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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 7 — Mar. 29, 2010
  • pp: 7092–7100
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Rigorous criterion for characterizing correlated multiphoton emissions

Hyun-Gue Hong, Hyunchul Nha, Jai-Hyung Lee, and Kyungwon An  »View Author Affiliations


Optics Express, Vol. 18, Issue 7, pp. 7092-7100 (2010)
http://dx.doi.org/10.1364/OE.18.007092


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Abstract

Strong correlation of photons, particularly in the single-photon regime, has recently been exploited for various applications in quantum information processing. Existing correlation measurements, however, do not fully characterize multi-photon correlation in a relevant context and may pose limitations in practical situations. We propose a conceptually rigorous, but easy-to-implement, criterion for detecting correlated multi-photon emission out of a quantum optical system, drawn from the context of wavefunction collapse. We illustrate the robustness of our approach against experimental limitations by considering an anharmonic optical system.

© 2010 Optical Society of America

1. Introduction

Recently, interest in the correlation effect has also been extended to multi-photon level in the context of multi-photon gateway, where a random (Poissonian) stream of photons can be converted into a bunch of temporally correlated n photons. In particular, Kubanek et al. demonstrated the operation of two-photon gateway to some extent using an optical cavity QED system [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

]. In view of all these efforts, it seems very crucial to have a theoretical framework that can appropriately characterize multi-photon correlations [13

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]

, 14

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008). [CrossRef]

, 15

15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009). [CrossRef]

, 16

16. L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999). [CrossRef]

], e.g., n-photon blockade effect, and desirably that can be efficiently tested in experiment.

Conventionally, correlation of photons is measured by the nth-order coherence functions introduced by Glauber [17

17. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963). [CrossRef]

], g (n)(0) = 〈a n a n〉/〈a an, where a (a ) is the annihilation (creation) operator of an optical field. However, it is noted in the recent experiments of cavity QED [2

2. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]

, 3

3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef] [PubMed]

, 12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

] that g (2)(0) is not effective to resolve the correlated two photon emission due to a huge bunching at the atom-cavity bare resonance overshadowing the two-photon resonance. Furthermore, as n goes beyond two, g (n)(0) or its simple variants [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

] contain more peaks at k= 1, ⋯,n - 1 photon resonances, irrelevant to genuine n-photon correlation, as to be shown below.

Instead one may take n-photon excitation peaks in 〈a n an〉 spectrum itself, e.g. in [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

, 13

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]

,14

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008). [CrossRef]

, 15

15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009). [CrossRef]

, 16

16. L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999). [CrossRef]

], as a confirming evidence of n-photon correlations. Rigorously speaking, however, the multi-photon resonant excitation peaks spectroscopically identified only uncover the energy-level structure of the system. Whether each peak in the bare coincidence 〈a n an〉 indicates relevant photon correlation must be checked very carefully. For example, the three-photon coincidence tends to increase in the spectrum without any correlation if the system possesses two-photon correlation since an uncorrelated emission added to a correlated pair may register another three-photon coincidence. Therefore, we need to consider a stricter physical context for characterizing correlated emission of n-photons. Although there have been several studies on higher-order photon statistics in view of nonclassicality [18

18. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990). [CrossRef] [PubMed]

, 19

19. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A , 2137–15 (1996). [CrossRef]

], none of them carries a clear interpretation as multiphoton correlation.

2. The criterion

In order to envision a generic, though not exhaustive, scenario where multiphoton correlations may arise, let us compare two systems, one with harmonic and the other with anharmonic level structure [Fig.1 (a)]. When a harmonic system with level spacing h̄ω 0 is excited by an external driving on resonance (ωL = ω 0), all energy levels are equally accessible. On the other hand, for an anharmonic system, if the external field is n-photon resonant with the nth level, other levels than nth would not be substantially addressed by the external field. As a result, the system could be excited to contain only n correlated quanta and further excitation would be prohibited—n-quanta (photon) blockade effect. We will apply a similar line of reasoning to emission, rather than excitation, process. Specifically, we construct a criterion to detect ‘pure’ n-photon correlated emission by incorporating two distinct features, (i) surge or rapid emission of photons up to n quanta and (ii) blockade beyond n, which can be applied to any quantum optical systems, not necessarily anharmonic ones.

