Coexistence of multi-photon processes and longitudinal couplings in superconducting flux qubits

Abstract

In contrast to natural atoms, the potential wells for superconducting flux qubit (SFQ) circuits can be artificially controlled. When the inversion symmetry of the potential energy is broken, it is found that the multi-photon processes can coexist in the multi-level SFQ circuits. Moreover, there are not only transverse but also longitudinal couplings between the external magnetic fields and the SFQs with the broken inversion symmetry. The longitudinal coupling would induce some new phenomena in the SFQs. Here we show that the longitudinal coupling can result in the coexistence of multi-photon processes in the SFQs in analogue to the multi-level SFQ circuits. We also show that the SFQs can become transparent to the transverse coupling fields when the longitudinal coupling fields satisfy the certain conditions. We further show that the quantum Zeno effect can also be induced by the longitudinal coupling in the SFQs. Finally we clarify why the longitudinal coupling can induce coexistence and disappearance of single- and two-photon processes for a driven SFQ, which is coupled to a single-mode quantized field. Comment: 10 pages, 6 figures

Full text

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arXiv:1003.1671v2 [quant-ph] 14 Apr 2010
Coexistence of multi-photon processes and longitudinal couplings in superconducting flux qubits
Yu-xi Liu,1,2Cheng-Xi Yang,3and Xiang-Bin Wang3
1Institute of Microelectronics, Tsinghua University, Beijing 100084, China
2Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China
3Department of Physics, Tsinghua University, Beijing 100084, China
(Dated: April 15, 2010)
Incontrast tonatural atoms, thepotential wellsfor superconducting fluxqubit (SFQ)circuitscanbeartificially
controlled. When the inversion symmetry of the potential energy is broken, it is found that the multi-photon
processes can coexist in the multi-level SFQ circuits. Moreover, there are not only transverse but also longitu-
dinal couplings between the external magnetic fields and the SFQs with the broken inversion symmetry. The
longitudinal coupling would induce some new phenomena in the SFQs. Here we show that the longitudinal
coupling can result in the coexistence of multi-photon processes in the SFQs in analogue to the multi-level
SFQ circuits. We also show that the SFQs can become transparent to the transverse coupling fields when the
longitudinal coupling fields satisfy the certain conditions. We further show that the quantum Zeno effect can
also be induced by the longitudinal coupling in the SFQs. Finally we clarify why the longitudinal coupling can
induce coexistence and disappearance of single- and two-photon processes for a driven SFQ, which is coupled
to a single-mode quantized field.
PACS numbers: 85.25.Cp, 32.80.Qk, 42.50.Hz.
I. INTRODUCTION
Superconductingquantum circuits possess discrete energy-
levels and behave like natural atoms that the transitions be-
tween different energy levels can be induced by the electro-
magnetic fields. Thus, many experiments realized in natural
atoms can also be demonstrated by using the superconduct-
ing quantum circuits, e.g., cavity quantum electrodynamics
(e.g., in Refs. [1, 2]), dressed states (e.g., in Refs. [3–7]), elec-
tromagneticallyinducedtransparencyand adiabatic control of
quantum states (e.g., in Refs. [8–12]), and cooling for the su-
perconductingqubits (e.g., in Refs. [13–15]), the sideband ex-
citations (e.g., in Refs. [16–18]). In contrast to natural atoms,
the potential wells of superconducting flux qubit (SFQ) cir-
cuits can be changedby adjusting externallyapplied magnetic
fields, this tunability makes the SFQ circuits have many fea-
tures which cannot be shown by the nature atoms.
Innaturalatoms,sincetheinversionsymmetryofthepoten-
tial energy is given by the nature and cannot be changed arti-
ficially. Thus each eigenstate has well-defined parity and the
electric-dipole transitions can only link two eigenstates which
have different parities. However, the inversion symmetry of
the potential energy for the SFQ circuits can be controlled by
the external magnetic flux. When the inversion symmetry is
adjusted to be broken, there is no well-defined parity for each
eigenstate of the multi-level SFQ circuits, and the microwave
induced transitions between any two energy levels are pos-
sible, thus the multi-photon and single-photon processes can
coexist for such multi-level systems. This coexistence of the
multi-photon processes can be easily understood by virtue of
an example using three-level SFQ circuits. That is, the transi-
tion between the groundstate and the second excited state can
be realized via two different pathes: (i) from the groundto the
second excited states via a single-photon process; or (ii) from
the ground state to the first excited state via a single-photon,
and then from the first excited state to the second excited state
via another single-photon process [19]. This means that the
single- and two-photon processes can realize the same goal:
the transition from the ground to the second excited states,
and thus single- and two-photon processes can coexist in such
system.
