1. INTRODUCTION

Controlling absorption and emission of single photons by single atoms is a central question of quantum optics. Single atoms are known to interact with a single photon [1,2], a property that can be exploited to develop efficient single-photon sources [3] or to fabricate devices showing nonlinearities at the single-photon level [4]. This requires, however, that the atom interacts ideally with a single mode of the electromagnetic field, so as to collect every emitted photon or to ensure the deterministic interaction of a single photon with the atom. Such an ideal system, called a “one-dimensional atom” [5], is a long-sought system to generate and manipulate quantum light [6].

Efficient atom–photon interfaces have been demonstrated with natural and artificial atoms using geometrical approaches [7], or controlling the emitter spontaneous emission [8,9] through inhibition—as demonstrated for emitters in very thin nanowires [10,11]—or through acceleration of spontaneous emission—as shown using various types of optical microcavities [12,13]. Such systems have led to the demonstration of ultra-efficient sources of indistinguishable single photons [14–17] for quantum computing [18–20], single-photon nonlinearities for photon–photon gates [4,21–24], and quantum memories [25].

All these systems are built using the paradigm of a two-level system (TLS) coupled to a cavity or to a waveguide. The spectral bandwidth of the emitter is then determined by the modified spontaneous emission rate that is engineered through the Purcell effect (for cavity-based structures) or dielectric screening (for nanowire-based structures). Hence, these one-dimensional atom structures present a fixed spectral bandwidth, which defines once and for all the temporal profile of the emitted photons, the memory time, and the temporal window for nonlinear gates. This limitation is inherent to a linear stationary system. It has been proposed recently [26] to overcome this limit by taking advantage of the nonlinearity of the system, thereby reconciling the possibility of long-time storage with a controlled photon release. Here, we address this issue by breaking time invariance introducing a time-varying modulation of the system’s bandwidth.

An atom with a tunable bandwidth would open many possibilities: one could make the bandwidth narrow for long-term quantum memories or broad for high-rate single-photon generation or fast photon–photon gates [27]. If this bandwidth tuning could be performed in real time, one could shape the temporal profile of the emitted photon [28] for efficient single-photon absorption or, more generally, shape the time–frequency structure of a single-photon wavepacket. Such tunability has been theoretically proposed through a tuning of the cavity decay rate $\kappa$ [29]. This proposition is well suited for microwave photons and superconducting quantum bits, where cavities with adjustable couplers can be defined on chip [30]; it was implemented to demonstrate the optimized absorption of single photons [31]. However, such an approach is challenging for most platforms, especially in the optical domain where tuning the cavity bandwidth has not been reported yet. One could then consider controlling the detuning between the atomic resonance and the cavity mode. Recently, the spectral engineering of single-photon emission has been demonstrated by fast modulation of an emitter’s frequency [32]. Alternatively, ultra-fast cavity detuning by the Kerr effect has been reported [33]. However, this is obtained at the expense of a reduced probability to couple to the cavity mode, the so-called “mode coupling” $\beta$-factor.

In the present work, we propose a new way of implementing an effective one-dimensional atom that offers bandwidth tunability with a mostly constant mode coupling to the cavity mode. We consider the system made of two close atoms mutually coupled through dipolar interaction [34–36] and coupled to the same cavity mode. In the bad cavity limit, the eigenstates of a system of two atoms with the same frequency are a subradiant (narrowband) and a superradiant (broadband) state. Upon detuning the atoms, the eigenstates become a superposition of subradiant and superradiant states [37] with a weight that depends on the detuning between the two atoms. The subradiant state with a narrow bandwidth is of interest for memory applications; however, observing and manipulating a subradiant state is a difficult task, precisely because it cannot be excited directly. Only a few works have reported their observation [36,38–40]. Here, we propose to introduce a real-time detuning between the two atoms to dynamically transform the dark state into a gray state. Remarkably, while the eigenstates present bandwidth tunability, their efficient coupling to the cavity mode remains mostly constant.

These unique features open many possibilities for optical quantum technologies where one can dynamically tune the system parameters during the emission or absorption of light, while remaining in the one-dimensional atom regime. We first demonstrate the generation of single-photon wavepackets with continuous variable encoding in the time–frequency domain, light states that have recently attracted strong interest for quantum computing and quantum sensing [41,42]. Single-photon wavepackets in the form of cat-like and compass states are generated, both presenting sub-Planck features. We then propose a bandwidth tunable quantum memory that stores short photon pulses and releases them with tunable bandwidth. Such a device shows great potential for quantum networks, be it for interfacing various quantum systems [43–45] or for synchronization [46–48].

The new concepts introduced here are applicable to all kinds of coupled natural or artificial atoms. We discuss experimental implementation in the last section, identifying the key limiting factors and comparing different candidate systems such as solid state emitters, molecules, and Rydberg atoms.

