Open Access
Add to BibSonomy
Abstract
As one of the typical quantum coherence phenomena, electromagnetically induced transparency (EIT) has been extensively applied to implement various quantum coherent manipulations, typically, e.g., optical quantum memories, photonic switches, and optical quantum computations, etc. By applying the input-output theory to the photonic transports through two cavities dispersively coupled by an artificial two-level atomic ensemble, we show here that the EIT-like effects could be observed. Particularly, the transparency windows and phase shift spectra of the transmitting photons could be engineered by manipulating the atomic levels in the ensemble to adjust the effective coupling strength between the cavities. As a consequence, the group delays of the transmitting photons can be manipulated by using the EIT-like effects. The proposal is demonstrated specifically with the experimental superconducting coplanar waveguide resonators coupled by the voltage-biased electrons on liquid Helium.
© 2022 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) of the light scattered by the driven atomic gas is a typical quantum interference effect [1–5]. Specifically, due to the application of the control field, the spectrum of the probing field shows a transparency window with the abnormal dispersion, resulting in a dramatic reduction of the group velocity of the probing field. This is the so-called slow light effect, which plays an important role in the optical quantum science and technology [6], typically for optical quantum memories [7], photonic switches [8,9], and optical quantum computation [3], etc.. Remarkably, with the EIT effect in the ultracold atoms, slow light with the group velocity of $17$m/s (which is seven orders of magnitude lower than the speed of light in a vacuum) has been recently demonstrated [10].
Analogously, the EIT-like effects have been demonstrated with various cavity systems, e.g., the cavity containing the multi-level atoms [11], coupled waveguides [12], and coupled cavities [13–15], etc. In these configurations, the actions of the control fields are replaced by the coupling structures such as the multi-level atoms, waveguide, and cavity, respectively. However, the slow light effects in these EIT-like systems have been paid relatively less attention [16], as the transparent windows of the probing fields in these EIT-like effects are usually uncontrollable. This limits the applications of the EIT-like effects.
To overcome such a difficulty, certain auxiliary systems have to be utilized. For example, an optomechanical can be introduced to adjust the power of the ’control’ field [16–20] and a tunable coupler is designed to control the coupling strength between the cavities [21–25]. Typically, the dynamic control of the group delay of the probe field up to about $1.2$ms is achieved by adjusting the pump power in an optomechanical double-ended cavity [16]. Alternatively, in this paper, we propose a novel configuration by introducing an atomic ensemble [26,27] to implement the control of the effective coupling strength between two cavities. As a consequence, the EIT-like effects with controllable transparency windows can be observed, and thus the group delay of the photons transmitting through such a cavity-atomic ensemble-cavity structure could be manipulated.
The paper is organized as follows. In Sec. 2, we first discuss how the group delay could be reached for the photons transmitting through an empty two-sided cavity. We also show that such a group delay is unchanged if the cavity contains a dispersively coupled atomic ensemble, which just shifts the center frequency of the transmitting photons. Differently, in Sec. 3, we show that if the dispersive atomic ensemble is introduced to implement the effective coupling between the distant cavities, then the transparency windows of the photons transmitting through the cavities can be observed. In such a cavity-atomic ensemble-cavity configuration, the group delays of the transmitting photons can be engineered by manipulating the eigenfrequency of the two-level atom in the ensemble. In Sec. 4, we specifically demonstrate the proposal with an artificial atomic ensemble generated by the voltage-biased electrons floating on liquid Helium. Finally, we summarize our results in Sec. 5.
2. Group delays of the photons transmitting through a two-sided cavity
In this section we first discuss how the light delay phenomenon appears when the photons transmit through a single two-side cavity. Furthermore, we discuss how the dispersively coupled atomic ensemble influences the transmission properties of the traveling wave photons.
2.1 Photonic transport through an empty two-sided cavity: the input-output relations
For simplicity, we first consider the configuration shown in Fig. 1, wherein the traveling wave photons transmitting through a two-sided cavity. Suppose that the dissipation rate of the cavity is $\Gamma$, then the Hamiltonian of the system can be written as $(\hbar =1)$ [28]:
Fig. 1. Schematic diagram of the traveling wave photons transmitting through a two-sided cavity. Here, the traveling wave photons incident from the left into the cavity, with the dissipation rate $\Gamma$, and output along the right side of the cavity.
