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We investigate electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) in a driven three-level superconducting artificial system which is a dressed-state system resulting from the coupling of a superconducting charge qubit (an artificial atom) and a transmission line resonator. In the frame of the dressed-state approach and steady-state approximation, we study the linear absorption of the dressed artificial system to a weak probe signal in depth. In light of the spectrum-decomposition method and some other restrictions, we obtain the explicit conditions for the dressed-state realization of EIT and ATS and present a corresponding “phase diagram”. In contrast to usual bare systems, these conditions given in the dressed system have an extra dependency on the qubit-resonator parameters. And by varying the qubit’s Josephson coupling energy we demonstrate a transition from EIT to ATS.
© 2015 Optical Society of America
Coherent effects induced by light-matter interaction at the quantum level lead to many striking and practical phenomena, such as electromagnetically induced transparency (EIT) [1, 2], lasing without inversion [3], coherent information storage [4], adiabatic population transfer [5]. As an important essential concept in quantum optics and atomic optics, dressed states [6] stemming from the mixing a two-level system with a resonant field have successfully been used to explain a lot of extraordinary optical behaviors of atomic systems, including those mentioned above.
Recently, dressed states have obtained a new testing ground and have attracted renewed interest in circuit quantum electrodynamics(QED) [7, 8], where superconducting qubits [9, 10] based on Josephson junctions behave as controllable artificial atoms and superconducting transmission line resonators play the role of conventional optical cavities. In circuit QED many novel optical phenomena previously inaccessible to usual cavity QED have been accomplished theoretically and experimentally, such as ultra-strong coupling regime [11], coexistence of single-and multi-photon processes [12, 13], quantum state collapse and revival due to the single-photon Kerr effect [14]. And in terms of the dressed-state approach, quantum information transfer [15] and dressed Zeno effect [16] are studied in superconducting circuits. In addition, Landau-Zener-Stückelberg interference in a dressed phase qubit [17] and dressed-state amplification in a flux qubit [18] are observed.
In this paper, we study the linear absorption spectrum of a weak probe field and discriminate EIT from Autler-Townes splitting (ATS) [19] in a driven three-level dressed superconducting artificial system derived from the interplay of a charge qubit and a transmission line resonator. EIT and ATS are two similar but distinct phenomena: they both exhibit a dip in the probe-field absorption profile. However, EIT is realized by Fano interference [20] between two atomic transitions, while ATS is simply due to the frequency shift induced by a strong control field. As EIT and ATS have different physical origin and potential applications, and these two effects are usually confused in some previous literatures, so to distinguish whether a dip in the absorption line is the character of EIT or ATS is a significant issue [21, 22]. Here in light of the spectrum-decomposition method introduced in [23, 24] and some other restrictions, we obtain the detailed conditions for the realization of EIT and ATS in the dressed three-level Λ-type artificial system. In contrast to the regular bare systems, these conditions described in the dressed system have an extra dependency on the qubit-resonator parameters. As a result, we present a dressed-state transition from EIT to ATS by tuning the energy spacing of the artificial atom in the solid-state circuit architecture.
