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Abstract
We demonstrate that light quanta of well-defined characteristics can be generated in a coupled system of three two-level atoms. The quantum nature of light is controlled by the entanglement structure, discord, and monogamy of the system, which leads to sub- and superradiant behavior, as well as sub-Poissonian statistics, at lower temperatures. Two distinct phases with different entanglement characteristics are observed with uniform radiation in one case and the other displaying highly focused and anisotropic radiation in the far-field regime. At higher temperatures, radiance witness is found to exhibit sub- and superradiant behavior of radiation intensity in the absence of entanglement albeit with non-zero quantum discord. This establishes the physical manifestation of quantum discord. It is also observed that the radiation intensity can be a precise estimator of the inter-atomic distance of a coupled system of two-level atomic systems. Our investigation shows, for the first time, the three body correlation in the form of a ‘monogamy score’ controlling the sub- and superradiant nature of radiation intensity.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light of desired nature is much in demand for both fundamental [1–4] and technological applications such as lithography [5], quantum key distribution [6–8], cryptography [9,10], target detection [11], in beating classical imaging limit [12], ultrafast quantum interferometry [13], to mention a few. At a fundamental level, single photon interference has been demonstrated with an emitted photon from entangled atoms [14]. A number of methods have been devised in producing entangled photons [15–17], quantum light of desired nature [18,19], characterizing [20], and transmitting them [21]. Study of coherent and incoherent sources of light, as well as sources generating single photons [7,22], manipulating entangled photons [3,4,8–11,13], and bunching-antibunching [23–26] of photons are subjects of intense investigation. In this regard, light emission from entangled sources is being studied with particular interest to unveil the role of nonlocal quantum correlations on spontaneous emission, as also its superradiant character [27], originally predicted by Dicke in 1954 [28]. Apart from a dramatic enhancement of intensity, the emitted radiation provides much room for controlling its property and can provide a precise estimator of the inter-atomic distance of coupled two-level atomic systems.
Dicke superradiance is the coherent spontaneous emission from a many-body system, owing its origin owing to the co-operative simultaneous interaction among its constituents, all of them experiencing a common radiation field [28]. The collective behavior of the ensemble arises from the coherent superposition and entanglement structure of the many-body wave function. The correlated structure can also show subradiant behavior due to destructive interference of the superposition states. It is interesting to note that some of the excited and ground states in the original study of Dicke are highly entangled [27]. Superradiance has attracted significant interest due to its possible applications, ranging from generation of X-ray lasers with high powers [29], short pulse generation [30] to self-phasing in a system of classical oscillators [31], to name a few. Super- and sub-radiant behavior has been investigated experimentally in many physical systems [32–36]. In particular, Dimitrova et al. [34] observed superradiance in a Bose-Einstein condensate (BEC). In yet another study, superfluidity of BEC along the axis of the ring cavity has been shown to yield superradiant scattered photons [36].
In the context of quantum information, it is of particular interest to explore how the behavior of the radiation field gets affected for a collection of atoms, when the quantum states are correlated in different ways. This allows for optical probing of quantum correlations (QCs) and aids in quantifying QCs that may be present in the system. It is well understood that depending on the nature of interactions of the multi-particle system, one can realize different types of entangled states [37,38] leading to different radiation characteristics. These atomic entangled states can find potential applications in quantum information processing [39], for generating different entangled quantum states of light for quantum communication [40] and quantum cryptography [41,42], among others.
Two particle entanglement has been well characterized both for pure and mixed states [43], using different measures viz., von Neumann entropy and concurrence. Recently, concurrence [44,45] and quantum discord [46] have been used for quantitatively characterizing entanglement governing the quantum phase transition occurring in an antiferromagnetic spin chain, consisting of weakly coupled dimers [43–46]. In comparison, the three particle entanglement is much less understood. It is known to exhibit stronger QCs as compared to the Bell states and also shows stronger non-locality [47]. The entanglement structure of multiparticle states of different type is yet to be completely understood [48]. Here, we investigate the effect of entanglement, quantum discord, and monogamy relations on sub and superradiance of three two-level atomic system.
