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Superluminal pulse propagation and amplification without inversion of microwave radiation via four-wave mixing in superconducting phase quantum circuits
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Laser Phys. Lett. 12 085202 (http://iopscience.iop.org/1612-202X/12/8/085202) View the table of contents for this issue, or go to the journal homepage for more
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Laser Physics Letters
Astro Ltd Laser Phys. Lett. 12 (2015) 085202 (9pp)
doi:10.1088/1612-2011/12/8/085202
Superluminal pulse propagation and amplification without inversion of microwave radiation via four-wave mixing in superconducting phase quantum circuits Z Amini Sabegh, A Vafafard, M A Maleki and M Mahmoudi Department of Physics, University of Zanjan, University Blvd, 45371-38791, Zanjan, Iran E-mail:
[email protected] Received 17 March 2015, Revised 13 May 2015 Accepted for publication 3 June 2015 Published 8 July 2015 Abstract
We study the interaction of the microwave fields with an array of superconducting phase quantum circuits. It is shown that the different four-level configurations i.e. cascade, N-type, diamond, Y-type and inverted Y-type systems can be obtained in the superconducting phase quantum circuits by keeping the third order of the Josephson junction potential expansion whereas by dropping the third order term, just the cascade configuration can be established. We study the propagation and amplification of a microwave field in a four-level cascade quantum system, which is realized in an array of superconducting phase quantum circuits. We find that by increasing the microwave pump tones feeding the system, the normal dispersion switches to the anomalous and the gain-assisted superluminal microwave propagation is obtained in an array of many superconducting phase quantum circuits. Moreover, it is demonstrated that the stimulated microwave field is generated via four-wave mixing without any inversion population in the energy levels of the system (amplification without inversion) and the group velocity of the generated pulse can be controlled by the external oscillating magnetic fluxes. We also show that in some special set of parameters, the absorption-free superluminal generated microwave propagation is obtained in superconducting phase quantum circuit system. Keywords: quantum coherence, superluminal pulse propagation, superconducting circuits, wave mixing (Some figures may appear in colour only in the online journal)
Introduction
energy difference scale of their energy levels and the system coupling with its environment. The energy scale in the SQC corresponds to the gigahertz and the coupling in SQC is also stronger than the natural atoms. Moreover, there are not usually well-defined selection rules for interaction of superconducting circuits with the microwave fields [3]. The superconducting charge, flux and phase quantum circuits based on the Josephson junctions are macroscopic in size but they show quantum mechanical behaviors such as having discrete energy levels, superposition of states and entanglement. Then, it have been extensively used as a powerful tool
It is well known that a superconducting quantum circuit (SQC), containing the Josephson junctions, has a set of quantized energy levels and behaves as an artificial atom. It has been used to establish a number of the atomic physics and the quantum optics phenomena [1]. A Josephson junction is a weak link between two superconducting bulks which is separated by an insulating layer and acts as a device with nonlinear inductance and no energy dissipation [2]. The main differences between the SQC and a natural atom lie on the 1612-2011/15/085202+9$33.00
1
© 2015 Astro Ltd Printed in the UK
Z Amini Sabegh et al
Laser Phys. Lett. 12 (2015) 085202
in quantum information processing [4, 5]. The generation, conversion, amplification and propagation of the microwave signals are important because of their potential applications in measuring and controlling the qubits in solid state quantum information processing. The atomic system has been employed to establish the several interesting nonlinear coherent optical phenomena, i.e. electromagnetically induced transparency (EIT) [6], Autler– Townes splitting [7], coherent population trapping [8], amplification without inversion [9], wave mixing [10], slow light [11] and fast light [12]. Such behaviors are not restricted to the atomic systems and the solid state systems can be also used to establish the quantum phenomena. The EIT-like phenomenon has been introduced in superconducting systems in rotating wave approximation [13] and beyond it [14]. The phase-sensitive properties of the closed-loop systems, established in the SQC, have been studied in Δ-type configuration and it was shown that EIT window can be controlled by relative phase of driving fields [15]. The Autler–Townes doublet has been experimentally measured in a superconducting charge [16] and phase qubits [17]. The EIT, decoherence, Autler–Townes and dark states have been theoretically investigated in two-tone driving of a three-level superconducting phase [18] and flux quantum circuit [19]. Direct observation of coherent population trapping has been reported in the Λ-type configuration of superconducting phase qubits [20]. Coherent microwave pulse control of quantum memory via slow light in SQC was studied and realization of the coherent storage and on-demand pulse retrieval are reported in this system [21]. The difference- and sum-frequency generation via three-wave mixing was investigated in the microwave regime using a single three-level superconducting flux quantum circuit [4]. The optical bistability in a V-type artificial atom, established in a SQC, was studied and it was shown that the ratio of the Josephson coupling energy to the capacitive coupling strength has a significant impact on creating optical bistability [22]. Controlling of the group velocity in a SQC was investigated and superluminal pulse propagation was reported in the presence of the absorption peak in the absorption spectrum [23] but, because of the attenuation, it is difficult to propagate the pulse in the presence of such absorption peak. In this letter, we investigate the propagation and amplification without inversion of microwave pulse in a superconducting phase quantum circuit (SPQC). We show that by changing the applied time-dependent magnetic flux amplitudes, the slope of dispersion can be switched from positive to negative which corresponds to the superluminal pulse propagation. Moreover, such propagation happens in a transparent window and it is accompanied by the gain doublet without appearance of any inversion population in energy levels. In the next step, we investigate the microwave generated pulse in the SPQC via four-wave mixing. The four-wave mixing has been investigated in atomic systems [24, 25]. We find that the stimulated microwave field is generated without any inversion of population and propagates in superluminal region. Then stimulated generation of superluminal microwave pulse propagation is obtained in the SPQC via four-wave mixing process.
