Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses
We study theoretically the behavior of the superconducting flux qubit subjected to the series of the electromagnetic pulses. The possibility of controlling the system state via changing the parameters of the pulse is studied. We calculated the phase shift in the tank circuit weakly coupled to the qu...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/7765 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Dynamic behavior of superconducting flux qubit excited by the series of electromagnetic pulses / A.S. Kiyko, A.N. Omelyanchouk, S.N. Shevchenko // Физика низких температур. — 2007. — Т. 33, № 2. — С. 1338-1341. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860149659001946112 |
|---|---|
| author | Kiyko, A.S. Omelyanchouk, A.N. Shevchenko, S.N. |
| author_facet | Kiyko, A.S. Omelyanchouk, A.N. Shevchenko, S.N. |
| citation_txt | Dynamic behavior of superconducting flux qubit excited by the series of electromagnetic pulses / A.S. Kiyko, A.N. Omelyanchouk, S.N. Shevchenko // Физика низких температур. — 2007. — Т. 33, № 2. — С. 1338-1341. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| description | We study theoretically the behavior of the superconducting flux qubit subjected to the series of the electromagnetic pulses. The possibility of controlling the system state via changing the parameters of the pulse is studied. We calculated the phase shift in the tank circuit weakly coupled to the qubit which can be measured by the impedance measurement technique. For the flux qubit we consider the possibility to estimate relaxation rate from the impedance measurements by varying the delay time between the pulses.
|
| first_indexed | 2025-12-07T17:51:22Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12, p. 1338–1341
Dynamic behavior of superconducting flux qubit excited
by a series of electromagnetic pulses
A.S. Kiyko, A.N. Omelyanchouk, and S.N. Shevchenko
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: kiyko@ilt.kharkov.ua
Received July 17, 2007
We study theoretically the behavior of the superconducting flux qubit subjected to a series of electro-
magnetic pulses. The possibility of controlling the system state via changing the parameters of the pulse is
studied. We calculated the phase shift in a tank circuit weakly coupled to the qubit which can be measured by
the impedance measurement technique. For the flux qubit we consider the possibility of estimating the relax-
ation rate from the impedance measurements by varying the delay time between the pulses.
PACS: 03.67.Mn Entanglement production, characterization, and manipulation;
74.25.Nf Response to electromagnetic fields;
85.25.Am Superconducting device characterization, design, and modeling;
85.25.Hv Superconducting logic elements and memory devices; microelectronic circuits.
Keywords: two-level systems, superconducting qubits.
1. Introduction
Quantum effects in mesoscopic superconducting cir-
cuits based on small Josephson junctions have attracted
renewed attention. It has been demonstrated that
Josephson devices at low temperature behave like quan-
tum two-level systems. Therefore, ideas developed in
atomic and molecular physics can be used for description
of artificially fabricated circuits of macroscopic size.
These concepts are stimulated further by the perspectives
to realize quantum bits (qubits) for quantum information
processing. Qubits are effective two-level quantum sys-
tems with externally controlled parameters. In the last de-
cade a large number of proposals for building the qubits
based on Josephson elements were proposed [1–4]. There
are three basic types of Josephson-junction circuits that
behave quantum-mechanically at low temperature. They
are charge [1], phase [2], and flux [3] qubits. All of them
can be fabricated with high precision with the help of
modern lithography and can be the basis of the quantum
computer. Promising for quantum computations is the 3JJ
flux qubit that consists of the superconducting loop with
three Josephson junctions [3]. This type of qubit is insen-
sitive to charge noise, and it was shown that it has a high
quality factor [5]. It was predicted that such systems
should exhibit various quantum-mechanical effects in-
cluding macroscopic quantum tunneling of the flux [6].
Indeed the predicted effects have been observed experi-
mentally [2,7,8]. The quantum dynamics in single qubits
was studied in [3,4,9].
In our work we study the dynamics of the flux qubit
subjected to a series of rectangular electromagnetic
pulses. We present the model we use for calculations of
the phase shift � in the resonant tank circuit based on the
density matrix approach. Next we analyze the case which
permits analytical solution and obtain small addition to
the � in the ground state as function of the relaxation rate
�R . For arbitrary parameters we solve equations numeri-
cally and compare obtained result with analytical
calculations.