2.1. Photon surge

Generally, the photo-detection rate 𝓡 is proportional to the intensity of the optical field under consideration, 𝓡 ∝ 〈𝓔^ -𝓔+〉, where the operators 𝓔± correspond to the positive- and the negative-frequency part of the field. Let us assume that a quantum system can be described by a pure steady state ∣Ψ〉s for simplicity, but our argument applies equally well to mixed states. If it has emitted n - 1 quanta, the wavefunction is collapsed to ∣Ψc (n-1)〉 = 𝓔^ n-1 +∣Ψ〉s conditioned on these emissions. The detection rate for the succeeding nth photon is then given by

𝓡nΨc(n1)𝓔̂𝓔̂+Ψc(n1)Ψc(n1)Ψc(n1)=𝓔̂n𝓔̂+n𝓔̂n1𝓔̂+n1
(1)

after the normalization of the conditional state ∣Ψc (n-1)〉. Specifically, if the emission out of the system is a bunch of highly correlated n-photons, the second photon will be emitted right after the first photon and the third photon after the second, and so on. This idea can be used to construct our criterion as follows.

Rk,k1𝓡k𝓡k1=𝓔̂k𝓔̂+k𝓔̂k2𝓔̂+k2𝓔̂k1𝓔̂+k12>1,(k=2,,n),
(2)

which must be satisfied for each k = 2,…,n.

2.2. Photon blockade

Rk,k1<1,(k=n+1,).
(3)

The fulfillment of all the surge and the blockade conditions in Eqs. (2) and (3) respectively constitutes our criterion for n-photon correlated emission. Note that the condition (3) coincides with the special case of the higher-order antibunching criteria introduced in [18

18. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990). [CrossRef] [PubMed]

] for the non-classicality of photon statistics, rather than the correlation effect.

In our criterion, it is crucial to use the conditional rates 𝓡k, rather than the bare rates 〈𝓔^ k - 𝓔^ k +〉, as the former takes into account the correlation between adjacent emissions in a stronger sense. However, the resulting criterion does not require any conditional measurements. Instead, the quantities Rk,k-1 in Eqs. (2) and (3) simply involve various photon-coincidence rates and we particularly note that the numerator and the denominator are in the same order of the field strength. It is thus given in an experimentally desirable form, that is, insensitive to the quantum efficiency of photodetectors.

2.3. Measure of multi-photon correlation

The conditions in Eqs. (2) and (3) may be used to define a quantitative measure of n of n-photon correlation as

𝓜nk=2nmax{Rkk11,0}k=n+1Ntrmax{Rkk111,0},
(4)

where N tr is a truncated excitation number to be taken appropriate to a given situation. 𝓜n quantifies the strength of the n-photon correlation by measuring the deviation of Rk,k-1 from unity in the surge and the blockade conditions of Eqs. (2) and (3), respectively, and returns a nonzero value only when all those conditions are satisfied. To experimentally obtain 𝓜n for a given system, one first measures the bare k-photon coincidence rates 〈𝓔^ k - 𝓔^ k +〉 for all k = 1,…, N tr. Then, each conditional rate Rk,k-1 defined by Eq. (2) is evaluated and plugged in to Eq. (4) to determine the value of 𝓜n.

2.4. Remarks

(a) Conventionally, multi-photon correlations have been discussed in terms of the Glauber coherence functions

g(n)𝓔̂(x1)𝓔̂(x2)𝓔̂(xn)𝓔̂+(xn)𝓔̂+(x2)𝓔̂+(x1)𝓔̂(x1)𝓔̂+(x1)𝓔̂(x2)𝓔̂+(x2)𝓔̂(xn)𝓔̂+(xn)

(xi: a general space-time point) [17

17. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963). [CrossRef]

]. The context of correlation in g (n) is, however, rather limited and we particularly note that g (n) compares the n-photon coincidence rate (numerator) only with the single-photon counting rates (denominator). Large (small) value of g (n) characterizes a bunching (antibunching) effect with no strict n-photon correlation that can emerge even in a classical scattering system, e.g. g (n) = n! for a thermal light (Hanbury-Brown–Twiss effect [21