Following Collett and Gardiner [28], we treat such a photonic scattering problem by the input-output theory. First, the Heisenberg equation for the operator $c_{m}(\omega _{m})$ reads
The solution to this equation may be written in two ways. In terms of initial conditions at time $t>t_{0}$ (i.e., the input field), it reads
Similarly, the Heisenberg equation for the cavity field operator $a$ can be written as
Substituting Eq. (3) into Eq. (5), we get
Under the usual Markov approximation, $k_{ac_{L}}$ and $k_{ac_{R}}$ can be assumed to be frequency-independence. As a consequence, an input field operator of the cavity can be introduced as
Applying the Fourier transformation: $x(t)=\int _{-\infty }^{+\infty }e^{-i\omega (t-t_{0})}x(\omega )d\omega /\sqrt {2\pi }$, with $x(t)=a(t), a_{in}^{(m)}(t), a_{out}^{(m)}(t)$, to Eq. (8) and Eq. (10), we get
For the photonic scattering configuration shown in Fig. 1, wherein the traveling wave photons input from the left side of the cavity (i.e., the input field from the right side vanishes and thus $a_{in}^{(R)}=0$), the transmission amplitude of the photons through the cavity can be expressed as
Similarly, the probability of a traveling wave photons being reflected by the cavity reads
The corresponding phase shift of the reflected photons can be expressed as
simply for $\gamma _{L}=\gamma _{R}=\gamma$ and $\Gamma =0$.Figure 2(a) shows the transmission coefficient $T_{e}(\omega )=|t_{e}(\omega )|^{2}$ versus the frequency $\omega$ of the traveling wave photons. It is seen that the transmission spectrum of the photons shows the standard Lorentzian shape: the maximal transmission probability is at the resonance point, i.e., $\omega =\omega _{0}$. Also, due to the dissipation of the cavity, the maximal transmission probability $T_{max}\neq 1$.
Fig. 2. The transmission spectrum (a) and phase shift spectrum (b) of the traveling wave photons transmission through the two-sided cavity. The relevant parameters are set as: $\gamma _{L}=\gamma _{R}=0.05\omega _{0}$, $\Gamma =0.01\omega _{0}$.
Interestingly, from Fig. 2(b), one can find that the phase shift spectrum of the transmitted photons changes sharply around the cavity frequency $\omega _0$. In fact, this behavior is related to the slow light effect [29], with the group delay [2,30]:
Here, $L$ is the length of the cavity, and $c=3\times 10^{8}$m/s is the speed of traveling light in a vacuum. We now investigate how the transmission properties of the traveling wave photons are influenced by the parameters of the driven cavity. Without loss of the generality, we assume that $\gamma =\gamma _{L}=\gamma _{R}$. Figure 3(a) shows that, for a fixed dissipation rate $\Gamma$, the width of the transmission peak increases with the effective coupling strength $\gamma$. Importantly, Fig. 3(b) shows that the group velocity of the photons through the cavity increases with the effective coupling strength $\gamma$. Specifically, for $\omega _{0}/2\pi =3.09839$GHz, $\gamma _{L}=\gamma _{R}=0.05$GHz, $\Gamma =0.01$GHz, and $L=19.158$mm [31], we have $\tau _e\approx 16.67$ns, and thus $v_{g}\approx 1.14509\times 10^{6}\ll c$. Physically, the stronger $\gamma$-parameter implies the lower quality factor of the cavity [32,33], and thus the shorter group delay of the transmitted traveling wave photons through the cavity. Therefore, the slow light effect is still very remarkable; even the photons transmit through an empty cavity. This phenomenon is usually paid less attention, although it has been experimentally demonstrated [34].
Fig. 3. (a) The transmission probabilities versus the effective coupling strength $\gamma =\gamma _{L}=\gamma _{R}$ between the traveling wave photons and cavity for different incidence photons frequencies. (b) The group velocity $v_{g}$ varies with the effective coupling strength $\gamma =\gamma _{L}=\gamma _{R}$. The relevant parameters are chosen as: $\omega _{0}/2\pi =3.09839$GHz, $L=19.158$mm, $\Gamma =0.01$GHz [31].
2.2 Transmission of the photons through a two-sided cavity with a atomic ensemble
As shown in the above subsection, the group delay (and thus the group velocity) of the photons transmitting through the empty cavity depends mainly on the effective coupling strength $\gamma _{L,R}$ between the travelling wave photons and the standing wave ones, and also the dissipation rate $\Gamma$ of the cavity. In this subsection, we discuss theoretically what happens when the cavity contains an atomic ensemble, as shown in Fig. 4. Under the rotating wave approximation, the Hamiltonian of the present system can be expressed as $(\hbar =1)$
Fig. 4. A schematic representation of the traveling wave photons transmitting through a two-side cavity embedded by a two-level atomic ensemble. The dashed green line represents the distribution of the electric field in the cavity. $\omega _{eg}$ and $\lambda$ are the eigenfrequency of the two-level atom and its interaction with the cavity, respectively.
