Electromagnetically induced transparency in circuit QED with nested polariton states

TL;DR

This paper demonstrates electromagnetically induced transparency (EIT) in a circuit QED system with nested polariton states, revealing quantum interference effects and negative group velocities in a superconducting qubit setup.

Contribution

It introduces a novel method to generate and observe EIT using nested polariton states in circuit QED, expanding the understanding of quantum interference in solid-state systems.

Findings

EIT observed in a circuit QED system with nested polariton states

Negative group velocities up to -0.52 km/s measured

Verification of EIT using Akaike's information criterion

Abstract

Electromagnetically induced transparency (EIT) is a signature of quantum interference in an atomic three-level system. By driving the dressed cavity-qubit states of a two-dimensional circuit QED system, we generate a set of polariton states in the nesting regime. The lowest three energy levels are utilized to form the $\Lambda$-type system. EIT is observed and verified by Akaike's information criterion based testing. Negative group velocities up to $-0.52\pm0.09$ km/s are obtained based on the dispersion relation in the EIT transmission spectrum.

Equations

$$\begin{array}{rl}& |1⟩=-\sin \left(\frac{{\theta }_0}{2}\right)\widetilde{|e,0⟩}+\cos \left(\frac{{\theta }_0}{2}\right)\widetilde{|g,0⟩},\\ & |2⟩=\cos \left(\frac{{\theta }_0}{2}\right)\widetilde{|e,0⟩}+\sin \left(\frac{{\theta }_0}{2}\right)\widetilde{|g,0⟩},\\ & |3⟩=-\sin \left(\frac{{\theta }_1}{2}\right)\widetilde{|g,1⟩}+\cos \left(\frac{{\theta }_1}{2}\right)\widetilde{|e,1⟩},\\ & |4⟩=\cos \left(\frac{{\theta }_1}{2}\right)\widetilde{|g,1⟩}+\sin \left(\frac{{\theta }_1}{2}\right)\widetilde{|e,1⟩},\end{array}$$

$$\begin{array}{rl}& F_{|D⟩}=\sqrt{⟨D|\rho |D⟩},\\ & =\sqrt{\frac{1}{2}\left[\cos 2\Theta ({\rho }_{11}-{\rho }_{22})-\sin 2\Theta ({\rho }_{21}+{\rho }_{12})+(1-{\rho }_{11})\right]},\end{array}$$

$$\ln (S_{21})=\ln (|S_{21}|)+i\varphi =i\frac{{\omega }_{p}L}{c}\left(1+\frac{1}{2}{\chi }_s\right)-{\alpha }_0+i{\varphi }_0,$$

$${\chi }_s^{\mathrm{EIT}}=\frac{A_{+}}{({\omega }_p-{\omega }_{+})-i({\Gamma }_{+}/2)}-\frac{A_{-}}{({\omega }_p-{\omega }_{-})-i({\Gamma }_{-}/2)},$$

$${\chi }_s^{\mathrm{ATS}}=\frac{A_1}{({\omega }_p-{\omega }_1)-i({\Gamma }_1/2)}+\frac{A_2}{({\omega }_p-{\omega }_2)-i({\Gamma }_2/2)},$$

Full text

Controllable interaction between electromagnetic quanta and discrete levels in a quantum system, i.e., light matter interaction, is the key to quantum information storage and processing in a quantum network Cirac et al. (1997); Kimble (2008). Consider a three-level atomic system driven by two coherent electromagnetic waves. The destructive interference between the two excitation pathways creates a transparency window for one of the drive fields and switches the system into a “dark state.” This phenomenon is called electromagnetically induced transparency (EIT) Fleischhauer et al. (2005). Recently, EIT has been harnessed for implementing different building blocks of a quantum network, such as all-optical switches and transistors Baur et al. (2014); Shomroni et al. (2014); Bajcsy et al. (2009); Chen et al. (2013); Souza et al. (2013), quantum storage devices Liu et al. (2001); Phillips et al. (2001); van der Wal et al. (2003); Kuzmich et al. (2003); Chaneliere et al. (2005), and conditional phase shifters Tiarks et al. (2016); Beck et al. (2016); Shahmoon et al. (2011); Ottaviani et al. (2003); Liu et al. (2016a). Despite this remarkable success, utilizing EIT and related effects at the single-photon and single-atom level with highly scalable devices is a formidable challenge that prevents realization of a practical quantum network Ma et al. (2017). A promising solution is to extend these techniques to the microwave domain using superconducting quantum circuits that are both scalable and enable deterministic placement of long-lived artificial atoms for the network nodes Kirchmair et al. (2013); Blais et al. (2004); Schuster et al. (2007); Koch et al. (2007).

