expansion. Employing our methods, we found an interesting new effect, which is amplification of spontaneous emission of thermal Af-atom states due to collective effects. We expect that this phenomenon can be observed in cavity experiments. 92 Vadeiko I. P. et al.

Acknowledgements. This work has been supported by the Academy of Finland (Project No. 44875). I. V. is grateful to the Centre for International Mobility (CIMO) and University of Jyvaskyla for financial support.

APPENDIX

1. Similarity Transformations. We look for similarity transformations Uk that diagonalize the Hamiltonian in different orders of a,

V£n) = VkV£\Ub\ (Al) where fc = 0,1,2,3. In Eq.(Al) only the terms of order n in the small parameter a are present. For fc = 0, the term Vfc_i should be replaced by the corresponding term in expansion of V in the series. Regrouping the terms, we obtain

(A2)

(A3) *± \ ** / /

V(2} = - (f )2 [*3H^3] ' (A4)

1 / (У. \ Г ~ — "I 2 \2/ L J' where

В = [S3SX + SXS3] , (A6) and 5, - ^ + ^- (A7)

2. The Zero-Order Transformation f/o- As is known from the theory of SU(2) algebra, the operator Sx can be diagonalized by the transformation

U0 = exp [| (5+ - 5_)] = exp [-*|sy] - (A8)

Employing this transformation, we obtain

(A9) An Algebraic Method to Solve the Tavis-Cummings Problem 93

3. The First-Order Transformation U\. It can readily be seen that the transformation

Ui = exp [oxDl] , Dl = -i [s3Sy + SyS3] (AlO) diagonalizes the operators in the first order. In the diagonalization, one needs the commutators

2 [S3, Dl] = B, [ [S3, Dl] , Dl] = 453 (> - 2S3 - ^ ,

2 [[[S3, Dl] , Dl] , Dl] = (45 - l) В - 8 [§3B + BS3 - S3SXS3] , (All)

2 2 [S3SZS3,D1] = 2Sf (5 - 253 ) Then up to third order in a,

_ 2ax i _

and

+ a3(|)2{(452-l)B-8j}, (A.1

where J = 53B + BS3 - S3SXS3.

l 3 2 2 2 2 йгУМ (у,) ~ = V™ + a (|) {5XS3 (2S - 45 - 5 ) +

2 2 + (25 -45|-S )535I}. (A.1

In the third order, the operator VQ 3* remains unchanged after the transformation, i. e., V/3* VQ . In order to calculate VQ up to third order, we take into account that

and find then that

(A15)

2 2 - 453 - 5, + zS - 45| - sx 94 Vadeiko I. P. et al.

4. The Second- and the Third-Order Transformations. To find the second-order trans- /o\ formation, we rewrite Vj in a symmetrized form,

2 2 V^ = -I (|) [LXS3 + ftl, + S3 (S - SI - l)] , (A16)

2 2 S+ + S where Lx = — — . The diagonalizing transformation is then given by

(A17)

Si - S2 where Ly = — - . Keeping the terms up to third order we obtain (see (A 13))

\ (f ) [L,S, + S3LX] . (A18)

The transformation (A 17) does not change the expressions given above for V/ and Vj , and we find that

/Q\ -,

Diagonalization of V2 can be performed in a similar way with an operator С/з = 3 exp —(a/4) 0. As there are no diagonal terms in V2 , which would contribute to the spectrum of the Hamiltonian, we do not give here the fairly complicated form of operator O. The final diagonal form for the interaction V is thus given by

2 V = ft53j ll ++ (|) [5555|| -- 33ff ((ff ++ 11)) ++ i ij I. . (A20)

To recapitulate, we introduced four transformations Uk, k = 0,1,2,3, which successively diagonalize the interaction operator in the Tavis-Cummings Hamiltonian up to third order with respect to the small parameter a = (1/2Пя)~ . with Пд the generalized Rabi frequency of Eq. (18), such that

V = UVU-\ U = U3U3UiUo. (A21)

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QUANTUM TELEPORTATION OF NUCLEAR MATTER AND ITS INVESTIGATION V. V. Ivanov a, B. F. Kostenko a'1, V. D. Kuznetsov a, M. B. Miller b, A. V. Sermyagin b a Joint Institute for Nuclear Research, Dubna 6 Institute of Physical and Technology Problems, Dubna, Russia Since its discovery in 1993, quantum teleportation (QT) is a subject for intense theoretical and experimental studies. Experimental demonstration of QT has so far been limited to teleportation of light. In this paper, we propose a new experimental scheme for QT of nuclear matter. We show that the standard technique of nuclear physics experiment could be successfully applied for teleportation of spin states of atomic nuclei. We claim that there are no theoretical prohibitions upon a possibility of a complete Bell measurement, therefore, the implementation of all the four quantum communication channels is at least theoretically possible. A general expression for scattering amplitude of two 1/2-spin particles is given in the Bell operator basis, and the peculiarities of Bell states registration are briefly discussed.

INTRODUCTION

Since its discovery in 1993, quantum teleportation (QT) is a subject of intensive theoretical and experimental studies [1]. If quantum computers will be really constructed, QT is expected to have a great influence on a future computation and communication hardware, presumably comparable with the impact of the radio network on modern technique. This is because the practical realization of quantum information processing requires, special quantum gates which cannot be performed through unitary operations, but may be constructed if one uses quantum teleportation as a basis primitive. Another important application of QT is a secure data transmission in a so-called quantum cryptography [2,3]. Besides a relevancy to the applications, QT presents also a new fundamental concept in quantum physics. Indeed, the very phenomenon of QT is based on Einstein-Podolsky-Rosen correlations, which have been correctly confirmed only for photons. The same is true for QT — only the entangled optical beams have been used to teleport quantum states up to now [4,5]. As far as the quantum information processing involves material particles with nonzero mass, teleportation of heavy matter is considered now to be the next important landmark on the way to obtain a complete set of quantum computation tools [6-8]. We propose here a new experimental scheme for QT of heavy matter states, which is based on the standard techniques of nuclear physics experiment and may be fulfilled in the next one or two years. To the best of our knowledge, other proposals require much more time for their implementation.

'Corresponding author. Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980, Dubna, Russia. Tel.: +007-096-21-64-069; fax: +007-096-21-65-145. e-mail: [email protected] (B. F. Kostenko). Quantum Teleportation of Nuclear Matter and Its Investigation 97

1. BASIC IDEAS

The first experimental demonstration of photon state teleportation had a great public resonance and was even elucidated in mass media. Indeed, not long ago only science-fiction authors ventured to use this term. Although no materials, but only quantum states, were really transmitted in those experiments, in fact, journalists were not very far from the truth using the term in the same sense as the science-fiction authors. If one takes into account the indistinguishableness of elementary particles of the same sort, the photons in our case, one can say that interchange of distant photon states is nothing else but teleportation of the matter itself. It is slightly more difficult to explain why the experiments mentioned above had produced a big impression on the specialists as well. The fact is that just not long before a no-cloning theorem for quantum states has been proved [9]. It states that it is not possible to construct a device that produces an exact copy of an arbitrary quantum system. This result forbids to use the methods of information copying technique traditionally employed for classical computers. The theorem was accepted as a serious obstacle encountered on the way of quantum computer elaboration. At that time nobody knew with certainty if it is possible to organize somehow the communication between different parts of a quantum computer, or the very idea of quantum information processing is quite wrong. The solution of the problem, very simple and amazing, was given in [1]. The idea can be explained as follows. Let us consider a quantum system consisting of three particles with half-integer spins and let us suggest that one of the particles is in the state

to be teleported. If the residuary two particles were prepared in the entangled state with zero total spin the state vector of the considered three-particle system is

1Ф123) = №l>l*23>- Using a so-called Bell's basis,

the state of three-particle system can be identically rewritten in the form

= ОзН Т2> + Ь\ Ь» + |Ф)(а| Та) - Ь\

+ I^X-al Ь) - Ь\ Та».+ |Ф^)(-а| Ь> + Ь\ Та»]- Now, let us assume that the observer succeeds in fulfilling the measurements discriminating different Bell's states for the system consisting of particles 1 and 3. Then carrying out such 98 Ivanov V. V. et al. a measurement and knowing the result, he will know certainly the state of the particle 2. Namely, in accordance with the modern quantum theory, the following correspondence should take place:

