Quantum dynamics of a two-level system under extrenal field
We present exact analytic solutions for non-linear quantum dynamics of twolevel system (TLS) subject to periodic-in-time external field. >n constructing the exactly solvable models, we use approach where the form of external perturbation is chosen to preserve n integrability constra...
Saved in:
| Published in: | Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
|---|---|
| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2011
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/28313 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Quantum dynamics of a two-level system under extrenal field / A. Korostil, Ju.Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 73-84. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859718761470230528 |
|---|---|
| author | Korostil, A. Korostil, Ju. |
| author_facet | Korostil, A. Korostil, Ju. |
| citation_txt | Quantum dynamics of a two-level system under extrenal field / A. Korostil, Ju.Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 73-84. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| description | We present exact analytic solutions for non-linear quantum dynamics of
twolevel
system (TLS) subject to
periodic-in-time external field. >n constructing the
exactly solvable models, we use
approach where the form of external perturbation
is chosen to preserve
n integrability constraint, which yields
single non-linear
differential equation for the ac-field. solution to this equation is expressed in terms
of Jacobi elliptic functions with three independent parameters that allows n to
choose the frequency, average value, and amplitude of the time-dependent field at
will. This form of the ac-drive is especially relevant to the problem of dynamics of
TLS charge defects that cause dielectric losses ?n superconducting qubits.
|
| first_indexed | 2025-12-01T08:57:10Z |
| format | Article |
| fulltext |
73 © �. Korostil, Ju.Korostil
1. ������� . .,
��
��� �.�. ������
��
�
������ ����
��� ����������-
����
.– �.: �
��
, 1970.– 280 �.
2. �������� �. . ����������
���
��
�
�.– �.: ����
� ����
, 1976.– 276 �.
3. ������� .�. ���
����
� �
�
�
��
��
���
�!������. �.: �
��
, 1972.–432 �.
4. ������� �.�., ������� �., �!�"�� �. � ��������" ����
" �����
� �
���
������-�
##�����
����" ��
����
�, ������
$%
"�� � �
�
�
" �������������
//
��
��
��
�
��
�
�
�"
�
�
.– 1987. –&.51.– ' 5. –(.867–871.
5. ������� �. )�����
����� ����
��
�
������� ����
���
�
� �
�
�
�
�
�
����������
" �
��� . ������#���� �
��. �-�
. ��". �
��, *
��, 1991.
�!��"��� 17.02.2011�.
+;* 72.25. 72.25.
�. Korostil, Ju.Korostil
QUANTUM DYNAMICS OF A TWO-LEVEL
SYSTEM UNDER EXTERNAL FIELD
We present exact analytic solutions for non-linear quantum dynamics of
two-
level system (TLS) subject to
periodic-in-time external field. >n constructing the
exactly solvable models, we use
approach where the form of external perturbation
is chosen to preserve
n integrability constraint, which yields
single non-linear
differential equation for the ac-field. � solution to this equation is expressed in terms
of Jacobi elliptic functions with three independent parameters that allows �n� to
choose the frequency, average value, and amplitude of the time-dependent field at
will. This form of the ac-drive is especially relevant to the problem of dynamics of
TLS charge defects that cause dielectric losses ?n superconducting qubits.
1. Introduction
The problem of
periodically-driven two-level system (TLS) appears in
many physical contexts including magnetism, superconductivity, structural glasses
and quantum information theory [1-7]. The interest in this �ld probl�m has been
revived recently due to advances in the field of quantum computing (see, e.g., [8-
12] and references therein). First of
ll,
qubit itself is
two-level system and the
question of its evolution under an exter�
l time-dependent perturbation is
obviously of interest. Also, the physical mechanism that currently limits coherence
particularly in superconducting qubits is believed to b� due to other types of
unwanted TLSs within the qubit, \whose charge dynamics under
periodic-in-time
electric field gives rise to dielectric losses directly probed in exper?m��t. [13,14].
In what follows, we mostly
��l� our solution to the latter charge TLS model, but
the general methods
nd some particular results of this work evidently can b�
74
��l?�d to
much broader range of problems.