2. TUNABLE ONE-DIMENSIONAL ATOM

To achieve a one-dimensional atom, it is necessary to favor the atom emission into a single mode. The mode coupling scales as $\beta = \frac{{{\Gamma _0}}}{{{\Gamma _0} + \gamma}}$, where ${\Gamma _0}$ (respectively, $\gamma$) is the spontaneous decay rate into the cavity mode (resp., other modes). The most common way to obtain a one-dimensional atom ($\beta \approx 1$) is thus to control its spontaneous emission. High $\beta$ can be obtained by either reducing $\gamma$, i.e., inhibiting the spontaneous emission in the other modes, or increasing ${\Gamma _0}$, i.e., accelerating the emission in the chosen mode. In both cases, changing the atom bandwidth ${\Gamma _0} + \gamma$ automatically modifies its mode coupling [49–52]. We hereafter introduce an effective one-dimensional atom where the system bandwidth and mode coupling are largely independent.

We consider the system consisting of two atoms represented by TLSs strongly coupled through a dipole–dipole interaction and coupled to the same cavity mode, as shown in Fig. 1(a). The TLSs have ground states $|{g_i}\rangle$ and excited states $|{e_i}\rangle$ ($i = 1,2$), with transition energies ${E_{|{e_i}\rangle}} - {E_{|{g_i}\rangle}} = \hbar {\omega _i}$ and a dipole–dipole coupling strength ${\Omega _{12}}$ [34]. When both TLSs are degenerate, ${\Delta _{12}} = \frac{{{\omega _1} - {\omega _2}}}{2} = 0$, the atomic eigenstates are the well-known entangled states, i.e., the subradiant antisymmetric state $| - \rangle = \frac{{|e,g\rangle - |g,e\rangle}}{{\sqrt 2}}$ and the superradiant symmetric state $| + \rangle = \frac{{|e,g\rangle + |g,e\rangle}}{{\sqrt 2}}$. Both states are energy split by

 

Fig. 1. (a) Two two-level systems (TLSs) are weakly coupled to the same cavity mode. They are coupled to each other through dipole–dipole interaction with a rate ${\Omega _{12}}$. Their frequency mismatch ${\Delta _{12}}(t) = {\omega _1} - {\omega _2}$ allows to dynamically control the emission linewdith of the system. (b) Energy diagram of the eigenstates with at most one excitation. The first and second parts of the kets correspond, respectively, to the atomic state and number of photons in the cavity. Double arrows represent coupling rates between the coupled atoms’ eigenstates and the cavity mode, single dotted arrows the decay rates into all other modes. Energy splittings are indicated in the energy scale. The cavity frequency is set here to the mean atomic frequency ${\omega _0} = \frac{{{\omega _1} + {\omega _2}}}{2}$. (c) Modulus of the coefficients $\mu$ and $\nu$ corresponding to the subradiant and superradiant components of the eigenstates as a function of $\delta = \frac{{{\Delta _{12}}}}{{{\Omega _{12}}}}$. See text for details.

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As shown by Eq. (2), ${\Delta _{12}}$ acts as an effective coupling between states $| + \rangle$ and $| - \rangle$. The system eigenstates $| -\! {\rangle _{{\rm eff}}}$ and $| +\! {\rangle _{{\rm eff}}}$ are then a superposition of both symmetric and antisymmetric states for which simple analytical expressions can be obtained when $\gamma \ll {\Omega _{12}},g$ [37]:

The effective antisymmetric eigenstate $| -\! {\rangle _{{\rm eff}}}$ couples to the cavity mode only through its symmetric component $\mu| + \rangle$ at a rate given by ${g_{| -\! {\rangle _{{\rm eff}}}}} =\mu\sqrt 2{g}$. This corresponds to its symmetric component, $\mu$, times the coupling rate of $| + \rangle$ with the cavity mode, $\sqrt 2g$. This coupling can be tuned through $\delta$ from no coupling at all ($\mu\approx 0$), i.e., a completely dark state, for $\delta = 0$, to the same coupling as a single emitter when $\delta \gg 1$ [$\mu\approx \frac{1}{{\sqrt 2}}$ from Eq. (5)]. The effective antisymmetric state can thus be seen as a single dipole with an adjustable dipole moment, where the control knob is the detuning between the TLSs. Note that this transition is obtained for a detuning ${\Delta _{12}}$ on the order of ${\Omega _{12}}$ whereas for a single emitter in a cavity, the detuning needs to be on the order of $\kappa$. The emission rate of the antisymmetric eigenstate into the cavity is then given by

Remarkably, the dipole–dipole interaction results in a modification of the emission rate in the cavity mode (${\Gamma _{| -\! {\rangle _{{\rm eff}}}}}$) and in all other modes (${\gamma _{| -\! {\rangle _{{\rm eff}}}}}$), both proportional to the symmetric part of the state ${\mu ^2}$. As a result, the mode coupling of the effective antisymmetric state to the cavity reads as

To illustrate this effect, we consider the following parameters that will be used in the following sections as well. We consider $\kappa = 20 g$ and $g = 33 \gamma$, corresponding to a system of high Purcell factor ${F_p} = 6.6$ in the weak coupling or bad cavity regime. A strong dipole–dipole coupling $\frac{{{\Omega _{12}}}}{\gamma} = 50$ is considered, rendering ${\gamma _{12}} = 0.99 \gamma$. This is obtained with a small distance $kd = 0.25$ between the two emitters, compatible with experimental realizations (see Section 5 for a discussion on experimental parameters).

