Stabilization of the excited states
The mechanism allowing to stabilize excited states of quantum systems is offered. The mechanism stabilization is similar to quantum Zeno’s effect. The difference consists that under system is not made supervision. Instead of it the system is periodically transferred in other quantum state. The speed...
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nasplib_isofts_kiev_ua-123456789-170222025-02-23T19:58:30Z Stabilization of the excited states Стабилизация возбужденных состояний Стабілізація збуджених станів Buts, V.A. Новые методы ускорения, сильноточные пучки The mechanism allowing to stabilize excited states of quantum systems is offered. The mechanism stabilization is similar to quantum Zeno’s effect. The difference consists that under system is not made supervision. Instead of it the system is periodically transferred in other quantum state. The speed, necessary for stabilization, of stabilizing transferring is determined. Предлагается механизм, позволяющий стабилизировать возбужденные состояния квантовых и классических систем. Механизм стабилизации аналогичен квантовому эффекту Зенона. Отличие заключается в том, что над системой не производится наблюдения. Вместо этого система периодически переводится в другое квантовое состояние. Определена необходимая для стабилизации скорость стабилизирующих переходов. Пропонується механізм, який дозволяє стабілізувати збуджені стани квантових та класичних систем. Механізм стабілізації аналогічний до квантового ефекту Зенона. Відмінність полягає у тому, що над системою не наглядають. Замість цього система періодично переводиться в інший квантовий стан. Знайдена необхідна для стабілізації швидкість переходів, які стабілізують систему. 2010 Article Stabilization of the excited states / V.A. Buts // Вопросы атомной науки и техники. — 2010. — № 3. — С. 90-93. — Бібліогр.: 3 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17022 en application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки Buts, V.A. Stabilization of the excited states |
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The mechanism allowing to stabilize excited states of quantum systems is offered. The mechanism stabilization is similar to quantum Zeno’s effect. The difference consists that under system is not made supervision. Instead of it the system is periodically transferred in other quantum state. The speed, necessary for stabilization, of stabilizing transferring is determined. |
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Article |
| author |
Buts, V.A. |
| author_facet |
Buts, V.A. |
| author_sort |
Buts, V.A. |
| title |
Stabilization of the excited states |
| title_short |
Stabilization of the excited states |
| title_full |
Stabilization of the excited states |
| title_fullStr |
Stabilization of the excited states |
| title_full_unstemmed |
Stabilization of the excited states |
| title_sort |
stabilization of the excited states |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2010 |
| topic_facet |
Новые методы ускорения, сильноточные пучки |
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https://nasplib.isofts.kiev.ua/handle/123456789/17022 |
| citation_txt |
Stabilization of the excited states / V.A. Buts // Вопросы атомной науки и техники. — 2010. — № 3. — С. 90-93. — Бібліогр.: 3 назв. — англ. |
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AT butsva stabilizationoftheexcitedstates AT butsva stabilizaciâvozbuždennyhsostoânij AT butsva stabílízacíâzbudženihstanív |
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2025-11-24T20:52:07Z |
| last_indexed |
2025-11-24T20:52:07Z |
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1849706443387699200 |
| fulltext |
STABILIZATION OF THE EXCITED STATES
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
The mechanism allowing to stabilize excited states of quantum systems is offered. The mechanism stabilization
is similar to quantum Zeno’s effect. The difference consists that under system is not made supervision. Instead of it
the system is periodically transferred in other quantum state. The speed, necessary for stabilization, of stabilizing
transferring is determined.
