True quantum chaos
It is shown, that the dynamic chaos is inherent for quantum systems not only in semi-classical approximation. As an example the especially quantum three-level system is considered. The value of external perturbation is analytically found, at which the regimes with dynamic chaos is realized. The poss...
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| Zitieren: | True quantum chaos / V.A. Buts // Вопросы атомной науки и техники. — 2008. — № 6. — С. 120-122. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-1107722017-01-07T03:03:26Z True quantum chaos Buts, V.A. Plasma electronics It is shown, that the dynamic chaos is inherent for quantum systems not only in semi-classical approximation. As an example the especially quantum three-level system is considered. The value of external perturbation is analytically found, at which the regimes with dynamic chaos is realized. The possible consequences of regimes with dynamic chaos in quantum systems are discussed. Показано, що динамічний хаос є характерним для квантових систем не тільки в квазікласичному наближенні. Як приклад розглянута сугубо квантова трьохрівнева система. Аналітично знайдена величина зовнішнього збурення, при якому реалізується режим з динамічним хаосом. Обговорюються можливі наслідки режимів з динамічним хаосом в квантових системах. Показано, что динамический хаос характерен для квантовых систем не только в квазиклассическом приближении. В качестве примера рассмотрена сугубо квантовая трехуровневая система. Аналитически найдена величина внешнего возмущения, при котором реализуется режим с динамическим хаосом. Обсуждаются возможные следствия режимов с динамическим хаосом в квантовых системах. 2008 Article True quantum chaos / V.A. Buts // Вопросы атомной науки и техники. — 2008. — № 6. — С. 120-122. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 05.45.Mt http://dspace.nbuv.gov.ua/handle/123456789/110772 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics Buts, V.A. True quantum chaos Вопросы атомной науки и техники |
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It is shown, that the dynamic chaos is inherent for quantum systems not only in semi-classical approximation. As an example the especially quantum three-level system is considered. The value of external perturbation is analytically found, at which the regimes with dynamic chaos is realized. The possible consequences of regimes with dynamic chaos in quantum systems are discussed. |
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Buts, V.A. |
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Buts, V.A. |
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Buts, V.A. |
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True quantum chaos |
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True quantum chaos |
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True quantum chaos |
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True quantum chaos |
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True quantum chaos |
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true quantum chaos |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2008 |
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Plasma electronics |
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http://dspace.nbuv.gov.ua/handle/123456789/110772 |
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True quantum chaos / V.A. Buts // Вопросы атомной науки и техники. — 2008. — № 6. — С. 120-122. — Бібліогр.: 3 назв. — англ. |
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Вопросы атомной науки и техники |
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AT butsva truequantumchaos |
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2025-07-08T01:06:21Z |
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2025-07-08T01:06:21Z |
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1837038863698100224 |
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120 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 6.
Series: Plasma Physics (14), p. 120-122.
TRUE QUANTUM CHAOS
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
E-mail: vbuts@kipt.kharkov.ua
It is shown, that the dynamic chaos is inherent for quantum systems not only in semi-classical approximation. As an
example the especially quantum three-level system is considered. The value of external perturbation is analytically
found, at which the regimes with dynamic chaos is realized. The possible consequences of regimes with dynamic chaos
in quantum systems are discussed.
PACS: 05.45.Mt
1. INTRODUCTION
Now there is paradigm in which have been formulated
in an obvious kind, that the evolutionary operator
describing dynamics systems with a regime with dynamic
chaos, should have two indisputable properties: 1) To be
stretching. 2) To be nonlinear. Certainly, these two features
are necessary for realization of dynamic chaos. However
concerning the second property (to be nonlinear) it is
necessary to give some explanations. Really, for example,
it is known, that the equations of the quantum mechanics
and Maxwell equations are the linear equations. However
at transition from the quantum equations to the classical
equations, and also at transition from the Maxwell equation
to the equations of geometrical optics we have got systems
of the nonlinear equations. Such equations can describe
regimes with dynamic chaos. Thus, now are known at least
two examples, when in linear systems at the certain
meanings of their parameters (which allow to pass to
classical consideration) the regimes with dynamic chaos are
possible. In work [1] is shown, that this situation is
considerably more widespread, that the regimes with
dynamic chaos is internally inherent in huge number of
linear systems. The results of the analysis of these features
for the quantum systems are presented in this article.
2. STATEMENT OF A PROBLEM.
THE BASIC EQUATIONS
Let's consider quantum system, which is described by
Hamiltonian:
0 1( )H H H t= + . (1)
Second item in the right part describes perturbation. The
wave function of system (1) obey to the Schrödinger
equation which decision we shall search as a series of
own functions of the unperturbed task for:
( ) ( ) exp( )n n n
n
t A t i tψ ϕ ω= ⋅ ⋅∑ , (2)
where /n nEω = h ; 0
n
n n ni H E
t
ϕ ϕ ϕ∂
= = ⋅
∂
h .