T2) -

The first of the above-mentioned states of particle 2 exactly coincides with the state that particle 1 possessed before the measurement. Thus, in this case, teleportation of the given quantum state is totally completed. For three other cases, three different unitary operations should be additionally performed to transform the obtained state of particle 2 into the required one. They could be accomplished either over particle 2, or over the observer's laboratory itself, but in both cases they are ordinary rotations1. The most important from the principal point of view is «disappearing» of the individual particle 1 in the place, which could be notified as «zone of scanning» (ZS), where the measurement preparing particles 1 and 3 in one of Bell's states is accomplished. Indeed, this measurement destroys the particle 1, in a sense that none of the two particles outgoing from ZS has definite properties of the particle 1 . They constitute a new pair of particles, which only as a whole has some quantum state, and the individual components of the pair are deprived of this property. Therefore, in some sense the particle 1 really disappears at ZS. Exactly at the same moment, the particle 2 obtains the properties particle 1 had in the beginning. Once it has happened, in view of the identity principle of elementary particles, we can say that the particle 1, disappearing at ZS, reappears at another location. Thus, quantum teleportation is accomplished. It can be seen as well that the no-cloning theorem is not violated. QT has several paradoxical features. In spite of the absence of contacts between objects (particles, photons) 1 and 2, 1 manages to pass its properties to 2. It may be arranged in such a way that the distance from 1 to 2 is large enough to prevent any causal signals between them! Furthermore, in contrast to the transportation of ordinary material cargo, when a delivery vehicle first visits the sender to collect a cargo from it, quantum properties are delivered in a backward fashion. Here, the proton 3 plays a role of the delivery vehicle, and one can see that 3 first interacts with the recipient 2, to prepare the entangled state \-ф2з), and only after that it travels to the sender (1) for the «cargo». Finally, to reconstruct the initial object completely, it is necessary to inform a receiver at the destination about a result of the measurement in ZS. This allows him to accomplish processing of the quantum signal (incoming with the particle 2) in a due manner. There- fore, one more channel of communication is needed for an ordinary or classical information transmission. Only receiving a message (using the classical communication line) that 1 and 3 form a new EPR-pair with zero total spin, an observer at destination may be sure that the

'Taking into account that orthogonal vectors in the spinor space correspond to oppositely oriented projections of spin in the usual one, it is easy to understand that the rotation angle of the laboratory is twice as many the angle of the spinor unitary operation for \2). Quantum Teleportation of Nuclear Matter and Its Investigation 99 properties of 2 are identical to those of 1 before the teleportation. In the case when 1 + 3 system has nonzero total spin, additional transformation of quantum signal is needed.

2. LAYOUT OF AN EXPERIMENT

One can convert the previous formulas into a concrete experimental layout for proton teleportation. In Fig. 1, the layout of an experiment on teleportation of spin states of protons from a polarized target, PHa, into the point of destination (target C) is shown. A proton beam po of the suitable energy within the range 20-50 MeV bombards a liquid hydrogen target, LH2 (which may be also replaced by an ordinary polyethylene foil). According to the known experimental data, the scattering in the LHa target corresponding to the scattering angle в ~ 90° at the c. m. s. occurs within an acceptable accuracy through the singlet intermediary state [10]. Thus, the outgoing protons pa and рз form a two-proton entangled system in the state |Фаз) that was introduced in the preceding section. One of the scattered protons, pa, then travels to the destination point (the target-analyzer C), while the other, рз, arrives to the point where teleportation is started, i. e., to the PHa target. The last one is used as a source of teleportated particles which have quite definite quantum state

determined by a direction of polarization. This direction could be chosen accidentally and, thus, unknown to the experimenters. In the case when scattering in the polarized PHa target occurs under the same kinematic conditions as in the LHa target (i.e., at the c. m. s. scattering angle в ~ 90°), the total spin of the particles pi and рз must also be equal to zero after collision, and they find themselves in the state Classical Classical communication/ ч communication l*). To detect the events, a re- line 0 Ш line movable circular modules F-l and F-2 (e.g., of the facility «Fobos») F-l could be used1. If all the above conditions are provided, the protons reaching the point К will suddenly receive the same spin projections as the protons in the polarized PHa target. Thus, if the coincidence mode of the detection of [Ф^) state is provided via any classical channel shown in Fig. 1, then a strong correlation has to take place between Space coordinate polarization direction in the PHa target and the direction of the deflection Fig. 1. Experiment on proton state teleporation of pa protons scattered in the carbon

'The 4тг spectrometer FOBOS is a setup at the Flerov Laboratory of Nuclear Reactions of JINR. 100 Ivanov V. V. et al. target C1. Therefore, at least for those events when particles 1 and 3 are registered to be in the state |Ф1з ), teleportation of protons could be experimentally fulfilled. Besides, if one succeeds to make a distance between the detectors F-l and F-2 to be sufficiently large and the difference between the moments of registration in F-1 and F-2 to be short enough, then it will be possible to meet the important criterion of the causal independence between the events of the «departure» of the quantum state from PH2 target and the «arrival» of this «cargo» to the recipient (proton p2) at the point K. This kind of measurements consists in recording signals entering two independent but strictly synchronized memory devices with the aim to select afterwards those events alone that appeared to be causally separated. Thus, the experimental setup shown in Fig. 1 also allows one, at least in principle, to fill the gap in the verification of the EPR-effect for heavy matter.

3. MORE EFFECTIVE EQUIPMENT FOR PROTON TELEPORTATION

In fact, it is possible to increase essentially the rate of teleportation, mainly due to the extension of the solid registration angle, if one makes use of another layout of experiment, shown in Fig. 2. In the figure, protons PQ from the primary beam incident onto a hydrogen target TI (polyethylene) and undergo the interaction which results in arising of two protons traveling toward targets T2 and Тз. These ones are located in such a way that only the protons with scattering angles of about 90° at c. m. s. (or nearly of 45° in the laboratory system) arrive at them both. Under this condition, PI and PQ should have the entangled quantum state with zero total spin value (i. e., they constitute the EPR-pair). T2 is the polarized target analogous to that mentioned in the previous section. Тз represents the analyzer for protons PQ, and it may be the carbon foil as well as another hydrogen polarized target. In the case displayed in picture, one more polarized target is used for this purpose. A new entangled two-particle state with zero total spin value can be knocked out as a result of proton-proton scattering within T2. The registration of proton in the detector Db selecting the scattering angles in the vicinity of 45° 1. s., should be the signature for this event. In that case, in accordance with the general idea of teleportation, «Alice» (the signal sender) must inform «Bob» (the receiver) that the quantum communication channel Ф~ has been realized.

A similar information about proton detection in the D2 indicates the fact of preparation of another two-particle state with the spin value 5 = 1 and spin projection Sz = 0. Accordingly, the Ф+ communication channel will be formed in the left shoulder of the layout. The existence of proton state teleportation can be easily proved by measuring the proton cross section in the polarized target T3. If teleportation takes place, indeed, protons PQ should be polarized, too, and this fact could be checked due to the dependence of nuclear cross section on the polarization of scattering particles. Contrary to the idea of teleportation, proposed by C. Bennett et al. [1], the spatial separation of different quantum channels in the present layout allows Bob to control the value of the

'Here, the carbon foil С plays a role of the polarization analyzer; i.e., one measures the asymmetry of the left-right counting rates to determine a spin state orientation of P2 before the scattering [10]. Quantum Teleportation of Nuclear Matter and Its Investigation 101

(NH3)T; T2(NH3)

Fig. 2. Layout of a more effective experiment teleported polarization by means of the Renninger-type measurements [11], i.e., without any perturbation of the teleported state. Thus, the information on the signal presence in the DI detector in conjunction with the absence of the signal in the detector DH makes Bob to be sure that the teleported particle has the same polarization as that of protons within the T2 target. The conjunction could be organized as follows: when the signals from the DI and D2 detectors enter controlling inputs of gate-units, Gate 1 and Gate 2, the reciprocal signals arise at the output of these gates corresponding to the realization of different Bell's states (indicated in the table seen at the bottom of Fig. 2)1. It is interesting to note that from classical physics viewpoint the target Тз should be located lower than it is shown in the figure, in order the protons PQ to incident onto it only after the complementary proton P" of the EPR-pair is detected within the DI detector. From

'it is worth noting that the represented layout is set specially for the experimental verification of possibility of the quantum state teleportation for heavy matter. In the case of «practical application» of quantum teleportation, neither the analyzer Тз, nor the detectors DH, 022 are necessary. What remains essential is to transmit from the Alice location to the Bob one, via the classical communication channel, the information about the results of measurements in the right side of the layout. 102 Ivanov V. V. et al. a naive point of view, only after that protons PQ acquire a certain state of spin. Nevertheless, according to the Copenhagen interpretation, no event can be considered existing until it is detected (by Dn and 022 in the case under consideration); the displayed dislocation of the targets is possible as well. A real polarized target, which permits one to carry out the above-considered experiment, is described in the Appendix.

