85, VOLUME NUMBER 12
PHYSICAL REVIEW LETTERS
18 SEPTEMBER 2000
Measurement of Wave Chaotic Eigenfunctions in the Time-Reversal Symmetry-Breaking Crossover Regime
Seok-Hwan Chung, Ali Gokirmak, Dong-Ho Wu, J. S. A. Bridgewater, E. Ott, T. M. Antonsen, and Steven M. Anlage
Department of Physics, University of Maryland, College Park, Maryland 20742-4111 (Received 27 October 1999) We present experimental results on eigenfunctions of a wave chaotic system in the continuous crossover regime between time-reversal symmetric and time-reversal symmetry-broken states. The statistical properties of the eigenfunctions of a two-dimensional microwave resonator are analyzed as a function of an experimentally determined time-reversal symmetry-breaking parameter. We test four theories of onepoint eigenfunction statistics and introduce a new theory relating the one-point and two-point statistical properties in the crossover regime. We also find a universal correlation between the one-point and twopoint statistical parameters for the crossover eigenfunctions.
PACS numbers: 05.45.Mt, 03.65.Sq, 11.30.Er, 84.40.Az
Many complex quantum systems whose underlying classical behavior is chaotic can be described by treating their Hamiltonian matrix elements as random numbers which fluctuate around zero with a Gaussian distribution. There are universal statistical properties of the eigenvalues and eigenfunctions of these random matrices which depend only on the symmetries of the Hamiltonian. For instance, random matrix theory has been shown to be consistent with the statistical properties of nuclei [1], molecules [2], and two-dimensional quantum dots [36]. In the case of twodimensional quantum mechanical systems with classically chaotic dynamics, there is a direct relation between the statistics of measured conductance values through quantum dots in the Coulomb blockade limit and the statistics of amplitudes of chaotic electron waves in confined systems [36]. When time-reversal symmetry is present, wave chaotic systems have statistical properties described by a Gaussian orthogonal ensemble (GOE) of random matrices [7]. As a magnetic field is applied, time-reversal symmetry (TRS) is lost in these systems, and the statistical properties are described by a Gaussian unitary ensemble (GUE) of random matrices. However, it is found in the semiclassical regime that the evolution between these types of symmetry is continuous and that a broad crossover of intermediate statistics exists [811]. It has been proposed that careful measurements of this crossover behavior provides a demanding test of the random matrix hypothesis [12], and we perform such a test in this paper. Prior experimental evidence for the existence of a crossover regime comes from statistical properties of eigenvalue spacings, which showed indications of a progression from GOE to GUE statistics as a function of a time-reversal symmetry breaking (TRSB) parameter [8,13,14]. Here we address the evolution of eigenfunctions of semiclassical wave chaotic systems from the TRS to the TRSB limits. A considerable theoretical literature has developed proposing detailed descriptions of eigenvector statistics in the crossover regime, although little experimental data are available to test these theories. These theories treat only 2482 0031-9007 00 85(12) 2482(4)$15.00
the evolution of the one-point statistical property of eigenfunction distribution, P jCj2 , which quantifies the degree of probability density, jCj2 , fluctuations in the eigenfunctions [4,1517]. No eigenfunction imaging experiment has explicitly demonstrated the crossover of eigenfunction statistics from GOE to GUE symmetry, to our knowledge. In addition, no work has addressed the question of which of the theories of one-point eigenvector distribution function best describes the crossover regime and no work has addressed the crossover properties of the two-point correlation functions. Hence this work forms an important testing ground for theories of wave chaotic systems and quantum dots based on random matrix theory, supersymmetry, and semiclassical techniques. The experimental arrangement used to create and measure the wave chaotic eigenfunctions has been described previously [14,18,19]. Briefly, a two-dimensional microwave cavity with walls defining a nonintegrable infinite square well potential is used to simulate the solutions to the two-dimensional Schrdinger equation in the semiclassical limit [20]. The cavity is a symmetry-reduced bow tie with dimensions shown in Fig. 1(a) [19]. A magnetized microwave ferrite incorporated into the cavity is used to break TRS. We image the probability density jC x, y j2 by measuring the electric fields in the standing wave pattern of the resonator and using