PACS: 03.65.Xp: 03.65.Yz
1. INTRODUCTION
The mechanism of stabilization of the excited states
offered in this work, is similar to quantum Zeno's effect
[1-2]. Therefore, to clarify the difference between of-
fered mechanism and Zeno's mechanism, we briefly
shall describe the contents of Zeno's effect. Let we have
a two-level quantum system. The zero level corresponds
to stationary, not excited state. The first level corre-
sponds to the excited state. Let now this system is under
action of resonant perturbation. We want to consider
how the crossing occurs from the zero level to the first
level and back. As known, such process is described by
the following simple system of the differential equa-
tions:
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 3. 90
1A 00 01i A V⋅ ⋅ =&h ; , (1) 1 10i A V A⋅ ⋅ =&h
where Ai − complex amplitudes of wave functions. The
square of the module of these amplitudes defines the
probability of a presence of the quantum system at the
correspond power level. The matrix elements of the
interaction and , generally, depend both on struc-
ture of considered system, and from the characteristics
of perturbation.
01V 10V
We shall be consider them equal, constant and real.
Let at the initial moment of time the quantum system is
in the excited state. Then the solutions of the equations
(1) will be functions:
( )1 cosA t= Ω ⋅ , (0 sin )A t= Ω ⋅ , (2)
where − Rabi frequency. /VΩ = h
The physical contents of the solutions (2) is such: if
at the initial moment of time the system was in the ex-
cited state, then after the expiration of time 2 /T π= Ω
she, with probability which is equal unit, will pass in the
basic, not excited state. Further the process will repeat,
but on each small interval of time this process has prob-
ability character.
Therefore, for further it is convenient all interval of
time to divide on small time intervals . T /t T nΔ =
Now we shall enter a new element - measurement of
a state of investigated system. Let at the moment of time
we somehow can estimate a state of our system.
Probability of that fact that she during the time
tΔ
tΔ will
pass from the excited state to basic state will be equal:
( 2( ) 1w t tΔ = − Ω⋅Δ ) . (3)
This formula already contains the important element
of quantum transitions. It consists in that that the veloc-
ity of transitions at small intervals of time is the less the
less are these intervals:
2/ 2dw w
dt
t= − ⋅Ω ⋅ . (4)
This result in the theory of quantum Zeno's effect
named as nonexponential law of disintegration (see, for
example, [2]). After the expiration of the following in-
terval of time we again include process of measurement.
The probability of detection of the originally excited
system in the initial state will be defined by formula:
( )( )22(2 ) 1w t t⋅ Δ = − Ω⋅Δ . (5)
Such formula reflects the fact of independence of quan-
tum transitions in each of time intervals . Eventually,
after the large number of measurements the probability
of a presence of system in the excited state will be ex-
pressed by the formula:
tΔ
( )( )2( ) 1
n
w n t t⋅ Δ = − Ω⋅Δ . (6)
Let us take the logarithm from the left and right
parts of expression (6). In result we shall get that in lim-
it of a large number of measurements during time , the
probability of detection of system in its initial excited
state go to unit:
T
( )2 2( ) exp / 1nw T T n →∞= −Ω ⎯⎯⎯→ . (7)
Thus, the process of supervision of the excited sys-
tem does not give for this system to pass from the ini-
tial excited state in any other state. This fact makes the
contents of quantum Zeno's effect.
Now it is easy to explain the basic contents of work.
It consists in the following. We refuse from procedure
of measurement. However we assume, that near to the
excited energy level of system there is an additional
level (system becomes three-level). At that, on quantum
system except perturbation which can transfer system
from excited state in stationary, not excited state the
additional external low-frequency perturbation acts.
And, the frequency of this perturbation is resonant in
relation to transitions between the excited state and new
additional energy level. Besides we shall require that the
Rabi frequency of transitions between the excited state
and new state should be much greater, than return time
of transition of the excited system in the not excited
stationary state. Thus we shall show that the period of
these additional transitions, in which the excited system
is involved, will play a role of an interval of time be-
tween supervision in quantum Zeno's effect. The large
this frequency the smaller probability of transition the
Series: Nuclear Physics Investigations (54), p.90-93.
mailto:vbuts@kipt.kharkov.ua
system from the excited state to the basic state (not ex-
cited stationary state).