2. The model
Our aim is to study the behavior of superconducting
3JJ flux qubit excited by a series of rectangular electro-
magnetic pulses. Flux qubit consists of a superconducting
loop with three junctions: two identical and one with the
parameters differing by factor �. For all calculations be-
low we take� equal to 0 8. .
The Hamiltonian of the flux qubit in the two-level ap-
proximation has the form [10,11]:
© A.S. Kiyko, A.N. Omelyanchouk, and S.N. Shevchenko, 2007
� � � ,H z x� � ��� �� (1)
where the diagonal term � is the bias and the off-diagonal
term � �exp( )E /EJ C is the tunneling amplitude bet-
ween the wells. Here �� x , �� z are Pauli matrices in the ba-
sis {| , | }
� �� of the current operator in the qubit: � �I I z� 0� ,
I 0 � I g /C
� �( , ) 2 , where IC is the critical current of the
qubit, g E /EJ C� , the explicit formula for the
�( , )g can
be found in Ref. 12. The eigenstates of �� z correspond
to the clockwise ( � | | )� z
� � �
� and counterclockwise
( � | | )� z �� � �� currents in the qubit. The bias
� � ��
�
�
�
�
�I f0 0
1
2
� (2)
is controlled by the dimensionless applied magnetic flux
f /x�� �0 through the qubit;�0 2� h/ e is the flux quan-
tum.
The magnetic flux consists of two components:
f f f tDC� �
~
( ) , (3)
which describe the adiabatically changing magnetic flux,
f DC , and the time-dependent component,
~
( )f t . We will
study the possibility to control the system state via the se-
ries of the rectangular pulses with the amplitude f A and
duration from t n T
n
1
( )
( )� � � to t n T T
n
2
( )
( )� � �� :
~
( ) [ ( ) ( )]
( ) ( )
f t f t t t tA
n n� � � �� � �
1 2
, (4)
where �( )t stands for the theta-function, T is the pulse du-
ration, and � is the delay between pulses (Fig. 1). The effect
of the pulse is in changing the level occupation probabili-
ties and to make them oscillating functions of time during
the pulse. It should be noted that in the basis {| , | }
� �� of the
current operator, which are not eigenstates of the
Hamiltonian, the probabilities oscillate both during and af-
ter the pulse.
We describe the system’s evolution with the Bloch
equation for the density matrix �� (� �1):
d
dt
i H
�
[ � , �] � �
�
� �� � � � . (5)
The impedance measurement technique [13,14] con-
sists in that the tank circuit probes the effective induc-
tance of the system via measuring the phase shift � be-
tween the voltage and current in the tank circuit. The
phase shift � is related to the Josephson inductance � of
the qubit as follows:
tan ,� � �k QL2 1
� (6)
�
� �
�� �
�
� �
�
�
1
0
0
0
1
� �
�
( � � )
I
f
I
fDC DC
zSp �� . (7)
Here M is the mutual inductance of the qubit with the tank
circuit; Q R C LT T T� / , and k M LLT� / are the quality
factor and the coupling coefficient for the tank circuit,
which consists of the inductor, LT , capacitor, CT , and resis-
tor, RT , connected in parallel (see in [15] for more details).
3. Excitation of the flux qubit with the series of
pulses
In this Section we study the excitation of the flux qubit
with the series of rectangular pulses. We start from the
general 1-qubit Hamiltonian that has the form of Eq. (1)
in the basis of states {| , | }
� �� , assuming
~
( )f t � 0. For a
flux qubit these states correspond to a definite direction
of the current circulating in the ring. First the time-inde-
pendent Hamiltonian is diagonalized in the basis of
eigenstates {| , | }�� �� with the rotation matrix �S:
�
cos ( ) sin ( )
sin ( ) cos ( )
S
/ /
/ /
�
�
�
�
��
�
�
��
� �
� �
2 2
2 2
,
with sin � �� � �� �/ 2 2 , cos � � �� �/ �2 2 .
For the calculation of the observable value, the phase
shift � in the tank circuit, according to Eq. (6), we need
the density matrix in the energy representation, where its
diagonal components are equal to the probability of the
system to be in the ground |�� or excited state |��.