(a) 600
U( ϕ)/h (GHz)
500
400 420
300
410
400
200
0
|2> |1>
Ω10
390
380 0.16
100
|3>
Ω32 Ω21
0.2
0.24
0.25
|0>
0.28
0.32
0.5
ϕ (b)
0.75
1
Figure 1. (a) Schematic of SPQC consisting of a Josephson junction in a superconducting circuit with a gate capacitance C, inductance L and Josephson junction inductance LJ, placed in an external magnetic flux Φext. (b) Potential energy and energy levels of SPQC system for L = 2LJ = 700 pH and Φdc = 0.54Φ0 versus φ between zero and unity. The time-dependent flux amplitudes applied to the transitions 0 − 1 , 1 − 2 and 2 − 3 are shown by Ω10, Ω 21 and Ω32, respectively.
Model and equations We introduce a SPQC consisting of a Josephson junction in a superconducting circuit with a gate capacitance C, inductance L and Josephson junction inductance LJ which is placed in an external magnetic flux as a driving element. The schematic of the system is shown in figure 1(a). The external magnetic flux consists of a constant term Φdc and an oscillating term Φrf (t ), so that Φext = Φdc + Φrf (t ). The Josephson junction energy is denoted by EJ = Φ20/(4π 2LJ ), where Φ0 = 2.07 × 10−15 Wb is quantum of the flux. Using the critical Josephson current, Ic = (2π /Φ0)EJ, the Josephson junction potential energy is given by EJ(1 − cos(2π Φ/Φ0)) where Φ stands for the flux variable. Then the classical Hamiltonian of the SPQC can be written as ⎛ Φ⎞ Q 2 (Φ − Φext )2 + − EJ cos⎜2π ⎟. H= (1) ⎝ Φ0 ⎠ 2C 2L
By assuming Φrf (t ) ≪ Φdc the Hamiltonian switches to 2
Z Amini Sabegh et al
Laser Phys. Lett. 12 (2015) 085202
⎛ Φ ⎞ ΦΦrf (t ) Q 2 (Φ − Φdc)2 + H= − EJ cos⎜2π ⎟ − , (2) ⎝ Φ0 ⎠ 2C 2L L
the commutation relation [Φ, Q] = iℏ. The creation (a†) and annihilation (a) operators for the harmonic oscillator can be written as
where 2π Φ/Φ0 stands for superconducting phase difference across the junction. The first term, in the right hand side of equation (2) is the circuit kinetic energy and two next terms are the circuit potential energy. The last term is time dependent part of the classical Hamiltonian. Then the potential energy of the circuit can be written as
Cω0 ⎛ Cω0 ⎛ iQ ⎞ † iQ ⎞ a= ⎜Φ + ⎟, a = ⎜Φ − ⎟, (9) Cω0 ⎠ 2ℏ ⎝ Cω0 ⎠ 2ℏ ⎝
where ω0 = 1/ LJ*C is the Josephson plasma frequency. The operators a and a† obey the commutation relation [a, a†] = 1 and H0 reads to
U (φ) = U0[(φ − φdc )2 − β cos(2πφ)] , (3)
where U0 = Φ20/(2L ), β = L /(2π 2LJ ), φ = Φ/Φ0 and φdc = Φdc/Φ0. Note that for φdc = 0.5 the potential energy mentioned in equation (3), is symmetric around φ = 0.5. In figure 1(b), we plot the potential energy for L = 2LJ = 700 pH and Φdc = 0.54Φ0 versus φ between zero and unity. Noting that two minima of the potential energy in this interval are close to φ = 0.22 and φ = 0.82, their locations are approximately given by φm1 = 0.22 +
H0 = ℏω0(a†a + 1/2) , (10)
