Open Access
Add to BibSonomy
Abstract
We experimentally study interactions between two microwave fields mediated by 3-level transmon artificial atom with two-photon processes. The transmon has good selection rule, preventing one-photon transition, but allowing two-photon transition from ground state(0) to 2nd excited state(2). By pumping a control tone in resonance to the transition between 1st(1) and 2nd excited state(2), we control the one-photon transparency for 0 to 1 transition and two-photon transparency for 0 to 2 transition. The results are explained by the Autler-Townes splitting induced by the control microwave. In addition, two possible microwave amplification processes involving two-photon processes are also studied. The 4-wave mixing scheme increases the transmission by 3% while 2-photon optical pumping produces a 11% narrowband increment. All these phenomena can be operated with control and probe tones in a narrow band.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of superconducting quantum circuits have led to the realization of quantum optics at microwave frequencies [1,2]. In particular, the architecture of building microwave transmission lines and micrometer-sized superconducting circuits provides a great enhancement in coupling strength between electromagnetic waves and quantum circuit, also called “artificial atom” [3,4]. Taking advantage of the strong couplings, people have demonstrated a broad range of quantum optical phenomena, such as single atom spectroscopy [5], dressed state [6,7], amplification of light [8], lasing [9], Mollow triplets, Autler-Townes splitting [10–13] and electromagnetically-induced transparency [14]. The large non-linearity in this system also allows us to observe versatile wave mixing effects [15–18]. Because the energy levels of the artificial atoms are nearly arbitrarily tunable in the GHz range, it ensures applications without much restrictions in operation frequency.
In this paper, we utilize a transmon-type artificial atom [19] in a co-planar waveguide(CPW) resonator to investigate interesting quantum optical processes particularly involving two-photon processes with two-tone experiment. The selection rule makes the transmon a $\Xi$-type atom such that the single photon transition from from ground state($|0\rangle$) to 2nd excited state($|2\rangle$) is forbidden [19]. Nevertheless, combined with 2-photon processes, we could observe effects of Autler-Townes splitting [20] and Mollow triplet [21] with the control tone. Taking advantage of these effects, we may allow the probe tone, which originally absorbed by the atom to transmit or be amplified. We can simply demonstrate by fixing the probe and control frequencies, and gradually tune the energy levels of the atom. In this process, we observed the appearance of 2-photon controlled transparency, 1-photon controlled transparency, 2-photon optical pumping and 4-wave mixing [17]. In experimental point of view, it is easier to modulate the magnetic field to tune the atom levels than modulate the frequencies, and as a consequence, leading to vast applications in quantum optics, such as the single photon router demonstrated by Hoi et al [12]. We also note that these 2-photon processes are easy to achieve by using $\mu$s-pulsed microwaves rather than continuous wave microwaves. In addition, these effects can be operated with the control and probe tone in a narrow band not larger than the anharmonicity of the transmon, suitable for applications with narrow band optics.
To see what are the optical effects we can demonstrate with a $\Xi$-type 3-level atom, let’s consider the Hamiltonian of a 3 level system under a driving:
Fig. 1. (a) The eigenvalues of $H_\textrm {eff}$ when $\Omega _c=0.4\alpha$ are plot by blue curves, while the results for zero driving $\Omega _c\sim 0$ are plot by crayon dash curves. (b) The experimental setup for pulsed control and probe tones. The optical micrographs of the transmons and chip are illustrated at the center of the plot. The scale bar is 50 $\mu$m.
2. Experimental methods
As illustrated by Fig. 1(b), the sample contains two transmon-type artificial atoms coupled to a co-planar waveguide(CPW) resonator. The artificial atoms and the resonator were made of Al and Nb, respectively on a Si substrate. The transmission-type resonator has a input port with a smaller coupling capacitance 8 fF and output port with a larger coupling capacitance, 64 fF. The designed resonator frequency is roughly 5.0 GHz. The transmon is coupled to the signal line through a large capacitance 28 fF. The total capacitance of the transmon, including those from junctions and interdigital structures is 40 fF. The transmons have the structure of a superconducting quantum interference device(SQUID) for tunable resonance under a magnetic field. The SQUID loops of the two transmons provide respectively the periods of $\Phi _1=2.12$ Gs and $\Phi _2=2.38$ Gs. The different period allows us to investigate one transmon at a time if biasing the other transmon to a much lower or higher frequency under study.
The sample was cooled down in a dilution refrigerator with a base temperature below 40 mK. The transmission of the sample is measured by using a commercial vector network analyzer(Agilent N5230A). For pulse measurement, the signal is first down-converted by a homemade homodyne IQ mixer and then sampled and digitized(Agilent M9703A). The pulsed data were confirmed by a commercial vector signal analyzer(Agilent M9392A). The pulsed pumping and probing signals were prepared by two microwave continuous wave generators mixed by ns short pulses from a two-channel arbitrary waveform generator(Agilent M9330).
