Emission of single photons in the weak coupling regime of the Jaynes Cummings model

A recently proposed variant of an unconventional photon blockade scheme is studied for a single emitter weakly coupled to a resonator mode. By controlling two weak coherent fields driving the emitter and the resonator mode, a strongly nonclassical output field is obtained, which is not only antibunched, but has vanishing higher photon number coincidences. For a given set of system parameters, the frequencies and strengths of the driving fields that yield such an output are given.

Going into the frame rotating at the probe frequency ω p , the Hamiltonian changes to in which � r = ω r − ω p , � 0 = ω 0 − ω p and � d = ω d − ω p . For ω p = ω d , the Hamiltonian becomes time independent: This case was briefly studied in 29 showing that, surprisingly, d can always be tuned to obtain UPB given an arbitrary p .
To see this, consider the quantum master equation, which includes dissipations in the emitter and resonator modes: in which κ and γ denote dissipation rates for the resonator and atom, respectively. Next, approximating the dissipative part by ignoring the cρc † parts of the dissipators, the above master equation reduces to the Schrödinger equation with a non-Hermitian Hamiltonian, which amounts to making changes � r → � r − iκ/2 ≡� r and � 0 → � 0 − iγ /2 ≡� 0 in Eq. (4) Now let us label the state as |n, α� , in which n indicates the number of excitations in the resonator mode and α = {g, e} denotes the ground and excited state of the atom, respectively. Then to a low number of excitations, a general state is given by In this basis, ignoring the contributions from the states with a higher number of excitations, the equations of motion are in which ˜ ≡˜ r +˜ 0 .
To solve for the steady state, one assumes c 0g ≈ 1 and solves for c 1g and c 0e ignoring the contributions from higher number of excitations 13 . This yields which can be substituted into the last two equations of motion to give The condition for vanishing two-photon excitation, c 2g = 0 , is then approximated by The solutions (12) www.nature.com/scientificreports/ thus give two values of d for which the two photon correlation function g (2) (0) vanishes. This condition, first derived in 29 , shows that a low value of g (2) (0) can be achieved in the weakly-driven regime if one carefully tunes the amplitude and phase of the field driving the atom.
Single driving field. Let us first consider the case in which d = 0 . In this case, antibunching can only be observed if both the coupling strength and one of the detunings are chosen carefully. From (12), one readily sees that these are With these parameters, destructive interference occurs between the pathways to reach |2, g� , resulting in an almost-vanishing second order correlation function For example, with r = 0 , p = 0.01 , and κ = γ = 1 , we have opt 0 = 0 and g opt = 1/ √ 2 , giving g (2) (0) ≈ 0.003 in the steady state with the average photon number n ≈ 4 × 10 −5 . These and all numerical results in this work are obtained by numerically solving Eq. (5) using QuTiP 31 . For a larger value of detuning r , the required value of g opt becomes larger as can be clearly seen from Eq. (14).
In 30 , UPB was shown to be related to Gaussian states: the fine tuning of the parameters correspond to choosing optimal values of squeezing in a generic Gaussian state. Because of this Gaussianity, the suppression from the Poisson photon number distribution is limited to the two-photon manifold for UPB and comes at the cost of enhanced probabilities for higher photon numbers 29 . This means that the output field in the UPB scheme is n-photon bunched for n ≥ 3 , i.e. g (n) (0) = �a †n a n �/�a † a� n > 1 . Similar behaviour is observed for the current weakly coupled case with a single driving field. For the above parameters, g (n) (0) = 13, 158, 903 for n = 3, 4, 5 respectively. Introducing the atom-driving field changes this-the higher photon number manifolds are also suppressed.
Two driving fields. With two driving fields, it is no longer necessary to tune the 'internal' values g and 0 .