91
2. STATEMENT OF A PROBLEM AND
BASIC EQUATIONS
Let's consider quantum system, which is described
by Hamiltonian:
0 1
ˆ ˆ ˆ ( )H H H t= + . (8)
Second addend in the right part describes perturbation.
The wave function of system (8) obeys to the Schröd-
inger equation, which solution we shall search as a row
of own functions of the not perturbed equation:
( ) ( ) exp( )n n n
n
t A t i tψ ϕ ω= ⋅ ⋅∑ , (9)
where /n nEω = h ; 0
ˆn
n ni H E
t
ϕ
nϕ ϕ
∂
= = ⋅
∂
h .
Let's substitute (9) in the Schrödinger equation and by
usual way we shall receive system of the connected eq-
uations for a finding of complex amplitudes nA :
( )n n m
m
i A U t A⋅ = ⋅∑&h m
dq
1
, (10)
where . 1
ˆ ( ) exp[ ( ) / ]n m m n n mU H t i t E Eϕ ϕ∗= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∫ h
Let’s consider more simple case – the case of harmonic
perturbation:
1 0 0 1
ˆ ˆ ˆ( ) exp( ) exp( )H t U i t U i tω ω= ⋅ + ⋅ .
Then the matrix elements of interaction will get the
following expression:
exp{ [( ) / ]}n m n m n mU V i t E E= ⋅ ⋅ − +h Ω
dq
,
. (11) ( ) ˆk
n m n k mV Uϕ ϕ∗= ⋅ ⋅∫
Let’s consider of the dynamic three-level system:
( 0 , 1 , 2 ). We shall consider that frequency of ex-
ternal perturbation and the own meanings of energy of
these levels satisfy to such relations:
1, 0m n= = , 0 1E E0ω = −h ; 2, 0m n= =
0 2 0( ) E Eω δ+ = −h 0δ ω<< E E, 1 2 1ω = −h 1~δ ω .(12)
These relations indicate that fact, that the frequency
0ω of external perturbation is resonant for transitions
between zero and first levels, and the frequency 1ω is
resonant for transitions between the first and second
levels. Using these relations in system (10), it is possi-
ble to be limited by three equations:
0 01 1 02 2 exp( )i A V A V A i tδ⋅ ⋅ = + ⋅ ⋅ ⋅&h ;
1 10 0 12i A V A V A⋅ ⋅ = +&h 2 ;
2 21 1 20 0 exp( )i A V A V A i tδ⋅ ⋅ = + ⋅ − ⋅ ⋅&h . (13)
The system of the equations (13) is that system,
which we shall analyze. The scheme of energy levels for
system (13) is represented in Fig.1.
3. DYNAMICS IN ABSENCE OF DETUNING
Let's consider, first of all, case, when detuning is
enough large and the terms in system of the equations
(13), which contain detuning, can be neglected. Besides
we shall consider that the matrix elements of direct and
return transitions are equal ( , ).Besides
we shall consider that the matrix elements of transitions
between the first and second levels are much more, than
matrix elements of transitions between zero and first
levels (
12 21V V= 10 01V V=
12 10/V V 1μ≡ >> ). For further it is convenient to
introduce dimensionless time 10 /V tτ = ⋅ h .