The coexistence [19] of the single- and two-photon pro-
cesses in the three-level SFQ circuits with the cyclic transi-
tion, which is also called as ∆-type transition in analogue to
the Ξ-, Λ-, and V -type transitions in atomic physics or quan-
tum optics [20], has been experimentally demonstrated via
a delicate superconducting qubit-resonator circuit [21]. The
three-level SFQ circuits with ∆-type transitions can be used
to generate single photon [22, 23] and cool superconducting
qubits [13, 14]. The analysis of the inversion symmetry for
the potential energy of the SFQs [19, 24] also shows that one
of SFQs cannot be at the optimal point when the frequency
matching method is used to control the coupling between two
SFQs. Thus an auxiliary circuit or a coupler [16, 25, 26] is
necessary to make that both of SFQs can be at their optimal
points [16, 24]. Afterwards, several theoretical works [27–
30] followed proposals in Refs. [16, 24] and studied how to
control two-flux-qubit coupling using an additional nonlinear
coupler, which results in experimental studies on the control-
lable coupling [31] and engineered selection rules for the tun-
able coupling [32].
Here, we will first explain why the transverse and longitu-
dinal couplings between the microwave fields and the SFQs
can coexist, and then mainly focus on some unexplored phe-
nomena resulted from the longitudinal coupling. The trans-
verse coupling between a single-mode microwave field and
the SFQ, which can be reduced to the Jaynes-Cummings
model in the rotating wave approximation, is well studied in
the quantum optics and atomic physics [20]. However, the
model with both the transverse and longitudinal couplings is
less studied. This is because electric-dipole induced longitu-
dinal coupling does not exist in the natural atoms with well-
defined inversion symmetry. But both couplings in the SFQs
can coexist when the inversion symmetry of the potential en-

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2
?
?
?
?
?
FIG. 1: (Color online) Comparisons of the reduced magnetic flux f-
dependent transition matrix elements ?i|I|j? with i ?= j between the
loop current in Eq. (3) with solid curves and that in Refs. [19, 35]
with dashed curves for the three lowest energy levels with i, j =
0, 1, 2. Here, the different transition matrix elements are denoted
by different colors as shown in the figure. We take α = 0.8 and
EJ = 40Ecin our numerical simulations.
ergy is broken. Therefore some results, which do not exist
in the Jaynes-Cummings model, can be obtained when both
couplings coexist. Moreover, we will also study the interac-
tionbetweenthedrivenSFQs andthelow frequencyharmonic
oscillator [15, 33] when the inversion symmetry of the SFQs
is broken.
Our paper is organized as below. In Sec. II, we will briefly
review the SFQ circuits. As the complementary and gener-
alization of Ref. [19], we will also clarify some points which
were notstudied in ourearlierliteratures. Forinstance, how to
take phase transformations so that the interaction between the
SFQ circuits and the external circuit can be described via the
product of the loop current of the SFQ circuits and the exter-
nal magnetic flux, how the multi-photon processes can coex-
ist in n-level systems when the inversion symmetry is broken.
In Sec. III, we will present a Hamiltonian on the transverse
and longitudinal couplings between the magnetic fields and
the SFQs, and show how the longitudinal coupling can in-
duce the coexistence of multi-photon processes in the SFQs.
We will also demonstrate the longitudinal coupling induced
dynamical quantum Zeno effect and the transparency of the
SFQs to the transverse coupling fields. In Sec. IV, the trans-
verse and longitudinal couplings between the driven SFQ and
the low frequency harmonic oscillator (e.g., an LC circuit) is
studied. We will explore the nature of the coexistence and
disappearance of the single- and two-photon processes in the
driven SFQs. Finally, we summary our results in Sec. V.
II.
