Electromagnetically induced interference in a superconducting flux qubit
Interaction between quantum two-level systems (qubits) and electromagnetic fields can provide additional coupling channels to qubit states. In particular, the interwell relaxation or Rabi oscillations, resulting, respective-ly, from the multi- or single-mode interaction, can produce effective crosso...
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Du, Lingjie Yu, Yang Lan, Dong 2017-05-30T12:15:10Z 2017-05-30T12:15:10Z 2013 Electromagnetically induced interference in a superconducting flux qubit / Lingjie Du, Yang Yu, Dong Lan // Физика низких температур. — 2013. — Т. 39, № 6. — С. 649–662. — Бібліогр.: 62 назв. — англ. 0132-6414 PACS: 03.67.Lx, 32.80.Xx, 42.50.Hz https://nasplib.isofts.kiev.ua/handle/123456789/118468 Interaction between quantum two-level systems (qubits) and electromagnetic fields can provide additional coupling channels to qubit states. In particular, the interwell relaxation or Rabi oscillations, resulting, respective-ly, from the multi- or single-mode interaction, can produce effective crossovers, leading to electromagnetically induced interference in microwave driven qubits. The environment is modeled by a multimode thermal bath, ge-nerating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the qubits and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic frequency of the bath. In addition, we demon-strate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi oscillations, Rabi-oscillation-induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the qubit, with more controllable parameters in-cluding the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation in qubits either with or without crossover structure. Thanks to Xueda Wen for useful discussions. This work was supported in part by the State Key Program for Basic Researches of China (2011CB922104, 2011CBA00205), the NSFC (91021003, 11274156), the Natural Science Foundation of Jiangsu Province (BK2010012), and PAPD. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Свеpхпpоводимость, в том числе высокотемпеpатуpная Electromagnetically induced interference in a superconducting flux qubit Article published earlier |
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Electromagnetically induced interference in a superconducting flux qubit Du, Lingjie Yu, Yang Lan, Dong Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Electromagnetically induced interference in a superconducting flux qubit |
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electromagnetically induced interference in a superconducting flux qubit |
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Du, Lingjie Yu, Yang Lan, Dong |
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Du, Lingjie Yu, Yang Lan, Dong |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Interaction between quantum two-level systems (qubits) and electromagnetic fields can provide additional coupling channels to qubit states. In particular, the interwell relaxation or Rabi oscillations, resulting, respective-ly, from the multi- or single-mode interaction, can produce effective crossovers, leading to electromagnetically induced interference in microwave driven qubits. The environment is modeled by a multimode thermal bath, ge-nerating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the qubits and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic frequency of the bath. In addition, we demon-strate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi oscillations, Rabi-oscillation-induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the qubit, with more controllable parameters in-cluding the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation in qubits either with or without crossover structure.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118468 |
| citation_txt |
Electromagnetically induced interference in a superconducting flux qubit / Lingjie Du, Yang Yu, Dong Lan // Физика низких температур. — 2013. — Т. 39, № 6. — С. 649–662. — Бібліогр.: 62 назв. — англ. |
| work_keys_str_mv |
AT dulingjie electromagneticallyinducedinterferenceinasuperconductingfluxqubit AT yuyang electromagneticallyinducedinterferenceinasuperconductingfluxqubit AT landong electromagneticallyinducedinterferenceinasuperconductingfluxqubit |
| first_indexed |
2025-11-27T07:03:05Z |
| last_indexed |
2025-11-27T07:03:05Z |
| _version_ |
1850806056443183104 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6, pp. 649–662
Electromagnetically induced interference
in a superconducting flux qubit
Lingjie Du1,2, Yang Yu1, and Dong Lan1
1National Laboratory of Solid State Microstructures and Department of Physics
Nanjing University, Nanjing 210093, China
E-mail: lingjie.du@rice.edu
yuyang@nju.edu.cn
2Department of Physics and Astronomy, Rice University, Houston, Texas 77251-1892, USA
Received November 23, 2012, revised December 22, 2012
Interaction between quantum two-level systems (qubits) and electromagnetic fields can provide additional
coupling channels to qubit states. In particular, the interwell relaxation or Rabi oscillations, resulting, respective-
ly, from the multi- or single-mode interaction, can produce effective crossovers, leading to electromagnetically
induced interference in microwave driven qubits. The environment is modeled by a multimode thermal bath, ge-
nerating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides
deeper understanding to the interaction between the qubits and their environment. It also supplies a useful tool to
characterize the relaxation strength as well as the characteristic frequency of the bath. In addition, we demon-
strate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand,
different from Rabi oscillations, Rabi-oscillation-induced interference involves more complicated and modulated
photon exchange thus offers an alternative means to manipulate the qubit, with more controllable parameters in-
cluding the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear
quantum phenomena and quantum state manipulation in qubits either with or without crossover structure.
PACS: 03.67.Lx Quantum computation architectures and implementations;
32.80.Xx Level crossing and optical pumping;
42.50.Hz Strong-field excitation of optical transitions in quantum systems; multiphoton processes;
dynamic Stark shift.
Keywords: quantum qubits, Rabi-oscillation-induced interference.
1. Introduction
Strongly driven transition at avoided crossings is an old
issue but plays an essential role in a variety of physical
phenomena. Cases of avoided crossings can be met in
many areas in physics, such as atomic-collision [1,2], la-
ser-atom interactions [3], optical atoms [4], ultracold-
molecules gas [5], Bose–Einstein condensate [6], and more
recently, qubits for quantum information processing [7–9].
A qubit is generally built from a quantum two-level system
(TLS), which is adjustable by external control fields. If the
external control parameter is changed so that a TLS tra-
verses the crossover, a non-adiabatic transition between
two states occurs. This transition is known as Landau–
Zener (LZ) transition [10], which can be used to enhance
the quantum tunneling rate [11,12], to prepare the quantum
states [13], and to control the gate operations [14]. When
the qubit is subjected to a strong periodic microwave field,
consecutive LZ transitions between two states at the cros-
sover may result in Landau–Zener–Stückelberg (LZS) in-
terference [15,16], which has been demonstrated in many
systems [17–28]. The strong driving could allow for fast
and reliable control of the qubits, paving a way of demon-
stration of macroscopic quantum coherence and implemen-
tation of practical quantum processor.
In practice, besides controlling fields, TLSs inevitably
couple with electromagnetic environments which can often
be described as a multimode thermal bath of harmonic os-
cillators [29], experiencing dephasing and energy relaxa-
tion simultaneously. Both processes contribute to decohe-
rence which may hinder qubit performance in quantum
information [30–34]. In general, the bath couples to the
qubit longitudinally, causing the dephasing related with the
environment property at low energies. The transverse inte-
© Lingjie Du, Yang Yu, and Dong Lan, 2013
mailto:yuyang@nju.edu.cn
Lingjie Du, Yang Yu, and Dong Lan
raction between the qubit and bath produces the interwell
relaxation between two lowest qubit states. In previous
works [16,35–37], the interwell relaxation is described
empirically and we will show additional interference is
usually overlooked. Therefore, the investigation on strong-
ly driven dynamics considering additional electromagnetic
degrees of freedom will clarify the environmental effect on
qubits. Furthermore, it is possible to improve the quantum
gate operation with LZS interference.
In this article, we study the strongly microwave driven
qubit in the presence of additional electromagnetic interac-
tions based on the general theories of TLSs [36–45]. It is
found that the interactions between qubits and electromag-
netic fields provide additional coupling channels to qubit
states, which will detour the evolution path to lead a new
mechanism of interference. We name the new mechanism
of interference as electromagnetically induced interference
(EII). Two typical electromagnetic systems: (i) a thermal
bath with multimode electromagnetic fields and (ii) a weak
single-mode field, are chosen to discuss interesting phe-
nomena and applications.