The addition of 2 more control parameters is enough to guarantee vanishing second order correlation function even in the weak coupling regime g κ, γ . Figure 1a shows the contour plot of g (2) (0) in the steady state as a www.nature.com/scientificreports/ function of g and p . The driving frequency is fixed to the first excited state (assuming ω r ≥ ω 0 ) of the JC model, ω − = (ω r + ω 0 − � 2 + 4g 2 )/2 , with � = ω r − ω 0 and will remain so unless otherwise stated. d = + d is chosen for reasons that will be explained below. The result makes sense intuitively. The larger the value of g the larger the nonlinearity, which in turn allows for a stronger drive strength until nonclassicality is lost. In other words, the strength of the driving field is limited by the value of the coupling strength, which sets the nonlinearity of the system. Unlike in the UPB scheme, higher order correlations are also suppressed as shown in Fig. 1b-d. Three photon coincidence as given by g (3) (0) has a smaller region of suppression compared to g (2) (0) , but higher order correlations are suppressed in wider regions. Higher order g (n) (0) 's not shown are also suppressed. For weak driving strengths as considered here, average photon number n ≪ 1 and the chance of observing n-photons decreases exponentially with n, but in view of possible generalization to single photon applications it is nice to have suppressed coincidence counting too all orders.
The fact that merely adding another driving field can make such a big change to all orders of correlation functions may seem surprising, but one should note that the additional driving field addresses the highly nonlinear two-level emitter as emphasized in 11 . In fact, in the bad cavity regime characterized by κ ≫ g 2 /κ ≫ γ , the emitter-driven system is commonly used as a single-photon source 2 . One can therefore understand the suppressed occurrences of two or more photons as resulting from a combination of non-classical light produced by driving the emitter and UPB which further suppresses the two-photon coincidence. To see this, let us consider the case in which only the atom is driven, i.e. p = 0 and d = 0.01 . This simplification allows one to obtain an analytic solution for higher-order correlation functions in a manageable form. For a given n the equations of motion (9) generalizes to (still ignoring the contributions from the higher number excitations) In the steady state the solutions are Repeatedly applying the second equation to itself, one obtains Using Eq. (10), nth order correlation function at zero delay can thus be approximated by We get r = 0 = −g for = 0 and assuming g ≪ γ , κ , the above equation can be further simplified to For γ = κ , g (n) (0) ≈ n−1 j=0 1/(1 + j) 2 . This formula yields g (2) (0) ≈ 1/4 , g (3) (0) ≈ 1/36 ≈ 0.027 , and g (4) (0) ≈ 1/576 ≈ 0.0017 for the set of parameters used in Fig. 1, except for d = 0.01 and p = 0 . These agree reasonably well with the numerically obtained results g (2) (0) = 0.26 , g (3) (0) = 0.03 , and g (4) (0) = 0.002 , for g = 0.1, where the differences are due to the terms ignored in going from Eqs. (19) to (20). The above formula shows that higher-order correlations are well-suppressed for n > 2 when the atom is driven and the atom-field coupling is weak, but that antibunching is not strong.
Upon increasing opt p , which is to be obtained from Eq. (12) after allowing p to be complex, a numerical calculation shows that g (2) (0) varies rapidly while higher-order correlations do not, as illustrated in Fig. 2. This shows that one can obtain good suppression of all nth order correlation functions for n ≥ 2, by controlling the magnitude of p . We can therefore understand the single-photon behaviour of the emitted photons as resulting from a combination of non-classical light produced by driving the emitter, and UPB which suppresses the two-photon coincidence. (16) iċ ng ≈ √ ngc n−1e + n� r c ng , (20) and it is evidently advantageous to drive the emitter more strongly. If ω p were chosen to be ω + = (ω r + ω 0 + � 2 + 4g 2 )/2 , such that the second excited state is driven, the situation is reversed. The magnitudes of ± d are merely reversed although their phases are changed. The values g (n) (0) = (0.145, 14.2, 200, 1580) are obtained for d = + d , and the lower values are obtained for d = − d . Therefore, ω p can be chosen to be either ω + or ω − as long as the correct d is used. This freedom only holds as long as the atomic transition is resonant to the cavity mode, i.e. ω r = ω 0 . If this were not the case, as in many experimental implementations, merely choosing the larger of ± d turns out to be insufficient. In addition, the correct resonator-driving frequency must be chosen, which in this case (Δ > 0) is ω − . For example, with ω p = ω + , ω r = 12 and the rest of the parameters remaining the same, neither + d nor . This asymmetry between ω − and ω + can be understood by noting that for � = ω r − ω 0 ≥ 0 the first excited state is predominantly in |0, e� , which is the preferred state to be driven (because we want to drive the atom, not the cavity). On the other hand, if � < 0 , then the situation is reversed and the privileged role is taken by the second excited state.