Our model is a dressed three-level superconducting artificial system arising from the interaction of a superconducting charge qubit (a Cooper-pair box or a transmon qubit) with a superconducting transmission line resonator. Working in the qubit’s eigenbasis and at the charge degeneracy point, the qubit-cavity interaction is described by the Jaynes-Cummings Hamiltonian (h̄ = 1) [25]
where a† and a are the creation and annihilation operators for the single-mode field with angular frequency ωr, σz = |e〉 〈e| − |g〉 〈g| and are the qubit operators defined by its ground state |g〉 and excited state |e〉, is the effective Josephson coupling energy with the external flux Φx and the flux quantum Φ0, ℘ is the the interaction strength of the qubit-resonator coupling.In terms of the subspaces {|e,n〉, |g, n + 1〉}, the exact diagonalization of the Jaynes-Cumming Hamiltonian H yields the excited eigenstates (dressed states)
and the ground state |g, 0〉 with corresponding excited-state-energy spectrums and ground-state energy . Here the rotation angle θn is given byIn addition, two microwave signals, a weak probe field ωp with Rabi frequency Ωp and a strong control field ωc with Rabi frequency Ωc, are applied to the superconducting artificial system. The interaction Hamiltonian between the qubit and two classical fields can be expressed as
Considering the dressing effect and neglecting the ground state |g, 0〉, we rewrite the bare qubit’s raising operator σ+ in the dressed-state picture Note from Eq. (7) that similar to the usual atomic system, the dressed qubit presents the ”selection rules”: the allowed transitions only exist among dressed states possessing two adjacent energy quanta in the transmission line resonator. Moreover, the transition coefficients can be controlled by tuning the qubit-resonator parameters EJ, ωr and ℘. For superconducting circuits this tunability is a remarkable advantage over atomic systems.By frequency selection, a dressed three-level Λ-type superconducting system can be obtained, as shown in Fig. 1. Specifically, denoting the notations |1〉 = |−, 1〉, |+〉 = |+, 0〉 and |−〉 = |−, 0〉 for the first three levels of the multi-level dressed qubit according to Eq. (4) and choosing the frequencies ωp and ωc of the two microwave fields to be nearly resonant with the transitions |1〉 〈−| and |1〉 〈+|, we can write a total Hamiltonian similar to that of a three-level atomic system interacting with two driving fields in the dressed-state basis
Here ξp = − sinθ1 cosθ0Ωp and ξc = − sinθ1 sinθ0Ωc are defined as modified Rabi frequencies for the probe and control fields in the dressed-state representation. In this case, only these two effective transitions are induced by the driving fields and all other transitions can be ignored in our discussion. In the interaction picture, the total Hamiltonian with the rotating-wave approximation is written as where Δp = (E−,1 − E−.0) − ωp is the detuning of the probe field, and the control field frequency ωc is assumed to be resonant with the energy spacing E−,1 − E+,0 (i.e. Δc = 0).
Fig. 1 Schematic of a dressed three-level Λ-type system with a probe field ωp and a control field ωc driving the |−, 1〉 ↔ |−, 0〉 and |−, 1〉 ↔ |+, 0〉 transitions.
Using the standard density matrix formalism and the steady-state approximation, we obtain the first-order matrix element associated with the probe transition,
where the initial population is assumed to be prepared in the dressed ground state |−〉 and the damping rates Γ1− and Γ+− are inserted phenomenologically. It is well known that the absorption of the probe field is quantified by the imaginary part of the matrix element and can be used to characterize EIT or ATS.Now we intend to investigate the detailed characters of the probe absorption in accordance with the spectrum-decomposition method and then display the specific conditions for discerning EIT from ATS. By decomposing the first-order linear response of the dressed three-level system to the probe field into two Lorentzians, can be rewritten as
where Δ± are two spectrum poles of the linear response , From [23, 24], we know there is a threshold factor Γ1− − Γ+− which is employed to divide two different coupling regimes, and EIT occurs in the weak coupling regime |ξc| < Γ1− − Γ+− while ATS occurs in the strong coupling regime |ξc| > Γ1− − Γ+−, respectively. However, it is well known that the probe field is substantially absorbed and no EIT dip is exhibited when the amplitude of the control field is comparable to or even smaller than that of the weak probe field. Thus the threshold factor is not completely sufficient condition to guarantee the achievement of EIT and ATS. The emergence of a dip in the absorption spectrum demands that the imaginary part of the matrix element must be a local minimum at the origin Δp = 0. Thus the second derivative of at Δp = 0 should be larger than zero and we have Combining this equation with the threshold factor Γ1− − Γ+−, we obtain the specific criterion for the appearance of an absorption peak and a dip (EIT or ATS), Because all this confusion about a dip in EIT and ATS comes from their overlapped regions and EIT only occurs in the area given in Eq. (15), in the above formulae we just consider the range Γ1− > 2Γ+− and neglect another ATS case in Γ1− ≤ 2Γ+−. Note that in the above expressions we introduce a new threshold factor to describe the EIT regime completely. Furthermore, different with the regular bare natural or artificial systems, these conditions obtained for EIT and ATS in the dressed system have an evident dependency on the rotation angles θ1 and θ0 and then have an implicit dependency on the parameters EJ, ωr and ℘ of the composite qubit-resonator superconducting system. We stress that considering the huge tunability of superconducting artificial atom, these dependencies possess important significance and can lead to flexible quantum physical phenomena in superconducting circuits compared with the conventional atomic systems with fixed parameters, as will be seen in later part of this paper.According to the aforementioned conditions given in Eqs. (14)–(16), we present a vivid “phase diagram” displaying the clear absorption peak, EIT and ATS regions separated by two reduced threshold factors 1 − Γ and as a function of the relative damping rate Γ = Γ+−/Γ1−, as shown in Fig. 2(a). The yellow (green) area illustrates the EIT (ATS) region while the purple area indicates a single peak in the probe absorption spectrum. To facilitate the following analysis, in Fig. 2(b) we plot the response profile of the coupling coefficient sinθ1 sinθ0 versus the Josephson coupling energy EJ to a given ℘ value.