We show that the relaxation induced coupling contri-
butes to asymmetrical transitions to form interferences, i.e.
relaxation induced interference (RII). We find that the re-
laxation can generate population inversion in a strongly
driven TLS through the interference, which violates the
intuitive picture. This work also clarifies the range of va-
lidity of previous results in atomic physics. Moreover, the
model presents deeper understanding to the effect of the
environment. It provides a useful tool for characterizing
the interaction of qubits and environment, e.g., the relaxa-
tion strength and the characteristic frequency of the bath.
Since the interwell relaxation is determined by the envi-
ronment, we turn to a more controllable case by applying a
weak single mode field which results Rabi oscillations.
Rabi oscillation can construct new controllable coupling
channels which act as effective level crossover. The inter-
ference, which we call Rabi-oscillation-induced interfe-
rence (ROII), is generated under the strong microwave
field. ROII offers a fully tunable way to operate qubits in
the strongly driven region. It is promising for further appli-
cation in the time-dependent interference which can mani-
pulate the qubit over short timescales. In addition, it also
supplies a testing ground for exploring nonlinear quantum
phenomena and quantum state manipulation, in the general
TLS without the crossover structure.
The paper is organized as follows. In Sec. 2, through
the perturbation theory, we calculate transition rates based
on the relaxation induced coupling. Then we present a dis-
cussion about relaxation induced interference. In Sec. 3,
we discuss the competition of two interferences, one is RII
and another is the primary LZS interference. By tuning the
relaxation strength and the characteristic frequency of en-
vironment bath, we show different interference patterns in
saturated and unsaturated cases. Finally, we compare our
model with the phenomenological relaxation theory. In
Sec. 4, we study Rabi-oscillation-induced interference in
the presence of the weak monochrome field. Then we dis-
cuss new interference mechanism and its application in
quantum state manipulation. In Sec. 5, we discuss EII in
superconducting qubits.
2. Relaxation induced interference
We consider a two-level system described by the Ha-
miltonian in the presence of the environment bath with a
continuum of electromagnetic modes in equilibrium at
temperature T (we set = = 1Bk )
†( ) 1= ( )
2 2 2x z i i zi
tH a a∆ ε
− σ − σ − λ + σ −∑
†( ) ,
2 i i x Bia a Hφ
− λ + σ +∑ (1)
where xσ and zσ are Pauli matrices, 0( ) = cost A tε ε + ω ,
= |1 1| | 0 0 |zσ 〉〈 − 〉〈 , = |1 0 | | 0 1 |xσ 〉〈 + 〉〈 . †= ( 1/2)B i iiH a a∑ω +
is the bath Hamiltonian, †
ia and ia denote the boson crea-
tion and annihilation operators corresponding to the fre-
quency iω for the bath, and iλ expresses the oscillator
coupling constant. φ represents the transverse interaction
strength. It should be mentioned that Eq. (1) can be applied
in a general TLS. If there is no crossover structure, the
tunnel coupling ∆ becomes zero.
Here, we replace the conventional expressions with the
noise operator model [36–39,46], where the environment
noise is assumed to follow the Gaussian approximation.
†= ( )i iiQ a a∑λ + is the environment operator, / 2xQ−φ σ
describes the transverse interaction between the qubit and
the multimode fields (the interwell relaxation) that causes
the energy exchange between the qubit and bath. / 2zQ− σ
describes the longitudinal interaction between the qubit
and the multimode fields (the dephasing process) dominat-
ed by the low-frequency energy from the bath. As a result
all averages can be expressed by the spectral density
1( ) = e ( ) (0) .
2
i tS dt Q t Q
∞
′ω
−∞
′ω 〈 〉
π ∫ (2)
Now we follow the process in Ref. 36. By taking an in-
teraction picture with respect to ,BH the Hamiltonian is
changed to
1
( ) ( ) ( )= ,
2 2 2 2x z z x
t Q t Q tH ∆ ε φ
− σ − σ − σ − σ (3)
where ( ) = e eiH t iH tB BQ t Q − . Then we make a transforma-
tion to a rotation frame, such that a state vector |ψ〉 in the
interaction picture can be expressed as 0| = ( ) | ,U t ′ψ〉 ψ 〉
with
0
0
( ) = exp [ ( ) ( )] ,
2
t
z
iU t Q d
σ τ + ε τ τ
∫
650 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
where expresses the time ordering and | ′ψ 〉 is the state
vector in the rotation frame.
Here we get the effective Hamiltonian
†
0
1( ) = exp ( ) [ ( ) ( )
2
t
H t i d U t U t+ −
′ − − ε τ τ ∆ +
∫
† ( ) ( ) ( )] |1 0 | H.c.U t Q t U t+ −+ φ 〉〈 + , (4)
where
0
( ) = exp ( )
2
tiU t Q d±
± τ τ
∫ .
Then we obtain the total density matrix of the qubit system
plus the bath at the time interval t, 1( ) = ( ,0)SB t U tρ ×
†
1(0) ( ,0),SB U t× ρ where the time evolution operator in the
laboratory frame can be expressed as
†1 21 1 2 0 1 1 2 20( , ) = e ( ) ( , ) ( ) eiH t iH tB BVU t t U t U t t U t− ,
and 1 2( , ) =VU t t
1
2
exp ( ) .
t
t
i dt H t
′ ′ ′−
∫
The system reduced density matrix is ( ) = Tr [ ( )],B SBt tρ ρ
where TrB [...] denotes the trace over the environmental
degrees of freedom. When the system-bath interaction
energy is small, we assume (0)SBρ factors as a direct
product (0) = (0) ,SB Bρ ρ ⊗ρ where /= e H TBB
−ρ is the
density matrix of the bath.
Due to the weak relaxation and tunnel coupling com-
pared with the dephasing rate 2Γ , we consider the time
interval t within which the change of qubit population is
slow, but t is much larger than the dephasing time
2 2= 1/T Γ . Then we make the approximation with
( ,0)VU t by performing a perturbation expansion:
0
( ,0) 1 ( )
t
VU t i dt H t′ ′ ′≈ − ∫ .
Strong dephasing eliminates the off-diagonal elements
of ( )tρ quickly and changes the density matrix equation to
the rate equation. Considering the small time interval, i.e.
00 ( ) 0t tρ , if the noise is frequency independent (white)
in low-frequency region, we have the rate equation (see the
Appendix)
00 10 10( ) =t Wρ + Γ , (5)
where quantum tunneling ∆ resulted Landau–Zener transi-
tion rate from state |1〉 to | 0〉 is
22
2
10 2 2
0 2
( / )
= ,
2 ( )
n
n
J A
W
n
Γ ω∆
ε + ω +Γ
∑ (6)
and 01 0 10 0( ) = ( )W Wε −ε , where 2 = (0)SΓ π . On the other
hand, the transition rate induced by the interwell relaxation
from state |1〉 to | 0〉 is
2
2
10 0= ( ),
4 n
n
AJ S nφ Γ −ε − ω
ω ∑ (7)
and 01 0 10 0( ) = ( )Γ ε Γ −ε . It is clear that the dephasing
would not influence the transition in Eq. (7). In order to
clarify the physical picture, we consider an important class
of spectral densities [29,45]
| |/
/
e( ) = .
1 e
c
TS
′− ω ω
′−ω
′αω′ω
−
(8)
The spectral density at negative (positive) frequencies
corresponds to the emission (absorption) of photons to
(from) the bath. Generally, the spectral density describes
the ability of energy exchange between the qubit and bath.