21. R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]

, 22

22. M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009). [CrossRef] [PubMed]

]). Another example of g (2) ≫ 1 with no rigorous two-photon correlation will be shown below in Sec.3.

akakak2ak2=d2αα2kP(α)d2αα2k4P(α)
(d2αα2k2P(α))2=ak1ak22.
(5)

The violation of the above inequality, which is nothing but the blockade condition Rk,k-1 < 1 in Eq. (3), is thus a clear signature of nonclassicality. So our criteria of multiphoton correlation can be fulfiled only by nonclassical sources. We emphasize that, in the so called multiphoton antibunching criteria in [18

18. C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990). [CrossRef] [PubMed]

, 19

19. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A , 2137–15 (1996). [CrossRef]

], the focus was made on how to reveal nonclassicality of the field by a mathematical approach based on the positive Glauber-P function, thus lacking a clear interpretation as multiphoton correlation.

3. Application: Cavity QED system

3.1. Model

To illustrate our criterion, we consider a cavity QED system—one of the well known anharmonic systems that can be implemented in various experimental platforms [2

2. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]

, 3

3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef] [PubMed]

, 4

4. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008). [CrossRef] [PubMed]

, 14

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008). [CrossRef]

]. A qubit (two-state atom, quantum dot, etc.) is coupled to a single mode field driven by a classical field. For simplicity we investigate the on-resonance case, ωA = ωcω 0, where ωA is the qubit transition frequency and ωc the cavity resonance frequency. The qubit-cavity system at coupling strength g is then described by the Hamiltonian

H=h¯ω0(aa+12σz)+ih¯g(aσ+),
(6)

where σ ± and σz are the Pauli pseudospin operators. The composite system has the ground state ∣0,g〉 with the energy E 0 = 0 and the polaritonic excited states Ψ±n=12(n,g±n1,e)withEn,±=nh¯ ω0±h¯gn(n=1,2) [See Fig.1 (b)]. Therefore, when the system is driven by an external field at frequency ωL, n-photon resonant absorption may occur [23

23. Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000). [CrossRef]

] at

nh¯ωL=En,±=nh¯ω0±h¯gn.
(7)
Fig. 1. (a) Energy-level diagram for (i) harmonic and (ii) anharmonic system. (b) Energy level structure for cavity QED system. (c) multiphoton coincidence rates 〈a n an〉 as a function of δ/g for 2κ/g = γ/g = 0.01 with 𝓔/κ = 0.1. The dotted vertical lines represent the locations of the multiphoton resonances, δ = ±g/√n throughout Figs. 1–3.

In practical situations, the qubit and the cavity field may interact with Markovian environments, which causes dissipation and decoherence to the system. The global evolution is then governed by the master equation ρ˙=1ih¯[HI,ρ]+γ(σρσ+12σ+σρ12ρσ+σ)+κ(2aρaaaρρaa), where γ (2κ) is the qubit (cavity) decay rate and the interaction Hamiltonian HI=h¯δ(aa+12σz)+ih¯g(aσaσ+)+ih¯𝓔(aa),, with the driving strength 𝓔 and the detuning δω 0 - ωL.

3.2. Correlation measures

Instead, if one measures the Glauber coherence function g (n) of the output, the result may characterize the correlation of emitted photons to some extent, but not in a full rigorous sense. In particular, g (n)(0) = 〈a n an〉/〈a an in Fig. 2(a) shows a large bunching at zero detuning δ = 0, which has nothing to do with genuine n-photon correlation as we will clearly show below. Close inspection of photon statistics reveals that the system does exhibit some non-classical behavior at δ = 0, e.g. the oscillation of conditional detection rate 𝓡k which peaks at even number of k, but it is not a rigorous n-photon correlation at any level n in view of our criterion. To get rid of this “cumbersome” resonance effect observed at δ = 0 that may overwhelm the other resonance peaks, Kubanek et al. introduced the differential correlation function, C (2)(0) = 〈a †2 a 2〉 - 〈a a2, that measures the absolute occurrence of two-photon excitation with respect to the single-photon excitation [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

]. The context in this correlation function, however, is insufficient just like g 2 (0) in general, although it was instrumental to identify the second resonant peak in [12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

]. Furthermore, a generalization to n-photon level, C (n)(0) = 〈a n an〉 - 〈aan for n ≥ 3, becomes hardly effective in identifying the higher-order peaks by the broadening effect in the realistic regime [Fig. 2 (a)–(c)].

