Signal amplification in a qubit-resonator system
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of the dressed qubit with the second probing signal can essentially change the...
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Karpov, D.S. Oelsner, G. Shevchenko, S.N. Greenberg, Ya.S. Il’ichev, E. 2018-01-10T14:25:54Z 2018-01-10T14:25:54Z 2016 Signal amplification in a qubit-resonator system / D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, E. Il’ichev // Физика низких температур. — 2016. — Т. 42, № 3. — С. 246–253. — Бібліогр.: 41 назв. — англ. 0132-6414 PACS: 42.50.Hz, 85.25.Am, 85.25.Cp, 85.25.Hv https://nasplib.isofts.kiev.ua/handle/123456789/128489 We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate the transmission amplitude of the probe signal through the resonator as a function of the qubit’s energy and the driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and demonstrated graphically. We present the influence of the signal parameters on the value of the amplification, and discuss the values of the qubit-resonator system parameters for an optimal amplification and attenuation of the weak probe signal. This work was partly supported by DKNII (Project No. M/231-2013), BMBF (UKR-2012-028), RFFR (No. 15-32- 50195/15), MES (8.337.2014/K). D.S.K. acknowledges the hospitality of IPHT (Jena, Germany) and NSTU (Novosibirsk, Russia), where part of this work was done. The authors are grateful to A.N. Omelyanchouk for useful discussions and comments. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Свеpхпpоводимость, в том числе высокотемпеpатуpная Signal amplification in a qubit-resonator system Article published earlier |
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| title |
Signal amplification in a qubit-resonator system |
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Signal amplification in a qubit-resonator system Karpov, D.S. Oelsner, G. Shevchenko, S.N. Greenberg, Ya.S. Il’ichev, E. Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| title_short |
Signal amplification in a qubit-resonator system |
| title_full |
Signal amplification in a qubit-resonator system |
| title_fullStr |
Signal amplification in a qubit-resonator system |
| title_full_unstemmed |
Signal amplification in a qubit-resonator system |
| title_sort |
signal amplification in a qubit-resonator system |
| author |
Karpov, D.S. Oelsner, G. Shevchenko, S.N. Greenberg, Ya.S. Il’ichev, E. |
| author_facet |
Karpov, D.S. Oelsner, G. Shevchenko, S.N. Greenberg, Ya.S. Il’ichev, E. |
| topic |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| publishDate |
2016 |
| language |
English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction
of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of
the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate
the transmission amplitude of the probe signal through the resonator as a function of the qubit’s energy and the
driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and
demonstrated graphically. We present the influence of the signal parameters on the value of the amplification,
and discuss the values of the qubit-resonator system parameters for an optimal amplification and attenuation of
the weak probe signal.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/128489 |
| citation_txt |
Signal amplification in a qubit-resonator system / D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, E. Il’ichev // Физика низких температур. — 2016. — Т. 42, № 3. — С. 246–253. — Бібліогр.: 41 назв. — англ. |
| work_keys_str_mv |
AT karpovds signalamplificationinaqubitresonatorsystem AT oelsnerg signalamplificationinaqubitresonatorsystem AT shevchenkosn signalamplificationinaqubitresonatorsystem AT greenbergyas signalamplificationinaqubitresonatorsystem AT ilicheve signalamplificationinaqubitresonatorsystem |
| first_indexed |
2025-11-25T21:47:38Z |
| last_indexed |
2025-11-25T21:47:38Z |
| _version_ |
1850560439285448704 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3, pp. 246–253
Signal amplification in a qubit-resonator system
D.S. Karpov1, G. Oelsner2, S.N. Shevchenko1,3, Ya.S. Greenberg4, and E. Il’ichev2,4
1B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Nauki Ave., Kharkov 61103, Ukraine
2Leibniz Institute of Photonic Technology, Jena, Germany
3V. N. Karazin National University, Kharkov, Ukraine
4Novosibirsk State Technical University, Novosibirsk, Russia
E-mail: karpov@ilt.kharkov.ua
Received November 18, 2015, published online January 26, 2016
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interac-
tion of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of
the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate
the transmission amplitude of the probe signal through the resonator as a function of the qubit’s energy and the
driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and
demonstrated graphically. We present the influence of the signal parameters on the value of the amplification,
and discuss the values of the qubit-resonator system parameters for an optimal amplification and attenuation of
the weak probe signal.
