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Quantum correlations of an atomic ensemble via a classical bath

Somewhat surprisingly, quantum features can be extracted from a classical bath. For this, we discuss a sample of three-level atoms in ladder configuration interacting only via the surrounding bath, and show that the fluorescence light emitted by this system exhibits non-classical properties. Typical realizations for such an environment are thermal baths for microwave transition frequencies, or incoherent broadband fields for optical transitions. In a small sample of atoms, the emitted light can be switched from sub- to super-poissonian and from anti-bunching to super-bunching controlled by the mean number of atoms in the sample. Larger samples allow to generate super-bunched light over a wide range of bath parameters and thus fluorescence light intensities. We also identify parameter ranges where the fields emitted on the two transitions are correlated or anti-correlated, such that the Cauchy-Schwarz inequality is violated. As in a moderately strong baths this violation occurs also for larger numbers of atoms, such samples exhibit mesoscopic quantum effects.

Initiated by Dicke [1], ensembles of few-level emitters interacting collectively with an environmental reservoir have been shown to be a source for many remarkable effects and applications [1,2,3,4,5,6,7,8,9,10]. In the recent past, this interest was renewed by the possible applications of such samples to quantum communications and logic devices. For example, the production of non-classical light has been a subject of several recent experiments [11,12]. On the other hand, it was demonstrated that long-time entanglement between two arbitrary qubits can be generated if they interact with a common bath [13]. Thus, quantum features can be induced through the interactions with a classical electromagnetic field. This is not obvious, as, usually, it is believed that an interaction with a large environmental reservoir leads to decoherence. Thermal light may also produce effects like ghost imaging or sub-wavelength interference [14], which otherwise are known to occur for entangled light [15]. These results are of especial interest, as non-classical driving fields are often hard to produce experimentally with adequate intensities. Therefore, schemes which allow to extract quantum features from atoms in an otherwise classical external setup are highly desirable.
In this Letter, we demonstrate that such a conversion scheme may be implemented with an ensemble of atomic few-level systems subject to a classical bath. For this, we discuss a three-level setup in ladder configuration, and show that the fluorescence light emitted spontaneously on the two transitions has non-classical properties. The bath could e.g. be a thermal bath for atoms with microwave transition frequencies, or incoherent broadband driving for the optical frequency region. In particular, we demonstrate that in a small sample of atoms, the mean number of atoms in the sample allows to switch the light emitted on one of the transitions from subto super-poissonian statistics, and from anti-to superbunching. Here, both super-and anti-bunched light with super-poissonian statistics can be produced. Larger samples, on the other hand, can be used to generate superbunched light over a wide range of bath parameters, thus enabling a control of the intensity of the strongly correlated light. We also identify parameter ranges where the fields emitted on the two transitions are correlated or anti-correlated. As an application for this, we show that the Cauchy-Schwarz inequalities are violated for a moderately strong reservoir and large samples, thus demonstrating a mesoscopic quantum effect.
The basic element of our study is a sample of N identical non-overlapping three-level radiators which interact with a classical reservoir, see Fig. 1. The emitters are located within a volume with linear dimensions smaller than the relevant emission wavelengths λ 12 , λ 23 , and densities low enough to avoid collisions (Dicke model) [1,2,3,4,5,6,7,8,9]. The excited atomic level |1 (|2 ) spontaneously decays to the state |2 (|3 ) with a decay rate 2γ 1 (2γ 2 ). The only external driving is via the surrounding bath, which induces transitions among the atomic levels. In the microwave region, the bath could be a thermal bath, where the rates are proportional to the mean thermal photon number at FIG. 1: The system setup. A sample of N atoms, each with three atomic states in ladder configuration, is confined to a region small as compared to the emission wavelengths λ. The whole setup is placed in a classical bath. The excited states decay spontaneously with rates 2γ 1(2) , giving rise to fluorescence lights with non-classical features.