3. Results and discussions
3.1 Transmon spectroscopy
The CPW resonator has a low quality factor($Q$) of about 250 due to the large coupling to the input and output ports, which gives the major contribution to this low $Q$ value [22]. Despite of the low $Q$ value, the resonator showed a strong coupling with the transmon, that can be quantified with the vacuum Rabi splitting of 80 MHz when the transmon is in resonance to the resonator. The large coupling $g$ ensures the superposed photon-atom state when the energy detuning is within $g/h=$40 MHz. The low finesse resonator allows the off-resonant probe microwaves to be introduced to the resonator. When the probing frequency is detuned from the resonator frequency, we may observe the transmon resonance for ground state $|0\rangle$ to the 1st excited state, $|1\rangle$, when sweeping the magnetic field. We would also like to note that in order to prevent the resonator is involved in the quantum optical operations, the control and probe tones were mostly kept in the range of 6.2 to 6.9 GHz. Because the operation was far detuned from the resonator, the transmission is rather small. As a comparison, the transmission change associated with transmon absorption is one order of magnitude smaller than that with the resonator. The similar experiment can be also conducted in a system that a transmon is coupled to an open transmission line.
Interestingly, if we apply a larger probing power by using pulse measurement, a small structure appear beside the main transmon resonance structure as shown in Fig. 2(a). The side structures appear symmetrical to external flux number threading through the SQUID loop, $f=0.5$ and its depth linearly depends on the microwave amplitude(Fig. 2(b)). We would like to note that the 2-photon process is more efficiently probed by using the pulse measurement due to a higher probe power is allowed. Because of the power dependence, we may judge it a signature of two-photon transition from ground state $|0\rangle$ to the 2nd excited state, $|2\rangle$. Indeed, because the energy level spacing $\hbar \omega _{21}$ is slightly smaller than $\hbar \omega _{10}$, the two-photon absorption will appear in a smaller $f$ values than the single photon absorption for $|0\rangle$ to $|1\rangle$. Doubling the frequency of probe tone which produces 2-photon absorption, we may deduce the energy spectrum for $\hbar \omega _{20}$ as illustrated in Fig. 2(c). For transmons, the anharmonicity $\alpha \equiv (\omega _{10}-\omega _{21})$ should be $\alpha =E_C/\hbar$ [19]. From the spectroscopy of $|0\rangle$ to $|1\rangle$ and $|0\rangle$ to $|2\rangle$ transitions, we may determine the important energy scales of the transmon as $E_C/h\sim$0.6 GHz and $E_J/h\sim$65 GHz(zero magnetic field). Moreover, we confirmed that the transition matrix element for transmon is very small for $|0\rangle$ to $|2\rangle$. Such a selection rule makes the transmon a $\Xi$-type atom [19]. The relevant 1-photon and 2-photon optical effects one can observe for a 3-level $\Xi$-type atom are briefly summarized in Table 1. $\Delta _1$ and $\Delta _2$ respectively note the detuning of probe and control tones for each effect.
Fig. 2. (a) The in-phase(I) voltage of the measurement pulse clearly shows the single photon resonance(1-ph) and two-photon(2-ph) resonance. (b) The depth of the 2-photon absorption scaled by that of the 1-photon absorption shows a linear dependence on probe amplitude. (c) The resonance of transmon associated with the transition of $\omega _{10}$, and 2-photon transition of $\omega _{20}$ as a function of biased flux. The curves are fitting results of standard transmon spectrum with $E_C/h=$0.6 GHz.
Table 1. A brief summary of the optical effects in a 3-level $\Xi$ atom. $\Delta =\omega _c-\omega _{21}$. CPT stands for the optical effect of controlled photon transparency.
