Single-photon character of light means a special antibunching quantum nature indicated by the normalized second-order correlation function g(2)(0)=0, which is of crucial importance in quantum communications [1,2] and optical quantum computing [3,4]. It is a requirement for a single-photon source and may be required for single modes of a multiphoton source, e.g., an entangled two-photon source. Fortunately, the simple resonance fluorescence from a single emitter, including atoms [5,6] and various atom-like systems [7–9], satisfies the perfect single-photon character. Naturally, fluorescent photons have always been a fundamental element in the fields of quantum optics, photonics, and quantum information science.

The conversion of the quantum state between flying photonic and stationary qubits is the basic physical process for quantum information storage and processing and is therefore crucial for long-distance quantum communications [10] and quantum networks [11,12]. To achieve efficient interfaces between photons and atoms, the photons are required to have a bandwidth smaller than the linewidth of atomic transitions [13–16]. In recent years, research on subnatural-linewidth photons based on atoms and quantum dots has received tremendous attention and gained significant progress [13,15–20].

However, it was recently demonstrated in Refs. [21–23] that for the resonance fluorescence from a two-level system, a single-photon character and subnatural linewidth cannot be satisfied simultaneously, which means that fluorescent single photons cannot have subnatural linewidth. These studies reveal a previously neglected point that the individual spectral analyzer or interferometer can only measure the properties of main fluorescent components but cannot faithfully reflect the nature of fluorescent single photons. Alternatively, the measurement scheme of adding a filter in front of the Hanbury Brown–Twiss (HBT) setup can be used to estimate the linewidth of single photons. According to Refs. [21–23], the perfect single-photon character of fluorescence can account for the destructive interference between the coherent and incoherent components. It is demonstrated that the spectral filtering with a bandwidth approaching the natural linewidth of emitter can significantly spoil the single-photon character of fluorescence due to the partial loss in the incoherent component. When most fluorescent components are focused on an ultranarrow coherent peak [16,17], the conclusions above remain valid [21–23], which can be demonstrated by the destruction of a single-photon response on the detector. Consequently, the linewidth of fluorescent single photons appears to be limited by the natural linewidth of an emitter. A natural question is whether it is possible to generate subnatural-linewidth single photons based on resonance fluorescence.

Here, we propose a universally applicable approach to generate single photons with subnatural linewidth using an atom or atom-like system. Specifically, we introduce a quantum emitter with a long-period transition loop, as shown in Fig. 1(a). We find that all the spectral components of fluorescence radiated by the emitter can be focused on a bandwidth that is smaller than the natural linewidth. Therefore, subnatural-linewidth single photons are achieved. Besides, the linewidth of single photons can be easily manipulated over a broad range by varying the intensities of external fields. Thereby, the single-photon linewidth several orders of magnitude smaller than the natural linewidth can also be readily obtained. Noteworthily, we also reveal the general condition to generate single photons with subnatural linewidth utilizing an atom or atom-like system that is the successive emission process of the target single photons totally dominated by the transition loop with a metastable state. This condition can be satisfied in various physical systems with Λ-shape and similar energy structures, which exhibits the high experimental feasibility of our proposed approach.

As shown in Fig. 1(a), a closed Λ-shape system is considered as the quantum emitter, where the transition |g⟩↔|e⟩ is driven by a laser field F1, and the transition |g⟩↔|a⟩ is driven by another coherent field F2. A transition loop composed of the transitions |g⟩→|e⟩→emission|a⟩ and |a⟩→|g⟩ is constructed in the emitter by these two external fields. The fluorescence from the transition |e⟩→|a⟩ is collected by the detector, which comprises a standard HBT setup and a spectral filter inserted between the emitter and the HBT setup. The detector is used to test whether subnatural-linewidth single photons are generated [21,23]. In the rotating frame at the driving frequencies, the Hamiltonian of the system consisting of the emitter and detector shown in Fig. 1(a) is given by H=Δeσee+Δaσaa+Δss†s+(Ωσeg+Ωrσga+gσeas+H.c).(1)Here the detector is modeled by a quantized harmonic oscillator with bosonic annihilation operator s. Δe and Δa (Δs) represent the frequency detunings between the transitions |g⟩↔|e⟩ and |g⟩↔|a⟩ (detector) and the external fields. Without loss of generality, we set Δe=Δa=Δs=0 for simplicity. Ω and Ωr, respectively, represent the Rabi frequencies of the external fields F1 and F2. g is the coupling coefficient between the detector and the detected transition |e⟩→|a⟩. Therefore, the evolution of the combined system composed of the emitter and detector is governed by the master equation ρ˙=−i[H,ρ]+γ1D[σge]ρ+γ2D[σae]ρ+κD[s]ρ,(2)where D[o]ρ≡oρo†−12ρo†o−12o†oρ is the Lindblad superoperator. γ1 and γ2 represent the natural linewidths of the decays |e⟩→|g⟩ and |e⟩→|a⟩, respectively. The detection bandwidth κ and its inverse reflect the frequency and time resolutions of detector, respectively, so that the constraints imposed by the uncertainty principle on the detection process are included in a self-consistent way.