the corresponding transition frequencies. In the optical frequency region, thermal excitations are negligible, and the bath can be realized by a pseudo-thermal bath, i.e. broadband incoherent driving fields perpendicular to the observation direction. Then, the rates depend on the field strength of the incoherent fields. In the usual mean-field, Born-Markov and rotating-wave approximations, the system is described by the following master equation [6,8]: Here, an overdot denotes differentiation with respect to time. For thermal baths, describes the strength of the incoherent pumping [16]. It is important to note that Eq. (1) contains collective atomic operators S ij = N k=1 |i kk j|, which describe populations for i = j, transitions for i = j, and which obey the commutation relation The steady-state limit of Eq. (1) can conveniently be evaluated with the help of coherent atomic states |N, n, m for the su(3) algebra, which denote a symmetric collective state of N atoms with n atoms in bare state |1 , m−n in bare state |2 , and N −m atoms in bare state |3 with 0 ≤ n ≤ N , n ≤ m ≤ N [6,7,8]. The diagonal elements P nm = N, n, m|ρ ss |N, n, m of the steady-state density operator ρ ss evaluate to: (2), the atomic expectation values can easily be evaluated. We now turn to our main interests in this study, the coherence properties of the collective fluorescent light generated on the transitions |1 → |2 and |2 → |3 . The photons emitted on the two transitions are distinguishable by their polarizations and frequencies, and can be detected e.g. by a pair of single-photon detectors or by atomic state detection [4,5]. The second-order coherence function is defined as [17,18]: where i, j ∈ {1, 2} with J 1 = S 21 and J 2 = S 32 . The quantity g (2) ij (τ ) can be interpreted as a measure for the probability for detecting one photon emitted on transition i and another photon emitted on transition j with time delay τ . g (2) ij (0) < 1 characterizes sub-poissonian, g (2) ij (0) > 1 super-poissonian, and g (2) ij (0) = 1 poissonian photon statistics. g (2) ij (τ ) > g (2) ij (0) is the condition for photon anti-bunching, whereas g (2) ij (τ ) < g (2) ij (0) means bunching. We further define super-bunching as bunching with g (2) ij (0) > 2 [19,20]. More specific, correlation functions with i = j describe the photon statistics of the fluorescence light emitted on a single atomic transition, and g (2) i =j (0) the cross-correlations between the photon emission on two different transitions. We also need to consider the fluorescence intensities G (1) i (0) = J + i J i s of the two transitions. For example, applications may require particularly strong or weak non-classical fields. On the other hands, in the microwave region, the signal from the sample competes with noise from the surrounding heat bath proportional ton i , which especially for small samples of atoms with low signal can render experimental verifications difficult. For bath parameters 0 < η 1 , η 2 < 1 and larger samples N ≫ 1 such that η N 1(2) → 0, one has G 2 (0) increases linearly with N . In the strong field limit (η 1 , η 2 → 1), one has G The first results are shown in Fig. 2, where the correlation function g 11 (0) of the light emitted on transition |1 → |2 is plotted against η 1 and η 2 for a sample of N = 150 atoms. It can be seen that photons with super-poissonian statistics are generated on this transition for moderate baths. In this region, the emitted light is super-bunched except for small values of η 2 , where the light is anti-bunched. This range of η 2 for anti-bunching depends on the number of atoms N and η 1 . In Fig. 2, for η 1 < 0.6, anti-bunching requires η 2 < 0.01. For smaller samples, however, the range for anti-bunching increases. If η 2 is small, then almost all atoms are in the ground state |3 due to collective effects, and on average at most one atom gets excited to |1 to emit light contributing to g (2) 11 (0). Thus the light is anti-bunched. For higher η 2 , more atoms can be excited to |1 simultaneously, and the 11 (0) for small numbers of atoms N in the sample. Solid, long dashed and short dashed curves are for η1 = η2 = 1, 0.5 and 0.1, respectively. Non-integer values for N are possible if N is a mean number of atoms, e.g. for atoms crossing a cavity field. Triangles (squares) denote bunching (anti-bunching). (b) g light is bunched. As the super-bunched photons are produced over a wide range of values for η i , the intensity of the generated light can be controlled via the bath parametersn 1 ,n 2 from very weak up to intense flux. In other words, by modifying the reservoir characteristics, we can obtain a low or an intense flux of strongly correlated photons. This setup is particularly suitable for microwave transitions, as then thermal baths with high values for η i can be achieved, and larger samples of atoms allow for a good signal-to-noise ratio (SNR). An optical realization, however, is also possible. We now turn to the "channel" around η 1 = 1, η 2 ≥ 1 in Fig. 2. For η 1 = η 2 = 1, one finds g (2) 11 (0) = 8(N − 1)(N + 4)/[5N (N + 3)], which for N → ∞ goes to 8/5. The corresponding limit for η 1 = 1, η 2 > 1 yields 6/5. This "channel" can be understood by noting that for these parameters the sample acts collectively, i.e. G (1) i (0) ∝ N 2 , (i ∈ {1, 2}), resulting in a close to coherent photon statistics that corresponds to a superfluorescent atomic sample [1,2,3]. It should be emphasized here that while a thermal reservoir or a direct incoherent pumping of the transitions only admits for values 0 < η 1 , η 2 ≤ 1, we have also used larger values for these parameters in Figs. 2,4. Such situations may occur if additional driving fields are applied to the sample of atoms, e.g. an incoherent repumping from the lower to the upper atomic states [8]. We stress, however, that our main results are obtained without such driving.
