Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
Landau–Zener–Stückelberg (LZS) interference of continuously driven superconducting qubits is studied. Going beyond the second order perturbation expansion, we find a time dependent stationary population evolution as well as unsymmetrical microwave driven Landau–Zener transitions, resulting from the...
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irk-123456789-1186512017-05-31T03:10:07Z Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits Du, Lingjie Lan, Dong Yu, Yang Сверхпроводимость, в том числе высокотемпературная Landau–Zener–Stückelberg (LZS) interference of continuously driven superconducting qubits is studied. Going beyond the second order perturbation expansion, we find a time dependent stationary population evolution as well as unsymmetrical microwave driven Landau–Zener transitions, resulting from the nonresonant terms which are neglected in rotating-wave approximation. For the low-frequency driving, the qubit population at equi-librium is a periodical function of time, owing to the contribution of the nonresonant terms. In order to obtain the average population, it is found that the average approximation based on the perturbation approach can be applied to the low-frequency region. For the extremely low frequency which is much smaller than the decoherence rate, we develop noncoherence approximation by dividing the evolution into discrete time steps during which the co-herence is lost totally. These approximations present comprehensive analytical descriptions of LZS interference in most of parameter space of frequency and decoherence rate, agreeing well with those of the numerical simula-tions and providing a simple but integrated understanding to system dynamics. The application of our models to microwave cooling can obtain the minimal frequency to realize effective microwave cooling. 2013 Article Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits / Lingjie Du, Dong Lan, Yang Yu // Физика низких температур. — 2013. — Т. 39, № 8. — С. 863–878. — Бібліогр.: 52 назв. — англ. 0132-6414 PACS: 03.65.Yz, 85.25.Dq, 85.25.Cp, 03.67.–a http://dspace.nbuv.gov.ua/handle/123456789/118651 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Сверхпроводимость, в том числе высокотемпературная Сверхпроводимость, в том числе высокотемпературная |
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Сверхпроводимость, в том числе высокотемпературная Сверхпроводимость, в том числе высокотемпературная Du, Lingjie Lan, Dong Yu, Yang Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits Физика низких температур |
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Landau–Zener–Stückelberg (LZS) interference of continuously driven superconducting qubits is studied. Going beyond the second order perturbation expansion, we find a time dependent stationary population evolution as well as unsymmetrical microwave driven Landau–Zener transitions, resulting from the nonresonant terms which are neglected in rotating-wave approximation. For the low-frequency driving, the qubit population at equi-librium is a periodical function of time, owing to the contribution of the nonresonant terms. In order to obtain the average population, it is found that the average approximation based on the perturbation approach can be applied to the low-frequency region. For the extremely low frequency which is much smaller than the decoherence rate, we develop noncoherence approximation by dividing the evolution into discrete time steps during which the co-herence is lost totally. These approximations present comprehensive analytical descriptions of LZS interference in most of parameter space of frequency and decoherence rate, agreeing well with those of the numerical simula-tions and providing a simple but integrated understanding to system dynamics. The application of our models to microwave cooling can obtain the minimal frequency to realize effective microwave cooling. |
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| author |
Du, Lingjie Lan, Dong Yu, Yang |
| author_facet |
Du, Lingjie Lan, Dong Yu, Yang |
| author_sort |
Du, Lingjie |
| title |
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits |
| title_short |
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits |
| title_full |
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits |
| title_fullStr |
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits |
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Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits |
| title_sort |
low-frequency landau–zener–stückelberg interference in dissipative superconducting qubits |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
| topic_facet |
Сверхпроводимость, в том числе высокотемпературная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118651 |
| citation_txt |
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits / Lingjie Du, Dong Lan, Yang Yu // Физика низких температур. — 2013. — Т. 39, № 8. — С. 863–878. — Бібліогр.: 52 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT dulingjie lowfrequencylandauzenerstuckelberginterferenceindissipativesuperconductingqubits AT landong lowfrequencylandauzenerstuckelberginterferenceindissipativesuperconductingqubits AT yuyang lowfrequencylandauzenerstuckelberginterferenceindissipativesuperconductingqubits |
| first_indexed |
2025-07-08T14:23:26Z |
| last_indexed |
2025-07-08T14:23:26Z |
| _version_ |
1837089013265072128 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8, pp. 863–878
Low-frequency Landau–Zener–Stückelberg interference
in dissipative superconducting qubits
Lingjie Du1,2, Dong Lan1, and Yang Yu1
1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University
Nanjing 210093, China
2Department of Physics and Astronomy, Rice University, Houston, Texas 77251-1892, USA
E-mail: lingjie.du@rice.edu
yuyang@nju.edu.cn
Received January 22, 2013, revised April 3, 2013
Landau–Zener–Stückelberg (LZS) interference of continuously driven superconducting qubits is studied.
Going beyond the second order perturbation expansion, we find a time dependent stationary population evolution
as well as unsymmetrical microwave driven Landau–Zener transitions, resulting from the nonresonant terms
which are neglected in rotating-wave approximation. For the low-frequency driving, the qubit population at equi-
librium is a periodical function of time, owing to the contribution of the nonresonant terms. In order to obtain the
average population, it is found that the average approximation based on the perturbation approach can be applied
to the low-frequency region. For the extremely low frequency which is much smaller than the decoherence rate,
we develop noncoherence approximation by dividing the evolution into discrete time steps during which the co-
herence is lost totally. These approximations present comprehensive analytical descriptions of LZS interference
in most of parameter space of frequency and decoherence rate, agreeing well with those of the numerical simula-
tions and providing a simple but integrated understanding to system dynamics. The application of our models to
microwave cooling can obtain the minimal frequency to realize effective microwave cooling.
PACS: 03.65.Yz Decoherence; open systems; quantum statistical methods;
85.25.Dq Superconducting quantum interference devices (SQUIDs);
85.25.Cp Josephson devices;
03.67.–a Quantum information.
Keywords: superconducting qubits, population evolution, effective microwave cooling.
1. Introduction
Superconducting devises based on Josephson tunnel
junctions have great importance in demonstrating quantum
phenomena at macroscopic scales and hold promise for
applications in quantum information [1–5]. In the studies
of these systems, crossovers between energy levels are
usually associated with a wide variety of interesting phe-
nomena, particularly in connection with Landau–Zener–
Stückelberg (LZS) interference [6–8].
Recently, a series of great progresses [9–20] have been
implemented to study the coherent dynamics of qubits in
the regime dominated by LZS interference. The strong
high-frequency microwave generates controllable popula-
tion transitions, allowing for fast and reliable control of
quantum systems. The delicate LZS interference patterns
also provide a way to calibrate the parameters of the qubit
and its interaction with the environment. On the other
hand, through continuous Landau–Zener (LZ) transitions,
the low-frequency microwave can be used to achieve an
effective temperature of qubit up to two orders of magni-
tude lower than the bath temperature through an analogue
of optical sideband cooling [21,22]. This microwave cool-
ing provides an effective means for preparing qubit state
with high fidelity and suppressing decoherence. Therefore,
understanding dynamics of LZS interference in strongly
driven superconducting qubits has both fundamental and
practical significances. A number of theoretical investiga-
tions [6,14,23–41] have been developed in driven quantum
two-level systems (TLS). These works generally addressed
the coherent phenomena in the high-frequency driving in
superconducting qubits, as reviewed recently in Ref. 6. In
the realistic situation, the influence of environment always
needs to be considered. Compared with natural atoms, the
solid state systems contain macroscopic degrees of free-
dom strongly coupled to environment bathes that decohere
the quantum states to be controlled. To address the driven
© Lingjie Du, Dong Lan, and Yang Yu, 2013
mailto:lingjie.du@rice.edu
Lingjie Du, Dong Lan, and Yang Yu
dissipative TLS, two typical methods usually taken into
account are rotating-wave approximation (RWA) [32,41]
and perturbation-RWA [19,27].
RWA provides a simple physical picture and good ap-
proximation for the stationary population. However, it lim-
its at the high-frequency region, i.e. 2 , ( / ),nJ Aω Γ ∆ ω
where n is the closest integer to 0 /ε ω and 0ε is the dc
detuning. The other method, perturbation-RWA, applies
the second order perturbation approximation based on
RWA and can be used in the parameter region with the
high frequency and strong decoherence, i.e. 2> ,ω ∆ Γ and
2 01,WΓ where 01W is microwave driven Landau–
Zener (MDLZ) transition rate from | 0〉 and |1 .〉 However
in the low-frequency driving where microwave cooling
takes effects, both methods would be inappropriate.
In this article, we focus on analyzing the stationary state
which is useful in spectroscopy experiments [9,10,12,15]
and microwave cooling [21] in the presence of decoherence.
Going beyond the second order perturbation and focusing on
the nonresonant terms which are dropped in RWA, we
present the real-time population evolution as well as un-
symmetrical MDLZ transitions in a strongly driven TLS of
the long-time dynamics. In the high-frequency case, i.e.