3.2 Controlled photonic transparency
As we have mentioned in the introduction section, three different fundamental microwave-controlled processes can be demonstrated. As illustrated in Fig. 3(a), the transition for $|0\rangle$ to $|1\rangle$ can be controlled by pumping the transition $|1\rangle$ to $|2\rangle$, so-called controlled photonic transparency(CPT). $\Delta _1=\omega _p- \omega _{10}$ and $\Delta _2(=\Delta )=\omega _c-\omega _{21}$ are respectively the detunings of the probe and control tunes. For demonstrating this, we used probe frequency $\omega _p/2\pi = 6.9$ GHz and control frequency $\omega _c/2\pi = 6.28$ GHz, which can be deduced from the energy spectra $\omega _p-\omega _c\sim \alpha$. Instead of tuning probe frequency, we introduce detuning by sweeping magnetic flux, which would both change the energy level spacings $\hbar \omega _{10}$ and $\hbar \omega _{21}$ but their difference $\omega _{10}-\omega _{21}=\alpha$ is always obeyed. In this operation scheme $\Delta _1=\Delta _2$, and every $10^{-2}$ change in $f$ produces the detuning by 0.725 GHz. When the control tone is absent, the transmission show single resonance feature of $|0\rangle$ to $|1\rangle$ transition at $\Delta _1=0$. As control amplitude increases, the resonance gradually splits into two, featuring Autler-Townes splitting in atomic physics as shown in Fig. 3(b). Similar experimental realization was reported by Hoi et al [12]. We would like to note that our 1-photon absorption is Fano-like so the transmission amplitude is not symmetric in positive and negative branches. As we will discuss later with an analytical model, the splitting is Rabi oscillation frequency $\Omega _c$ of the driving, and linearly depends on control tone amplitude. When the Rabi frequency is greater than 300 MHz, the Autler-Townes splitting is not observable because the detuning $\Delta _2>\alpha$ exceeds the anharmonicity.
Fig. 3. (a) Controlled photon transparency for $|0\rangle$ to $|1\rangle$ transition. The control tune produces the Autler-Townes splitting with the Rabi frequency $\sqrt {2}\Omega _c$. (b) The transmission amplitude of the probe microwave with $\omega _p/2\pi$= 6.9 GHz. When there is a continuous pumping microwave with a frequency $\omega _c/2\pi$= 6.28 GHz, the resonance featuring $|0\rangle$ to $|1\rangle$ transition split into two and the splitting is larger when the pumping power is larger. The Rabi frequency $\Omega _c$ is linearly propositional to the control tone amplitude.
Under the same microwave control, we may probe the transition from $|0\rangle$ to $|2\rangle$ using 2-photon process as illustrated by Fig. 4(a). In order to probe the resonant 2-photon transition, a different scheme that $\omega _c=\omega _p-\frac 12\alpha$ is applied and the detuning are redefined as $\Delta _1=2\omega _p-\omega _{20}=2\Delta _2(=2\Delta )$. Table 1 summarizes the definition of detuning $\Delta _1$ and $\Delta _2$ for each effect. To set the resonance condition, we let $\omega _c/2\pi =6.6$ GHz and $\omega _p/2\pi =6.9$ GHz. Figures 4(b) and 4(c) respectively demonstrate the homodyne in-phase(I) and quadruture(Q) voltages of the probe nearby the 2-photon absorption $|0\rangle$ to $|2\rangle$ in absent and in presence of the control tone. Again, Autler-Townes splitting presents in the 2-photon spectroscopy with control tone on. In this case, the resonance should occur when the system is detuned to $\Delta _1=\pm \Omega _c$. Notice that the detuning in the plot is $\Delta _1/2$ so the splitting shown is $\Omega _c/4\pi =$ 0.11 GHz. By studying Rabi frequencies in different control tone amplitude, we obtained the dependencies for 1-photon for $\omega _c/2\pi =$6.28 GHz and 2-photon processes for $\omega _c/2\pi =$6.6 GHz as shown in Fig. 4(d). The consistency in Rabi frequencies observed in this plot confirms our proposed operation schemes in Figs. 3(a) and 4(a). When the transmon energy levels are further detuned to $\omega _{10}/2\pi \sim \omega _p/2\pi =$ 6.9 GHz($\omega _{10}=\omega _p-\frac 12\Delta _1+\frac 12\alpha$), the probe tone becomes in resonance to $|0\rangle$ to $|1\rangle$ transition , and the operation scheme shown in Fig. 3(a) is recovered. Indeed, one may see that with control on, resonance shift as marked by blue arrow and crayon arrow with the amount of about 0.20 GHz. The accurate values for the shift of the 1-photon and 2-photon absorption will be given in the next paragraph.
Fig. 4. (a) Controlled photon transparency for $|0\rangle$ to $|2\rangle$ two-photon transition. (b)(c) The transmission in-phase signal (b) and quadrature-phase (c) of the probe microwave with $\omega _p/2\pi = 6.9$ GHz. When there is a continuous pumping microwave with a frequency $\omega _c/2\pi =6.6$ GHz, the resonance featuring $|0\rangle$ to $|2\rangle$ transition(black arrow) split into two(red arrows). The detuning plotted here is $\Delta _1/2$. Because the probing tone is two-photon process, the splitting is $\Omega _c/4\pi =0.11$ GHz. (d) The Rabi frequencies determined from 1-photon and 2-photon CPT agree with each other.