To this end, three-level superconducting artificial atoms have been used to demonstrate coherent population trapping (CPT) Kelly et al. (2010) and Autler-Townes splittings (ATS) Baur et al. (2009); Sillanpää et al. (2009); Abdumalikov et al. (2010); Novikov et al. (2013); Hoi et al. (2013); Suri et al. (2013); Cho et al. (2014). However, conclusive evidence of EIT in these simple systems eluded researchers as it is difficult to find a superconducting quantum system with metastable states and lifetimes that satisfy its stringent requirements Anisimov et al. (2011); Sun et al. (2014); Peng et al. (2014); Liu et al. (2016b). Recently, progress has been made in a circuit quantum electrodynamics (QED) system that exploits qubit coupling to a single-mode cavity Novikov et al. (2016). In that experiment, one leg of the $\Lambda$-type system is dipole forbidden, requiring that it be driven with a two-photon transition. The small photon scattering cross section of this two-photon transition hinders applications such as single atom quantum memory Specht et al. (2011), all-optical switching and routing of a single photon gated by another single photon Shomroni et al. (2014), single photon-photon cross phase modulation Hoi et al. (2013), and vacuum induced transparency Tanji-Suzuki et al. (2011). On the other hand, high scattering cross sections have been observed in a dipole allowed transition of an artifacial atom coupled to one dimensional waveguide Astafiev et al. (2010). Thus, implementing a $\Lambda$-type system with all dipole allowed transitions in a circuit QED system is highly desirable for building a quantum network with microwave photons.

In this Letter, we report the first observation of EIT using all dipole allowed transitions in a $\Lambda$-type system implemented with superconducting quantum circuit. Our scheme is based on a theoretical proposal Gu et al. (2016) that utilizes polariton states generated with a rf biased two-level system coupled to a resonator. Here, we realized the polariton states in a transmon–cavity system and achieved a metastable state with a long lifetime. Moreover, we were able to tune the polariton states to establish a $\Lambda$-type system that can be driven with control- and probe-fields through dipole allowed transitions. Note that due to the transmission geometry of our cavity where nominally the signal is transmitted on resonance, the observed experimental signal is actually electromagnetically induced absorption (EIA). However, our EIA and conventional EIT have identical underlying physics of quantum interference. Conventional EIT spectra can be observed if a hanger resonator geometry is used. We retain the nomenclature of quantum optics and use the term “EIT” for the rest of the Letter. From our EIT data, a large transmission extinction (99.73 %) of the probe field is observed and high dark state preparation fidelity ($>$ 99.39 %) is inferred. To our best knowledge, the EIT transmission extinction of 99.73 % is the highest one that has been observed to date in the circuit QED system. Our EIT scheme opens up new possibilities for realizing scalable devices that utilize single-photons and single-atoms for constructing EIT as a building block of a quantum network in the microwave domain.

Our experiment is performed on the device that consists of a concentric transmon capacitively coupled to a $\lambda/2$ microstrip resonator with a coupling strength $g/2\pi=74$ MHz, as shown in Fig. 1(a). The transmon comprises a single Al/AlOx/Al Josephson junction shunted by a capacitor consisting of a superconducting island and a surrounding ring. The Josephson junction is fabricated with an overlap technique Wu et al. (2017). The transmon has a resonance frequency $\omega_{q}/2\pi=5.648$ GHz between its lowest two levels and an anharmonicity $\alpha/2\pi=262.5$ MHz. The coherence times are measured to be $T_{1}=35~{}\mu$s and $T_{2}^{*}=22.5~{}\mu$s. The fundamental mode of the resonator is at $\omega_{r}/2\pi=6.485$ GHz with a internal quality factor $Q_{i}=(1.2\pm 0.2)\times 10^{6}$ and a loaded quality factor $Q=7,900$ dominated by the strong coupling to the microwave feedline at the output port.