�n� of the key metrics of
superconducting qubit is the quality factor,
which is defined as
ratio of the real and imaginary parts of the dielectric
response function, ( )� � , evaluated at the resonant frequency of the
��rresponding LC-circuit, Re ( ) / Im ( )r rQ � � � �� . Very high values of the
quality factor are required for the qubit to be operational. However, existing
experiments consistently show significant dielectric 1osses that occur ?n
n
amorphous dielectric (e.g.., ?n Al2O3) used as
barrier ?n the Josephson junctions.
It is believed that the losses are primarily due to the presence of charge two-level
system defects ?n the barrier and/or the contact interfaces, which respond to
n �(
electric field in the LC-resonator. It is still unclear what the physical origin of
these defects is, but an �
�l� work of Phillips [13] as well as very recent
comprehensive density functional theory studies point to the OH-rotor defects as
very likely source of the dielectric losses. &he determination of the physical
origin and the properties of the TLSs responsible for the dielectric loss is
investigated in the presented work.
The usual theoretical approach to calculating the quality factor and more
generally the full dielectric response function, ( )� � , involves
formal mapping
of charge d�namics ?n
double-well potential onto the problem of "spin"
dynamics ?� an �( field, described b� the "spin" Hamiltonian
( ) ( ) / 2, ( ) 2( ,0, ( ))t TLSH t b t b t d E t� �� � � � � �
where � denotes the B
ul? matrices
nd ( )b t is an effective "magnetic field" that
drives TLSs, with � , t�
nd TLSd being the TLS energy splitting, the tunneling
amplitude between its two states, and the TLS dielectric moment correspondingly
and ( )E t is the �( electric field. � linear analysis within the canonical TLS
predicts that the dielectric function due to identical TLSs is peaked at the
frequency, 2 2
t �� � � . Ad-hoc inclusion of 1T and 2T relaxation processes and
the assumption about random distribution of TLS energy-splitting and tunneling
(typically assumed to be uniform and long-uniform correspondingly) l�
d to the
quality factor 01 ( / )x
cQ E E� , with 2x , 0E being the
m�l?tude of an
��>?�d �( electric field
nd CE is
critical value of the amplitude which also
encodes the information �n the strength of the relaxation processes (see, e.g., [5]).
Both formulas are used widely ?n interpreting experimental data
nd probing
energetic of the relevant TLS defects.
While this 1inear ana1ysis is
fine
�proximation to describe
majority of
regimes current1y studied experimentally, the existing experiments are certain1y
75
�
�
bl� and some do access non-linear regimes as well, where the energy of the
applied e1ectric fie1d is ��mparable or 1arger than the re1evant TLS energies.
��nc�, this non-perturbative regime is of c1ear experimental and theoretica1
interest. More important1y studies of ���1inear dynamics m
� provide another
effective means to probe the properties of TLS.
FIG. 1: Schematic representation of an OH-rotor two-level system in an �l20�
oxide. [16.17]. Here, the role of the generalized variable is assigned to the angle
defined as an angle between the OH-bond and an axis perpendicular to the verti�
l
�l� bond. At low enough temperatures, the phase space an isolated rotor is
reduced to the two-states corresponding to the minima of the double-well potential!
( )V
. Application of external ac-field parametrically coupled to the rotor's dipole
moment induces oscillations between the two minima.
The mathematica1 formu1ation of the non-linear TLS dynamics problem
studied ?n this paper is deceptive1y simp1e. We will solve the Schrödinger
equation for a spinor wave-function
1 ( ) , ,
2ti b t
�
�
�
�
�
�
� � � � � � � � �
� �
that describes
ha1f-integer spin subject to
periodic in time magnetic field of the
form, ( 0 2( ,0, ( ))tb t f t� � , where t� is
constant describing the coupling between
the two states and the function ( ) ( )f t f t T� � , describes the time dependent
perturbation. Despite the simp1icity of the formu1ation, the prob1em is generally
unso1v
b1� ?n ana1ytic form for most cases of practica1 interest. The origin of this
surprising fact �
n be understood if we introduce
new function
( ) ( ) / ( )R t t t� �� �� , which reduces the matrix Schrödinger equation to the Riccatti
equation
2
( ) 2 (1 )it tR fR R�� � � � � .
It is
non-linear differential equation that has known analytic solution in a very
76
limited number of cases (not that case of a monochromatic perturbation is not one
of them).