Recently, Wiegner et al. [27] have investigated the sub and superradiant characteristics of an N-atom system in a generalized W-state of the form $\dfrac {1}{\sqrt {n}} |j,~ n-j\rangle$, with $j$ atoms in the excited state and $(n - j)$ atoms in the ground state, where the role of entanglement has been highlighted for pure states. In another study, the effect of quantum discord on the sub and superradiant intensities in a system of X-type quantum states has been investigated [49], without taking into account the effect of finite temperatures. In the present study, we carry out a systematic investigation of the sub and superradiant properties of a dipole-coupled system of three two-level atoms and explore the effect of transition frequency and coupling, on the resulting radiation pattern. The behavior of the radiation field pattern as a function of concurrence and quantum discord is probed for gaining a physical understanding of the effect of different QCs on the emitted light. The role of QCs in producing highly collimated light, as well as completely uniform illumination is illustrated. The connection of entanglement on far field radiation pattern is demonstrated for line configuration. The intensity pattern can be used to determine the inter-atomic distance. Further, it is found that intensity increases with the monogamy of entanglement. The photon-photon correlation, as a function of system parameters, is found to yield sub and super-Poissonian characteristics, which can be controlled.
The paper is organized as follows. In Sec. II, the Hamiltonian for the system of three identical two-level atoms interacting via dipole-dipole coupling is introduced, using pseudo spin variables. The measure of various quantum correlations and radiance witness are also defined in Sec. II. We have also presented the exact expression for intensity and photon-photon correlation in Sec. II. We present the characteristics of the intensity of the emitted radiation from the three two-level atoms arranged in line configuration at a non-zero temperature in Sec. III. Finally, we conclude with a summary of the results and direction for future research.
2. Theory and model
We consider a system of three coupled identical atoms, where the excited state $ |e_i\rangle $ and the ground state $ |g_i\rangle$, $i = 1,~ 2,~ 3$ are separated by an energy interval $\hbar \omega$. The Hamiltonian for the system of three identical two-level atoms coupled through dipole-dipole interaction is given by,
2.1 Thermal density matrix
The thermal density matrix of coupled system of three identical two-level atoms is provided in Eq. (4). For simplicity, we consider the transition frequencies of all the three atoms to be the same, $\omega _{1}=\omega _{2}=\omega _{3}=\omega$ and the nearest neighbor dipole-dipole interactions $\Omega _{12}=\Omega _{23}=\Omega$ and $\Omega _{13}=0$. By diagonalizing the Hamiltonian $H$, we can obtain all the eigenvalues $\epsilon _{i}$ and their corresponding eigenstates $|\psi _{i} \rangle$. The eigenvalues, $\epsilon _i$, in the line configuration are
The density matrix elements for this system are symmetric, i.e., $\rho _{ij}=\rho _{ji}$.
2.2 Correlation measures
In this subsection, a brief explanation of the various measures of the quantum correlation that are investigated has been provided.
2.2.1 Concurrence
Concurrence is a measure of entanglement [52] defined as,
where $\lambda _i$’s are the square root of the eigenvalues of the non-Hermitian matrix $R=\rho \tilde {\rho }$ (or eigenvalues of Hermitian matrix $R\equiv \sqrt {\sqrt {\rho }\tilde {\rho }\sqrt {\rho }}$) in decreasing order with $\tilde {\rho }$ being given by Here asterisk denotes the complex conjugate and $\sigma _y$ is the Pauli matrix and the matrix $\tilde {\rho }$ is known as the spin flip matrix.2.2.2 Quantum discord
Quantum discord [53,54] is defined as a measure of the quantum correlations present in a system. The difference arises because of the role played by measurement on the system. It can be computed for specific reduced density matrices of the form X-type (see Eq. (9)) [55].