PACS: 42.50.Hz Strong-field excitation of optical transitions in quantum systems; multiphoton processes; dy-
namic Stark shift;
85.25.Am Superconducting device characterization, design, and modeling;
85.25.Cp Josephson devices;
85.25.Hv Superconducting logic elements and memory devices; microelectronic circuits.
Keywords: dressed states, superconducting qubit, amplification.
1. Introduction
Quantum optical effects with Josephson-junction-based
circuits have been intensively studied for the last decade.
In particular, such systems are interesting as two-level arti-
ficial atoms (qubits) [1–5]. Quantum energy levels and
quantum coherence are inherent to qubits and provide the
basis for studying fundamental quantum phenomena. It is
important to note that qubits can be controlled over a wide
range of parameters [1,6–12] and they have unavoidable
coupling to the dissipative environment.
The ability of stimulated emission and lasing in super-
conductive devices has been actively studied during the
last several years both theoretically [13–18] and experi-
mentally [19–24]. The work is underway on using these
phenomena as basis for a quantum amplifier of signals near
the quantum limit. This paper was motivated by several
recent publications where the amplification of the input
signal was observed in systems with nanomechanical reso-
nators [8,25], with waveguide resonators [7,21,25–30] and
the concept of the amplifiers was discussed [31–36].
A key value of the qubit-resonator system in the
experiment is the transmission coefficient of the signal
through the resonator. This transmission coefficient
depends on different parameters. The speed and direction
of the energy exchange is determined by relaxation rates.
The variation of the coupling strength allows to change the
width of resonance. The change of the driving amplitude
and the magnetic flux (for flux and phase slip qubit; for
charge qubit this quantity is the applied voltage) allows to
find an acceptable point on the resonance line comparative
to other parameters. In the paper we consider how the
amplification and attenuation of the input signal depend on
the parameters of the system. The general idea is to find
values and there relationship for the parameters of the
system in order to make the amplification maximal.
In addition to Ref. 13 here we systematically study the
impact of such parameters as coherence time, resonator
losses, coupling and other. Also we demonstrated how tem-
perature influences the transmission coefficient. Besides we
show the universality of the doubly-dressed approach for
two-level systems. We compare the appearance of the am-
plification-attenuation phenomena in both flux and phase-
slip qubits [37].
The paper is organized as follows Sec. 2 contains a
description of the studied system which is a qubit coupled
© D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, and E. Il’ichev, 2016
Signal amplification in a qubit-resonator system
to the two-mode /2λ waveguide resonator. Section 3 is
devoted to the evolution of the qubit-resonator system
which is described by a Lindblad equation. We analyze the
solution of the Lindblad equation in Sec. 4. and Sec. 5.
Section 6 concludes the paper.
2. The qubit-resonator system
The studied system consists of a quantum resonator
(transmission-line resonator with the length = /2L λ ) [38]
and a two-level system, the superconducting flux qubit. The
qubit interacts with two harmonics in the resonator: first
probing signal with frequency pω close to the first harmonic
of the resonator and the second signal is a high amplitude
driving signal with frequency dω close to the third harmonic
of the resonator. Such system is analogous to the one studied
recently experimentally in Refs. 6, 13, 39.
The qubit located in the middle of the resonator ( = 0x )
is coupled only to the odd harmonics m, for which the
current is defined by ( ) = cos /mI x I mx Lπ (see Fig. 1). The
transmission-line resonator runs from /2L− to /2.L We
consider the interaction of the qubit and two-mode resonator
according to doubly-dressed approach, as in Ref. 13.