First of all we consider a special situation in Eq. (7),
where the microwave amplitude = 0A , then Eq. (7) is
changed to the downward and upward relaxation rates
2 2
01 0 10 0= ( ), = ( ).
4 4
S Sφ φ
Γ ε Γ −ε (9)
Seen from the form of ( ),S ′ω the bath satisfies the Eins-
tein relation with a larger downward relaxation rate
/010 01/ = e .T−εΓ Γ (10)
01 10( )Γ Γ corresponds to the process that qubit is spon-
taneously relaxed down (up) from state | 0〉 to |1〉 (from
Fig. 1. (Color online) (a) Schematic energy diagram of a driven
two-level system. The dotted curve represents the strong driving
field cosA tω . The field through the tunnel coupling ∆ forms a
LZS interference, exchanging photons with the qubit. (b) Quan-
tum tunnel coupling exists between states | 0〉 and |1〉 . The inte-
raction between a qubit and an electromagnetic system (such as
the environment bath or a single-mode electromagnetic field)
would form new couplings between the two states.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 651
Lingjie Du, Yang Yu, and Dong Lan
state |1〉 to | 0〉 ) accompanied by emitting (absorbing)
photons with energy 0ε to (from) environment bath. Equa-
tion (8) demonstrates that for the low temperature
0.5 cT ω there is a maximum downward relaxation rate
near the detuning 0 = cε ±ω . Then at the characteristic fre-
quency cω , the bath can absorb photons from the qubit
fastest. For the high temperature, the downward relaxation
rate will decrease monotonically with the detuning. From
Eqs. (8) and (10), we can know with the detuning ap-
proaching to zero, relaxation rates become constant and
equal, while with the detuning larger than cω , transition
rates gradually become zero. Therefore even if the temper-
ature does not satisfy the relation 0.5 cT ω , there still
exist the downward relaxation and smaller upward relaxa-
tion, mostly near the appropriate detuning 0 = cε ±ω .
Hence the interwell relaxations can asymmetrically couple
two states in the qubit at the detuning 0 cε ≈ ±ω . Serving
as beam splitters, the relaxations at the detuning 0 cε ≈ −ω
can split state |1〉 with a larger possibility than | 0〉 as in-
dicated in Fig. 2(a).
In the presence of microwave, i.e. 0A ≠ , the stimulated
transition will change the population distribution. Substi-
tuting ( )S ′ω in Eq. (8) into Eq. (7), we get
| |/2 02 0
01 ( )/0
( )e
= .
4 1 e
n c
n n T
n
nAJ
− ε − ω ω
− ε − ω
ε − ωφ α Γ ω −
∑ (11)
The term 01 10( )Γ Γ can be understood from Fig. 2,
where the qubit state decaying (exciting) processes with
01 10( )Γ Γ are shown. For the decaying process in Fig. 2(a),
the qubit emits a photon with energy 0ε . Part of this energy
is absorbed by the microwave field and the residue is ab-
sorbed by the bath. If the energy absorbed by the field is
larger than 0ε , i.e. 0>nω ε , the bath would emit photons
with energy 0nω− ε to the field (not notable in Fig. 2).
01Γ reaches maximum when the energy 0ε is larger than
the energy absorbed the field, and the residue energy ab-
sorbed by the bath is just cω . It is similar to the exciting
process in Fig. 2(b). 10Γ reaches maximum when the
energy provided by the field is larger than the energy 0ε ,
and the residue energy absorbed by bath is just cω .
Furthermore, Eqs. (7) and (11) show that the micro-
wave amplitude can modulate the transition through the
asymmetric couplings to form interference. In order to re-
veal the physical picture, we further simplify the spectral
density in Eq. (8) as
, =
( ) =
0, .
c c
c
S
′αω ω ω
′ω ′ω ≠ ω
(12)
It is clear that here the bath can only absorb photons
with the characteristic frequency cω . Moreover there is
only downward relaxation, which means the splitters at
0 = cε ω or c−ω will work only for | 0〉 or |1〉 , respective-
ly. Substituting Eq. (12) into Eq. (7), we obtain transition
rates
22
0
01
0
22
0
10
0
, = ,
=
4 0, ,
, = ,
=
4 0, .
n cc
c
n cc
c
AJ n
n
AJ n
n
ε ω + ωφ αω Γ ω
ε ≠ ω + ω
ε −ω + ωφ αω Γ ω
ε ≠ −ω + ω
(13)
At the detuning 0 = cnε ω±ω , transition rates with the
probability 2 2 ( / ) / 4c nJ Aφ αω ω are modulated by the am-
plitude. As shown in Fig. 2(a), when the system is initia-
lized in state |1〉 , the microwave sweeps the state along |1〉
until it reaches the detuning A with 0 = cε −ω at time 1t .
Then state |1〉 is split into the superposition of states | 0〉
and |1〉 , collecting different phases in two paths. After
evolving independently, they are driven back to A (time 2 )t .
|1〉 is split into | 0〉 and |1〉 again, while | 0〉 is not split. The
relative phase difference accumulated by states | 0〉 and |1〉
Fig. 2. (Color online) Schematic energy diagram of relaxation
induced interference: (a) refers to the transition from state |1〉 to
| 0〉 through A; (b) refers to the transition from state | 0〉 to |1〉
through B. The dashed red upward arrows mark the multiphoton
absorption from the driving microwave field while the dashed
green downward arrows mark the multiphoton release to the mi-
crowave. The dotted arrows describe the energy released or ab-
sorbed by the bath. Effectively, the resonant conditions are
= cnω ε + ω (from state |1〉 to | 0 )〉 and = cnω ε −ω (from state
| 0〉 to |1 )〉 . The blue region I describes effective phase differ-
ence contributing to the interference at A. The yellow region II
expresses the phase eliminated by the bath. The green region III
expresses the effective phase difference contributing to the inter-
ference at B.
652 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
(marked with the blue region I plus yellow region II) re-
sults in the interference. We call this case relaxation induc-
ed interference.
However, according to the simplified spectral densities
in Eq. (12), the second term in Eq. (A.10) in Appendix can
be changed to
( )
2
( )( )2 0 1 21 2
0 0
e ,
4
t t
i n t tA cn
n
dt dt d J ε + ω+ω −
ω
φ α ′ω∑∫ ∫ ∫ (14)
which describes that the multimode fields transversely
produce a phase accumulation ci tω in the transition from
state |1〉 to | 0〉 . Then the effective phase difference contri-
buting to the interference at A is qa∆θ (blue region I) since
the bath eliminates part of the phase (yellow region II).
At A, the interference phase
2
|0 |1
1
= ( ),
t
qa c
t
dt 〉 〉∆θ ε − ε +ω∫
where |1 = ( ) / 2t〉ε −ε and |0 = ( ) / 2t〉ε ε , is modulated by
the amplitude. The interference condition = 2qa n∆θ π or
2 nπ + π would make the interference effect maximum or
minimum. The total phase gained over one period referring
to A is
|0 |1 0= ( ) 2 ( ) /A c cdt 〉 〉θ ε − ε +ω = π ε +ω ω∫ .
Then the qubit state is driven to the detuning B where
0 = cε ω and a similar procedure happens with an effective
phase difference contributing to the interference at B
(green region III). The total phase gained over one period
referring to B is
|1 |0 0= ( ) = 2 ( ) /B c cdt 〉 〉θ ε + ω − ε π −ε +ω ω∫ .