4. BELL OPERATOR BASIS

In the experiments that were carried out up to now, only one quantum channel corresponding to the registration of Bell's state |Ф13~ ) has been used. In this connection, the question arises as to whether it is possible to involve other channels correlating with the states |Ф1з~ ), \Ф(?) and |Ф(+>>. To answer this question, let us consider the general expression for the scattering amplitude of two particles, not necessarily identical ones, with the spin value 1/2,

f = A + B(Si • A)(S2 • A)

Using a relation

in the case of the coordinate system to be fixed in the following way:

A || x, ц || y, v || z, the expression for / can be represented in the form Л с U il D where S = Si + 82, s = Si — 83. It can be shown that the scattering operator / can be now expressed through the Bell transition operators by making use of the formulas

Sx=

sy = i

Sz =

Sz=

and the decomposition of the unity i = Рф_ + Д,+ + Рф_ + РФ+. As a result, one obtains

сРФ_ + сгРФ+ + ESZ + Fsz, (1) Quantum Teleportation of Nuclear Matter and Its Investigation 103 where . B+C+D , B + C a = A -- - - , b = a + — - — , C+D , B+D

In the case E = F = 0, the expression (1) is the usual spectral decomposition for the operator / that allows one to interpret / as an operator of some quantum observable. Therefore, to register a definite Bell state, one has to find such experimental conditions at which all coefficients but one of a, b, c, or d in the expression (1) turn into zero. For these purposes, the type and energy of colliding particles, as well as the angle which scattered particles are recorded at, could be altered. Since the number of necessary conditions formulated above is less than the number of free coefficients in (1), it is clear that the registration of each Bell's state is possible at least theoretically. The directions which spin projections of the scattered particles should be measured along for detecting the states |Ф(+)), |Ф(~^) and |Ф(+)) form three orthogonal spatial vectors. This follows from the relations

where ej are the normalized orthogonal states with the definite values of spin and its projections, ei = |l,0), e,= |l,l), es = |l,-l), which transform in accordance with 3-vector representation of the rotational group. It is clear that spatial rotations at the angle тг/2, corresponding to e* —+ ±ел-, represent the group of permutation for the Bell's states considered (putting aside an unimportant phase factor -1). Thus, the possibility of registration of the |ф(+>) state also opens the way to register two other states |Ф(+)), |Ф(-)) by means of change to тг/2 of the direction along which the spin projection is measured. For identical spin 1/2 particles the scattering operator (1) has some additional symmetries, so that in c. m. s. one has

o(0) = а(тг-в), Ъ(0) = -Ь(тг - в), с(в) = -с(тг - в), d(6) = -d(ir - в), Е(в) = Е(тг - в), F(9) = Р(к - в).

For nucleon-nucleon scattering, we have F = 0 as the total spin squared of such a system is conserved and the last two terms in (1) describe the transitions between Bell's state with different S2. Thus, e. g., for two identical nucleons at в = тг/2, one obtains

The experimental identification of Bell's states |Ф(~)} and |ф(+>) is rather simple due to the characterization of these states by the definite values of total spin and its projections 104 Ivanov V. V. el al.

(|S| = 0, Sz = 0, and |S| = 1, Sz = 0, respectively). The result of the spin projection measurement for particles 1 and 3 is

Szl — ±2' 5z3 = =F-, for any choice of the z-axis direction, provided their initial state is |Ф(~)). For particles in the |Ф(+)) state such correlations take place only if the spin projections are measured along a definite axis n. If the axis of measuring is deflected at an angle 9 from 2 this direction, the probability to have Szi + SZ3 = 0 will decrease as cos 9. One may expect that at the energies considered, there is a scattering angle interval corresponding to I = 1 and, therefore, to the |Ф(+)} final state of two protons. It seems more difficult to identify states |Ф^} and Ф'+)). In this case, it is necessary first to find out a direction n' (which is perpendicular to n) for which measurements of spin projections give either Sz\ = 1/2 and Szs = 1/2 or Szl = —1/2 and 52з — —1/2 with the same probability p — 0.5. Now the measurement of the spin projection of the particle 2 allows one to determine which of two possible states, |Ф^') or |Ф1з ), the scattering has really occurred into.

CONCLUSION

In recent years, there has been an explosive growth in new experimental techniques that can be adapted in future quantum computers and communication channels. Among them are: neutron and electron interferometry, photon down-conversion in nonlinear crystals, special ultrafast lasers and micromasers that can create few-photon states, trapping of individual atoms, atomic interferometry, Josephson tunnel junctions, cavity quantum electrodynamics, quantum dots in solids, etc. Some of these results are now used to test quantum theory itself for the purpose of determining its limits and for the verification of the alternative interpretations of quantum mechanics which have been developed instead of the traditional Copenhagen one over the years. The proposed technique of quantum teleportation is the first one which suggests to use methods of nuclear physics including nucleon-nucleon scattering and appropriate methods of event registration. The project presented in this paper has the following nearest objectives:

• Experimental setup for realization of nuclear teleportation.

• Experimental proof of quantum teleportation phenomenon for nuclear matter over Ф<~) Bell's information transmission channels.

• Theoretical and experimental proof of possibility of other ways of quantum information transmission for nuclear matter states, in particular, using ф(+) transmission channel.

Although each novel way of quantum state engineering with single heavy particles is valu- able itself, we point out some distinguished properties of this approach. The first preference concerns the rate of teleportation. Our estimations show that the present nuclear physics experimental device allows one to teleport at least 105 protons per second without any additional Quantum Teleportation of Nuclear Matter and Its Investigation 105

refinements. Another peculiarity of the method is very small space-time region of event localization — inside the atomic nuclei. These advantages may be useful not only in the future quantum computers, but also for investigations of some theoretical puzzles today. The most intriguing of them are: how and when does the modern theory of quantum measurements break down (presumably somewhere in the middle between micro- and macrolevel), has the environment-induced wave packet collapse any sense, or is presence of the observer's mind necessary, indeed? Among future objectives of the project, one may point out the following: • A test of Bell's inequalities by the measurement of the spin correlations in low-energy proton-proton scattering, which meets the requirement of space-like character for the intervals between the successive events of proton spin projection measurements. • Measurements of the wave packet collapse duration and its spatial localization with a maximally possible precision. Practically, this objective may be achieved by maximal increase of teleportation rate and decrease of size of the teleporting equipment. Acknowledgements. We are grateful to Prof. I. Antoniou (Solvay Institutes for Physics and Chemistry, Brussels), Prof. V. V. Belokurov and Prof. 0. A. Khrustalev (Moscow State University) for helpful discussions and support. Our special gratitude to Dr. Yu. F. Kiselev (Joint Institute for Nuclear Research, Dubna) who provided us the Appendix. One of us (B. F. K.) acknowledges the financial support of the Russian Foundation for Basic Research, project No. 02-01-00606.

Appendix POLARIZED HYDROGEN TARGET

At present, two types of solid polarized nuclear targets exist, which differ as to the type of refrigerator used for their cooling. Namely: 1. Targets with refrigerator of evaporator type with 3He or 4He. 2. Targets of frozen type with the dissolution of 3He in 4He. Targets of the first type operate at temperatures 0.3-1.2 K. Their evaporation cycle of cooling allows one to attain the greater rate of the cold generation (up to 20-50 J/mol), so that the targets are resistant to the greater particle intensity (up to 1011 pps). They are distinguished by a high reverse velocity of polarization (depending on temperatures of the cooling agent). With the account of conditions of the teleportation experiment, the best variant of the target scheme is the American installation E143 (SLAC). In this scheme, a leakage of nuclear polarization is reduced not by the temperature lowering, but by means of increasing the magnetic field up to 5 T. As a result, when operating with ammonia NHs and liquid 4He, the authors of the method managed to reach a remarkable result: 90 % polarization of the protons under a specific RF-power of the order of 12 mW/cm3. As it has been in the original construction, our evaporation refrigerator is supposed to contain a wide-aperture magnet from two or three pairs of superconducting Helmholtz coils (see Fig. 3), intended for getting a magnetic field of necessary homogeneity on targets with diameter and length of about 30 mm. The refrigerator represents an ordinary helium cryostat (4.2 K) for a magnet of that type the Oxford Instruments Company (OIK) produces. In the area of the nuclear target, a temperature of about 1 К could be provided with the use of a rather simple cooling plunger. In the case of thin nuclear targets, the requirements on 106 Ivanov V. V. et al.

