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Transient absorption and lasing without inversion in an artificial molecule via Josephson coupling energy
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Laser Physics Letters
Astro Ltd Laser Phys. Lett. 12 (2015) 035201 (8pp)
doi:10.1088/1612-2011/12/3/035201
Transient absorption and lasing without inversion in an artificial molecule via Josephson coupling energy Hamid Reza Hamedi Institute of Theoretical Physics and Astronomy, Vilnius University A. Gostauto 12, LT-01108 Vilnius, Lithuania E-mail:
[email protected] Received 28 May 2014, revised 20 September 2014 Accepted for publication 10 January 2015 Published 28 January 2015 Abstract
This letter investigates the dynamical behavior of the absorption in a superconducting quantum circuit with a tunable V-type artificial molecule constructed by two superconducting Josephson charge qubits coupled with each other through a superconducting quantum interference device. It is found that the ratio of the Josephson coupling energy to the capacitive coupling strength provides an extra controlling parameter for manipulating transient absorption behaviors. It is also realized that in the presence of an incoherent pumping field, lasing without inversion can be obtained just through the joint effect of the Josephson coupling energy and the capacitive coupling strength. Results may provide some new possibilities for solid-state quantum information science. Keywords: transient behavior, lasing without inversion, superconducting quantum circuit, artificial molecule, Josephson coupling energy (Some figures may appear in colour only in the online journal)
1. Introduction
and transient-amplification without inversion are widely investigated [36–45]. The evolutional optical behaviors of a four-level double-control tripod-configuration EIT system based on the transient solution to the equation of motion of the probability amplitudes of the atomic levels was considered by Shen [36]. Zhang et al compared the steady and transient optical responses of a four-level system and a three-level system, dressed by two and three laser fields, respectively [38]. We proposed an open atomic scheme to study the transient evolution of the atomic response with applications to lasing with and without population inversion [45]. On the other hand, recently, a novel system was shown to be capable of implementing basic quantum optical systems. Circuit quantum electrodynamics (cQED) [46, 47] is an onthe-chip counterpart of cavity QED systems, that employs a quantized microwave mode held in a Co-planar waveguide resonator (CPW) (substituting the standing-wave optical cavity) and a Cooper pair box (CPB) (instead of a two-level atom trapped in the cavity). Compared with conventional optical implementations, this solid-state architecture offers
Recently, many kinds of quantum optical phenomena based on the quantum interference and coherence in atomic systems have been extensively studied, such as optical switching [1, 2], optical bistability [3–8], Kerr nonlinearity [9–14], gain without inversion [15, 16], four-wave mixing (FWM) [17–20], optical soliton [21], atom localization [22–24], electromagnetically induced transparency (EIT) [25–28], and so on [29–32]. In fact, taking advantage of EIT, one can readily control the optical response and related absorption of weak laser light. The EIT effect has found applications in quantum information science, such as photon information storing and releasing in an atomic assemble [33], correlated photon pairs’ generation [34] and even the entanglement of remote atomic assembles [35], which form the building blocks of quantum communication and quantum computation. Transient properties of the weak probe field via quantum interference such as transient-absorption, transient-dispersion 1612-2011/15/035201+8$33.00
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© 2015 Astro Ltd Printed in the UK
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Laser Phys. Lett. 12 (2015) 035201
unprecedented tunability and scalability which are leading to flexible quantum optics in electronic circuits, such as offering long coherence time to implement the quantum gate operations [48], huge tunability and controllability by external electromagnetic fields [49]. As an on-chip realization of cavity QED, circuit QED has reproduced many quantum optical phenomena, including Kerr and cross-Kerr nonlinearities [49], the Mollow triplet [50], Autler–Townes effect [51], EIT [48, 52–54] and optical bistability [55]. Circuit quantum electrodynamics (QED) [56] based on Josephson junctions, where a transmission line resonator plays the role of the cavity and superconducting qubit [57, 58] behaves as an artificial atom to replace the natural atom, has recently become a new test-bed for quantum optics. As we mentioned, transient properties of traditional atomic and nanostructure configurations have being extensively discussed. Nevertheless, to the best of our knowledge, the transient behaviors in superconducting circuits based on mesoscopic Josephson junctions are still rarely reported, which motivates us for this work. We propose a three-level V-type artificial molecule constructed by two superconducting charge qubits which are coupled to each other through a large capacitor. Two upper states are coupled to the ground level through a weak probe and a strong control field, to build the V-shaped configuration which depends on the energy separation of Josephson coupling energy of two charge qubits. We aim to investigate the dynamical evolution of the artificial molecule through the density matrix formalism. We show that the transient properties of this system depend strongly on the ratio of the Josephson coupling energy to the capacitive coupling strength, which compared with traditional atomic schemes, can be introduced as a new freedom for modifying the optical properties. In addition, we find that in the presence of an incoherent pump field, this new dependence can lead to the lasing without population inversion in the artificial medium, which makes our scheme much more practical than the other atomic counterparts. Our letter is organized as follows: in section 2, the model and equations are presented. Results and physical discussion are given in section 3. And in section 4 some simple conclusions are given.