In the limit of the vanishing coupling between the emitter and detector (i.e., g→0), the detector can be regarded as a passive object [22,24]. Therefore, it is beneficial to study the nature of the emitter before considering the detection response for this quantum source. In the weak excitation regime of the emitter (i.e., Ω,Ωr≪γ1,γ2), the evolution rate of the excited state is much greater than that of the ground states. Therefore, the excited state can be adiabatically eliminated, which induces effective decays between the ground states, i.e., Leffρg=γ1*D[σgg]ρg+γ2*D[σag]ρg.(3)The first term denotes the dephasing of the state |g⟩, and the second term denotes the effective decay |g⟩→|a⟩ (see Appendix A for details). The effective linewidths satisfy γi*=4γi|Ω|2(γ1+γ2)2,(4)with i=1,2, which can be reduced to γ*=|Ω|2/γ with γ1=γ2=γ.

Therefore, the emitter in Fig. 1(a) can be regarded as an effective two-level system driven by a coherent field, as shown in Fig. 1(b). The effective decay |g⟩→|a⟩ corresponds to a two-photon process in the original level system, where the emitter absorbs the first photon by the transition |g⟩→|e⟩ driven by the laser, and subsequently emits the second photon by the spontaneous decay |e⟩→|a⟩. Remarkably, different from the normal two-level system, the linewidth of the effective two-level system is determined by the ratio of the laser intensity to the emitter’s natural linewidth and thus can be set artificially.

The intensity of fluorescence from the transition |e⟩→|a⟩ in the steady-state limit is determined by the steady-state population of the excited state ⟨σee⟩=2Ω2Ωr2/(Ω4+2Ωr2(2γ2+Ω2+2Ωr2)) (see Appendix B). The corresponding fluorescence spectrum is described by Sea(ωs)=γ2Re∫0∞dτlimt→∞⟨σea(t)σae(t+τ)⟩eiωsτ, where ωs=ω−ωL denotes the fluorescence frequency in the rotating frame at the driving frequency ωL. One can see from Figs. 2(a)–2(c) that different spectral structures are exhibited for different Rabi frequencies Ωr, which can be understood through the effective two-level system shown in Fig. 1(b). As mentioned above, the effective decay |g⟩→|a⟩ corresponds to the two-photon process composed of the transition |g⟩→|e⟩ driven by laser and the spontaneous decay |e⟩→|a⟩. It can be seen that the parameters adopted in Figs. 2(a) and 2(c) correspond to the effective weak excitation regime, i.e., Ωr≪γ*. Therefore, the fluorescence spectrum of the effective decay |g⟩→|a⟩ exhibits a single-peak structure, whose spectral width is determined by the effective linewidth γ*. Because the laser frequency is constant, the transition |e⟩→|a⟩ exhibits the same spectral structure as that in the effective decay |g⟩→|a⟩ (see Appendix C for details).

Similarly, the parameters adopted in Fig. 2(b) correspond to the effective strong excitation regime, i.e., Ωr≫γ*. Therefore, the fluorescence spectrum of the effective decay |g⟩→|a⟩ exhibits a Mollow-like triplet [25,26], where the spectral widths of the three peaks depend on γ* and the two side peaks are located at ωL±2Ωr (the proof is in Appendix C). Accordingly, the transition |e⟩→|a⟩ in the original three-level system exhibits the same Mollow-like triplet (see Appendix C for details). Note that all of the fluorescent components are almost focused on a bandwidth determined by the effective linewidth γ* and the sidebands in ωL±2Ωr. This bandwidth is smaller than the natural linewidth of the emitter and can even be narrowed arbitrarily by reducing the intensities of the two external fields, as shown in Fig. 2(c).