with the eigenvalue equation H0 n 0 = ℏω0(n + 1/2) n 0. We can shift all the energy levels in order to get the zero ground state energy. The energy levels of this Hamiltonian are the same as the case of the harmonic oscillator. However, using equations (3) and (6) in equation (2), reads to
φdc − πβ sin(0.44π ) − 0.22 , 1 + 2π 2β cos(0.44π )
ΦΦrf (t ) Φ3 H = H0 − − , (11) L 2Φ0L*
1 1 1 ⎛ 3⎞ 1 . + + 3 ⎜φdc − ⎟ + 3 2 ⎝ ⎠ 4π β 2π 2π β 4 16π 6β2 (4) φm2 = 0.75 −
where L* = (3LJ /2π )[sin(0.44π ) + 2π cos(0.44π )(φm1 − 0.22)]−1. The matrix form of the Hamiltonian in the basis of the perturbed states allows us to have various four-level systems in SPQCs. Considering a specific value for gate capacitance [18], the Hamiltonian of the four-level artificial atom interacting with microwave fields is given by
The condition for existence of the second minimum is 3 1 φdc > − πβ − 3 . (5) 4 8π β
H = H0 −
Now, we choose the first minimum, φm1, and expand the potential energy as U (Δφ) = (φm1 − φdc )2 − β cos(0.44π ) + 2πβ sin(0.44π )(φm1 − 0.22) U0 + [1 + 2π 2β cos(0.44π ) − 4π 3β sin(0.44π )(φm1 − 0.22)] (Δφ)2 −
4π 3β [sin(0.44π ) + 2π cos(0.44π )(φm1 − 0.22)](Δφ)3, 3
(6) where Δφ = φ − φm1. Dropping the first constant term, shifting the reference point to the first minimum φm1 and neglecting the third term in the vicinity of φ = φm1, we obtain
ΦΦrf (t )
L ⎡ ⎤ 1 −0.0064g(t ) ⎥ g(t ) 0.02g(t ) 0 ⎢ 2 ⎢ ⎥ ⎢ 1 ⎥ 0.06g(t ) g(t ) g(t ) ω10 ⎢ ⎥ 2 ⎥, = ℏ⎢ ⎢ ⎥ 3 g(t ) g (t ) ω10 + ω 21 ⎢ 0.02g(t ) ⎥ 2 ⎢ ⎥ ⎢ ⎥ 3 g(t ) ω10 + ω 21 + ω32 ⎥ ⎢⎣−0.0064g(t ) 0.06g(t ) ⎦ 2
(12)
where ω10, ω 21 and ω32 are the central frequencies of corresponding transitions. Moreover the coupling parameter is defined by
Φ2 U (Φ) = , (7) 2LJ*
Φrf (t ) g (t ) = − . (13) L ℏCω0
−1
where LJ* = LJ[(2π 2β )−1 + cos(0.44π ) − 2π sin(0.44π )(φm1 − 0.22)] . Then the time independent part of the classical Hamiltonian becomes as
We ignore the small terms proportional to 0.0064 g(t ) in the Hamiltonian. For different choosing of the external magnetic flux, we obtain the different systems such as cascade, N-type, diamond, Y-type and inverted Y-type quantum systems. Let us consider the external magnetic flux to be
Q2 Φ2 . H0 = + (8) 2C 2L J*
To quantize the Hamiltonian of equation (8), we consider Φ and Q as the canonical conjugate variables which satisfy
Φ rf (t ) = Φ10 cos(ω10t ) + Φ21 cos(ω 21t ) + Φ32 cos(ω 32t ), (14)
3
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Laser Phys. Lett. 12 (2015) 085202
which can excite the three transitions 0 − 1 , 1 − 2 and 2 − 3 to establishing the cascade type system, as shown in figure 1(b). The parameters ω10, ω 21 and ω32 are the frequencies of the applied external microwave fields. Thus, the coupling parameter is given by
where Ω10 = g10 / 2 , Ω 21 = g21 and Ω32 = 3/2 g32 are the Rabi frequencies of applied fields. The detuning of applied fields with corresponding transitions are denoted by Δ10 = ω10 − ω10, Δ21 = ω 21 − ω 21, Δ32 = ω32 − ω32. Next, we consider the external magnetic flux as