2.2.3 Monogamy of negativity
Negativity was first introduced by Vidal and Werner [56]. It is based on the trace norm of the partial transpose $\rho ^{T_A}$ of the state $\rho$ (mixed or pure) and is equal to the sum of modulii of only the the negative eigenvalues of $\rho ^{T_A}$. Negativity can be expressed mathematically as
where $\rho ^{T_A}$ is the partial transpose of $\rho$ and $\lambda '_i$s are negative eigenvalues of $\rho ^{T_A}$. Now, consider the density matrix $\rho$ which acts on $H_A\otimes H_B \otimes ~.~.~.~\otimes H_X$. The decomposition of $\rho$ can be expressed as The partial transpose of A ($A\in H_A$) can now be defined as We next describe the essentials of monogamy of negativity. Monogamy of quantum correlations was first established by Coffman, Kundu, and Wootters [57] for three qubit pure state using squared concurrence. This was later generalized by Osborne and Verstraete [58]. Unlike classical correlations, the shareability of quantum correlations is restricted by the non-classical properties of the quantum system. For example, in a tripartite quantum state $\rho _{ABC}$, if two parties A and B are maximally correlated, then neither A nor B can share quantum correlation with the third party, C. The monogamy of negativity denoted by $\tau _{A:BC}$ is defined by It may be noted that $N_{A:BC}$ corresponds to entanglement between subsystem A and subsystem BC.2.3 Analytical expression of intensity and photon-photon correlation
We investigate the intensity emitted by three atom system, in the far field zone, i. e., $|\vec {r}|>>d$; where $d$ is spacing between the atoms and $\vec {r}$ denotes the position of the detector to record the photons emitted by the atoms in the far field regime. The positive frequency component of the electric field operator [2,27] is given by,
Thus, the characteristics of the intensity depends on the incoherent terms $\langle \hat {S}^{+}_{i}\hat {S}^{-}_{i}\rangle$, the non-vanishing of the dipole moments $\langle \hat {S}^{+}_{i}\rangle$, and the QCs of the form $\langle \hat {S}^{+}_{i}\hat {S}^{-}_{j}\rangle - \langle \hat {S}^{+}_{i}\rangle \langle \hat {S}^{-}_{j}\rangle$. We now take into account the thermal effects, where, at finite temperature, the expectation value of an observable $\left \langle\hat {A}\right \rangle$ takes the form
where $\rho$, the thermal density matrix of the system is given in Eq. 4.In the line configuration, the dipole coupled system of three identical two-level atoms are placed symmetrically along a line with equal spacing $d$ between adjacent atoms. For this topology, $\phi _{j}$, the relative optical phase accumulated by a photon emitted at $\vec {R}_{j}$ and detected at $\vec {r}$ is
with $k=\frac {2\pi }{\lambda }$ and $\lambda$, the wavelength of the emitted radiation. The exact expression for the intensity from the system of three atoms arranged along a line, which is obtained by combining Eqs. (17) to (19) is given byWe next derive the normalized second order photon-photon or intensity-intensity correlation, defined by
Therefore,
2.4 Radiance witness R
The radiance witness $R$ is defined as
In the study presented here for the case of three atoms, the collective intensity $\langle E^{-}E^{+}\rangle _N=\langle E^{-}E^{+}\rangle _3$ is given in Eq. (20). In the ensuing sections, the intensity pattern resulting from the three atoms in a line configurations, as a function of the system parameters, as well as the temperature and observation angle is presented.
3. Intensity characteristics of the line-configuration
In earlier work, the role of entanglement on super and subradiant behavior for the three-atom system, with a zero net dipole moment, was studied [27]. Here, we have generalized this study, exhibiting the presence of QCs and their physical effect for the three-atom system. Figure 1 illustrates the periodic variation in the intensity from super to subradiant behavior as a function of the ratio of transition frequency and fixed dipole coupling ($\frac {\omega }{\Omega }$) and observation angle for two temperatures. This reflects the subtle interference effects present in the three particle system. For high $\frac {\omega }{\Omega }$ and at low temperatures, a phase with uniform light emission is seen, separated from a non-uniform intensity with periodic modulations. The uniform phase (emission) of radiation arises when both entanglement and discord vanish, as is evident from Figs. 1(a) and 2(b). A smooth crossover connects the two phases. The uniform phases vanish at higher temperatures as seen in Fig. 1(b).