First, if the cavity and atomic ensemble is resonantly coupled, the Heisenberg equations of motion, associated with the variables $a$ and $b$, can be expressed as
Consequently, the input-output relations between input (output) operators $\widehat {a}_{in}^{(m)}(\omega )$ $(\widehat {a}_{out}^{(m)}(\omega ))$ read
Fig. 5. Transmission/Reflection spectra of traveling photons transmitting through the resonantly coupled cavity-atomic ensemble system and the relevant phase shifts. Here, the dashed red lines are the situations of the empty cavity scattering, shown in Fig. 2. The parameters are chosen as $\gamma _{L}=\gamma _{R}=0.05\omega _{0}$, $\Gamma =0$, $g_{a}=0.1\omega _{0}$.
Next, let us consider the situation wherein the coupling between the cavity and atomic ensemble is dispersive [39,40]. Without loss of the generality, the atomic decay is neglected for simplicity. Under the large detuning limit, i.e., $|\Delta _{a}|=|\omega _{eg}-\omega _{0}|\gg \lambda$ (In general, $\omega _{0}>\omega _{eg}$ [39,40].), and by performing the unitary transform $U=exp[g_{a}(ab^{\dagger}-ba^{\dagger})/\Delta _{a}]$ to the Hamiltonian (20), we obtain the following dispersive Hamiltonian [41]:
It is seen that, the existence of the dispersive atomic ensemble just shifts the center frequency of the cavity. Above, we have assumed that the two-level atom in the cavity is initially at its ground state $|g\rangle$.
Following the above subsection, the transmission probability of the cavity reads
with the amplitudeAlso, the phase shift of the transmitted photons can be obtained as
Similarly, the reflection probability of the cavity can be expressed as
with the amplitudeThe phase shift of the reflected photons can be calculated as
Fig. 6. (a)Photonic transmission spectra: $T_a(\omega )=|t_a(\omega )|^{2}$ and (b) the phase shift of the transmitted light. Here, the dashed red lines are the situations for the empty cavity scattering shown in Fig. 2. The parameters are chosen as $\gamma _{L}=\gamma _{R}=0.05\omega _{0}$, $\Gamma =0$, $g_{a}^{2}/\Delta _{a}=0.1\omega _{0}$.
3. Controllable group delay of the transmitted photons through two cavities coupled dispersively by an atomic ensemble
In order to achieve the manipulation of the group delay for the photons through the cavities, in this section, we investigate the transport of the photons transmitting sequentially through a cavity-atomic ensemble-cavity system, which is simply in Fig. 7, and discuss how to control the EIT-like effect [2] by engineering the effective cavity-cavity interactions.
Fig. 7. The driven two cavities coupled via an atomic ensemble. Here, $g_{l,r}$ represents the coupling strength between the left (right) and the two-level atomic ensemble, which acts as a controllable coupler between two cavities.
3.1 EIT-like effect in a two-cavity system embed by an atomic ensemble
Again, under the rotating wave approximation the Hamiltonian of the present system can be expressed as
Furthermore, in order to achieve the coupling between the two cavities, we adiabatically eliminate the degrees of freedom of the atomic ensemble [41]. Indeed, under the large detuning limit, i.e., $|\Delta _{j}|=|\omega _{eg}-\omega _{j}|\gg \lambda _{j}$, we can perform the unitary transform $\widetilde {U}=exp[g_{l}(a_{l}b^{\dagger}-ba_{l}^{\dagger})/\Delta _{l}+g_{r}(a_{r}b^{\dagger}-ba_{r}^{\dagger})/\Delta _{r}]$ to the Hamiltonian (34), and get [41]:
Here, $\omega _{j}^{\prime }=\omega _{j}-g_{j}^{2}/\Delta _{j}$ is the stark shift of the resonance frequency of the $j$th cavity and
the effective coupling strength between the two cavities. Above, we have neglected the Hamiltonian of the atomic ensemble: $\omega _{eg}^{\prime }b^{\dagger }b$ with $\omega _{eg}^{\prime }=\omega _{eg}-g_{l}^{2}/\Delta _{l}-g_{r}^{2}/\Delta _{r}$, as it is now adiabatically eliminated. Eq. (36) indicates that, mediated by the atomic ensemble, two cavities are indirectly coupled together with the controllable effective coupling strength $g$. Interestingly, such a parameter can be controlled by engineering the detunings $\Delta _j$ between the $j$th cavity and the atomic level $\omega _{eg}$ of the ensemble. This provides an effective approach to control the transparency window in the EIT-like effect for implementing the manipulations of the group delay of the transmitting photons. Again, the input-output theory is used to treat the present scattering problem of the traveling wave photons.The Hamiltonian of the present driven two-cavity system can now be expressed as
Next, it is easily seen that, the Heisenberg equation for the mode operator $a_{l}$ in the left cavity:
For the convenience, we introduce the input field operator of the left cavity:
Analogously, the input and output field operators of the right-cavity field can be defined as