the analogy between the Helmholtz and Schrdinger equations in two dimensions [18]. Previous results have established that GOE [21] and GUE statistical properties of both eigenvalues [14,22] and eigenfunctions [18] are seen in the limit of zero and large nonreciprocal phase shift in the magnetized ferrite [14], respectively. We have found that the nonreciprocal property of the ferrite, hence the degree of TRSB, is a function of frequency of the eigenmode in a relatively narrow range of frequency [18]. This fortuitous property creates a series of eigenmodes of similar energy spanning the GOE to GUE crossover regime. Making use of this property, we identified a crossover of D3 spectral rigidity statistics from the GOE to GUE limits in earlier experimental work [14]. 2000 The American Physical Society
VOLUME 85, NUMBER 12
PHYSICAL REVIEW LETTERS
18 SEPTEMBER 2000
FIG. 1. Experimentally determined two-dimensional probability amplitude eigenfunctions plotted as log10 jC x, y j2 Acavity , derived from microwave resonator eigenmodes. The cavity has a symmetry-reduced bow-tie shape of area Acavity and is loaded with a magnetized ferrite on the left side. The images are taken at (a) 11.73 GHz (GOE), ( b) 10.79 GHz (crossover), and (c) 11.05 GHz (GUE). The antenna are located at (18.0, 15.5) and (45.5, 26.5) cm.
functions, it may make a contribution to the statistical measures of crossover behavior presented below [26]. As mentioned above, the distribution of jCj2 values is a simple means of identifying the symmetry of the corresponding Hamiltonian. In the GOE limit one finds the Porter-Thomas distribution function of eigenfuncp tion values [27], PGOE y 2py e2y 2 , where 2 y jCj Acavity , and Acavity is the area of the cavity. In the GUE limit the distribution function is simply PGUE y e2y [27,28]. To enhance the subtle differences between these two distribution functions, we plot P log10 y versus log10 y in Fig. 2 [15]. Note that PGUE log10 y peaks at a higher value and falls off more quickly, while PGOE log10 y has a shorter and broader distribution. From this it is clear that GOE eigenfunctions have more large (black in Fig. 1) and small (white) jCj2 fluctuations, consistent with our qualitative observations of Fig. 1. Also shown in Fig. 2 are distribution functions of eigenmodes in the crossover regime. Each data set is an average of five eigenfunctions of similar degree of TRSB (as discussed below). There is a smooth variation of the distribution of eigenfunction fluctuations as TRS is broken. The exact form of this variation is not predicted by theory. We have found four theories for the GOE ! GUE crossover probability density distribution function Pcrossover (y): that due to Brickmann et al. [Pb y , b [ 1, 2 ] [15], Zyczkowski and Lenz [Pb y , b [ 1, 2 ] [16], Sommers and Iida [P y , [ 0, ` ] [17], and Falko and Efetov [PX y , X [ 0, ` ] [4]. The subscript denotes the symmetry-breaking parameter used by the authors. In each case, the lower limit of the TRSB parameter (b, b, , X) corresponds to GOE statistics, and the upper limit to GUE statistics. All of the theories
In this paper we systematically examine the crossover eigenfunctions, and, for the first time, quantify the degree of TRSB with an experimentally determined parameter. Figure 1 shows three eigenmodes jC x, y j2 of the microwave resonator with different degrees of TRS, but of similar energy. Figure 1(a) shows a GOE-limit eigenmode, Fig. 1(c) shows a GUE-limit eigenmode, while Fig. 1(b) shows a mode of intermediate statistics. The probability amplitude is plotted as log10 jCj2 Acavity and presented in three shades to accentuate the differences between the GOE and GUE characteristics of the eigenfunctions. Note that the GOE mode is distinguished by the tall sharp fluctuations of jCj2 (dark areas) and the abundance of low jCj2 regions (white areas) between the spikes [23]. The GUE mode on the other hand has smaller fluctuations of jCj2 and the peaks are more spread out, showing an abundance of intermediate value jCj2 regions (gray areas) [23]. The crossover eigenfunction [Fig. 1(b)] is a mixture of coexisting regions showing GOE-like and GUE-like properties. Although we see no obvious signs of scarring [24,25] in any of the eigen-
1.0 0.8 P(Log(v)) 0.6 0.4 0.2 0.0 -1.5
b b b b b
= = = = =
1.59 1.28 1.16 1.11 1.05
2.00(GUE)
1.59 1.28 1.16 1.11 1.05
1.00(GOE)
-1.0
-0.5 0.0 Log(v)
0.5
1.0
FIG. 2. Probability amplitude distribution functions plotted as jCj2 Acavity . Solid lines P log10 y vs log10 y , where y represent theoretical distribution functions in the GOE, GUE, and crossover regimes described by the Zyczkowski and Lenz theory Pb y . Also shown are averages of distribution functions for data in the crossover regime.