Microwave input

02 5-40 cm

Fig. 3. Low-temperature refrigerator with nuclear targets and particle detectors: 1 — primary hydrogen target; 2,3 — nuclear hydrogen polarized targets; 4, 5, 6, 7 — particle detectors (Di, Dz, Dn and 022 from Fig. 2) homogeneity are not very strict. Preliminary negotiations with QIC have displayed that the magnet could even be of a more simple design. Contrary to the standard El43 device, it is planned to implement the circulation contour (of the type used in the frozen targets). This innovation has to allow cooling of the target with 4He as well as with 3He, and this, in turn, allows one to achieve a very high polarization, practically nearly 100%. The arrangement of primary hydrogen target and polarized targets (which were shown above in Fig. 2) in the low-temperature refrigerator is depicted in Fig. 3.

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Ю. Ц. Оганесян и др. Кинематическая сепарация и массовый анализ тяжелых ядер отдачи (На английском) 5 Т. И. Михайлова и др. Динамические эффекты на фазе сближения тяжелых ионов (На английском) 13 В. Б. Злоказов VACTIV-DELPHI программа для обработки гамма-спектров в режиме графического диалога 25 М. И. Адамович и др. Облучение ядерных фотоэмульсий в пучке релятивистских ядер 6Не и 3Н (На английском) 29 Е. А. Виноградов и др. Оптические и структурные исследования PLZT х/65/35 (х = 4,8 %) сегнетокерамики, облученной сильноточным импульсным пучком электронов 39 И. А. Шелаев Осциллирующий заряд 48 X. В. Клапдор-Клайнгротхаус, А. Дейц, И. В. Кривошеина Первые свидетельства безнейтринного двойного бета-распада (На английском) 57

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Б. Ф. Костенко и др. Ядерная телепортация (предложение эксперимента) (На английском) 32 М. И. Широков О ненаблюдаемости «склеивающих» бозонов в модельной теории поля 42 A. К. Каминский Расчет дифференциальных эффективных сечений ионизации в быстрых ионно-атомных столкновениях 47 B. А. Артемьев О взаимодействии ультрахолодных нейтронов вблизи поверхности твердых тел 56 X. Л. Ядав и др. Использование ^-факторов для исследования структуры высокоспиновых состояний нейтроноизбыточных изотопов диспрозия (На английском) 66 Б. 3. Белашев, М. К. Сулейманов Поиск периодичностей в экспериментальных данных методами авторегрессионной модели 77

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В. А. Карнаухов и др. Тепловая мультифрагментация, ядерный туман и критическая температура для фазового перехода жидкость-газ (На английском) 5 К. А. Свешников и др. Топологические и нетопологические решения в киральной модели мешка с конституентными кварками (На английском) 14 В. Ю. Батусов и др. Сравнение данных высокоточных метрологических измерений модуля № 8 адронного тайл-калориметра ATLAS лазерным (ОИЯИ) и фотограмметрическим (ЦЕРН) методами (На английском) 36 Л. С. Ажгирей и др. Калибровка пучкового поляриметра на синхрофазотроне ОИЯИ 51 О. А. Займидорога Аномальное поведение А-, f-зависимостей и фазы парциальной волны когерентного образования радиального резонанса тг (1300) 59 Б. 3. Белашев, М. К. Сулейманов О некоторых обратных задачах ядерной физики 63 Указатель статей 111

В. Н. Пенев, А. И. Шкловская Модификация ядерной среды при промежуточных энергиях (На английском) 73 Н. С. Борисов и др. Экспериментальное исследование ЛГЛГ-рассеяния с поляризованными частицами на ускорителе Ван-де-Граафа Карлова университета. «Проект AW-взаимодействия» (На английском) 86 Б. Чиладзе и др. Передний детектор спектрометра ANKE. Сцинтилляционные и черенковские годоскопы (На английском) 95

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В. В. Хрущев, С. В. Семенов О спектрах масс псевдоскалярных мезонов в релятивистской модели независимых кварков (На английском) 5 У. Ашманскас и др. Вершинный триггер установки CDF: предварительные результаты второго сеанса тэватрона (На английском) 12 А. Артиков и др. Подсистема счетчиков «miniskirt» установки CDF II (На английском) 25 А. Абызов и др. Метод профильных функций эффективности для изучения эффективности срабатывания ячеек дрейфовых камер (На английском) .40 Ю. А. Троян и др. Резонансы в системе тг+тг~-мезонов в реакции пр —> пртг+тг"

при Рп = 5,20 ГэВ/с: поиск, результаты прямого наблюдения, интерпретация (На английском) — 53 А. Мензионе и др. Работы по улучшению временного разрешения времяпролетного детектора установки CDF (На английском) 61 Л. Л. Енковский, Б. В. Струминский О распределении частиц при очень больших множественностях (На английском) 67 112 Указатель статей

О. Пухов и др. Автоматизация контроля системы мюонных сцинтилляционных счетчиков CDF II 72 Е. П. Шабалин и др. Исследование эффектов радиолиза материалов холодного замедлителя быстрыми нейтронами (На английском) 82

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PARTICLES AND NUCLEI, LETTERS. 2002 PAPER INDEX

Yu. Ts. Oganessian et al. Kinematic Separation and Mass Analysis of Heavy Recoiling Nuclei 5

T. I. Mikhailova et al. Dynamical Effects Prior to Heavy Ion Fusion 13

V. B. Zlokazov VACTIV-DELPHI Graphical Dialog Based Program for the Analysis of Gamma-Ray Spectra (In Russian) 25 M. I. Adamovich et al. Irradiation of Nuclear Emulsions in Relativistic Beams of 6He and 3H Nuclei 29 E. A. Vinogradov et al. Optical and Structural Study of PLZT x/65/35 (x = 4,8%) Ceramics Irradiated by High-Current Pulsed Electron Beam (In Russian) 39 I. A. Shelaev The Oscillating Charge (In Russian) 48 H. V. Klapdor-Kleingrothaus, A. Deitz, I. V. Krivosheina First Evidence for Neutrinoless Double Beta Decay 57

V. K. Lukyanov et al. Role of the Coulomb Distortion in Form-Factor Calculations for 12C with Alpha-Clusterization and Nucleon-Nucleon Correlations 114 Paper index

V. P. Ladygin, S. Nedev Optimization of the Set-Up for the Investigation of the Light-Nuclei Spin Structure at the Internal Target of the Nuclotron 13 I. M. Sitnik et al. Precessing Deuteron Polarization 22 F. V. Tkachov Approaching the Parameter Estimation Quality of Maximum Likelihood via Generalized Moments (In Russian) 28 I. N. Izosimov et al. 156 Determination of the Total Energy QEC for Ho (T1/2 ~ 56 min) /3+/EC Decay Using the Total Absorption 7-Ray Spectrometer 36 A. Yu. Bonushkina, V. V. Ivanov, G. B. Pontecorvo Selection of Signal Events in the DUBTO Experiment 39 S. B. Borzakov, Ts. Panteleev, A. V. Strelkov A Search for Dineutron in Interaction of Neutrons with the Deuterons (In Russian) 45 A. A. Bel'kov et al. Tracking Performance of the HERA-B Outer Tracker PC Chambers 51 V. N. Bychkov et al. Construction and Manufacture of Large Size Straw-Chambers of the COMPASS Spectrometer Tracking System 64 M. Grypeos et al. The Charge Form Factor and the Nucleon Momentum Distribution of |He and Their Centre-of-Mass Correction 74

No.3[112]

N. A. Chernikov On the History of Lobachevsky's Discovery of the Non-Euclidean Geometry (In Russian) 5 A. K. Managadze et al. A Complanar Emission of Particles in Nuclear Interaction for EO > Ю16 eV Detected in the Stratosphere (In Russian) 19 F. G. Lepekhin 10B Nucleus Fragment Yields (In Russian) 25 B. F. Kostenko et. al. Nuclear Teleportation (Proposal for an Experiment) 32 Paper index 115

M. I. Shirokov Invisible «Glue» Bosons in Model Field Theory (In Russian) 42 A. K. Kaminsky Calculation of the lonization Differential Effective Cross Sections in Fast Ion-Atom Collisions (In Russian) 47 V. A. Artemiev Interactions of Ultracold Neutrons near Surface of Solids (In Russian) 56 H. L. Yadav et al. g Factors as a Probe for High-Spin Structure of Neutron-Rich Dy Isotopes 66 B. Z. Belashev, M. K. Suleimanov Search for Periodicities in Experimental Data by the Autoregressive Model Methods (In Russian) 77