Next we introduce the time-dependent terms into the
time-independent Hamiltonian. Making use of the trans-
formation � ( ) � � ( ) �H t S H t S� �1 , we get the Hamiltonian � ( )H t
in the energy representation for the flux qubit [16]:
� ( ) �
~
( ) (cos � sin � )H t
E
I f t / Ez z x� � � �
�
� �
2
2 0 0� �� �� ,
(8)
� �E � �2 2 2� . (9)
The time evolution of the density matrix, which can be
taken in the form � ( � � � � )� � � �� � � �1 2X Y Z /x y z , is described
by the equation of motion (5). Initial condition for the den-
sity matrix in {| , | }�� �� basis is X Y( ) ( )0 0 0� � , Z( )0 1� ,
which corresponds to the ground state of the system. Sol-
ving the system of equations for X t( ), Y t( ), Z t( ) with
Dynamic behavior of superconducting flux qubit excited by the series of electromagnetic pulses
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12 1339
0
2 4 6
0.2
0.4
0.6
0.8
1.0
T
f(
t)
/f
A �
t
Fig. 1. Series of pulses
phenomenologically introduced dephasing and relaxation
rates �� and �R :
dX
dt
E h t Y X
dY
dt
h t Z E h t
� � �
� � �
( ( ) cos ( )) ,
( ) sin ( ) ( (
� �
�
�
�
�
) cos ( )) ,
( ) sin ( ) ( ( )) ,
�
�
�X Y
dZ
dt
h t Y Z ZR
�
� � � �
�
� 0
(10)
with h t I f t( )
~
( )� 2 0 0� , we obtain the probability of occu-
pation of the upper level |�� (the excited state) P t� �( )
� � ��22 1 2( ) ( ( ))t Z t / . We calculate the density matrix in the
flux basis making use of the transformation � � � �� �flux �
�S S 1 and
obtain the probability of the current to be circulating in the
clockwise direction P tL( ), according to:
P t X t Z tL( ) ( sin ( ) ( ) cos ( ) ( ))� � �
1
2
1 � � . (11)
Averaging the P tL( ) over t we calculate phase shift� ac-
cording to Eq. (6). For arbitrary values of the parameters of
applied perturbation the system of equations (10) can be
solved numerically. In the limiting case of one pulse (that is
� � ) and without relaxation processes taking into account
obtained results the solution can be found in Ref. 17.
Before presenting the numerical results, consider the
limiting case which permits the analytical solution:
T TR R!! � !!�
, ,� � �� 1 . (12)
In this case we can neglect the decay rates �R and �� in
(10) during the excitation time T and assume that after the
pulse during the delay time � the system is returned to the
ground state. Periodically repeating this process, we will
have the input of relaxation processes in the time-aver-
aged characteristics of the system.
With the assumptions (12) and for zero temperature
the solution of the equations (10) can be found for the two
time intervals: during the pulse (0 ! !t T ) and after the
pulse (T t T! ! � �).
The solution for 0 ! !t T is the following:
X t
AC
A C
A C t
Y t
A
A C
1 2 2
2 2 2
1
2 2
2 1
2
( ) sin ,
( ) sin (
�
�
��
�
�
�
�
�
� �
�
A C t
Z t
A C
C A A C t
2 2
1 2 2
2 2 2 21
�
�
�
� �
) ,
( ) ( cos ( ) ,
(13)
where A h� � sin ,� C E h� � �� cos ,� h I f A� 2 0 0� ; and
for T t T! ! � �:
X t t T X T E t T
Y T E
2 1
1
( ) exp[ ( )][ ( ) cos ( ( ))
( ) sin (
� � � � �
�
� �
�
�
( ))],
( ) exp[ ( )]( ( )) .
t T
Z t t T Z TR
�
� � � � �2 11 1�
Let the duration of the pulse T equals to � / A C2 2� ,
which corresponds to the one cycle of excitation during
the time T .