g(t ) = g10 cos(ω10t ) + g21 cos(ω 21t ) + g32 cos(ω32t ), (15)
Φ rf (t ) = Φ20 cos(ω 20t ) + Φ21 cos(ω 21t ) + Φ31 cos(ω 31t ), (21)
where the applied time-dependent flux amplitudes are g10 = −
Φ10 , L ℏCω0
g21 = −
Φ21 , L ℏCω0
g32 = −
to establish the N-type system. The Hamiltonian of the system in the interaction picture is given by
Φ32 . L ℏCω0
⎛ ∂U † ⎞ H͠ = U †HU + iℏ⎜ ⎟U ⎝ ∂t ⎠ ⎡ ⎤ Ω 20 0 0 ⎥ ⎢ 0 2 ⎢ ⎥ Ω Ω31 ⎥ 21 ⎢ 0 Δ21 − Δ20 ⎢ 2 2 ⎥, = ℏ⎢ ⎥ Ω Ω 21 ⎢ 20 0 ⎥ −Δ20 2 ⎢ 2 ⎥ ⎢ ⎥ Ω31 0 −Δ30 ⎥ ⎢⎣ 0 ⎦ 2
(16)
We are going to use the density matrix formalism to investigate the dynamical properties of the SPQC system. Then we should construct the density matrix in the interaction picture as ρ = U †ρS U where ρS is the density matrix in the Schrödinger’s picture. The following unitary matrix ⎡1 ⎤ 0 0 0 ⎢ ⎥ −iω10t 0 e 0 0 ⎥, U=⎢ (17) −i(ω10 + ω 21)t 0 e 0 ⎢0 ⎥ ⎢⎣ 0 0 0 e−i(ω10 + ω21+ ω32)t ⎥⎦
Φ20 L ℏCω0
), Ω
21
=−
Φ21 , L ℏCω0
and Ω32 = 0.06
− (system. ). Equation (22) explains the N-type quantum Φ31 L ℏCω0
transforms the operators from Schrödinger’s to the interaction picture. Then we get ⎡ ρ∼00 ρ∼01 ρ∼02 ρ∼03 ⎤ ⎢∼ ∼ ∼ ∼ ⎥ ⎢ ρ10 ρ11 ρ12 ρ13 ⎥ = ρ (18) ⎢ ρ∼ ρ∼ ρ∼ ρ∼ ⎥, ⎢ ∼20 ∼21 ∼22 ∼23 ⎥ ⎣ ρ30 ρ31 ρ32 ρ33 ⎦
To establish the diamond type quantum system, we consider Φrf (t ) = Φ10 cos(ω10t ) + Φ20 cos(ω 20t ) + Φ31 cos(ω31t ) (23) + Φ32 cos(ω32t ).
Then the Hamiltonian can be written as
where the matrix elements are given by ρ∼00 = ρ00 , ρ∼01 = e−iω10t ρ01, ρ∼02 = e−i(ω10 + ω21)t ρ02 , ρ∼03 = e−i(ω10 + ω21+ ω32)t ρ03, ρ∼11 = ρ11, ρ∼12 = e−iω21t ρ12 , ρ∼13 = e−i(ω21+ ω32)t ρ13 , ρ∼22 = ρ22 , ρ∼33 = ρ33.
(
where Ω 20 = 0.02 −
(22)
⎛ ∂U † ⎞ H͠ = U †HU + iℏ⎜ ⎟U ⎝ ∂t ⎠ ⎡ ⎤ Ω10 Ω 20 0 ⎢ 0 ⎥ 2 2 ⎢ ⎥ Ω31 ⎢ Ω10 −Δ ⎥ 0 10 ⎢ 2 ⎥ 2 = ℏ⎢ ⎥, Ω 20 Ω32 ⎢ ⎥ 0 −Δ20 2 ⎢ 2 ⎥ ⎢ ⎥ Ω31 Ω32 −(Δ20 + Δ32) ⎥ ⎢⎣ 0 ⎦ 2 2
(19)
The Hamiltonian in the interaction picture can be also followed by ⎛ ∂U † ⎞ H͠ = U †HU + iℏ⎜ ⎟U ⎝ ∂t ⎠ ⎡ ⎤ Ω10 0 0 ⎢ 0 ⎥ 2 ⎢ ⎥ Ω 21 ⎢ Ω10 −Δ ⎥ 0 10 ⎢ 2 ⎥ 2 = ℏ⎢ ⎥ , (20) Ω 21 Ω32 ⎢ 0 ⎥ −(Δ10 + Δ21) 2 2 ⎢ ⎥ ⎢ ⎥ Ω32 0 −(Δ10 + Δ21 + Δ32) ⎥ ⎢⎣ 0 ⎦ 2
where Ω10 = −
(−
Φ31 L ℏCω0
Φ10 , L 2ℏCω0
) and Ω
32
=
(
Ω 20 = 0.02 −
(
3/2 −
Φ32 L ℏCω0
Φ20 L ℏCω0
).
(24)
), Ω
31
= 0.06
The Y-type quantum system is also established by considering
Φ rf (t ) = Φ10 cos(ω10t ) + Φ21 cos(ω 21t ) + Φ31 cos(ω 31t ), (25) 4
Z Amini Sabegh et al
Laser Phys. Lett. 12 (2015) 085202
Here γij = γji is the pure inter-level dephasing rate and σij = i j . The parameter Γij is the inter-level relaxation rate of the transition i → j . Then the equations of motion for the density matrix elements are given by
and ⎛ ∂U † ⎞ H͠ = U †HU + iℏ⎜ ⎟U ⎝ ∂t ⎠ ⎡ ⎤ Ω10 0 0 ⎢ 0 ⎥ 2 ⎢ ⎥ Ω 21 Ω31 ⎢ Ω10 −Δ ⎥ 10 ⎢ 2 ⎥ 2 2 = ℏ⎢ ⎥, Ω 21 ⎢ 0 ⎥ 0 −(Δ10 + Δ21) 2 ⎢ ⎥ ⎢ ⎥ Ω31 0 −(Δ10 + Δ31) ⎥ ⎢⎣ 0 ⎦ 2
where Ω10 = −
(−
Φ31 L ℏCω0
).