The transmon-cavity system is well in the dispersive regime with a dispersive shift $\chi/2\pi=1.54$ MHz. The eigenlevels are described by the dispersive Jaynes-Cummings ladder as shown in Fig. 1(b). The resonance frequencies are $\omega_{q}-\chi$ for the $\widetilde{\left|g,0\right\rangle}\leftrightarrow\widetilde{\left|e,0\right\rangle}$ transition, and $\omega_{q}-3\chi$ for the $\widetilde{\left|g,1\right\rangle}\leftrightarrow\widetilde{\left|e,1\right\rangle}$ transition, where $\widetilde{\left|g,n\right\rangle}$ ($\widetilde{\left|e,n\right\rangle}$) denotes the qubit ground (excited) state with $n$ photons in the resonator. The tilde indicates that these levels are singly-dressed states, i.e., they are transmon states slightly dressed with resonator photons.

The polariton states are generated by injecting a strong microwave drive field through the input coupler to doubly dress the Jaynes-Cummings states. In particular, if the drive frequency $\omega_{d}$ is in the so-called nesting regime, $\omega_{q}-3\chi<\omega_{d}<\omega_{q}-\chi$, the resulting eigenstates $\left|2\right\rangle$ and $\left|3\right\rangle$ will be nested in between the eigenstates $\left|1\right\rangle$ and $\left|4\right\rangle$ Koshino et al. (2013); inomata2014microwave; inomata2016single.

We use the set of polariton states $\left|1\right\rangle$, $\left|2\right\rangle$, and $\left|3\right\rangle$ to form a $\Lambda$-type system [Fig. 2(d)]. In the driven two-level-system model, these polariton states can be approximated as

[TABLE]

where the mixing angles $\theta_{0}$ and $\theta_{1}$ are given by, $\tan(\theta_{0})=\Omega_{d}/[(\omega_{q}-\chi)-\omega_{d}]$ and $\tan(\theta_{1})=\Omega_{d}/[\omega_{d}-(\omega_{q}-3\chi)]$ Gu et al. (2016).

Eq. (1) shows that the $\left|1\right\rangle\leftrightarrow\left|3\right\rangle$ and $\left|2\right\rangle\leftrightarrow\left|3\right\rangle$ transitions are mainly cavity-like transitions, while $\left|1\right\rangle\leftrightarrow\left|2\right\rangle$ is a qubit-like transition. These properties can be revealed by calculating the decay rate $\gamma_{ij}$ of $\left|i\right\rangle\rightarrow\left|j\right\rangle$ transition, which can be approximated as, $\gamma_{31}=\gamma_{c}\sin^{2}\left[(\theta_{0}+\theta_{1})/2\right]$, $\gamma_{32}=\gamma_{c}\cos^{2}\left[(\theta_{0}+\theta_{1})/2\right]$, and $\gamma_{21}=\gamma_{q}\cos^{4}\left(\theta_{0}/2\right)$, where $\gamma_{c}$ is the cavity decay rate and $\gamma_{q}$ is the qubit decay rate Gu et al. (2016). Thus, the decay rate of $\left|3\right\rangle\rightarrow\left|1\right\rangle$ transition ($\gamma_{31}$) can be tuned to be comparable with the decay rate of $\left|3\right\rangle\rightarrow\left|2\right\rangle$ transition ($\gamma_{32}$), while extending the metastable state lifetime ($1/\gamma_{21}$) even beyond the qubit lifetime. These two effects are key to achieve EIT in our superconducting circuit system.

