Single-photon emission of two-level system via rapid adiabatic passage

In this paper, we present a high quality single-photon source based on the two-level system undergoing rapid adiabatic passage (RAP). A trigger strategy (sweet region) is suggested to optimize the single-photon emission and explain a counter-intuitive phenomenon on the optimal parameters. The RAP strategy of single-photon source is robust against control error and environmental fluctuation.


TLS Model
The Hamiltonian of the TLS can be described by 23 where ε(t) is the time-dependent detuning, Δ is the coupling strength between the ground state |g〉 and excited state |e〉 . σ x and σ z are the Pauli matrices and we set ħ = 1 throughout this manuscript. In this work, we consider the case that ε(t) is modulated via 23 0 where A is modulation amplitude, ω denotes modulation frequency and ε 0 represents static detuning, respectively.

Photon counting statistics via Generating Functions formula
The time evolution of density matrix ρ(t) of single quantum system obeys the Liouville-von Neumann equation, where  is the superoperator including the spontaneous emission rate Γ which is caused by the vacuum fluctuation.
By employing the generalized Bloch vector notation [11][12][13] , the generating function equations can, by employing Eqs 3, 4 and 5, be written as where The probabilities of emitted n photons in time interval [0, t] can be extracted from s t 2 ( , )  as follows [11][12][13]  The factorial moments of photon emission can be simply obtained by taking derivatives with respect to s evaluated at s = 1 11-13 , namely Thereinto, the average photon emission number 〈 N〉 = 〈 N (1) 〉 is given by r = 1 (the first factorial moment). Meanwhile, the Mandel's Q parameter (2) ( 1) 2 (1) follows immediately, which characterizes the statistical properties of emitted photons. The case of Q < 0 is called sub-Poissonian distribution (anti-bunching behavior) and Q > 0 is named super-Poissonian distribution (bunching behavior). Particularly, a perfect single-photon emission results in Q = − 1.

Single-photon emission via trigger strategy of RAP
Based on the photon counting statistics of generating function approach, one can study the properties of photon emission from TLS undergoing level-crossing. Before giving the quantitative results, we analyze the RAP strategy of single-photon emission and its optimal control region by LZ formula. Supposing that TLS is initialized at its ground state |g〉 , the detuning sweeps with a finite velocity v and passes through level-crossing. The population inversion occurs in the vicinity of level-crossing, and the asymptotic excited state population P g→e is, from LZ formula, given by Scientific RepoRts | 6:32827 | DOI: 10.1038/srep32827 g e LZ where the adiabaticity parameter 2 divides LZ problem into two regimes: non-adiabatic passage with δ  1 and near-adiabatic passage with δ > ∼ 1.
In this work, we focus on the near-adiabatic passage to generate near-complete population inversion, which is crucial to trigger single-photon emission.
Considering the harmonic sweeping, the sweep velocity in level-crossing vicinity is given by 23 One can, by using Eqs 12 and 13, distinguish between the non-adiabatic passage ω ∆  A 2 and the near-adiabatic passage ω < ∆  A 2 . In addition, the time t LZ of LZ transition measures the population inversion speed. t LZ , as an important parameter in LZ transition, is given by 23 Eq. 14 means that t LZ is a monotonically decreasing function of v.
At each level-crossing vicinity, the ground state |g〉 is triggered to excited state |e〉 in near-adiabatic regime. Whereafter, |e〉 will emit a photon due to spontaneous emission rate Γ , and go back to |g〉 before next trigger event. The cyclic repeats will construct a sequential photon-emitter with trigger frequency 2ω. To present good quality single-photon emission, the population inversion should approach unit for each passage event.
Meanwhile, the inversion time should be short to minimize the effect of Γ . This control condition can be noted as rapid and adiabatic passage. Namely, (i)For adiabatic feature, from LZ formula of Eq. 11, it is necessary to slow down the sweep velocity v to obtain near-complete inversion between |g〉 and |e〉 .
(ii)For rapid feature, the time t LZ of LZ transition should satisfy ⋅ Γ  t 1 LZ to protect LZ process against strong Γ .
This indicates that the above contradict requirements seek a sophisticated sweep velocity v. Here we can notice as follows The first one gives the boundary of adiabatic feature, as faster v will lead to incomplete inversion. And the last one gives the boundary of rapid feature, as lower v or longer t LZ makes no significant improvement in population inversion. Based on the above analysis, the optimal control region is suggested as follows to ensure rapid and adiabatic feature, and we mark Eq. 15 as the sweet region. In this paper, the sweet region plays a central role in analyzing single-photon emission. Also, in this region, t LZ · Δ gives the same order of sequential pulses control with π-pulse. Meanwhile, ∆ Γ  is necessary to ensure population inversion in both RAP and π-pulse scheme.
In the following, we show the numerical results of single-photon emission via the photon counting statistics of the generating function formula, and verify the RAP strategy of single-photon generation in the sweet region.