(a)
E J 2 ,C J 2
E J 1 ,C J 1 Cm
Cb 2
C b1
Vg2
V g1
Cg2
Cp Vg2
C g1
Cp V g1
Vp
(b)
d
ωdc c
ωdb
ωca
b
ωba a Figure 1. (a) Schematic diagram of the two-coupled-qubit circuit. Black bars show Cooper pair boxes. (b) Schematic of the allowed transition paths of the coupled charge qubits operated at the coresonance point.
c b
r
2. Theoretical model and equations Figure 1(a) illustrates the schematic of the circuit we study [58, 59], which consists of two charge qubits that are electrostatically coupled by an on-chip capacitor Cm. Both qubits have a common pulse gate but separate dc gates, probes and reservoirs. The pulse gate has nominally equal coupling to each box, while both reservoirs are kept grounded. External controls that we have in the circuit are the dc probe voltages Vb1 and Vb2, dc gate voltages Vg1 and Vg2, and pulse gate voltage Vp. The information on the final states of the qubits after manipulation comes from the pulse-induced currents measured in the probes. Each charge qubit has a superconducting quantum interference device (SQUID) ring geometry biased by an external flux. The Hamiltonian of coupled qubits reads [54, 58]
Ωp
Ωc
a
Figure 2. Schematic of V-type three-level artificial molecule.
H = Ec1(n1 – ng1)2 – EJ1 cos + Ec2(n2 – ng2 )2 – EJ2 cos + Em (n1 – ng1) (n2 – ng2 )
(1)
Here, the first four terms describe two independent qubits, whereas the last term shows the interaction between the qubits due to the electrostatical coupling of the capacitor. In this equation, EJ1 and EJ2 represent the effective Josephson coupling energy for the corresponding SQUID, 2
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Laser Phys. Lett. 12 (2015) 035201 −3
1
x 10
1
(a)
0.9
X=1 X=3 X=10
X=1 X=3 X=10
0.8
0.7
0.7
0.6
0.6 aa
0.5
ρ
Absorption
0.8
(b)
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0.2
0.52
0.1
0.1
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0 0
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4
γt
6
8
0.54
9.4
0 0
10
9.6
2
9.8
4
γt
10
6
8
10
−6
0.8
(c)
5
X=1 X=3 X=10
0.7
4.5
3.5
0.5
X=1 X=3 X=10
cc
3
0.4
ρ
bb
(d)
4
0.6
ρ
x 10
0.3
2.5 2
0.48
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0 0
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γt
0.5
10
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0 0
10
−6
4.8
x 10
4.7 4.6 4.5
6
7
8
2
9
4
γt
10
6
8
10
Figure 3. (a) Transient evolution of probe absorption for different values of X. (b), (c), (d) Transient evolution of population distributions for different values of X. The selected parameter is Ωc = 5γ . The other parameters are γca = γ , γba = 1.5γ , γcb = 2γ , Γb = 1.5γ , Γc = γ , Δc = Δp = r = 0, Ω p = 0.01γ .
where Θ1 and Θ2 are the phases of the SQUID. Moreover,
EJ1 = EJ2 = EJ, Ec1 = Ec2 = Ec, CJ1 = CJ2 = CJ, C∑ 1 = C∑ 2 = C∑ ) .