Φ10 , L 2ℏCω0
Ω 21 = −
Φ21 L ℏCω0
Ω ρ∼˙00 = −i 10 (ρ∼10 − ρ∼01) + Γ10ρ∼11, 2 ⎡ ⎤ Ω Ω ρ∼˙01 = −i⎢ 10 (ρ∼11 − ρ∼00 ) + Δ10ρ∼01 − 21 ρ∼02 ⎥ ⎣ 2 ⎦ 2 1 − (Γ10 + γ10 )ρ∼01, 2 ⎡Ω ⎤ Ω Ω ∼ ˙ ρ02 = −i⎢ 10 ρ∼12 + (Δ10 + Δ21)ρ∼02 − 21 ρ∼01 − 32 ρ∼03⎥ ⎣ 2 ⎦ 2 2 1 − (Γ21 + γ20 )ρ∼02 , 2 ⎡Ω ⎤ Ω ρ∼˙03 = −i⎢ 10 ρ∼13 + (Δ10 + Δ21 + Δ32)ρ∼03 − 32 ρ∼02 ⎥ ⎣ 2 ⎦ 2 1 − (Γ32 + γ30 )ρ∼03, 2 ⎡Ω ⎤ Ω ρ∼˙11 = −i⎢ 10 (ρ∼01 − ρ∼10 ) + 21 (ρ∼21 − ρ∼12 )⎥ − Γ10ρ∼11 + Γ21ρ∼22 , ⎣ 2 ⎦ 2 ⎡Ω ⎤ Ω Ω ρ∼˙12 = −i⎢ 21 (ρ∼22 − ρ∼11) + Δ21ρ∼12 + 10 ρ∼02 − 32 ρ∼13⎥ ⎣ 2 ⎦ 2 2 1 − (Γ10 + Γ21 + γ21)ρ∼12 , 2 ⎡Ω ⎤ Ω Ω ∼ ˙ ρ13 = −i⎢ 10 ρ∼03 + (Δ21 + Δ32)ρ∼13 + 21 ρ∼23 − 32 ρ∼12 ⎥ ⎣ 2 ⎦ 2 2
(26)
and Ω31 = 0.06
Finally, the inverted Y-type system is also obtained for
Φ rf (t ) = Φ20 cos(ω 20t ) + Φ21 cos(ω 21t ) + Φ32 cos(ω 32t ) (27)
and
⎛ ∂U † ⎞ H͠ = U †HU + iℏ⎜ ⎟U ⎝ ∂t ⎠ ⎡ ⎤ Ω 20 0 0 ⎢ 0 ⎥ 2 ⎢ ⎥ Ω 21 ⎢ 0 Δ −Δ ⎥ (28) 0 21 20 ⎢ ⎥ 2 = ℏ⎢ ⎥, Ω 20 Ω 21 Ω32 ⎢ ⎥ −Δ20 2 2 ⎢ 2 ⎥ ⎢ ⎥ Ω32 0 −(Δ20 + Δ21) ⎥ ⎢⎣ 0 ⎦ 2
(
where Ω 20 = 0.02 − =
(
3/2 −
Φ32 L ℏCω0
).
Φ20 L ℏCω0
),
Ω 21 = −
Φ21 L ℏCω0
1 (Γ10 + Γ32 + γ31)ρ∼13, 2 ⎡Ω ⎤ Ω ρ∼˙22 = −i⎢ 32 (ρ∼32 − ρ∼23) + 21 (ρ∼12 − ρ∼21)⎥ + Γ32ρ∼33 − Γ21ρ∼22 , ⎣ 2 ⎦ 2 ⎡Ω ⎤ Ω ρ∼˙23 = −i⎢ 32 (ρ∼33 − ρ∼22 ) + Δ32ρ∼23 + 21 ρ∼13⎥ ⎣ 2 ⎦ 2 1 − (Γ21 + Γ32 + γ32 )ρ∼23, 2 ρ∼00 + ρ∼11 + ρ∼22 + ρ∼33 = 1. (31) The SPQC absorption and amplification properties as well as the group velocity of weak microwave probe field can be obtained from the linear susceptibility, χ = F d10 2 /(Vε0ℏΩ10)ρ10, which is the response of the system to the applied fields. Here F, d10, ε0 and V stand for the optical confinement factor, the dipole moment vector, the vacuum permittivity and the volume of the single SPQC, respectively. Real and imaginary parts of the susceptibility are considered as the dispersion and absorption responses of the system. In our notation, the positive value of the imaginary part corresponds to the absorption, while the negative value shows the gain. The transition 0 − 1 is excited by a weak microwave probe field for readout the dispersion, absorption and pulse propagation in the system. In the special conditions Δ10 < < Γ, γij = 0, Γ10 = Γ32 = Γ, Γ21 = 2Γ and for the weak probe field, Ω10 < < Γ, the following analytical expression is obtained for the probe transition coherence −
and Ω32
It is worth noting that, in second order approximation, only the cascade four-level type system can be established in SPQC and the different four-level quantum systems are obtained by keeping the third order of the Josephson junction potential expansion which exceeds the symmetry of the parabola harmonic oscillator Hamiltonian. Here, we drop the third order of interaction and consider a ladder type SPQC. The master equation for density matrix operator in the interaction picture is given by ∂ρ i ∼ = − [H͠ , ρ] + L [ρ], (29) ∂t ℏ ∼ where the operator L [ρ], indicating the decaying between the quantum states, can be written as [26, 27] Γi, i − 1 (2σi − 1, i ρσi, i − 1 − σiiρ − ρσii ) 2 i ∈{1,2,3} (30) γij σiiρσjj. − ∑ i, j ∈{0,1,2,3}; i ≠ j 2 ∼ L [ρ ] =
∑
5
Z Amini Sabegh et al
Laser Phys. Lett. 12 (2015) 085202
ρ10 =
2 2 2 ) + Ω32 (Ω221 + Ω32 )] 2Ω10 [4Γ 4 − Γ2(Ω221 − 4Ω32 2
−3
Δ10
2.5
2 Γ2(2Γ2 + Ω221 + Ω32 ) (32) 2 2 i(2Γ + Ω32)Ω10 − . 2 2 2 Γ (2Γ2 + Ω221 + Ω32 )
