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Abstract
We investigate photonic transport through fiber optical cavities under tunable fast modulation, which is induced by modulating atom-cavity coupling in a cavity quantum electrodynamics (CQED) system. The modulation bandwidth exceeds the timescales of all other system processes, such as cavity decay and atom-cavity coupling, and allows control of the dynamics of photonic transport through the cavity array. The transmission as a function of the modulation frequency clearly shows enhancement peaks in single cavity and two coupled-cavity cases. In particular, in the coupled-cavity case, the position of the enhancement peak is shifted due to the delocalized cavity field in the coupled system.
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1. Introduction
Energy transfer is a ubiquitous and essential physical phenomenon, important in systems ranging from common electrical devices to biological systems. Quantum random walk [1,2] is a simple model to describe coherent quantum transport in a structured system, which can produce dramatically different results compared to its classical counterpart. The key feature of the quantum properties is interference in the wave function. For example, quantum transport in static disorder can result in an important consequence called Anderson localization, in which propagation is totally inhibited [3,4]. The consequences of static disorder have been studied in various physical systems, such as microwave [5,6] and photonic transport [7–11].
A new perspective on transport physics comes from considering both static and dynamic disorder. Motivated by the progress of experimental studies on energy transport in photosynthetic protein complexes [12–14], many theoretical models, which incorporate the detail of the static and dynamic disorders due to the system-environment coupling, have been proposed and analyzed [15]. These studies have pointed out that coupling to the environment can enhance the coherent energy transport even in the complicated energy landscape like in the actual photosynthetic system [16–19]. In addition to the studies on the natural photosynthetic systems, artificial quantum systems, such as superconducting qubits [20] or trapped ions [21–23], have been utilized as experimental platforms with tunable environment noise, and optical networks based on optical fibers [24,25] or photonic integrated circuits [26,27] can be used to study quantum transport of photons.
Introducing photon-photon interactions adds an additional layer of interest, allowing us to probe the influence of many-body quantum dynamics on transport [28–30]. However, optical nonlinearities in standard optical materials are not enough to induce meaningful interaction between single photons. To overcome this difficulty, we consider a cavity quantum electrodynamics (CQED) system [31]. CQED provides an ideal experimental platform, in which cavity-enhanced atom-photon interactions mediate effective nonlinear interaction between photons [32]. Recent advances have allowed coupling between multiple optical-fiber-based CQED units, which has opened the door to embedded CQED systems in optical fiber networks [33,34]. In addition, the atom-photon interaction in the CQED system provides a way to modulate the photonic transport in a cavity array by introducing control laser fields for atoms at each cavity, whose modulation bandwidth can cover typical timescales of the transport. The ability to implement dynamic modulation covering the transport timescale gives CQED a unique functionality compared to purely optical networks, such as integrated photonic circuits, where disorders are encoded on the programable but static network structure for the each photon transmission [27].
In this paper, we investigate the photon transfer efficiency through a coupled-cavities system under controlled modulation by utilizing fiber-based CQED. The modulation bandwidth covers the time scale of the atom-cavity coupling, the photon lifetime in the cavities, and the coupling rate between the cavities. The wide bandwidth modulation allows us to change the dynamics of the photonic transport through the system. We observe a significant enhancement of the transmission assisted by the external modulation field, and find good agreement with numerical calculations without any fitting parameters. In addition to the external modulation, we study the system responses in different cavity coupling conditions by tuning the mirror reflectance between the cavities. The main contribution to the transport depends on the coupling rate between the cavities. For weak coupling, localized modes in each cavity are important, while for strong coupling, a mode delocalized over all coupled cavities is the dominant contribution. We observe the different responses to the modulation in these two regimes in numerical and experimental studies.
2. Model of photonic transport
As schematically shown in Fig. 1, we consider the transport of photons through a 1-dimensional coupled-cavities system. The standard approach to forming a quantum-optical model of the photonic transport dynamics is to use a master equation with a density operator $\rho$, which includes the cavity modes and the interacting atoms. The master equation in the Lindblad form ($\hbar =1$) is given by:
Fig. 1. Schematic diagram of the coupled-cavities system. Each cavity contains an atomic ensemble (red ovals), which couples to the cavity mode at the rate of $g_i$. In order to introduce the modulation, we apply an external laser field to induce an AC Stark shift (yellow arrows).