The Autler-Townes splitting would present when $\Delta$ is small so that the eigenvalues of Eq. (2) can be solved as $\lambda _0\sim 0$ and
if $\alpha \gg \Omega _c, \Delta$. If the atom is perfectly tuned $\Delta =0$, the levels split by $\sqrt {2}\Omega _c$ and the probe will be in resonance if $\omega _p=\omega _c+\alpha \pm \frac {\sqrt {2}}{2}\Omega _c$ for 1-photon absorption and $2\omega _p=2\omega _c+\alpha \pm \frac {\sqrt {2}}{2}\Omega _c$ for 2-photon absorption. If we set the condition as our 1st scheme $\omega _p-\omega _c=\alpha$, and vary the energy levels, the resonance conditions will satisfy Eq. (3) under the constraint $\lambda =\omega _p-\omega _c=\alpha$. The associated resonances will occur at two conditions $\Delta _\pm =\pm \Omega _c/2$ so the splitting in $\Delta _1(=\Delta _2=\Delta )$ would be $\Omega _c$. For 2-photon probe, the scheme $\omega _p-\omega _c=\frac 12\alpha$ is used and the resonance would be at $\Delta _1=2\Delta _2=2\Delta =\pm \Omega _c/2$. Meanwhile, the 1-photon resonances would present at a far blue-detuned condition, Put in the values $\Omega _c\sim$ 0.2 GHz, $\alpha \sim$ 0.6 GHz, we expect that the resonances in Fig. 4(b) should occur at $\Delta _1\sim$ 0.2 and 0.7 GHz, consistent with the experimental results. We will revisit these results in Sec. 3.5 with an approach calculating time evolution of density matrix.We would like to note that recent theoretical works suggest the objective criteria for justifying electromagnetically induced transparency(EIT) from ATS though the experimental setup may look similar [23,24]. Regarding the system preparation, damping rates and control tone intensity are the important parameters to govern these conditions. Regarding the justification on the experimental data, Akaike’s information provides additional test ATS vs. EIT for three-level atomic systems. Nevertheless, in our $\Xi$ type system the damping rate from $|1\rangle$ to $|0\rangle$ is much greater than $|2\rangle$ to $|0\rangle$, so we may exclude the possibility of EIT in the 1-photon or 2-photon probe cases. Indeed, EIT is rather delicate, exists only for a weak control field and more likely occurs in a $\Lambda$-type atom.
3.3 4-wave mixing
In addition to the controlled transparency phenomena originated by the Autler-Townes splitting, two microwave tones may lead to microwave amplification in the model given by Eq. (2). Firstly, when the system is tuned to $\omega _{10}\sim \omega _c(\Delta =\alpha )$, a possible 4-wave mixing may build up as illustrated in Fig. 5(a) [17]. In this scenario the control tone is in resonance with $|0\rangle$ to $|1\rangle$ transition and produces split levels to form a 4-level system, the so-called Mollow triplet in spectroscopy. Meanwhile, the probe tone is in resonance to the transition from the highest level to the lowest one, $\omega _p\sim \omega _{10}+\Omega _c$ as the control tone serves as the optical pumping between two other resonance transitions. As a result, an emission from the highest level to the lowest level is stimulated to enhance the probe tone transmission. There is the emission of an associated idler tone $\omega _i=2\omega _c-\omega _p$ to complete the the 4-wave mixing. In this scheme, the detuning obeys $\Delta _1=\Delta _2=\Delta -\alpha$.
Fig. 5. (a) The 4-wave mixing scheme. (b)The comparison for different control tone power at $\omega _c/2\pi =$ are 6.57 GHz. At the highest power, $\Omega _c/2\pi \sim$ 0.25 GHz. The detuning scan provides 4 different regions for interesting quantum optical phenomena: 2-photon CPT around zero detuning; 1-photon CPT between 0.15 to 0.3 GHz detuning.; 4-wave mixing from 0.4 to 0.55 GHz detuning; 2-photon optical pumping around 0.3 GHz detuning(indicated by the vertical arrow).