Hamiltonian of the system is
†
tot =
2
qb
z rH a a
δω
σ + δω +
( ) ( )† † † ,pg a a a a+ σ + σ + ξ +
(1)
where ,zσ ,xσ yσ , 1= ( )
2 x yiσ σ − σ are the Pauli’s
operators in the doubly-dressed basis; = /qb pEδω ∆ −ω
is
the detuning of the doubly-dressed qubit; =E∆
2 2= = Rε + ∆ Ω
is the Rabi frequency of the dressed
qubit;
0
1g g
E E
ε ∆
=
∆ ∆
is the renormalized coupling; =E∆
2 2
0= ε + ∆ ; 0 0 0= 2 ( / 0.5)p xIε Φ Φ Φ − where xΦ is the
external magnetic flux applied to the qubit loop; pI is the
persistent current in the qubit loop; 0Φ is the flux
quantum; ∆ is the energy separation between two levels at
the degeneracy point 0 = 0ε ; r r pδω = ω −ω is the detuning
of the resonator; 34dA N g= 〈 〉 is the normalized
amplitude of the driving signal, given by the average
number of photons N in resonator of the third harmonic.
The dressed bias ε and the tunneling amplitude ∆ are
defined by the driving frequency dω and amplitude dA
either in the weak-driving regime, at d dA < ω ,
, /2 ,d dE A Eε = ∆ − ω ∆ = ∆ ∆
(2)
or in the strong-driving regime, where the energy bias is
defined by the detuning from the k -photon resonance,
( )kε → ε , and the renormalized tunneling amplitude is
defined by the oscillating Bessel function, ( )k∆ → ∆ , as
following
( ) ( ) 0
0
= , = .k k d d
d k
d
k A
E k J
E
ω ε
ε ∆ − ω ∆ ∆ ε ω ∆
(3)
In the doubly-dressed representation the Hamiltonian (1)
is written for the energy states | 0〉 and |1〉 , where we
omitted constants terms; we have used the rotating-wave
approximation. In Fig. 2 we explain the processes which
take place in the qubit-resonator system.
3. Evolution of the system
One possible method to describe the evolution of an
open system is a solution of the Lindblad equation. In our
case we rewrite it in the dressed basis similar to Ref. 13
and take into account finite temperature [40]:
Fig. 1. (Color online) Diagrammatic representation of the system
under study: a qubit placed in a waveguide resonator. The qubit
interacts with a two-mode resonator. The first signal has an
amplitude dξ and a frequency pω . The second signal has an
amplitude dA and a frequency dω . A measurable (probing)
signal at the output has an amplitude different from the input
values. t is transmission amplitude.
Fig. 2. (Color online) The interaction between a qubit and a
resonator can be described in frame of the dressed states. (a) A
high-amplitude signal dA interacts with a two-level system
(qubit). (b) Energy levels are modified. The qubit can be
described in terms of the dressed states, in other words we obtain
a dressed two-level system with energy distance proportional to
the amplitude dA of the third-harmonic. (c) The dressed qubit
interacts with probe signal. (d) The amplitude of the output signal
is increased or weakened.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 247
D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, and E. Il’ichev
tot= [ , ] [ ],i
i
d i H
dt
ρ − ρ + Λ ρ∑
(4)
{ }† †1[ ] = , ,
2↓ ↓
Λ ρ Γ σρσ − σ σ ρ
(5)
{ }† †1[ ] = , ,
2↑ ↑
Λ ρ Γ σ ρσ− σσ ρ
(6)
( )[ ] = ,
2 z z
ϕ