We measure the transition frequencies between the polariton states by performing two-tone spectroscopy with a polariton drive and a weak probe field. The drive frequency and the probe power are fixed at $\omega_{d}/2\pi=5.6466$ GHz and P${}_{p}=-163.15$ dBm respectively, while scanning the drive strength and the probe frequency. The probe transmission, defined as the ratio of the probe output complex amplitude to the input complex amplitude $S_{21}\equiv V_{\mathrm{out}}/V_{\mathrm{in}}=|S_{21}|e^{i\phi}$, was measured by a vector network analyzer (VNA). Our definition of $S_{21}$ includes all round-trip amplification and attenuation, where $\phi$ has been corrected for electric delay. As shown in Fig. 2(b), there are four transmission peaks near the resonator frequency. The four peaks correspond to, from low to high frequencies, $\omega_{23}$, $\omega_{13}$, $\omega_{24}$, and $\omega_{14}$ respectively [Fig. 2(c)], where $\omega_{ij}$ denotes the energy difference between the polariton states $\left|i\right\rangle$ and $\left|j\right\rangle$. The spacing between the first and second (first and third) transmission peaks, which corresponds to the splitting between levels $\left|1\right\rangle$ and $\left|2\right\rangle$ ($\left|3\right\rangle$ and $\left|4\right\rangle$), widens as the drive strength increases. This is consistent with the expected AC Stark shift drawn as the black dashed curves in Fig. 2(b). Another crucial feature of the spectrum is that, as the polariton drive strength increases, the height of the $\omega_{23}$ and $\omega_{14}$ peaks decreases, while the height of $\omega_{13}$ and $\omega_{24}$ peaks increases. This behavior agrees with the change of the transition probabilities between polariton states predicted in reference Gu et al. (2016).

In this system, EIT is demonstrated by a suppression of transmission for a weak probe field on resonance with one leg of a $\Lambda$-system, while a control field addressing the other leg [Fig. 2 (d)]. The $\Lambda$-system is established by a polariton drive field with frequency $\omega_{d}/2\pi=5.6466$ GHz and strength $\Omega_{d}/2\pi=1.46$ MHz. The resultant $\Lambda$ levels have $\gamma_{31}/2\pi=0.35$ MHz and $\gamma_{32}/2\pi=0.47$ MHz, which are much larger than $\gamma_{21}/2\pi=2.74$ kHz. The control field frequency $\omega_{c}/2\pi=\omega_{23}/2\pi=6.4828$ GHz and the probe strength $\Omega_{p}/2\pi=62$ kHz are fixed, while we scan the control field strength $\Omega_{c}$ and the probe frequency $\omega_{p}$. The probe transmission ($S_{21}$) measured by the VNA is shown in Fig. 3(a)&(b). With our parameters, the theoretical condition of EIT is given by $\Omega_{c}/2\pi<\gamma_{c}/2\pi=0.82$ MHz [black dash-dotted line in Fig. 3(a)] Gu et al. (2016). Under this condition, we observe a transmission suppression window around $\omega_{p}=\omega_{13}$ with the largest suppression 25.66 dB [Fig. 3(c)&(d)], which means about 99.73% of power of the original transmitted probe field is suppressed. However, as the control field strength exceeds the EIT boundary, the transmission for $\omega_{p}>\omega_{13}$ in Fig. 3(a) is becoming smaller and completely disappears above $\Omega_{c}/2\pi=2.8$ MHz instead of changing to an ATS lineshape. This behavior is most likely due to excess cavity population, above a single photon, due to the strong control field.

Quantum interferences in a driven $\Lambda$-system create a dark state, which is transparent to the probe field. The fidelity of the dark state preparation is an important metric for a EIT-based quantum memory Ma et al. (2017). With our experimental parameters, we inferred the dark state fidelity defined as Li et al. (2011)

[TABLE]

where the dark state $\left|D\right\rangle=\cos{\Theta}\left|1\right\rangle-\sin{\Theta}\left|2\right\rangle$ and the mixing angle $\Theta=\mathrm{tan}^{-1}(\Omega_{p}/\Omega_{c})$. The density matrix $\rho$ is calculated by numerically solving a Lindblad master equation of a driven $\Lambda$-system, including decay rates $\gamma_{ij}$ Johansson et al. (2013). At the EIT boundary ($\Omega_{p}/2\pi=0.82$ MHz, $\Omega_{c}/2\pi=0$ MHz), the dark state fidelity is calculated to be 99.39 %. Note that we switched the role of the probe and the control fields to simulate the fidelity when the dark state is essentially the polariton $\left|2\right\rangle$ state and the main infidelity source is its decay rate $\gamma_{21}$.