No.4[113]

V. A. Karnaukhov et al. Thermal Multifragmentation, Nuclear Fog and Critical Temperature for the Liquid-Gas Phase Transition 5 K. Sveshnikov et al. Topological and Nontopological Solutions for the Chiral Bag Model with Constituent Quarks 14 V. Batusov et al. Comparison of ATLAS Tilecal Module No. 8 High-Precision Metrology Measurement Results Obtained by Laser (JINR) and Photogrammetric (CERN) Methods 36 L. S. Azhgirey et al. Calibration of the Beam Polarimeter at the JINR Synchrophasotron (In Russian) 51 O. A. Zaimidoroga Anomalous A-, t-Dependence and the Phase Behaviour of Partial Wave of Coherent Production of Radial State n (1300) (In Russian) 59 B. Z. Belashev, M. K. Suleimanov About Some Inverse Problems of Nuclear Physics (In Russian) 63 VI. Penev, A. Shklovskaia The Nuclear Matter Modification at Intermediate Energies 73 116 Paper index

N. S. Borisov et al. Experimental Research of the NN Scattering with Polarized Particles at the VdG Accelerator of Charles University. Project «NN Interactions» 86 B. Chiladze et al. The Forward Detector of the ANKE Spectrometer. Scintillation and Cherenkov Hodoscopes 95

No.5[114]

V. V. Khruschev, S. V. Semenov On the Mass Spectra of the Pseudoscalar Mesons in the Relativistic Independent Quark Model 5 W. Ashmanskas et al. CDF Silicon Vertex Tracker: Tevatron Run II Preliminary Results 12 A. Artikov et al. The «Miniskirt» Counter Array at CDF II 25 A. Abyzov et al. Efficiency Profile Method to Study the Hit Efficiency of Drift Chambers 40 Yu. A. Troy an et al. Resonances in the System of тг+тг~ Mesons from np —> np тг+тг~ Reaction at Pn = 5.20 GeV/c: Search, Results of Direct Observations, Interpretation 53 A. Menzione et al. Activity on Improving Performance of Time-of-Flight Detector at CDF 61 L. L. Jenkovszky, B. V. Struminsky On Very High Multiplicity Distributions 67 O. Pukhov et al. Automatization of the Monitoring and Control of the Muon Scintillation Counters at CDF II (In Russian) 72 E. P. Shabalin et al. Study of Fast Neutron Radiation Effects in Cold Moderator Materials 82

No.6[115]

L. V. Bobyleva et al. Wien Filter Using in Exploring on Low-Energy Radioactive Nuclei (In Russian) 5 H. L. Yadav, К. С. Agarwal Memory Effects in Dissipative Nucleus-Nucleus Collision 12 Paper index 117

A. M. Kotzinian, O. Yu. Shevchenko, A. N. Sissakian Topological Charge and Topological Susceptibility in Connection with Translation and Gauge Invariance 29 V. S. Barashenkov, M. Z. Yuriev Solutions of the Multitime Dirac Equation 38 B. Z. Belashev, M. K. Suleimanov The Maximum Entropy Technique. System's Statistical Description (In Russian) 44 M. D. Shafranov, T. P. Topuria The Charged Conductor Inside Dielectric. Solution of Boundary Condition by Means of Auxiliary Charges and the Method of Linear Algebraic Equations (In Russian) 51 V. V. Efimov et al. Optical Constants of the TGS Single Crystal Irradiated by Power Pulsed Electron Beam 65 Yu. N. Pokotilovski Interatomic Pair Potential and n-e Amplitude from Slow Neutron Scattering by Noble Gases 72 T. P. Topuria, M. D. Shafranov The Two-Coordinate Multiwire Proportional Chamber of the High Spatial Resolution (In Russian) 79 Yu. S. Lutostansky et al. Neutron Drip Line in the Region of O-Mg Isotopes 86 Письма в ЭЧАЯ. 2003. № 1[117] Particles and Nuclei, Letters. 2003. No. 1[117]

ИМЕННОЙ УКАЗАТЕЛЬ К ЖУРНАЛУ «ПИСЬМА В ЭЧАЯ» за 2002 т.

Абызов А. №5, с. 40 Брож Я. — №4, с. 86 Авдеев С. П. №4, с. 5 Бромберг К. — №5, с. 72 Агарвал К. С. №6, с. 12 Будагов Ю. А. — №4, с. 36 Адамович М. И. №1,с.29 — №5, с. 12 Ажгирей Л. С. №4, с. 51 — №5, с. 25 Анагностатос Г. С. №2, с. 5 Будзановски А. — №4, с.5 Анзари А. №3, с.66 Бычков В. Н. — №2, с. 64 Антонов А. Н. №2, с. 5 Вальтер М. — №5, с. 40 Аракелян С. Г №5, с. 53 Ватага Е. — №5, с.61 Артемьев В. А. №3, с. 56 Вила И. — №5, с. 72 Артиков А. № 5, с. 25 Вильгельм И. — №4, с. 86 №5, с. 72 Виноградов Е. А. — №1, с. 39 Ашманскас У. №5, с. 12 — №6, с. 65 Балдин А. М. №1,с.29 Вирясов К. С. — №2, с. 64 Барашенков В. С. №6, с. 38 Волков А. — №4, с. 95 Батусов В. Ю. №4, с. 36 Волков В. И. — №2, с. 22 Белашев Б. 3. №3, с. 77 Гайде Ж.- К. — №4, с. 36 №4, с. 63 Галкин В. И. — №3,с. 19 №6, с. 44 Гейер Р. — №2, с. 64 Беллеттини Дж. №5, с. 25 Гиапитзакис Я. — №2, с.5 Белозеров А. В. №1, с.5 Глаголев В. — №5, с. 12 Белфортэ С. №5, с. 12 Голиков В. В. — №5, с. 82 Бельков А. А. №2, с.51 Гомез Г. — №5, с. 72 №5, с. 40 Гончарова Л. А. — №3, с. 19 Беляев А. В. №5, с. 53 Горбачева Н. М. — №2, с. 64 Берек Г. №1, с.5 Горшков В. А. — №1, с.5 Бобылева Л. В. №6, с.5 Грейпеос М. — №2, с. 74 Бонюшкина А. Ю. №2, с. 39 Гуревич Г. М. — №4, с. 86 Борзаков С. Б. №2, с. 45 Гусаков Ю. В. — №2, с. 64 Борисов Н. С. №4, с. 86 Дейц А. — №1, с. 57 Ботвина А. С. №4, с.5 Джахар И. Р. — №3, с. 66 Бочкарев О. В. №4, с.5 Ди Торо М. — № 1, с. 13 Браднова В. №1, с.29 Долежал 3. — №4, с. 86 Брида И. №1, с.5 Донати С. — №5, с. 12 Именной указатель 119