Let's substitute (2) in the equation Schrödinger and in the
ordinary way we shall get system of the connected
equations for finding of complex amplitudes:
( )n n m m
m
i A U t A⋅ = ⋅∑&h , (3)
where 1( ) exp[ ( ) / ]n m m n n mU H t i t E E dqϕ ϕ∗= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∫ h .
Let's consider the most simple case of harmonic
perturbation: 1( ) exp( )H t U i t= ⋅ Ω . Then the matrix
elements of interaction will get the following expression:
exp{ [( ) / ]}n m n m n mU V i t E E= ⋅ ⋅ − +Ωh ,
nm n mV U dqϕ ϕ∗= ⋅ ⋅∫ (4)
Let's consider dynamics of three-level system
( 0 , 1 , 2 ) (see Fig.1). We'll consider that frequency of
external perturbation and the own meanings of energy of
these levels satisfy to the ratio:
1, 0m n= = , 1 0E EΩ = −h , 2, 0m n= = ,
2 0( ) E EδΩ+ = −h , δ << Ω . (5)
These ratios show on that fact, that the frequency of
external perturbation is resonant for transitions between
zero and first levels, and the energy of the third level is
slightly differs from energy second one. Using these ratios
in system (3), it is possible to left only three equations:
0 01 1 02 2 exp( )i A V A V A i tδ⋅ ⋅ = + ⋅ ⋅ ⋅&h ; 1 10 0i A V A⋅ ⋅ =&h ;
2 20 0 exp( )i A V A i tδ⋅ ⋅ = ⋅ − ⋅ ⋅&h . (6)
Let matrix elements of interaction for direct and opposite
transitions are equal ( 0 0 , ( 1;2)i iV V i= = ). Then from (6)
we find the following connection between squares of
complex amplitudes nA :
( )2 2 2
0 1 2 2 0 22 sind A A A A A
d
μ δτ
τ
⎡ ⎤− − = ⋅ ⋅⎣ ⎦ , (7)
where tτ = Ω⋅ , /δ δ≡ Ω , 02 /i iVμ = ⋅ ⋅Ωh . From this
ratio follows, that if the third level coincides with second
(two-level system, 0δ = ), the system (6) has only one
degree of freedom. The development of dynamic chaos in
such system is impossible. Below we shall see, that this
difference in energy between second and third levels
( )δh define the distance between nonlinear resonances.
For further analysis of dynamics of complex amplitudes
( )iA τ it is convenient present them as:
( ) ( ) exp( ( ))i iA a iτ τ ϕ τ= . (8)
Here ia , iϕ - real amplitudes and real phases. Substituting
(8) in (6) for a finding of the real amplitudes and phases,
we shall get the following system of the equations:
( ) ( )0 1 1 2 2 1sin sina a aμ μ= ⋅ ⋅ Φ + ⋅ ⋅ Φ& ,
( )1 1 0 sina aμ= − ⋅ ⋅ Φ& , ( )2 2 0 1sina aμ= − ⋅ ⋅ Φ& ,
( ) ( )0 1 2
1 2 1
1 0 0
cos cosa a a
a a a
μ μ
⎛ ⎞ ⎛ ⎞
Φ = − − Φ + Φ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
& , (9)
( ) ( )0 2 1
1 2 1 1
2 0 0
cos cosa a a
a a a
μ μ δ
⎛ ⎞ ⎛ ⎞
Φ = − − Φ + Φ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
& ,
121
where 1 0 1 2 0,ϕ ϕ ϕ ϕ δτΦ ≡ − Φ ≡ − + .
From first three equations of this system follows such
integral: 2 2 2
0 1 2a a a const+ + = . The system of the
equations (9) is nonlinear. In general, dynamics of such
system can be chaotic.
3. CRITERION OF DYNAMIC CHAOS
OCCURENCE
It is significant to find analytical conditions, at which
fulfilled the dynamics of quantum system (9) will be
chaotic. For this purpose in the beginning we shall
assume, that there are only two levels - zero and first.
Third level is absent ( 2 0A = ). In this case from system
(9) it is possible to get the following equation for the
phaseΦ :
( ) ( ) ( )
2 22 2 2 22
0 1 0 11
2 2
0 1
sin 2
2
a a a a
a a
μ ⎡ ⎤+ + −
⎢ ⎥Φ = − Φ
⎢ ⎥
⎣ ⎦
&& . (10)
The equation (10) represents the equation of a
mathematical pendulum. Minimal width of a nonlinear
resonance can be estimated by value 1~ μΔ . Let's
consider now situation, when the first level is absent
( 1 0A = ). Similarly to the previous case, from system (9)
it is possible to get the equation for a phase 1Φ . This
equation
also represents the equation of nonlinear oscillator. The
analysis of this equation gives the following estimation of
minimal width of a nonlinear resonance: 1 2~ μΔ . It is
natural to expect, that when the nonlinear resonances will
be overlapped, i.e. when the condition ( )1 2μ μ δ+ > will
be executed, dynamics of system (9) will be chaotic.