The statistical property of the fluorescence emitted by the transition |e⟩→|a⟩ can be reflected by the normalized second-order correlation gea(2)(τ)=limt→∞⟨σea(t)σea(t+τ)·σae(t+τ)σae(t)⟩/⟨σee(t)⟩2, as shown in Fig. 2(d). It reveals that the correlation gea(2)(τ) remains a value very close to zero even when the delay τ is significantly larger than the lifetime of the emitter. In contrast, the normalized second-order correlation function of the two-level system increases rapidly as the delay increases. This feature of the correlation function can be understood according to the transition loop indicated in Fig. 1(a). The emitter is driven from the ground state |g⟩ to the excited state |e⟩, and then enters into the ground state |a⟩ after emitting the first photon from the spontaneous decay |e⟩→|a⟩. The transitions above can be equivalent to the decay in the effective two-level system, and the corresponding decay rate is determined by γ*. After a long time determined by Ωr, the emitter is driven from state |a⟩ to state |g⟩. Subsequently, the emitter can be excited again by the laser and then emit the second target photon. Consequently, it is understandable that the correlation function gea(2)(τ) remains zero for a long delay whose duration is determined by the effective decay rate γ* and the Rabi frequency Ωr (the proof is in Appendix D).

The single-photon character is the global property of fluorescence including all spectral components. According to Refs. [21,23], a single-photon character can be maintained when detected only if the bandwidth of the detector (or filter) is much larger than the linewidth of the fluorescent single photons, thus including the dominant fluorescent components. Therefore, to verify the realization of the subnatural linewidth single photons, we consider the response of the fluorescence from transition |e⟩→|a⟩ on the detector with a certain bandwidth, which can be described by the zero-delay two-photon correlation [24] gs(2)(0)=limt→∞⟨s†(t)s†(t)s(t)s(t)⟩⟨s†(t)s(t)⟩2.(5)

In Fig. 3, we show the correlation gs(2)(0) as a function of the detection bandwidth and the intensities of the external fields. It reveals that when the detection bandwidth approaches the natural linewidth of the emitter, an excellent single-photon response for the fluorescence can appear on the detector, i.e., gs(2)(0)→0. In contrast, for the fluorescence from a normal two-level system in the weak excitation regime where the main fluorescent components are focused on an ultranarrow coherent peak, the two-photon correlation on the detector deviates significantly from zero at the same bandwidth, as shown in Fig. 3(b), implying the destruction of the single-photon character of fluorescence [21–23]. According to the earlier discussion on the spectral and statistical properties of the fluorescence from transition |e⟩→|a⟩, the physical origin of the excellent single-photon response on the detector can be understood. In the weak excitation regime, almost all the spectral components of fluorescence are concentrated on a bandwidth that is significantly smaller than the natural linewidth of the emitter. Therefore, when the detection bandwidth approaches the natural linewidth of the emitter, all spectral components of the fluorescence are proportionally responded to by the detector. Accordingly, the perfect single-photon character of the fluorescence is well preserved. In turn, the excellent single-photon response for the quantum source on the detector also confirms the previous conclusion that the dominant components of fluorescence are concentrated on a bandwidth significantly smaller than the natural linewidth of the emitter.

Besides, one can see in Fig. 3(a) that even if the detector bandwidth is much smaller than the natural linewidth of the emitter (i.e., κ≪γ), the excellent single-photon response on the detector can still be maintained by manipulating the external fields. Therefore, it can be concluded that the subnatural-linewidth single photons are achieved. Moreover, the linewidth of the single photons can be manipulated easily by adjusting the intensities of external coherent fields, which dramatically facilitates the experimental implementation of this single-photon scheme, and has potential applications in extensive quantum technology fields.