Note that $a_{in}^{(r)}(\omega )=0$ for the present photon scattering configuration. Thus, from Eq. (50) we get $a_{r}(\omega )=a_{out}^{(r)}(\omega )/\sqrt {\chi _{r}}$. Then, with Eq. (49b), we have
Substituting Eq. (52) into Eq. (51), we get the amplitude
On the other hand, by substituting the Eq. (47) and Eq. (49a) into Eq. (46) we get the amplitude
It is seen from Fig. 8 that the desired EIT-like phenomena appear in the present coupled cavities mediated by an atomic ensemble. We have assumed here that the resonance frequencies of the two cavities are the same, i.e., $\omega _{l}=\omega _{r}=\omega _{0}$, and the effective coupling strength between the cavities is $g=0.1\omega _0$ for the simplicity. The dissipations of the cavities are negligible. For the scattering of the single cavity with a dispersively coupled atomic ensemble discussed in section 2.2, we have shown that the existence of the atomic ensemble only shifts the center frequency of the peak of the cavity transmission; from $\omega _0$ for the empty cavity to $\omega _a=\omega _{0}-g_{a}^{2}/\Delta _{a}$ for the cavity with the atomic ensemble. However, for the present coupled-cavity system, the transmission/reflection spectrum of the photons shows two peaks with the center frequencies being $\omega _{0}-2g$ and $\omega _{0}$, respectively. Between them, a transparent window centered at $\omega _{cen}=\omega _{0}-g=\omega _{0}-g_{a}^{2}/\Delta _{a}$ appears. Also, the solid blue line in Fig. 8(b) shows that the phase shift spectra change sharply in the transparent window regions. This is the typically EIT-like effect, which is assisted with the experimental observations [2,3]. Above, we set $g_{l}=g_{r}=g_{a}, \omega _{l}=\omega _{r}=\omega _{0}$ for the simplicity.
Fig. 8. (a) The transmission spectrum $\widetilde {T}(\omega )$ and (b) the phase shift of the transmitted photons $\widetilde {\varphi }_{T}(\omega )$; (c) the reflection spectrum $\widetilde {R}(\omega )$ and (d) the phase shift one of the reflected photons $\widetilde {\varphi }_{R}(\omega )$. For the comparison, the spectra of the photons through a cavity embed by an atomic ensemble, discussed in subsection 2.2, is represented again here as the dashed red lines. Here, the parameters are typically chosen as, $\omega _{l}=\omega _{r}=\omega _{0}$, $g_{l}=g_{r}$, $g=0.1\omega _{0}$, $\chi _{l}=\chi _{r}=0.05\omega _{0}$, $\Gamma =0$, and the other parameters are the same as in Fig. 6.
Figure 9 shows how the dissipation of the cavity influence the EIT-like phenomena appeared in the present coupled cavities mediated by an atomic ensemble. For the simplicity, we assume that the dissipation rates of the two cavities are identical, i.e., $\Gamma _l=\Gamma _r=\Gamma$. It is seen from Fig. 9(a) that, for a given dissipation parameter, i.e., $\Gamma =0.01\omega _{0}$, the widths of the transmitted peaks increase with the effective coupling strengths between the travelling- and standing photons, but the width and also the center frequency of the transparent window are unchanged. Figure 9(b) further shows that, the width and also the center frequency of the transparent window are still unchanged for the different dissipations, once the effective coupling strengths between the travelling- and standing photons are given. This indicates that the behaviors of the appeared EIT-like phenomena are controllable by manipulating the effective coupling strength between the cavities, via engineering the eigenfrequency of the atoms. In the following subsection we investigate how to achieve such a manipulation to implement the group delay controls of the transmitted photons through the coupled cavities.
Fig. 9. (a) Transmission coefficient versus different effective coupling strength $\chi$ between the traveling wave photons and cavity, we set $\chi _{l}=\chi _{r}=\chi$, $\Gamma =0.01\omega _{0}$. (b) Transmission coefficient as function of the dissipation of the cavity field, $\chi _{l}=\chi _{r}=0.15\omega _{0}$. The other parameters are set: $\omega _{l}=\omega _{r}=\omega _{0}$, $g_{l}=g_{r}$, and $g=0.2\omega _{0}$.
3.2 Controllable group delay of the photons through the two-cavity system
Following subsection 3.1, the dispersion relation in the phase shift spectrum $\widetilde {\varphi }_T(\omega )$ of the transmitted photons through the coupled cavities cause the group delay [2,30]
Given the EIT-like effects appear due to the effectively cavity-cavity coupling mediated by the dispersively coupled atomic ensemble, the group velocity of the transmitted photons through such a coupled-cavity system could be controlled by adjusting detunings $\Delta _{l,r}$ and couplings $g_{l,r}$, between the atoms and the cavities, to manipulate the effectively cavity-cavity coupling $g$. This is basically different from the situation discussed in subsection 2.2, wherein the dispersively coupled atomic ensemble just shifts the center frequency of the transmitted photons through a single cavity, but does not change its group velocity.