> ,ω ∆ the result of perturbation approach can fully cover
those of RWA and perturbation-RWA. In addition, the per-
turbation approach can be applied to larger parameters
space, at which other methods are inappropriate.
Then we move on lower frequency region, i.e. < .ω ∆ In
this parameter region, microwave cooling begins to take
effects, motivating us to give a thorough understanding of
the dynamics. The steady dynamics shows a periodic time
dependence of qubit population, which is from the nonreso-
nant terms and cannot be explained by perturbation-RWA.
Then we focus on the average stationary population. Al-
though one can use the perturbation approach, it is cumber-
some thus difficult to get the final solutions. Therefore, we
use the average perturbation approximation (APA), which is
based on perturbation approach through the average over
one driving period. Interestingly, the analytical expression in
APA has the same form with that of perturbation-RWA. If
the frequency further decreases, APA cannot correctly give
the qubit population. We thereby propose the noncoherence
approximation (NCA). The incoherent evolution is divided
into discrete incoherent time steps, from which we can
obtain the analytical solution. It is found that when the
frequency is much smaller than the decoherence rate, the
time dependent and average population calculated by NCA
agree quantitatively with those of the numerical simula-
tion. At last, we extend the analytical results of APA and
NCA to the multi-level system. We discuss microwave
cooling [21] and obtain the minimal frequency of the effec-
tive microwave cooling.
Through these approximations of perturbation ap-
proach, we present simple and comprehensive descriptions
in the average stationary population in most of parameter
space of frequency and decoherence rate. The agreement
with numerical results indicates that the picture captures
the underlying physics, thus giving a better understanding
of the behavior of system dynamics. The models and me-
thods we used here are also valid for other TLSs with simi-
lar structures.
The article is organized as follows. In Sec. 2, we intro-
duce the basic model and approximations. In Sec. 3, as-
suming high order perturbation in the long-time dynamics,
we present the perturbation approach which considers the
nonresonant terms. Then with this approach we discuss the
high-frequency driving case. In Sec. 4, we discuss the av-
erage stationary population in the low frequency. Through
APA, we obtain the average population in the steady case
under the low-frequency driving. In Sec. 5, for the ex-
tremely low frequency, we demonstrate that NCA can be
used to get the analytical solution, which is useful to de-
scribe the system dynamics. In Sec. 6, we extend our mod-
el to multi-level flux qubit and discuss the frequency range
of effective microwave cooling.
2. Basic model
For a superconducting flux qubit [42,43] with three
Josephson tunnel junctions, if the external flux bias
00.5 ,Φ ≈ Φ where 0 = /2h eΦ is the flux quantum, a
double-well landscape of the potential energy parameterized
by the dc flux detuning 0= 0.5dcδΦ Φ − Φ exists as shown
in Fig. 1. In the millikelvin temperature 10 mK, a series
of diabatic states are localized in different wells. States | 0〉
and |1〉 are the lowest levels in the left and right wells, re-
spectively. They are characterized by different flowing di-
rections of the persistent currents, namely clockwise and
anticlockwise. The states in each well are separated by large
energy spacing on the order of 20 GHz [15], while the
coupling strength between two wells is small compared with
the energy spacing. As the qubit is driven with a microwave
= sin ,ac rf tΦ Φ ω where rfΦ is the microwave amplitude,
the time dependent flux detuning is
( ) = = sin .dc ac dc rft tδΦ δΦ +Φ δΦ +Φ ω
Fig. 1. (Color online) (a) Schematic picture of the double-well
potential of a superconducting flux qubit. States |1〉 is in the
right well; | 0〉 is in the left well. (b) Schematic energy diagram
of the flux qubit. Red solid curve represents the microwave
sin .rf tΦ ω The vertical dashed line marks a particular static
flux detuning .dcδΦ
864 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
Then the time dependent energy detuning of states | 0〉 and
|1〉 can be described as
0( ) = sin ,t A tε ε + ω
where 0 0 1= (| | | |) dcm mε + δΦ is the dc energy detuning
of states | 0〉 and |1 ,〉 0 1= (| | | |) rfA m m+ Φ is the micro-
wave energy amplitude, and = ( )/i im dE dΦ Φ is the di-
abatic energy-level slope of state | i〉 in units of frequency
per flux. For the initial time = 0,t the qubit is at the de-
tuning 0(0) = .ε ε
We start from a periodically driven TLS coupled to an
environment bath. Such a process can be described by
Hamiltonian (we set = = 1)Bk [26,27,40,44,45]
int
( )= ,
2 2x z B
tH H H∆ ε
− σ − σ + + (1)
where xσ and zσ are Pauli matrices, =|1 1| | 0 0 |zσ 〉〈 − 〉〈
and = |1 0 | | 0 1 |,xσ 〉〈 + 〉〈 intH is the system-environment
interaction Hamiltonian, and BH is the environment Ha-
miltonian. We assume the qubit is embedded in the bath in
equilibrium at temperature T with the density matrix of
environment /= e .H TBB
−ρ The system-environment inte-
raction Hamiltonian is written as
int = /2,zH Q− σ (2)
where Q is an operator acting on the environment.
Before investigating the system dynamics in the pres-
ence of the environment bath, it is worth recalling main
features of its free time evolution which is described by
Hamiltonian Eq. (1) without intH and .BH By a gauge
transformation, the Hamiltonian is brought to the form
( )
1 = e |1 0 | H.c. ,
2
tH φ∆
− 〉〈 + (3)
where
0
( ) = ( ) .
t
t i dφ − ε τ τ∫
Using Bessel functions, we have
/( ) ( /2)0e = e ei t iAt in t
n
AJ− ε − ωφ ω +π
ω
∑
and find
( ) /2 /02 = e |1 0 | H.c.
2
i t n t in iA
n
AH J − ε − ω + π − ω∆ − 〉〈 + ω
∑
(4)
With the relation / /2e = ( / )e ,iA in
nJ Aω πω∑ we consider
parameters close to the n-photon resonance, where
0=nω ε and rely on RWA to neglect the fast oscillating
terms with k n≠ for the high frequency .ω Therefore,
when the system is initially in state |1〉 we have the popu-
lation in state | 0〉 [6,9,32]
2 2
00 2 2 2
0
( / )1=
2 ( / ) ( )
n
n
J A
J A n
∆ ω
ρ ×
∆ ω + ε − ω
∑
2 2 2
0[1 cos ( / ) ( ) ].nJ A n t× − ∆ ω + ε − ω (5)
This approximation neglects all the nonresonant terms
which oscillate very fast in the time scale of system dy-
namics, so the high frequency is needed to keep the validi-
ty of RWA, i.e. > ( / )nJ Aω ∆ ∆ ω [6,29,32]. We com-
pare the real-time numerical solution with the analytical
results in Eq. (5), shown in Fig. 2. RWA has the high accu-
racy under the high-frequency driving. However, when the
frequency is low, i.e. < ,ω ∆ the difference between nu-
merical and RWA results are not negligible. It should be
noted that in Fig. 2(a) there exist two kinds of oscillations.
One agrees with Eq. (5) while another as shown in the
panel in Fig. 2(a) is from the nonresonant terms in Eq. (4).
When the frequency is small just as shown in Fig. 2(b), the
effect of nonresonant terms could not be neglected. In next
sections, we will further show the influence of these nonre-
sonant terms on system dynamics.
In the realistic solid-state system, the decoherence
which is always strong needs to be taken into account. In
this case, the Bloch equations are usually used to provide
the numerical stimulation, which read
1= [ , ] [ ],i Hρ − ρ + Γ ρ (6)
where [ ]Γ ρ is the dissipative term [46,47]
Fig. 2. (Color online) Population in state |1〉 as a function of time,
assuming that the system is initially in state |1 .〉 In both figures,
0 /2 = 90ε π MHz, /2 = 90∆ π MHz, and /2 = 5A π GHz. The
microwave frequency is /2 = 90ω π MHz (a) and 10 MHz (b),
respectively. The blue solid line shows the numerical stimulations
and the black dotted line shows the RWA results.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 865
Lingjie Du, Dong Lan, and Yang Yu
10 11 01 00 2 01
2 10 10 11 01 00
[ ] = ,
−Γ ρ + Γ ρ −Γ ρ
Γ ρ −Γ ρ Γ ρ −Γ ρ
(7)
where 2 2= 1/TΓ is the decoherence rate, 2T is the decohe-
rence time, ijΓ is the interwell relaxation rate from state
| i〉 to | ,j〉 /010 01= e ,T−εΓ Γ 1 10 011/ = max{ , },T Γ Γ and
1T is the interwell relaxation time. After the long-time
system dynamical evolution, whatever the initial state is,
the temporal oscillation in Fig. 2 is eliminated by the de-
coherence and the qubit population will converge to a
stationary solution. We make RWA with the Eq. (6), neg-
lect the fast oscillating terms, and obtain the probability
in state | 0〉 [6]
2 2
2
102 2
2 0
00 2 2
2
01 102 2
2 0
( / )
2[ ( ) ]
= .