1ωh
0ωh
0
1
2
Fig.1. Schema of energetic levels
Take into account these reasons the system of the
equations (13) becomes elementary simple:
0 1iA A=& , 1 0 2iA A Aμ= +&
2 1A, iA μ=& . (14)
Let at the initial moment of time ( ) the consid-
ered quantum system is on first, excited level. Then, as
it is easy to see, the solution of system (14) will be func-
tions:
0t =
( )0
1 sinA t
i
μ
μ
= ⋅
⋅
, ( )1 cosA tμ= ⋅ , ( )2 sinA i μ= − ⋅t .(15)
From the solution (15) follows, that than there will
be the more parameter μ , the there will be less prob-
ability, that the system from the excited state will pass
in not excited, stationary state. It is necessary to say a
little words about parameter μ . Physically this parame-
ter defines the relation of number of quantum of low-
frequency perturbation which is responsible for transi-
tions between the first and second levels to number of
quantum of high-frequency perturbation which defines
transitions between the first and zero levels. Than there
will be more this ratio, the there will be less a probabil-
ity that the excited system will pass in the not excited
state. The account of influence of the terms containing
detuning can be made by numerical methods. Such
analysis was carried out. He has shown, that presences
even enough large detuning only little changes the re-
sult. And, the more size of parameter μ , the smaller
appears this influence. In a Fig.2 is given characteristic
dependence of module of amplitude 0A from time, and
in Fig.3 – time dependence of module of amplitude 1A
when there is detuning, which was equal 0.1. Parameter
μ at this calculation was equal 100.
At the initial moment of time the system practically
was at the first excited level. The dependence of the
module of amplitude 2A from time does not differ prac-
tically from dependence submitted in a Fig.3. One can
see from these figures that initial probability to be sys-
tems in the not excited state (at a zero level) during the
time practically does not vary. She is very small and in
all cases inversely to a square of parameter μ . Shall
note, that result is a little varies if parameter detuning to
increase up to 0.5.
Fig.2.
Fig.3.
4. TRANSITIONS AT SPONTANEOUS
RADIATION
In the previous section we have considered quantum
system, the transitions between which levels were
caused by induced processes. The probabilities of in-
duced processes are considerably large than spontane-
ous, however spontaneous transitions can be occur on
all underlying energy levels. Such transitions, generally,
can be much. There is a question, as the process of sta-
bilization of the excited state will proceed in this case.
Whether he can be realized in this case? For the answer
to the put question we shall consider a concrete case of
stabilization excited state in synchrotron. At this we
shall use results stated in [3]. Thus, we shall consider
synchrotron radiation. In difference to consideration in
[3], we shall study not two-level system, but three-level
system. And, we shall assume, that on considered sys-
tem acts the external periodic perturbation causing in-
duce resonant transitions between two top levels (be-
tween the level one and the level two in Fig.1). The
transitions between level 1 and zero level occur sponta-
neously. System of the equations describing dynamics
of amplitudes of wave functions for such system can be
represented as:
( )0 1 exp /i A V A i E t+= − ⋅Δ ⋅&h h ,
( )1 0 12exp /i A V A i E t V A−= ⋅Δ ⋅ +&h h 2⋅
1
x x
x
,
2 21i A V A= ⋅&h . (16)
Here , ,
, operators U are defined in [3].
3
0 1V U dψ ψ+ + += ∫ 3
1 0V U dψ ψ− + −= ∫
3
12 1 2V U dψ ψ+= ∫ ±
Let's note only, that the operator U is proportional
to the operator of birth, and the operatorU is propor-
tional to the operator of destruction. The operator U de-
fines potential of external periodic perturbation.
+
−
We shall consider that the transitions between levels
one and two are caused by induce processes and occur
during the times which are considerably smaller times
of spontaneous transitions. In this case last two equa-
tions can be considered independently from first equa-
tion. Besides, taking into account, that transitions be-
tween two top levels are induce, the matrix elements of
transitions upwards and downwards are equal to each
other ( 12 21V V= ). In this case solution for amplitudes of
wave functions will look like:
( )1 cosA t= Ω ⋅ , ( )2 sinA i t= − Ω ⋅ , (17)
where 12 /VΩ = h − Rabi frequency.