Дреосси Д. №5, с. 25 Комаров В. — №4, с. 86 Дронов В. А. №1, с. 29 Копенкин В. В. — №3, с. 19 Дугинова Е. В. №4, с.5 Короткое С. П. — №1,с.5 Дымов С. №4, с. 95 Костенко Б. Ф. — №3, с. 32 Енковский Л. Л. №5, с. 67 Котельников К. А. — №3,с.19 Еремин А. В. №1, с.5 Кох Р. — №4, с. 95 Ефимов В. В. №1, с. 39 Коцинян А. М. — №6, с. 29 №6, с. 65 Краснов В. А. — №1,с.29 Жданов А. А. №4, с.51 Кривошеина И. В. — №1, с.57 Жмыров В. Н. №4, с.51 Кубик П. — №4, с. 86 Жуков И. А. №2, с. 64 Кузнецов В. Д. — №3,с.32 Журавлев Н. №4, с. 95 Кузнецов И. В. — №6, с.5 Займидорога О. А. №4, с. 59 Кузьмин Е. А. — №4, с.5 Залиханов Б. №4, с. 95 Кузьмин Е. С. — №4, с. 86 Зарубин П. И. №1,с.29 Кулагин Е. Н. — №5, с. 82 Зверев М. В. №6, с. 86 Куликов А. — №4, с. 95 Земляная Е. В. №2, с.5 Куликов С. А. — №5, с. 82 Злобин Ю. Л. №2, с. 64 Кутрулос X. — №2, с. 74 Злоказов В. Б. №1, с.25 Ладыгин В. П. — №2, с. 13 Иванов В. В. №2, с. 39 — №4, с.51 №6, с. 65 Лазарев А. Б. — №4, с. 86 Иванов М. П. №4, с. 86 Ланев А. В. — №2, с.51 Иванченко И. М. №2, с. 64 — №5, с. 40 Иерусалимов А. П. №5, с. 53 Ларионова В. Г. — №1, с. 29 Изосимов И. Н. №2, с. 36 Лассер К. — №4, с. 36 Ильгнер К. №2, с. 64 Легар Ф. — №4, с.51 Инкагли М. №5, с.25 — №4, с. 86 Кабаченко А. П. №1, с.5 Лепехин Ф. Г. — №3,с.25 КадревД. Н. №2, с.5 Ливийский В. В. — №2, с.64 Казимов А. А. №2, с. 36 Лукьянов В. К. — №2, с.5 Калинников В. Г. №2, с. 36 Лукьянов С. М." — №6, с. 86 Калмыков А. В. №1,с.39 Луханин А. А. — №4, с.86 Каманин Д. В. №3, с. 32 Лысан В. М. — №2, с.64 Каминский А. К. №3, с. 47 Лютостанский Ю. С. — №6, с. 86 Карнаухов В. А. №4, с.5 Ляблин М. В. — №4, с. 36 Карози Р. №5, с. 12 Маврин Б. Н. — №1, с. 39 Карпенко Н. Н. №2, с.51 Малахов И. Ю. — №4, с. 14 Карч В. №4, с.5 Маллот Г. — №2, с.64 Каушик М. №3, с. 66 Малышев О. Н. — №1, с.5 Кекелидзе Г. Д. №2, с. 64 Манагадзе А. К. — №3, с. 19 Кириллов Д. А. №2, с. 22 Матафонов В. Н. — №4, с. 86 Клапдор- Матвеев Д. А. — №2, с.64 Клайнгротхаус X. В. №1,с.57 Мачарашвили Г. — №4, с. 95 Клевцова Е. А. №6, с. 65 Мелихов В. В. — №5, с. 82 Коваленко А. Д. №1, с. 29 Мензионе А. — №5, с. 25 Кодыш П. №4, с. 86 — №5, с. 61 120 Именной указатель

Мерзляков С. №4, с. 95 — №5, с. 72 Миллер М. Б. №3,с.32 Ракобольская И. В. — №3, с. 19 Минашкин В. Ф. №1,с.39 Римарциг Б. — №4, с. 95 Мираллес-Верже Л. №4, с. 36 Ристори Л. — №5, с. 12 Михайлов И. Н. № 1, с. 13 Роганова Т. М. — №3, с. 19 Михайлова Т. И. №1, с. 13 Родионов В. К. — №4, с.5 Мишин С. В. №2, с. 64 Рубинская Л. А. — №1, с.5 Молодцова И. В. № 1, с. 13 Рукояткин П. А. — №1, с. 29 Неганов А. Б. №4, с. 86 Русакова В. В. — №1, с. 29 Недев С. №2, с. 13 Русакович Н. А. — №4, с. 36 Несси М. №4, с. 36 Сагайдак Р. Н. — №1, с.5 Ниорадзе М. №4, с. 95 Салманова Н. А. — №1, с. 29 Новиков Е. А. №2, с. 64 Саркар С. — №5, с. 12 Новикова Н. Н. №1, с. 39 Свешников К. А. — №4, с. 14 №6, с. 65 Свешникова Л. Г. — №3,с. 19 Норбек Э. №4, с.5 Свирихин А. И. — №1, с.5 Оганесян Ю. Ц. №1, с.5 Сейфарт X. — №4, с. 95 Ойшлер X. №4, с.5 Семенов А. — №5, с. 12 Орлова Г. И. №1, с.29 Семенов С. В. — №5, с.5 Оседло В. И. №3, с. 19 Сермягин А. В. — №3, с. 32 Пантелеев Ц. №2, с. 45 Сиколенко В. В. — №1, с. 39 Паулетта Дж. №5, с. 25 — №6, с. 65 №5, с. 72 Сисакян А. Н. — №2, с. 64 Пенев В. Н. №4, с. 73 — №4, с. 36 Пензо А. №5, с. 25 — №5, с. 12 Пенионжкевич Ю. Э. №6, с. 86 — №6, с. 29 Перельштейн О. Э. №6, с.5 Ситник И. М. — №2, с. 22 Перельштейн Э. А. №6, с. 5 Скрибано А. — №5, с. 25 Пересадько Н. Г. №1, с. 29 Скрыпник А. В. — №1, с. 39 Петрус А. №4, с. 95 Смирнова Е. М. — №1, с.5 Печеное В. Н. №5, с. 53 Солнышкин А. А. — №2, с. 36 Пешехонов В. Д. №2, с. 64 Спасова К. — №2, с.5 Писарев И. Л. №4, с. 86 Спинелла Ф. — №5, с. 12 Пискунов Н. М. №2, с. 22 Спиридонов А. А. — №2, с. 51 Плеханов Е. Б. №5, с. 53 — №5, с. 40 Плис Ю. А. №2, с. 22 Стефанини А. — №5, с. 25 №4, с. 86 Столетов Г. Д. — №4, с.51 Покотиловский Ю. Н. №6, с. 72 Стрелков А. В. — №2, с. 45 Полухина Н. Г. №3, с. 19 Струминский Б. В. — №5, с. 67 Понтекорво Д. Б. №2, с. 39 Сулейманов М. К. — №3, с. 77 Попеко А. Г. №1, с.5 — №6, с.44 Прокофьев А. Н. №4, с. 51 — №4, с. 63 Прокошин Ф. №5, с. 61 Ткачев Ф. В. — №2, с. 28 №5, с. 72 Токар С. — №5, с. 61 Пунзи Д. №5, с. 12 Топилин Н. Д. — №4, с. 36 Пухов О. №5, с. 25 Топурия Т. П. — №6, с. 51 Именной указатель 121

№6, с. 79 Троян А. Ю. №5, с. 53 Чиладзе Б. — №4, с. 95 Троян Ю. А. №5, с. 53 Чириков-Зорин И. — №5, с. 25 Тютюнников С. И. №1, с. 39 Члачидзе Г. — №5, с. 12 №6, с. 65 — №5, с. 25 Упсилантис К. •№2, с. 74 Чохели Д. — №5, с.25 Усов Ю. А. •№4, с.86 — №5, с. 72 Федоров А. Н. №4, с. 86 ЧулковЛ. В. — №4, с.5 Федоров С. М. №4, с. 14 Шабалин Е. П. — №5, с. 82 Фесслер М. •№2, с. 64 Шафранов М. Д. — №6, с.51 Фиори И. №5, с. 12 — №6, с. 79 Халили М. Ф. •№4, с. 14 Швейда Я. — №4, с. 86 Харламов С. П. №1, с. 29 Шебеко А. — №2, с. 74 Хартманн М. •№4,с.95 Шевченко О. Ю. — №6, с. 29 Хрущев В. В. •№5,с.5 Шелаев И. А. — №1,с.48 ХубуаД. И. №4, с. 36 Шилов С. Н. — №4, с.86 Хулъсберген В. •№5, с. 40 Широков М. И. — №3,с.43 Цветков А. И. №4, с. 86 Шкловская А. И. — №4, с. 73 Челноков М. Л. •№1,с.5 Шляйхерт Р. — №4, с. 95 Чепигин В. И. №1,с.5 Шокин В. И. — №2, с.64 Червелли Ф. •№5,с.25 Эссер Р. — №4, с.5 Черников Н. А. •№3,с.5 Юрьев М. 3. — №6, с. 38 Черны Я. №4, с. 86 Ядав X. Л. — №3,с.66 Чернявский М. М. №1, с. 29 — №6, с. 12 Черри А. №5, с. 12 Яковлев В. А. — №1, с. 39 Черри К. №5, с. 61 — №6, с. 65 Письма в ЭЧАЯ. 2003. № 1[116] Particles and Nuclei, Letters. 2003. No. 1 [116]