ϕ
Γ
Λ ρ σ ρσ −ρ
(7)
( ) { }† †
th
1[ ] = 1 ,
2
n a a a aκ
Λ ρ κ + ρ − ρ +
{ }† †
th
1 , ,
2
n a a aa + κ ρ − ρ
(8)
2
1 th
1= 1
4
n
E↓
ε Γ Γ − + ∆
( )
22
1 th
11 1 ,
4 2
n
E E
ϕΓ ε ∆ + Γ + + + ∆ ∆
(9)
2
1 th
1= 1
4
n
E↑
ε Γ Γ + + ∆
(10)
( )
22
1 th
11 1 ,
4 2
n
E E
ϕΓ ε ∆ + Γ + − + ∆ ∆
( )
2 2
1 th
1= 2 1
2
n
E Eϕ ϕ
∆ ε Γ Γ + + Γ ∆ ∆
(11)
where 1
th = exp[ ] 1k
B
n
k T
− ν
−
is the thermal photon number in
the resonator; kν is density frequency distribution for
thermal photons; Bk is Boltzmann constant; T is
thermodynamic temperature of the system; ρ is the density
matrix; [ ]ϕΛ ρ
is the dressed phase relaxation of the
dressed qubit; 1Γ and ϕΓ are the qubit relaxation and
dephasing rates; [ ]↓Λ ρ
is the relaxation from | 0〉 to |1〉
level; [ ]↑Λ ρ
is the excitation from |1〉 to | 0〉 level. The
analysis of the difference between the rates [ ]↑Λ ρ
and
[ ]↓Λ ρ
shows availability of the inverse population in the
system (Figs. 4 and 8). The equation of motion for the
expectation value of any quantum operator A:
tot tot= [ , ] Tr ( [ ]),i
i
d A i A H AH
dt
〈 〉
− 〈 〉 + Λ ρ∑
(12)
where = Tr ( )A A〈 〉 ρ , [ , ] = Tr ([ , ] )A H A H〈 〉 ρ , the trace is
over all eigenstates of the system; and here H is the
Hamiltonian of the system in the doubly-dressed basis
Eq. (1). For the expectation values of the operators ,a † ,a
† ,σ ,σ and zσ we obtain the so-called Maxwell-Bloch
equations:
,p
r
d a
i a ig i
dt
ξ
′ ′= − δω − σ −
(13)
= ,qb z
d
i ig a
dt
σ
′− δω σ + σ
(14)
( )† †2zd
i g a a
dt
σ
= − σ − σ −
(15)
,z+ −−Γ σ −Γ
where
( )= ,± ↓ ↑Γ Γ ± Γ (16)
/2,r r i′δω = δω − κ (17)
qb 2= .qb i′δω δω − Γ (18)
The Eqs. (13)–(15) were solved in our previous work [10]
in the small photon number limit ( 1n〈 〉 << ). In general, the
system of equations is infinite, but it may be factorized
† †=aa a a〈 〉 etc. This approximation can be used in
the limit of strong perturbation, when the average number
of photons in the system is substantially greater than unity
( 1n〈 〉 >> ). In this way, we simplify Eqs. (13)–(15):
2= ,p qb
z qb r
a
g
′ξ δω
−
′ ′σ + δω δω
(19)
*
†
2 * *
= ,p qb
z qb r
a
g
′ξ δω
−
′ ′σ + δω δω
(20)
( )† †= 2 2 .p
z i a a a a+ −
ξ
Γ σ + Γ − + κ
(21)
The transmission amplitude of the signal is defined by
the formula [19,40]
| | | | .
2 p
t aκ
= 〈 〉
ξ
(22)
The dynamics of the two-level system coupled to a two-
mode quantum resonator can be described as the solution
of the Eqs. (19)–(21) in the limit of large photon numbers
in the resonator. Such description offers a satisfactory
explanation of the experiments with different qubits [6,39].
This is demonstrated below.
4. Amplification and attenuation of the probe signal
Consider Eqs. (19)–(21) in the limit of the weak probing
signal pξ . We obtain the asymptotic solution for z〈σ 〉 :
0 = .z
−
+
Γ
〈σ 〉 −
Γ
(23)
248 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3
Signal amplification in a qubit-resonator system
A solution can also be found in the limit of large
amplitudes of the probing signal pξ :
( )2
2
2
2
= ,
qb
z
g
∞
κ Γ + δω
〈σ 〉 −
Γ
(24)
where 2 =
2
↓ ↑
ϕ
Γ + Γ
Γ Γ +
.