Verification about sweet region
Here, we show the high quality of single-photon emission can be obtained at the sweet region of Eq. 15. Figure 2 shows the distribution of the Mandel's Q parameter (left) and single-photon emission probability (right) after the first trigger event. One can find the position of the high quality single-photon emission is located in the sweet region of Eq. 15, namely, Δ 2 /4 < v = Aω < Δ 2 . That is, the region between red and blue lines in Fig. 2. Here, the best trace is along the curve of v ~ Δ 2 /2 (the green lines in the figure). Besides, along this sweet region, one can improve the quality of single-photon emission by increasing Δ . Obviously, the extremum of Q and p 1 is located in the end of sweet region. Table 1 shows the typical values of Q, p 0 , p 1 and p 2 , and their optimal parameters under different modulation frequency ω. One can find the optimal locations perform as expectation: They follow a sophisticated sweep velocity v around Δ 2 /2 and located in the end of sweet region. Noticeably, although small ω means that the interval between each trigger event is long enough to achieve distinguishable photon emission 30 . However, Table 1 indicates a counter-intuitive result that smaller ω does not give the better single-photon emission. Actually, in the sweet region, Δ is limited by ω A . Such a small ω will lead to low Δ , which causes an inferior single-photon emission. To obtain the optimal modulation frequency, the sweet region should get the limitation of Δ (Δ = 15Γ   in Table 1) to reduce the impact of Γ . Meanwhile, the ω should be kept as small as possible to ensure distinguishable photon emission. As the expectation of our trigger strategy, we find when ω ranges from 9 to 13 (not shown in the table), the quality of single-photon emission is better than that of other modulation frequencies. Hence, the trigger strategy of sweet region by Eq. 15 can help optimizing the modulation frequency as well.
Besides, one can find that, in the sweet region, Q and p 1 show a quite smooth feature, especially along A/Γ . This feature implies that the RAP scheme is robust.
To analyze the robust features of single-photon emission based on RAP scheme, in the following, we compare the RAP and π-pulse schemes in the same laser coupling strength Δ and the same trigger frequency 2ω. In the π-pulse scheme, we turn off modulation of detuning and switch on and off laser field in fixed duration.

Control Error
As mentioned above, the RAP scheme is not sensitive to control parameter. However, in the π-pulse scheme, a fixed-area laser pulse is needed. The deviation of pulse strength will result in incomplete population inversion apparently. Here, we assume the optimal laser strength is defined by Δ 0 and study the deviation effect on single-photon emission, in which the deviation is measured by log 2 (Δ /Δ 0 ). Figure 3 shows the Mandel's Q parameter as a function of log 2 (Δ /Δ 0 ) and time t for the RAP scheme (left) and the π-pulse scheme (right). Naturally, the optimal laser strength gives the minimal value of Q in both schemes. Noticeably, the deviation log 2 (Δ /Δ 0 ) causes obvious change in the Mandel's Q parameter in the π-pulse scheme, as shown in the right panel of Fig. 3. Especially, the values of Q indicate that the emitted photons have a strong super-Poissonian distribution when Δ = 2Δ 0 . In this case, the area of laser field is 2π rather than π, so that TLS is driven back to ground state |g〉 by laser pulse. Nevertheless, the RAP scheme shows its resilience against this error from laser strength (see the left panel of Fig. 3).

Spectral Diffusion Process
TLS may undergo perturbations from stochastic fluctuations of the surrounding. Here we employ the telegraph model of the surrounding to model the spectral diffusion, namely, the static detuning ε 0 (t) hops back and forth between two states ε + and ε − with rate constant R in both directions 11,12 . Based on the telegraphy model of the spectral diffusion, we have 11,12 Table 2. The single-photon emission of RAP and π-pulse scheme † . † The data are extracted after each trigger events in the stable interval. The "Order" means the order of trigger events. One can find the Mandel's Q parameter remains stable and average photon emission number 〈 N〉 jumps after each trigger event for both schemes. However, the RAP scheme shows a better quality of single-photon emission. The parameters are Γ = 20 MHz, Δ = 5Γ , A = 110Γ , ω = 3 MHz, ε − = 0, ε + = 5Γ and R = 10 MHz.
, and  ± is the same with  in Eq. 6 but ε 0 is replaced by ε ± , respectively. The physical moments of photon counting statistics under the influences of the surrounding fluctuations can be extracted from Eq. 16.
Here we show the RAP scheme is also robust against spectral diffusion process but the π-pulse scheme isn't. Table 2 shows the photon emission statistics of the RAP and the π-pulse schemes for the same control parameters under spectral diffusion process. One can find the Mandel's Q parameter for the RAP scheme is bigger negatively than that for the π-pulse scheme. The accumulated average photon emission number 〈 N〉 shows a high quality of single-photons feature for the former scheme. Specifically, in second trigger event, p 2 dominates the dynamics of the RAP scheme. As a comparison, p 1 makes a non-negligible contribution in the π-pulse scheme.
In conclusion, we present the RAP scheme to generate the high quality single-photon emission. The RAP scheme could trigger single-photons sequentially by periodical modulation of detuning. The sweet region, namely the optimal region of the single-photon emission has been suggested. To achieve better single-photon emission, the counter-intuitive modulation mode was given, which has been interpreted by our trigger strategy. Also, it has been shown that the RAP scheme is robust against control errors and spectral diffusion process.