Then we obtain:
X t t T
AC
A C
E t T
Z t
A
2 2 2
2
2
1
2
( ) exp( ( )) cos ( ( )) ,
( )
� � �
�
�
� �
� ��
2
2 2A C
t TR
�
� �exp( ( )) .�
Taking into account the inequalities (12), we obtain for
the time-averaged values Z and X the following expres-
sions:
X
AC
A C E
�
� �
2
2 2 2 2
�
� �
�
� �( )
, (14)
Z
A
A C R
� �
�
1
2 12
2 2 � �
. (15)
Before substituting these values in the (11) and (16) we
estimate the contribution of the X and Z into the phase
shift � for the parameters we use for calculation. Our
evaluation indicates that the contribution of the X is about
three orders lower than that of the Z, so we neglect the
term containing X in Eq. (11). We calculate the small ad-
dition to the phase shift � due to the relaxation process
� ( )1 . At the point f /�1 2 we obtain after some algebra:
�
�
( ) ( )
(( ) )
.1
2
0
2
0 0
3
0 0
2 2
2 1
�
�
k QL f I
I f
A
A R�
�
� � � �
(16)
Hence from the measurement of the phase shift� at the
point f /�1 2 it is possible to estimate the relaxation rate
according to Eq. (16). The behaviour of � ( )1 at the point
f /�1 2 as the function of the product ��R is presented in
Fig. 2. For the calculations we used the same parameters
as for the numerical calculations below.
1340 Fizika Nizkikh Temperatur, 2007, v. 33, No. 12
A.S. Kiyko, A.N. Omelyanchouk, and S.N. Shevchenko
0 20 40 60 80 100
0.01
0.02
0.03
0.04
�
"
Fig. 2. The addition to the phase shift �" for the flux qubit ex-
cited by the series of rectangular pulses due to the relaxation
process in the point f � 0 5.
Now we study the excitation of the flux qubit by the series
of rectangular pulses numerically. Namely we calculate the
phase shift in the tank circuit by making use of the solution of
the Eqs. (6) and (10). Fig. 3 is plotted for the following set of
parameters for the qubit: I 0 0� � � 200 GHz, � �14. GHz,
k Q LI /2
0 0
32 10( ) � � # � ; the excitation
~
( )f t was considered
to be the series of the pulses with � � !! �T � 1, � � �0 5 1. � ,
� � �2 1� (from upper to down) and the decay rates
� � �R � � �� 0 1. GHz. We observe that at � � ��1 the reso-
nances disappear with increasing delay time �. This can be
used in practice for relatively simple estimation of the decay
rates by changing the delay time between the pulses.
Next we compare the theoretically (dashed) and numeri-
cally (marked by points) calculated curves for the phase shift
� calculated under the assumption (12) from the (14) and
(15). For the numerical calculations we use the following pa-
rameters of the pulse: T � 0 5. , � �100, f A � 0 005. and of the
decay rates � � �R � � �� 01. GHz. Such values of �, T and
�R ,� correspond to the limiting case which we considered
previously (12), and one can see very good agreement in
Fig. 4.
4. Conclusion
Dynamics of a flux qubit subjected to a series of rectan-
gular electromagnetic pulses. We investigated the changes
of the tank circuit phase shift� for the single qubit that ap-
pears due to excitation by the pulses. It was demonstrated
that the response of the tank circuit essentially depends on
the relation between the decay rates �R ,� and the delay
time �, which may be used for the estimation of �R by mea-
suring the phase shift � as a function of the delay time �.
We would like to thank M. Grajcar and E. Il’ichev for
helpful discussions. A.N.O. acknowledges the partial fi-
nancial support of DFG grant. S.N.S. acknowledges the
financial support of INTAS under the Fellowship Grant
for Young Scientists (No. 05–109–4479).
1. Y. Nakamura, Y.A. Pashkin, and J.S. Tsai, Nature (Lon-
don) 398, 786 (1999).
2. J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo, and J.E.
Lukens, Nature (London) 406, 43 (2000).
3. J.E. Mooij, T.P. Orlando, L. Levitov, Lin Tian, Caspar H.
van der Wal, and Seth Lloyd, Science 285, 1036 (1999).
4. D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C.
Urbina, D. Esteve, and M.H. Devoret, Science 296, 886
(2002).
5. C.H. van der Wal, F.K. Wilhelm, C.J.P.M. Harmans, and
J.E. Mooij, Eur. Phys. J. B31, 111 (2003); L. Tian, S.
Lloyd, and T.P. Orlando, Phys. Rev. B65, 144516 (2002).
6. A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149, 374
(1983).
7. J. Clarke, A.N. Cleland, M.H. Devoret, D. Estive, and J.M.
Martinis, Science 239, 992 (1988).
8. C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N.
Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd, and
J.E. Mooij, Science 290, 773 (2000).
9. I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, and J.E.
Mooij, Science 299, 1869 (2003).
10. Y. Makhlin, G. Sch�n, A.Shnirman, Rev. Mod. Phys. 73,
357 (2001).