Figure 2 first illustrates how the emission rate into the cavity mode can be widely tuned from no emission at all, to almost the Purcell enhanced rate ${\Gamma _0}$, while maintaining a high mode coupling $\beta$. The inset shows the Purcell enhanced spontaneous emission rate of the subradiant state ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}}$, normalized to the rate of a Purcell enhanced single emitter in a cavity ${\Gamma _0} = \frac{{4{g^2}}}{\kappa}$. It is plotted as a function of the detuning ${\Delta _{12}}$ between the two TLSs. ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}}$ goes from zero for ${\Delta _{12}} = 0$ and increases with the detuning to reach $0.65 {\Gamma _0}$ for ${\Delta _{12}}/\kappa = 0.25$, evidencing a widely adjustable linewidth. In the same range of parameters, the main panel in Fig. 2 shows a mostly constant mode coupling $\beta$, with a maximum decrease of 2%, proving the persistence of the one-dimensional atom character over the whole detuning range. It is important to emphasize that the ability to modify the emission linewidth into the cavity mode comes from a modulation of the dipole moment of the effective antisymmetric state, as set by the parameter $\mu$ of Eq. (5), which is modified by the detuning ${\Delta _{12}}$. To summarize, for any system in the Purcell regime, the emission bandwidth through the cavity mode of the subradiant state can be tuned from zero, for a completely antisymmetric state, to almost the Purcell enhanced emission rate of a single emitter ${\Gamma _0}$ with no significant change in the $\beta$-factor.

 

Fig. 2. Mode coupling $\beta$ of the effective atom to the cavity mode as a function of the detuning between the two TLSs over the cavity linewidth. Inset: ratio of the spontaneous emission rate ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}}$ of the effective subradiant state $| -\! {\rangle _{{\rm eff}}}$ normalized to the spontaneous emission rate ${\Gamma _0}$ of a single TLS in the cavity. The cavity frequency is set to the frequency of the subradiant state at zero detuning: ${\omega _c} = {\omega _{| - \rangle}} = {\omega _0} - {\Omega _{12}}$, which corresponds to the frequency of a single emitter shifted by the dipole–dipole coupling strength. See text for details on the parameters used in this simulation.

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An alternative route to control the emitter bandwidth is to detune a single emitter from the cavity [33,62]. If the Purcell factor is large, this can lead to significant modification of the bandwidth. While our approach is more complex, as it involves two emitters, it offers a number of attractive features: the mode coupling $\beta$ remains constant, the required tuning amplitude is on the order of ${\Omega _{12}}$ which is smaller than the cavity width $\kappa$, and the maximum lifetime is limited only by the non-radiative decay processes.

As shown in the next sections, this system opens many possibilities, such as the full engineering of the temporal profile of single-photon wavepackets or the possibility to develop a versatile quantum memory. A straightforward application concerns, for instance, an important challenge for quantum networks, where high communication rates require short photon pulses, and quantum relays need long-term quantum memories, resulting in an unavoidable bandwidth mismatch.

3. TIME–FREQUENCY ENCODED SINGLE-PHOTON SOURCES

The system introduced here allows not only to control the spectral bandwidth of atomic emission, but also its phase and amplitude, and can eventually be used to obtain a full shaping of the single-photon wavepackets [63–65] at constant high-mode coupling. Next, we discuss how to obtain bright single-photon sources with the information encoded in the time–frequency domain [66].

A. Tunable Emission Dynamics

In the following, we consider that one can dynamically control the detuning ${\Delta _{12}}(t)$ between the two emitters. We model the spontaneous emission of the system restricted to the case of single excitation. In the collective basis, we consider the states $\{{|g,g,0\rangle ;| - ,0\rangle ;| + ,0\rangle ;|g,g,1\rangle} \}$, where the first and second parts of the ket correspond, respectively, to the two atomic states and number of photons in the cavity. In this basis, the evolution of the mean values of single operators are described by a set of closed equations [67]:

Figure 3 illustrates the potential of ${\Delta _{12}}(t)$ dynamical tuning to control the time of emission and the emission rate. As seen in Fig. 3(a), we consider that the system is in the subradiant state $| - \rangle$ at $t = 0$ and that ${\Delta _{12}} = 0$. The scheme to obtain this initial state will be detailed in Section 4. The population of this subradiant state is uncoupled from the cavity and decays only slowly into the leaky modes at a rate $\gamma - {\gamma _{12}}$, which is not visible on the time scale in Fig. 3. To turn on the spontaneous emission, ${\Delta _{12}}(t)$ is set to a finite value, as illustrated in Fig. 3(a). This induces an effective coupling of the subradiant state with the symmetric, superradiant, state while keeping the phase relations between the two emitters. The superradiant component $\mu| + \rangle$ whose amplitude increases with $\delta$, and hence with the detuning, introduces a coupling to the cavity mode, as seen in Eq. (4). The only requirement is to operate adiabatically, i.e., ${\dot \Delta _{12}} \ll \Omega _{12}^2$, so that the subradiant state $| - \rangle$ is converted only into the effective antisymmetric state $| -\! {\rangle _{{\rm eff}}}$. Otherwise, if the applied detuning is too fast, the excitation will also be transferred to the symmetric state $| +\! {\rangle _{{\rm eff}}}$, and will be emitted rapidly without the possibility to control the emission. Most importantly, such an adiabatic coherent process ensures the indistinguishability of the emitted light wavepackets.