In Fig. 10, we plot the transmitted probability $\widetilde {T}(\omega )$ and phase shift $\widetilde {\varphi }_{T}(\omega )$ of the photons through the present cavity-cavity configuration change with the effective cavity-cavity coupling strength $g$ and the frequency of the driven photons, typically for $\omega _{l}=\omega _{r}=\omega _{0}$, $g_{l}=g_{r}$, $\chi _{l}=\chi _{r}=0.05\omega _{0}$, and $\Gamma =0.01\omega _{0}$. One can see that the width of the transparent window increases with the increase of the effective coupling strength $g$. The frequency interval between the double peaks in the transmission spectrum is kept as $2g$ with the central frequency $\omega _{cen}=\omega _{0}-g$ of the transparent window. This indicates clearly that the EIT-like phenomena shown in the present configuration are controllable, and the behaviors of the transparent window can be manipulated by adjusting the relevant physical parameters; $g_{l,r}$ and $\Delta _{l,r}$, to control the cavity-cavity effective coupling strength $g$.
Fig. 10. Variation of photon transmission spectrum (a) and phase shift (b) with effective coupling strength $g$. Here, $\omega _{l}=\omega _{r}=\omega _{0}$, $g_{l}=g_{r}$, $\chi _{l}=\chi _{r}=0.05\omega _{0}$, $\Gamma =0.01\omega _{0}$.
Consequently, with Eqs. (57– 58), we can investigate how to modulate the group delay and group velocity of the transmitted photons through two coupled cavities (with the same eigenfrequency $\omega _l=\omega _r=\omega _0$) by controlling the cavity-cavity effective coupling strength $g$, as shown in Fig. 11. Typically, for the experimental parameters: $\omega _{l}/2\pi =\omega _{r}/2\pi =\omega _{0}/2\pi =3.09839$GHz, $\chi _{l}=\chi _{r}=\gamma _{L}=\gamma _{R}=0.05$GHz, $\Gamma =0.01$GHz, $g=0.01$GHz [31], the group delay of the photons transmitting through the present cavity-cavity configuration is enhanced as $\widetilde {\tau }\approx 52.83$ns, which is significantly larger than the value of $\tau \approx 16.67$ns for the photons transmitting through an empty cavity (or with a dispersively-coupled atomic ensemble). Phenologically, the stronger cavity-cavity effective coupling strength $g$ results in the smaller group delay and thus the weaker slow light effect. Extremely, without the cavity-cavity coupling, i.e., $\tilde {\tau }\rightarrow \infty$ for $g=0$, the photons cannot be sequentially transmitted through the cavities. Practically, the dissipations of the cavities and the coupling strengths between the cavities and the baths always exist, i.e., $\Gamma,\chi _{l}$ are not equivalent to zero. Therefore, with the effective coupling strength $g$, between the cavities, decreases, the group delay $\tilde {\tau }$ increases. As a consequence, if $g=0$, the delay is maximal, i.e., $\tilde {\tau }_{max}=4/(\chi _l+2\Gamma )$. Furthermore, one can see easily that, for the ideal case with $\Gamma =\chi _{l}\rightarrow 0$, we naturally have $\tilde {\tau }\rightarrow \infty$, if $g=0$.
Fig. 11. The effective coupling strength between the two cavities influence on group delay of the probe field $\widetilde {\tau }$ (a) and the group velocity $\widetilde {v}_{g}$ (b). The relevant parameters are chosen as $\omega _{l}/2\pi =\omega _{r}/2\pi =\omega _{0}/2\pi =3.09839$GHz, $g_{l}=g_{r}$, $\chi _{l}=\chi _{r}=0.05$GHz, $\Gamma =0.01$GHz, $L=40$mm.
4. Physical implementation with the atomic ensemble generated by the electrons on liquid Helium
As one of the possible implementations of the above proposal, we now investigate how to implement the group delay of the microwave through a pair of transmission line resonators coupled by the atomic ensemble generated by the surface state electrons on liquid Helium. In fact, the electrons on liquid Helium behave as Hydrogen-like atoms [42]. Applying an electric field $E_{z}$ perpendicular to the surface of the liquid Helium, the potential $U(z)$ of the surface state electron can express as [43–45]
Here, $\Lambda =e^{2}(\varepsilon -1)/4(\varepsilon +1)$ with $e$ being the electron charge, $\varepsilon$ being the dielectric constant of liquid helium and $\varepsilon _{0}$ the vacuum permittivity. $z$ is the coordinate perpendicular to the surface of the liquid Helium. For the weak biased electric fields, the first order energy correction to the $n$th level is given by a linear Stark shift [46]:
Fig. 12. (a) Schematic of the two superconducting coplanar waveguides (CPW) resonators are coupled to an atomic ensemble generated by the electrons on liquid Helium. The solid red line represents the distribution of the electric field in the CPW resonators [47,48]. (b) Cross-sectional view of atomic ensemble, generated by the electrons on liquid Helium. The electrons are trapped by the potential generated by the electric bias via the microelectrodes. (c) The transition frequency $\omega _{eg}/2\pi$ of the two-level atom versus the external electric field $E_{z}$. (d) Cross-sectional view of the electromagnetic field distribution around a superconducting CPW resonator [49]. Here, the gray part indicates the silicon substrate and ${\mathbf S}$ is the poynting vector with the direction perpendicular to the paper facing inward. The red and blue arrows indicate the electric ${\mathbf E}$ and magnetic fields ${\mathbf B}$ distributions around the CPW resonator, respectively.