MULTI-PHOTON PROCESSES IN MULTI-LEVEL SFQ
CIRCUITS
THEORETICAL MODEL AND COEXISTENCE OF
In this section, we first briefly review the theoretical model
on the interaction between the SFQ circuit with three Joseph-
son junctions and the externally applied time-dependent mag-
netic flux. In the meanwhile, as the complementary and gen-
eralization of the results in Ref. [19], we will clarify some
points which were not studied in the former literatures (e.g.,
in Refs. [19, 34]). Then we will summarize the selection rules
and discuss the coexistence of multi-photon processes in the
multi-level systems.
A. Hamiltonian and phase transformations
Let us consider a superconducting flux qubit (SFQ) cir-
cuit, which is composed of a superconducting loop with three
Josephson junctions. As in Ref. [19] and Ref. [34], the two
larger junctions are assumed to have equal Josephson ener-
gies EJ1 = EJ2 = EJand capacitances CJ1 = CJ2 = CJ.
While for the third junction, the Josephson energy and the ca-
pacitance are assumed to be EJ3 = αEJand CJ3 = αCJ,
with α < 1. We assume that a static magnetic flux Φeand
a time-dependent magnetic flux Φ(t) are applied through the
superconducting loop. In this case, the Hamiltonian can be
given by
H =
P2
2Mp
p
+
P2
2Mm
m
+ U(ϕp,ϕm) + IΦ(t).
(1)
The potential energy U(ϕp,ϕm) of the SFQ circuit is defined
as
U(ϕp,ϕm) = 2EJ(1 − cosϕpcosϕm)
+ αEJ[1 − cos(2πf + 2ϕm)],
(2)
with the reduced magnetic flux f = Φe/Φ0and the magnetic
flux quantum Φ0. The third term IΦ(t) in Eq. (1) plays the
similar role as the electric-dipole interactions between the na-
ture atoms and the electric fields, and describes the interaction
betweentheSFQcircuitandthetime-dependentmagneticflux
provided by the external circuit. The parameter I in Eq. (1)
denotes the loop current of the SFQ circuit given by [24]
I =
αI0
2α + 1[2sinϕpcosϕm− sin(2πf + 2ϕm)],
(3)
when the time-dependent magnetic flux Φ(t) = 0, here I0=
2πEJ/Φ0. We note that Eq. (3) is different from that in the
former literatures (e.g., in Refs. [19, 35]). This difference re-
sults from different phase transformations
φp =
1
2(φ1+ φ2) +
1
2(φ2− φ1),
2πα
(1 + 2α)
Φ(t)
Φ0
,
(4)
φm =
(5)

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with the superconducting phase differences φ1and φ2of the
two identical Josephson junctions. We also use the phase
constraint condition for superconducting phase differences φi
(with i = 1, 2, 3) of the three Josephson junctions as
− φ1+ φ2+ φ3+2πΦe
Φ0
+2πΦ(t)
Φ0
= 0,
(6)
when Eqs. (2) and (3) are derived.
In the former literatures (e.g., in Refs. [19, 35]), the sec-
ond term in Eq. (4) for the phase transformations has been
neglected, thus I in Eq. (1) is the supercurrent operator of
the third Josephson junction. However, the phase transforma-
tions in Eqs. (4) and (5) are more appropriate than that used
in former literatures (e.g., in Refs. [19, 35]) when the time-
dependent magnetic flux is considered. Because in this trans-
formation,the interactionbetween the SFQ circuit and the ex-
ternal circuit can be explained by the product of the loop cur-
rent I of the SFQ circuit [24] and the external magnetic flux
Φ(t) provided by the external circuit. This is more reason-
able than the product of the supercurrent operator of the third
Josephson junction and the external magnetic flux Φ(t).
Physically, the interaction Hamiltonian between the SFQ
circuitandtheexternalmagneticfluxΦ(t) intheformerlitera-
tures(e.g.,in Refs.[19, 35])is just anapproximatedresult that
the displacement currents in the Josephson junctions are sim-
ply neglected when the loop current is calculated. Therefore,
the loop current is simply taken as one supercurrent of three
Josephson junctions. However, here, the interaction Hamilto-
nian in the fourth term of Eq. (1) is an exact result that the
displacement current for each junction is considered when the
loop current is calculated.