Fig. 2. (a)–(c) The conventional correlation functions g (n) (0) by Glauber and C (n) (0) by Kubanek et al. [12]. (d)–(f) The quantitative measure 𝓜n together with conditional relative rates Rk,k-1. The truncation numbers used are (d) N tr = 4, (e) and (f) N tr = 5. In all plots, 2κ/g=γ/g=𝓔/K = 0.1.

In contrast, our criterion not only detects correlated emission in a well-defined context, but also provides a practical tool to identify the multi-photon resonance structure of a system in realistic situations. In Figs. 2(d), 2(e), and 2(f), we plot the quantitative measure 𝓜n of Eq. (4) for n = 2,3, and 4 along with various rates Rk,k-1 which are ingredients for constructing the corresponding 𝓜n. Remarkably, 𝓜n yields a positive value only in the spectral vicinity of the resonant peaks δ = ±g/ √n. The “spurious” peak at δ = 0 disappears by our criterion, which rigorously confirms that this seeming “resonance” indeed does not represent pure n-photon correlated emission.

Fig. 3. Comparison between g (n)(0) (black dotted curve) and our measure 𝓜n (red solid curve) for (a) n = 2 and (b) n = 3, with the truncation numbers (a) N tr - 4 and (b) N tr = 5, respectively. The driving intensity is rather high, 𝓔/κ = 1, with the coupling condition 2κ/g=γ/g=0.1.

3.3. Large driving field

As we increase the pump strength to obtain more substantial signal, the coincidence spectrum usually becomes difficult to resolve due to the saturation of the system. Our method is, however, still useful for moderately strong pumping owing to the rigorous context established in it. To demonstrate this merit, we have considered the case of a large driving field 𝓔κ = 1 in Fig. 3, together with the realistic coupling γ/g = 2κ/g = 0.1. Due to the intensity-dependent broadening effect, the Glauber function g (n)(0) no longer shows noticeable marks of resonance, except for the peak at δ = 0 overwhelming the entire shape in the spectrum. On the other hand, our measure 𝓜n identifies a clear signature of multi-photon correlations under the same condition. This capability would allow one to increase the pump strength to some extent, and thereby easing the difficulty of having to measure higher-order coincidence than n [i.e., blockade conditions in Eq. (3)] to identify n-photon correlation in our method. Furthermore, we have also checked that other possible broadening effects, e.g. atomic motion in the cavity, do not degrade the capability of our criterion for characterizing multiphoton correlations. We attribute this robustness against experimental imperfections to the rigorous context established with the measure 𝓜n.

4. Conclusion

In conclusion, we have devised an easy-to-implement criterion for detecting correlated multi-photon emission, imposing surge and blockade requirements in photoemission processes. A quantitative measure 𝓜n has been derived from the correlation context between successive photon emissions in the framework of wavefunction collapse. Our criterion applies to any quantum optical systems, including the ones with anharmonic structure (cavity QED systems, multi-level atoms, etc.).

We have illustrated our method can efficiently detect multi-photon correlations at the resonant peaks of the cavity QED system in contrast to the existing correlation functions. Note that the anharmonic spectrum which scales as √n is a clear signature of quantum nature of light field [24

24. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990). [CrossRef] [PubMed]

, 25

25. H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996). [CrossRef] [PubMed]

], and it thus has been of considerable interest for long but experimentally verified only recently [13

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]

, 14

14. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008). [CrossRef]

, 15

15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009). [CrossRef]

]. In the optical cavity-QED system [3

3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef] [PubMed]

, 4

4. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008). [CrossRef] [PubMed]

, 12

12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

, 13

13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]

], it becomes harder to directly observe this anharmonicity in higher-orders due to less strong coupling than in the microwave circuit QED system, but our method remarkably makes it possible to clearly pick up the √n-dependence despite experimental limitations. We anticipate that our conceptually rigorous approach can also be useful in addressing correlation effects in other quantum systems beyond optics.