Fig. 2 (a) Peak, EIT and ATS regions divided by two reduced threshold factors 1 − Γ and . The yellow (green) region denotes the EIT (ATS) regime while the purple area represents single absorption peak. (b) Coupling coefficient sinθ1 sinθ0 of the control field as a function of the Josephson coupling energy EJ for ℘ = 0.008ωr.
On the basis of the above discussion, Fig. 3 presents the evolution of the linear absorption and two Lorentzians Im[β±(Δp)] as a function of the probe detuning Δp for various values of the Josephson coupling energy EJ. The first three subgraphs are plotted with the same parameters except for EJ and correspond to three different point (Γ = 0.2) in the purple, yellow and green regions in Fig. 2(a) respectively. In Figs. 3(a) and 3(b), two Lorentzians are located at the origin: one positive and the other negative, and their sum displays a single absorption peak for EJ = 1.025ωr in Fig. 3(a) and leads to a dip for EJ = 1.005ωr in Fig. 3(b). The dip is due to the strong quantum destructive interference between the two Lorentzians and is a feature of EIT phenomenon. In Fig. 3(c) with EJ = 0.95ωr, two positive Lorentzians centered at different positions result in a dip, which originates from a gap between the two Lorentzians and is deemed to be an ATS character. Thus, we achieve a transition from single absorption peak to EIT again to ATS only by tuning the Josephson coupling energy EJ in the dressed superconducting three-level system. We also show that as Γ decreases, quantum destructive interference becomes more strong and the more deep EIT dip is exhibited, as seen in Fig. 3(d).
Fig. 3 Imaginary part of the matrix element (red solid line) and two Lorentzians β+(Δp) (blue dashed line) and β−(Δp) (black dashed-dotted line) as a function of the probe field detuning Δp for different values of the Josephson coupling energy EJ. (a) EJ = 1.025ωr and Γ+− = 0.2Γ1−, (b) EJ = 1.005ωr and Γ+− = 0.2Γ1−, (c) EJ = 0.95ωr and Γ+− = 0.2Γ1− and (d) EJ = 1.005ωr and Γ+− = 0.01Γ1−. The other parameters are ℘ = 0.008ωr and Ωc = 1.5Γ1−.
Finally, we emphasize that the dressed states of superconducting charge qubit have been directly probed in recent experimental reports [26, 27]. Besides, the dressed-state model has been successfully applied to the superconducting circuits. For instance, microwave amplification of a strongly driven flux qubit in a coplanar waveguide resonator has been observed and theoretical predictions obtained in the dressed-state picture are in good agreement with the experimental results [18]. So our proposal for dressed-state realization of EIT and ATS in circuit QED is well-founded and is feasible in the current technology.
In summary, we have presented an alternative scheme to study EIT and ATS in a dressed three-level superconducting Josephson system stemming from the interaction of a superconducting charge qubit with a transmission line resonator. Based on a new threshold factor we have obtained the completely sufficient conditions for the realization of EIT and ATS in the dressed-state frame and have given an objective “phase diagram” for visualization according to these conditions. Different with the conventional bare natural or artificial systems, these conditions obtained in the dressed system have an evident dependency on the parameters of the composite qubit-resonator system. And by varying the qubit’s Josephson coupling energy, we display a transition from EIT to ATS for the dressed-state artificial system. This exhibits the superiority of tunability of superconducting circuits again in contrast with the usual atomic systems. Our investigation on dressed-state EIT and ATS would be useful for analysis of many relevant phenomena due to abundant applications of dressed-state approach in atomic optics and quantum optics.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under the Grant No. 11274132.
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