According to Eq. (36), the effective coupling strength $g$ between the two CPW resonators can be controlled by adjusting the detuning between the resonance frequencies of the CPW resonators and the transition frequency of the two-level Hydrogen-like atom. As shown in Fig. 13, we find that the group delay of the transmitting microwave increases with the increase of the transition frequency of the two-level Hydrogen-like atom. Typically, for the typical parameters: $\omega _{l}/2\pi =\omega _{r}/2\pi =6.399$GHz, $N=10^{4}$, $\lambda _{j}=4.8$MHz, $\chi _{l}=\chi _{r}=\gamma _{L}=\gamma _{R}=5$MHz, $\Gamma =10^{-6}$GHz, $\omega _{eg}/2\pi =130$GHz [50,51], the group delay for the photons transmitting through a single CPW resonator is $\tau \approx 199.96$ns. While, the group delay of the microwave photons transmitting through the two CPW resonators mediated by the dispersively coupled atomic ensemble is $\widetilde {\tau }\approx 519.72$ns. Such a typical EIT-like enhanced effect of the microwave group delay could be experimentally tested with the current low temperature microwave superconducting technique.
Fig. 13. (a) The group delay of $\widetilde {\tau }$ as a function of the transition frequency of the two-level atoms $\omega _{eg}/2\pi$. (b) The group velocity $\widetilde {v}_{g}$ versus the transition frequency of the two-level atoms $\omega _{eg}/2\pi$. Here, we set $N=10^{4}$, $\lambda _{j}=4.8$MHz, $\chi _{l}=\chi _{r}=5$MHz, $\omega _{l}/2\pi =\omega _{r}/2\pi =6.399$GHz, $\Gamma =10^{-6}$GHz, $L=20$mm.
5. Conclusion
Given the EIT effects have been demonstrated with various atomic gases and played important roles in quantum information processing, in this paper, we discussed the EIT-like effects with controllable transparency windows in coupled cavities by using the input-output theory. By exactly solving the transmission probability and phase shift spectra of traveling photons transmitting through an empty cavity (or with a dispersively-coupled atomic ensemble) and two cavities coupled by an atomic ensemble, we investigated the group delays of the transmitting photons. To achieve the controllable group delay for the photons transmitting through the coupled cavities, we propose a possible way to manipulate the effective coupling strength between the cavities by engineering the atomic levels of the atomic ensemble. As a consequence, the effective coupling strength $g$ between the two cavities, in such a cavity-atomic ensemble-cavity system, can be effectively engineered. Therefore, the group delay of the transmitted photons is controllable; the group delay decreases with the increase of the value of $g$. Due to the flexible controllability of the transparency windows, the approach to implement EIT-like effects with the controllable light delays could be applied to quantum information storages, at least theoretically.
Probably, one of the challenges for experimental demonstrations is how to engineer the eigenfrequency of the two-level atom in an ensemble. Given the engineering of the eigenfrequencies of the NV centers in diamond have been realized [52], it is believed that the engineering of coupling strength between two distant cavities coupled via an atomic ensemble for adjusting the group delay of the transmitting photons is achievable. Anyway, our proposal might provide an effective approach to control the group velocity of the photons transmitting through the coupled-cavity system, basing on the EIT-like phenomena with the controllable transparency windows.
Funding
National Natural Science Foundation of China (11974290).
Disclosures
The authors declare no conflicts of interest.