As shown in Fig. 5(b), $\omega _c/2\pi =$6.57 GHz are selected to control the probe light $\omega _p/2\pi =$6.9 GHz. This condition is closed to show a resonant 2-photon CPT, namely $\omega _p-\omega _c=\alpha /2$. The control amplitude were chosen $\Omega _c/2\pi \sim$ 0.25 GHz at the control tone power =$-1$ dBm, which is confirmed by the 2-photon CPT. At a greater $\Delta =$0.4 - 0.55 GHz, we observed a significant increase in transmission amplitude comparing to the no-control case. With the $\Delta$ value, the 6.57 GHz control tone may drive the atom with $\omega _{10}/2\pi = 6.62 -6.77$ GHz and probe tone is in resonance to the 4-wave mixing effect that $\omega _p/2\pi =6.9$ GHz $\sim (\omega _{10}+\Omega _c)/2\pi =6.87-7.02$ GHz. The increment in IQ amplitude due to the 4-wave mixing is $\sim$3% (4 mV out of the base line 130 mV for the perfect transmission). Wen et al [17] reported a pronounced optical gain by 4-wave mixing of a transmon atom. The small change in our case is likely due to unoptimized parameters since the effect of 4-wave mixing is optimally operated at $\Delta \sim \alpha$. Other reasons may be the large relaxation rate for our transmons and the structure of a CPW resonator we used. We also like to mentioned that theoretical [25] and experimental [17] works demonstrated that when the control tone is off-resonant, there will be one peak with amplification and one dip with attenuation in transmission if sweeping the probe frequency. The feature of attenuation associated with the amplification was neither observed in experiment nor displayed by our calculation in Sec. 3.5, probably due to the operation scheme of tuning energy level. Further work may be conducted to clarify this point.
3.4 Optical pumping by 2-photon absorption
The second amplification process is 2-photon optical pumping. In Fig. 5(b), we can see a small peak structure is built when the atom energy is detuned to $\Delta \sim \alpha /2$, which gives $\omega _p\sim \omega _{10}$. The peak only appears when the control tone power is very large, and originates from the optical pumping from $|0\rangle$ to $|2\rangle$ by using 2-photon absorption as illustrated in Fig. 6(a). By using a coherent probe tone in resonance to $|0\rangle$ to $|1\rangle$ transition, one can expect the microwave amplification due to stimulated emission by probe tone. We note that the spontaneous emission from $|1\rangle$ to $|0\rangle$ has a random phase and cannot contribute to the homodyne signals. Similar to the 4-wave mixing, the spontaneous relaxation from $|2\rangle$ to $|1\rangle$ with rate $\Gamma$ is involved to complete the cycle. The resonance condition requires the choice $\omega _p-\omega _c=\alpha /2$. When $\omega _p$ and $\omega _c$ are fixed, the detunings should obey $\Delta _1=\Delta _2/2=\Delta -0.5\alpha$. To demonstrate the 2-photon pumping, we used synchronized pumping and probing pulses(Fig. 6(b)) with a different pair of tones $\omega _p/2\pi =$5.1 GHz and $\omega _c/2\pi =$4.7 GHz. Figure 6(c) shows the real-time scan of I/Q voltage measured by the digitizer. At the optimal condition, the change in I/Q due to pumping pulse is roughly 11% as one can find in Fig. 6(d). The value is referred to the transmission amplitude(30 mV) when the probe is tuned away from the resonant absorption. Because of the 2-photon process, the optical pumping has a lower efficiency. Though the current result does not change to system to a gain medium, yet it paves the way to the population inversion and probable masing if the relaxation rate $\Gamma$ is smaller than the pumping rate.
Fig. 6. (a) Two-photon optical pumping process. The population of state $|1\rangle$ is increased by spontaneous relaxation from $|2\rangle$. (b)(c) The real-time scan of the transmission I and Q signals on resonance (c) when the control and probe tones are gated as shown in (b). Here $\omega _p/2\pi =$5.1 GHz and $\omega _c/2\pi =$4.8 GHz. (d) The transmission amplitude(blue) and gain(red) as a function of detuning $\Delta _1$. The emission is increased by 17% on resonance.