The Eqs. (19) and (21) were solved in the limit of large
photon numbers in the system ( )dA >> ∆ . We obtain two
extremes of the transmission coefficient: amplification (the
driving signal energy is transferred to the probing signal)
and attenuation (here vice versa the probing signal energy
is pumped to the driving signal). Consider the case of full
reflection of the probing signal, | | = 0t . Then Eq. (19) is
simplified
= 0.
2
↓ ↑
ϕ
Γ + Γ
Γ +
(25)
The left part of Eq. (25) consists of only positive functions,
which in all experimental parameters space do not come to
zero. The full reflection of the probing signal is impossible.
A major effect in studied system is inverse population.
Practically, it is the difference between excitation and
relaxation processes in the dressed qubit,
1= .
E↓ ↑
ε
Γ −Γ Γ
∆
(26)
The inverse population in the system arises when the
relaxation < 0−Γ
or
.dE∆ < ω (27)
Figure 3 is plotted for the following parameters
/ = 3.7h∆ GHz, 1/2g π = 0.8 MHz, /2rω π = 2.5 GHz,
3 ,d rω = ω / 2 = 30κ π kHz, p = 0.05ξ κ, =p rω ω ,
/ = 7dA h GHz. The transmission coefficient sharp changes
in the value at the magnetic flux about 0 / = 4.7hε GHz and
0 / = 8.9hε GHz. In the former case, the amplitude of the
transmission signal increases. In the latter case, the
transmission signal attenuates.
The amplification of the signal takes place in the system
when the Rabi frequency RΩ is close to the resonator
frequency (see Fig. 4(a),(c)). We obtain the resonant
exchange of energy between the probing signal and the
dressed states. The direction of the energy transfer is
specified by the difference between dissipative rates of the
states (see Fig. 4(b)).
The analysis of Eqs. (19) and (21) demonstrates that we
can considerably effect on it by varying of the relaxation
coefficient and the amplitudes of the probing and driving
signals. In Fig. 3 it is demonstrated how the variation of the
relaxation coefficient effects on the amplification and the
Fig. 3. (Color online) The normalized transmission amplitude as
a function of the normalized magnetic flux 0 /hε . We obtain (a)
an amplification and (b) an attenuation of the input signal. The
transmission coefficient strongly depends on relaxations terms.
Parameters for (a): 1/2 = 4.8Γ π , 6, 8 MHz and /2 = 0.15ϕΓ π ,
2.8, 2 MHz for black, red, and green curve,respectively; parameters
for (b): 1/2 = 0.8Γ π , 4.8, 6 MHz and /2 = 0.2ϕΓ π , 2, 2.8 MHz
for black, red, and green curve, respectively.
Fig. 4. (Color online) A schematic of the processes in the doubly-
dressed system. (a) The black solid line is the Rabi frequency
= /R E hΩ ∆ of the dressed qubit. The blue dashed line is the
resonator frequency rω . In the points of the intersection, denoted
by A and B, the dressed system and the resonator exchange the
energy. The dressed system influences on the passed signal. The
output signal increases or decreases. The type of the process
depends on the population of the energy states (see (b) and (c)). If
the relaxation down ↓Γ
is smaller than the excitation ↑Γ
, it will
be an amplification of the transmitted signal ( < 0−Γ , point A1).
When the opposite situation is realized ( > 0−Γ ), it will be an
attenuation. (b) The difference between relaxations from level
| 0〉 to level |1〉 and from level |1〉 to level | 0〉 . According to
this graphics, we expect an inverse population in the system. (c)
Transmission amplitude as a function of the magnetic flux 0.ε
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 249
D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, and E. Il’ichev
attenuation in the system. Such results were experimen-
tally demonstrated in papers [22–26]. The Figs. 5 and 6
demonstrate impact of the coherence time on transmis-
sion amplitude.