First, we consider the simplest case of a single cavity ($N=1$). When $\delta _{{\rm a},1}=0$ and $\omega _{\rm a}^0=\omega _{{\rm c},1}\sim \omega _{\rm p}$ and the atom-cavity coupling $g_1$ is larger than any dissipation rate such as $\kappa _{\rm in,\ out}$, $\kappa _1^{\rm loss}$ or $\gamma$, the probe transmission at the resonance of the cavity is suppressed due to the hybridization of the photonic and atomic states and the split of the energy eigenstates as shown in Fig. 2(a) and (b). In this work, we control and modulate $\delta _{{\rm a},i}$ through a light shift by an additional laser field (Fig. 1). Fast intensity modulation of the light shift laser by using an electro-optic modulator (EOM) enables us to realize the modulation bandwidth that covers the typical time scale in the system, such as $g_i$, $\kappa _{\rm in,\ out}$, and $\nu _{\rm i,j}$. To compare the effect of the static and dynamical modulation, we perform a time-domain numerical calculation based on the master equation (Fig. 2(c)-(f)) with QuTip open source software [35]. In this calculation, the initial state is zero photons in the cavity and the atom is in its ground state, and the weak probe pulse is launched at $t=0$. As shown in Fig. 2(c), larger $\delta _{{\rm a},1}$ results in a larger cavity photon number in the static shift case in equilibrium. In the dynamical modulation case, the modulation frequency is a crucial parameter for the transmission property. When the modulation frequency is much smaller than other typical time scales of the system, the cycle-averaged transmission should be equal to the average of the static case with a corresponding $\delta _{{\rm a},1}(t)$ (black dashed lines in Fig. 2(d)). In the higher frequency limit, the system no longer responds to the modulation, and the transmission converges to the result of a static case where $\delta _{{\rm a},1}$ matches the average of the modulation (black dotted line in Fig. 2(f)). On the other hand, when the modulation frequency is close to the frequency difference between the probe laser and the system resonance frequency, which corresponds to $g_1$ in this case, the cavity photon number is driven by the modulation as shown in Fig. 2(e). This transmission enhancement can be understood by considering the atomic state in the presence of the modulation laser. Under the modulation, the atomic energy spectrum has sidebands spaced by the modulation frequency. The atomic states in this sideband interact with the cavity mode to create dressed states. Therefore, for sufficiently small modulation depth, when the modulation frequency compensates the frequency mismatch with the probe laser, the system including the light shift laser will have a dressed state that resonates with the probe laser.
Fig. 2. Transmission spectra and temporal variation of the cavity photon number under modulation. a Energy diagrams of a cavity, an atom, a CQED system and a CQED system with a modulation. In this figure, only single-cavity systems are considered, and the subscript indicating the cavity number $i$ has been removed for clarity. $|n\rangle (n=0,1)$ is a Fock state of the cavity photon number $n$, and $|g (e)\rangle$ is a ground (excited) state of the atom. $|\pm \rangle =(|e,0\rangle \pm |g,1\rangle )/\sqrt {2}$ are dressed states of the atom-cavity system. b Cavity transmission spectra with and without an atomic resonance frequency shift $\delta _{{\rm a}}$. Blue and red curves correspond to $\delta _{{\rm a}}=0$ and $0.2g$, respectively. c-f Temporal variations of the cavity photon number. In this calculation, $\omega _{\rm p}$ equals $\omega _{\rm c}$ and $\omega _{\rm a}^0$. Blue and red curves in c are results with static shift of $\delta _{{\rm a}}=0$ and $0.2g$, respectively. For the time dependent atomic resonance frequency shift, we use a sinusoidal function $\delta _{{\rm a}}^0(1+\sin {\omega _{\rm mod}t})$, and $\delta _{{\rm a}}^0=0.2g$. The modulation frequencies are $\omega _{\rm mod}=0.2g$ (c), $g$ (d), and $2g$ (e). Dashed lines in c and d represent the result with a sinusoidal average of static detuning results. Dotted lines in d and e represent the result with a static shift with $\delta _{{\rm a}}=0.2g$. The small difference between these results is due to the nonlinearity of the transmission as a function of $\delta _{{\rm a}}$. Black circles in d-f correspond to the cycle average at the last part of the oscillation in each result. Throughout these calculation, the cavity QED parameters $(\kappa, \gamma )/g=(0.2, 0.5)$ are used and the mean cavity photon number ($\langle \hat {a}^\dagger \hat {a} \rangle$) is normalized at the value with $\delta _{{\rm a}}=0$.