( / )
( )
n
n n
J A
n
J A
n
∆ ω Γ
+Γ
Γ + ε − ω
ρ
∆ ω Γ
+Γ +Γ
Γ + ε − ω
∑ (8)
To ensure Eq. (8) is valid, we need not only the high fre-
quency, i.e., > ( / ),nJ Aω ∆ ∆ ω but also the high cohe-
rence, i.e., 2.ω Γ If the decoherence is strong 2 ,ω Γ
the population 00ρ may even be larger than 1 and this sta-
tionary solution is obviously not reasonable.
In order to describe the dynamics in the strong deco-
herence, Berns et al. [19] proposed the perturbation-
rotating-wave approximation. Under the Gaussian white
low-frequency noise, based on the second order expan-
sion perturbation, we have MDLZ transition rate from
state | 0〉 to |1〉
22
2
01 2 2
0 2
( / )
= .
2 ( )
n
n
J A
W
n
Γ ω∆
ε − ω +Γ
∑ (9)
Correspondingly, 10 0 01 0 01 0( ) = ( ) = ( ).W W Wε −ε ε Then
for the long-time dynamics, we employ the rate equations,
in which the qubit population in state | 0〉 and |1 ,〉 i.e., 00ρ
and 11,ρ follow
00 01 01 00 10 10 11= ( ) ( )W Wρ − + Γ ρ + + Γ ρ ,
00 111 = .ρ +ρ (10)
Since MDLZ transition rate is constant in the stationary
case, the equations would lead to the constant population,
i.e. 00 11= = 0,ρ ρ and the population in state | 0〉 becomes
2 2
2
102 2
2 0
00 2 2
2
01 102 2
2 0
( / )
2[ ( ) ]
= .
( / )
( )
n
n n
n
J A
n
J A
n
∆ ω Γ
+Γ
Γ + ε − ω
ρ
∆ ω Γ
+Γ +Γ
Γ + ε − ω
∑
∑
(11)
In order to ensure the validity of this approach [19,27],
the high frequency and strong decoherence, i.e., 2> ,ω ∆ Γ
and 2 01,WΓ are necessary, which greatly restrict the
application of this approximation. In next sections, we will
introduce approximations to extend the quantitative analy-
sis to larger parameters space.
3. Perturbation approach
In order to get analytical results in larger parameters
space, we go back to Hamiltonian (1) and take the pertur-
bation approach. First of all, we follow the process in
Refs. 27 and 44, and the complete time evolution operator
in the laboratory picture can be expressed as
11 2 0 1 1 1 1 2( , ) = e ( ) ( ) ( , )iH tB VU t t U t U t U t t− ×
† † 22 21 0( ) ( )e ,iH tBU t U t× (12)
where
0
0
( ) = exp[ ( ) ]
2
t
z
iU t dτε τ σ∫ ,
1
0
( ) = exp[ ( ) ]
2
t
z
iU t d Qτ τ σ∫ ,
1
1 2
2
( ) = e e , ( , ) = exp[ ( )]
t
iH t iH tB B V
t
Q t Q U t t i d V− − τ τ∫ ,
0( ) = ( )exp[ ( )
2 n
n
AV t J i t n t∆
− − ε − ω +
ω∑
†
11] ( ) |1 0 | ( ) H.c.
2
in Ai U t U tπ
+ − 〉〈 +
ω
,
and is the time ordering operator. Let SBρ express the
total density matrix of the system plus environment bath,
which can be described as †( ) = ( ,0) (0) ( ,0).SB SBt U t U tρ ρ
Then the system reduced density matrix is defined
as ( ) = Tr { ( )},B SBt tρ ρ where Tr {...}B denotes the trace
over the environmental degrees of freedom. We assume
at the initial time the density matrix is separable, i.e.,
(0) = (0) ,SB Bρ ρ ⊗ρ and the system is initialized in state
| 0 ,〉 i.e., (0) = | 0 0 | .ρ 〉〈
We perform a perturbation expansion of VU in ( ),V τ
which reads
0 0 0
( ,0) 1 ( ) ( ) ( )
t t
VU t i d V d V d V
τ
′ ′≈ − τ τ − τ τ τ τ +∫ ∫ ∫
0 0 0
( ) ( ) ( ) ...
t
i d V d V d V
′τ τ
′ ′ ′′ ′′+ τ τ τ τ τ τ +∫ ∫ ∫ (13)
For the strong decoherence, i.e., 2 01( / ) ,nJ A WΓ ∆ ω
we can consider the expansion to the second order in a time
interval 21/ < 1/ [ ( / )]nt J AΓ ∆ ω [27,45]. However, in the
high-coherence region where 2 < ( / ),nJ AΓ ∆ ω instead of
considering the time interval < 1/[ ( / )],nt J A∆ ω we need
take account of a time interval t much larger than the de-
coherence time. Hence this time interval t is also much
larger than 1/[ ( / )].nJ A∆ ω Moreover, for the low frequency
866 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
( / ) < ,nJ A∆ ω ω ∆ in order to calculate MDLZ transi-
tion in the stationary case, even if 1/[ ( / )]nJ A∆ ω is much
longer than the decoherence time, the time interval
< 1/[ ( / )]nt J A∆ ω is still not sufficiently long relative to
the driving period. As a result we consider 1/ ,t ω
1/[ ( / )],nJ A∆ ω 21/ .Γ It is clear that the second order per-
turbation expansion would be insufficient [27,45]. Consi-
dering high order expansion terms, we obtain the popula-
tion in state |1〉 (see Appendix)
____________________________________________________
1
† † † †
11 1 2 3 4 4 3 2 1
0 0 0 0 0 0
( ) Tr { 1| ( ) (0) ( ) |1 } 2 Tr { 1| [ ( ) (0) ( )] ( ) ( ) |1 }
ttt t t t
B SB B SBt d d V V dt dt dt dt V t V t V t V t′ ′ρ ≈ τ τ 〈 τ ρ τ 〉 − 〈 ρ 〉 −∫ ∫ ∫ ∫ ∫ ∫
2
† † †
1 4 4 2 3 3 2 1
0 0 0 0
Tr { 1| ( )[2 (0) ( ) ( )] ( ) |1 } ...
ttt t
B SBdt dt V t dt dt V t V t V t− 〈 ρ 〉 +∫ ∫ ∫ ∫ (14)
_______________________________________________
For the short time interval, i.e., < 1/[ ( / )],nt J A∆ ω we
use the second order perturbation expansion and can keep
only the first term in Eq. (14) expressing the population in
state |1〉 transferred from | 0 ,〉 just as discussed in Ref. 27.
As the time goes on, the population is not completely in
state | 0 .〉 The first term does not consider the effect of the
reduced population in state | 0〉 such that it cannot charac-
terize the population transferred from state | 0〉 to |1〉 any
more. Higher order terms in ( ,0)VU t need to be consi-
dered, which result into the second as well as third term in
Eq. (14). In the second term,
†
3 4 4 3
0 0
( ) (0) ( )
t t
SBdt dt V t V tρ∫ ∫
describes the population transferred from state | 0〉 to |1〉
in the time interval .t Then the second term characterizes
the transition of population from state |1〉 to | 0〉 based on
this part of population in |1 .〉 Similarly,
2
† †
2 3 3 2
0 0
2 (0) ( ) ( )
tt
SBdt dt V t V tρ∫ ∫
in the third term describes the reduced population in state
| 0〉 in the time interval t . The third term describes the
decrease of population transferred from state | 0〉 to |1〉
because of the reduced population in state | 0 .〉 When the
time interval t becomes further larger, more terms will
appear in Eq. (14) as a ripple effect and we can also extend
the integration limit t to .∞
Then we only discuss the first term in Eq. (14) as an ex-
ample, and other terms can be treated with a similar proce-
dure. We rewrite the first term of Eq. (14) to
2
1 2 1 1 2 2
0 0
( ) ( ) /2 /20 1 0 2
,
( ) = ( ) ( ) ( ) ( ) ,
4
= ( ) ( )e ,
t t
i m t i n t in im
n m
n m
y t dt dt K U t U t U t U t
A AK J J
+ +
− + + −
ε − ω − ε − ω + π − π
∆
〈 〉
ω ω
∫ ∫
∑
(15)
where
0
( ) = exp[ ( / 2) ( ) ]
t
U t i Q d± ± τ τ∫ .