 

Fig. 3. Controllable spontaneous emission. (a) Applied detuning $\delta = {\Delta _{12}}(t)/{\Omega _{12}}$ as a function of time. (b) Total population of the two TLSs. (c) Emitted power through the cavity (scaled). Solid blue lines correspond to a maximal detuning of $\delta = 0.3$ and dotted yellow lines to a maximal detuning of $\delta = 1.6$. The vertical line across the three panels indicates the passage from the totally dark state $| - \rangle$ to the partially radiant effective antisymmetric state $| -\! {\rangle _{{\rm eff}}}$ due to the applied detuning. The cavity is taken resonant to the final $| -\! {\rangle _{{\rm eff}}}$ state in both cases: ${\omega _c} = {\omega _0} - \sqrt {\Delta _{12}^2 + \Omega _{12}^2}$.

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The emission rate through the cavity mode depends on the symmetric portion of the $| -\! {\rangle _{{\rm eff}}}$ state and hence on the final value of the detuning to which the TLSs are brought. In Fig. 3, a non-zero detuning is applied around $t = 0.7/{\Gamma _0}$, with a rise time equal to $0.28/{\Gamma _0}$. Two examples are shown: the solid blue case with final detuning ${\Delta _{12}}/{\Omega _{12}} = 0.3$ decays slower than the dotted yellow case with final detuning ${\Delta _{12}}/{\Omega _{12}} = 1.6$. The more detuned the TLSs are at the end, the more radiant $| -\! {\rangle _{{\rm eff}}}$ becomes and the faster is the decay, as illustrated in Fig. 3(b). This in turn modifies the temporal profile, and thus the bandwidth, of the emitted photon wavepacket, as shown in Fig. 3(c).

The present approach requires a dynamical tuning of ${\Delta _{12}}$ that can be implemented for various atomic platforms considering the current state of the art, as discussed in the last section.

B. Generation of Cat-Like State and Compass State Single-Photon Wavepackets

We extend the control of the emission process to achieve complex pulse shaping of the single photon’s amplitude. One can indeed obtain complex time structures for the emitted single-photon wavepackets by turning on or off the coupling with the cavity mode through ${\Delta _{12}}(t)$ during the emission process. This will in turn modify the emission rate of the effective subradiant state $({{\Gamma _{| -\! {\rangle _{{\rm eff}}}}} + {\gamma _{| -\! {\rangle _{{\rm eff}}}}}})(t)$, as indicated in Eqs. (6) and (7). The radiated intensity in the cavity mode at an instant $t$ is then given by [68]

By optimizing the spontaneous emission rate temporal profile, one can obtain arbitrary amplitudes of the emitted single photon’s wavepacket. Analytical approximations can be obtained for simple desired output shapes, and numerical optimization can be performed for any general shape. The limitations are: a maximum modulation rate ${\dot \Delta _{12}} \lt \Omega _{12}^2$ given by adiabaticity, and a maximum emission rate ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}} \lt {\Gamma _0}$ that depends on the Purcell enhancement of the system.

As a first example, we show how to separate the single photon in different time bins. For this, we apply two ${\Delta _{12}}(t)$ pulses separated in time, as illustrated in Fig. 4(a). The mean value of the emitted power then shows two distinct peaks, as shown in Fig. 4(c), corresponding to a time bin encoding of the information. The time interval between the peaks $\Delta t$ can be chosen at will, and the duration of each peak $\tau$ is controlled by the maximum value of ${\Delta _{12}}$ that is applied. Two different examples are depicted with a waiting time $\Delta t = 5/{\Gamma _0}$ (in dashed red) and $\Delta t = 11/{\Gamma _0}$ (in solid blue). Note that to obtain the same emitted power for the two peaks, the maximum applied detuning is higher for the second peak. We also assume that one can control not only the emitters’ detuning ${\Delta _{12}}(t)$ but also the mean frequency of the two emitters ${\omega _0}(t)$, as shown in Fig. 4(b).

 

Fig. 4. Time bin encoding of a single photon. (a) Applied detuning sequence $\delta (t)$ and (b) overall frequency shift ${\omega _0}(t)$. (c) Corresponding emitted power by the cavity. Two pulse sequences are considered with a waiting time $\Delta t$ between each pulse of $5/{\Gamma _0}$ (dashed red) and $11/{\Gamma _0}$ (solid blue). (d) Population of the collective state $| -\! {\rangle _{{\rm eff}}}$. (e) Density power spectrum of the power emitted by the cavity. (f) Corresponding Wigner–Ville function of the emitted electric field: left, waiting time of $5/{\Gamma _0}$, and right, waiting time of $11/{\Gamma _0}$.