Ec1,2 = 4e C∑ 2,1 / 2 C∑ 1C∑ 2 −
In this situation, one may introduce EJ and Em as
(
2
Cm2
) are
defined as the effective Cooper-pair charging energies for the qubits, C∑ i = Cgi + CJi + Cm, (i = 1, 2) are the sum of all capacitances connected to the corresponding island, while ni and ngi = CgiVgi / 2e, (i = 1, 2) are the number operator of excess Cooper-pairs on the island and the normalized gate induced charge. Also, Em = 4e 2Cm / 2 C∑ 1C∑ 2 − Cm2 is the capaci-
(
H=
)
1 1 1 EJ1σz1 + EJ2σz2 + Emσx1σx2, 2 2 2
Em = ϖ sin 2φ,
(3a)
EJ =
ϖ cos 2φ, 4
(3b)
where
ϖ = (Em2 + (4EJ )2 ) .
(4)
Therefore, from these equations, one can define parameter φ as
tive coupling energy between the charge qubits. The Hamiltonian of the superconducting system in the eigenbasis of the qubits and in the vicinity of one degeneracy point (ngi ∈ [0, 1] ) can be written as [53, 58].
φ=
⎛ 4E ⎞ 1 arc cot ⎜ J ⎟ . ⎝ Em ⎠ 2
(5)
Therefore, the level spacing of the coupled-qubit system can be tuned by changing the Josephson coupling energy EJ. Here we note that due to selection rules for superconducting qubits, arbitrary transitions among the four states are not allowed, as can be seen in figure 1(b). Accordingly, both transitions a ↔ d and b ↔ c are forbidden due to matrix
(2)
where σ are a set of Pauli matrices acting on the eigenstates of qubits. Without loss of generality, one can consider a special case that the two superconducting charge qubits are identical (i.e. 3
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Laser Phys. Lett. 12 (2015) 035201 −4
6
x 10
5
1
(a)
X=1 X=3 X=10
0.7
2
0.6 aa
3
1 0 −1
x 10
8
−3
6
−4 0
5
0.51
0.3
10
−2
0.5 0.4
−5
12
X=1 X=3 X=10
0.8
ρ
Absorption
4
(b)
0.9
0.505
0.2 10
15
10 γt
0.1
20
15
0.5 16
0 0
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5
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10 γt
19
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1 0.9
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X=1 X=3 X=10
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X=1 X=3 X=10
1
0.6 cc
0.8
0.5
ρ
bb
(d)
1.2
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ρ
x 10
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−6
1.235
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0.491 16
0 0
1.23
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x 10
17
5
18
19
1.22 16
17
18
19
20
20
10 γt
15
0 0
20
5
10 γt
15
20
Figure 4. (a) Transient evolution of probe absorption for different values of X. (b), (c), (d) Transient evolution of populations for different values of X. The selected parameter is Ωc = 10γ . The other parameters are the same as figure 2.
Where in this equation, ℧p = Ω p cosφ , and ℧c = Ωc cos φ , and the Rabi frequencies of the probe and control fields are Ω p = ℘caE p / 2, Ωc = ℘baEc / 2 . The amplitude of probe and control fields are defined by Ep and Ec respectively, while the corresponding transition dipole moments are ℘ja (j = c, b ) . Also, Δp = ωca − ω p, and Δc = ωba − ωc are the probe and control detuning parameters. The dynamics of this system can be described by the density-matrix approach
elements a σxi d = 0 and b σxi c = 0, (i = 1, 2) , whereas the other transitions with nonzero matrix elements are allowed. By selecting the three levels with the lowest eigenenergies (i.e. a , b , c ), a three level V-type system is obtained, as shown in figure 2. Now, the three level system interacting with a weak probe field and a strong control field is considered. The total Hamiltonian of the V-type artificial system is Hv =
∑
i = a, b, c
+Ωc e
Ei i i −
−iωct
1 2 2
cos φ[Ω p e−iωpt c a
b a + H.c.