We should also note that the transformations applied in
Eqs. (4) and (5) do not change the basic results on the se-
lection rules and the adiabatic control of the quantum states
that were studied in Ref. [19] with neglecting displacement
currents. This can be very easily verified by using Eq. (2) and
Eq. (3), that is: (i) when f = 1/2 which is called as an opti-
mal point, the potential energy in Eq. (2) is even function of
the variables φpand φm, the supercurrentin Eq. (3) is the odd
function of the variables φmand φp, therefore the potential
energy and the supercurrent have the well-defined symmetry;
(ii) when f ?= 1/2, the inversion symmetries for both the po-
tential energy and the supercurrent in Eq. (2) and Eq. (3) do
not exist. Therefore, when the f = 1/2 or f ?= 1/2, the con-
clusions in (a) and (b) are the same as those in Refs. [19, 35].
However due to the different expressions of the loop currents
in Eq. (3) and in Refs. [19, 35], the transition matrix elements
will be re-normalized. Below we will further clarify this con-
clusion via the discussions on the selection rules and numeri-
cal calculations for the transition matrix elements.
B. Selection rules and coexistence of multi-photon processes in
n-level systems
As necessary supplementary and generalization of the re-
sults in Ref. [19] for the microwave induced transitions be-
tween two different energy levels, we now rewrite the Hamil-
TABLE I: Comparison between SFQ circuits and natural atoms for
dipole moments, parities, symmetry, and selection rules.
Atoms Dipole moments
∝ e−
Parities
Odd
Symmetry Selection rules
Well-
defined
Natural atoms
→
r
Has
SQC (f = 1/2)
∝ −sin(2ϕm)
+2 sin(ϕp)cos(ϕm)
OddWell-
defined
Has
SQC (f ?= 1/2)
∝ −sin(2ϕm+ 2πf)
+2 sin(ϕp)cos(ϕm)
No parity BrokenNo
n
n-3 energy levels
2
1
0
FIG. 2: (Color online) Schematic diagram for the coexistence of dif-
ferent photon transition processes when the inversion symmetry of
the potential is broken, i.e., f ?= 1/2. In this case, all transitions
between any two energy levels are possible, i.e., there is no forbid-
den transition. For example, the loop formed by three red arrow lines
(which link to ground state, the second excited state, and the nth ex-
cited state) denote a coexistence of single- and two-photon processes
for the n + 1 level system. Of course, the coexistence of single- and
two-photon processes can also be formed by theground, first excited,
and second excited states. However, the loop (formed by the green
arrow lines which link the ground, second, third, until nth excited
states) denotes the coexistence of the single- and n-photon processes
for the n + 1 level system. This schematic diagram also shows that
many different photon processes can coexist in the SFQ circuit with
the broken inversion symmetry.
tonian in Eq. (1) using eigenstates {|i?, i = 0, ···n} of the
SFQ circuits as the basis
H =
?
i
?ωii|i??i| +
n
?
i,j=0
Iij(f)|j??i|Φ(t).
(7)
with “dipole” matrix elements Iij(f) = ?i|I|j? and the eigen-
value ?ωiiof the eigenstate |i?. Equation (2) clearly shows
that the reduced magnetic flux f = Φe/Φ0determines the
symmetry of the potential energy of the SFQ circuits.
As discussed above for f = 1/2, the potential energy in
Eq. (2) and the loop current in Eq. (3) have inversion sym-
metries, and also all eigenstates of the SFQ circuit have well-
defined parities. Because the loop current I in Eq. (3) is an
odd function of the variables φp and φm, Therefore at the
point f = 1/2, the SFQ circuit has the same selection rules as

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4
the natural atoms, the microwave-induced transition can only
link two states which have different parities. However the
symmetry is broken when f ?= 1/2, the selection rules of the
SFQ circuits do not exist, the microwave induced transitions
between any two energy levels are possible. The comparison
of the selection rules between the SFQ circuits and natural
atoms is summarized in Table I.
To compare the results on the reduced magnetic flux f-
dependent transition matrix elements using the loop current
in Eq. (3) and that in Refs. [19, 35], the transition matrix el-
ements ?i|I|j? versus the reduced magnetic flux is plotted in
Fig. 1 for the three lowest energy levels with i, j = 0, 1, 2
and i ?= j. Numerical results in Fig. 1 also clearly show that
there are the same transition rules induced by the loop cur-
rent operator used either in Eq. (3) or in Ref. [19]. However,
as shown in Fig. 1, the amplitudes of the transition matrix
elements are a little different for two different loop current
expressions.