The stable interference can be constructed with the reso-
nant condition Aθ or = 2B nθ π . It should be mentioned
that this resonant condition is different from the interfe-
rence condition, playing the role of stabilizing the interfe-
rence effect. Therefore, the resonant transition from state
|1〉 to | 0〉 (from state | 0〉 to |1〉) would occur with the
relation shown in Eq. (13). For the actual spectral density
as shown in Eq. (8), there would be additional interference
resonant peak width which is on the order of cω . There-
fore, in order to form clear interference fringes, the micro-
wave frequency should be larger than the characteristic
frequency.
Generally, the interwell relaxation provides additional
couplings between states | 0〉 and |1〉 to form extra interfe-
rences. In the presence of tunnel coupling ∆, together with
the primary LZS interference, the two ones would simulta-
neously determine the final qubit population. In addition,
due to asymmetrical upward and downward relaxation
transitions, there are interesting phenomena which will be
discussed in next section.
3. Competition of two interferences
In order to study RII clearly, firstly we assume there is
no crossover structure, i.e. = 0∆ , to separate this interfe-
rence from the primary one. Then the stationary population
in state | 0〉 in RII is
10 0
10 0 01 0
( )
= ,
( ) ( )
S
Γ ε
Γ ε + Γ ε
(15)
where 01Γ and 10Γ are determined by Eq. (11). In the case
cω ω , at the detuning 0 = cnε ω−ω , multiphoton reso-
nant fringes of the transition rate 10 0( )Γ ε are observable
in Fig. 3(a). Due to asymmetric transition rates, population
inversion also emerges. The modulation of the amplitude
results that interference fringes are the Bessel dependence
on the driving amplitude. Moreover, the peak width of
spectral density yields different fringes from those of the
primary LZS interference. In the other case < cω ω , the
resonant peak width in RII is larger than ω . Therefore, the
individual resonances and Bessel function dependence are
no longer distinguishable and merge into a continuous band
[Fig. 3(b)]. Population inversion also vanishes. In Fig. 3(a)
the temperature is larger than the characteristic frequency
while in Fig. 3(c) the temperature is much smaller than the
characteristic frequency. Besides, although in the high tem-
perature there is not maximum downward relaxation, it is
obvious that patterns in Fig. 3(a) and (c) are similar, which
agrees the discussion in Sec. 3.
3.1. Population dynamics
Then we return to the case in the presence of ∆. In or-
der to describe the time evolution of the qubit population,
we employ rate equations, in which the population obey
01 01 00 10 10 11 00( ) ( ) ( ) ( ) = ( ),W t W t t− + Γ ρ + + Γ ρ ρ
00 11( ) ( ) = 1,t tρ +ρ (16)
where 01W and 10W are defined by Eq. (6), 01Γ and 10Γ
are defined by Eq. (11). Therefore, RII and the primary one
would contribute to the final qubit population together. The
population in state | 0〉 is
10 0 10 0
00
10 0 10 0 01 0
( ) ( )
( ) =
( ) 2 ( ) ( )
W
p t
W
ε + Γ ε
+
Γ ε + ε + Γ ε
0 10 0 10
10 0 10 0 01 0
( )
tanh
2 ( ) 2 ( ) ( )
W
T W
ε ε + Γ + − × Γ ε + ε + Γ ε
{ }10 0 10 0 01 0exp [ ( ) 2 ( ) ( )] .W t× − Γ ε + ε + Γ ε (17)
Here we assume at the initial time the qubit is in equili-
brium with its environment.
If the system dynamics time is long enough that the po-
pulation is stationary, the population in state | 0〉 can be
described by
10 0 10 0
00
10 0 10 0 01 0
( ) ( )
= .
( ) 2 ( ) ( )
W
p
W
ε + Γ ε
Γ ε + ε + Γ ε
(18)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 653
Lingjie Du, Yang Yu, and Dong Lan
RII, dependent on the characteristic frequency cω , would
greatly influence the final population, and hence the cha-
racteristic frequency cω would determine the final popula-
tion distribution. 10Γ 01( )Γ is on the order of 2
cω φ α ,
which is always smaller than 10W 01( )W at the region
0>A ε . Therefore the stationary population is dominated
by the symmetric transition rates 10W and 01W thus near-
ly 0.5. At the region 0<A ε , 10W 01( )W can be neglected
so that RII would dominate the population.
In the case cω ω , for the increasing relaxation
strength, i.e. from 2 2
2/cφ αω ∆ Γ to 2 2
2/cφ αω ∆ Γ ,
at the region 0>A ε population inversion can be more pro-
nounced [Figs. 4(a) and (b)] while at the region 0<A ε the
population would be nearly 0.5 as indicated by Fig. 3(a). In
Fig. 3. (Color online) The stationary population of relaxation induced interference. The pattern is obtained from Eq. (15). (a) The characte-
ristic frequency /2 = 0.05cω π GHz with the temperature 20 mK. Features of population inversion and periodical modulation are notable.
(b) The characteristic frequency /2 = 6cω π GHz with the temperature 20 mK. (c) The characteristic frequency /2 = 0.05 GHzcω π with the
temperature 2·10–5 mK. The driving frequency / 2 = 0.6ω π GHz.
Fig. 4. (Color online) Calculated final qubit population versus energy detuning and microwave amplitude. (a) The stationary interfe-
rence pattern in the weak relaxation situation. The parameters we used are the driving frequency / 2 = 0.6ω π GHz, the dephasing rate
2 / 2Γ π = 0.06 GHz, the couple tunneling / 2 = 0.013∆ π GHz, 2 =αφ 0.0002, the temperature is 20 mK, and the characteristic fre-
quency / 2 = 0.05cω π GHz. The periodical patterns of RII can be seen, although not clear. (b) The stationary interference pattern in the
strong relaxation situation with 2 = 0.02αφ and / 2 = 0.05cω π GHz. Since the relaxation strength is stronger, the periodical interfe-
rence patterns are more notable. (c) The stationary interference pattern in the weak relaxation situation with 2 = 0.000002αφ and
/ 2 = 6cω π GHz. (d) The stationary interference pattern in the strong relaxation situation with 2 = 0.0002αφ and / 2 = 6cω π GHz.
(e) The unsaturated interference pattern in the weak relaxation situation. The system dynamics time = 0.5t µs. The characteristic fre-
quency / 2 = 0.05cω π GHz, 2 = 0.0002αφ . (f) The unsaturated interference pattern in the weak relaxation situation. The system dy-
namics time = 0.5t µs. The characteristic frequency / 2 = 6cω π GHz, 2 = 0.000002αφ . (g) The unsaturated interference pattern in the
strong relaxation situation. The system dynamics time = 0.5t µs. The characteristic frequency / 2 = 0.05cω π GHz, 2 = 0.02αφ .
(h) The unsaturated interference pattern in the strong relaxation situation. The dynamics time = 0.5t µs. The characteristic frequency
/ 2 = 6cω π GHz, 2 = 0.0002αφ . The other parameters used in these figures are the same with those in Fig. 4(a).
654 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
the case < cω ω , with the relaxation strength 2
cφ αω in-
creasing, at the region 0<A ε the population would be
nearly 0 as indicated by Fig. 3(b) while at the region
0>A ε the visibility of RII improves, but with no popula-
tion inversion, as shown in Figs. 4(c) and (d).
For the short system dynamics time, i.e. 2
2 /t Γ ∆ ,
the population in state | 0〉 is described by Eq. (17). The
relaxation strength determines the time the bath needs to
excite the qubit, and becomes more important in determin-
ing the qubit population. For the weak relaxation strength,
i.e. 2 2
2/cφ αω ∆ Γ , the bath has no enough time to ex-
cite the qubit. With the characteristic frequency increasing,
there is little difference between interference patterns, as
shown in Figs. 4(e) and (f). For the strong relaxation
strength, i.e. 2 2
2/cφ αω ∆ Γ , RII emerges, and results in
clearer difference between interference patterns for various
characteristic frequencies [Figs. 4(g) and (h)].
