Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise
We investigate the nearest level spacing statistics of open chaotic wave systems. To this end we derive the spacing distributions for the three Wigner ensembles in the one-channel case. The theoretical results give a clear physical meaning of the modifications on the spacing distributions produced by the coupling to the environment. Based on the analytical expressions obtained, we then propose general expressions of the spacing distributions for any number of channels, valid from weak to strong coupling. The latter expressions contain one free parameter. The surmise is successfully compared with numerical simulations of non-Hermitian random matrices and with experimental data obtained with a lossy electromagnetic chaotic cavity.
It is by now well established that classical chaos mani- fests itself in generic statistical properties of the corre- sponding quantum or wave systems. In the ideal case of closed systems, i.e., those whose coupling to the environ- ment can be neglected, the spectral and spatial statistics are well described by the random matrix theory (RMT) Mehta (2004). The statistics of chaotic wave systems coincide with the Gaussian orthogonal ensemble (GOE) if time-reversal symmetry (TRS) holds; with the Gaussian unitary en- semble (GUE) if TRS is broken; and with the Gaussian symplectic ensemble (GSE) for TRS systems with spin-1=2 interactions. Among all the statistical quantities used to analyze the spectral properties of closed systems, the nearest level spacing distribution PÃ°sÃ is certainly the most referred one (see Ref. Stöckmann (1999) for a review). The spacings distributions, surmised by Wigner using the 2-level approximation Wigner (1951), reads
where is the Dyson index labeling GOE (), GUE (), and GSE (). These distributions were first used to describe spectral statistics of heavy nuclei and later successfully employed to describe spectral statistics of a wide range of closed and weakly open chaotic systems (see Guhr et al. (1998) and references therein).
However, for systems of current interest such as quantum dots Beenakker (1997), nuclear compounds reactions Mitchell et al. (2010), micro-lasers cavities Shinohara et al. (2009); Bogomolny et al. (2011) or microwave billiards Stöckmann (1999); Hemmady et al. (2005); Poli et al. (2010); Dietz et al. (2010), the coupling to the environment must be explicitly taken into account. The open systems are then characterized by a set of resonances embedded in the continuum given by the poles of the -matrix Okolowicz et al. (2003). The poles are, in turn, the complex eigenvalues of the effective Hamiltonian:
where the Hermitian part is the Hamiltonian of the closed system giving rise to real energy levels and the anti-Hermitian part models the coupling to the environment in terms of scattering channels. The matrix contains the coupling amplitudes that connect the th level to the th scattering channel. The eigenvalues of the effective Hamiltonian are complex: , where and are, respectively, the eigenenergies and the resonance widths of the open system. Applying RMT to the effective Hamiltonian (2) (see Kuhl et al. (2005); Fyodorov et al. (2005) for recent reviews), the Hermitian part of is described by a Gaussian ensemble of the appropriate symmetry and the coupling amplitudes are considered as independent random Gaussian variables Savin et al. (2006); real for GOE, complex for GUE, and real quaternion for GSE, with zero mean and covariance: . Here is the coupling strength.
Using the above formalism, much progress has been achieved in understanding the statistical properties of the widths from weak Alt et al. (1995) to strong coupling Sommers et al. (1999); Kuhl et al. (2008) as well as the statistics of eigenvectors Schomerus et al. (2000); Poli et al. (2009a). However, to the best of our knowledge, there are still not general analytical expressions for the distributions of the spacings for open chaotic systems. This is so, even if analytical results for interesting particular cases have been reported Stöckmann and Šeba (1998); Mizutori and Zelevinsky (1993); Shchedrin and Zelevinsky (2011) and progresses have been done in analyzing the spacing distribution of the resonances in the complex plane Grobe et al. (1988); Fyodorov et al. (1997). What is missing then is the counterpart of the Wigner surmise for open chaotic wave systems. In this Letter, we achieve this goal by deriving analytically the probability distributions of the spacings for the 3 Gaussian ensembles for the 2-level model and the one-channel case. Then, we extend those results to the -level model and to any number of channels considering the coupling strength as a free parameter.
To derive the spacing distributions we start with the joint energy distribution , first obtained for GOE by Sokolov and Zelevinsky Sokolov and Zelevinsky (1989) and then for the 3 ensembles by Stöckmann and Šeba Stöckmann and Šeba (1998). In analogy with the derivation of the Wigner surmise, we assume that the 2-level approximation holds. Specializing Eq. (4.4) of Stöckmann and Šeba (1998), the joint energy distribution reads
where fixes the mean level of the closed system: . and respectively, correspond to the weak and strong coupling regimes. Changing to variables and in (3) and integrating over one gets
To go further let us consider the 3 ensembles separately.
