1. Introduction

Since the discovery of quantum mechanics, the interactions between (artificial) atoms and photons have become an important research area in both fundamental physics and practical applications [1–6]. Quantum interference [7–10,14,15] especially plays a substantial role for the diversified manifestations of the light-matter interactions [16–18]. In principle, enhancement, reduction, or no change of transition probability may happen due to the interference between amplitudes from alternative pathways. The photon fields can thus control the optical response of the matter and the quantum interference effect can occur as phenomena of the electromagnetically induced transparency (EIT) [8–13] and the Fano resonance [14,15]. These abnormal optical responses resulting from quantum interference have been observed in atomic and molecular systems [19,20], photonic crystals [21], solid-state systems [22], microresonators [23], metamaterials [24] and electronic systems [25]. How to tune the quantum coherence has then become a very important issue. In the past few decades, the Landau–Zener–Stückelberg (LZS) [26] interference has received a lot of attention for serving as a feasible way to control the quantum interference by periodically changing the relative phase between quantum states [27].

In this paper, we investigate the interaction between a three-level atom and two photon fields with different frequencies as depicted in Fig. 1. We assume that the intensity of one light field (control beam, $\Omega _c$) is much stronger than the other (probe beam, $\Omega _p$). Due to interactions between the strong control beam and the atom, the interacting atom becomes a $``\rm {dressed}"$ one. Then it would have great impact on the quantum states of the probe beam which becomes a $``\rm {dressed}"$ field through interacting with the dressed atom. Henceforth, the dispersion relation and associated physical behaviors of the dressed probe photon would be quite different from those of free photons [28]. Through the well-known Feynman’s diagrammatic method in field theory of QED, we calculate the dispersion relation and transmission of the dressed probe field. We find significant tunable quantum interferences when the frequency of the control beam is around the resonant regimes of the atom. For example, the asymmetric patterns and locations of Fano resonances and the widths of EIT windows ($\propto \Omega _c^2$) can be adjusted by tuning the intensities of the two photon fields, and the frequency of the control field. With the amplitude modulated control beam, i.e., the so called Landau-Zener-Stückelberg (LZS) type source, we find that the signatures of quantum coherence shows oscillating behavior in the transmission of the probe photon. We further find that the destructive interference from a sequence of Landau-Zener transitions can lead to a wide EIT window in time. In this paper, we use natural units by putting $\hbar = c= 1$ as is widely adopted in field theory literature, $e.g.$, Ref. [29].

 

Fig. 1. A schematic diagram of the bare-state $\Lambda$-type three-level EIT/Fano system: ground state $|1\rangle$; metastable state $|2\rangle$; excited state $|3\rangle$. The probe field of frequency $\omega _p$ governs the $|1\rangle \rightarrow |3\rangle$ transition with the electric current density $J_p$, and the control field of frequency $\omega _c$ drives the $|2\rangle \leftrightarrow |3\rangle$ transition with the electric current density $J_c$.

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2. Model formulation

Here we assume the EM waves to propagate in $x$-axis and they are uniform in the transverse. Therefore, in radiation gauge ($\nabla \cdot \vec {A}=0$), the vector potential $\vec {A}$ can be written as $\vec {A}(x)=(0,A_y(x),A_z(x))$. The photonic annihilation operator $A$, and the photonic creation operator ${A}^{\dagger }$ are defined as,

t-matrix. To handle the hard-core interaction in Eq. (4), we can follow the calculations done in Ref. [30] by using the method of binary collision [31,32]. We add up all the repeated scatterings between different fields $c_\alpha$ and $c_\beta$ (ladder diagrams in Fig. 2(a)) to get a finite effective coupling ($u_0$) between them at low energy [30]. For brevity, we shall not repeat calculations parallel to those shown in Ref. [30]; and similar results are obtained: The effective coupling $u_0=\frac {1}{2}(\varepsilon _{\alpha }+\varepsilon _{\beta }-P_0)$ with the assumption that the total collision energy $P_0\ll (\varepsilon _{\alpha }+\varepsilon _{\beta })$ for long wavelength scatterings. The modification to the self-mass is $\Sigma _\alpha (\omega )=\frac {1}{4}(\varepsilon _{\alpha }-\omega )+$ constant $C_1$. It follows that all three (bare) energy levels $\varepsilon _{\alpha }$’s are shifted by the same amount of constant $\frac {4}{5} C_1$ due to the hard-core interaction, and this can be incorporated into the theory by redefining the shifted energy levels as the renormalized (physical) energy levels $\nu _{\alpha }$ as what is obtained in Ref. [30].