PARTICLES AND NUCLEI, LETTERS. 2002 AUTHOR INDEX

Abyzov A. No. 5, p. 40 Bromberg C. — No. 5, p. 72 Adamovich M. I. No. 1, p. 29 Broz J. — No. 4, p. 86 Agarwal К. С. No. 6, p. 12 Budagov J. — No. 4, p. 36 Anagnostatos G. S. No. 2, p. 5 — No. 5, p. 12 Ansari A. No.3, p. 66 — No. 5, p. 25 Antonov A. N. No. 2, p. 5 Budzanowski A. — No. 4, p. 5 Arakelian S. G. No. 5, p. 53 Bychkov V.N. — No. 2, p. 64 Artemiev V. A. No.3, p. 56 Carosi R. — No. 5, p. 12 Artikov A. No. 5, p. 25 Cerny J. — No. 4, p. 86 No. 5, p. 72 Cerri A. — No. 5, p. 12 Ashmanskas W. No. 5, p. 12 Cerri C. — No. 5, p. 61 Avdeyev S.P. No. 4, p. 5 Cervelli F. — No. 5, p. 25 Azhgirey L. S. No. 4, p. 51 Chelnokov M. L. — No. 1, p. 5 Baldin A. M. No. 1, p. 29 Chepigin V. I. — No. 1, p. 5 Barashenkov V. S. No. 6, p. 38 Chernikov N. A. — No.3, p. 5 Batusov V. No. 4, p. 36 Chernyavsky M. M. — No. 1, p. 29 BelashevB.Z. No.3, p. 77 Chiladze B. — No. 4, p. 95 No. 4, p. 63 Chirikov-Zorin I. — No. 5, p. 25 No. 6, p. 44 Chlachidze G. — No. 5, p. 12 Belforte S. No. 5, p. 12 — No. 5, p. 25 Beljaev A. V. No. 5, p. 53 Chokheli D. — No. 5, p. 25 Bellettint G. No. 5, p. 25 — No. 5, p. 72 Belozerov A. V. No. l,p.5 ChulkovL.V. — No. 4, p. 5 Bel'kov A. A. No. 2, p. 51 Deitz A. — No. 1, p.57 No. 5, p. 40 Di Того М. — No. 1, p. 13 Berek G. No. 1, p. 5 Dolezal Z. — No. 4, p. 86 Bobyleva L. V. No. 6, p. 5 Donati S. — No. 5, p. 12 Bochkarev O. V. No. 4, p. 5 Dreossi D. — No. 5, p. 25 Bonushkina A. Yu. No. 2, p. 39 Dronov V. A. — No. 1, p. 29 Borisov N. S. No. 4, p. 86 Duginova E. V. — No. 4, p. 5 Borzakov S. B. No. 2, p. 45 Dymov S. — No. 4, p. 95 Botvina A. S. No. 4, p. 5 Efimov V.V. — No. 1, p. 39 Bradnova V. No. 1, p. 29 — No. 6, p. 65 Brida I. No. l,p.5 Esser R. — No. 4, p. 95 Author Index 123

Faessler M. • No. 2, p. 64 KirillovD.A. — No. 2, p. 22 Fedorov A. N. -No. 4, p. 86 Klapdor-Kleingrothaus H. V. — No. 1, p. 57 Fedorov S. •No. 4, p. 14 Klevtsova E. A. — No. 6, p. 65 Fiori I. •No. 5, p. 12 Koch R. — No. 4, p. 95 Galkin V. I. • No. 3, p. 19 Kodys P. — No. 4, p. 86 Gayde J. C. •No. 4, p. 36 Komarov V. — No. 4, p. 95 Geyer R. No. 2, p. 64 Kopenkin V.V. — No.3, p. 19 Giapitzakis J. •No. 2, p. 5 Korotkov S. P. — No. 1, p. 5 Glagolev V. •No. 5, p. 12 Kostenko B. F. — No.3, p. 32 Golikov V.V. • No. 5, p. 82 Kotelnikov K. A. — No. 3, p. 19 Gomez G. No. 5, p. 72 Kotzinian A. M. — No. 6, p. 29 Goncharova L. A. No. 3, p. 19 Koutroulos C. — No. 2, p. 74 Gorbacheva N. M. No. 2, p. 64 Kovalenko A. D. — No.l, p. 29 Gorshkov V. A. No. 1, p. 5 Krasnov V. A. — No. 1, p. 29 Gousakov Yu. V. No. 2, p. 64 Krivosheina I. V. — No.l, p. 57 Grypeos M. No. 2, p. 74 Kubik P. — No. 4, p. 86 Gurevich G. M. No. 4, p. 86 Kulagin E. N. — No. 5, p. 82 Hartmann M. No. 4, p. 95 Kulikov A. — No. 4, p. 95 Hulsbergen W. No. 5, p. 40 Kulikov S. A. — No. 5, p. 82 Ilgner C. No. 2, p. 64 Kuzmin E. A. — No. 4, p. 5 Incagli M. No. 5, p. 25 Kuzmin E. S. — No. 4, p. 86 Ivanchenko I. M. No. 2, p. 64 Kuznetsov I. V. — No. 6, p. 5 Ivanov M. P. No. 4, p. 86 Kuznetsov V. D. — No.3, p. 32 Ivanov V. V. No. 2, p. 39 Ladygin V. P. — No. 2, p. 13 No. 6, p. 65 — No. 4, p. 51 Izosimov I. N. No. 2, p. 36 Lanyov A. V. — No. 2, p. 51 Jakharl.R. No. 3, p. 66 — No. 5, p. 40 Jenkovszy L. L. No. 5, p. 67 Larionova V. G. — No.l, p. 29 Jerusalimov A. P. No. 5, p. 53 Lasseur C. — No. 4, p. 36 Joukov I. A. No. 2, p. 64 Lazarev A.B. — No. 4, p. 86 Kabachenko A. P. No.l, p. 5 Lehar F. — No. 4, p.51 KadrevD.N. No. 2, p. 5 — No. 4, p. 86 Kalinnikov V. G. No. 2, p. 36 Lepekhin F. G. — No.3, p. 25 Kalmykov A. V. No. 1, p. 39 Livinski V.V. — No. 2, p. 64 Kamanin D. V. No. 3, p. 32 Lukhanin A. A. — No. 4, p. 86 Kaminsky A. K. No. 3, p. 47 Lukyanov S. M. — No. 6, p. 86 Karcz W. No. 4, p. 5 Lukyanov V. K. — No. 2, p. 5 Karnaukhov V. A. No. 4, p. 5 Lutostansky Yu. S. — No. 6, p. 86 Karpenko N. N. No. 2, p. 51 Lyablin M. — No. 4, p. 36 Kaushik M. No. 3, p. 66 Lysan V. M. — No. 2, p. 64 Kazimov A. A. No. 2, p. 36 Macharashvili G. — No. 4, p. 95 KekelidzeG.D. No. 2, p. 64 Malakhov I. — No. 4, p. 14 Khalili M. No. 4, p. 14 Mallot G. — No. 2, p. 64 Kharlamov S. P. No. 1, p. 29 Malyshev 0. N. — No.l, p. 5 Khruschev V.V. No. 5, p. 5 Managadze A. K. — No.3, p. 19 Khubua J. No. 4, p. 36 Matafonov V. N. — No. 4, p. 86 124 Author Index