Therefore, to solve for TLS dynamics driven by
specific n�nequilibrium
fie1d is equiva1ent to generating
particu1ar solution to the Ricatti equation
corresponding to the perturbation. C1ear1y this is
challenging mathematica1 task
and this observation partially exp1ains the current deficit of exact mathematical
resu1ts. The difficu1ties in obtaining exact solutions have 1ed to the emergence of
severa1 perturbative approaches, used ?n particu1ar to characterize re1axation and
dephasing rates in qubits as
function of driving amp1itude. These ana1yses
provide very usefu1 physica1 insights and correct1y describe the physics if the
time-dependent perturbation is weak, but it is a1so c1ear that there exist non-linear
effects beyond perturbation theory and it is desirab1e to have exact resu1ts to
access this qua1itative1y different physics.
The mathematica1 approach that we use to obtain exact resu1ts is to exact1y
solvab1e Hamiltonians of specific form re1evant to the prob1em of interest. � key
observation in our ana1ysis is that finding
Hami1tonian corresponding to
given
solution is much easier than solving the Schrödinger equation with
given
Hamiltonian. In some genera1ized sense, the two procedures are re1ated to �n�
another much like differentiation re1ates to integration. &� see this, it is useful to
consider the evo1ution operator, or the S -matrix, which re1ates the initia1 state at
0t � to
fina1 state at 0t � as follows, ( ) ( ) (0)t S t� � � . >n the absence of
re1axation process the time-evo1ution is unitary and it satisfies the Schrödinger
equation,
( ) ( ) ( )ti S t H t S t� � .
If we choose an arbitrary S -matrix,
2
1exp ( )
2
S t SU�
�� � � � �� �
� �
,
we �
n immediate1y reconstruct the corresponding Hamiltonian that gives rise to
such evo1ution as follows ( ) ( ) ( )tH t i S t S t�� � . Using this method, �n� �
n
generate
n infinite number of exact non-equilibrium solutions and explicit models.
These solutions m
� b� of importance to physics of NMR, to the question of physi-
�
1 imp1ementation of gate operations �n
qubit as well as of some mathematica1
interest. Neverthe1ess without additiona1 constraints such ana1yses wou1d
generally produce Hami1tonians of litt1e importance to the prob1em of dynamics
of TLS charge defects.
� very usefu1 insight that allows us to constructive1y narrow down the range
of re1evant dynamica1 systems comes from the mathematically re1ated prob1em
of far-from-equi1ibrium superconductivity. It is well-known that the reduced BCS
Hamiltonian is a1gebraically equiva1ent to
n interacting XY-spin mode1 ?n
n
effective "inhomogeneous" magnetic fie1d ?n the z-direction, whose profi1e is
dictated b� the bare sing1e partic1e-energy dispersion. Far from equilibrium,
dynamics of
given ��derson pseudospin is determined b�
n effective time -
77
dependent se1f-consistent fie1d of other pseudo-spins that it interacts with. In
m
n� cases (determined b� specific initial conditions), this BCS se1f-consistency
constraint dynamically se1ects
specific order-parameter, such that the dynamics
of essentially infinite number of spins is equiva1ent to the dynamics of few spins
�n1�.
For specia1 sets of initia1 conditions, these spins move ?n unison and
therefore the se1f-consistent "magnetic fie1d" (or superconducting order parameter
?n the 1anguage of BCS theory) ?s periodic in time. The reduced BCS model ?s
integrable and there exists
very elegant prescription for ��nstructing exact non-
equilibrium solutions to it. These solutions contain, ?n particular, exact s�?n
dynamics in
periodic time-dependent field that �
n b� expressed ?n terms of
elliptic functions. >n this paper, we generalize such
n�m
l�us soliton solutions to
encompass
wider range of time dependencies relevant to the problem of TLS
dynamics, which ?s of our primary interest.