In the following, we discuss the special case of a small collection of atoms. This is of particular interest for an experimental verification in the optical region, whereas in the microwave region the small number of atoms makes it hard to obtain a decent SNR against the thermal background. In Fig. 3(a), we show g (2) 11 (0) against the number of atoms in the sample for different values of η 1 = η 2 . Consider, for example, an experimental setup with an atomic beam passing through a low quality cavity. Then, depending on the mean number of atoms which are simultaneously inside the cavity and on the bath parameters, switching between sub-and super-poissonian statistics or PSfrag replacements FIG. 4: The cross-correlation function g 12 (0) against bath parameters η1 = η2. The number of atoms in the sample is (i) N = 2, (ii) N = 50, and (iii) N = 150. (a) η1 = η2, (b) η2 chosen such that the SNR on both transitions is above 10.
anti-bunching and super-bunching of the emitted light can be observed [12]. In the figure, squares (triangles) denote anti-bunching (bunching) of the emitted photons. Note that together with a super-poissonian statistics, both bunching and anti-bunching can be observed. Only collective states |N, 1, m with a single atom in state |1 may lead to anti-bunching. All other states |N, n > 1, m contribute to bunching. The total system behavior depends on the ratio of these two contributions. With increasing bath strength and increasing number of atoms, more bunching states |N, n > 1, m are available and populated, such that the switching from anti-bunching to super-bunching occurs. Some examples for g (2) 11 (τ ) versus delay time τ are shown in Fig. 3(b). Curve (i) shows super-bunching for N = 15 and η 1 = η 2 = 0.6. Starting from an initial value close to 4, the correlation function drops rapidly to unity with increasing τ . Example (ii) shows anti-bunching with super-poissonian photon statistics and large intermediate values of g (2) 11 (τ ). The maximum value of the correlation function can be further increased, however, at the cost of intensity. As reference, (iii) shows an evolution for the single-atom case.
The cross-correlations g (2) i =j (0) also show non-classical behavior. For an atomic sample in a weak bath, g 12 (0) is much larger than unity as shown in Fig. 4, indicating super-poissonian light statistics, which is accompanied by strong correlation between the fluorescence light radiated on both atomic transitions, i.e. cross super-bunching. The reason is that then atoms which decay from |1 to |2 also decay further to |3 with a high probability. For stronger baths, however, larger samples exhibit bunched sub-poissonian light. Then lim {η1,η2}→1 g (2) 12 (0) = 4(N + 2)(N + 4)/[5N (N + 3)], with limit 4/5 < 1 for N → ∞. In this case, atoms decaying from |1 to |2 are repumped by the bath rather than decaying further to |3 .
The other cross-correlation, g 21 (0), is below unity for a weak bath, as a transition |2 to |3 cannot directly be followed by a transition |1 to |2 without an extra excitation. For small samples (N < 8), or for medium samples (N ∼ 40) with strong bath on transition 1 (η 1 → 1), the emitted light is anti-bunched, thus showing collective cross anti-bunching. If both transitions are driven strongly, one finds cross anti-bunching with G For larger samples and smaller η 1 , the light is anti-bunched for low values of η 2 , but switches to bunched light with increasing η 2 .
As an application for the non-classical features, we now show that the light emitted from the sample of atoms violates the Cauchy-Schwarz inequalities (CSI) [21]. The CSI are violated if χ 1(2) = g (2) 11 (0)g (2) 22 (0)/[g (2) 12(21) (0)] 2 < 1, i.e., if the cross-correlations between photons emitted on two different transitions are larger than the correlation between photons emitted from the individual levels. Fig. 5 shows the violation of the CSI function for moderately strong baths. Within the Dicke model, this violation is present for any number of atoms in the sample, thus demonstrating a mesoscopic quantum effect. In addition, χ 1 is always smaller than unity for N ≤ 3 and the entire range of 0 ≤ η 1 , η 2 ≤ 1 (lim {η1,η2}→1 χ 1 = 4[(N − 1)/(N + 2)] 2 ), while χ 2 is larger than unity for any number of atoms and for any values of η 1 , η 2 . The CSI violation can best be observed in the optical region, as low values of η i are favorable, and as it is more difficult to obtain a decent SNR in the microwave region.
In summary, we have demonstrated quantum features in the fluorescence light of a sample of atoms driven only by a surrounding classical bath. We discussed both thermal baths with microwave atomic transition frequencies and pseudo-thermal baths for optical transition frequencies as realizations of the bath. For small samples, a change of the mean number of atoms in the sample induces sensitive switching between sub-or super-poissonian statistics and anti-bunching or superbunching of the light emitted on one of the transitions. For appropriate bath parameters, even mesoscopic samples exhibit anti-bunching. As an application, we have shown that the Cauchy-Schwarz inequalities are violated in our system over a wide range of parameters. † Permanent address: Technical University of Moldova, Physics Department, Ştefan Cel Mare Av. 168, MD-2004 Chişinȃu, Moldova.