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agree in these two limiting cases, but they disagree for intermediate parameter values in the crossover regime 2 [17]. For instance, Pb y xb y , a generalized chisquare distribution suggested by Brickmann et al. [15], shows a maximum of the distribution function Pb y at y 1 for all values of b, whereas the other theories show a peak in Pb,,X y which occurs for y , 1 in the crossover regime [see, e.g., the solid lines in Fig. 2 for Pb y ]. It should be noted that, strictly speaking, none of the theories for the crossover behavior is precisely to applicable this experiment. The theory of Brickmann et al. proposed an interpolation of the chi-square distribution with no microscopic justification. Zyczkowski and Lenz performed a more detailed calculation in which two chi-square distributions were convolved together. Sommers and Iida addressed the issue of properly treating the fluctuations of the matrix elements but had to perform an energy averaging to arrive at the final distribution function. The work of Falko and Efetov uses a model appropriate for disordered systems and was not explicitly derived for ballistic billiard systems such as ours. Despite the fact that none of these theories exactly corresponds to the experimental situation, they are remarkably accurate in their description of the data. The probability distribution functions P y for 64 experimental eigenmodes have been fit to the four theories of crossover statistics. It is found that the Zyczkowski and Lenz theory produces the best fits for these eigenfunctions, although the difference in fit quality for the other theories is not statistically significant. Figure 2 shows averaged distribution functions for groups of five eigenfunctions with similar values of the Zyczkowski and Lenz crossover parameter b, as well as best fits to the averaged Pb y . Notice that significant changes in P(y) already occur for a small deviation of b from 1, consistent with Falko and Efetovs prediction that a small amount of magnetic flux quickly moves the system away from GOE statistics [4]. Also note that the peak of the experimental distribution function occurs at y , 1 for crossover eigenfunctions, demonstrating 2 that the generalized xb y distribution is not a correct description of the crossover data. These theories, as well as random matrix theory [12], do not predict how the crossover parameters evolve with nonreciprocal phase shift in the ferrite and do not even relate their crossover parameters to those of other theories. Therefore we proceed empirically and define a simple experimental measure of the degree of TRSB. We have noticed a strong asymmetry of the forward S12 f and reverse S21 f complex transmission coefficients of the microwave cavity for TRSB eigenmodes [19] and correlated this asymmetry with the degree of TRSB derived from statistical analysis of the associated eigenfunctions. We define the experimental time-reversal asymmetry parameR R ter A jjS12 j 2 jS21 jj df jS12 j 1 jS21 j df, where the integrals are carried out over one resonant peak (between neighboring minima in jS12 f j) in the frequency 2484