2. General framework for constructing ����t solutions
>n this paper, we derive
family of exact solutions for the non-dissipative
TLS dynamics subject to
n external ac-field. The m
?n ingredient of our approach
?s
s���?
l ansatz for the TLS's dynamics that corresponds to periodic-in-time but
non-monochromatic external fields. Before proceeding to the specific ansatz, let us
first ?ntroduce
general algebraic framework of exact solutions. We are interested
in solving the non-equilibirum Schrödinger equation for the spinor
( ) ( ) ( ), ,ti t H t t
�
�
�
�
�
� � � � � � � �
� �
(1)
where the Hamiltonian ?s ( ) (1/ 2) ( )H t b t �� � . �s mentioned in the introduction,
instead of solving Eq. (1) for the wave-function, we �
n consider the Schrödinger
equation for the evolution operator that relates the initial and f?n
l states,
( ) ( ) (0)t S t� � � . This equation for the S-matrix has the form identical to Eq. (1)
( ) ( ) ( ), (0) 1ti S t H t S t S� � � (2)
but now it ?s
n equation for the matrix function ( )S t , which belongs to the two-
dimensional representation of the SU(2) group, while the Hamiltonian �"pressed ?n
terms of SU(2)2 generators belongs to the two-dimensional representation of the
su(2) algebra.
Note that the form of Eq. (2) ?s such that it m
� b� generalized to
n arbitrary
s�?n or equivalently to
n arbitrarydimensional representation of SU(2) or it �
n b�
viewed as
n equation of motion ?n the abstract group such that
( ) ( ) (2), ( ) exp( ( ) ) (2)abs abs abs absH t b t J su S t i t J SU� � � � � � � � ,
where absJ are the ��rresponding generators. Therefore,
solution of the problem
in
particular representation, ?.�., an explicit form of ( )t� , immediately gives the
corresponding solutions in all other representations (e.g.,
two-Ievel system dy-
78
namics uniquely determines
"d-level system" dynamics ?n the same field). This
TLS problem that we are interested ?n corresponds to the two-dimensional
generators 2 (1/ 2)J� ��� with �� ( ( , , )x y z� � ) being the familiar Pauli matrices.
The problem of determining the solution, ( )t� , from the magnetic field time-
dependence ( )b t ?s
complicated �n�, but the inverse problem ?s almost trivial.
Indeed, if we select
specific S-matrix (defined uniquely b� the choice of
specific function, ( )t� , the Hamiltonian will read
( ) ( ) ( )tH t i S t S t�� � , (3)
where
( ) exp ( )
2
iS t t �
�� � �� �
� �
. (4)
Using the algebraic identities for the B
uli matrices, we obtain the corresponding
magnetic field
( ) sin (1 cos )[ ]b t n n n n
� �
� � � � � � � � , (5)
where ( ) | ( ) | ( )t t n t� � � , with | ( ) | 1n t � . Note that �n� �
n generate exactly-
solvable models b� s?m�l� picking
n arbitrary ( )t� dependence and using Eq. (3)
to find the corresponding Hamiltonian. However, without guidance or luck, such
n
nalysis would generally produce complicated non-equilibrium fields that have
little to do with
n underlying physical problem. Let us however mention here that
this procedure m
� b� of interest to quantum computing an general, b��
use the
time-evolution governed b�
n S-matrix �
n b� viewed as
"gate operation" �n the
s�?n (if the TLS/spin corresponds to
qubit rather than to
defect within
qubit).
�� picking "trajectories," ( )t� , �n the algebra that start ?n the origin, #.
.
(0) 0� � , but end at
s��cific point at
time T , �n� �
n immediately determine
the non-equilibrium magnetic pulse, ( )b t , or
�l
ss of such pulses, that will give
rise to
desired gate operator ( ) exp ( )
2
iG S t t �
�� � � � �� �
� �
.
Let us note here that the function, ( )t� , contains ��mplete information about
the solution to the original probl�m, Eq. (1), including the overall quantum phase
��umulated b� the wave-function during the time evolution (
s we shall s��
below, this phase ?s of particular interest to the problem of dielectric response of
TLSs ?n superconducting qubits). �n interesting question ?s whether and how
purely quantum phase can be restires from a solution of the corresponding classic
Bloch equations that are usually considered in this context. Let us recall that a
classical mapping can be achieved by introducing the average magnetic moment,
( ) ( ) ( )
2
m t t t��� � � . (6)
79
Therefore 2 ( ) 1/ 4m t � and the classical equations of motion for the spin moment
follow from
� �1( ) ( ) ( ), ( )
2t m t t H t t��� � � �
and yield the familiar result
( ) ( ) ( )t m t b t m t� � � (7)
Let us recall that these Bloch equations are
saddle point of quantum spin
dynamics, much ?n the same way that Newton's equations of motion governed b�
the force, [ ( )]V r�� , represent
saddle point of the action describing
quantum
particles ?n the potential, ( )V r , and therefore do not contain direct information
about quantum intformation and tunneling effects. Similarly, Eqs. (7) do not
directly contain the quantum phase and to determine it �n� has to go back to the
Schrödinger equation. Another more abstract way to see this is b� noticing that.