The detection response can also be understood based on the effective level system. For a normal two-level system, when the detection bandwidth is smaller than or approaches the natural linewidth, the destructive interference between the coherent and incoherent components of the fluorescence is destroyed because a fraction of the incoherent component is removed by filtering [21–23]. Consequently, the single-photon character is spoiled. Similarly, when the detection bandwidth κ is smaller than or approaches the effective linewidth γ* of the effective level system shown in Fig. 1(b), the single-photon character of the fluorescence is spoiled, as shown in Fig. 3(b). According to the earlier discussion, we can infer that only when κ≫γ*+4Ωr can the single-photon character be maintained.

In addition, according to the effective level system shown in Fig. 1(b), we see that when Ωr∼γ* or Ωr≫γ*, the re-excitation rate of the state |g⟩ approaches or exceeds the decay rate γ* at which the photon is emitted by the effective decay |g⟩→|a⟩. Therefore, the emitted photons, which are actually the photons from the decay |e⟩→|a⟩, exhibit a clustered temporal distribution [i.e., gea(2)(τ)>1], as shown by Fig. 6(b) in Appendix D. Since the second-order correlation gea(2)(τ) of the emitter’s transition is linked to the two-photon correlation gs(2)(0) on the detector, this temporal effect of photons is detected for a proper detection bandwidth, resulting in a bunching effect in the detector [i.e., gs(2)(0)>1], as shown by the red solid line in Fig. 3(b) (see Appendix D for details).

According to Fig. 1(a), we can see that a long-period transition loop composed of the transitions |g⟩→|e⟩→emission|a⟩ and |a⟩→|g⟩ exists in the emitter and dominates the successive emissions from the transition |e⟩→|a⟩. Specifically, the emitter in the excited state |e⟩ emits the first photon from the transition |e⟩→|a⟩ and enters into the state |a⟩. Next, the emitter is driven to the state |g⟩ by the coherent field F2, and then is driven to the excited state |e⟩ by the laser field F1. Subsequently, the second photon from the transition |e⟩→|a⟩ is emitted. Noteworthily, this transition loop totally dominates the successive emissions from the transition |e⟩→|a⟩ because the probabilities of the other transition loops are negligible. In addition, due to Ωr≪γ, the state |a⟩ is the metastable state or the so-called shelving state [27–30], which can remain for a time much longer than the lifetime of the excited state. Therefore, the time interval between two successive photons from the transition |e⟩→|a⟩ is sufficiently long compared to the lifetime of the emitter.

According to the Heisenberg uncertainty principle, when the detection bandwidth κ is smaller than the natural linewidth of the emitter, the indeterminacy in the time resolution of detection 1/κ is larger than the natural lifetime of the emitter. Within a large delay determined by indeterminacy 1/κ, the normalized probability of reemission from the transition |e⟩→|a⟩ can remain very close to zero, as shown in Fig. 2(c), so an excellent single-photon response arises on the detector with a narrow bandwidth.

It is the direct indication of the subnatural-linewidth single photons to induce an excellent single-photon response on the detector with a bandwidth approaching or smaller than the natural linewidth of emitter. In terms of time, the detection bandwidth κ corresponds to the indeterminacy 1/κ in time, and then the decrease of single-photon linewidth means the increase of the average time interval between single photons. In fact, in the spectrum of fluorescence (i.e., fluorescent single photons), the contributions of multiple transition loops may be included. Among them, the transition loops without a metastable state would contribute to broad spectral components, leading to the broad linewidth of single photons. Therefore, we can conclude that the general condition for generating subnatural-linewidth single photons is that the successive emission process of the target photons is totally dominated by the transition loop with a metastable state. The validity of this condition is comprehensible by means of the detection response and can be verified by the various emitter systems shown in Appendices F and G.

The linewidth of a single-photon wavepacket corresponds to its frequency distribution range Δω. According to the uncertainty relation ΔωΔt≥1, the time distribution range of the single-photon wavepacket satisfies Δt≥1/Δω. This means that if all the frequency components of a single photon are concentrated within a width Δω, at most one photon can exist within the time range 1/Δω. Therefore, for a field satisfying single-photon statistics, the number of photons per unit time must satisfy Is≡1Δt≤Δω.(6)According to Eq. (6), the single-photon linewidth determines the upper limit of the single-photon emission rate. Therefore, when the single-photon linewidth is small, the single-photon emission rate of the emitter would inevitably be small, regardless of the type of emitter.