 

Fig. 2. (a) Diagrammatic expansion of the effective coupling $u_0$ which is the sum of the repeated and continuous scatterings between two different fields $c_\alpha$ and $c_\beta$. The external legs are only for the eyes, the black dot $``\bullet "$ is $U$ in Eq. (4), and the internal double-line or single line are only used to represent different index of the electronic bare propagator of field $c_\gamma$, $\widetilde \Delta ^{(\pm )}_{(0),\gamma }$’s, $\gamma$=1,2,3 (Eq. (5)). (b) Diagrammatic expansions of the self-mass $\Sigma ^{(2)}_{\alpha }(\omega )$ (Eq. (9)) for the renormalized propagators $\widetilde \Delta ^{(\pm )}_{(2),\alpha }$ (Eq. (8)) of the dressed electronic field $c_\alpha$. The internal straight line represents electron propagator $\widetilde \Delta ^{(\pm )}_{(1),\alpha }$ (Eq. (7)). The curvy line represents the propagator of free photon $\widetilde {G}_{(0)}$ (Eq. (3)). (c), (d) Diagrammatic representations of self-masses $\Pi ^{(2)}(\omega )$ (Eq. (11)) and $\Pi ^{(4)}(\omega )$ (Eq. (12)) of the renormalized propagator of the dressed probe photon, $\widetilde {G}(k,\omega )$ (Eq. (10)) are shown in (c) and (d), respectively. The solid black lines represent the renormalized electronic propagators $\widetilde \Delta ^{(\pm )}_{(2),\alpha }$, and the (dashed) curvy external legs are only for the eyes.

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Electron propagator. It follows that the energy differences between atomic energy levels are unchanged. In the mean time, the renormalized propagators of the electron field $c_\alpha$ become

Photon propagator. Under the illumination of the control beam with frequency $\omega _c$ and amplitude $J_c$, the propagator of the dressed photon $\widetilde {G}( k,\omega )$ from the probe beam can be calculated from the Feynman diagrams (Figs. 2(c) and 2(d)), and it satisfies the following Dyson’s equation,

 

Fig. 3. The dispersion relation of free photon, $K_{\omega }c$ versus $\omega /\nu _0$, shown as the straight green dotted line. The real and imaginary parts of $K_{\omega }c$ of the dressed probe photon are represented as the blue solid and red dashed lines, respectively. Here we assign the three energy levels $\nu _1$, $\nu _2$, $\nu _3$, to be $\nu _0$, 3$\nu _0$, 4$\nu _0$, respectively, with $\nu _0$ an appropriate energy scale denoting the transition frequency between $|2\rangle$ and $|3\rangle$. In plotting this figure, we choose $g^2=1/137$ and $\delta =8\times 10^{-3}$, while the frequency and intensity of the control beam are $\omega _c/\nu _0=2.9$ and $J_c=9/\nu _0$, respectively. The intensity of the probe beam is $J_p=3/\nu _0$. The inset shows the real part of the index of refraction $n$. All the above-mentioned parameters are shown in arbitrary unit.

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The index of refraction $n(\omega )$ can also be obtained [33] (here we put $c$ back),

 

Fig. 4. The relative transmission $T(\omega )$ of dressed probe photon. Here, we use the same parameters as those in Fig. 3 except the $J_c$: $J_c= 17/\nu _0$ (red dashed line), $J_c= 9/\nu _0$ (black solid line), and $J_c= 0$ (blue dotted line).