1.5 1
Re (ρ10)
0.5 0 −0.5 −1 Ω21=1
−1.5
Ω21=5
−2
Ω21=8
−2.5 −10
−5
0
∆
5
10
10
−3
c c = . vg = ω ∂χ ′ (ω) 1 (33) ng 1 + χ ′ (ω) + 2
(a)
2
According to equation (32) the slope of dispersion can be switch from positive to negative for small probe detuning. The imaginary part is negative for all value of the Rabi frequencies which is corresponding to a gain in the SPQC. Such behavior is similar to the four-level cascade type atomic system. In a dispersive medium the different frequency components of a pulse will experience different refractive index and then each frequency component in the pulse travels at different velocity. The group velocity of a pulse, velocity of the pulse peak, in a dispersive medium is determined by the slope of dispersion. We introduce the group index ng = c /vg where the group velocity vg at the frequency ω is given by
2
x 10
0
∂ω
x 10
−0.5
According to equation (33), the group velocity of the pulse in a dispersive medium can exceed the velocity of light in vacuum (c), leading to the superluminal pulse propagation. It is worth noting that the group velocity can be different from the information velocity and then such propagation does not violate the special relativity principle of Einstein. In our notation the negative slope of dispersion corresponds to the anomalous dispersion, while the positive slope shows the normal dispersion. Here we consider an array of many SPQCs on a line and investigate the weak microwave pulse propagation along the array.
(b)
−1
10
Im (ρ )
−1.5 −2 −2.5 −3 −3.5
Ω21=1
−4
Ω21=5 Ω =8 21
−4.5 −10
Results and discussions
−5
0
∆
5
10
10
Now, we are interested in summarize the numerical results of equation (22). We consider a weak probe field which is applied to the transition 0 − 1 and investigate the absorption and propagation of the probe field. We scale all frequency parameters by Γ10. Figure 2 shows the real (a) and imaginary (b) parts of the susceptibility versus probe detuning for different values of the Rabi frequency of applied microwave fields. Used parameters are Γ10 = Γ = 14 π MHz, Γ21 = 1.57Γ, Γ32 = Γ, γ10 = Γ, γ20 = 2.29Γ, γ21 = 2.57Γ, γ30 = 2.43Γ, γ31 = 1.71Γ, γ32 = Γ, [18] and the fields parameters are Δ32 = 0, Δ21 = 0, Ω10 = 0.01Γ, Ω32 = Γ, Ω 21 = Γ(solid), 5Γ(dashed), 8Γ(dash– dotted). An investigation on figure 2 shows that for Ω 21 = Γ the slope of dispersion around zero probe detuning is positive and it accompanies by a gain dip. But by increasing the Rabi frequency of the applied microwave fields the slope of dispersion switches to the negative corresponding to the superluminal pulse propagation. It is worth to note that such propagation is accompanying by a gain doublet with negligible value around zero probe detuning. Then the gain-assisted superluminal pulse propagation is obtained in the SPQC system. The numerical results are in good agreement with equation (32).
Figure 2. The real (a) and imaginary (b) parts of the susceptibility versus probe detuning for different values of the Rabi frequency of applied microwave fields. Used parameters are Γ10 = Γ, Γ21 = 1.57Γ, Γ32 = Γ, γ10 = Γ, γ20 = 2.29Γ, γ21 = 2.57Γ, γ30 = 2.43Γ, γ31 = 1.71Γ, γ32 = Γ, Δm = 0, Δc = 0, Ω10 = 0.01Γ, Ω32 = Γ, Ω 21 = Γ (solid), 5Γ (dashed), 8Γ (dash–dotted).