3. Results
3.1 Single cavity case
To study the effect of the modulation experimentally, we constructed a setup as schematically shown in Fig. 3(a). Details of the setup are described elsewhere [33]. Here, we briefly summarize the features of the fiber cavity and the laser system. We use an optical-fiber-based cavity, which consists of two fiber Bragg grating (FBG) mirrors embedded in a standard optical fiber and a thin fiber part, whose diameter is 400 nm, and length is 3 mm, in the middle of the cavity. The thin part is sandwiched by tapered regions, which are adiabatically connected to the standard single-mode optical fiber. The evanescent field of the propagation mode at the thin part is used as an interface for atoms located in the vicinity of the fiber surface. We use laser-cooled Cs atoms, and the cavity is resonant to the Cs D2 line (F=4$\leftrightarrow$F’=5). To optically trap the atoms, we use the so-called two-color evanescent fields optical trap [36]. We choose two magic wavelengths (937 nm laser for the attractive potential and 688 nm laser for the repulsive potential) for the optical trap [37,38]. We use a standing wave of the red-detuned 937 nm laser to form an optical lattice. Since only a single atom can be trapped at each lattice site due to the collisional blockade, collisions between the trapped atoms are suppressed [33]. In addition to the trapping lasers, we add another laser to induce a controllable light shift for the modulation, whose wavelength is 900 nm. The light shift laser is superimposed with the trapping lasers through a dichroic mirror and coupled into the optical fiber cavity (Fig. 3(a)). Since the stopbands of the FBGs are away from 900 nm, the light shift laser passes through the fiber cavity without loss. We use an EOM to control the intensity of the light shift laser, and the modulation bandwidth is up to 10 GHz.
Fig. 3. Transmission enhancement in a single cavity system. a Simplified sketch of the experimental setup. The laser cooled Cs ensemble is coupled to the cavity mode at the thin fiber part inside a vacuum chamber. The fiber Bragg grating mirrors (FBG1 and FBG2) are temperature stabilized to control the reflectance. The transmitted probe laser through the fiber cavity is detected after the filter array, which removes stray light. For the attractive trapping potential, 937 nm laser is coupled to the fiber from both sides to create a standing wave, and for the repulsive potential, 688 nm laser is coupled along the modulation laser. Both optical trapping lasers are not shown in this figure for clarity. Vacuum Rabi splittings without (b) and with constant light shift (c). Blue dots are experimental results, and red solid lines are fitted curves. d Ratio of the resonant transmission with time dependent modulation to the modulation free case, as a function of modulation frequency. Blue dots are experimental observations and the error bar represents standard error of the mean. Vertical dashed green line corresponds to the $\bar {g}_1'$. Red dots are results of numerical simulation without any fitting parameters, and the line between them is a guide to the eyes.