For the GOE case, introducing in (4) the new variables and , the integration over can be done. With the proper normalization constant, the distribution yields
where is the modified Bessel function of first kind.
For the GUE case, the exponential prefactor can be written in terms of the partial derivatives of and , where and are generated by and , respectively. The distribution can then be written as
The integrations over and are now straightforward and give rise to
where is the exponential integral. Operating (7) with and including the normalization constant we finally arrive at
where . Applying exactly the same method to GSE, one gets
Note that the distributions (5), (8) and (Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise) tend to the Wigner surmise (1) in the limit of vanishing coupling . The Gaussian tail , proper of the closed systems (1), remains and the Wigner prefactor now becomes some function of : . These expressions reveal also that the effects of the coupling show up only in the prefactor. Thus, the main modifications occur at small spacings.
In particular, there appears a finite probability for null spacings: . In other words, the level repulsion, a main feature of the Wigner surmise, is suppressed for open systems. This effect is associated to the phenomenon of attraction of the levels along the real axis induced by the coupling to the environment Poli et al. (2009b). For the GUE case is a polynomial containing the Wigner term and a constant term. For the GSE case, in addition to the Wigner term and the constant term, a quadratic term appears too. The presence of this quadratic term can be viewed as a consequence of the breaking of the TRS, since it characterizes the behavior at small spacings for closed systems with broken TRS.
It is important to note that in the limit of strong coupling we obtain a Gaussian distribution . This behavior, intrinsic of the 2-level approximation, does not correspond to the -level model where the Wigner surmise is recovered Stöckmann and Šeba (1998). Indeed, the 2-level model expressions are valid only up to the intermediate coupling regime. However, we will show that if is considered now as a free parameter i.e., an effective coupling strength called , the distributions (5), (8) and (Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise) are valid for any coupling strength and, moreover, for any number of channels. To confirm the validity of our proposal and to analyze how evolves with the coupling and the number of channels, we carried out numerical simulations of non-Hermitian random matrices defined by (2). The effective coupling strength was determined by a least-square algorithm; being fixed by the usual normalization condition . The simulations were performed with ensembles of 150 non-Hermitian of size varying the mean width in the range and for , 3, 5 and 10.
A comparison between the expressions (5), (8) and (Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise) and the numerical results is presented in Fig. 1. For all coupling values and number of channels analyzed the confidence level is larger than 99.5%. We stress that the excellent agreement with the numerical simulations confirms that the Wigner generalizations Eqs. (5), (8) and (Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise), being exact for the 2-level model and , are excellent effective formulas for arbitrary coupling strength, number of levels and channels.
Using the best fit procedure mentioned above for the effective coupling strength, we present in Fig. 2 the evolution of for the GOE case as a function of the mean width. We can distinguish two regimes. For weak enough coupling (), increases quite rapidly with the mean width. For strong coupling, the reverse occurs and we go back to the Wigner surmise as predicted in Stöckmann and Šeba (1998) for the one-channel case. Note that the approach to the Wigner surmise also occurs for (and ), albeit extremely slow. A similar behavior was observed for the GUE and GSE cases.
Furthermore, we compare the theoretical prediction for GOE, Eq. (5), with recent experimental data obtained for an open chaotic 2D microwave cavity displaying TRS in the closed limit. The data were provided to us kindly by U. Kuhl Kuhl et al. (2008). In that experiment, the coupling to the environment is produced by the finite conductivity of the wall (modeled by fictitious open channels) and one antenna attached to the cavity to perform the measurements. Fig. 3 shows the experimental distribution of the spacings for two different ranges of frequency i.e., for two different coupling strengths and number of coupling channels. The agreement is good with a confidence level larger that in both cases.
In summary, we have proposed general expressions for the distributions of the nearest level spacings of open chaotic wave systems, where the internal Hamiltonian belongs to one of the three well-known classes of Gaussian Ensembles. We stress again that the expressions are exact for and and otherwise excellent approximations considering the coupling strength as a free parameter. The expressions indicated that coupling to the environment modified, with respect to the Wigner surmise, the form of the cofactors from to some function of . Our predictions, valid from the weak to the strong coupling regime were shown to be in excellent agreement with numerical simulations of non-Hermitian random matrices and with experimental data from a microwave cavity. In essence, the set of distributions (5), (8) and (Nearest level spacing statistics in open chaotic systems: A generalization of the Wigner Surmise) may be viewed as the extension of the Wigner surmise to open chaotic systems.