As shown in Table I and Fig. 1, when the inversion sym-
metry of the potential energy is broken (i.e., f ?= 1/2), all
transition matrix elements are not zero, thus the transitions
between any two levels are allowed. In this case, the single-
photon and n-photonprocesses can coexist for a (n+1)-level
SFQ circuit. That is, for a n + 1 level system, the transition
from the ground state |0? to the n excited state |n? can be re-
alized by either the single-photon process (|0? → |n?) or the
n-photon processes (|0? → |1? → ···|n − 1? → |n?). Simi-
larly, many different photon processes can also coexist in the
case with the broken inversion symmetry. The coexistence of
single- and n-photonprocesses has been schematically shown
in Fig. 2. When n = 2, we have the three-level SFQ circuit
as discussed in Ref. [19], then the single- and two-photon can
coexist. Itshouldbenotedthatthetransitionsbetweentwoen-
ergylevels shouldobeythe selection rules at the optimal point
f = 1/2. The photontransitionprocessesforthe (n+1)-level
SFQ circuit have been schematically shown in Fig. (3) for the
case f = 1/2.
In the above, we mainly complement some analysis on the
basic properties of the multi-level SFQ circuits. In view of
the big progress for the experimental studies on SFQs [36–
39], below we will focus on some unexplored features for the
SFQs when the inversion symmetry of the potential energy is
broken.
III. NEW PHENOMENA INDUCED BY LONGITUDINAL
COUPLINGS BETWEEN SFQS AND EXTERNAL
MAGNETIC FLUXES
A.Theoretical model on the couplings between SFQs and
time-dependent magnetic fluxes
Let us consider the case of the two energy levels for the
SFQs, i.e., n = 1, in this case, the Hamiltonian in Eq. (7)
is reduced to that of the superconducting flux qubits (SFQs),
driven by the time-dependent external magnetic flux.
Fig. 4, the matrix elements of the loop current I of the SFQs
for the two energy levels |0? and |1? is plotted versus the re-
In
(a)
(b)
n
n
n-3 energy levels
n-3 energy levels
2
2
1
1
0
0
FIG. 3: (Color online) The Schematic diagram for the photon transi-
tion processes when the inversion symmetry of the potential energy
is well-defined, i.e., f = 1/2. For the convenience of the discus-
sions, let us assume that the microwave induced transition between
energy levels |n? and |n + 1? is possible for the n + 1 level system.
Therefore, it is clear that the transition between the state |0? and the
state |2? is prohibited. In (a), if the transition between the state |2?
and the state |n? is prohibited, then the transition between the state
|n? and the state |0? is also prohibited; In (b), if the transition be-
tween the state |2? and the state |n? is allowed, then the transition
between the state |n? and the state |0? is also allowed, however tran-
sition between the states |0? and |2? is forbidden. In both figures,
the sign “×” denotes that the electric-dipole-like microwave induced
transition is prohibited. Because the parities for those states are the
same.
?
?
?
?
FIG. 4: (Color online) The reduced magnetic flux f-dependent loop
current ?i|I|i? for the two lowest eigenstates |0? and |1?. Here, we
also take the typical numbers α = 0.8 and EJ = 40Ec in our nu-
merical simulations.
duced magnetic flux f. Fig. 4 shows a well-known result that
the loop current I in the ground |0? and the first excited |1?
states are zero at the symmetric point f = 1/2. However,
once the symmetry is broken, the loop current for both states
are not zero. Therefore, for a SFQ interacting with the time-
dependent magnetic flux, we have the following Hamiltonian
H =
1
?
i=0
?ωii(f)|i??i| + [I01(f)|0??1| + I10(f)|1??0|]Φ(t)
+ [I00(f)|0??0| + I11(f)|1??1|]Φ(t).
(8)

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5
Here, we write ωii(f) and Iij(f) (with i, j = 0, 1) to empha-
size the f-dependent parameters.
Let us now first discuss the interaction between the SFQs
and the classical magnetic flux Φ(t) using the Hamiltonian in
Eq. (8) when f = 1/2. The above analytical analysis together
with Fig. 1 and Fig. 4 show
I00(f = 0.5) = I11(f = 0.5) = 0,
(9)
and
I10(f = 0.5) = I01(f = 0.5) ?= 0.