For the low frequency, i.e., ( / ) < ,nJ A∆ ω ω ∆ the
terms in K with n m≠ would oscillate not fast enough,
leading the invalidity of RWA. Assuming the white Gaus-
sian low-frequency noise, we rewrite Eq. (15) to
2
( )( ) | |0 1 2 2 1 21 2
0 0
( ) = e
4
t t
i m t t t t
m
y t dt dt ε − ω − −Γ −∆
×∑∫ ∫
/22 2
0
{ ( ) ( ) ( )e }.in t in
m n m m
n
A A AJ J J ω + π
+
≠
× +
ω ω ω∑ (16)
For the long-time dynamics, we extend the integration lim-
its of Eq. (16) to ,∞ and have [27]
22
2
2 2
0 2
( / )
( ) = ( ),
2 ( )
n
n
J A
y t f t
n
Γ ω∆
+
ε − ω +Γ
∑ (17)
2
1 2
0, 0 0
( ) = ( ) ( )lim4
t t
n m m
tn m
d A Af t dt dt J J
dt +
→∞≠
∆
×
ω ω∑ ∫ ∫
( ) /2 | |0 2 2e ,i m in t inε − ω τ+ ω + π −Γ τ×
where 1 2= .t tτ − The first term in Eq. (17) is the familiar
result in perturbation-RWA. For convenience we change
the form of ( )f t to
2
0,
( ) = ( ) ( )
4 n m m
n m
A Af t J J+
≠
∆
×
ω ω∑
( ) ( /2) /2 | |0 2
0
e ,lim
t t
i m in in
t t
d d d
dt
′ε − ω τ+ ω τ −τ + π −Γ τ
→∞ −
′× τ τ∫ ∫
(18)
where 1 2= ( )/2t t′τ + and = min{2 ,2 2 }.t t′ ′τ − τ Therefore
we have the leading term given by
2
2
2 2
0, 0 2
cos ( /2)
( ) ( ) ( ) .
2 ( )
n m m
n m
n tA Af t J J
m
+
≠
Γ ω + π∆
≈
ω ω ε − ω +Γ
∑ (19)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 867
Lingjie Du, Dong Lan, and Yang Yu
Then we consider other terms in Eq. (14) and follow the
above process, obtaining
11 0 0 11 0 11( ) ( , ) ( , ) ( ) ( , ) ( )t y t y t t y t tρ ≈ ε − ε ρ − −ε ρ =
01 00 10 11= ( ) ( ) ( ) ( ),W t t W t tρ − ρ (20)
where 01W 10( )W is MDLZ transition rate from state | 0〉
to |1〉 (|1〉 to | 0 ).〉 We emphasize that in the derivation of
Eq. (20), all the expansion terms in Eq. (14), not only the
notable terms, have the contribution. Equation (20) gives
the evolution of the diagonal part of the system reduced
density matrix under the “Markovian” approximation (see
Appendix). It should be noted that we take the relaxation
phenomenonally and hence it does not emerge in Hamilto-
nian and rate equations, while in the calculation of popula-
tion we will add it. Furthermore, MDLZ transition rates
obtained are the combination of Lorentzian line shapes and
time dependent,
2
2
01 2 2
, 0 2
cos ( /2)
( ) = ( ) ( ) ,
2 ( )
n m m
n m
n tA AW t J J
m
+
Γ ω + π∆
ω ω ε − ω +Γ
∑
2
2
10 2 2
, 0 2
cos ( /2)
( ) = ( 1) ( ) ( ) ,
2 ( )
n
n m m
n m
n tA AW t J J
m
+
Γ ω + π∆
−
ω ω ε − ω +Γ
∑
(21)
where we use the relation ( ) = ( 1) ( ).m
m mJ x J x−− Differ-
ent from Eq. (9), MDLZ transition rates in Eq. (21) are
unsymmetrical. Comparing Eqs. (21) and (9), we find that
their difference is the additional time-dependent oscillating
terms ( ),f t which comes from the nonresonant terms. The
nonresonant terms also correspond to the weak oscillation
in Fig. 2(b), due to the coherence evolution. It is clear that
the coherence in the nonresonant terms results into un-
symmetrical MDLZ transitions, i.e., 01 10( ) ( ).W t W t≠ In
addition, although the coherence is preserved in the nonre-
sonant terms, nonresonant terms with different oscillation
frequencies counteract the resonant terms which would
become less obvious. Therefore, not only the decoherence
but also the low-frequency driving would eliminate the
multiphoton resonant fringes.
Then we have the population in state | 0〉
00 01 10 01 10
0
= exp[ ( ( ) ( ) )]
t
dt W t W t′ ′ ′ρ − − −Γ −Γ ×∫
10 10
0
[ ( ( ) ]
t
dt W t′ ′× + Γ ×∫
01 10 01 10
0
exp[ ( ( ) ( ) ) 1].
t
dt W t W t
′
′′ ′′ ′′× − − −Γ −Γ +∫ (22)
Similarly, by following the above procedure, we can
extend perturbation approach to higher levels and different
noise sources, which were previously explored under high-
frequency driving [26,27]. Especially, MDLZ transition
between higher levels can be used to analyze microwave
cooling with the maximal efficiency [22].
Under the large amplitude driving, the coherence of
LZS interference comes from two aspects. The first one is
traditionally from the prepared initial states. The off di-
agonal elements of the density matrix describe this part of
coherence, which is usually needed to be preserved for the
application of quantum computation. The second one is the
coherence induced by the crossover which acts as a beam
splitter [9,16]. The microwave drives the state through the
crossover, splitting the state into the superposition of states
| 0〉 and |1 ,〉 and the system gains the coherence again. For
a stationary case, the first kind has disappeared fully while
the second one is created incessantly. However, it should
also be noted that usually the coherence induced by the
crossover only exists between two consecutive passing
through the crossover within one period. There is no cohe-
rence for the processes in arbitrary periods.
To make this method reasonable, as discussed in Ap-
pendix, the coherent evolution produced by the crossover
cannot produce remarkable population change in the sta-
tionary case, which means ( / )nJ A∆ ω has little effect in
one driving period. If the tunnel coupling strength ∆ is
large and the decoherence is weak, the coherence will pers-
ist for a long time, producing remarkable coherent popula-
tion change. Therefore, the validity of this method depends
on two factors. The first one is the high frequency, i.e.,
( / ),nJ Aω ∆ ω which makes coherent dynamics not have
enough time to cause population change. The second one is
the strong decoherence, which makes the coherence disap-
pear quickly and produce little effect on the population,
i.e., 2 > ( / ).nJ AΓ ∆ ω If one of factors is satisfied, the me-
thod would be available.
For the high frequency, i.e., > ( / ),nJ Aω ∆ ∆ ω the
time-dependent terms in Eq. (21) oscillate fast and can be
neglected just like that in RWA. Therefore MDLZ transition
Eq. (21) returns to symmetrical perturbation-RWA result
Eq. (9) and is constant. In addition, as shown in the deriva-
tion, Eq. (9) can be further extended to the incoherent region
2<ω Γ and the high coherence 2 < ( / ).nJ AΓ ∆ ω Further-
more, in the condition, i.e., 2 ,ω Γ ( / )nJ A∆ ω the statio-
nary population Eq. (11) in perturbation-RWA is equivalent
to Eq. (8) in RWA. To validate the perturbation-RWA result
in the two regions, we have numerically solved the Bloch
equations to compare with the analytical results obtained by
Eq. (9) on the steady occupation probability in state | 0 ,〉 as
illustrated in Fig. 3. Here we consider the frequency
/2 = 90ω π MHz with different decoherence rates. The ana-
lytical results agree well with the numerical simulations.
For the low frequency, i.e., < ,ω ∆ the nonresonant terms
will be comparable with the oscillating terms and cannot be
neglected. In this case, perturbation-RWA result is not availa-
ble and Eq. (21) must be needed. As discussed above, if
the decoherence rate is small, i.e., 2 01< ( / ) ,nJ A WΓ ∆ ω
868 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
the strong coherent population evolution may break pertur-
bation approach. Therefore in this region, for smaller fre-
quency, perturbation approach can be used in stronger de-
coherence. When the decoherence is enough strong, i.e.,
2 > ( / ),nJ AΓ ∆ ω the coherence generated by the crossover
would have no effects.
When the state is driven with the low frequency
through the crossover ,∆ the small sweeping velocity
makes the effect of nonresonant terms released. Nonreso-
nant terms generate the periodic change of MDLZ transi-
tion rate with the same period of the driving, which
makes the interwell relaxation participate in the dynamics
and cause the change of the qubit population, i.e.,
00 11= 0,ρ ρ ≠ in the stationary case. Then the population
would also oscillate with the frequency .ω As shown in
Fig. 4, in the steady case with large amplitude 0> ,A ε for
the driving frequencies 10 and 1 MHz, the population in
state | 0〉 exhibits a periodic change [Fig. 4(a) and (b)]
Fig. 3. (Color online) Stationary population in state | 0〉 as a function of dc energy detuning ε0 in the presence of decoherence. In all
figures, /2 = 90ω π MHz, /2 = 90∆ π MHz, the interwell relaxation time 1 = 1/(2 0.00005)T π× ns, and the temperature 50 mK. The
driving amplitude is given by /2 = 5A π GHz (a)–(c) and 0.01 GHz (d)–(f). The decoherence rate is given by 2 /2 =Γ π 3 MHz (a) and
(d), 110 MHz (b) and (e), and 1050 MHz (c) and (f). The red solid line shows the analytical result from Eq. (11). The open circles give
the numerical results. The inset shows clearly the accordance between analytical and numerical results.