Fig. 1. The variation of radiation intensity as a function of $\frac {\omega }{\Omega }$ and observation angle is depicted for two different temperatures for (a) $k_BT=5\times 10^{-3}\hbar \Omega$ and (b) $k_BT=\hbar \Omega$. Panels $c$ and $d$ show the radiation intensity as a function of observation angle and ratio of emission wavelength and fixed inter-atomic spacing ($\frac {\lambda }{d}$) at ‘$\frac {\omega }{\Omega }=1$’ for (c) $k_BT=5\times 10^{-3}\hbar \Omega$ and (d) $k_BT=\hbar \Omega$, clearly showing the interference effect.
Fig. 2. Panels $a$ shows the variation of QCs as a function of temperature ($\times \frac {\hbar \Omega }{k_B}$) for $\frac {\omega }{\Omega }=1$. Panels $b$ shows the variation of QCs as function of $\frac {\omega }{\Omega }$ for $k_BT=5\times 10^{-3}\hbar \Omega$. Here $C$ stands for concurrence while $QD$ and $QC$ stand for quantum discord and quantum correlation, respectively.
The behavior of intensity as a function of observation angle and $\frac {\lambda }{d}$ at fixed $\frac {\omega }{\Omega }~(=1)$ for different temperatures is shown in Fig. 1(c) and 1(d). Sub and superradiant nature of radiation is observed at all observation angles (except for $\theta =-\pi , ~0,~\pi$). For $\theta =\frac {\pi }{2}$, the intensity observed is super-radiant in the vicinity of $\frac {\lambda }{d}=\frac {2}{5}$, $\frac {\lambda }{d}=\frac {2}{3}$, and $\frac {\lambda }{d}=2$ only. This result is significant in light of the fact that it provides a method to find the inter-atomic distance of equally spaced array of atoms. For example, for given emission wavelength $\lambda$ the superradiant intensity will be observed at specific angles. As $\lambda$ and $\theta$ are known and from Fig. 1, the relation between $\frac {\lambda }{d}$ and $\theta$ can be used to estimate $d$. Therefore, on observing the emitted photons at different observation angles, one can infer about the inter-atomic distance of the system. It is evident from Fig. 1(c) and (d) that the behavior of intensity remains same at higher temperature, albeit with reduced intensity. In [27], authors have shown that the contour plot of intensity where super-radiant intensity can be observed for large value of $\frac {kd}{\pi }=\frac {2d}{\lambda }$. In contrast to that, we have plotted the contour plot of intensity with respect to $\frac {\lambda }{d}$ and found that super-radiant intensity can be found for $\frac {\lambda }{d}\leq 2.5$ only, for dipole coupled systems at lower temperatures. It can be seen in Fig. 5(c). The white color in Figs. 1(c), 1(d), 4, 5(c), and 5(d) is the artifact and it has the same color as its nearest surrounding.
Fig. 3. Panels $a$ and $b$ show intensity variation with respect to monogamy score ($\tau _{1:23}$) of negativity at $k_BT=5\times 10^{-3}\hbar \Omega$ for (a) $\frac {\lambda }{d}=2$ and (b) $\frac {\lambda }{d}=\frac {2}{3}$, showing the increase of intensity with increase in monogamy score.
Fig. 4. Panels $a$ and $b$ show photon-photon correlation for $\frac {\lambda }{d}=2$ as function of $\frac {\omega }{\Omega }$ and observation angle for (a) $k_BT=5\times 10^{-3}\hbar \Omega$ and (b) $k_BT=\hbar \Omega$. Panels $c$ and $d$ show photon-photon correlation for $\frac {\omega }{\Omega }=1$ as function of wavelength of emitted radiation and observation angle for (c) $k_BT=5\times 10^{-3}\hbar \Omega$ and (d) $k_BT=\hbar \Omega$, clearly indicating the sub and super-Poissonian statistics.
Fig. 5. The variation of radiance witness as a function of $\frac {\omega }{\Omega }$ and observation angle is depicted for two different temperatures for (a) $k_BT=5\times 10^{-3}\hbar \Omega$ and (b) $k_BT=\hbar \Omega$. Panels $c$ and $d$ show the radiance witness as a function of observation angle and ratio of emission wavelength and fixed inter-atomic spacing ($\frac {\lambda }{d}$) at ‘$\frac {\omega }{\Omega }=1$’ for (c) $k_BT=5\times 10^{-3}\hbar \Omega$ and (d) $k_BT=\hbar \Omega$, clearly showing the sub ($R<0$) and superradiance ($R>0$) effect.