Matveev D. A. No. 2, p. 64 Popeko A. G. No. 1, p. 5 Mavrin B.N. No. 1, p. 39 Prokofiev A. N. No. 4,p. 51 Melikhov V.V. No. 5, p. 82 Prokoshin F. No. 5,p.61 Menzione A. No. 5, p. 25 No. 5,p. 72 No. 5, p. 61 Pukhov 0. No. 5, p. 25 Merzlyakov S. No. 4, p. 95 No. 5,p. 72 Mikhailov I.N. No. 1, p. 13 Punzi G. •No. 5,p. 12 Mikhailova T. I. No. 1, p. 13 Rakobolskaya I. V. No. 3, p. 19 Miller M. В. No. 3, p. 32 Rimarzig B. •No. 4,p. 95 Minashkin V. F. No. 1, p. 39 Ristori L. No. 5,p. 12 Miralles Verde L. No. 4, p. 36 Rodionov V. K. No. 4,p. 5 Mishin S. V. No. 2, p. 64 Roganova Т. М. •No.3, p. 19 Molodtsova I. V. No. 1, p. 13 Rubinskaya L. A. No. 1, p. 5 Nedev S. No. 2, p. 13 Rukoyatkin P. A. - No. 1, p. 29 Neganov A.B. No. 4, p. 86 Rusakova V.V. No. 1, p. 29 Nessi M. No. 4, p. 36 Rusakovich N. •No. 4,p. 36 Nioradze M. No. 4, p. 95 SagaidakR.N. No. 1, p. 5 Norbeck E. No. 4, p. 5 Salmanova N. A. No. 1, p. 29 Novikov E. A. No. 2, p. 64 Sarkar S. •No. 5,p. 12 Novikova N. N. No. 1, p. 39 Schleichert R. •No. 4,p. 95 No. 6, p. 65 Scribano A. -No. 5,p. 25 Oeschler H. No. 4, p. 5 Semenov A. •No. 5,p. 12 Oganessian Yu. Ts. No. 1, p. 5 Semenov S.V. -No. 5,p. 5 Sermyagin A.B. - No. 3, p. 32 Orlova G. I. No. 1, p. 29 Seyfarth H. -No. 4,p. 95 Osedlo V. I. No.3, p. 19 Shabalin E. P. -No. 5,p. 82 Panteleev Ts. No. 2, p. 45 Shafranov M. D. - No. 6, p.51 Pauletta G. No. 5, p. 25 - No. 6, p. 79 No. 5, p. 72 Shebeko A. - No. 2, p. 74 Pechenov V.N. No. 5, p. 53 Shelaev I. A. - No. 1, p. 48 Penev V. No. 4, p. 73 Shevchenko O. Yu. - No. 6, p. 29 Penionzhkevich Yu. E. No. 6, p. 86 Shilov S. N. -No. 4,p. 86 Penzo A. No. 5, p. 25 Shirokov M. I. - No. 3, p. 42 Perelstein E. A. No. 6, p. 5 Shklovskaia A. -No. 4,p. 73 Perelstein О. Е. No. 6, p. 5 Shokin V. I. - No. 2, p. 64 Peresadko N. G. No. 1, p. 29 Sikolenko V. V. - No. 1, p. 39 Peshekhonov V.D. No. 2, p. 64 - No. 6, p. 65 Petrus A. No. 4, p. 95 Sissakian A. N. -No. 2,p. 64 Pisarev I. L. No. 4, p. 86 -No. 4,p. 36 Piskunov N. M. No. 2, p. 22 -No. 5,p. 12 PlekhanovE.B. No. 5, p. 53 - No. 6, p. 29 Plis Yu.A. No. 2, p. 22 Sitnikl.M. - No. 2, p. 22 No. 4, p. 86 Skripnik A. V. -No. 1,p. 39 Pokotilovski Yu. N. No. 6, p. 72 Smirnova E. M. -No. 1,p. 5 Polukhina N. G. No.3, p. 19 Solnyshkin A. A. -No. 2,p. 36 Pontecorvo G. B. No. 2, p. 39 Spasova K. - No. 2, p. 5 Author Index 125

Spinella F. No. 5, p. 12 Vila I. — No. 5, p. 72 Spiridonov A. A. No. 2, p. 51 Vinogradov E. A. — No. 1, p. 39 No. 5, p. 40 — No. 6, p. 65 Stefanini A. No. 5, p. 25 Viriasov K. S. — No. 2, p. 64 Stoletov G. D. No. 4, p. 51 Volkov A. — No. 4, p. 95 Strelkov A.V. No. 2, p. 45 Volkov V.I. — No. 2, p. 22 Struminsky B. V. No. 5, p. 67 Walter M. — No. 5, p. 40 Suleimanov M. K. No.3, p. 77 Wilhelm I. — No. 4, p. 86 No. 4, p. 63 Yadav H.L. — No.3, p. 66 No. 6, p. 44 — No. 6, p. 12 Svejda J. No. 4, p. 86 Yakovlev V.A. — No.l, p. 39 Sveshnikov K. No. 4, p. 14 — No. 6, p. 65 Sveshnikova L. G. No.3, p. 19 Yeremin A. V. — No.l, p.5 Svirikhin A. I. No. 1, p. 5 Ypsilantis K. — No. 2, p. 74 Tiutiunnikov S.I. No.l, p. 39 Yuriev M. Z. — No. 6, p. 38 No. 6, p. 65 Zaimidoroga O. A. — No. 4, p. 59 Tkachov F. V. No. 2, p. 28 Zalikhanov B. — No. 4, p. 95 TokarS. No. 5, p. 61 Zarubin P. I. — No.l, p.29 Topiline N. No. 4, p. 36 Zemlyanaya E. V. — No. 2, p. 5 Topuria T. P. No. 6, p. 51 Zhdanov A. A. — No. 4, p. 51 No. 6, p. 79 Zhmyrov V. N. — No. 4, p. 51 Troyan A. Yu. No. 5, p. 53 Zhuravlev N. — No. 4, p. 95 Troyan Yu. A. No. 5, p. 53 Zlobin Yu.L. — No. 2, p. 64 Tsvetkov A. I. No. 4, p. 86 Zlokazov V.B. — No.l, p.25 Usov Yu.A. No. 4, p. 86 Zverev M. V. — No. 6, p. 86 Vataga E. No. 5, p. 61 ПРАВИЛА ОФОРМЛЕНИЯ АВТОРСКОГО ОРИГИНАЛА СТАТЬИ В ЖУРНАЛ «ПИСЬМА В ЭЧАЯ»

1. Оригинал статьи предоставляется автором в двух экземплярах. Необходимо предо- ставить дискету с текстовым файлом в формате LaTex-2e (не следует вводить свои макро- команды) и файлами рисунков, таблиц, подрисуночных подписей. Оригинал должен включать все необходимые элементы статьи, иметь сквозную нумерацию страниц и быть подписан всеми авторами. 2. На первой странице статьи указывается индекс УДК, название статьи, инициалы и фамилии авторов на двух языках (русском и английском), место работы. Затем следует аннотация на русском и английском языках, включающая характери- стику основной темы, цели работы и ее результаты. В аннотации указывают, что нового несет в себе данная статья. Аннотация должна быть краткой, 5-8 строк. 3. В формулах все буквы латинского алфавита, обозначающие физические величины, набирают светлым курсивным шрифтом (Е, V.mvL др.). Векторы следует набирать прямым полужирным шрифтом, без стрелок сверху. Шрифтом прямого светлого начертания набирают следующие обозначения: чисел по- добия (Ar, Re и др.); функций (sin, arcsin, sh и др.); условных математических сокращений (max, min, opt, const, idem, lim, Ig, In, log, del, exp) и др. Латинские буквы в индексах набирают строчным курсивом, кроме сокращений, в

том числе и от фамилий (их набирают шрифтом прямого начертания): fabc, ^ C/Yult, ^-cfr(?>P)> x mm • Буквы русского алфавита в индексах используют, когда отсутствуют стан- дартизованные международные индексы, и набирают строчным шрифтом прямого начер-

тания: [7ф — фазное напряжение; Р, — мощность возбуждения. Символы химических элементов набирают шрифтом прямого светлого начертания: Cl, Fe. 4. Рисунки должны быть четкими и качественными. Желательно, чтобы файл рисунка был подготовлен в формате .eps (Encapsulated PostScript). He рекомендуется предостав- лять цветные рисунки. Рисунки должны быть одного масштаба, их максимальная ширина 13,5 см, максимальная высота 19,5 см (с учетом подписи). Размеры небольших рисунков не должны превышать 7 см по ширине. Не следует загромождать рисунок ненужными деталями: надписи выносятся в под- пись, а на рисунке заменяются арабскими цифрами или буквами. Если рисунок предостав- лен на отдельном листе, то необходимо указать фамилии авторов, название статьи и номер рисунка. 5. Библиографические ссылки приводят в конце статьи в порядке их упоминания в тек- сте под рубрикой «Список литературы». Ссылки на неопубликованные работы не допус- каются. Ниже следуют примеры оформления: • книги: Кокорева Л. В. Проектирование банков данных. М.: Наука, 1998. 241 с. • статьи из сборника: Быстрицкий В. М. и др. Исследование температурной зависимости скорости образования ме- зомолекул dd\i в газообразном дейтерии // Мезоны в веществе: Тр. Междунар. симпоз. Дубна, 1977. С. 199-205. • статей из журналов: Афанасьев Ю. В. и др. Лазерное инициирование термоядерной реакции в неоднородных сфери- ческих мишенях // Письма в ЖЭТФ. 1975. Т. 21, вып. 2. С. 150-155. Barbashov В. М., Pestov I. В. On Spinor Representations in the Weyl Gauge Theory // Mod. Phys. Lett. A. 1997. V. 12, No. 26. P. 1957-1968. • препринта и сообщения: Тяпкин А. А. Экспериментальные указания о существовании тахионов, полученные при иссле- довании черепковского излучения. Препринт ОИЯИ Д1 -99-292. Дубна, 1999. 4 с. Muzychka Al. Yu., Pokotilovski Yu. N., Geltenbort P. Search for an Anomalous Transmission of Ultra- cold Neutrons Through Metal Foils. JINR Commun. E3-98-18. Dubna, 1998. 10 p.

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