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As can be seen in Fig. 3, the dispersion relation of the dressed probe photon $K_\omega$ is nonlinear with energy gaps around which the momenta are complex corresponding to attenuated waves of absorbed photons. It should be noticed that the shape of $K_\omega$ and especially the locations of gaps depend largely on the frequency $\omega _c$ and the electric current densities of the control and probe beams ($J_c$ and $J_p$). Therefore, the relative rate of transmission of the dressed probe photon $T(\omega )$ shows different performances in Fig. 4. Taking the black solid line in Fig. 4 as an example that the corresponding electric current density and frequency of the control beam are $J_c=9/\nu _0$ and $\omega _c/\nu _0=2.9$. Note that $\nu _0$ is an appropriate energy scale denoting the transition frequency between $|2\rangle$ and $|3\rangle$. The results show an apparent Fano resonance around probe photon frequency $\omega /\nu _0=0.7$, a less apparent one is around $\omega /\nu _0=1.22$, and a window of EIT roughly located between $\omega /\nu _0=0.8$ and $\omega /\nu _0=1.1$. Comparing the dispersion relation in Fig. 3 with the transmission curve shown as the black solid line in Fig. 4, it can be found that slow photons appear around the Fano resonance frequencies. Moreover, the differences between the red dashed line ($J_c=17/\nu _0$) and the blue dotted line ($J_c=0$) (same $\omega _c$ and $J_p$) shown in Fig. 4 indicate that the asymmetric patterns and locations of Fano resonances, and the widths of EIT windows are modified by tuning the electric current density of the control beam $J_c$. When the control beam is absent, there usually exists Lorentzian lineshapes in the scattering/absorption spectra. However, the blue-dotted curve in Fig. 4 shows a non-Lorentzian behavior around $\omega =0.9 \nu _0$. This is because the amplitude of the next-to-leading order diagram [Fig. 2(d)] cannot be ignored compared with that of the leading-order diagram [Fig. 2(c)] therein, the net effect of both leading-order and next-to-leading order scattering processes therefore leads to this non-Lorentzian lineshape in transmission. This has been demonstrated more clearly in Fig. 5 which shows the influences of adjusting the intensities (amplitudes) of the control and probe beams on the transmission. For example, for fixed frequencies of the control and probe beams, the (relative) transmission $T(\omega )$ can be varied from $T(\omega )<1$ to $T(\omega )>1$ when we tune the amplitude of the control beam (probe beam) and keep the amplitude of the probe beam (control beam). The varying quantum interference can also be clearly seen in Fig. 5. For example, when $5J_p=10/{\nu _0}$ in the upper x-scale (which coincides with $J_c= 10/{\nu _0}$ in the lower x-scale ), the lineshape of $T_{p1} (T_{p2})$ shows the constructive (destructive) interference.

 

Fig. 5. (a) (Lower x-scale) The relative transmission $T(\omega )$ of dressed probe photon versus the electric current density of the control beam $J_c$. The probe frequency $\omega _p/\nu _0=0.5$ ($T_{c1}$, the black solid line) and $\omega _p/\nu _0=0.6$ ($T_{c2}$, the red solid line), and the amplitude of the probe beam is fixed as $J_p=1/\nu _0$. (b) (Upper x-scale) The relative transmission $T(\omega )$ of dressed probe photon versus 5 times of amplitude of probe beam ($5J_p$). The probe frequency $\omega _p/\nu _0=0.5$ ($T_{p1}$, the black dashed line) and $\omega _p/\nu _0=0.6$ ($T_{p2}$, the red dashed line), and the amplitude of the control beam is fixed as $J_c=14.6/\nu _0$. Other parameters like $\omega _c$, $g$, and $\delta$ are the same as those in Fig. 3.

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3. Landau-Zener-Stückelberg interference

We now study the effect of periodic driving to this three-level system, that is, to consider the control beams with auxiliary waves of oscillatory amplitudes. The resulting quantum interference is called the Landau-Zener-Stückelberg (LZS) interferometry [26]. The amplitude of the oscillatory control beam can be written as

The leading order contributions to the modifications of transmission amplitudes of dressed probe photons (frequency $\omega$) with the two sideband waves $J_\pm (t)$ on the control beam are represented diagrammatically in Fig. 6. It involves the processes of wave $J_c(t)$ as the source (sink), and the sidebands $J_\pm (t)$ as the sink (source). Note that higher order terms of $\gamma$ would be ignored for a sufficiently small $\gamma$. The analytical expression of the probability amplitude represented by Fig. 6 is

Therefore, similar to Eq. (15), we can derive the relative transmission of probe beam with frequency $\omega$ at time t, $T(\omega ;t)$.