Therefore, from the short time dynamics, we can obtain
the information of the relaxation strength. Furthermore,
from the long time dynamics, the effect of the characteris-
tic frequency emerges.
3.2. Phenomenological relaxation theory and discussions
If we take an average approximation with 1/ siniA tω ω −
2/ siniA t− ω ω in the second term of Eq. (A.5) in Appen-
dix, the final expressions of transition rates in Eq. (7) will
be changed to
2 2
10 0 01 0= ( ), = ( ).
4 4
S Sφ φ
Γ −ε Γ ε (19)
Then assuming 01 = constΓ , we obtain /010 01= e .T−εΓ Γ
This phenomenological theory is consistent with the exist-
ing experimental results [35]. Then we substitute Eq. (19)
into Eqs. (17) and (18), obtaining the stationary and unsa-
turated population.
Now we discuss the valid region of the approximation.
When the relaxation is weak, i.e. 2
01 2/ ,Γ ∆ Γ the qubit
population in state | 0〉 in the short time dynamics
2
2( /t ≈ Γ ∆ ) can be calculated as shown in Fig. 5(a). For
the model of RII, the weak relaxation strength
2 2
2/cφ αω ∆ Γ in the short time dynamics 2
2( / )t ≈ Γ ∆
would lead to the population in state | 0〉 as shown in Figs.
4 (e) and (f). There is almost no difference between two
theories in the weak relaxation and short time dynamics.
For the long driving time, the stationary situation will
be reached. Shown in Fig. 5(b) is the population in state
| 0〉 obtained from the phenomenological theory. It is no-
ticed that the stationary situation for the phenomenological
theory does not agree with the long time dynamics of RII,
as shown in Figs. 5(b), 4(a), and 4(c). But further from the
detail comparison in Fig. 5(c) we can find that actually in
an intermediary time scale the phenomenological theory is
still a good approximation. However, if the time further
increases, the difference between two models will be re-
markable.
Then we consider the strong relaxation, i.e.
2
01 2/Γ ∆ Γ . Although the relaxation is in the same or-
der of magnitude with 2
2/∆ Γ , it can still be much smaller
than 2Γ . Shown in Figs. 5(d) and (e) are the results of the
phenomenological theory in the short time dynamics
Fig. 5. (Color online) (a) The unsaturated interference pattern obtained by phenomenological relaxation theory. The system dynamics time
is 0.5 µs, 01 / 2 =Γ π 0.000008 GHz. (b) The stationary interference pattern obtained by the phenomenological relaxation theory.
01 / 2 = 0.000008 GHz.Γ π (c) Comparison of the results of two theories. The blue dashed line expresses the population in | 0〉 with the
characteristic frequency / 2 = 6cω π GHz, the microwave amplitude is fixed at 8 GHz, 2 = 0.000002φ α and the system dynamics time
= 13t µs, the red dotted line expresses the population with the characteristic frequency / 2 = 0.05cω π GHz, the microwave amplitude is
fixed at 8 GHz, 2 = 0.0002φ α , and the system dynamics time =t 16 µs. The black line uses the stationary result with the phenomenolog-
ical relaxation theory with 01 / 2 = 0.000008Γ π GHz, and the microwave amplitude is fixed at 8 GHz. (d) The unsaturated interference
pattern obtained by phenomenological relaxation theory. The evolution time is 0.5 µs, 01 / 2 = 0.001 GHz.Γ π (e) The stationary interfe-
rence pattern obtained by phenomenological relaxation theory. 01/2 = 0.001 GHz.Γ π Other parameters are identical with those of Fig. 4(a).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 655
Lingjie Du, Yang Yu, and Dong Lan
2
2( / )t ≈ Γ ∆ and in the stationary case, respectively.
Comparing Fig. 5(d) with Figs. 4(g) and (h), we can find
the disagreement of two models is large in the short time
dynamics. In the stationary case, the disagreement is larg-
er, as shown in Figs. 4(b), 4(d) and 5(e). This indicates that
the phenomenological theory is only a good approximation
of RII in the weak relaxation case.
On the other hand, the results of the phenomenological
theory in Eq. (19) follow the Einstein relation with a larger
downward relaxation rate. If we use Eqs. (6) and (19) in
rate equations Eq. (16), larger downward relaxation rate
would make more population exist at lower state and popu-
lation inversion would not emerge. This is the result of
well-known Einstein spontaneous and stimulated emission
(ESSE) theory.
Actually, for the weak driving A ω , the transition
rates Eq. (11) would be changed to the forms of Eq. (19).
For the strong periodic driving, Eq. (19) would be replaced
by Eq. (11). The relaxation described by Eq. (11) will form
the asymmetric transitions and produce population inver-
sion, breaking the Einstein relation, as well as ESSE theory.
As shown in Figs. 3 and 4, at some detunings, the upward
relaxation is larger than the downward one and the popula-
tion at higher state is larger than 0.5. Therefore, in the pre-
sence of the environment bath, ESSE theory is only justified
in the weak driving. Since RII is not dependent on the cross-
over, these conclusions are applicable to a general TLS.
It should be mentioned Wilson et al. [47] demonstrated
population inversion can occur in a TLS of a single Coop-
er-pair box in the dressed state picture. In their system,
weak driving, not strong driving will produce population
inversion. Furthermore, population inversion in their sys-
tem is dependent on the tunnel coupling (crossover). With-
out the crossover structure, i.e. = 0∆ , population inversion
will not emerge. These results are different from ours.
4. Rabi-oscillation-induced interference
In above sections, we investigate RII which is produced
by the interaction of the qubit and electromagnetic bath.
The bath can be considered as multimode electromagnetic
fields. However, the characteristic frequency and coupling
strength are dominated by the environment thus uncontrol-
lable, which restrict the applications of EII. Therefore, in
this section we will introduce a more controllable situation.
For a clear physical picture, we apply the monochrome
high-frequency weak field with only the longitudinal com-
ponent, which would interact with the qubit as shown in
Fig. 6. The resulted Rabi oscillations has been demonstrat-
ed in artificial mesoscopic systems, such as superconduct-
ing qubits, serve as a basic method to manipulate quantum
states [48,49].
In the following discussions, although the effect of the
inevitable environment bath still needs to be considered,
for a clear physical picture we will replace RII by the phe-
nomenological relaxation theory discussed in Sec. 3, not
treating it as multimode fields as discussed in former sec-
tions. Then the Hamiltonian (1) is changed to
( ) ( )= ,
2 2 2 2x z z z B
t t QH H∆ ε ε
− σ − σ − σ − σ +
(20)
where ( ) = cos ( )t A tε ω
describes the weak field, A and
ω are the amplitude and frequency of the weak field. Now
we make a transformation to an interaction picture with
respect to ( ) / 2B zH t− ε σ . Therefore, the Hamiltonian is
rewritten to
2
( ) ( )= ( / )e
2 2 2
in t
z z n
t Q tH J A − ω
+
ε ∆ − σ − σ − ω σ − ∑
( / )e
2
in t
nJ A ω
−
∆ − ω σ ∑
. (21)
Due to the weak driving, i.e. <A ω
, we only consider the
case = 1n ± in the sum of the above expression. Now the
Hamiltonian is
1
2
( / )( ) ( )= [e e ]
2 2 2
i t i t
z z
J At Q tH − ω ω
+
∆ ωε
− σ − σ − − σ −
1( / )
[e e ] ,
2
i t i tJ A ω − ω
−
∆ ω
− − σ
(22)
where the last two terms would lead to familiar Rabi oscil-
lations [42]. Then we repeat the process in Appendix. Con-
sidering the initial condition (0) |1 1|,ρ = 〉 〈 we have the
population in state | 0〉
1
2
( )2 2
1
00 1 2
0 0
( / )
( ) = e
4
t
t
i dt tJ At dt dt
ε τ τ∫∆ ω
ρ ×∫ ∫
† 1 11 1Tr { 0 |{[ ( ) ( )(e e )] } |1 1|i t i t
B U t U t ω − ω
− + −× 〈 − σ 〉〈
† 2 22 2{[ ( ) ( )(e e )] } | 0 }.i t i t
B U t U t − ω ω
+ − +⊗ ρ − σ 〉 (23)
Fig. 6. (Color online) Schematic energy diagram of a strongly
driven two-level system interacting with a weak single-mode
field. The green solid curve represents the weak field, forming
effective coupling between states | 0〉 and |1〉 .