Data availability
Date underlying the results presented in the paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]
2. Y.-C. Liu, B.-B. Li, and Y.-F. Xiao, “Electromagnetically induced transparency in optical microcavities,” Nanophotonics 6(5), 789–811 (2017). [CrossRef]
3. X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017). [CrossRef]
4. S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64(10), 1107–1110 (1990). [CrossRef]
5. K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66(20), 2593–2596 (1991). [CrossRef]
6. G. Wendin, “Quantum information processing with superconducting circuits: a review,” Rep. Prog. Phys. 80(10), 106001 (2017). [CrossRef]
7. L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017). [CrossRef]
8. D. Aghamalyan, J.-B. You, H.-S. Chu, C. E. Png, L. Krivitsky, and L. C. Kwek, “Tunable quantum switch realized with a single $\mathrm {\Lambda}$-level atom coupled to the microtoroidal cavity,” Phys. Rev. A 100(5), 053851 (2019). [CrossRef]
9. P. Ginzburg and M. Orenstein, “Photonic switching in waveguides using spatial concepts inspired by eit,” Opt. Express 14(23), 11312–11317 (2006). [CrossRef]
10. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
11. A. B. Ali and A. B. Ziauddin, “Cavity electromagnetically induced transparency with rydberg atoms,” Laser Phys. Lett. 15(2), 025201 (2018). [CrossRef]
12. C. Ciret, M. Alonzo, V. Coda, A. A. Rangelov, and G. Montemezzani, “Analog to electromagnetically induced transparency and autler-townes effect demonstrated with photoinduced coupled waveguides,” Phys. Rev. A 88(1), 013840 (2013). [CrossRef]
13. K. Di, C. Xie, and J. Zhang, “Coupled-resonator-induced transparency with a squeezed vacuum,” Phys. Rev. Lett. 106(15), 153602 (2011). [CrossRef]
14. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef]
15. G. Li, X. Jiang, S. Hua, Y. Qin, and M. Xiao, “Optomechanically tuned electromagnetically induced transparency-like effect in coupled optical microcavities,” Appl. Phys. Lett. 109(26), 261106 (2016). [CrossRef]
16. D. Tarhan, S. Huang, and O. E. Müstecaplioğlu, “Superluminal and ultraslow light propagation in optomechanical systems,” Phys. Rev. A 87(1), 013824 (2013). [CrossRef]
17. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011). [CrossRef]
18. A. Sohail, R. Ahmed, C. S. Yu, T. Munir, and F. e Alam, “Tunable optical response of an optomechanical system with two mechanically driven resonators,” Phys. Scr. 95(4), 045105 (2020). [CrossRef]
19. H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5(3), 031305 (2018). [CrossRef]
20. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013). [CrossRef]
21. A. Baust, E. Hoffmann, M. Haeberlein, M. J. Schwarz, P. Eder, J. Goetz, F. Wulschner, E. Xie, L. Zhong, F. Quijandría, B. Peropadre, D. Zueco, J.-J. G. Ripoll, E. Solano, K. Fedorov, E. P. Menzel, F. Deppe, A. Marx, and R. Gross, “Tunable and switchable coupling between two superconducting resonators,” Phys. Rev. B 91(1), 014515 (2015). [CrossRef]
22. M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78(10), 104508 (2008). [CrossRef]
23. F. Wulschner, J. Goetz, F. R. Koessel, E. Hoffmann, A. Baust, P. Eder, M. Fischer, M. Haeberlein, M. J. Schwarz, M. Pernpeintner, E. Xie, L. Zhong, C. W. Zollitsch, B. Peropadre, J.-J. G. Ripoll, E. Solano, K. G. Fedorov, E. P. Menzel, F. Deppe, A. Marx, and R. Gross, “Tunable coupling of transmission-line microwave resonators mediated by an rf squid,” EPJ Quantum Technol. 3(1), 10 (2016). [CrossRef]
24. B. Peropadre, D. Zueco, F. Wulschner, F. Deppe, A. Marx, R. Gross, and J. J. García-Ripoll, “Tunable coupling engineering between superconducting resonators: From sidebands to effective gauge fields,” Phys. Rev. B 87(13), 134504 (2013). [CrossRef]
25. Y. Hu, Y.-F. Xiao, Z.-W. Zhou, and G.-C. Guo, “Controllable coupling of superconducting transmission-line resonators,” Phys. Rev. A 75(1), 012314 (2007). [CrossRef]
26. Z.-B. Feng, “Robust quantum state transfer between a cooper-pair box and diamond nitrogen-vacancy centers,” Phys. Rev. A 91(3), 032307 (2015). [CrossRef]
27. Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85(2), 623–653 (2013). [CrossRef]
28. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985). [CrossRef]
29. Q. Xu, J. Shakya, and M. Lipson, “Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency,” Opt. Express 14(14), 6463–6468 (2006). [CrossRef]