 

Fig. 6. Diagrammatic representation of the processes for the leading orders of transmission of dressed photon with the LZS control beam. Here $J^*_i=J^*_c$, $J_j=J_\pm$, or $J^*_i=J^*_\pm$, $J_j=J_c$, and $\omega _c\approx \nu _3-\nu _1$, $\omega \approx \nu _3-\nu _2$ will be adopted in the numerical calculations shown in Figs. 7 and 8.

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Numerical results of $T(\omega ;t)$ are depicted in Figs. 7 and 8. In Fig. 7, the (relative) transmission $T(\omega ;t)$ shows more cycles of Fano resonance and EIT due to the LZS interference. For most ranges of frequencies, the (relative) transmission is modified to be oscillatory in time due to the oscillatory LZS control beam (black solid line and blue dashed line in Fig. 8). It is interesting to note that when the frequency of the probe beam is larger than a certain value, e.g., $\omega _p/\nu _0=1.3$ in Fig. 8, $T(\omega ;t)$ maintains almost a constant in time. Specifically, the (relative) transmission rate $T(\omega ;t) \approx 1$ as $\omega _p/\nu _0=1.3$ at which EIT occurs (red dotted line in Fig. 8). Figures 7 and 8 indicate that with the interference of the Stückelberg phase accumulated between Landau-Zener transitions, the Fano resonance and EIT can occur in an alternating way. More importantly, with appropriate frequencies of the probe photons, the transmissions become almost stationary in addition to a wide EIT window in time.

 

Fig. 7. The relative transmission $T(\omega ;t)$ of dressed probe photon with LZS type control beam observed at different times $t=0$ (red solid line) and $t=7.5~(\nu _0^{-1})$ (blue dashed line). The parameters of the oscillatory part of the control beam defined in Eq. (16) are $\gamma =0.05$, $\Omega /\nu _0=0.2$. The black dotted line is used for comparison, and it indicates the relative transmission when the control beam does not contain the LZS oscillation component ($\gamma =\Omega =0$). Here, $J_c=13/\nu _0$, $J_p=3/\nu _0$, and other parameters like $\omega _c$, $g$, and $\delta$ are the same as those in Fig. 3.

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Fig. 8. The relative transmission $T(\omega ;t)$ of dressed probe photon with LZS type control beam versus observation time $t$/$\pi$. Here, the control beam frequency $\omega _p/\nu _0=1$ (black solid line), $\omega _p/\nu _0=1.225$ (blue dashed line), $\omega _p/\nu _0=1.3$ (red dotted line), and $\omega _p/\nu _0=1.35$ (green dotted dashed line). Other parameters like $\omega _c$, $J_c$, $J_p$, $g$, and $\delta$ are the same as in Fig. 7.

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As shown in Figs. (4)–(8), the (relative) transmission can go beyond unity. This is because in this work, we adopted a field theoretical approach [Eq. (1)–Eq. (6)] which deals with a many-body system. As bosonic fields, photon fields can have particle numbers more than one, and we study multi-photon scattering processes such that the transmission of the dressed probe beam can be greater than unity under some conditions.

4. Conclusions

Fano resonance and EIT are well-known signatures of interference in multi-level quantum systems. However, in this paper, we show that the quantum interference in a three-level atom is not only frequency-dependent but also tunable through adjusting the intensities of control and probe fields. We also present that the alternating Fano/EIT lineshapes appear when the LZS interferometry is introduced to the control fields. Besides, under the condition that the frequency of the probe beam is larger than a certain value, the system turns out to be transparent for a relatively long time.

The theoretical scheme on tuning quantum interference we studied in this work can be experimentally realized in many three-level quantum systems, such as superconducting qubits, gate-control quantum dots, and trapped atoms, and microcavity systems. Since the scheme has the potential tuning the quantum interference, the results of tunable dispersion and EIT (Fano)-window can therefore be applied to the studies of lossless light-transmission, light delay and storage [34–36], quantum sensing, the optical quantum memory [37–41] in the nanostructures, the optical nonlinearity [42–44], the high-dimension quantum entanglement, and the adiabatic quantum computing [45,46].