domain. This parameter is easily evaluated, does not require an image of the eigenmode [29], and (as shown below) can be considered a measure of the magnetic flux through the cavity causing TRSB. We find that the one-point correlation function parameters (b, b, , X) are strongly correlated with our experimental measure of TRSB, A. Figure 3 shows the four distribution function crossover parameters b, b, 2 1 1 1 1 , 2X 1 1 X 1 1 plotted vs A, for groups of similar eigenfunctions. Note that the parameters and X have been mapped to the interval (1,2) in an ad hoc way, simply for comparison with the other statistical parameters. We see that all four statistical parameters describe a smooth and universal transition from GOE to GUE statistics as the asymmetry of the transmission characteristic, A, increases [30]. The reduction of the parameter values beyond A 0.25 is likely due to difficulties in properly calculating A for the highly distorted microwave resonance curves S12 f encountered in the GUE limit. One can also employ two-point statistical correlation functions to quantify the crossover from TRS to TRSB behavior of the eigenfunctions. It has been shown that 2 C kr jC 0 C r j2 1 1 cJ0 kr [31], where k is the wave number of the eigenmode, J0 is the zeroorder Bessel function, and c 2 for GOE and c 1 for GUE eigenfunctions [18,28]. Eigenfunctions with a smaller value of c show a more smeared out appearance [Fig. 1(c)]. We find that the value of c changes smoothly between 2 and 1 for the experimental crossover eigenfunctions. To investigate the self-consistency of these results, we have examined the relationship between the one-point correlation function parameters and the two-point parameter c [determined by fitting C kr ], as shown in Fig. 4. Again we notice that there is a universal behavior of the
2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.0
b (2+1) / (+1) (2X+1) / (X+1)
0.1 0.2 0.3 Asymmetry Parameter, A
0.4
FIG. 3. Plot of crossover parameters b, b, 2 1 1 1 1 , and 2X 1 1 X 1 1 versus the experimentally determined TRSB parameter A. The GOE limit is in the lower left, while the GUE limit is in the upper right of the figure. The error bars show the typical standard deviation of the data points.
VOLUME 85, NUMBER 12
PHYSICAL REVIEW LETTERS
18 SEPTEMBER 2000
2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8
[(2-b)/b]
2
1.0 0.5 0.0 0.0
0.5 C-1
1.0
smooth crossover in a consistent and universal manner and that a single TRSB parameter must describe the crossover behavior. We acknowledge assistance from Paul So and Karol Zyczkowski. This work has been supported by NSF NYI Grant No. DMR-9258183, the STEP program, and by the Maryland Center for Superconductivity Research.
b (2+1) / (+1) (2X+1) / (X+1)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Two-Point Correlation Parameter, C
FIG. 4. Plot of crossover parameters b, b, 2 1 1 1 1 , and 2X 1 1 X 1 1 versus the two-point correlation function parameter c. The GUE limit is in the upper left, while the GOE limit is in the lower right of the figure. The error bars show the typical standard deviation of the data points. The inset shows a correlation between the one-point statistical parameter b and the two-point parameter c in the GOE to GUE crossover regime.