Hint = Δp c
c + Δc b
a + ℧c b
i [H , ρ] + {Γρ} , ℏ
(8)
where Γρ denotes the relaxation terms. Substituting the interaction Hamiltonian given by equation (7) into equation (8), one can easily write the density-matrix equations of motion elements of the system as
The probe and the control fields are near resonant with the transitions a ↔ c and a ↔ b , respectively, and, other transitions can be subsequently ignored. Thus, in the interaction picture, and under the rotating-wave approximation (RWA), the Hamiltonian Hint of the artificial system reads 1 b − (℧p c 2
ρ˙ = −
(6)
ρ˙ca = − ( iΔp + γca − r / 2 ) ρca + i℧p ( ρaa − ρcc ) − i℧cρcb ,
a + H.c.
(
)
ρ˙cb = − i(Δp − Δc ) + γcb ρcb − i℧cρca + i℧pρab ,
(7) 4
(9a) (9b)
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Laser Phys. Lett. 12 (2015) 035201
ρ˙ba = − (iΔc + γba ) ρba + i℧c (ρaa − ρbb ) − i℧pρbc ,
−3
(9c)
4
ρ˙aa = Γcρcc + Γbρbb − i℧p (ρac − ρca ) − i℧c (ρab − ρba ) − rρaa , (9d) (9e)
ρ˙cc = −Γcρcc + i℧p (ρac − ρca ) + rρaa ,
(9f)
(a)
∆ =0 c
3.5
∆ =3γ c
∆c=6γ
3
∆ =9γ
2.5 Absorption
ρ˙bb = −Γbρbb + i℧c (ρab − ρba ) ,
x 10
where, Γb and Γc are the relaxation rates for the levels b and c respectively. moreover, γca, γba and γcb show the total dephasing rates of the off-diagonal terms. We include the effects of an indirect incoherent pump mechanism from lower level a to upper level c [60].
c
2 1.5 1 0.5 0 −0.5 0
3. Results and discussion In this section, we investigate the transient evolution of the artificial molecule response from different respects, by using the numerical result from the density matrix equation of motions ρij . It should be noted that although the system under consideration is similar to the conventional three level V-type atomic systems, in our configuration the set of density matrix equations has an extra dependence on the Josephson coupling energy EJ as well as the capacitive coupling strength Em through the parameter φ. Thus, the transient behavior of the artificial molecule can be readily controlled and even be modified by tuning these parameters. As known, the gain-absorption for the probe field on transition c → a is proportional to the imaginary part of ρca which can be obtained from eqaution (9). The system exhibits absorption for the probe field just when Im (ρca ) > 0, while, the probe laser will be amplified if Im (ρca ) < 0. Moreover, when Im (ρ31) < 0 and ρcc > ρaa the lasing without inversion can be obtained. We now present the numerical results of equation (9) through figures 2–7. The selected parameters are γca = γ , γba = 1.5γ , γcb = 2γ , Γb = 1.5γ , Γc = 1γ , where γ / 2π = 1.5 MHz [54], which for simplicity, all the parameters are scaled in the unit of γ. For more convenience, in equation (5), the ratio of the Josephson coupling energy EJ to the capacitive coupling strength Em is defined as X = EJ / Em, so that equation (5) can be rewritten as φ = 1 / 2arccot (4X ) . We now aim to explore how the parameter X, which can be attributed to the joint effect of Josephson coupling energy as well as the capacitive coupling strength, modifies the transient evolution, when all other parameters are fixed. Transient evolution of the gain–absorption for various values of parameter X = EJ / Em is depicted for Ωc = 5γ and Ωc = 10γ in figures 3 and 4, respectively. For the case Ωc = 5γ , the probe absorption never manifests periodic gain and absorption; the absorption exhibits an oscillatory behavior in a short time and finally reaches a positive steady-state value, which corresponds to the probe absorption, as shown in figure 3(a). To have an estimate of the population contributed in probe absorption, the population distribution of the V-type artificial system is presented for the set of parameter conditions in figure 3(a). It can be seen that the population
2
4
γt
6
8
10
−4
x 10
(b)
18
∆ =0.5γ p
16
∆p=γ
14
∆ =2γ p
∆ =3γ p
Absorption
12 10 8 6 4 2 0 −2
0
2
4
γt
6
8
10
Figure 5. Transient evolution of probe absorption for different values of (a) Δc , and (b) Δp . The parameter values are (a) Ωs = 5γ , X = 10, and the other parameters are the same as figure 2.