11. M. Grajcar A. Izmalkov, E. Il’ichev, Th. Wagner, N. Ouk-
hanski, U. H�bner, T. May, I. Zhilyaev, H.E. Hoenig,
Ya.S. Greenberg, V.I. Shnyrkov, D. Born, W. Krech, H.-G.
Meyer, Alec Maassen van den Brink, and M.H.S. Amin,
Phys. Rev. B69, 060501 (2004).
12. Y.S. Greenberg, A. Izmalkov, M. Grajcar, E. Il’ichev, W.
Krech, H.G. Meyer, M.H.S. Amin, and A. M. van den Brink,
Phys. Rev. B66, 214525 (2002).
13. R. Rifkin and B.S. Deaver, Jr., Phys. Rev. B13, 3894 (1976).
14. E. Il’ichev, V. Zakosarenko, L. Fritsch, R. Stolz, H.E. Hoenig,
H.-G. Meyer, M.G�tz, A.B. Zorin, V.V. Khanin, A.B. Pavo-
lotsky, and J. Niemeyer, Rev. Sci. Instr. 72, 1882 (2001).
15. S.N. Shevchenko, to be published.
16. S.N. Shevchenko, A.S.Kiyko, A.N. Omelyanchouk, W. Krech,
Fiz. Nizk. Temp. 31, 752 (2005) [Low Temp. Phys. 31, 564
(2005)].
17. A.S. Kiyko, A.N. Omelyanchouk, and S.N. Shevchenko, in:
Proceedings of MS+S Conference, Kanagawa, Japan (2006).
Dynamic behavior of superconducting flux qubit excited by the series of electromagnetic pulses
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12 1341
0.48 0.5 0.52
–1
0
1 a
ta
n
�
0.48 0.5 0.52
–1
0
1
b
ta
n
�
0.48 0.5 0.52
–0.5
0
0.5
ta
n
� c
� �x 0/
Fig. 3. The phase shift � for the flux qubit excited by the series
of rectangular pulses with: T � �� 0 5. (a), T � 0 5. , � � 5 (b),
T � 0 5. , � � 20 (c).
0.48 0.49 0.5 0.51 0.52
–0.3
–0.2
–0.1
0
�
� �x 0/
Fig. 4. Comparison of the theoretical (dashed) and numerical
(marked by points) curves for the ecxitation with the series of
the pulse with parameters: T � 0 5. , � �100.
|
| id | nasplib_isofts_kiev_ua-123456789-7765 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:51:22Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Kiyko, A.S. Omelyanchouk, A.N. Shevchenko, S.N. 2010-04-12T13:05:13Z 2010-04-12T13:05:13Z 2007 Dynamic behavior of superconducting flux qubit excited by the series of electromagnetic pulses / A.S. Kiyko, A.N. Omelyanchouk, S.N. Shevchenko // Физика низких температур. — 2007. — Т. 33, № 2. — С. 1338-1341. — Бібліогр.: 17 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/7765 We study theoretically the behavior of the superconducting flux qubit subjected to the series of the electromagnetic pulses. The possibility of controlling the system state via changing the parameters of the pulse is studied. We calculated the phase shift in the tank circuit weakly coupled to the qubit which can be measured by the impedance measurement technique. For the flux qubit we consider the possibility to estimate relaxation rate from the impedance measurements by varying the delay time between the pulses. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Сверхпроводимость, в том числе высокотемпературная Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses Article published earlier |
| spellingShingle | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses Kiyko, A.S. Omelyanchouk, A.N. Shevchenko, S.N. Сверхпроводимость, в том числе высокотемпературная |
| title | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| title_full | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| title_fullStr | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| title_full_unstemmed | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| title_short | Dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| title_sort | dynamic behavior of superconducting flux qubit excited by a series of electromagnetic pulses |
| topic | Сверхпроводимость, в том числе высокотемпературная |
| topic_facet | Сверхпроводимость, в том числе высокотемпературная |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7765 |
| work_keys_str_mv | AT kiykoas dynamicbehaviorofsuperconductingfluxqubitexcitedbyaseriesofelectromagneticpulses AT omelyanchoukan dynamicbehaviorofsuperconductingfluxqubitexcitedbyaseriesofelectromagneticpulses AT shevchenkosn dynamicbehaviorofsuperconductingfluxqubitexcitedbyaseriesofelectromagneticpulses |