The experimental sequence starts with collecting and cooling Cs atoms by using a standard magneto-optical trapping technique. We use the $D_2$-line F=4$\leftrightarrow$F’=5 transition for cooling and the F=3$\leftrightarrow$F’=4 transition for repumping. After 100 ms loading time, we further cool and load the atoms into the evanescent field optical trap by reducing the total laser intensity. The cooling duration is 40 ms and the temperature of the atoms is about 20 $\mu$K. First, we perform transmission spectroscopy without the modulation laser to measure the coupling rate $\bar {g}_1$ between the ensemble atoms and cavity mode. We sweep the probe laser frequency $\pm$18 MHz in 2 ms, and measure the transmission. The input probe power is about 300 pW. After the first sweep, we optically pump the atoms into the F=3 state, which is off-resonant to the cavity, to realize an effectively empty cavity condition, and perform the second sweep to obtain the empty-cavity transmission spectrum, which is used to confirm the cavity resonance condition ($\omega _{{\rm c},1}=\omega _{\rm a}^0$). Figure 3(b) shows the observed transmission spectrum with atoms and a theoretical curve with $\bar {g}_1=2\pi \times 5.7$ MHz as a fitting parameter, which corresponds to an effective atom number of about 20 [33]. In addition, we perform the same spectroscopy with constant intensity light shift laser, whose power is 110 $\mu$W. From the observed spectrum as shown in Fig. 3(c), the shift is calculated as 2.75 MHz from the fitted curve. The measured shift shows reasonable agreement with the numerically estimated shift of 2.70 MHz. For this calculation, the propagation mode of the light shift laser and the atomic position in the optical trap are considered. It should be noted that the atom-cavity coupling $\bar {g}_1'$ for the fitted curve in Fig. 3(c) is $2\pi \times 7.1$ MHz, which is slightly larger than $\bar {g}_1$. It is presumably due to the additional attractive potential introduced by the light shift laser, which drags the atoms to the fiber surface and increases the atom-cavity coupling rate. In the following calculation we use $\bar {g}_1'$ for the atom-cavity coupling rate.
Next, we compare the effect of the static and time-dependent light shift on the transmission. We consider the ratio of the probe transmission under the static light shift to the time-dependent modulation. In the experiment, we use a sinusoidal intensity modulation with variable modulation frequency, which on average becomes the same intensity as in the static case, and the static light shift is applied at all other times. The frequency of the probe laser is fixed at the empty cavity resonance ($\omega _{\rm p}=\omega _{{\rm c},1}=\omega _{\rm a}^0$), and the pulse widths are 0.2 ms. We measure the transmission of the sequential probe pulses without and with the light shift modulation. Figure 3(d) shows the observed enhancement ratio as a function of the modulation frequency and the numerical calculation result. These two results show good agreement without any fitting parameters. Notably, the peak position of the enhancement corresponds to $\bar {g}_1'/(2\pi )$, which corresponds to the detuning between the probe laser and the resonance frequency of the dressed state of the system.
3.2 Coupled cavity case
The response of the transmission to the external modulation is modified when we consider the system with coupled cavities. Figure 4 shows a sketch of the coupled cavity setup. We added an FBG mirror (FBG3) to the system shown in Fig. 3(a). The coupling rate between the cavities $\nu _{12}$ is controlled by tuning the temperature of the FBG in the middle (FBG1). The condition of the coupled-cavities system can be classified into three-regimes, (i) $\nu _{12}\ll \kappa (=\kappa _1+\kappa _2)$, (ii) $\nu _{12}\sim \kappa$, (iii)$\nu _{12}\gg \kappa$. In the regime of $\nu _{12}/\kappa \ll 1$, the coupling between the cavities is negligible compared to the photon lifetime in each cavity. Hence, the cavity modes do not interfere with each other, and the transmission property can be approximated by the product of each single cavity response. In the opposite limit, $\nu _{12}/\kappa \gg 1$, the coupling does not play a significant role, and the cavity-array should be considered as a single extended cavity. Therefore, the transmission property is again described by the single but longer cavity result. In the intermediate regime, $\nu _{12}/\kappa \sim 1$, the cavity modes should be described by the superposition of each cavity mode,