30. Z. Wu, R.-H. Luo, J.-Q. Zhang, Y.-H. Wang, W. Yang, and M. Feng, “Force-induced transparency and conversion between slow and fast light in optomechanics,” Phys. Rev. A 96(3), 033832 (2017). [CrossRef]
31. S.-L. Zhang, H.-J. Li, L.-F. Wei, Y.-R. Fang, Y.-W. Wang, P. J. Zhou, Q. Wei, C.-H. Cao, and X.-Z. Xiong, “Microwave transmissions through superconducting coplanar waveguide resonators with different coupling configurations,” Chin. Phys. Lett. 30(8), 088401 (2013). [CrossRef]
32. M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff, “Coplanar waveguide resonators for circuit quantum electrodynamics,” J. Appl. Phys. 104(11), 113904 (2008). [CrossRef]
33. J. Zmuidzinas, “Superconducting microresonators: Physics and applications,” Annu. Rev. Condens. Matter Phys. 3(1), 169–214 (2012). [CrossRef]
34. H. Y. Yao, N. C. Chen, T. H. Chang, and H. G. Winful, “Frequency-dependent cavity lifetime and apparent superluminality in fabry-pérot-like interferometers,” Phys. Rev. A 86(5), 053832 (2012). [CrossRef]
35. W. L. Yang, Y. Hu, Z. Q. Yin, Z. J. Deng, and M. Feng, “Entanglement of nitrogen-vacancy-center ensembles using transmission line resonators and a superconducting phase qubit,” Phys. Rev. A 83(2), 022302 (2011). [CrossRef]
36. B. M. Garraway, “The dicke model in quantum optics: Dicke model revisited,” Phil. Trans. R. Soc. A 369(1939), 1137–1155 (2011). [CrossRef]
37. I. Diniz, S. Portolan, R. Ferreira, J. M. Gérard, P. Bertet, and A. Auffèves, “Strongly coupling a cavity to inhomogeneous ensembles of emitters: Potential for long-lived solid-state quantum memories,” Phys. Rev. A 84(6), 063810 (2011). [CrossRef]
38. M. Zhang, H. Y. Jia, J. S. Huang, and L. F. Wei, “Strong couplings between artificial atoms and terahertz cavities,” Opt. Lett. 35(10), 1686–1688 (2010). [CrossRef]
39. M. Stammeier, S. Garcia, T. Thiele, J. Deiglmayr, J. A. Agner, H. Schmutz, F. Merkt, and A. Wallraff, “Measuring the dispersive frequency shift of a rectangular microwave cavity induced by an ensemble of rydberg atoms,” Phys. Rev. A 95(5), 053855 (2017). [CrossRef]
40. S. Garcia, M. Stammeier, J. Deiglmayr, F. Merkt, and A. Wallraff, “Single-shot nondestructive detection of rydberg-atom ensembles by transmission measurement of a microwave cavity,” Phys. Rev. Lett. 123(19), 193201 (2019). [CrossRef]
41. A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69(6), 062320 (2004). [CrossRef]
42. C. C. Grimes, T. R. Brown, M. L. Burns, and C. L. Zipfel, “Spectroscopy of electrons in image-potential-induced surface states outside liquid helium,” Phys. Rev. B 13(1), 140–147 (1976). [CrossRef]
43. M. Zhang, H. Y. Jia, and L. F. Wei, “Jaynes-cummings models with trapped electrons on liquid helium,” Phys. Rev. A 80(5), 055801 (2009). [CrossRef]
44. P. M. Platzman and M. I. Dykman, “Quantum computing with electrons floating on liquid helium,” Science 284(5422), 1967–1969 (1999). [CrossRef]
45. Y. Z. Kovdrya, “One-dimensional and zero-dimensional electron systems on liquid helium (review),” Low. Temp. Phys. 29(2), 77–104 (2003). [CrossRef]
46. M. I. Dykman, P. M. Platzman, and P. Seddighrad, “Qubits with electrons on liquid helium,” Phys. Rev. B 67(15), 155402 (2003). [CrossRef]
47. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, “Circuit quantum electrodynamics,” Rev. Mod. Phys. 93(2), 025005 (2021). [CrossRef]
48. H. Li, Y. Wang, L. Wei, P. Zhou, Q. Wei, C. Cao, Y. Fang, Y. Yu, and P. Wu, “Experimental demonstrations of high-q superconducting coplanar waveguide resonators,” Chin. Sci. Bull. 58(20), 2413–2417 (2013). [CrossRef]
49. R. Barends, N. Vercruyssen, A. Endo, P. J. de Visser, T. Zijlstra, T. M. Klapwijk, P. Diener, S. J. C. Yates, and J. J. A. Baselmans, “Minimal resonator loss for circuit quantum electrodynamics,” Appl. Phys. Lett. 97(2), 023508 (2010). [CrossRef]
50. G. Yang, A. Fragner, G. Koolstra, L. Ocola, D. A. Czaplewski, R. J. Schoelkopf, and D. I. Schuster, “Coupling an ensemble of electrons on superfluid helium to a superconducting circuit,” Phys. Rev. X 6, 011031 (2016). [CrossRef]
51. G. Koolstra, G. Yang, and D. I. Schuster, “Coupling a single electron on superfluid helium to a superconducting resonator,” Nat. Commun. 10(1), 5323 (2019). [CrossRef]
52. M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, “The nitrogen-vacancy colour centre in diamond,” Phys. Rep. 528(1), 1–45 (2013). [CrossRef]