Eqs. (7) describe the motion �n
two-dimensional (�loch) sphere, 2( )m t S� ,
while the original quantum problem Eq. (2) describes motion �n
three-
dimensional sphere since 3( ) (2)absS t SU S� .
Now let us recall that there exists the Hopf fibration such that
2(2) / (1)SU U S� , which summarizes the fact that classical equations, n
m�l� Eqs.
(7), represent quantum motion modulo the U(1) phase dynamics. Fortunately, this
phase dynamics �
n generally b� restored from exact dependence of the ( )m t
solution, albeit ?n
non-l��
l way. &� see this, we �
n write the magnetization ?n
terms of the 5-matrix as follows
1( ) (0) ( ) ( ) (0)
2
m t S t S t�� �� �� � � ! ,
where (0)� and the corresponding (0) (0)( / 2) (0)m ��� � � are initial conditions
for the wave-function and �loch magnetization, correspondingly. Using again the
well-known identities for the Bauli matrices, we find the evolution matrix for the
�loch equations, as follows ( ) ( ) (0)m t R t m� �" "� , as follows
( ) cos (1 cos ) sinm t n n n� �" � " �"# #$ % �� �� � � � (8)
This three-dimensional matrix describes a rotation, ( ) (3)R t SO� , and can be
represented equivalently as
� �
0
( ) exp ( ) , 0
0
z y
z x
y x
e e
R t t L L e e
e e
��
� �
� �� � � �� �
� ��� �
, (9)
where 0(3) (2)L s so� belong to the three-dimensional vector representation of
the (2)so algebra. They are related to the “usual” spin 1 representation (where 3
zJ
is diagonal) via simple linear transform.
80
Therefore, we see that if we known an arbitrary solution to the Bloch
equation, ( )m t we can at least in principle restore the function, ( )t� , (see, Eqs.
(9) and (4)), which uniquely determines the entire quantum solution. It also
suggests that if we choose an arbitrary dynamics function on
sphere we m
� b�
able to restore the quantum Hamiltonian that would give rise to it, via mappings
( ) ( ) ( )m t R t S t H& & & . However, the second step ?n this chain of transforms
involves effectively calculating
1ogarithm of the rotation matrix, which due to
complicated "analytic" structure of this matrix-logarithm function requires
careful calculation n�n-l��
l ?n time.
The sequent Sections are devoted to constructing exactly solvable periodic-in-
time Hamiltonians based on a specific anzats for the classical Bloch
“magnetization”, ( )m t . It further involves
restoration of the corresponding
quantum (1)U phase v?
straightforward integration. More specifically, we
reverse ' the following Hamiltonian
( )t x zH f t� �� � � , (10)
where ( ) ( )f t f t T� � ) is
periodic function, with
n arbitrary period, fT . Our
solution below also allows tuning of the average splitting, ( )
fTf t� �' � , and the
AC field amplitude, 2| ( ) |f TA f t �� ' � � . As mentioned in the introduction, this
problem is of great importance to the physical problem of externally-driven TLS
dynamics in superconducting qubits (with t� corresponding to tunneling between
the wells, � to
splitting of energy levels in
double-well potential, and fT and
fA being the period and the amplitude of the AC-electric field ��rr�spondingly).
�ur "guess" for the relevant ansatz for the Bloch "magnetization," ( )m t , is
based �n
set of formal solutions discovered ?n the related problem of quenched
dynamics of fermionic superfluids [19-21,24,25]. Formally, the quenched
dynamics of each individual Cooper pair is described b� the Bogoliubov-de
Gennes Hamiltonian, which is essentially
spin Hamiltonian that reduces to (10)
after the unitary transformation x z� �& and z x� �& � . with t� corresponding
to
single particle energy l�v�l and ( )f t to the superfluid order parameter.