(10)
Therefore, in Eq. (8), the two coupling terms become into
Iii(f = 0.5)|i??i|Φ(t) = 0,
(11)
with i = 0, 1, which means that there is no the longi-
tudinal coupling between the time-dependent magnetic flux
and the SFQ at the optimal point. There are only coupling
terms I01(f = 0.5)(|0??1| + |1??0|)Φ(t) in Eq. (8), called as
the transverse coupling between the time-dependentmagnetic
flux and the SFQ. Therefore, under the rotating wave approx-
imation, the Hamiltonian in Eq. (8) at the optimal point can
further be reduced to the Jaynes-Cumming model, which has
been extensively explored in the quantum optics and the cir-
cuit QED system.
When f ?= 1/2, all elements Iij(f) with i = 0, 1 are not
zero, the interaction Hamiltonianbetween the time-dependent
magnetic flux and the SFQs includes both transverse and lon-
gitudinal couplings, which are less studied. This longitu-
dinal coupling can induce some unusual phenomena which
will be explored below. For the convenience of the discus-
sions, our studies below just consider the case of the longitu-
dinal and transverse couplings between one driving classical
field and the SFQs, however all of discussions in the subsec-
tionsIIIB,IIIC,andIIIDcanbeappliedtothecasewithmore
driving fields.
B. Longitudinal coupling induced coexistence of multi-photon
processes in SFQs
For the convenience of the discussions, using the relations
in Eqs. (9) and (10), the Hamiltonian in Eq. (8) can be rewrit-
ten as
H = ?ωq
2σz+ ?(λxσx+ λzσz)cos(ω0t),
with σz = |1??1| − |0??0| and σx = |0??1| + |1??0|. Here,
we assume the magnetic flux Φ(t) in Eq. (8) to be Φ(t) =
Φcos(ω0t), and then λx= ΦI01and λz= ΦI11.
If the SFQ works at the optimal point (i.e., f = 0.5), then
I11(f = 0.5) = 0 which implies the longitudinal coupling
constant λz = 0. In this case, Eq. (12) becomes into a stan-
dard Hamiltonian of a driven SFQ, and there is only single-
photon resonant transition in the SFQ induced by the external
magnetic flux with the condition ωq= ω0.
When the reduced magnetic flux deviates from the optimal
point, i.e., f ?= 0.5, there are both the transverse and longi-
tudinal couplings between the SFQ and the external magnetic
(12)
……
0
1
n- photon
process
1- photon
process
2-photon
process
FIG. 5: (Color online) Schematic diagram for the longitudinal cou-
pling induced coexistence of multi-photon processes in the SFQs
flux. In contrast to the case of only the single-photon process
for the transverse coupling between the SFQ and the exter-
nal magnetic flux, the longitudinal coupling can result in the
coexistence of the multi-photon processes in the SFQs. To
demonstrate this, we now apply a unitary transformation
U(t) = exp
?
−i
2
?
ω0t + 2λz
ω0sinω0t
?
σz
?
(13)
to Eq. (12), and thus the Hamiltonianin Eq. (12) becomesinto
H = ?ωq− ω0
2
σz+ ?
?
n
?λne−inω0tσ++ h.c.?,(14)
under the rotating wave approximation. The effective Rabi
frequency in Eq. (14) is given by
λn= λxJn
?2λz
ω0
?
,
(15)
which depends on both λzand ω0with the Bessel functions
Jn(2λz/ω0) of the first kind. When Eq. (14) is derived, we
use the relation
exp
?
i2λz
ω0
sin(ω0t)
?
=
n=∞
?
n=−∞
Jn
?2λz
ω0
?
exp[inω0t].
(16)
We note if the SFQ works at the optimal point, then λz= 0
and the Bessel functions
Jn?=0
?2λz
ω0
?2λz
ω0
= 0
?
?
= 0,
(17)
Jn=0
= 0= 1.
(18)
In this case, Eq. (14) is reduced to the usual Jaynes-Cumming
model for externally driven two-level system, and only de-
scribes the single-photon resonant transition with the condi-
tion ωq= ω0. The single-photon process is characterized by
the term for n = 0 in Eq. (14) with the amplitude
λ0= λxJ0
?2λz
ω0
= 0
?
= λx.
(19)