Acknowledgments

HN is grateful to H. J. Carmichael for helpful discussions and remarks. This work was supported by NRL and WCU Grants. HN was supported by the NPRP grant 08-043-1-011 from Qatar National Research Fund.

References and links

1.

D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007) [CrossRef]

2.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]

3.

K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef] [PubMed]

4.

B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A photon turnstile dynamically regulated by one atom,” Science 3191062–1065 (2008). [CrossRef] [PubMed]

5.

M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006); M. J. Hartmann and M. B. Plenio, “Strong photon nonlinearities and photonic Mott insulators,” Phys. Rev. Lett. 99, 103601 (2007). [CrossRef]

6.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006). [CrossRef]

7.

D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805(R) (2007). [CrossRef]

8.

N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, “Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities,” Phys. Rev. A 77, 031803(R) (2008). [CrossRef]

9.

D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008). [CrossRef]

10.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997). [CrossRef]

11.

L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992). [CrossRef] [PubMed]

12.

A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, “Two-photon gateway in one-atom cavity quantum electrodynamics,” Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]

13.

I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr, and G. Rempe, “Nonlinear spectroscopy of photons bound to one atom,” Nat. Phys. 4, 382–385 (2008). [CrossRef]

14.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system,” Nature (London) 454, 315–318 (2008). [CrossRef]

15.

L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,” Nat. Phys. 5, 105–109 (2009). [CrossRef]

16.

L. Horvath, B. C. Sanders, and B. F. Wielinga, “Multiphoton coincidence spectroscopy,” J. Opt. B: Quantum Semiclassic. Opt. 1446–451 (1999). [CrossRef]

17.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963). [CrossRef]

18.

C. T. Lee, “Higher-order criteria for nonclassical effects in photon statistics,” Phys. Rev. A 411721–1723 (1990). [CrossRef] [PubMed]

19.

D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A , 2137–15 (1996). [CrossRef]

20.

H. J. Carmichael, R. J. Brecha, and P. R. Rice, “Quantum interference and collapse of the wavefunction in cavity QED,” Opt. Comm. 82, 73–79 (1991). [CrossRef]

21.

R. Hanbury-Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]

22.

M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, “Higher-order photon bunching in a semiconductor microcavity,” Science 325, 297–300 (2009). [CrossRef] [PubMed]

23.

Y.-T Chough, H.-J. Moon, H. Nha, and K. An, “Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder,” Phys. Rev. A 63, 013804 (2000). [CrossRef]

24.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990). [CrossRef] [PubMed]

25.

H. J. Carmichael, P. Kochan, and B. C. Sanders, “Photon correlation spectroscopy,” Phys. Rev. Lett. 77, 631–634 (1996). [CrossRef] [PubMed]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.4180) Quantum optics : Multiphoton processes
(270.5290) Quantum optics : Photon statistics
(270.5580) Quantum optics : Quantum electrodynamics

ToC Category:
Quantum Optics

History
Original Manuscript: February 3, 2010
Revised Manuscript: March 16, 2010
Manuscript Accepted: March 20, 2010
Published: March 23, 2010

Citation
Hyun-Gue Hong, Hyunchul Nha, Jai-Hyung Lee, and Kyungwon An, "Rigorous criterion for characterizing correlated multiphoton emissions," Opt. Express 18, 7092-7100 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7092