The corresponding group index behaviors versus probe detuning are shown in figure 3. Used parameters are same as in figure 2. It is clear that for small values of the Rabi frequencies the group index around Δ10 = 0, is greater than unity which means the group velocity is less than c. But by increasing the Rabi frequency the group index becomes less than unity or even negative leading to the group velocity greater than c or negative group velocity, respectively. The amplification without inversion is another scenario which we are following in the SPQC system. Our analytical result in equation (32) shows that the probe field will amplify for all values of the Rabi frequency of applied fields. We are going to investigate the inversion in the population of energy 6
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Laser Phys. Lett. 12 (2015) 085202
0.02
7
10
x 10
(a)
Ω =1 Γ
0.015
21
Ω =5 Γ 21
8
Ω21=8 Γ
0.01 0.005
g
4
Re (ρ30)
n −1
6
×3
0 −0.005
2
−0.01 −0.015
0
Ω=2.1 Ω=5
−0.02 −10
−2 −10
−5
0
5
∆
−5
10
0
∆
5
10
0.005
Figure 3. The corresponding group index behaviors versus probe
(b)
detuning corresponding to figure 2. Used parameters are the same as in figure 2.
0
ρ −ρ
00
−0.005
7
−0.9997
6
−0.9997
Im (ρ30)
11
8
Ω=2.1 Ω=5
−0.01
−0.015 5 Ω
32
−0.9998
−0.02
4
−0.9998
3 2 1
−0.025 −10
−0.9999
−0.9999 1
2
3
4
Ω
5
6
7
−5
0
∆
5
10
Figure 5. The real (a) and imaginary (b) parts of ρ30 versus Δ = Δ10 + Δ21 + Δ32 for Δ21 = Δ32 = 0, Ω10 = Ω 21 = Ω32 = 2.1Γ (solid), 5Γ (dashed). Other parameters are the same as in figure 2.
8
21
imaginary (b) parts of ρ30 versus the Δ = Δ10 + Δ21 + Δ32, for Δ21 = Δ32 = 0, Ω10 = Ω 21 = Ω32 = 2.1Γ(solid), 5Γ(dashed). Other parameters are same as in figure 2. We find that the system shows the gain at the frequency ω FWM via four-wave mixing process. Moreover the slope of dispersion for stimulated generated pulse becomes negative via the changing microwave pump tones feeding the system, so that for small values of the Rabi frequencies the slope of dispersion around zero detuning is negative. An investigation on the absorption spectrum shows that the generated superluminal pulse at the central frequency ω FWM is accompanied by a gain doublet. Then the gain-assisted superluminal pulse propagation is obtained for stimulated generated microwave field via four-wave mixing process in the SPQC system. Note that the four-wave mixing in second order nonlinear optical processes cannot occur in usual atomic system due to the electric-dipole selection rules.
Figure 4. The population difference of probe transition versus the Rabi frequencies. Used parameters are the same as in figure 2.
levels. The population difference of probe transition versus Rabi frequency is plotted in figure 4. Used parameters are same as in figure 2. It is found that the population inversion does not happen and amplification without inversion is established in this system. In the next step, we are interested in investigating the four-wave mixing in the SPQC system. We apply three microwave fields to the corresponding transitions as shown in figure 1(b). By annihilation of three photons, a fourth microwave photon with frequency ω FWM = ω10 + ω 21 + ω32 is generated via four-wave mixing. We focus on the thirdorder susceptibility medium response in the frequency ω FWM which is corresponding to ρ30, the coherence term of transition 0 − 3 . In figure 5, we plot the real (a) and 7
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Laser Phys. Lett. 12 (2015) 085202
Conclusion
0 −0.1
We have studied the interaction of the microwave fields with an array of the SPQCs. It was shown that the different four-level configurations can be established when the third order term of the Josephson junction potential expansion is included. We have investigated the generation and propagation of the microwave pulse in the four-level cascade SPQC systems. We found that by increasing the microwave pump tones feeding the system, the normal dispersion switches to the anomalous accompanied by the gain doublet. It was shown that the stimulated generated pulse via four-wave mixing propagates in superluminal condition and can be controlled by the external oscillating magnetic fluxes. Moreover, we found that the generation of superluminal microwave pulse appears without any inversion of the population of the transition states. The establishing of the absorption-free superluminal microwave pulse propagation and amplification of microwave field without inversion as well as the stimulated superluminal pulse propagation via four-wave mixing can be used to control of the SQCs and to quantum information transfer in these systems.
−0.2
ρ33−ρ00
−0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1
0
1
2
3
4
Ω
5
6
7
8
Figure 6. The population difference of states 0 and 3 versus
Rabi frequency in zero detuning of the applied fields for Ω10 = Ω 21 = Ω32 = Ω. Other used parameters are the same as in figure 2.