oscillates back and forth among the states and finally reaches the steady-state values, as shown in figures 3(b)–(d). Similar curves are now displayed in figure 4 when we chose Ωc = 10γ . Figure 4(a) indicates that the transient gain is obtained for a few times. Also, the oscillatory frequency and amplitude of the gain–absorption coefficient curves increases, and then it oscillates to a steady-state value. Figures 4(b)–(d) show the population distribution for each state. It can be seen that the curves oscillate more intensely with respect to the population distribution curves of figure 3. Also, after oscillatory behaviors, the population distribution of states reaches a steady-state value, so that populations are almost trapped in states a and b . This is an expected result which can be due to applying a strong control field between states a and b . Here, two important points must be mentioned. Firstly, as can be observed from both figures 3(a) and 4(a), by increasing the parameter X to the further values, the steady-state absorption values reduce dramatically. Secondly, it is realized that increasing the parameter X leads to the reduction of the population in state a (figures 3(b) and 4(b)), while the population 5
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Laser Phys. Lett. 12 (2015) 035201 −4
−3
1.5
x 10
18
(a)
x 10
16
(a)
14 12 Absorption
Absorption
1
0.5
10 8 6 4 2 0
0 0
1 0.9
2
4
γt
6
8
(b)
ρ
1 aa
0.9
ρbb
0.8
Population
0.5 0.4
0.2
0.1
0.1 6
8
8
(b)
0 0
10
10
ρaa ρ
bb
ρ
cc
0.4
0.2
γt
6
0.5
0.3
4
γt
0.6
0.3
2
4
0.7
0.6
0 0
2
0.8
ρcc
0.7 Population
−2 0
10
2
4
γt
6
8
10
Figure 6. Transient behaviors of (a) probe field absorption and (b) population distribution. The parameter values are Ωc = 1.5γ , Δc = γ , r = 0.9γ and X = 1. The other parameters are the same as figure 2.
Figure 7. Transient behaviors of (a) probe field absorption and (b) population distribution. The parameter values are Ωc = 1.5γ , Δc = γ , r = 0.9γ and X = 10. The other parameters are the same as figure 2.
in both levels b , c increases (figures 3(c), (d) and 4(c), (d)). Therefore, the joint effect of EJ and Em has a major role on the population distribution of each state. Obviously, by increasing parameter X, the steady-state population value decreases for state a , while it increases for states b and c . That is to say, further values of X transfer more population to the upper states, while less population will be retained in lower state a and thus, results in a reduction in probe absorption. The impact of the probe and control detuning parameters on transient absorption spectra is displayed in figure 5. It can be easily seen that the gain–absorption coefficients are very sensitive to the detuning parameters Δp and Δc , and increasing the detuning parameters leads to a significant increase of the steady-state probe absorption. As a matter of fact, the possible lasing without population inversion in traditional V-type atomic systems has been previously reported in the literature [61, 62]. In the following, we present details of our numerical results for lasing without inversion in our artificial system through figures 6
and 7. In the following numerical calculations, we assume Ωc = 1.5γ , Δc = γ , X = 1, while other parameters are set to be the same as figure 2, and plot the transient curves in the presence of incoherent pumping field r = 0.9γ . A typical transient evolution of probe absorption as well as population distribution is plotted in figure 6(a). It is deduced that for the chosen parameters, the probe absorption increases for a short time, then steeply descends and eventually reaches a positive steadystate value. In this situation, one can see from figure 6(b) that most of the population settles in level a . Therefore, the probe field withstands absorption, and population inversion never appears for this set of parameters. When the other parameters are chosen to be fixed, the impact of ratio X = EJ / Em on the transient behavior of probe absorption and population distribution is explored in figure 7. It is illustrated that for X = 10, after a short oscillation, the probe absorption finds a negative steady-state value (figure 7(a)), corresponds to probe amplification. Furthermore, most of the population still remains in level a (figure 7(b)). Thus, 6
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