3.5 Theoretical calculation for transmission
The associated transmission properties of the 3 resonance cases can be elaborated by knowing the time evolution of the density matrix $\rho$. Taking into account the dissipation modeled by the Lindblad formalism, the density matrix of the system should follow the master equation
The approach given by Kockum et al. [26] leads to a systematic calculation of the transmission. According to the linear response theory, the susceptibility can be calculated as
Here $\Sigma _p=(\Sigma _++\Sigma _-)$ and $\Sigma _-=\sum _{m=1}^2\sqrt {m}|m-1\rangle \langle m|$ and $\Sigma _+=\Sigma _-^{\dagger }$. To calculate the ensemble average, we apply the following formula, which is stated by quantum regression theorem [27,28]. $\rho _A(t)$ would satisfy Eq. (4) with the initial condition $\rho _A(0)=\rho _s A(0)$, in which $\rho _s$ is the static solution of density matrix. Similarly, $\langle B(t)A(0) \rangle =Tr(B(0)\rho _A'(t))$, and $\rho _A'(t)$ is the solution to Eq. (4) with the initial condition $\rho _A'(0)=A(0)\rho _s$. The susceptibility allows us to calculate the microwave transmission by which can be compared with the experimental results. We assumed zero temperature in our calculation, in view of the fact that the transmon temperature should not generate much thermal excitation to the 5 GHz atom level, which is equivalent to a temperature of 0.2 K.Figure 7(a) shows the intensity plot of the change in transmission amplitude $\Delta T=|T|-1$ as a function of normalized detuning $\Delta /\alpha$ and normalized probe frequency $\Delta \omega /\alpha \equiv (\omega _p-\omega _c)/\alpha$. Here the Rabi frequencies of the driving microwave are assumed $\Omega _{10}=\Omega _{21}/\sqrt {2}=\Omega _c=0.2\alpha$. The value is similar to the typical ones we used in experiment. The relaxation and dephasing rates are assume $\Gamma _1=2\Gamma _\phi =0.05\alpha$. The transmission amplitude in general shows an resonant absorption feature that colored in red. To assist the understanding, we also draw the resonance absorption conditions for $\omega _p=\omega _{10}$ and $\omega _p+\omega _c=\omega _{20}$ as the green dashed lines. One can clear see three avoided crossing at $\Delta /\alpha =0$, 0.5 and 1, that respectively reflect the features of (1) Autler-Townes effect, (2) 2-photon pumping and (3) 4-wave mixing. The 2-photon pumping is in resonance with driving tone and probe tone and present an emission feature that colored in blue. The later two features are more clearly shown in Fig. 7(b).
Fig. 7. (a) The change in transmission amplitude $\Delta T$ of a weak probe as a function of normalized detuning $\Delta /\alpha$ and probe frequency $\Delta \omega /\alpha$ when the 3-level atom is under driven by $\Omega _c/\alpha =0.2$ . The red color presents a resonance absorption while the blue color presents the resonant emission. Arrows (1), (2), and (3) respectively note the feature of ATS, 2-photon pumping and 4-wave mixing. Dissipation $\Gamma _1=2\Gamma _\phi =0.05\alpha$ (b) A blow-up of the intensity plot to show the 2-photon pumping and 4-wave mixing. (c) $\Delta T$ as a function of $\Delta /\alpha$ at the probe frequency $\Delta \omega /\alpha =$ 1, 0.8 and 0.36, respectively corresponding to the resonance conditions for processes (1), (2) and (3). The features due to 2-photon pumping and 4-wave mixing can bring amplification of the transmission. The curves are vertically shifted for clarity.
Figure 7(c) illustrates $\Delta T$ when the probe frequency is fixed at the 3 different resonance conditions at $\Delta \omega /\alpha =1.0$, 0.8 and 0.36. In the vicinity of $\Delta /\alpha =0$, we may observe the split transmission peaks due to the Autler-Townes effect. When the system blue-detuned, the transmission can be amplified due to the 2-photon pumping and 4-wave mixing process.
The framework of linear response theory cannot treat the case when the probe microwave is strong. To study the 2-photon spectroscopy, we consider the 3 level system under driven by two classical fields:
Figure 8(a) illustrates the (static) density matrix data as a function of $\Delta /\alpha$ at various probe frequency $\Delta \omega /\alpha$=0.5 to 1. Here $\Omega _c/\alpha =0.2$ and $\Omega _p/\alpha =0.04$. The dissipation rates are $\Gamma _1=2\Gamma _\phi =0.01\alpha$. The 1-photon and 2-photon processes are zero-detuned at $\Delta \omega /\alpha$=1 and 0.5, respectively. One can see that the splitting in the two cases are the same $\sim \Omega _c$. The grey dashed curve and solid curves are respectively the 1-photon and 2-photon resonances deduced from Eq. (3). Also one can see the optical amplification due to 2-photon pumping at $\Delta /\alpha =0.5$ and $\Delta \omega /\alpha =0.6$. Figure 8(b) is the plot of static $\textrm {Im}\rho _{02} \Gamma _1/\Omega _p$ for the 2-photon probe resonance at various probe amplitude $\Omega _p/\alpha$. The greater probe amplitude, the significant absorption. The resonance absorption at 0.115 has a linear dependence of $\Omega _p$ showing the clear feature of 2-photon absorption. The optical amplification due to 4-wave mixing can be found in Fig. 8(c). When the control tone $\Omega _c/\alpha >0.4$ , the 1-photon probe transmission presents a gain feature at $\Delta /\alpha =1.2 -1.3$. The presence of idler tone for the 4-wave mixing process is best identified by the Fourier transform of the density matrix element, $\rho _{01}(\omega )$ as illustrated in Fig. 8(d). The static result ($\omega =0$) is identical to the curve shown in Fig. 8(c), showing resonance absorption due to ATS and resonance gain due to 4-wave mixing. The large Fourier component at $\omega /\alpha =-0.5$ is the direct response from the control tone at the frequency of $\omega _c$. At $\Delta /\alpha =1.2$, where the 4-wave mixing is expected to occur, there is an additional Fourier component at $\omega /\alpha =-1.0$, signifying the idler tone satisfying $\omega _i=2\omega _c-\omega _p$.