656 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
We accept the rotating-wave approximation with the weak
and strong fields, respectively, obtaining
( )
2 2
21
00 1 2
0 0
( / )
( ) =
4
t t
A
n
n
J At dt dt J ω
∆ ω
ρ ×∑∫ ∫
( )( ) † †0 1 2 2 2 1 1e ( ) ( ) ( ) ( )i n t t U t U t U t U tε + ω+ω −
+ − − +× 〈 〉 +
( )
2 2
21
1 2
0 0
( / )
4
t t
A
n
n
J A dt dt J ω
∆ ω
+ ×∑∫ ∫
( )( ) † †0 1 2 2 2 1 1e ( ) ( ) ( ) ( ) .i n t t U t U t U t U tε + ω−ω −
+ − − +× 〈 〉 (24)
If the noise spectral density is dominated by the white
low-frequency noise, we have
22 2
21
00 2 2
0 2
( / )( / )
( ) =
2 ( )
n
n
J AJ At
n
Γ ω∆ ω
ρ +
ε + ω+ω +Γ
∑
22 2
21
2 2
0 2
( / )( / )
,
2 ( )
n
n
J AJ A
n
Γ ω∆ ω
+
ε + ω−ω +Γ
∑
(25)
where 2Γ has the same definition with Eq. (6). Consider-
ing the small time interval, i.e. 00 10( )t Wρ ≈ , we obtain the
transition rate from state |1〉 to | 0〉
22 2
21
10 2 2
0 2
( / )( / )
=
2 ( )
n
n
J AJ AW
n
Γ ω∆ ω
+
ε + ω+ω +Γ
∑
22 2
21
2 2
0 2
( / )( / )
,
2 ( )
n
n
J AJ A
n
Γ ω∆ ω
+
ε + ω−ω +Γ
∑
(26)
with 01 0 01 0( ) = ( )W Wε −ε . In order to ensure the validity
of Eq. (26), the frequency of the weak field ω needs to be
much larger than the driving frequency ω of the strong
field.
In the absence of the strong field, i.e. = 0A , the Hamil-
tonian in Eq. (20) is used to describe Rabi oscillations at
the detunings 0 =ε ±ω with the Rabi frequency 1( / ).J A∆ ω
Rabi oscillations replaces the primary tunnel coupling ∆ at
the detunings 0 = 0ε with two couplings at the detuning
A′ ( 0 =ε ω ) and B′ ( 0 =ε −ω ), 1= = ( / )A B J A′ ′∆ ∆ ∆ ω .
Unlike the interwell relaxation, Rabi oscillations can con-
struct symmetrical couplings between states |1〉 and | 0〉 ,
serving as a beam splitter working for both |1〉 and | 0〉 .
Under the strong field the couplings at A′ and B′ can form
effective crossovers to produce interferences, as shown in
Figs. 7(a) and (b). We call this case Rabi-oscillation-
induced interference. If 2>ω Γ , the detunings A′ and B′
are well separated. It is important to notice that the position
Fig. 7. (Color online) (a) and (b) Schematic energy diagram of Rabi-oscillation-induced interference: (a) describes the transition from
state |1〉 to | 0〉 ; (b) describes the transition from state | 0〉 to |1〉 . (c), (d), and (e) The interference pattern of population in state | 0〉
obtained from Eqs. (26), (27), and (28), respectively. The parameters used here are /2 = 2ω π GHz, / = 0.9,A ω 01/2 = 0.000008Γ π GHz
and the temperature is 20 mK. Other parameters of the qubit are identical with Fig. 4(a).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 657
Lingjie Du, Yang Yu, and Dong Lan
and strength of couplings can be controlled by the frequen-
cy and amplitude of the weak field. Moreover, the expo-
nent forms in Eq. (24) are similar with that of Eq. (14).
That means the weak field also produces a phase accumula-
tion ,i tω which would influence the interference phase and
total phase accumulation, just as the multimode fields do.
First of all, we discuss the transition from state |1〉 to
| 0〉 [Fig. 7(a)]. When the initial state is |1〉 , the microwave
sweeps the state along |1〉 (black solid arrow). As reaching
the detuning B′, state |1〉 is split into the superposition of
states |1〉 and | 0〉 , collecting different phases in two paths.
When reaching the detuning A′, state |1〉 is split into |1〉
and | 0〉 again. Evolving independently, states |1〉 and | 0〉
accumulate the same relative phase difference correspond-
ing to couplings A′ and B′, marked with the region con-
taining the blue region I, and the yellow region II. Howev-
er, the effective phase difference contributing to the
interference at A′ is the blue region I since the weak field
eliminates part of the phase (yellow region II). In a similar
way, the effective phase difference contributing to the in-
terference at B′ is the region containing the blue region I,
the yellow region II, and the green region III. The total
phase A′θ gained over one period referring to A′ is
02 ( ) /π ε +ω ω , while the total phase B′θ gained over one
period referring to B′ is 02 ( ) /π −ε +ω ω . The resonant
transition from state |1〉 to | 0〉 would occur at
0 = nε ω−ω or 0 = nε ω+ω .
The transition from state | 0〉 to |1〉 is similar [Fig. 7(b)].
The resonant transition from state | 0〉 to |1〉 would occur
at 0 = nε ω−ω or 0 = nε ω+ω . This result is the same
with that of transition from state | 0〉 to |1〉 , making that
there is no population inversion in this case.
The energy conservation is satisfied for these transitions
as shown in Fig. 7(a) and (b). For the transition from state
|1〉 to | 0〉 , if the transition is through the coupling at A′,
besides n photons with frequency ω absorbed, the total
energy 0=nω ε +ω is larger than the energy spacing 0ε so
that the excess energy ω flows into the weak field. It is
similar for other cases.
In order to describe the evolution of qubit population in
the presence of two couplings, we employ rate equations,
in which the qubit population obeys Eq. (16). But here,
10W and 01W are described by Eq. (26), with 01/2Γ π =
0.000008= GHz, /010 01= e T−εΓ Γ , and the temperature
= 20T mK. Figure 7(c) is the calculated qubit population
versus energy detuning and microwave amplitude. To un-
derstand this complicate pattern, firstly we consider the
only effect of the coupling at A′ with transition rates be-
tween states |1〉 and | 0〉
22 2
21
10 01 2 2
0 2
( / )( / )
= = .
2 ( )
n
n
J AJ AW W
n
Γ ω∆ ω
ε + ω−ω +Γ
∑
(27)
Substituting Eq. (27) into rate equations, we have the
contour plot of the qubit population in state | 0〉 as func-
tions of energy detuning and microwave amplitude, shown
in Fig. 7(d).