� realization of each particular form of the superfluid order parameter
dynamics in
steady state �
n b� unambiguously determined b� the initial
conditions using the exact ?�tegrability of BCS model. Note that
self-consistency
condition for the order parameter provides
limitation �n the set of functions for
which the corresponding problem is integrable and for some initial conditions
periodicin-time self-consistent dynamics, ( )f t , �
n b� realized. While in ourTLS
problem, there is n� natural selfconsistency constraint, such insights and
constraints from the BCS problem help us narrow down the range of possible
ansatze to restore reasonable physical Hamiltonians, which are also exactly
81
solvable b� construction. In what follows, we generalize the solution analysis of
the paper [16] and find a general soliton configuration, characterized by three
independent parameters, which we denote as (� and a� . For the physical problem
of interest, this conveniently implies that some, generally speaking, non-trivial
combination of these parameters will determine the arbitrary frequency, amplitude,
and the dc-component of the field. Due to the periodicity, we can generally
represent the AC-perturbation as a Fourier series
1
( ) cos( )f fn
n
f t A f n t� �
)
�
� � * . (11)
Note that for certain specific choices of the parameters ,a(� the leading
coefficient 1 nf f ( (2,3,...)n � and �n� recovers the limit of
monochromatic
AC-field, albeit in the regime of weak driving 11 max{ , }fA f �� . Therefore, our
n�n-l?n�
� analysis contains the standard lin�
� response results as
simple special
case.
3. Non-dissipative dynamics of the ac-drived TLS
Further we provide the details �n the derivation of the exact solution for the
TLS dynamics. We devote the special attention to the analysis of the (1)U phase
of the \wave function. We also elucidate the relations b�twecn the parameters of
�u� solution and the amplitude, phase and the dc-component of the external field,
which m
� be useful for experimental applications of �ur theory.
We n�w focus �n the Schrödinger equation for the half-integer spin ?n the
magnetic field, ( ) 2( ,0, ( ))tb t f t� � . When written ?n terms of spinor components,
it has the form
( ) ,
( ) .
t
t
i f t
i f t
� � �
� � �
�
� � �
�
� � �
� � �
� � �
(12)
The corresponding Bloch equation is
( ) 2( ,0, ( ) ( ))tm t f t m t
�
� � � . (13)
Let us now make the following anzats for its exact solution [25]:
.
2 , , ( )x y zm D Cf m B f m Af t F� � � � � . (14)
From two of the Eqs. (13) we find 2 tA B� � and B C� . Thus among five
parameters ?n (14) �nl� three
�� independent: ,P B and D . The equation for the
external field, ( )f t , �
n b� obtained from (14) using the condition 2 1/ 4m � . This
resulting equation for the function ( )f t acquires the form
.
2 4 2
2 1 34 8 4f f c f c f c� � � � � (15)
82
where coefficients jc
�� given b� some combinations of parameters ,B D
and F (see Eqs. (30) below). Equation (15) �
n b� cast to
m��� symmetric form,
using another set of parameters a� and (� , which
�� chosen to b� positive and
�� related to coefficients jc as
2 2 2 2 2
1 2
2 2 2 2
3
1( ), ( 2 ),
4 4
1 ( )( 2 ).
4
a
a
a a
c c
c
� � � �
� �
�
� � � � � � � � � � � �
� � � �� � � �
(16)
Without loss of generality and to be more specific we also assume � �� + � for the
remainder of this paper, while a� can be assigned an arbitrary value. By virtue of
expressions (16) equation (15) now reads
.
2 2 2 2 24 ( ) (a af f f� �� � � �� � � � � � � � � ! ! , (17)
Below we will make several transformations that allow us to reduce (17) to
n
equation for the Weierstrass elliptic function. Firstly, let us introduce
function,
( )y t ,
2( ) 1 ,
( ) af t
y t�
� �
� � � � �, -
!