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The corresponding energy spectral density is plotted in Fig. 4(e). As expected by Fourier transform, the two peaks in the time domain result in a frequency comb. The overall width is proportional to the inverse of the duration of one peak, $1/\tau$, which is equal for the two different cases here. The interval between the fast frequency oscillations is proportional to the inverse of the waiting time $1/\Delta t$. For the case where $\Delta t = 11/{\Gamma _0}$, one obtains fast oscillations and narrow peaks.

A more detailed insight into the generated single-photon wavepacket can be obtained by calculating the associated Wigner–Ville function [69], also called the chronocyclic Wigner function [70]. Its expression is given for a detection between $[{0,T}]$ by

It corresponds to the Fourier transform of the electric field correlations as a function of time and is displayed in Fig. 4(f) for the two different waiting times $\Delta t$. One recognizes a photonic cat-like state. The oscillations visible at the mean time between the two emission peaks result from the interference between the probability to emit a photon at a time $t - \tau /2$ with the probability to emit a photon at $t + \tau /2$. They present time–frequency sub-Planck features, subtle features of quantum states that have recently been proposed for advanced quantum sensing applications [42].

This first example illustrates the versatility of the tunable one-dimensional atom for the generation of light in two time bins. It can naturally be extended to more complex states of light since the effective atomic system presents a tunable lifetime and the phase of the state is controlled at the atomic level. This is illustrated in Fig. 5 where a set of four ${\Delta _{12}}(t)$ Gaussian pulses, separated in time, is applied as plotted in Fig. 5(a). The pulses are $0.17/{\Gamma _0}$ wide and are applied at $t{\Gamma _0} = \{0.3, 1.1,1.5, 2.3\}$. The amplitude of each pulse is increased each time so as to compensate for the smaller remaining emitter population. In parallel, the mean frequency of the emitters, ${\omega _0}(t) = \frac{{{\omega _1} + {\omega _2}}}{2}$, is shifted to four different values as shown in Fig. 5(b). As seen in Fig. 5(c), a compass state, corresponding to a four-legs time–frequency superposition, is generated. It shows an even richer structure in the time–frequency domain, similar to the higher class of state recently introduced for sensing applications in optomechanics [71].

 

Fig. 5. Generation of a compass state. The emission of such a state is obtained by applying (a) a four-pulse detuning $\delta (t) $, and at the same time (b) an overall frequency shift of the mean atomic frequency ${\omega _0}(t)$. The cavity frequency is fixed at ${\omega _c} = {\omega _0}(0) - {\Omega _{12}}$. (c) Wigner–Ville function of the emitted electric field for a one-photon state.

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Such a structure is also ideal for continuous variable time encoding of the quantum information, a new field of quantum computing that exploits the continuous degrees of freedom of time encoding to enlarge the Hilbert space of computation [41]. Until now, the generation of a continuous variable at the single-photon level was limited to ultra-low source efficiency, whether based on parametric down-conversion [72–75] or on the weak coherent excitation of an atomic system [76,77]. The approach introduced here, based on the dynamical control of a tunable one-dimensional atom, allows generating arbitrary time-encoded photons with near-unity efficiency.

In principle, the arbitrary control of the single-photon wavepacket is possible with such an approach, with high-frequency modulation limited by the adiabatic condition. This requires the possibility to independently adjust two knobs, the detuning ${\Delta _{12}}(t)$ and the central frequency ${\omega _0}(t)$, to reach the desired combination of emission rate and frequency control. As an example, full control of the wavepacket amplitude can be obtained, together with a constant emission frequency, by adjusting ${\Delta _{12}}(t)$ to modify the emission rate while adequately adapting ${\omega _0}(t)$ to suppress the frequency chirp. In addition, adjusting ${\omega _0}$ while maintaining ${\Delta _{12}}$ constant would allow modifying the frequency of the single-photon wavepacket, at a constant emission rate. In practice, the implementation of such detuning and average frequency control is system dependent (see Section 5), and would ideally require the ability to shift either or both emitter frequencies.

4. BROADBAND AND VERSATILE QUANTUM MEMORY

Another application of the present architecture is single-photon storage and bandwidth conversion. Many systems have been explored to develop efficient quantum memories for quantum networks, most of them based on atomic vapors or ion-doped crystals [78]. They present very narrow bandwidths that prevent efficient storage of short photon pulses, the latter being naturally desirable for high-rate quantum networks. To date, the development of broadband single-photon memories has been scarcely addressed. Such memory has recently been demonstrated using Raman assisted transition in warm atomic vapors: the proposed protocol offers interesting possibilities for releasing photons of improved quantum purity [79]. However, so far, the storage times remain in the few nanoseconds timescale [80]. Here, we show that by dynamically tuning the emission lifetime of the subradiant state, one can achieve the storage of single-photon pulses with a pulse duration three orders of magnitude shorter than the storage time. Naturally, building on the results of the previous section, the quantum information could also be released with a highly tunable temporal profile, independent of the incoming single-photon profile.