Note that the structure and dips position of the doublet gain can be understand via the four dressed states of the SPQC system which can be written as
References [1] You J Q and Nori F 2011 Nature 474 589 [2] Josephson B D 1964 Rev. Mod. Phys. 36 216 [3] Liu Y-X, You J Q, Wei L F, Sun C P and Nori F 2005 Phys. Rev. Lett. 95 087001 [4] Liu Y-X, Sun H-C, Peng Z H, Miranowicz A, Tsai J S and Nori F 2014 Sci. Rep. 4 7289 [5] Clarke J and Wilhelm F K 2008 Nautre 453 1031 [6] Harris S E, Field J E and Imamoglu A 1990 Phys. Rev. Lett. 64 1107 [7] Autlerand S H and Townes C H 1955 Phys. Rev. 100 703 [8] Arimondo E and Orriols G 1976 Nuovo Cimento Lett. 17 333 [9] Scully M O, Zhu S Y and Gavrielides A 1989 Phys. Rev. Lett. 62 2813 [10] Berman P R and Xu X 2008 Four-wave mixing in a Λ system Phys. Rev. A 78 053407 [11] Hau L V, Harris S E, Dutton Z and Behroozi C H 1999 Nature 397 594 [12] Wang L J, Kuzmich A and Dogariu A 2000 Nature 406 277 [13] Murali K V R M, Dutton Z, Oliver W D, Crankshaw D S and Orlando T P 2004 Phys. Rev. Lett. 93 087003 [14] Ian H, Liu Y-X and Nori F 2010 Phys. Rev. A 81 063823 [15] Joo J, Bourassa J, Blais A and Sanders B C 2010 Phys. Rev. Lett. 105 073601 [16] Baur M, Filipp S, Bianchetti R, Fink J M, Göppl M, Steffen L, Leek P J, Blais A and Wallraff A 2009 Phys. Rev. Lett. 102 243602 [17] Sillanpää M A, Li J, Cicak K, Altomare F, Park J I, Simmonds R W, Paraoanu G S and Hakonen P J 2009 Phys. Rev. Lett. 103 193601 [18] Li J, Paraoanu G S, Cicak K, Altomare F, Park J I, Simmonds R W, Sillanpää M A and Hakonen P J 2011 Phys. Rev. B 84 104527 [19] Sun H-C, Liu Y-X, Ian H, You J Q, Il’ichev E and Nori F 2014 Phys. Rev. A 89 063822
λ1 = −λ2 ⎞ 2⎛ 2 2 2 4 2 2 2 2 2 2 = ⎜ Ω10 + Ω 21 + Ω32 + Ω10 + 2 Ω10(Ω 21 − Ω32) + (Ω 21 + Ω32) ⎟, 4 ⎝ ⎠ λ3 = −λ 4 ⎞ 2⎛ 2 2 2 4 2 2 2 2 2 2 = ⎜ Ω10 + Ω 21 + Ω32 − Ω10 + 2Ω10(Ω 21 − Ω32) + (Ω 21 + Ω32) ⎟. 4 ⎝ ⎠
(34)
Each of the two dips in the gain doublet are approximately located at Δ = λ1 and Δ = λ2, however they are slightly shifted by including the pure inter-level dephasing rates. Finally we are looking for the amplification without inversion of the generated four-wave mixing pulse. In figure 6 we plot the population difference states 0 and 3 versus the Rabi frequencies for zero detuning of the applied fields. Used parameters are same as in figure 2. It is clear that the inversion of population does not appear in the transition 0 − 3 and stimulated generation of superluminal pulse propagation happens without inversion of population in SPQC system. The population difference of transition 0 − 3 for the parameters γij = 0, Γ10 = Γ32 = Γ, Γ21 = 2Γ, Ω10 = Ω 21 = Ω32 = Ω, Δ10 = Δ21 = Δ32 = 0, is given by ρ33 − ρ00 =−
72Γ 8 + 264Γ 6Ω2 + 268Γ 4Ω4 + 65Γ2Ω6 . (35) 72Γ 8 + 336Γ 6Ω2 + 442Γ 4Ω4 + 258Γ2Ω6 + 40Ω8
This expression reveals that for the conditions mentioned above, the population inversion cannot establish in transition 0 −3 . 8
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[24] Mahmoudi M and Evers J 2006 Phys. Rev. A 74 063827 [25] Berman P R and Xu X 2008 Phys. Rev. A 78 053407 [26] Carmichael H 1993 An Open Systems Approach to Quantum Optics (Berlin: Springer) [27] Breuer H P and Petruccione F 2002 The Theory of Open Quantum Systems (New York: Oxford University Press)
[20] Kelly W R, Dutton Z, Schlafer J, Mookerji B, Ohki T A, Kline J S and Pappas D P 2010 Phys. Rev. Lett. 104 163601 [21] Leung P M and Sanders B C 2012 Phys. Rev. Lett. 109 253603 [22] Hamedi H R 2014 Laser Phys. 24 115203 [23] Qiu T-H and Yang G-J 2012 Chin. Phys. B 21 104205
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