(18)
which satisfies the following equation
2
4( )( )( 1),
tdy y a y a y x
dx a a
�
� �
� �
�
� � � � � �� �
� �
(19)
where 2 /( 2 )aa( � � �� � � � � ( � . Now, Eq. (19) �
n b� easily reduced to
well-
known equation for the Weierstrass elliptic function b� rescaling the parameters
v?
the transformation
1
( ) ( )
3
a ay x Z x � �� �
� � (20)
so that
2
1 2 34( )( )( )dZ Z e Z e Z e
dx
� � � � �� �
� �
, (21)
where parameters je satisfy the following conditions 1 2 3e e e� � and 1 2e e� � .
3 0e� � . Coefficients je
�� determined b� the parameters a� and (� . The
specific expressions f�� the coefficients je , however, depend �n the relative values
of the initially introduced set of �
rameters. Solution of the equation (21) is
1 2
( ')( ) ( '), ' KZ x x x
e e
.��/ � �
�
, (22)
83
where ( )x/ is a Weierstrass elliptic unction, K is a complete elliptic integral of
the first kind and 1 2 1 3' ( )( )e e e e. � � � . Function ( )Z x is a doubly-periodic
function with the period along the physical time axis determined by, 2l �� , where
21 '� .� � is a modulus of elliptic functions. Combining (22) with Eqs. (20) and
(18) allows us to express ( )f t in terms of elliptic functions. Expression for ( )f t
can b� compactly written in terms of Jacob_ elliptic functions. Just as it is the case
for the parameters je , the particular form of the resulting expression depends �n
the relation between a� and (� .
All cases considered here are summarized b� the following compact
expression for the function, ( )f t , \written ?n terms of Jacobi elliptic function as
following
2
2
( , ) 1
( )
( , ) 1 a
sn zf t
sn z
0 .
0 .
�
�
�
�
� � � �
�
, (23)
where variable z is
2 2
1 3
4 ( ) ,
2 1[( 2 ) ]( )
f f
a
KT A
e e
0 0.
0
� � �
�� �
�� �
� � � ��� � � � � � � �
. (26)
Lastly, the average value of the function ( )f t over its period is
( ) 1 ( , )
( ) af t
K
0 0 0
0 . �
0 0 .
� � � �
�
� �
� �� �
' �� � 1 � �� �, -
!
(27)
with ( )K . and ( , )0 .1 being a complete elliptic integral of the first and third
kind correspondingly. As we have already mentioned, quantity (27) describes the
dc-component of the external field. One can view Eqs. (26, 27) as the definition of
yet another set of parameters fA , 2 /f fT� 2� and ( )f t� �' � , which allows us
to cast �"ternal field ( )f t into the form given b� (11). The dependence of the
parameters of the external field, ( )f t , on the ratio /� �� � allows to determine the
limits of strong and weak ac-driving. In particular, we can see that the regime of
the strong ac-driving should b� achieved for moderate values of a� and
/ 0.2� �� � .
Expressions (23,,24,25) constitute our m
?n results. Quite generally, our solution
�����s��ts the superposition of m�n��hr�m
t?� waves with frequencies integer
�ult?�l�s of 2 /f fT� 2� , The solution (23) �
n b� reduced to the mono�h���
t?�
wave with frequency 2 �� when 0a� � and .� �� �
1. U. Weiss, Q
ant
m dissipative systems (World Scientific, Singapore, 2008), 3rd Ed.
2. . J. Leggett et �l., Rev. Mod. Phys. 59, 1 (1987).
3. $. W. Anderson, �. Halperin and �. Varma, Phil. Mag. 25, 1 (1972).
84 © B.�. ~�
%����
�
4. R. F. Akhmet’yanov,� V. O. Ponomarev, O. A. Ponomarev, and E. S. Shikhovtseva,
Theor. Math. Phys., 149, N1 (2006).
5. �. �. Yu and $. W. Anderson, Phys. Rev. � 29, 6165 (1984).
6. S. Hunklinger and . �. Raychaudhur.i A�orpho
s Solids:Low-Temperature
Properties, edited by W.A. Philips (Springer, Berlin, 1981).
7. V. Lubchenko and $. G. Wolynes, J. Chem. Phys. 119 (17) 9088 (2002).
8. �. . Nielsen, %. L. Chuang, Q
ant
m Computations and Quantum Information
(Cambridge Univ. Press, (
�bridge, 2002).
9. �. D. Gerardot and $. Ohberg, Science 326, 1489 (2009).