Fig. 4. Population in state | 0〉 as a function of time in the presence of decoherence, assuming that the system is initially in state | 0 .〉 In
all figures, 0 /2 = 4950ε π MHz. The amplitude is given by /2 = 5A π GHz (a)–(c), 0.01 GHz (d). The microwave frequency is given
by /2 = 10ω π MHz (a), 90 MHz (c), 1 MHz (b) and (d). The decoherence rate is 2 /2 = 110Γ π MHz. Other parameters are identical
with those in Fig. 3. The inset shows clearly the steady periodic change of population.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 869
Lingjie Du, Dong Lan, and Yang Yu
while for the frequency 90 MHz the population is always
constant [Fig. 4(c)]. On the other hand, at the small am-
plitude, the curve becomes smooth again [Fig. 4(d)]. We
can approximate ( / ) = ( / !)( /2 ) .n
nJ A n A∆ ω ∆ ω Small am-
plitude makes ( / )nJ A∆ ω much smaller than ω such that
the high-frequency case Eq. (9) can be used. This case can
also be understood simply: when the amplitude is small as
illustrated, the state does not pass through the crossover
and LZ transition would not occur, resulting little change
of population in the steady case.
4. Average perturbation approximation
As discussed in Refs. 21 and 22, the low frequency ω ,
i.e. ( / ) < ,nJ A∆ ω ω ∆ can produce the optimal micro-
wave cooling which is used to realize the active cooling of
qubit and other artificial devises through MDLZ transi-
tions. Although the low frequency makes the qubit popula-
tion time-dependent in the stationary case, we can consider
the average population over time. For the high frequency,
the average population is just the steady population de-
scribed by Eq. (11) while for the low frequency, the aver-
age population can be obtained by perturbation approach.
However, it is complicate and not convenient to calculate
the average population with Eq. (21). Therefore, we will
employ average perturbation approximation (APA) based
on perturbation approach in this section.
Then, we go back to the expression of ( ),f t Eq. (19).
In the stationary case, considering the time in one period,
i.e., < < ,kT t kT T+ for the term ( , , )M n m t in ( ),f t we
have
2
2 2
0 2
cos ( /2)
( , , ) = ( ) ( ) ,
( )
n m m
n tA AM n m t J J
m
+
Γ ω + π
ω ω ε − ω +Γ
0n ≠
(23)
where ( , , ) = ( , , /2 )M n m kT t M n m kT T n t′ ′+ + + with an
even number n and ( , , ) = ( , , )M n m kT t M n m kT T t′ ′+ + −
with an odd number n. Therefore, in one period of time,
the above relation leads to
( ) = 0,lim
kT T
kT kT
dt f t
+
→∞
′ ′∫
and we obtain the average MDLZ transition rates by aver-
aging Eq. (21) over one period in the stationary case
10 01 01
1= ( )lim
kT T
kT kT
W W dt W t
T
+
→∞
′ ′〈 〉 〈 〉 ≈ =∫
22
2
2 2
0 2
( / )
= .
2 ( )
n
n
J A
n
Γ ω∆
ε − ω +Γ
∑ (24)
It should be mentioned that although Eq. (24) coincides
with the transition rate Eq. (9) in perturbation-RWA, the
latter is the stationary transition rate under the high-
frequency driving while the other is the average transition
rate through averaging over the driving period under the
low-frequency driving. Having obtained the transition rate,
we can describe the population of system dynamics using
the rate equations, where the average population ii〈ρ 〉 obey
00 01 01 00 10 10 11= ( ) ( ) ,W W〈ρ 〉 − 〈 〉 + Γ 〈ρ 〉 + 〈 〉 + Γ 〈ρ 〉
00 111 = .〈ρ 〉 + 〈ρ 〉 (25)
After the long-time system evolution, we have
00 11= = 0,〈ρ 〉 〈ρ 〉 and the average qubit population in
state | 0〉 can be easily solved from Eq. (25):
00 01 10 01 10 01= ( )/(2 ).W W〈ρ 〉 〈 〉 + Γ 〈 〉 + Γ + Γ (26)
The rate equations used here are different from Eq. (20).
Under the low-frequency driving, Eq. (20) describes the
real-time dynamics with the periodic change of population.
However, Eq. (26) neglects the population change within
one period as shown in Figs. 4(a) and (b) and focuses on
the scale of one period, in which the average population
would be constant. If APA is valid, the change of qubit
population must be slow on the scale of the driving period,
i.e. 01> .Wω 〈 〉 If ω is larger than 01 ,W〈 〉 the population
change in one period is so small that
01 00 00 01 00 01/ / =
t T t T
t t
W dt T W dt T W
′ ′+ +
′ ′
ρ ≈ 〈ρ 〉 〈ρ 〉〈 〉∫ ∫
and the rate equations can be changed to the form of
Eq. (25).
In order to show the region of validity for APA, we
compare the numerical solutions with analytical results.
The average transition rate 01W〈 〉 is probably less than
5⋅2π MHz and generally the same order with ( / ).nJ A∆ ω
For several decoherence rates, we select the frequency
1 MHz and obtain the average steady population, shown
in Fig. 5. The deviation between the numerical and analyti-
cal results cannot be neglected and APA is not proper. Fig-
ure 4(b) shows the real-time population in state | 0〉 with
the numerical stimulation under the frequency 1 MHz.
Comparing with the result of frequency 10 MHz in Fig. 4(a),
due to 01< Wω 〈 〉 the population change in Fig. 4(b) is
large, demonstrating the invalidity of APA (Fig. 5).
Moreover, as shown in Figs. 5(a)–(c), at the frequency
1 MHz, when the amplitude is less than the dc detuning,
which means the state does not reach the crossover, APA
exhibits the good agreement with the numerical results.
This is more clearly seen in Figs. 5(d)–(f) with a small
amplitude /2 = 10A π MHz. Since the population change in
the steady case is little [Fig. 4(d)], APA is still available. In
Fig. 6, for the frequency 10 MHz, the analytical results
show very good agreement for several decoherence rates,
changing from the rate less than ( / )nJ A∆ ω to the one
much larger than ∆ . When the amplitude is larger than the
dc detuning, the population is near 0.5, which also pro-
870 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
vides a way to determine the location of crossover with the
low frequency [Figs. 6(a)–(d)]. It should be noted that with
the decoherence 2 /2 =Γ π 3 MHz and the frequency 10
MHz, the agreement is not very good at the detuning
0 .Aε ≈ The high coherence makes the average transition
rate large. As discussed in Ref. 22, at the detuning 0 ,Aε ≈
the transition rate reaches maximum. Therefore the fre-
quency would be less than the average transition rate, i.e.,
01< ,Wω 〈 〉 and results into the disagreement.
5. Noncoherence approximation
For the extremely low frequency, i.e., <ω
01( / ) ,nJ A W< ∆ ω 〈 〉 APA becomes insufficient just as
shown in Fig. 5. Therefore in order to analyze this region
conveniently, here we develop an analytical method based
on noncoherence approximation (NCA), which will give
an intuitive and clear physical picture.
When the driving frequency is high and coherent, LZS
interference and multiphoton effect are clear and Bessel
Fig. 5. (Color online) Average stationary population in state | 0〉 as a function of dc energy detuning ε0 in the presence of decoherence. In
all figures, /2 = 1ω π MHz, and 0 /2 = 4950ε π MHz. The driving amplitude is given by /2 = 5A π GHz (a)–(c), 0.01 GHz (d)–(f). The
decoherence rate is given by 2 /2 =Γ π 40 MHz (a) and (d), 110 MHz (b) and (e), and 1050 MHz (c) and (f). Other parameters are identical
with those in Fig. 3. The red solid line is the analytical result from Eq. (26). The open circles give the numerical results.