Consider Eqs. (19)–(21) at the resonance point ( = pE∆ ω )
when the detuning of the resonator = 0rδω . Then Eqs. (19)
and (20) are simplified. We take into account that the
average of the operator z〈σ 〉 under weak probing signal is
given by Eq. (23). For small deviations = dEε ∆ − ω
the
transmission amplitude is given by the following formula
2 2
0 1
2
1
8
= 1 ,
g
t
E Q
ε ε
−
∆ ∆Γ κ
(28)
where th th
1 1
1 3= ( ) 2 .
2 2
Q n nϕ ϕΓ Γ + + + + Γ Γ
The Eq. (28) allows to roughly estimate effect of the
system parameters on the transmission amplitude t in the
first approximation. The nonzero temperature leads to the
decrease of the amplification. The qubit relaxation 1Γ and
dephasing ϕΓ rates should be small, then we have system
with long coherent time. We can use the asymptotic of the
Bessel function in the limit of the high amplitude of the
driving signal dA :
2 3
( )
3
0
( ) 1 .k d
dd
k E
AA
ω ∆
∆ ∝ ∝
ε
(29)
The transmission coefficient t is related to the driving
amplitude dA (see Eq. (28)).
Consider the impact of the coupling 1g between resonator
and qubit on the transmission. From Eqs. (19) and (22) one
can estimate its value for optimal amplification. In particular,
at the resonance point, where = pE∆ ω , we find that the
transmission coefficient is maximal when 2
2 .g Γ κ
For
small deviations ε , we estimate the coupling, at which the
transmission amplification is optimal,
( )
2
2
1 1
0
3 2 .dg ϕ
ω
κ Γ + Γ ε
(30)
The above estimates of the system parameters for
optimal amplification can be useful for both qualitative
theoretical analysis and for analyzing respective
experimental results.
Fig. 5. (Color online) Normalized transmission amplitude as a
function of the resonator detuning. The dash-dot line and the
solid line are plotted for / = 35dA h GHz, and the dash line is
plotted for / = 0dA h . The parameters of the system are same as
for Fig. 3. The relaxations rates are 1/2 = 9Γ π MHz,
/2 = 4.8ϕΓ π MHz. The dash-dot and dash lines are plotted for
= 0T K, the solid line is plotted for = 0.1T K.
Fig. 6. (Color online) (a) Measured normalized transmission
amplitude of a probe signal applied at the fundamental mode
frequency /2 = 2.5rω π GHz, while the qubit energy bias ε and
the driving amplitude dA are varied. The latter is applied in the
third harmonic of the resonator. The probing power takes a value
of –127 dBm. (b) Calculation results from (22). The calculation is
carried out by splitting the bias axes in parts where the kth.
resonance is dominant but under consideration the energy level
shift also induced by the neighboring resonances. The parameters
of the qubit-resonator system, / 3h∆ ≈ GHz, 1g /2 = 4π MHz,
1/2 = 0.75Γ π MHz, /2 = 30ϕΓ π MHz, and /2 = 22κ π kHz were
defined by separate experiments.
250 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3
Signal amplification in a qubit-resonator system
5. Amplification with phase-slip qubit
We consider in this section the situation, where there is
a so-called phase-slip qubit coupled to the transmission-
line resonator. Our aim is to clarify similarities and
distinctions from the previously considered case, where we
had a flux qubit coupled to the resonator.