10. J. �. Mooij, &. $. Orlando, L. Levitov, L. &#�n, �.H., van der. Wal and S. Lloyd,
Science, 285, 1036 (1999).
11. %. Chiorescu, '. Nakamura, �. J. $. �. Harmans, and J. �. Mooij, Science 299, 1869
(2003).
12. J. �. Martinis et �l., Phys. Rev. Lett. 95, 210503 (2005).
13. W. . Phillips, J. of Low &�m�. Phys. 11, 757 (1973).
14. *. Paik and �. D. Osborn, ���l. Phys. Lett. 96, 072505 (2010) .
15. E.A. Yuzbashyan, M. Dzero, Phys. Rev. Lett. 96 230404 (2006).
16. E.A. Yuzbashyan, Phys. Rev. B 78 184507 (2008).
�!��"��� 31.03.2011�.
+;* 358.4 : 656.7
B.�. ~�
%����
�, �.����.�., ������, �+�+, . *
��
���
� ��
� ����
����� ���� �������
��� ����� ��
The assessment mechanism to coordinate the planning system. The possibility of
well-coordinated system with large external and internal factors.
���������� ���!"�#$. ��
�
�� �����
������������" �
��� �
���
��
"��
��� ���
���
���� �� ���!��
�
� ������� ���!
����� ���
������
���
���$%
" ���������
�
���
����" �����
�. ;��� � �� , ��� �
��
��
����
" �
��
�
�" �����
� ���
� �� ���
��
�
�
��
���
����
����
�
��� � . �
%� ����� ��� ����
�� � ������
��������$
�#��
�
�
#
����
", ��
�$%
" �
�������
�� ��!�
���" �����
�
�
���
���$%
"
���������
�. � ��
��
����
" �
��
�
�" ���
����
�
��
���
� �����
�
��������
�
�
����
� �� ���
�
�
���
����
���� � �����$ �������
�� #
��� , ��� ��
�
����
$� ��������� ���!�� �, ����
���� �
�$%
�� � �
���� �
��
�
����������������
. )�������� �
�
�
���
� ��
����
� ���
�
�
, ���� ��!�
�� ��������
���������
������
�, �����
� ��
���
� � ����� �������
�
, ���
�#��
�
�
�
������, �
������" �����
�
����
� , ����
����� �����. ��
����
�����
|
| id | nasplib_isofts_kiev_ua-123456789-28313 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | XXXX-0067 |
| language | English |
| last_indexed | 2025-12-01T08:57:10Z |
| publishDate | 2011 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Korostil, A. Korostil, Ju. 2011-11-09T16:38:26Z 2011-11-09T16:38:26Z 2011 Quantum dynamics of a two-level system under extrenal field / A. Korostil, Ju.Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 73-84. — Бібліогр.: 16 назв. — англ. XXXX-0067 https://nasplib.isofts.kiev.ua/handle/123456789/28313 72.25. 72.25 We present exact analytic solutions for non-linear quantum dynamics of twolevel system (TLS) subject to periodic-in-time external field. >n constructing the exactly solvable models, we use approach where the form of external perturbation is chosen to preserve n integrability constraint, which yields single non-linear differential equation for the ac-field. solution to this equation is expressed in terms of Jacobi elliptic functions with three independent parameters that allows n to choose the frequency, average value, and amplitude of the time-dependent field at will. This form of the ac-drive is especially relevant to the problem of dynamics of TLS charge defects that cause dielectric losses ?n superconducting qubits. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Quantum dynamics of a two-level system under extrenal field Article published earlier |
| spellingShingle | Quantum dynamics of a two-level system under extrenal field Korostil, A. Korostil, Ju. |
| title | Quantum dynamics of a two-level system under extrenal field |
| title_full | Quantum dynamics of a two-level system under extrenal field |
| title_fullStr | Quantum dynamics of a two-level system under extrenal field |
| title_full_unstemmed | Quantum dynamics of a two-level system under extrenal field |
| title_short | Quantum dynamics of a two-level system under extrenal field |
| title_sort | quantum dynamics of a two-level system under extrenal field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/28313 |
| work_keys_str_mv | AT korostila quantumdynamicsofatwolevelsystemunderextrenalfield AT korostilju quantumdynamicsofatwolevelsystemunderextrenalfield |