Acknowledgements.We thank, U. Kuhl who supplied to us the experimental data of the microwave cavity, R. Mendez-Sanchez and D. V. Savin for fruitful discussions and T. H. Seligman and the Centro International de Ciencias (Cuernavaca) where the idea to this paper was developed. G. A. L.-A. acknowledges partial support from CONACYT Ref. No. 51458.
- Mehta (2004) M. L. Mehta, Random Matrices (Elsevier, San Diego, USA, 2004).
- Stöckmann (1999) H.-J. Stöckmann, Quantum Chaos: an introduction (Cambridge University Press, Cambridge, U. K., 1999).
- Wigner (1951) E. P. Wigner, in Proc. Cambridge Philos. Soc (1951), vol. 47, p. 790.
- Guhr et al. (1998) T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Phys. Rep. 299, 189 (1998).
- Beenakker (1997) C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
- Mitchell et al. (2010) G. E. Mitchell, A. Richter, and H. A. Weidenmüller, Rev. Mod. Phys. 82, 2845 (2010).
- Shinohara et al. (2009) S. Shinohara, M. Hentschel, J. Wiersig, T. Sasaki, and T. Harayama, Phys. Rev. A 80, 031801 (2009).
- Bogomolny et al. (2011) E. Bogomolny, N. Djellali, R. Dubertrand, I. Gozhyk, M. Lebental, C. Schmit, C. Ulysse, and J. Zyss, Phys. Rev. E 83, 036208 (2011).
- Hemmady et al. (2005) S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, Phys. Rev. E 71, 056215 (2005).
- Poli et al. (2010) C. Poli, O. Legrand, and F. Mortessagne, Phys. Rev. E 82, 055201(R) (2010).
- Dietz et al. (2010) B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A. Weidenmüller, Phys. Rev. E 81, 036205 (2010).
- Okolowicz et al. (2003) J. Okolowicz, M. Ploszajczak, and I. Rotter, Phys. Rep. 374, 271 (2003).
- Kuhl et al. (2005) U. Kuhl, H.-J. Stöckmann, and R. L. Weaver, J. Phys. A 38, 10433 (2005).
- Fyodorov et al. (2005) Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers, J. Phys. A 38, 10731 (2005).
- Savin et al. (2006) D. V. Savin, O. Legrand, and F. Mortessagne, Europhys. Lett. 76, 774 (2006).
- Alt et al. (1995) H. Alt, H. D. Gräf, H. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt, and H. A. Weidenmüller, Phys. Rev. Lett. 74, 62 (1995).
- Sommers et al. (1999) H.-J. Sommers, Y. V. Fyodorov, and M. Titov, J. Phys. A 32, L77 (1999).
- Kuhl et al. (2008) U. Kuhl, R. Höhmann, J. Main, and H.-J. Stöckmann, Phys. Rev. Lett. 100, 254101 (2008).
- Schomerus et al. (2000) H. Schomerus, K. M. Frahm, M. Patra, and C. W. J. Beenakker, Physica A 278, 469 (2000).
- Poli et al. (2009a) C. Poli, D. V. Savin, O. Legrand, and F. Mortessagne, Phys. Rev. E 80, 046203 (2009a).
- Stöckmann and Šeba (1998) H. Stöckmann and P. Šeba, J. Phys. A 31, 3439 (1998).
- Mizutori and Zelevinsky (1993) S. Mizutori and V. G. Zelevinsky, Z. Phys. A 346, 1 (1993).
- Shchedrin and Zelevinsky (2011) G. Shchedrin and V. V. Zelevinsky, preprint arXiv:1112.4919v2 (2011).
- Grobe et al. (1988) R. Grobe, F. Haake, and H.-J. Sommers, Phys. Rev. Lett. 61, 1899 (1988).
- Fyodorov et al. (1997) Y. V. Fyodorov, B. Khoruzhenko, and H.-J. Sommers, Phys. Rev. Lett. 79, 557 (1997).
- Sokolov and Zelevinsky (1989) V. V. Sokolov and V. G. Zelevinsky, Nucl. Phys. A 504, 562 (1989).
- Poli et al. (2009b) C. Poli, B. Dietz, O. Legrand, F. Mortessagne, and A. Richter, Phys. Rev. E 80, 035204(R) (2009b).