The storage mechanism relies on sending a single-photon wavepacket on the tunable one-dimensional atom. An intrinsic issue when using subradiant states for storage is the difficulty to populate the state. This problem is overcome here using a three-step process as shown in Fig. 6(a). First the system parameters are tuned to a non-zero value for the detuning ${\Delta _{12}}$, allowing to absorb the incoming excitation, and excite the effective subradiant state $| -\! {\rangle _{{\rm eff}}}$. More specifically, ${\Delta _{12}}$ is chosen to match ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}}$ with the bandwidth of the light pulse that one wants to store. The photon is then absorbed by tuning back the two emitters into resonance to adiabatically go from the effective subradiant state $| -\! {\rangle _{{\rm eff}}}$ to the dark state $| - \rangle$. This photon absorption process is the time-reversed of the emission process starting with the system in the ground state $|g,g,0\rangle$ at $t = 0$ and dynamically tuning ${\Delta _{12}}(t)$ so as to progressively bring the system into the fully antisymmetric dark state $| - \rangle$ where it presents the minimal coupling to light. Finally, release of the single photon is obtained by applying a detuning ${\Delta _{12}}$ between the two TLSs. Controlling the temporal and spectral shapes of the emission is then possible as explained in the previous section.

 

Fig. 6. Broadband memory operation. (a) Atomic population normalized by the average number of photons in the pulse $\langle n\rangle$ (solid blue) and applied detuning $\delta (t) $ (dashed green). The storage process takes place in three steps: first, the two atoms are detuned so as to absorb the incoming photon, the excitation is then stored in the subradiant state by bringing the two atoms into resonance, and finally the photon is released by detuning again the two atoms at $t = 35/{\Gamma _0}$. The pump is resonant with the $| -\! {\rangle _{{\rm eff}}}$ state. During storage, the excitation leakage takes place at a rate ${\gamma _ -} = \gamma - {\gamma _{12}}$ (dotted brown). (b) Close-up of figure (a) at short time delays. The Gaussian profile of the incoming wavepacket is depicted by the black dotted line.

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While the incoming pulse is a single-photon Fock state, it is easier to calculate this storage process using a coherent state $|\alpha \rangle = \sum\nolimits_k \frac{{{\alpha ^k}}}{{k!}}|k\rangle$ with a very low mean photon number $\langle n\rangle = |\alpha {|^2} = 0.01 \ll 1$. Only the one-photon component excites the atom so that the system response reflects the response to a single-photon Fock state when normalizing the atomic population by the probability of having one photon [81,82]. The system excitation is done via the cavity mode through the Hamiltonian of Eq. (2): ${H_p}(t) = - i{{\cal E}_p}(t)({a^\dagger} - a)$, where ${{\cal E}_p}$ is the field amplitude coupled to the cavity mode.

Figure 6 shows the evolution of the atomic population corresponding to a single-photon Fock state sent on the system over the whole sequence. The single-photon wavepacket to store presents a Gaussian temporal profile with a FWHM of ${\tau _{{\rm pulse}}} = 0.55/{\Gamma _0}$, i.e., a photon lifetime of $0.55/({F_P}\gamma) \approx {\gamma ^{- 1}}/10$. The system parameters are set so that the incoming photon is resonant with the $| -\! {\rangle _{{\rm eff}}}$ state:

The cavity frequency is fixed at ${\omega _c} = {\omega _0} - {\Omega _{12}}$. To achieve the highest absorption probability, the $| -\! {\rangle _{{\rm eff}}}$ state linewidth must be equal to 2/3 of the incident single-photon pulse width [28,83], which corresponds here to a temporal linewidth of $0.37/{\Gamma _0}$. The emitters are then adiabatically brought into resonance: ${\Delta _{12}}(t) \to 0$ during the storage process to bring the $| -\! {\rangle _{{\rm eff}}}$ state into the dark state $| - \rangle$. The photon will be stored for a characteristic time of $1/{\gamma _ -} = 100/\gamma$ and can be recovered by setting ${\Delta _{12}}(t)$ to a non-zero value.

Considering the present parameters, we obtain a 68% storage efficiency, given by the maximum flip inversion of the atomic system with single-photon pulse excitation. To stay within 5% of such maximum efficiency, we find that the error in the start time of the detuning gate ${\Delta _{12}}(t)$ has to be smaller than a fifth of the single-photon pulse. If the TLSs are brought into resonance too early, the excitation stops before maximum population is achieved, and if the TLSs are brought into resonance too late, spontaneous emission takes place before reaching the dark state. This value already represents a very high storage probability, considering all current memory architectures where typical values are in the tenths of percent range [84]. The maximum storage efficiency is in part limited by the chirp introduced by the detuning of the subradiant mode during the modification of its bandwidth, as shown in Eq. (13). As mentioned above, this chirp could in principle be compensated for, and full absorption of arbitrary photon shapes can be envisioned. The absorption process being the reverse of the emission process discussed in Section 3, the same constraints would apply for complete control of the absorption process: both ${\Delta _{12}}$ and ${\omega _0}$ need to be independently controlled, and the frequency range is limited due to the adiabaticity condition for the high frequencies.