one-point crossover parameters in their dependence on c [30]. This demonstrates that the crossover behavior is shared by both single-point and two-point statistical properties of the eigenfunctions, in agreement with the prediction of the supersymmetric nonlinear s model [4,5]. To develop a relationship between one-point and twopoint eigenfunction properties, P consider a superposition of plane waves [31], C ak eik?x , with fixed wave number, k jkj, but with random directions and amplitudes. Writing ak jak jeiuk , GOE corresponds to jak j ja2k j and Dk uk 1 u2k zero (to make C real), while GUE corresponds to Dk random in 0, 2p . Allowing general Dk in the crossover regime leads to the two-point result for C kr given above with P P c 1 1 j jak j ja2k j exp iDk jak j2 j2 and to the one-point result of Zyczkowski and Lenz, Pb y , with 22b c 1 1 b 2 . To test this prediction, we plot c 2 1 22b 2 as an inset to Fig. 4. The data points fit to a versus b straight line of slope close to 1, demonstrating remarkable agreement between the one-point and two-point statistical parameters in the crossover regime. To summarize, we have established both GOE (TRS) and GUE (TRSB) properties of wave chaotic eigenfunctions and shown that there is a continuum of eigenfunctions with intermediate statistics between the two time-reversal symmetry states. We have found that three of the four theories of one-point eigenfunction statistics P(y) in the crossover regime adequately describe our data. We have introduced a simple experimental quantity which measures the degree of TRSB and have shown that the statistical properties of the eigenfunctions evolve smoothly with an increase of this parameter. Finally we demonstrate that the one-point and two-point correlation functions describe the
[1] R. U. Haq, A. Pandey, and O. Bohigas, Phys. Rev. Lett. 48, 1086 (1982). [2] T. H. Zimmermann et al., Phys. Rev. Lett. 61, 3 (1988). [3] R. A. Jalabert, A. D. Stone, and Y. Alhassid, Phys. Rev. Lett. 68, 3468 (1992). [4] V. I. Falko and K. B. Efetov, Phys. Rev. B 50, 11 267 (1994). [5] V. I. Falko and K. B. Efetov, Phys. Rev. Lett. 77, 912 (1996). [6] A. M. Chang et al., Phys. Rev. Lett. 76, 1695 (1996). [7] An exception to this statement is discussed by F. Leyvraz, C. Schmit, and T. H. Seligman, J. Phys. A 29, L575 (1996). [8] M. V. Berry and M. Robnik, J. Phys. A 19, 649 (1986). [9] A. D. Stone, Phys. Rev. B 39, 10 736 (1989). [10] N. Dupuis and G. Montambaux, Phys. Rev. B 43, 14 390 (1991), and references therein. [11] G. Lenz and K. Zyczkowski, J. Phys. A 25, 5539 (1992). [12] O. Bohigas et al., Nonlinearity 8, 203 (1995). [13] J. B. French et al., Ann. Phys. (N.Y.) 181, 198 (1988). [14] P. So et al., Phys. Rev. Lett. 74, 2662 (1995). [15] J. Brickmann, Y. M. Engel, and R. D. Levine, Chem. Phys. Lett. 137, 441 (1987). [16] K. Zyczkowski and G. Lenz, Z. Phys. B 82, 299 (1991). [17] H.-J. Sommers and S. Iida, Phys. Rev. E 49, R2513 (1994). [18] D. H. Wu et al., Phys. Rev. Lett. 81, 2890 (1998). [19] A. Gokirmak et al., Rev. Sci. Instrum. 69, 3410 (1998). [20] H.-J. Stckmann and J. Stein, Phys. Rev. Lett. 64, 2215 (1990). [21] V. N. Prigodin et al., Phys. Rev. Lett. 75, 2392 (1995). [22] U. Stoffregen et al., Phys. Rev. Lett. 74, 2666 (1995). [23] The leftmost four inches of the cavity are not imaged due to the interference of the scanning magnet with the ferrite. [24] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984); L. Kaplan and E. J. Heller, Phys. Rev. E 59, 6609 (1999). [25] T. M. Antonsen et al., Phys. Rev. E 51, 111 (1995). [26] L. Kaplan, Phys. Rev. Lett. 80, 2582 (1998). [27] C. E. Porter and R. G. Thomas, Phys. Rev. 104, 483 (1956). [28] V. N. Prigodin, Phys. Rev. Lett. 74, 1566 (1995). [29] We have also evaluated a more comprehensive R asymmetry R parameter B jjC12 j2 2 jC21 j2 j dAcavity jC12 j2 1 2 jC21 j dAcavity , which makes use of a pair of entire eigenfunction images. We find that the quantities A and B are strongly correlated. [30] Note that the failure of the data points to reach the full GOE and GUE limits in Figs. 3 and 4 is due to the finite experimental resolution for the probability amplitude: 0.05 # y # 10 [18]. [31] M. V. Berry, in Chaotic Behavior of Deterministic Systems, edited by G. Iooss, R. H. G. Helleman, and R. Stora (NorthHolland, Amsterdam, 1991), p. 171.
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