To analyze the emission rate of the subnatural-linewidth single photons, we first consider the normal two-level system for comparison. A two-level system is the most general fluorescence emitter, whose single-photon emission rate can be expressed as ITLS=Γ4ΩT2Γ2+8ΩT2,(7)where Γ and ΩT denote the natural linewidth of the two-level system and the Rabi frequency of the laser field, respectively. In the weak excitation regime (i.e., ΩT≪Γ), the single-photon emission rate is ITLS=4ΩT2Γ≪Γ.(8)With an increase in ΩT, the single-photon emission rate increases and gradually approaches Γ. In the strong excitation regime (i.e., ΩT≫Γ), the single-photon emission rate reaches the maximum value ITLS=Γ2,(9)which is determined by the natural linewidth Γ.

Moreover, we know that when ΩT∼Γ, the sidebands in the fluorescence are not resolved, and thus, the bandwidth of all fluorescent components (i.e., the linewidth of fluorescent single photons) is approximately equal to the natural linewidth Γ. As shown by the blue dashed line in Fig. 4, the single-photon emission rate also approaches Γ when ΩT∼Γ. Therefore, we can conclude that for the two-level emitter, the single-photon emission rate can almost reach the upper limit determined by the single-photon linewidth according to Eq. (6).

For the emitter shown in Fig. 1(a), the emission rate IN=γ2⟨σee⟩ of the subnatural-linewidth single photons can be obtained as (see Appendix B) IN=γ2Ω2Ωr2Ω4+2Ωr2(2γ2+Ω2+2Ωr2).(10)For the weak excitation regime we focus on, i.e., Ω,Ωr≪γ, one can understand the emission rate of the subnatural-linewidth single photons by means of the effective two-level system similar to that in the previous sections. Specifically, Eq. (10) can be reduced to a function of the effective linewidth γ* and the Rabi frequency Ωr to get IN=γ*2Ωr2γ*2+4Ωr2.(11)Further, in the effective weak excitation regime (i.e., Ωr≪r*), the single-photon emission rate is IN=2Ωr2γ*≪γ*.(12)In the effective strong excitation regime (i.e., Ωr≫γ*), the single-photon emission rate reaches the maximum value IN=γ*2,(13)which is determined by the effective linewidth γ*.

One can deduce from Eqs. (11)–(13) and Fig. 4 that the rules followed by the emission rate of the subnatural-linewidth single photons are very similar to that in the two-level system, except that the natural linewidth Γ and the Rabi frequency ΩT are replaced by the effective linewidth γ* and the Rabi frequency Ωr. Measured in terms of the actual single-photon linewidth (i.e., providing γ*=Γ), the emitter of subnatural-linewidth single photons we propose has almost the same emission rate as the most general two-level emitter. For a visual demonstration, we show the ratio IN/ITLS as a function of the Rabi frequency in the inset of Fig. 4. It reveals that in the (effective) weak excitation regime, the emission rate of the subnatural-linewidth single photons can reach half of that of the two-level emitter. As the Rabi frequency increases, the emission rate of the subnatural-linewidth single photons gradually reaches the maximum value as that in two-level emitter, as demonstrated in Eqs. (9) and (13).

Significantly, similar to the case in the two-level emitter, the sidebands of the fluorescence are not resolved when Ωr∼γ*, and thus the linewidth of fluorescent single photons is largely determined by the effective linewidth γ*. Besides, the single-photon emission rate also approaches γ* when Ωr∼γ*, as shown by the red solid line in the region with blue background in Fig. 4. Therefore, we can conclude that for the subnatural-linewidth single photons we propose, the emission rate can almost reach the upper limit determined by its single-photon linewidth γ*, according to Eq. (6). Moreover, since γ*=|Ω|2/γ is determined by the intensity of the external field, the performance of single photons can be conveniently manipulated while maintaining the maximum emission rate.

The constraint between the single-photon linewidth and the emission rate can be understood in terms of the uncertainty principle followed by single-photon wave packets, as described in Section 4.A. Besides, this constraint also exhibits in the detection process. The narrow bandwidth κ of the detector means a long response time 1/κ. For multiple incident photons occurring within this response time, the detector cannot distinguish the order of these photons, but consider that they are simultaneously incident. Therefore, to ensure the single-photon response on the detector with a narrow bandwidth, the average time interval of successive single photons should be much longer than the response time of the detector, which limits the number of photons per unit time.