Fig. 6. (Color online) Average stationary population in state | 0〉 as a function of dc energy detuning ε0 in the presence of decoherence. In
all figures, /2 = 10ω π MHz and 0 /2 = 4950ε π MHz. The driving amplitude is given by /2 = 5A π GHz (a)–(d), 0.01 GHz (e)–(h). The
decoherence rate is given by 2 /2 =Γ π 40 MHz (a) and (e), 110 MHz (b) and (f), 1050 MHz (c) and (g), 3 MHz (d) and (h). Other parame-
ters are identical with those in Fig. 3. The red solid line is the analytical result from Eq. (26). The open circles give the numerical results.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 871
Lingjie Du, Dong Lan, and Yang Yu
Fig. 7. (Color online) Average stationary population in state | 0〉 as a function of dc energy detuning ε0 in the presence of decohe-
rence. In all figures, /2 = 1ω π MHz and /2 = 5A π GHz. The decoherence rate is 2 /2 = 3Γ π MHz (a), 40 MHz (b), 110 MHz (c),
and 1050 MHz (d). Other parameters are identical with those in Fig. 3. The red solid line shows the analytical result in Eq. (30) through
noncoherence approximation. The open circles give the numerical results.
functions can give a convenient picture. However, the de-
crease of frequency makes LZS interference gradually dis-
appear. Therefore the description with Bessel functions
would not be proper for the extremely low frequency and
we return back to the sinusoidal description.
We go back to Eq. (16) and have
/22
0 2
0 /2
( ) = exp{ ( sin ) | |}
4
t t
t
dy t d d i dt A t
dt
′τ +τ
′− τ −τ
∆ ′ ′ ′τ τ ε + ω −Γ τ =∫ ∫ ∫
2
0 2
0
= exp{ ( sin ) | |}
4
t t
t
d i dt A t
+τ∆ ′ ′τ ε + ω −Γ τ +∫ ∫
02
0 2exp{ ( sin ) | |}.
4
t
t t
d i dt A t
− −τ
∆ ′ ′+ τ ε + ω −Γ τ∫ ∫
(27)
Because of the strong decoherence, the extreme low
frequency usually is much less than the decoherence rate.
The decoherence makes the qubit lose coherence quickly,
even though the state is split by the crossover and obtains
the coherence again in each period. Hence we can approx-
imately replace sin t′ω to sin .tω Then we follow the pro-
cedure in Sec. 3, and have MDLZ transition rates
2
2
10 01 2 2
0 2
( ) = ( ) = .
2 ( sin )
W t W t
A t
Γ∆
ε + ω +Γ
(28)
Equation (28) is actually the result of Eq. (21) in NCA
limit. Interestingly, MDLZ transition rates become symme-
trical. The expression can also be understood as macros-
copic resonant tunneling (MRT) [44,46,48] model under
the sinusoidal driving with the extremely low frequency.
When the state is driven slowly to pass through each de-
tuning, we divide the evolution into discrete time steps
within which the coherence lost totally at the correspond-
ing detunings [49]. Furthermore there is no coherence to
connect time steps and the phase accumulation [9] has not
existed. Since each time step is much larger than the deco-
herence time, we can approximately consider that MRT
occurs at the detuning with the tunneling transition rate
Eq. (28). Hence, the dynamics is changed from LZS inter-
ference to the periodically driven MRT and we have the
rate equations
00 01 01 00 10 10 11= [ ( ) ] [ ( ) ] ,W t W tρ − + Γ ρ + + Γ ρ
00 111 = ,ρ +ρ (29)
where 01W is defined by Eq. (28) and 01 10( )Γ Γ has the
same definition with that in Eq. (7). In the evolution
Eqs. (29), the periodic change makes the previous treat-
ment 00 11= = 0ρ ρ incorrect. When the initial state is
| 0 ,〉 we have the population in state | 0〉
00 01 01 10 10 10
0 0
= exp[ ( 2 ( ) )]{ [ ( ) ]
t t
dt W t dt W t′ ′ ′ ′ρ − −Γ −Γ +Γ ×∫ ∫
01 01 10
0
exp[ ( 2 ( ) )] 1}.
t
dt W t
′
′′ ′′× − −Γ −Γ +∫ (30)
872 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
The limit of perturbation approach also has function on
NCA, which requires the strong decoherence, 2 >Γ
01( / ) ,nJ A W> ∆ ω 〈 〉 where 01W〈 〉 is defined by Eq. (24).
Moreover NCA needs the extremely low frequency com-
pared with the strong decoherence. Therefore, in this re-
gion, this method is reasonable with the condition
01 2< < .Wω 〈 〉 Γ In the frequencies where the average per-
turbation theory fails, we use Eq. (30) to reconsider the av-
erage population in the steady situation as shown in Fig. 7.
At the decoherence rate 2 /2 = 3Γ π MHz, the agreement is
not good, because the decoherence rate is comparable with
the driving frequency. However, for other decoherence
rates, the numerical stimulations agree well with the analyti-
cal results. Furthermore, we consider the real-time evolution
and compare it with the numerical stimulation (Fig. 8). As
the frequency is comparable or larger than the decoherence
rate, NCA will be insufficient [Fig. 8(b)].
From perturbation-RWA to NCA, we can find the tran-
sition from quantum to classic effect of the microwave
with the frequency decreasing. For the high-frequency
driving, being in a coherent state and with high photon
numbers, the microwave behaves as the quantized electro-
magnetic field exchanging photons with the qubit. At the
same time, the large microwave amplitude forms LZS in-
terferometry through the crossover, modulating the intensi-
ty of the n-photon resonance. In this case MDLZ transition
Eq. (9) is based on perturbation-RWA and the nonresonant
terms in the transition rate can be neglected. With the fre-
quency further decreasing, the decoherence becomes
stronger relative to the driving frequency and the nonreso-
nant terms are comparable with the resonant ones, which
makes the photons resonances as well as the modulation
from LZS interferometry vanish gradually. Then we have
to use Eq. (21) to describe MDLZ transition. For the low
frequency, in the scale of one driving period, the resonant
information can still be notable and we can use APA to
obtain the average population. For the extremely low fre-
quency compared with the decoherence, the microwave
can finally be treated as a classic field and the dynamic
process is changed to the periodically driven MRT as
shown in Eq. (30).
To sum up, we plot the average stationary population
obtained from analytical calculation and numerical stimu-
lation as shown in Fig. 9(a). The different approximations
of perturbation approach cover most of parameter space.
Figure 9(b) is the detail of Fig. 9(a) at the decoherence rate
2 /2 = 110Γ π MHz. We show the applicable regions for
these approximations in the average stationary population
in Fig. 9(c). The circles mark where perturbation-RWA
holds. In the red shaded region I, a high-frequency cohe-
rent field is used to realize multiphoton resonance [9,19].
With the frequency further decreasing, we enter
01 < .W〈 〉 ω ∆ In the stationary state the qubit population
exhibits the periodic oscillation which cannot be explained
using perturbation-RWA. Now APA is applicable, marked
with blue squares region III in Fig. 9(c). In the green
shaded region II, the low-frequency driving is able to gen-
erate microwave cooling and the optimal microwave cool-
ing was realized by Valenzuela et al. [21]. Then for the
extremely low frequency 01 2< < ,Wω 〈 〉 Γ APA is also
unsuccessful. NCA can describe the behavior, as marked
by the black ellipses. In the blue shaded region III, which
is not covered by any before mentioned approximation, the
high coherence and low frequency make it possible and
convenient to realize adiabatic LZ process or LZS interfe-
rence by a single crossing [16,50,51] with large amplitude.
It should be emphasized that the region in Fig. 9(c) would
change with the amplitude just as shown in Fig. 5. For the
amplitude larger than the dc detuning, the small frequency
locates in NCA region while for the amplitude less than the
dc detuning, this frequency may move to APA region.
6. Multilevel system
Our approach gives a simple but effective physical pic-
ture, which enables us easily to extend the discussion to a
multilevel flux qubit [Fig. 10(a)] [15,27]. In this section,
based on the approximations we will discuss the lower
limit of driving frequency in effective microwave driven
cooling.
Fig. 8. (Color online) Population in state | 0〉 as a function of
time in the presence of decoherence, assuming that the system is
initially in the state | 0 .〉 In both figures, /2 = 1ω π MHz,
0 /2 = 4125ε π MHz, and /2 = 5A π GHz. The decoherence rate
is given by 2 /2 = 1050Γ π MHz (a) and 3 MHz (b). Other para-
meters are identical with those in Fig. 3. The red line shows the
results from the noncoherence approximation. The open circles
and dotted line give the numerical results.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 873
Lingjie Du, Dong Lan, and Yang Yu
In Ref. 21, it was found that a lower driving frequency
realized better cooling effect and at 5 MHz the optimal cool-
ing appeared. The driving microwave frequency used is so
low that one does not need the high-frequency microwave
generator and microwave line to realize active cooling. On
the other hand, generally, if the microwave frequency is too
low, there would not be the cooling effect on the qubit.