The system introduced here can thus be used to store short single-photon pulses of lifetimes around ${\gamma ^{- 1}}/10$ for around $100{\gamma ^{- 1}}$, as shown in Fig. 6(a), surpassing by almost two orders of magnitude the current state of the art [80,85].

We note that tunable single-photon memories have recently attracted attention for their ability to re-emit the photon with a frequency and bandwidth different from the initial ones [84]. Indeed, frequency and bandwidth conversion is a valuable tool for building quantum networks. A bandwidth modulation of three orders of magnitude of the incoming bandwidth was recently demonstrated using a Raman transition of a rubidium atom in a cavity with a conversion efficiency of 22% [27]. With the scheme proposed here, a bandwidth modulation on the same order can be obtained combined with high efficiency.

5. CHALLENGES AND IMPLEMENTATION

In this section, we discuss a number of challenges for the experimental implementation of the ideas introduced here, and we then review current candidate platforms that can be used.

A. Dephasing and Non-Radiative Decay

Pure dephasing and non-radiative decay were not included in our model so that the linewidths were limited only by radiative decay.

Regarding pure dephasing, we need to distinguish between correlated and uncorrelated dephasing between the two emitters. Indeed, due to the close proximity of the emitters coupled through dipole–dipole interaction, the dephasing processes can happen simultaneously for both TLSs without affecting the phase relation between them [86]: such a correlated dephasing would not affect the phenomena described here. On the contrary, the presence of uncorrelated pure dephasing would affect the phase relation between the two emitters and hence destroy any collective state once an individual dephasing mechanism takes place. This means that all emission and storage processes are limited to a time $t\lesssim \frac{1}{{{\gamma ^*}}}$, where ${\gamma ^*}$ is the pure dephasing rate of one emitter. This limits the application to systems with near-Fourier-limited emission spectra.

The performance of the emission process is also limited by the presence of non-radiative decay. The intrinsic non-radiative decay of the emitters ${\gamma _{{\rm NR}}}$ remains unchanged by the optical dipole–dipole coupling between them, limiting both the lifetime and linewidth of the subradiant state. Note that this also impacts the beta factor of Eq. (8), with ${\gamma _{{\rm NR}}}$ coming as an additional term in the denominator.

Additionally, if the emitters are in the near field of nanophotonic structures, the dipole–dipole interaction is not only given by its form in a homogeneous medium dominated by the $1/{d^3}$ terms but is also mediated by the structure. Extrinsic non-radiative decay and decay into non-collected modes will occur. This modifies the exact values of $\gamma$ and ${\gamma _{12}}$, and thus of ${\gamma _ +}$ and ${\gamma _ -}$, the respective emission rates of the symmetric and antisymmetric states into the non-collected modes. However, when considering microcavities with a characteristic length $\lambda /n$, large compared to the dipole–dipole distance on the order of $\lambda /20n$, the physical processes at the heart of our proposal will remain.

B. Distance Between Emitters

Generation of entangled states due to dipole–dipole coupling has been observed for two molecules separated by 12 nm in the visible [87]. It has also been observed for Rydberg atoms separated by 10 µm at a frequency of 9.6 GHz [36]. Near-field coupling between the emitters depends on the sub-wavelength separation between the emitters. The dipole–dipole coupling strength considered throughout this paper corresponds to two emitters separated by $d = \lambda /25n$, where $n$ is the refractive index of the material. Increasing the distance will decrease both the coherent and dissipative coupling constants ${\Omega _{12}}$ and ${\gamma _{12}}$. The first scales as ${\Omega _{12}} \propto {(\lambda /d)^3}$, and since the tuning of the subradiant emission rate is governed (through $\mu$) by the normalized detuning $\delta = {\Delta _{12}}/{\Omega _{12}}$ [see Eq. (5)], the smaller the coupling strength ${\Omega _{12}}$, the more sensitive will the system be to interatomic detuning, i.e., the more will the decay rate of the subradiant state change for a given shift in detuning. Thus, as long as the detuning between the two emitters ${\Delta _{12}}$ can be finely controlled compared to ${\Omega _{12}}$, the system’s linewidth can be finely tuned. However, increasing the distance between the two emitters will decrease the storage time. Indeed, the storage time $1/{\gamma _{| - \rangle}}$ depends directly on the dissipative coupling rate ${\gamma _{12}}$ as $1/{\gamma _{| - \rangle}} = 1/(\gamma - {\gamma _{12}}) \simeq 1/(5\gamma {(2\pi nd/\lambda)^2})$.

For an emission wavelength at 800 nm in air, $\lambda /25$ corresponds to $d = 32\;{\rm nm} $. This can be achieved, for example, by positioning colored centers in h-BN [88]. Ion implantation has also allowed positioning colored centers in diamond with a precision of 30 nm [89]. Likewise, terrylene molecules can be placed at a distance of 12 nm using nanoimprint techniques, with an expected yield of up to 34% [90]. Epitaxially grown quantum dots (QDs) self-assembled on top of each other with atomically controlled distances during the growth process are also promising candidates. Care must be taken to avoid tunneling coupling of the charge carriers that can take place for distances up to 15 nm [91]. Additionally, for these systems, the point-like dipole approximation can no longer be valid, as discussed in [92], but the modification to the emitters’ linewidths is negligible even for $d = \lambda /25n$. Dipole–dipole coupling between laterally positioned QDs has also been demonstrated [93]. Finally, another system is Rydberg atoms where strong dipole–dipole coupling has been reported recently in the microwave regime, which allows to reach distances $d/\lambda \sim 1/3000$ [36].