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References

  1. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, "A single-photon transistor using nanoscale surface plasmons," Nat. Phys. 3, 807-812 (2007) [CrossRef]
  2. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, "Coherent generation of non-classical light on a chip via photon-induced tunnelling and blockade," Nat. Phys. 4, 859-863 (2008). [CrossRef]
  3. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, "Photon blockade in an optical cavity with one trapped atom," Nature 436, 87-90 (2005). [CrossRef] [PubMed]
  4. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, "A photon turnstile dynamically regulated by one atom," Science 319, 1062-1065 (2008). [CrossRef] [PubMed]
  5. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, "Strongly interacting polaritons in coupled arrays of cavities," Nat. Phys. 2, 849-855 (2006). [CrossRef]
  6. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, "Quantum phase transitions of light," Nat. Phys. 2, 856-861 (2006). [CrossRef]
  7. D. G. Angelakis, M. F. Santos, and S. Bose, "Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays," Phys. Rev. A 76, 031805(R) (2007). [CrossRef]
  8. N. Na, S. Utsunomiya, L. Tian, and Y. Yamamoto, "Strongly correlated polaritons in a two-dimensional array of photonic crystal microcavities," Phys. Rev. A 77, 031803(R) (2008). [CrossRef]
  9. D. E. Chang, V. Gritsev, G. Morigi, V. Vuletic, M. D. Lukin, and E. A. Demler, "Crystallization of strongly interacting photons in a nonlinear optical fibre," Nat. Phys. 4, 884-889 (2008). [CrossRef]
  10. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, "Strongly interacting photons in a nonlinear cavity," Phys. Rev. Lett. 79, 1467-1470 (1997). [CrossRef]
  11. L. Tian and H. J. Carmichael, "Quantum trajectory simulations of two-state behavior in an optical cavity containing one atom," Phys. Rev. A 46, R6801-R6804 (1992). [CrossRef] [PubMed]
  12. A. Kubanek, A. Ourjoumtsev, I. Schuster, M. Koch, P. W. H. Pinkse, K. Murr, and G. Rempe, "Two-photon gateway in one-atom cavity quantum electrodynamics," Phys. Rev. Lett. 101, 203602 (2008). [CrossRef] [PubMed]
  13. I. Schuster, A. Kubanek, A. Fuhrmanek, T. Puppe, P. W. H. Pinkse, K. Murr and G. Rempe, "Nonlinear spectroscopy of photons bound to one atom," Nat. Phys. 4, 382-385 (2008). [CrossRef]
  14. J. M. Fink, M. G¨oppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, "Climbing the Jaynes-Cummings ladder and observing its √ n nonlinearity in a cavity QED system," Nature (London) 454, 315-318 (2008). [CrossRef]
  15. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, "Nonlinear response of the vacuum Rabi resonance," Nat. Phys. 5, 105-109 (2009). [CrossRef]
  16. L. Horvath, B. C. Sanders, and B. F. Wielinga, "Multiphoton coincidence spectroscopy," J. Opt. B: Quantum Semiclassic. Opt. 1, 446-451 (1999). [CrossRef]
  17. R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963). [CrossRef]
  18. C. T. Lee, "Higher-order criteria for nonclassical effects in photon statistics," Phys. Rev. A 41, 1721-1723 (1990). [CrossRef] [PubMed]
  19. D. N. Klyshko, "Observable signs of nonclassical light, " Phys. Lett. A  213, 7-15 (1996). [CrossRef]
  20. H. J. Carmichael, R. J. Brecha, and P. R. Rice, "Quantum interference and collapse of the wavefunction in cavity QED," Opt. Commun. 82, 73-79 (1991). [CrossRef]
  21. R. Hanbury-Brown and R. Twiss, "Correlation between photons in two coherent beams of light," Nature 177, 27-29 (1956). [CrossRef]
  22. M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam, "Higher-order photon bunching in a semiconductor microcavity," Science 325, 297-300 (2009). [CrossRef] [PubMed]
  23. Y.-T. Chough, H.-J. Moon, H. Nha, and K. An, "Single-atom laser based on multiphoton resonances at far-off resonance in the Jaynes-Cummings ladder," Phys. Rev. A 63, 013804 (2000). [CrossRef]
  24. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, "Vacuum Rabi splitting as a feature of linear-dispersion theory: Analysis and experimental observations," Phys. Rev. Lett. 64, 2499-2502 (1990). [CrossRef] [PubMed]
  25. H. J. Carmichael, P. Kochan, and B. C. Sanders, "Photon correlation spectroscopy," Phys. Rev. Lett. 77, 631-634 (1996). [CrossRef] [PubMed]

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