Fig. 8. (a) The imaginary part of static solution of density matrix element $\rho _{02}$(solid curve) and $-\rho _{01}$ (dashed curve)as a function of $\Delta /\alpha$ for $\Omega _c=0.2$ and $\Omega _p=0.04$ at various $\Delta \omega /\alpha$ values. The curves are vertically shifted by the $\Delta \omega /\alpha$ values for clarity. For comparison, the analytical solutions for 1-photon and 2-photon probe are plotted with grey dashed curves and solid curves, respectively. Here the dissipation rates are $\Gamma _1=2\Gamma _\phi =0.01\alpha$. When $\Delta \omega /\alpha =0.6$, the $\rho _{01}$ curve becomes a peak at $\Delta /\alpha =0.5$ because of the amplification of 2-photon pumping. (b) $\textrm {Im}\rho _{02} \Gamma _1/\Omega _p$ as a function of $\Delta /\alpha$ at $\Delta \omega /\alpha =0.5$ show the large absorption under a large probe amplitude. Inset shows the $\Omega _p$ dependence of the resonance absorption at $\Delta /\alpha =0.115$. It shows a linear dependance, reflecting the feature of 2-photon absorption. (c) $-\textrm {Im}\rho _{01} \Gamma _1/\Omega _p$ as a function of $\Delta /\alpha$ at various $\Omega _c/\alpha$ values shows that the 1-photon probe could be amplified due to 4-wave mixing when $\Omega _c/\alpha$ is greater than 0.4. Here $\Delta \omega /\alpha =0.5$, $\Omega _p/\alpha =0.04$ and $\Gamma _1=2\Gamma _\phi =0.01\alpha$. (d) The spectrum(Fourier transform) of $-\textrm {Im}\rho _{01}$ for $\Delta \omega /\alpha =0.5$, $\Omega _c/\alpha =0.5$, $\Omega _p=0.04$ and $\Gamma _1=2\Gamma _\phi =0.01\alpha$. The large Fourier component at $\omega /\alpha =-0.5$ is due to the control driving at $\omega _c$. The 4-wave mixing is featured by the additional Fourier component at $\omega /\alpha =-1$, which corresponds to an emission of $\omega _i=2\omega _c-\omega _p$.
4. Conclusion
In conclusion, we have studied several quantum optical phenomena in a 3-level transmon atom. Because of the tunability of the transmon energy, we can study 1-photon CPT, 2-photon CPT, 2-photon optical pumping and 4-wave mixing using the same probe and control tones. These processes can be fully analyzed by a $3\times 3$ effective Hamiltonian under the microwave driving. The microwave amplification by using 2-photon process and 4-wave mixing are compared. The 4-wave mixing provides a larger bandwidth and a smaller amplitude increment of 3% while the 2-photon optical pumping a much narrow bandwidth with an increment of 11%. Both results can find realistic applications such as signal amplification and possible maser.
Funding
Ministry of Science and Technology, Taiwan (106-2112-M-005-007, 107-2112-M-005-001, 108-2112-M-005-007).