To confirm that the suppression of transmission is due to EIT, as opposed to ATS, Akaike’s information criterion (AIC) based testing was performed. The AIC-based testing calculates the weight of each fitting model based on the goodness of the fitting with the constraint that sum of the weights is unity Anisimov et al. (2011). Originally, the AIC-based testing was proposed to fit the susceptibility, $\chi_{s}$ Anisimov et al. (2011). To use this criterion, we derive the relationship between the measured $S_{21}$ and a generic susceptibility $\chi_{s}$ as Jackson (1999)

[TABLE]

where $L$ is the effective distance the microwave travels through the chip, $c$ is the speed of light, $\alpha_{0}$ is the attenuation of the cables, and $\phi_{0}$ is a frequency-independent initial phase offset. For EIT, the susceptibility takes the form, the difference between two Lorentzians Anisimov et al. (2011),

[TABLE]

and for ATS, it takes the form of the sum,

[TABLE]

where $\omega_{j}$, $A_{j}$, and $\Gamma_{j}$ are the center frequency, magnitude, and width of the $j$th Lorentzian, respectively. In comparison to reference Anisimov et al. (2011), the different negative signs in front of the $i(\Gamma_{j}/2)$ terms in Eq. (4) and (5), are due to the transmission geometry of the circuit. The model functions for EIT or ATS are then obtained by substituting either $\chi_{s}^{\mathrm{EIT}}$ or $\chi_{s}^{\mathrm{ATS}}$ for the $\chi_{s}$ in Eq. (3).

We fit the probe transmission $S_{21}$ data to both EIT and ATS models to extract the AIC-based testing weights to validate the observations was from EIT Anisimov et al. (2011). For each model, $\ln(|S_{21}|)$ and $\phi$ were fit simultaneously to assure the Kramers-Kronig relations. The transmission data at $\Omega_{c}/2\pi=0.82$ MHz and its fits of both models are shown in Fig. 4(a)&(b). Qualitatively, at this control field strength, the data fits significantly better to the EIT model than to the ATS model. Furthermore, the weights of EIT and ATS models for different control field strengths are plotted in Fig. 4(c). For control field strength $\Omega_{c}/2\pi<0.2$ MHz, both the EIT and ATS weights approach 0.5 due to the presence of noise and the relatively small size of transmission suppressions. In the range of $0.2~{}\mathrm{MHz}<\Omega_{c}/2\pi<2.8~{}\mathrm{MHz}$, the EIT weights are subtantially larger than the ATS weights, indicating strong EIT signatures. The maximum EIT weight happens around $\Omega_{c}/2\pi=0.82$ MHz, which is in agreement with the theoretical EIT boundary. For control field strength $\Omega_{c}>2.8$ MHz, the control field excites resonator photons and drives the system out of the nesting regime. Therefore, there is neither EIT nor ATS characters and results in equal weights of 0.5.

We also investigated the backward light phenomenon due to the giant dispersion of EIT Novikov et al. (2016). We calculated the time $\tau_{g}$ for the probe field to traverse the device at different control field strengths by using $\tau_{g}=-\mathrm{d}\phi/\mathrm{d}\omega_{p}$, where $\phi$ is obtained from the fittings of the EIT model [Fig. 5(a)]. The group velocity of the probe can then be calculated by $v_{g}=l/\tau_{g}$, where $l=10.3$ mm is the distance between the input and output coupler of the device. The largest inferred negative group velocity is $v_{g}=-0.52\pm 0.09$ km/s, further pushing the boundaries of slow light, compared to that reported in reference Novikov et al. (2016).

In conclusion, polariton states in the nesting regime have been generated with a transmon cQED system. The transmission spectra were measured and agree with theoretical predictions. We utilized three levels of nested polariton states to form a $\Lambda$-type transition. A robust EIT signature with all dipole allowed transitions was observed in a superconducting system for the first time. Our results constitute an important step towards scalable quantum network with propagating microwave photons.