Condition of occurrence of local instability is convenient
to rewrite as:
01V δ> h or 01 / 1K V δ≡ >h . (11)
We shall assume now, that the conditions for
realization of dynamic chaos are executed. In this case the
investigated system will wander on three power levels. It
is interesting to give estimation for transition time, which
is necessary for the system to pass from one level to
another. For an estimation of time of transition in a
stochastic regime we shall assume, that in this regime any
correlation are absent. Then, for example, for value of an
average square of the real amplitude it is possible to get
the following estimation: 2 2 2
1 0~a aμ τ⋅ ⋅ . Thus, the
average time of transitions between levels in a stochastic
regime appears about a square of time of transition in a
regular regime: ( )22
01~ ( ) ~ /ch r Uτ τ ⋅Ωh . In a stochastic
regime it is possible diffuse of a quantum system along
energetic levels. Thus the time of diffuse in energy space
on value EΔ can be estimated by value:
( )2
01~ ( / ) /D E Uτ Δ Ω Ωh h . Let's note, that the time of
excitation of energy levels from a zero level on levels
with energy in a vicinity 0E E+ Δ by manyphoton
excitation will be inverse proportional to a square of a
compound matrix element: 2~ 1/ Hτ . Here H -
compound matrix element, which is equal to the sum of
products of separate matrix elements determining
transitions between intermediate (often virtual) levels.
Each matrix element is small value. Therefore, practically
always the time of transitions caused by stochastic
instability much less of the time of transitions, induced by
manyphoton processes. Thus, as soon as the conditions
for development of stochastic instability are executed, the
processes connected to her, will determine transitions
between levels, when the frequency of external
perturbation is much less than distance between levels
( EΔ >> Ωh ).
5. NUMERICAL RESULTS
System of the equations (6) and system of the
equations (9) were investigated numerically. It is
naturally, that dynamics of real and imaginary parts of
complex amplitudes nA was regular (system (6)).
Dynamics of the real amplitudes ia and phasesΦ and 1Φ
at performance of conditions for overlapping of nonlinear
resonances was chaotic: the spectra were wide, the
correlation functions quickly fell down, the main
Lyapunov parameters were positive. For an illustration of
chaotic regimes in figures 2-5 are represented:
dependence on time of amplitude 1a and phase Φ (fig. 2
and fig. 3), correlation function for a phase Φ (fig. 4),
and also distribution of the main Lyapunov parameters on
a phase plane ( 1,a Φ ). The dependences submitted in
these figures, are received at 0.1δ = , 1 0.2μ = , 2 0.2μ = .
Let's notice that despite of that fact, that dynamics of
functions ia and iϕ are chaotic, the dynamics of function
cosi ia ϕ⋅ - is regular.
Fig.1. Three-level system
Fig.2. Time evolution of amplitude a1
0
1
2
1 0E EΩ = −h
122
Fig.3. Time evolution of phase Φ 1
Fig.5. Main Lyapunov index
Fig.4. Correlation function for Φ
5. CONCLUSIONS
Thus, in quantum systems the regimes with dynamic
chaos can be realized. At that, it is necessary to note, that
the phenomenon of quantum chaos for a long time was
intensively studied (see, for example, [2,3]). But in this
cases, however, all authors have emphasized, that the
quantum chaos is not true chaos, that in quantum chaos
those quantum systems are studied, which parameters
allow the semi-classical description and which in a
classical limit have a regimes with dynamic chaos.
For this reason many authors take the name of quantum
chaos in inverted commas. In this work is investigated the
true quantum system. Its parameters are those that do not
allow semi-classical consideration. For this reason it is
possible to name a circle of such phenomena as true
quantum chaos.
The author thanks K.N. Stepanov for useful debates
and discussion of the results.
REFERENCES
1. V.A Buts. Chaotic dynamics of the linear systems //
Electromagnetic waves and electron systems. 2006,
v. 11, № 11, p.65-70.
2. M. Tabor. Chaos and Integrability in Nonlinear
Dynamics. New York. 1988, p.318.
3. G.M. Zaslavsky. Stochasticity of dynamical systems.
М.: “Science”, 1984, p.271 .
Article received 22.09.08.
ИСТИННЫЙ КВАНТОВЫЙ ХАОС
В.А.Буц
Показано, что динамический хаос характерен для квантовых систем не только в квазиклассическом
приближении. В качестве примера рассмотрена сугубо квантовая трехуровневая система. Аналитически
найдена величина внешнего возмущения, при котором реализуется режим с динамическим хаосом.
Обсуждаются возможные следствия режимов с динамическим хаосом в квантовых системах.
ІСТИННИЙ КВАНТОВИЙ ХАОС
В.А.Буц
Показано, що динамічний хаос є характерним для квантових систем не тільки в квазікласичному
наближенні. Як приклад розглянута сугубо квантова трьохрівнева система. Аналітично знайдена величина
зовнішнього збурення, при якому реалізується режим з динамічним хаосом. Обговорюються можливі
наслідки режимів з динамічним хаосом в квантових системах.
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