To understand the intensity profile, for the given system, it is imperative to know the variation of QCs with temperature and transition frequency. In Fig. 2, panel $a$ shows the behavior of concurrence [52] and quantum discord [53–55] as a function of temperature for $\frac {\omega }{\Omega }=1$, while panel $b$ shows the variation of QCs as a function of $\frac {\omega }{\Omega }$ for $k_BT=5\times 10^{-3}$. The intensity pattern at small $k_BT$ and small values of $\omega /\Omega$ observed in Fig. 1 is predominantly due to the high amount of QCs present in the system. This result also confirms that even for $k_BT>\hbar \Omega$, the superradiant behavior is present, albeit with reduced intensity in the absence of concurrence but with non-zero discord as is evident from Figs. 1 and 2. This explicates the physical significance of quantum discord.
The variation of intensity as a function of monogamy score [57,58] of negativity ($\tau _{1:23}$) [56] is depicted in Fig. 3 at $k_BT=5\times 10^{-3}\hbar \Omega$. It is observed that, for higher monogamy score of negativity, superradiance is stronger. It represents shareability of entanglement (QCs) among entities, thus, as monogamy increases, the shareability of entanglement increases and thereby quantum coherence increases, explaining the superradiant intensity. This clearly shows the relevance of monogamy relations in a physical scenario. This may find application in quantum cryptography, as higher is the intensity observed from a system of array of atoms ($\geq 3$), more secure it is.
The behavior of photon-photon correlation ($g^2(0)$) as a function of observation angle and $\frac {\omega }{\Omega }$/$\frac {\lambda }{d}$ is depicted in Fig. 4. It can be clearly seen that photon statistics at higher temperature ($k_BT=\hbar \Omega$) is mostly super-Poissonian. It is evident from Fig. 4(c) that for “$\frac {\omega }{\Omega }=1$ and $k_BT=5\times 10^{-3}\hbar \Omega$” intensity pattern follows sub-Poissonian behavior for all values of $\frac {\lambda }{d}$ and $\theta$. It is to be noted that photons emitted from entangled sources display quantum nature at lower temperatures while at higher temperatures classical behavior is expected and show bunching of photons, even if quantum discord does not vanish. It is evident that photon-statistics can be controlled parametrically.
Figure 5(a) and (b) exhibit the radiance witness as a function of ratio of transition frequency and fixed dipole coupling ($\frac {\omega }{\Omega }$) and observation angle for two temperatures while Fig. 5(c) and (d) show the radiance witness as a function of observation angle and $\frac {\lambda }{d}$ at fixed $\frac {\omega }{\Omega }~(=1)$ for different temperatures. The value of $R<0$ indicates the subradiant behavior of radiance and $R>0$ corresponds to enhanced or superradiant behavior of radiance. In Fig. 5(a) $R=0$ corresponds to uncorrelated radiance which means all the quantum correlations are zero, as can be clearly seen in Fig. 2(b).
4. Conclusion
In conclusion, the emitted radiation from a coupled three particle system is shown to be a rich source of light of desired characteristics. Radiation intensity can be both uniform and highly focused in the far field regime in a controlled manner. The highly focused light owes its origin to quantum correlations and can find application for lithography [5] and other technological applications. We have obtained the exact expression for the radiation intensity and photon-photon correlation in the far field domain for three atoms at finite temperature. The photon-photon correlation demonstrates the sub-Poissonian (anti-bunching) and super-Poissonian (bunching) statistics of emitted photons and the fact that it can be controlled by tuning the system parameters. The effect of quantum correlations on emitted light, viz., concurrence, quantum discord, and monogamy score is explicitly demonstrated. The radiative behavior shows dramatic variation as a function of concurrence, quantum discord, and monogamy score of negativity, revealing the role of distinct QCs, thereby providing an optical probe for studying the quantum characteristics of emitting sources. At higher temperatures, radiance witness was found to exhibit sub and superradiant behavior of radiation intensity in the absence of entanglement albeit with non-zero quantum discord. This establishes the physical manifestation of quantum discord. Apart from revealing the physical signature of entanglement and quantum discord on the behavior of light, our investigation shows for the first time, the effect of three body correlation in the form of ‘monogamy score’ on sub and superradiance. Conditions under which hyper-radiance can be achieved is under investigation and also an extension of this study to open quantum systems.