Then we only consider the effect of coupling at B′ with
transition rates between states |1〉 and | 0〉
22 2
21
10 01 2 2
0 2
( / )( / )
= = .
2 ( )
n
n
J AJ AW W
n
Γ ω∆ ω
ε + ω+ω +Γ
∑
(28)
With Eq. (28), we obtain the population in state | 0〉 as
shown in Fig. 7(e). Having addressed two couplings sepa-
rately, we compare the results with Fig. 7(c) and find that
they match very well. The accordance gives a clear physi-
cal picture that interferences based on couplings at A′ and
B′ construct the final interference pattern.
Rabi oscillations changes the primary coupling at
0 = 0ε to two couplings which are controllable in the posi-
tion and coupling strength. Then two couplings result into
ROII so as to form a complicate structure shown in
Fig. 7(c). If we further increase the amplitude of the weak
field, multiphoton Rabi oscillations would emerge and
more couplings can be formed.
For EII, the longitudinal monochrome field forms the
channels which depend on the tunnel coupling ∆, such as
Rabi oscillations in Eq. (26). On the contrary, the trans-
verse multimode fields form the channels which are inde-
pendent on ∆, such as the interwell relaxation. Therefore,
if we employ a transverse monochrome field, the qubit-
field interaction will emerge in off-diagonal terms of the
Hamiltonian Eq. (1) and Rabi oscillations can still be pro-
duced but independent on ∆. As a result, even in the sys-
tem without the double well structure, such as a phase qu-
bit, ROII can still be produced.
5. Superconducting qubits
So far we have studied EII in an idealized TLS. We
now consider the situation in the realistic system. An ex-
ample of such case is a superconducting flux qubit, which
have been studied considerably both theoretically and ex-
perimentally [8,50].
The flux qubit is incorporated in a superconducting
loop with three small-capacitance Josephson junctions.
When the external magnetic flux bias through the loop
00.5Φ ≈ Φ , the classical potential energy has a double well
structure parameterized by 0= 0.5dcδΦ Φ − Φ and the ki-
netic part of Hamiltonian provides quantum tunnel coupl-
ing ∆ between two wells. In the millikelvin temperature
20 mK, energy-level quantization emerges and the
states localized in each well are separated by the energy
intervals on the order of the plasma frequency pω , which
is usually much larger than ∆. 0 = 2 p dcIε δΦ is the dc
energy detuning of states | 0〉 and |1〉 , with pI the maxi-
mum persistent current in the loop. The two lowest states
in each well, | 0〉 and |1〉 , form an effective two-level sys-
tem, which can be manipulated by application of micro-
wave pulses.
658 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
For a periodically driven flux qubit with the two-lowest
states | 0〉 and |1〉 in the right and left well, respectively,
(Fig. 1) as we drive the qubit with the microwave
= cosac rf tΦ Φ ω , where rfΦ is the microwave flux ampli-
tude, the time-dependent magnetic flux ( ) = dc actδΦ δΦ +Φ
controls the qubit and the time-dependent energy detuning
of states | 0〉 and |1〉 can be described as
0( ) = cost A tε ε + ω . = 2 p rfA I Φ is the microwave ampli-
tude in the unit of magnetic flux. To produce ROII, we can
apply another weak field as shown in Sec. 4, with extra ex-
ternal magnetic flux = cos ( )rf tΦ Φ ω
and = 2 p rfA I Φ
. This
is similar with circuit quantum electrodynamics (C-QED)
where qubits are strongly coupled to quantum oscillators
[51–53], especially the spectroscopy [54–56]. However
C-QED is also hard to change the parameters and here the
applied monochrome weak field can realize more flexible
control.
In a recent experiment [57], two longitudinal strong mo-
nochrome fields were applied simultaneously on the flux
qubit, just similar with the discussion in Sec. 4. Interesting
phenomena different from our results were observed, since
the amplitudes of two fields were modified together and
there did not exist the stable coupling resulted by Rabi os-
cillations. Furthermore, the frequencies in two fields were
comparable. Therefore, the phenomenon predicted here does
not emerge in that experiment. In another experiment [17],
two longitudinal weak monochrome fields were applied on
the flux qubit. In this case, two fields interacted with the
qubit and the energy transfer described above emerged.
But the small amplitude made the interference we predict-
ed invisible. Nevertheless, both experiments demonstrate
the feasibility of our model.
ROII offers an alternative means to manipulate the qu-
bit. It can be applied further in the time-dependent interfe-
rence which can coherently manipulate the qubit state on
the Bloch sphere with a single crossing [24,58] or conti-
nuous driving [20,57]. In addition, if the strong periodic
driving is further replaced to the linear adiabatic passage,
the interference is changed to Stark-chirped rapid adiabatic
passages [59,60] which could serve as elementary logic
gates for quantum computing. The adiabatic passages
should be very slow compared to Rabi oscillations and fast
compared to the coherence time. However, in our model,
with a single crossing or continuous driving, the interfe-
rence nature [18] allows faster manipulation than adiabatic
passages, which is important for the short coherence time.
In addition, if we replace the multimode eletromagnetic
field with the quantized single-mode field, such as micro-
wave resonators [61], and optical cavity [62], the situation
would be more interesting. Caused quantum noise would
not only influence xσ to produce RII, but also have effects
on zσ to cause unconventional Mach–Zehnder interfe-
rence just as shown in Ref. 34. This case can be used to
produce controllable cooling and population inversion.
6. Conclusion
We have proposed and comprehensively studied elec-
tromagnetically induced interference in quantum two-level
systems based on the couplings formed by different elec-
tromagnetic systems.
First of all, we consider an electromagnetic bath which
is familiar and unavoidable. We show the transverse mul-
timode fields form the interwell relaxation, which can
asymmetrically couple two qubit states at the detuning
0 = cε ±ω . In the presence of the strong driving, the relaxa-
tion induced couplings can generate the interference, i.e.
RII. Considering the rate equations, we show that the pri-
mary LZS interference resulted by ∆ and RII interact to-
gether.
Our model provides a deeper understanding to the ef-
fect of the environment and shows the way to determine
the relaxation strength and characteristic frequency of the
environment bath. For the weak relaxation, we note that
the phenomenological relaxation theory can be a good ap-
proximation. But for the strong relaxation, the phenomeno-
logical theory cannot be applicable. Moreover we show
that RII can break the ESSE theory under the strong driv-
ing. As a result, population inversion can be produced in a
strong driven TLS by the relaxation and the ESSE theory
can only be justified under the weak driving.
Furthermore, we consider a longitudinal monochrome
weak field, which interacts with the qubit through Rabi
oscillations. We show Rabi oscillation changes the primary
tunnel coupling into two symmetric beam splitters at
0 =ε ±ω . Different from RII, ROII here can be controlled
in the position and strength of the beam splitters by the
amplitude and frequency of the weak field. ROII provides
an alternative way to manipulate qubits in the strongly dri-
ven region, and characterize the qubit as well as the exter-
nal fields. It provides the potential of further application in
the time-dependent interference which can coherently ma-
nipulate the qubit state on the Bloch sphere with a single
crossing or continuous driving. The models used here can
be extended to other systems with double well such as
quantum dots [22], or even without double well such as
phase qubits. Relative experiments [58] have been already
performed to test some predictions of this work.