and the transmission property depends on the relationship between the superposition states of the two cavities and the modulation.Fig. 4. Transmission enhancements in the different coupled-cavities regimes. a Simplified schematics of the experiment. Transmission spectra of the coupled-cavities system with and without atoms in $\nu _{12}/\kappa \ll 1$ (b) and $\nu _{12}/\kappa \sim 1$ (c) regimes. The spectra are taken under the constant light shift condition. Blue (red) dots are experimental observations without (with) the atoms, and solid curve is a fitted curve to evaluate $\nu _{12}$ and $\bar {g}_2'$. The count rate for the red dots and the red solid curve in b are multiplied by three for ease of comparison. Ratios of the transmission with and without the modulation in $\nu _{12}/\kappa \ll 1$ (d) and $\nu _{12}/\kappa \sim 1$ (e) regimes. Dotted orange line in e corresponds to the $\sqrt {\bar {g_2}'^2+\nu _{12}^2}/(2\pi )$. Other color scheme and data representations are the same as in Fig. 3(d)
To experimentally investigate the effect of $\nu _{12}$, we first compare empty cavity transmission spectra with different $\nu _{12}$ (Fig. 4(b) and (c)). Here the resonance frequencies of the two cavities are set to coincide ($\omega _{{\rm c},1}=\omega _{{\rm c},2}=\omega _{{\rm c}}=\omega _{\rm a}^0$). We use a theory based on the transfer matrix method to analyze the observation [39]. In the $\nu _{12}/\kappa \ll 1$ condition, we see a single peak spectrum (blue dots and curve in Fig. 4(b)). In contrast, when $\nu _{12}$ is slightly larger than $\kappa$, two peaks, which correspond to the two cavity normal modes in Eq. (5), are resolved on the observed spectrum (blue dots and curve in Fig. 4(c)). From the fitted curves in Fig. 4(b) and (c), the $\nu _{12}/2\pi$ are (1.4, 4.7) MHz, respectively, and $\kappa /2\pi$ is 2.8 MHz. The coupling of the atoms introduces Rabi splitting to the transmission spectra (red dots and curve in Fig. 4(b) and (c)). Like in the single cavity case, the coupling rate ($\bar {g}_2'/(2\pi )$) under the static light shift and its light shift ($\delta _{{\rm a},2}$) are evaluated as 6.4 MHz and 1.6 MHz, respectively.
We again investigate the transmission enhancement under the time-dependent modulation in the coupled-cavities case. In the case of $\nu _{12}/\kappa \ll 1$, the enhancement peak position corresponds to the $\bar {g_2}'/(2\pi )$ as shown in Fig. 4(d). This behavior is similar to the single cavity case, which can be interpreted as the input photon passes through independent two single cavities with different transmission probabilities. On the other hand, in the case of $\nu _{12}/\kappa \sim 1$, the transmission property against the modulation frequency shows a different peak position in both of the numerical and experimental results (Fig. 4(e)). In this regime, the cavity mode is no longer a product state of each cavity mode, but a superposition state. In particular, the coupled-cavity QED system has three normal modes [34,39], and in this transmission experiment, we can probe two symmetric modes, which have resonance frequencies of $\pm \sqrt {\bar {g_2}'^2+\nu _{12}^2}/(2\pi )$. As shown in Fig. 4(e), the orange dotted line, which corresponds to $\sqrt {\bar {g_2}'^2+\nu _{12}^2}/(2\pi )$, well describes the shifted peak position in the $\nu _{12}/\kappa \sim 1$ regime. The deviation compared to the result in the $\nu _{12}/\kappa \ll 1$ case is due to the delocalization of the cavity mode and hence, the local modulation affects the transmission property of the whole system due to the coupling between the connected sub-systems.
4. Conclusions and outlook
In conclusion, we have developed a coupled-cavities QED system that has a wide modulation bandwidth by utilizing an intensity-modulated laser to control the light shift of the coupled atoms. We successfully observed transmission enhancement due to external modulation in the single cavity and the two-coupled-cavities situations, and the experimental observations show reasonable agreement with the numerical simulation results. It should be noted that the modulation bandwidth that covers the photon lifetime in the system is a key element to study the effect of the modulation on the photonic transport. The demonstrated real time fast modulation provides an important difference compared to the previous studies with the optical systems with slow modulation frequencies [24,25] or spatial encoding of dephasing or disorder effect along the propagation direction in waveguide arrays [26,27]. A great benefit of our approach is that the coupling of the coupled-cavities system can be tuned in order to study the localized and the delocalized modes of complex networks with different connection strengths, such as photosynthetic protein complexes. To study quantum many-body transport physics, a single excitation injection, instead of using the weak coherent laser, is essential. CQED with a single trapped atom or the heralded single-photon generation scheme with ensemble atoms may be suitable for this direction.
Funding
Japan Science and Technology CREST (JPMJCR1771); Japan Society for the Promotion of Science KAKENHI (18H05207); Japan Science and Technology PRESTO (JPMJPR1662).
Acknowledgments
S. K. thanks D. White for critical reading of the manuscript and valuable comments , and A. Goban for valuable discussions and insightful comments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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