C. External Control

External control of the emitters’ frequencies is important for two reasons. First, we have shown that the subradiant state linewidth ${\Gamma _{| -\! {\rangle _{{\rm eff}}}}} + {\gamma _{| -\! {\rangle _{{\rm eff}}}}} = 2{\mu ^2}({\Gamma _0} + \gamma)$ depends directly on the normalized detuning $\delta = {\Delta _{12}}/{\Omega _{12}}$ through its symmetric component $\mu$. Active control of the emission and absorption rate, and hence on the amplitude of the emitted or absorbed photon wavepacket, is made possible with a single external control of ${\Delta _{12}}$. This in turn modifies the emission and absorption frequency due to a shift of the subradiant state frequency ${\omega _{| -\! {\rangle _{{\rm eff}}}}} = {\omega _0} - \sqrt {\Delta _{12}^2 + \Omega _{12}^2}$, on the order of ${\Omega _{12}}$. Full phase and amplitude shaping of the emission and absorption process can thus be obtained by compensating for the frequency chirp, with a second external control acting on the average atomic frequency ${\omega _0}$. The high-frequency modulation is limited only by the adiabatic condition ${\dot \Delta _{12}} \lt \Omega _{12}^2$. Fast control of the emitters can be implemented with electrical or optical Stark shifts, which have been demonstrated in a number of experiments with SiV [32], QDs [94,95], with color centers in h-BN [96] and in diamond [89,97], with molecules [87,98], and with atoms [36]. Electrical control in the picosecond timescale has been demonstrated for QDs [99] and could be extended for other solid state systems. The optical Stark shift is particularly suited here for simultaneous control of the detuning and the mean frequency of the emitters. Indeed, it is possible to use both a blue-detuned and a red-detuned optical beam, optimizing the detuning and intensity of each beam to reach the proper combination of frequency shifts for both emitters, while avoiding photon absorption. The coherent control by light shift proposed in this paper has been demonstrated with Rydberg atoms [36] with a microwave transition at 9.6 GHz. It was shown that the dipole–dipole interaction between the atoms separated by 10 µm induced a level splitting ${\Omega _{12}}$ between symmetric and antisymmetric entangled states on the order of 10 MHz. By focusing a laser with a wavelength of 1005 nm and a beam waist of 3.4 µm on a single atom, it was possible to generate a selective Rydberg level shift allowing to perform coherent control of the state. In that experiment, the addressing beam was modulated with an acousto-optic modulator with a rise time of 10 ns, much shorter than other time scales of the system. In the next section, we present values of maximum tuning range to linewidth ratios $\Delta _{12}^{{\rm max}}/\gamma$ for different single photon emitters.

D. Implementations

Rapid progress has been obtained in the optimization and control of single-photon emitters in recent years [100]. The different requirements to reach tunable subradiant states have been independently addressed in different platforms. We present in Table 1 the list of emitters that are promising candidates to implement our proposal. We have focused on the most challenging parameters: Fourier-limited linewidth to preserve collective effects, the sub-wavelength control of their position for near-field coupling, and the frequency tunability to compensate for inhomogeneous broadening and for active linewidth control. These are mainly solid state emitters, which are more easily integrated into photonic cavities [6], and can reach high brightness with good Purcell enhancements [17]. Other platforms more mature in the direct inter-emitter coupling such as Rydberg atoms [36] can also be considered. In these systems, atoms are trapped with optical tweezers, and since they operate in the microwave regime, deep subwavelength distances $d/\lambda \lt 3000$ and strong dipole–dipole couplings have been demonstrated [36]. The remaining challenge is to insert the system into a cavity and perform gigahertz photon detection.

Table 1. Single-Photon Emitter Candidates for Realizing Tunable Subradiant States^a

View Table

6. CONCLUSION

We have proposed a scheme to obtain a one-dimensional atom of tunable bandwidth by taking advantage of the interplay between collective emission of two emitters and weak coupling to a cavity. This is achieved by a dynamical control of the subradiant state that, when the emitters are detuned, acts as an effective dipole that can be externally modified in real time. This system offers exciting possibilities for single-photon generation and storage, with a versatility that could be of use for many applications in quantum technologies. Efficient single-photon sources with continuous variable encoding in the time–frequency domain could be used for both quantum sensing [104] and quantum computing [41]. The proposed scheme can also solve the bandwidth mismatch in quantum networks where broadband single photons can be stored on a time scale a thousand times longer than their pulse duration with an efficiency up to 68%. Both storage and bandwidth control would finally allow efficient bandwidth conversion of single photons. Overall, the present system could act as a junction useful in any kind of hybrid quantum network, with the possibility to emit, store, and modify single photons of a wide variety of quantum systems, from solid state emitters such as semiconductor QDs to Rydberg atoms.