Therefore, there exists a minimum frequency to produce the
effective microwave cooling. Considering the three-level
system consisting of an “internal” qubit (|1〉 and | 0 )〉 and
an “internal” oscillator like state | 2 ,〉 as shown in Fig. 10(a),
we rewrite the rate equations (29) to three-level form
00 01 01 02 00= [ ( ) ( )]W t W tρ − + Γ + ρ +
10 10 11 20 22[ ( ) ] ( ) ,W t W t+ + Γ ρ + ρ
11 01 01 00 10 10 11 21 22= [ ( ) ] [ ( ) ] ,W t W tρ + Γ ρ − + Γ ρ + Γ ρ
00 11 221 = ,ρ +ρ +ρ (31)
where 21Γ is the intrawell relaxation from state | 2〉 to |1 .〉
For the low frequency as discussed in Sec. 4, APA can
be applicable and we have
2 2
20 2 21
20 02 2 2
0 2 21
( /2) ( / )
( ) = ( ) = ,
2 ( ) ( /2)
n
n
J A
W t W t
n
∆ Γ +Γ ω
′ε − ω + Γ + Γ
∑ (32)
and
2 2
01 2
01 10 2 2
0 2
( / )
( ) = ( ) = ,
2 ( )
n
n
J A
W t W t
n
∆ Γ ω
ε − ω +Γ
∑ (33)
where
0 0 2 20= (| | | |)( ),dcm m′ε + δΦ −Φ 0 0 1= (| | | |) ,dcm mε + δΦ
and 20Φ is the flux detuning where the crossover 20∆ is
reached. It should be mentioned that Eqs. (32) and (33)
actually express the average transition rate. Then for the
extremely low frequency as discussed in Sec. 5, we have
2
20 2 21
20 02 2 2
2 21
/2
( ) = ( ) = ,
2 ( ) ( /2)
W t W t
t∗
∆ Γ + Γ
ε + Γ + Γ
(34)
Fig. 9. (Color online) (a) Population in state | 0〉 of TLS described by Fig. 1 as functions of microwave frequency and decoherence rate.
0 /2 = 4950ε π MHz and /2 = 5A π GHz. Other parameters are the same with those in Fig. 3. The red surface with dashed line is ob-
tained from the average perturbation approximation. The yellow surface with solid line is resulted from the noncoherence approxima-
tion. The blue solid circles are from the data of numerical stimulation. (b) The detail of (a) at the decoherence rate 2 /2 = 110Γ π MHz.
(c) Regions of validity for different approximations. The axes are the driving frequency and decoherence rate, respectively. Perturba-
tion-RWA region is described by the condition >ω ∆ and it is marked by the red circles. The average perturbation approximation re-
gion is described by the condition 10 < <W〈 〉 ω ∆ and it is represented by the blue squares. The noncoherence approximation region is
shown by the condition 10 2< <Wω 〈 〉 Γ and it is marked by the black ellipses.
874 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
and
2
01 2
01 10 2 2
2
( ) = ( ) =
2 ( )
W t W t
t
∆ Γ
ε + Γ
(35)
where
0 1( ) = (| | | |) ( )t m m tε + δΦ
and
0 2 20( ) = (| | | |)[ ( ) ].t m m t∗ε + δΦ −Φ
Here, we focus the extremely low-frequency region. Since
the analytical solution is too complex to extract a clear
physical picture, we do not write it out explicitly here. Fig-
ure 10(b) shows the population in state | 0〉 at the dc flux
detuning dcδΦ = 0.05 mΦ0 as functions of frequency and
amplitude. The thick dashed line marks the population at
equilibrium with the temperature 50 mK. It can be seen
that when the frequency is less than 0.1 MHz, microwave
cooling would be ineffective.
In the extremely low frequency, the interwell relaxation
changes the qubit population within one period as shown in
Eq. (31). When the state is driven through the crossover
20 ,∆ LZ transition would transfer the population from
state | 0〉 to | 2 ,〉 which then quickly relaxes into state |1 .〉
As the states is driven back to the initial detuning, the in-
terwell relaxation warms the qubit again if the driving fre-
quency is less than the repopulation rate of the warm
process, i.e., 10 01,Γ +Γ which is approximately 102Γ at
the small dc detuning. The analysis agrees with the result
in Fig. 10(b).
7. Conclusion
In this article we have studied the population evolution
of a strongly driven superconducting qubit of the long-time
dynamics. First of all, we use perturbation approach, which
considers high order perturbation expansion and nonreso-
nant terms, to obtain the time dependent unsymmetrical
MDLZ transitions. Moreover, we point out that this ap-
proach has a high validity in the case where the influence
of coherence resulted from ∆ is small with the high fre-
quency or strong decoherence. Then for different frequen-
cies, we apply several approximations to discuss the sys-
tem dynamics.
For the high frequency, i.e., > ,ω ∆ MDLZ transition
rate in perturbation approach is constant and leads to per-
turbation-RWA result. In previous works [9,19] perturba-
tion-RWA has been proven successful in the coherent re-
gion under the strong decoherence, i.e., 2> ,ω ∆ Γ and
2 01.WΓ Besides this region, we show perturbation-
RWA result can also be used in the incoherent case where
the frequency is smaller than the decoherence rate,
2 > ,Γ ω and even the high-coherence region 2 01< .WΓ
With the frequency further decreasing to the region of
the optimal microwave cooling, 10 < ,W〈 〉 ω ∆ since the
coherence has effect on the nonresonant terms and nonre-
sonant terms are comparable with the resonant terms, the
qubit population exhibits periodic oscillation which cannot
be addressed by perturbation-RWA. In this case, we study
the average stationary population in the low-frequency driv-
ing. Then we employ APA by averaging MDLZ transition
rate of perturbation approach with time. Although the ana-
lytical expression is the same with that of perturbation-
RWA, they have different physical pictures. In perturbation-
RWA, nonresonant terms have no effects and can be neg-
lected, while in APA the nonresonant terms result oscilla-
tions and can be averaged over to obtain the average popula-
tion. Therefore in APA, we do not take account into the real-
time dynamics, but consider one period as the unit time.
Fig. 10. (Color online) (a) Schematic energy diagram of a multi-
level flux qubit. Red solid curve represents the microwave .acΦ
The dashed black line marks a particular static flux detuning
.dcδΦ State | 0〉 is in the left well; |1〉 and | 2〉 are in the right
well. The red solid path describes the cooling with the popula-
tion in state | 0〉 transferred to |1〉 . The locations of the cros-
sovers 01∆ and 20∆ are 0 mΦ0 and 8.4 mΦ0, respectively. The
diabatic energy-level slope 0 1| | (| |)m m = 2π×1.44 GHz/mΦ0,
and 2| |m = 2π×1.09 GHz/mΦ0. 01/2 =∆ π 0.013 GHz, 02 /2 =∆ π
= 0.09 GHz. The parameters used above are from the experiment
[15]. In calculation, 2 /2 =Γ π 0.06 GHz, 21/2 =Γ π 0.1 GHz [27],
and the interwell relaxation time 1 = 1/(2 0.00005)T π× ns (b). The
yellow surface with solid line shows the population in state | 0〉
versus microwave frequency and amplitude at dc flux detuning
= 0.05dcδΦ mΦ0. The temperature is 50 mK. The dashed line
indicates the population in equilibrium at this detuning.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 875
Lingjie Du, Dong Lan, and Yang Yu
Then in the extremely low frequency, i.e., 10 2< < ,Wω 〈 〉 Γ
where APA has failed, we propose NCA by dividing the
evolution into discrete time steps during which the cohe-
rence is lost totally. We show the dynamic process equiva-
lent to the periodically driven MRT, giving a convenient
and intuitive picture. In addition, in the multilevel system,
based on this approximation we examine the possibility of
employing the extremely low frequency driving as a useful
tool to realize microwave cooling and prove that there ex-
ists a lower limit of frequency for effective microwave
cooling. The different approximations of perturbation ap-
proach cover most of parameter space when describing the
average population [Fig. 9(c)].
Our theory offers the effective analytical description of
the driven dissipative TLS in the stationary case, including
real-time and average situation. The good agreement of our
theory with the numerical results shows the validity of the
approximations used. By shifting these approximations, we
also show the transition from quantum regime to classic
regime about the interaction between qubit and microwave,
with dynamics changing from LZS interference to periodi-
cally driven MRT. This could enhance the understanding
to the influence of decoherence on the qubit. Furthermore,
our results can be applied to understand the behavior of
other systems with similar structure. Especially, our theory
in the low frequency can be used to analyze the optimal
microwave cooling of qubit as well as other artificial de-
vice such as nano mechanical resonators.
Acknowledgment
Thanks to S.O. Valenzuela, S.N. Shevchenko and
M.H.S. Amin for useful discussions.
This work was supported in part by the State Key Pro-
gram for Basic Researches of China (2011CB922104,
2011CBA00205), the NSFC (11274156, 91021003), the
Natural Science Foundation of Jiangsu Province
(BK2010012), and PAPD.