The coherent quantum phase slip has been discussed
theoretically in Refs. 37, 41, and demonstrated experi-
mentally in Ref. 39. It describes a phenomenon exactly
dual to the Josephson effect; whereas the latter is a
coherent transfer of charges between superconducting
leads, the former is a coherent transfer of vortices or fluxes
across a superconducting wire. The similar behavior of the
coherent quantum phase slip to Josephson junction allows
to consider it as a part of the qubit-resonator system. The
quantum phase slip process is characterized by the
Josephson energy sE , which couples the flux states,
resulting in the Hamiltonian, Refs. 37, 41
( )1= 1 1 ,
2 s NH E N N N N E N N− + + + + (31)
which is dual to the Hamiltonian of a superconducting
island connected to a reservoir through a Josephson
junction; N is the number of the fluxes in the narrow
superconducting wire, 2
ext 0= ( ) /2N kE N LΦ − Φ is the
state energy, extΦ is an external magnetic flux, kL is the
length of the nanowire. The ground and excited states can
be related to the flux basis: = sin cos 1
2 2
g N Nα α
+ +
and = cos sin 1
2 2
e N Nα α
− + , where the mixing angle
Fig. 7. (Color online) The transmission amplitude for PSQ-
resonator system at the value of the driving amplitude,
corresponding to the maximal amplification. The dash-dot line
and the solid line are plotted for / 35dA h = GHz, and the dash
line is plotted for / 0dA h = . Parameters for the calculations are the
following: / =E h∆ 4.9 GHz, /2 2.4rω π = GHz, /2 =κ π 30 kHz.
Note that the half-width at half-maximum of the transmission line
decreases under pumping. The dash-dot and dash lines is plotted
for = 0T K, the solid line is plotted for = 0.1T K.
Fig. 8. (Color online) Processes taking place in the qubit-resonator
system. (a) The energy levels of the qubit. The one, two and three
photon resonances are marked by the arrows. (b) The maximum
value of the excited state probability of the dressed qubit corresponds
to the resonance condition p=E n∆ ω
for integer n . (c) The qubit
interaction with driving high-amplitude signal leads to a renorma-
lization of the energy levels of the qubit. (d) The inverse population is
typical for the system with various relaxation times between the
dressed levels. (e) It corresponds to the energy transfer from the
dressed states in the probe signal. When the opposite situation is
realized, the amplitude of the probe signal reduces.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 251
D.S. Karpov, G. Oelsner, S.N. Shevchenko, Ya.S. Greenberg, and E. Il’ichev
is 0arctan /sEα = ε ; the energy splitting between the
ground and excited states is 2 2
0 s=E E∆ ε + . In the rotating
wave approximation, the effective Hamiltonian of the
system resonantly driven by a classical microwave field
with amplitude cos ( / )d Eξ ∆ is RWA = .
2 zH Ω
σ
Such
Hamiltonian coincides with the Hamiltonian of the flux
qubit in the RWA up to the notations [2]. The interaction
between the PSQ and the two-mode resonator can be
described by Hamiltonian (1). We demonstrate the
transmission coefficient for a real experimental PSQ in
Fig. (7). We use data which corresponds to Ref. 39.
Equations (19)–(21) are also applicable for the PSQ-
resonator system. The doubly-dressed approach is useful
instrument for description of the quantum behavior of the
different mesoscopic systems.
6. Conclusions
We studied the evolution of the doubly-driven qubit-
resonator system. We demonstrated the possibility of a
large amplification of the input signal and the ability of
almost full reflection of the probe signal in the system.
The value of the transmitted signal depends on all the
system parameters, of which the coupling coefficient g1
and the relaxation rates κ and 1,ϕΓ are the most
influential. The numerical simulation of the different
qubit-resonator systems with using of real experimental
parameters allows to estimate the optimal parameter
range for this samples. In particular, we have found that
for both amplification and attenuation the following
parameter values are optimal: 4 2
1/ 10 –10g − −∆ , pκ ξ ,
2 1
1/ 10 –10− −Γ ∆ , and 2 1/ 10 –10− −
ϕΓ ∆ . The tempera-
ture noise (non-zero temperature) diminishes the
transmission amplitude.
This work was partly supported by DKNII (Project No.
M/231-2013), BMBF (UKR-2012-028), RFFR (No. 15-32-
50195/15), MES (8.337.2014/K). D.S.K. acknowledges the
hospitality of IPHT (Jena, Germany) and NSTU (Novo-
sibirsk, Russia), where part of this work was done. The
authors are grateful to A.N. Omelyanchouk for useful
discussions and comments.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 3 253
1. Introduction
2. The qubit-resonator system
3. Evolution of the system
4. Amplification and attenuation of the probe signal
5. Amplification with phase-slip qubit
6. Conclusions
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