Acknowledgements
Mithilesh K. Parit acknowledges discussion with Dr. Chiranjib Mitra and Department of Science and Technology, New Delhi, India, for providing the DST-INSPIRE fellowship during his stay at IISER Kolkata.
References
1. G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light, (Cambridge University Press, 2010).
2. G. S. Agarwal, Quantum Optics (Cambridge University Press, 2013).
3. A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, “Photons yield to peer pressure,” Nature 412(6850), 887–890 (2001). [CrossRef]
4. A. Zeilinger, “Light for the quantum. Entangled photons and their applications: a very personal perspective,” Phys. Scr. 92(7), 072501 (2017) and references therein. [CrossRef]
5. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000). [CrossRef]
6. M. Curty, T. Moroder, X. Ma, and N. Lutkenhaus, “Non-Poissonian statistics from Poissonian light sources with application to passive decoy state quantum key distribution,” Opt. Lett. 34(20), 3238–3240 (2009). [CrossRef]
7. M. Schiavon, G. Vallone, F. Ticozzi, and P. Villoresi, “Heralded single photon sources for QKD applications,” Phys. Rev. A 93(1), 012331 (2016). [CrossRef]
8. A. Poppe, A. Fedrizzi, R. Ursin, H. R. Bohm, T. Lorunser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express 12(16), 3865–3871 (2004). [CrossRef]
9. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum Cryptography with Entangled Photons,” Phys. Rev. Lett. 84(20), 4729–4732 (2000). [CrossRef]
10. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum Cryptography Using Entangled Photons in Energy-Time Bell States,” Phys. Rev. Lett. 84(20), 4737–4740 (2000). [CrossRef]
11. A. R. Usha Devi and A. K. Rajagopal, “Quantum target detection using entangled photons,” Phys. Rev. A 79(6), 062320 (2009). [CrossRef]
12. M. Padgett, E. Toninelli, T. Gregory, and P. A. Moreau, “Beating classical imaging limits with entangled photon,” Proc. SPIE 10934, 109341R (2019). [CrossRef]
13. J. P. W. MacLean, J. M. Donohue, and K. J. Resch, “Ultrafast quantum interferometry with energy-time entangled photons,” Phys. Rev. A 97(6), 063826 (2018). [CrossRef]
14. G. Araneda, D. B. Higginbottom, L. Slodicka, Y. Colombe, and R. Blatt, “Interference of Single Photons Emitted by Entangled Atoms in Free Space,” Phys. Rev. Lett. 120(19), 193603 (2018) and references therein. [CrossRef]
15. M. Hunault, H. Takesue, O. Tadanaga, Y. Nishida, and M. Asobe, “Generation of time-bin entangled photon pairs by cascaded second-order nonlinearity in a single periodically poled $LiNbO_3$ waveguide,” Opt. Lett. 35(8), 1239–1241 (2010). [CrossRef]
16. M. Rezai, J. Wrachtrup, and I. Gerhardt, “Polarization-entangled photon pairs from a single molecule,” Optica 6(1), 34–40 (2019). [CrossRef]
17. K. Y. Spasibko, D. A.. Kopylov, T. V. Murzina, G. Leuchs, and M. V. Chekhova, “Ring-shaped spectra of parametric downconversion and entangled photons that never meet,” Opt. Lett. 41(12), 2827–2830 (2016). [CrossRef]
18. R. Ma, W. Liu, Z. Qin, X. Su, X. Jia, J Zhang, and J. Gao, “Compact sub-kilohertz low-frequency quantum light source based on four-wave mixing in cesium vapor,” Opt. Lett. 43(6), 1243–1246 (2018). [CrossRef]
19. A. M. Branczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White, “Engineered optical nonlinearity for quantum light sources,” Opt. Express 19(1), 55–65 (2011). [CrossRef]