Appendix. Derivation of Eq. (14)
In this Appendix, we will accept the straightforward
physical picture for intuitive understanding. Assuming that
the system is initialized in state | 0 ,〉 first of all, we try to
calculate the population in state | 0 ,〉 i.e., 00 ( ).tρ In a short
time interval, i.e., < 1/[ ( / )],nt J A∆ ω considering the per-
turbation expansion of VU in ( )V τ up to the second order
term, we have
____________________________________________________
†
00 ( ) = Tr { 0 | ( ) | 0 } = Tr { 0 | ( ,0) (0) ( ,0) | 0 }B SB B V SB Vt t U t U tρ 〈 ρ 〉 〈 ρ 〉 =
† †
0 0 0 0
= Tr { 0 | (0) | 0 } Tr { 0 | ( ) ( ) (0) | 0 } Tr { 0 | (0) ( ) ( ) | 0 }.
t t
B SB B SB B SBd d V V d d V V
τ τ
′ ′ ′ ′〈 ρ 〉 − τ τ 〈 τ τ ρ 〉 − τ τ 〈 ρ τ τ 〉∫ ∫ ∫ ∫ (A.1)
We substitute ( )V t into Eq. (A.1) and have
† † † †
00 1 2
0 0 0 0
( ) = 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
t d d K U U U U d d K U U U U
τ τ
− + + − − + + −′ ′ ′ ′ ′ ′ρ − τ τ 〈 τ τ τ τ 〉 − τ τ 〈 τ τ τ τ 〉 =∫ ∫ ∫ ∫
† †
1
0 0
= 1 2 ( ) ( ) ( ) ( ) ,
t
d d K U U U U
τ
− + + −′ ′ ′− τ τ 〈 τ τ τ τ 〉∫ ∫ (A.2)
where
( ) ( ) /2 /20 01
,
= ( ) ( )ei m i n in im
n m
n m
A AK J J ′ε − ω τ− ε − ω τ + π − π
ω ω∑
and
( ) ( ) /2 /20 02
,
= ( ) ( )e .i m i n in im
n m
n m
A AK J J ′− ε − ω τ+ ε − ω τ − π + π
ω ω∑
In the above derivation, we also obtain
† †
0 0 0 0
Tr { 0 | ( ) ( ) (0) | 0 } Tr { 0 | (0) ( ) ( ) | 0 }.
t t
B SB B SBd d V V d d V V
τ τ
′ ′ ′ ′〈 τ τ τ τ ρ 〉 = 〈 τ τ ρ τ τ 〉∫ ∫ ∫ ∫
Moreover
0 0
2Tr { 0 | ( ) ( ) (0) | 0 }
t
B SBd d V V
τ
′ ′〈 τ τ τ τ ρ 〉∫ ∫ just describes the reduced population in state | 0〉 in the short time
interval.
Then we turn to high order expansion. It should be mentioned for simpleness in the below derivation we only consider
the first four terms in Eq. (13) and higher order terms can be treated with the similar process. We have
876 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits
† †
11
0 0
( ) = Tr { 1| ( ,0) (0) ( ,0) |1 } = Tr { 1| ( ) (0) ( ) |1 }
t t
B V SB B SBVt U t U t d d V V′ ′ρ 〈 ρ 〉 τ τ 〈 τ ρ τ 〉 −∫ ∫
1 2
† † †
4 1 2 3 4 3 2 1
0 0 0 0
Tr { 1| ( ) (0) ( ) ( ) ( ) |1 }
t tt t
B SBdt dt dt dt V t V t V t V t− 〈 ρ 〉 −∫ ∫ ∫ ∫
1 2
†
1 2 3 4 1 2 3 4
0 0 0 0
Tr { 1| ( ) ( ) ( ) (0) ( ) |1 } ...
t tt t
B SBdt dt dt dt V t V t V t V t− 〈 ρ 〉 +∫ ∫ ∫ ∫ (A.3)
Considering that the last two terms in Eq. (A.3) is equivalent, we can write Eq. (A.3) as
1 2
† † † †
11 4 1 2 3 4 3 2 1
0 0 0 0 0 0
( ) = Tr { 1| ( ) (0) ( ) |1 } 2 Tr { 1| ( ) (0) ( ) ( ) ( ) |1 }
t tt t t t
B SB B SBt d d V V dt dt dt dt V t V t V t V t′ ′ρ τ τ 〈 τ ρ τ 〉 − 〈 ρ 〉 =∫ ∫ ∫ ∫ ∫ ∫
1 2 2
† † † †
1 2 3 4 4 3 2 1
0 0 0 0 0 0
Tr { 1| ( ) (0) ( ) |1 } 2 Tr { 1| ( ) (0) ( ) ( ) ( ) |1 }
t t tt t t
B SB B SBd d V V dt dt dt dt V t V t V t V t′ ′= τ τ 〈 τ ρ τ 〉 − 〈 ρ 〉 −∫ ∫ ∫ ∫ ∫ ∫
1 2
† † †
1 4 2 3 4 3 2 1
0 0 0 0
2 Tr { 1| ( ) (0) ( ) ( ) ( ) |1 } ...
t tt t
B SBdt dt dt dt V t V t V t V t− 〈 ρ 〉 +∫ ∫ ∫ ∫ (A.4)
Then we find that the system population rate change is given by
† †
11
0 0
( ) = Tr { 1| ( ) (0) ( ) |1 } Tr { 1| ( ) (0) ( ) |1 }
t t
B SB B SBt d V t V d V V t′ρ τ 〈 ρ τ 〉 + τ 〈 τ ρ 〉 −∫ ∫
2 2 2
† † † † † †
2 3 4 4 3 2 2 3 4 4 3 2
0 0 0 0 0 2
2 Tr { 1| ( ) (0) ( ) ( ) ( ) |1 } 2 Tr { 1| ( ) (0) ( ) ( ) ( ) |1 }
t t tt t t
B SB B SB
t
dt dt dt V t V t V t V t dt dt dt V t V t V t V t− 〈 ρ 〉 − 〈 ρ 〉 −∫ ∫ ∫ ∫ ∫ ∫
1 2
† † †
1 2 3 3 2 1
0 0 0
2 { 1| ( ) (0) ( ) ( ) ( ) |1 } ...
t tt
B SBdt dt dt Tr V t V t V t V t− 〈 ρ 〉 +∫ ∫ ∫ (A.5)
Combining the equivalent terms in Eq. (A.5), we have
2 2
† † † †
11 2 3 4 4 3 2
0 0 0 0
( ) = Tr {2 1| ( ) (0) ( ) |1 } Tr {2 1| [ ( ) (0) ( )] ( ) ( ) |1 }
t tt t
B SB B SBt d V t V dt dt dt V t V t V t V tρ τ〈 ρ τ 〉 − 〈 ρ 〉 −∫ ∫ ∫ ∫
1 2
† † †
1 2 3 3 2 1
0 0 0
Tr {2 1| ( )[2 (0) ( ) ( )] ( ) |1 } ...
t tt
B SBdt V t dt dt V t V t V t− 〈 ρ 〉 +∫ ∫ ∫ , (A.6)
_______________________________________________
where
2 2
†
3 4 4 3
0 0
( ) (0) ( )
t t
SBdt dt V t V tρ∫ ∫
describes the population transferred from state | 0〉 to |1〉
in the time interval 2.t Comparing with Eqs. (A.1) and
(A.2), we can find that the second term in Eq. (A.6) cha-
racterizes the population rate transferred from state |1〉 to
| 0〉 based on the population transferred from state | 0 .〉
Similarly
1 2
† †
2 3 3 2
0 0
2 (0) ( ) ( )
t t
SBdt dt V t V tρ∫ ∫
describes the reduced population in state | 0〉 in the time
interval 1t and the third term in Eq. (A.6) describes the
reduced population rate from state | 0〉 to |1〉 due to the
decrease of population in state | 0 .〉 Then we replace the
integration limit 2t in Eq. (A.6) to t, which is similar with
the widely known Markovian approximation dealing with
the population transition caused by the coupling to the bath
[52]. Here, different from the Markovian approximation, the
approximation is used in the population transition caused by
the tunnel coupling .∆ To make this approximation valid,
the system should be independent on the past, which means
the coherent population change resulted from ∆ should be
little. Integrating with Eq. (A.6) we can obtain Eq. (14).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8 877
Lingjie Du, Dong Lan, and Yang Yu
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878 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 8
http://dx.doi.org/10.1038/ncomms1050
Lingjie Du1,2, Dong Lan1, and Yang Yu1
2. Basic model
3. Perturbation approach
Then we have the population in state
4. Average perturbation approximation
5. Noncoherence approximation
We go back to Eq. (16) and have
6. Multilevel system
7. Conclusion
Acknowledgment
Appendix. Derivation of Eq. (14)
We substitute into Eq. (A.1) and have
In the above derivation, we also obtain
Considering that the last two terms in Eq. (A.3) is equivalent, we can write Eq. (A.3) as
Then we find that the system population rate change is given by
Combining the equivalent terms in Eq. (A.5), we have
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