7. Appendix
The system reduced density matrix can be expressed as
†
1 1( ) = Tr [ ( ,0) (0) ( ,0)]B Bt U t U tρ ρ ⊗ρ =
{ †
1 2 1 1
0 0
1= Tr [ ( ) ( )
4
t t
B dt dt U t U t+ −
∆ +
∫ ∫
}
1
0
( )
†
1 1 1( ) ( ) ( )]e |1 0 | H.c. (0)
t
i d
U t Q t U t
− ε τ τ
+ −
∫
+ φ 〉〈 + ρ ⊗
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6 659
Lingjie Du, Yang Yu, and Dong Lan
{ † †
2 2 2 2 2[ ( ) ( ) ( ) ( ) ( )]B U t U t U t Q t U t+ − + −⊗ρ ∆ + φ ×
}
2
0
( )
e |1 0 | H.c. .
t
i d− ε τ τ
∫
× 〉〈 + (A.1)
The strong dephasing makes the off-diagonal elements of
( )tρ decay completely [37], and hence we focus on the
diagonal part of the density matrix. Then we assume the
initial condition (0) = |1 1|ρ 〉〈 , and obtain 00 ( )tρ
†
00 1 2 1 1
0 0
1( ) = Tr { 0 |{[ ( ) ( )
4
t t
Bt dt dt U t U t− +ρ 〈 ∆ +∫ ∫
1
0
( )
†
1 1 1( ) ( ) ( )] e | 0 1|} |1 1|
t
i d
U t Q t U t
ε τ τ
− +
∫
+ φ × 〉〈 〉〈 ⊗
† †
2 2 2 2 2{[ ( ) ( ) ( ) ( ) ( )]B U t U t U t Q t U t+ − + −⊗ ρ ∆ + φ ×
2
0
( )
e |1 0 |} | 0 }
t
i d− ε τ τ∫
× 〉〈 〉 =
1
2
( )
2 † †
1 2 2 2 1 1
0 0
1= e ( ) ( ) ( ) ( )
4
t
t
i dt t
dt dt U t U t U t U t
ε τ τ
+ − − +
∫
〈∆ +∫ ∫
2 † †
2 2 2 1 1 1( ) ( ) ( ) ( ) ( ) ( )U t Q t U t U t Q t U t+ − − ++ φ +
† †
2 2 1 1 1( ) ( ) ( ) ( ) ( )U t U t U t Q t U t+ − − ++φ∆ +
† †
2 2 2 1 1( ) ( ) ( ) ( ) ( ) .U t Q t U t U t U t+ − − ++∆φ 〉 (A.2)
We can calculate the first two average terms in Eq. (A.2)
by expanding † ( )U t± to the second order in ( )Q t , with
† †
2 2 1 1( ) ( ) ( ) ( ) 1U t U t U t U t+ − − +〈 〉 ≈ +
2 1 2
0 01
1 ( ) ( ) ( ) ( ) ,
2
t t t
t
dt dt Q t Q t dt Q t Q t
′′ ′ ′′ ′ ′ ′ ′′+ 〈 〉 − 〈 〉
∫ ∫ ∫ (A.3)
† †
2 2 2 1 1 1 2 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .U t Q t U t U t Q t U t Q t Q t+ − − +〈 〉 ≈ 〈 〉
(A.4)
Since the last two terms in Eq. (A.2) are neglected under
the average approximation compared with the other two
terms, we obtain
2
† †
00 1 2 2 2 1 1
0 0
( ) = ( ) ( ) ( ) ( )
4
t t
t dt dt U t U t U t U t+ − − +
∆
ρ 〈 〉 ×∫ ∫
1 1
2 2
( ) ( )2
1 2 2 1
0 0
e e ( ) ( ) .
4
t t
t t
i d i dt t
dt dt Q t Q t
ε τ τ ε τ τ∫ ∫φ
× + 〈 〉∫ ∫ (A.5)
The first term in Eq. (A.5) has been discussed in Ref. 35,
where
{† †
2 2 1 1( ) ( ) ( ) ( ) = exp ( )U t U t U t U t d S+ − − + ′ ′〈 〉 ω ω ×∫
1 2( ) 21 2 1 2( ) ( )
e 1 2 sin cos / .
2 2
i t t t t t ti′ω − ′ ′ω − ω + ′× − − ω
(A.6)
If the noise is frequency independent (white) in low-fre-
quency region /2 1/t′ω π , ( ) (0)
'
S Sω ≈ , Eq. (A.6) can be
integrated to yield
| |† † 2 2 12 2 1 1( ) ( ) ( ) ( ) = e ,t tU t U t U t U t −Γ −
+ − − +〈 〉 (A.7)
where 2 = (0)SΓ π . Then we focus on the second term in
Eq. (A.5) which is related to the interwell relaxation. With
the Bessel functions
sin
e = e ,
Ai t in t
n
n
AJ
ω ωω
ω ∑ (A.8)
where ( / )nJ A ω are Bessel functions of the first kind, for
the high frequency driving ( / )nJ Aω ∆ ω , where n is the
closest integer to 0 /ε ω , we apply the rotating-wave ap-
proximation to make
( )
sin sin1 2 ( )2 1 2e e ,
A Ai t i t in t tA
n
n
J
ω − ω ω −ω ω
ω≈ ∑ (A.9)
where we neglect the nonresonant terms with exponent
oscillating quickly with high frequency comparing to the
time scale of the dynamics 1/ [ ( / )]nt J A∆ ω . Hence we
have
( )
2
( )( )2 0 1 200 1 2
0 0
( ) = e
4
t t
i n t tA
n
n
t dt dt J ε + ω −
ω
∆
ρ ×∑∫ ∫
† †
2 2 1 1( ) ( ) ( ) ( )U t U t U t U t+ − − +× 〈 〉 +
( )
2
( )( )2 0 1 21 2 2 1
0 0
e ( ) ( ) .
4
t t
i n t tA
n
n
dt dt J Q t Q tε + ω −
ω
φ
+ 〈 〉∑∫ ∫
(A.10)
With the inverse Fourier transformation
( )2 12 1( ) ( ) = e ( ),i t tQ t Q t d S′− ω −′ ′〈 〉 ω ω∫ (A.11)
we find
( )
2
( )( )2 0 1 200 1 2
0 0
( ) = e
4
t t
i n t tA
n
n
t dt dt J ε + ω −
ω
∆
ρ ×∑∫ ∫
† †
2 2 1 1( ) ( ) ( ) ( )U t U t U t U t+ − − +× 〈 〉 +
( )
2
( )( )2 0 1 21 2
0 0
e ( ).
4
t t
i n t tA
n
n
dt dt d J S′ε + ω+ω −
ω
φ ′ ′+ ω ω∑∫ ∫ ∫
(A.12)
660 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 6
Electromagnetically induced interference in a superconducting flux qubit
For the time interval t, which is much larger than the
dephasing time, we extend the integration limits t to ∞ and
obtain
( )
2
( ) | |2 0 200 ( ) = e
4
i nA
n
n
t d J
∞
ε + ω τ−Γ τ
ω
−∞
∆
ρ τ +∑∫
( )
2 ( )2 0e ( )4
i n iA
n
n
d J d e S
∞
ε + ω τ ′ω τ
ω
−∞
φ ′ ′+ τ ω ω =∑∫ ∫
( ) ( )
22 2
22
02 2
0 2
/
= ( ),
2 4( )
n A
n
n n
J A
J S n
n ω
Γ ω∆ φ
+ −ε − ω
ε − ω +Γ
∑ ∑
(A.13)
where 1 2= .t tτ −
Acknowledgment
Thanks to Xueda Wen for useful discussions. This work
was supported in part by the State Key Program for Basic
Researches of China (2011CB922104, 2011CBA00205),
the NSFC (91021003, 11274156), the Natural Science
Foundation of Jiangsu Province (BK2010012), and PAPD.
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