20. J. M. Lukens, O. D. Odele, D. E. Leaird, and A. M. Weiner, “Electro-optic modulation for high-speed characterization of entangled photon pairs,” Opt. Lett. 40(22), 5331–5334 (2015). [CrossRef]
21. Z. H. Xiang, J. Huwer, R. M. Stevenson, J. S. Szymanska, M. B. Ward, I. Farrer, D. A. Ritchie, and A. J. Shields, “Long-term transmission of entangled photons from a single quantum dot over deployed fiber,” Sci. Rep. 9(1), 4111 (2019). [CrossRef]
22. X. Zhang, C. Xu, and Z. Ren, “High fidelity heralded single-photon source using cavity quantum electrodynamics,” Sci. Rep. 8(1), 3140 (2018) and references therein. [CrossRef]
23. R. Brouri, A. Beveratos, J. P. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. 25(17), 1294–1296 (2000). [CrossRef]
24. G. Messin, J. P. Hermier, E. Giacobino, P. Desbiolles, and M. Dahan, “Bunching and antibunching in the fluorescence of semiconductor nanocrystals,” Opt. Lett. 26(23), 1891–1893 (2001). [CrossRef]
25. Z. Chen, Y. Zhou, and J. T. Shen, “Photon antibunching and bunching in a ring-resonator waveguide quantum electrodynamics system,” Opt. Lett. 41(14), 3313–3316 (2016). [CrossRef]
26. E. Suarez, D. Auwärter, T. J. Arruda, R. Bachelard, P. W. Courteille, C. Zimmermann, and S. Slama, “Photon-antibunching in the fluorescence of statistical ensembles of emitters at an optical nanofiber-tip,” New J. Phys. 21(3), 035009 (2019). [CrossRef]
27. R. Wiegner, J. von-Zanthier, and G. S. Agarwal, “Quantum-interference-initiated superradiant and subradiant emission from entangled atoms,” Phys. Rev. A 84(2), 023805 (2011). [CrossRef]
28. R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93(1), 99–110 (1954). [CrossRef]
29. R. Bonifacio, N. Piovella, and B. W. J. McNeil, “Superradiant evolution of radiation pulses in a free-electron laser,” Phys. Rev. A 44(6), R3441–R3444 (1991). [CrossRef]
30. E. Prat, L. Florian, and S. Reiche, “Efficient generation of short and high-power x-ray free-electron-laser pulses based on superradiance with a transversely tilted beam,” Phys. Rev. Spec. Top. Accel. Beams 18(10), 100701 (2015). [CrossRef]
31. P. Sprangle and T. Coffey, “New sources of high-power coherent radiation,” Phys. Today 37(3), 44–51 (1984). [CrossRef]
32. M. Gross, C. Fabre, P. Pillet, and S. Haroche, “Observation of Near-Infrared Dicke Superradiance on Cascading Transitions in Atomic Sodium,” Phys. Rev. Lett. 36(17), 1035–1038 (1976). [CrossRef]
33. A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015). [CrossRef]
34. I. Dimitrova, W. Lunden, J. Amato-Grill, N. Jepsen, Y. Yu, M. Messer, T. Rigaldo, G. Puentes, D. Weld, and W. Ketterle, “Observation of two-beam collective scattering phenomena in a Bose-Einstein condensate,” Phys. Rev. A 96(5), 051603 (2017). [CrossRef]
35. D. Bhatti, R. Schneider, S. Oppel, and J. von Zanthier, “Directional Dicke Subradiance with Nonclassical and Classical Light Sources,” Phys. Rev. Lett. 120(11), 113603 (2018). [CrossRef]
36. F. Mivehvar, S. Ostermann, F. Piazza, and H. Ritsch, “Driven-Dissipative Supersolid in a Ring Cavity,” Phys. Rev. Lett. 120(12), 123601 (2018). [CrossRef]
37. A. Biswas and G. S. Agarwal, “Preparation of W, GHZ, and two-qutrit states using bimodal cavities,” J. Mod. Opt. 51(11), 1627–1636 (2004). [CrossRef]
38. D. D. B. Rao, P. K. Panigrahi, and C. Mitra, “Teleportation in the presence of common bath decoherence at the transmitting station,” Phys. Rev. A 78(2), 022336 (2008). [CrossRef]