Acknowledgments
Fruitful discussions with M. C. Chung and I. C. Hoi are acknowledged. This work is financially supported by the Ministry of Science and Technology, Taiwan under grant Nos. 106-2112-M-005-007 and 107-2112-M-005-001.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474(7353), 589–597 (2011). [CrossRef]
2. 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]
3. J.-T. Shen and S. Fan, “Coherent single photon transport in a one-dimensional waveguide coupled with superconducting quantum bits,” Phys. Rev. Lett. 95(21), 213001 (2005). [CrossRef]
4. M. Devoret, S. Girvin, and R. Schoelkopf, “Circuit-qed: How strong can the coupling between a josephson junction atom and a transmission line resonator be?” Ann. Phys. (Berlin, Ger.) 16(10-11), 767–779 (2007). [CrossRef]
5. O. C. Astafiev, A. M. Zagoskin, A. A. Abdumalikov Jr., Y. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance fluorescence of a single artificial atom,” Science 327(5967), 840–843 (2010). [CrossRef]
6. C. Wilson, T. Duty, F. Persson, M. Sandberg, G. Johansson, and P. Delsing, “Coherence times of dressed states of a superconducting qubit under extreme driving,” Phys. Rev. Lett. 98(25), 257003 (2007). [CrossRef]
7. K. Koshino, H. Terai, K. Inomata, T. Yamamoto, W. Qiu, Z. Wang, and Y. Nakamura, “Observation of the three-state dressed states in circuit quantum electrodynamics,” Phys. Rev. Lett. 110(26), 263601 (2013). [CrossRef]
8. O. V. Astafiev, A. A. Abdumalikov, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Ultimate on-chip quantum amplifier,” Phys. Rev. Lett. 104(18), 183603 (2010). [CrossRef]
9. O. V. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449(7162), 588–590 (2007). [CrossRef]
10. M. A. Sillanpää, J. Li, K. Cicak, F. Altomare, J. I. Park, R. W. Simmonds, G. S. Paraoanu, and P. J. Hakonen, “Autler-townes effect in a superconducting three-level system,” Phys. Rev. Lett. 103(19), 193601 (2009). [CrossRef]
11. A. A. Abdumalikov, O. V. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010). [CrossRef]
12. I.-C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki, B. Peropadre, and P. Delsing, “Demonstration of a single-photon router in the microwave regime,” Phys. Rev. Lett. 107(7), 073601 (2011). [CrossRef]
13. I.-C. Hoi, A. F. Kockum, T. Palomaki, T. M. Stace, B. Fan, L. Tornberg, S. R. Sathyamoorthy, G. Johansson, P. Delsing, and C. M. Wilson, “Giant cross kerr effect for propagating microwaves induced by an artificial atom,” Phys. Rev. Lett. 111(5), 053601 (2013). [CrossRef]
14. J. Long, H. S. Ku, X. Wu, X. Gu, R. E. Lake, M. Bal, Y.-x. Liu, and D. P. Pappas, “Electromagnetically induced transparency in circuit quantum electrodynamics with nested polariton states,” Phys. Rev. Lett. 120(8), 083602 (2018). [CrossRef]
15. A. Y. Dmitriev, R. Shaikhaidarov, V. N. Antonov, T. Hönigl-Decrinis, and O. V. Astafiev, “Quantum wave mixing and visualisation of coherent and superposed photonic states in a waveguide,” Nat. Commun. 8(1), 1352 (2017). [CrossRef]
16. T. Hönigl-Decrinis, I. V. Antonov, R. Shaikhaidarov, V. N. Antonov, A. Y. Dmitriev, and O. V. Astafiev, “Mixing of coherent waves in a single three-level artificial atom,” Phys. Rev. A 98(4), 041801 (2018). [CrossRef]
17. P. Y. Wen, A. F. Kockum, H. Ian, J. C. Chen, F. Nori, and I. C. Hoi, “Reflective amplification without population inversion from a strongly driven superconducting qubit,” Phys. Rev. Lett. 120(6), 063603 (2018). [CrossRef]
18. A. Y. Dmitriev, R. Shaikhaidarov, T. Hönigl-Decrinis, S. E. de Graaf, V. N. Antonov, and O. V. Astafiev, “Probing photon statistics of coherent states by continuous wave mixing on a two-level system,” Phys. Rev. A 100(1), 013808 (2019). [CrossRef]
19. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the cooper pair box,” Phys. Rev. A 76(4), 042319 (2007). [CrossRef]
20. S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703–722 (1955). [CrossRef]
21. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. 188(5), 1969–1975 (1969). [CrossRef]
22. 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]
23. P. M. Anisimov, J. P. Dowling, and B. C. Sanders, “Objectively discerning autler-townes splitting from electromagnetically induced transparency,” Phys. Rev. Lett. 107(16), 163604 (2011). [CrossRef]
24. H.-C. Li, G.-Q. Ge, and H.-Y. Zhang, “Dressed-state realization of the transition from electromagnetically induced transparency to autler-townes splitting in superconducting circuits,” Opt. Express 23(8), 9844–9851 (2015). [CrossRef]
25. J. Mompart and R. Corbalán, “Lasing without inversion,” J. Opt. B: Quantum Semiclassical Opt. 2(3), R7–R24 (2000). [CrossRef]
26. A. F. Kockum, M. Sandberg, M. R. Vissers, J. Gao, G. Johansson, and D. P. Pappas, “Detailed modelling of the susceptibility of a thermally populated, strongly driven circuit-qed system,” J. Phys. B: At., Mol. Opt. Phys. 46(22), 224014 (2013). [CrossRef]
27. I. Rau, G. Johansson, and A. Shnirman, “Cavity quantum electrodynamics in superconducting circuits: Susceptibility at elevated temperatures,” Phys. Rev. B 70(5), 054521 (2004). [CrossRef]
28. P. D. Blocher and K. Mølmer, “Quantum regression theorem for out-of-time-ordered correlation functions,” Phys. Rev. A 99(3), 033816 (2019). [CrossRef]