To quantify the suppression of higher-order photon coincidences let us adopt a measure called n − norm , defined as ||(g (k) )|| n = n n+1 k=2 [g (k) ] 32 . It measures the distance in the correlation space between the given source and an ideal single-photon source. 4 − norm s are adopted in this work, but higher-order norms are also suppressed. Figure 3a-d show that out of the four different combinations with ω p = ω ± and d = ± d , the combination (ω − , � + d ) gives the best result as explained above. It is not shown but g (2) (0) is always suppressed and the corresponding figures are qualitatively similar to Fig. 1a. Figure 4a gives a close-up view of Fig. 3d, which shows that (ω + , � − d ) can in fact yield good single-photon characteristics in a limited region of the parameter space. Note the changes in the range of p and the color coding. However, the figure shows that it is very difficult to obtain good single-photon characteristics for g ≪ γ , κ . Figure 4b is the corresponding figure for ω r = ω 0 , showing a much wider region of suppressed ||(g (k) )|| 4 . It is exactly the same as the one obtained using (ω − , � + d ) . From these facts we see that non-zero detuning breaks the symmetry between (ω + , � − d ) and (ω − , � + d ) , which can again be explained by the relative fraction of |0, e� in the addressed state.
To sum up, to obtain good single-photon characteristics one must: (1) choose the driving frequency such that the state |0, e� is predominantly driven and (2) choose the driving strength such that the emitter-driving field is the stronger of ± in Eq. (13). Then the strong nonlinearity of the atom guarantees that the photons are emitted one by one in a large region of parameter space. If a 'wrong' choice is made, two-photon coincidences are still suppressed but higher-order ones are not, as in UPB.
Lastly, let us briefly consider asymmetries in κ and γ . The bad-cavity regime characterized by κ ≫ g 2 /κ ≫ γ is often invoked for the purpose of creating single photons. The emitter is driven, which mainly emits into the leaky cavity mode. Even in this regime, additional cavity-driving field improves the quality of output single photons. For example, for κ = 1, g = 0.1, γ = 0.01, ω r = ω 0 = 10 , ω p = ω − , and | d | ≈ 0.11 , the first three values of  13)). On the other hand, if the atom decays much faster, i.e., γ ≫ g, κ the driving field strengths need to be reduced and the suppression for higher order photon coincidences are not so good. For γ = 1, κ = g = 0.1 and  Figure 5 shows how ||g (n) || 4 changes as a function of g for p = 0.01 (solid blue curve) and p = 0.001 (dashed red curve). The values differ significantly for g < 0.5 . For p = 0.01 it is interesting that ||g (k) || 4 gets bigger first before it decreases, indicating strong multi-photon bunching.
In summary, a generalization of a scheme based on UPB proposed recently in 29 has been investigated in detail. It is shown that unlike in UPB, the output field has suppressed probabilities for all n-photon coincidences for n ≥ 2 . The main reason for the suppression can be attributed to the strong nonlinearity of the 2-lv system, upon which the interference effects of UPB are added to reduce the two-photon coincidences further. One aspect not discussed in this work is the time-variation of the correlation functions. In the conventional UPB scenario, the coupling strength J between the resonators need to be large for weak nonlinearities. This causes the second order correlation function to oscillate rapidly on the time scale of order 1/J. Such rapid oscillations are absent in the system investigated in this work, because there is no large parameter involved. That the correlation functions change on the time scale of 1/γ and 1/κ has been checked by numerical calculations.
As a future work it would be interesting to investigate whether the proposed scheme can be generalized to produce an on-demand single photon source which typically requires a multi-level emitter. Due to the small coupling strength required and the ease of parameter tuning, such a scheme would prove to be very useful.