Various emitters with Λ or Λ-like structures can be used to generate single photons with subnatural linewidth. For instance, as a system extensively investigated and employed in quantum technologies [31–33], the single R87b atom loaded into an optical dipole trap is suitable for the implementation of our proposed approach. Specifically, the closed hyperfine transition Fg=1→Fe=0 of the D2 line of R87b is driven by a circularly polarized light resonantly, and the transitions between the ground-state Zeeman sublevels are simultaneously driven by the magnetic field perpendicular to the direction of light propagation. Therefore, single photons with subnatural linewidth can be emitted (see Appendix F for details). In addition, our approach can be implemented based on the Λ-level structure contained in the charged GaAs and InGaAs quantum dots in a Voigt magnetic field [34,35], where a transition between an electron-spin ground state (|↑⟩ or |↓⟩) and trion excited state (|↓↑⇑⟩ or |↓↑⇓⟩) can be driven by a laser resonantly, and the coherent transition between electron-spin ground states can be realized by a magnetic field or a stimulated Raman process. Both the frequencies and polarizations of the target single photons and the driving field can be different in our approach. Therefore, based on the frequency and polarization filtering, as well as the existing experimental techniques [36], the laser background can be excluded efficiently to improve the quality of the single photons.

Noteworthily, the reduction of the linewidth of subnatural-linewidth single photons is relative to the natural linewidth of a given emitter. According to the general condition for generating subnatural-linewidth single photons, our study can be readily implemented based on various physical systems (e.g., atom, quantum dot, ion, molecule, and nitrogen-vacancy center), whose natural linewidths vary over a wide range from MHz to THz [37]. Therefore, the specific linewidth and emission rate of the subnatural-linewidth single photons, which depend on the specific kind of emitter, also vary over wide ranges. For instance, with the parameters shown in Fig. 2(a), subnatural-linewidth single photons with a bandwidth of 1/10 natural linewidth are generated. In this case, if the single-photon scheme is implemented based on a quantum dot with a typical natural linewidth on the order of GHz, the linewidth and thus emission rate of the subnatural-linewidth single photon will be about 100 MHz, which are significantly larger than the linewidth and emission rate of the single photons emitted by an atom with a typical natural linewidth on the order of MHz. Therefore, we can conclude that “subnatural linewidth” does not denote “narrow linewidth”, but rather breaks the limit imposed by natural linewidth on the linewidth of fluorescent single photons. In practice, subnatural-linewidth single photons may have a large linewidth and emission rate.

We propose an approach to focus all the spectral components of fluorescence on the subnatural linewidth, and thus the single photons with subnatural linewidth can be emitted under the premise of inheriting the perfect single-photon character of fluorescence. Thereby, the limit imposed by the natural linewidth of the emitter’s transition on the linewidth of fluorescent single photons is broken. Compared to the constant natural linewidth of an emitter, the single-photon linewidth can be easily manipulated over a broad range by varying the intensities of the external fields. Further, we reveal that the general condition to generate single photons with subnatural linewidth using the atom or atom-like system is that the successive emission process of the target photons is totally dominated by the transition loop with a metastable state. Benefiting from this revelation, this proposed approach can be generalized in various physical systems with Λ-shape and similar energy structures. Since the distribution of single-photon wave packets in time and frequency must satisfy the uncertainty relation, the upper limit of the emission rate imposed by the single-photon linewidth is inevitable, both for the conventional single photons and for the subnatural-linewidth single photons. However, we demonstrate that the emission rate of the subnatural-linewidth single photons can reach its theoretical upper limit.

In addition, the single photons with a narrow or even ultranarrow linewidth, which have a long coherence time, can also be obtained by reducing the intensities of external fields. For the excellent tunability of the single-photon linewidth and emission rate, it is possible to find the best trade-off between these parameters according to the requirement in a specific situation. Consequently, we see that the capability to manipulate the linewidth and emission rate of single photons in various atomic and atom-like systems can be significantly enhanced based on our research. This research will significantly facilitate the insight into the quantum nature of single photons and resonance fluorescence, and has extensive applications in quantum optics, photonics, and quantum technologies due to the outstanding performance of subnatural and narrow linewidth single photons.

Acknowledgment. He-Bin Zhang would like to thank Jizhou Wu for his valuable discussions.