For the analysis of efficiency of spontaneous transi-
tions we should use the first equation of system (16). At
this in the usual theory of perturbation instead function
1A substitute a constant which is equal to unit. It corre-
sponds to that fact, that at the initial moment of time the
system is in the excited condition at the first level. On
logic of things, we should substitute in this equation the
solution (17). If we shall act thus, the result of stabiliza-
tion will be absent, only the spectra of spontaneous ra-
diation will slightly change. Actually it is necessary to
take into account that the investigated system at a level
one exists only during the limited intervals of time,
which size is order 12~ /VτΔ h . During each of these
intervals the system can spontaneously pass to a zero
level. The probability of such transition will be propor-
tional to a square of size of this interval of time. Taking
into account, that each of such transitions is casual
process, and also that fact, that all these casual transi-
tions are independent from each other, we shall receive
result, which is described in introduction. Thus, it is
possible to expect, that when frequency of transitions
between the first and second levels (the Rabi frequency)
will be much greater, than return time of life of the ex-
cited state one, the process synchrotron radiation will be
put down. It is interesting to estimate size and character-
istics of external perturbation which can result in such
suppression. It is known (see, for example, [3]), that the
time of life of the excited state can be estimated by the
formula:
2
0
Rt
r mc γ
⋅
Δ =
⋅ ⋅
h , (18)
where − classical radius of electron; − radius of
electron orbit in synchrotron.
0r R
If as an example we shall take cm,
MeV the time of life will be about
Thus, for observation of suppression of syn-
chrotron radiation it is necessary, that the Rabi fre-
quency for transitions between the first and second lev-
els was much more, than .
100R =
2 500E mc γ= ⋅ =
910 sec.−
910
Let for definiteness she is equal . Then, those
conclusions of the theory, which was used by us above
(theory of perturbation) it is necessary, that the fre-
quency of perturbation was greater, than . It is clear
also, that the large Rabi frequency and higher the fre-
quency perturbation the more appreciable will be effect
of suppression of synchrotron radiation.
1010
1110
CONCLUSIONS
Thus, it is possible to formulate the following rec-
ommendations for stabilization of the excited quantum
systems. First of all, it is necessary to know time of life
92
93
of this excited state. Further it is necessary to pick up
the appropriate energy levels located not too far from an
excited state. Further it is necessary to pick up perturba-
tion which frequency will correspond to transitions be-
tween the excited state and this additional level. The
intensity of this perturbation should be such, that the
appropriate Rabi frequency was as greater as possible in
comparison with return time of life of the excited sys-
tem. It is necessary to say, that such stabilization of
quantum systems is similar to stabilization of classical
systems. This mechanism is similar to those which ap-
pear in movements gyroscope and in the Kapitsa turned
pendulum. For this reason it is possible to name offered
effect of stabilization as effect of a quantum gyroscope
(whirligig).
REFERENCES
1. E.C.G. Sudarshan, B. Misra. The Zeno’s paradox in
quantum theory // Journal of Mathematical Physics.
1977, v.18, №4, с.756-763.
2. L.А. Halfin. Zeno’s quantum effect // UFN. 1990,
v.160, №10, p.185-188 (in Russian).
3. А.А. Sokolov, I.М. Теrnov. Relativistic electron.
M.: «Science», 1974, p.391 (in Russian).
Статья поступила в редакцию 26.10.2009 г.
СТАБИЛИЗАЦИЯ ВОЗБУЖДЕННЫХ СОСТОЯНИЙ
В.А. Буц
Предлагается механизм, позволяющий стабилизировать возбужденные состояния квантовых и классиче-
ских систем. Механизм стабилизации аналогичен квантовому эффекту Зенона. Отличие заключается в том,
что над системой не производится наблюдения. Вместо этого система периодически переводится в другое
квантовое состояние. Определена необходимая для стабилизации скорость стабилизирующих переходов.
СТАБІЛІЗАЦІЯ ЗБУДЖЕНИХ СТАНІВ
В.О. Буц
Пропонується механізм, який дозволяє стабілізувати збуджені стани квантових та класичних систем.
Механізм стабілізації аналогічний до квантового ефекту Зенона. Відмінність полягає у тому, що над систе-
мою не наглядають. Замість цього система періодично переводиться в інший квантовий стан. Знайдена не-
обхідна для стабілізації швидкість переходів, які стабілізують систему.
|