39. D. Porras and J. I. Cirac, “Collective generation of quantum states of light by entangled atoms,” Phys. Rev. A 78(5), 053816 (2008). [CrossRef]
40. M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature (London) 438(7069), 837–841 (2005). [CrossRef]
41. A. Kalachev and S. Kroll, “Coherent control of collective spontaneous emission in an extended atomic ensemble and quantum storage,” Phys. Rev. A 74(2), 023814 (2006). [CrossRef]
42. A. Kalachev, “Quantum storage on subradiant states in an extended atomic ensemble,” Phys. Rev. A 76(4), 043812 (2007). [CrossRef]
43. V. Vedral, “Quantum entanglement,” Nat. Phys. 10(4), 256–258 (2014) and references therein. [CrossRef]
44. C. Mitra, “Spin chains: Long-distance relationship,” Nat. Phys. 11(3), 212–213 (2015). [CrossRef]
45. D. Das, H. Singh, T. Chakraborty, R. K. Gopal, and C. Mitra, “Experimental detection of quantum information sharing and its quantification in quantum spin systems,” New J. Phys. 15(1), 013047 (2013). [CrossRef]
46. H. Singh, T. Chakraborty, P. K. Panigrahi, and C. Mitra, “Experimental estimation of discord in an antiferromagnetic Heisenberg compound $Cu(NO_3)_2.2.5H_2O$,” Quantum Inf. Process. 14(3), 951–961 (2015). [CrossRef]
47. D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’s Theorem, Quantum Theory, and Conceptions of the Universe (Kluwer Academics, 1989), pp. 73–76.
48. V. S. Bhaskara and P. K. Panigrahi, “Generalized concurrence and partial transpose for pure continuous variable systems of arbitrary degrees of freedom using Lagrange’s identity and wedge product,” Quantum Inf. Process. 16(5), 118 (2017) and references therein. [CrossRef]
49. S. Q. Tang, J. B. Yuan, L. M. Kuang, and X. W. Wang, “Quantum-discord triggered superradiance and subradiance in a system of two separated atoms,” Quantum Inf. Process. 14(8), 2883–2894 (2015). [CrossRef]
50. S. Banerjee, V. Ravishankar, and R. Srikanth, “Dynamics of entanglement in Two-Qubit Open System Interacting with a Squeezed Thermal Bath via Dissipative interaction,” Ann. Phys. 325(4), 816–834 (2010). [CrossRef]
51. S. Banerjee, V. Ravishankar, and R. Srikanth, “Entanglement dynamics in two-qubit open system interacting with a squeezed thermal bath via quantum nondemolition interaction,” Eur. Phys. J. D 56(2), 277–290 (2010). [CrossRef]
52. W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Phys. Rev. Lett. 80(10), 2245–2248 (1998). [CrossRef]
53. H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88(1), 017901 (2001). [CrossRef]
54. L. Henderson and V Vedral, “Classical, quantum and total correlations,” J. Phys. A: Math. Gen. 34(35), 6899–6905 (2001). [CrossRef]
55. S. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77(4), 042303 (2008). [CrossRef]
56. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002). [CrossRef]
57. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61(5), 052306 (2000). [CrossRef]
58. M. N. Bera, R. Prabhu, A. Sen(De), and U. Sen, “Characterization of tripartite quantum states with vanishing monogamy score,” Phys. Rev. A 86(1), 012319 (2012). [CrossRef]
59. M.-O. Pleinert, J. Von Zanthier, and G. S. Agarwal, “Hyperradiance from collective behavior of coherently driven atoms,” Optica 4(7), 779–785 (2017). [CrossRef]
60. J. P. Xu, S. L. Chang, Y. P. Yang, S. Y. Zhu, and G. S. Agarwal, “Hyperradiance accompanied by nonclassicality,” Phys. Rev. A 96(1), 013839 (2017). [CrossRef]
61. Q. Bin, X. -Y. Lu, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99(3), 033809 (2019). [CrossRef]
