How T-invariance violation leads to an enhanced backscattering with increasing openness of a wave-chaotic system
HHow T -invariance violation leads to an enhanced backscattering with increasingopenness of a wave-chaotic system Ma(cid:32)lgorzata Bia(cid:32)lous, Barbara Dietz, ∗ and Leszek Sirko † Institute of Physics, Polish Academy of Sciences,Aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE,Lanzhou University, Lanzhou, Gansu 730000, China (Dated: September 30, 2020)We report on the experimental investigation of the dependence of the elastic enhancement, i.e.,enhancement of scattering in backward direction over scattering in other directions of a wave-chaoticsystem with partially violated time-reversal ( T ) invariance on its openness. The elastic enhance-ment factor is a characteristic of quantum chaotic scattering which is of particular importance inexperiments, like compound-nuclear reactions, where only cross sections, i.e., the moduli of the as-sociated scattering matrix elements are accessible. In the experiment a quantum billiard with theshape of a quarter bow-tie, which generates a chaotic dynamics, is emulated by a flat microwavecavity. Partial T -invariance violation of varying strength 0 ≤ ξ (cid:46) M of open channels, 2 ≤ M ≤
9, whilekeeping the internal absorption unchanged. We investigate the elastic enhancement as function of ξ and find that for a fixed M it decreases with increasing time-reversal invariance violation, whereas itincreases with increasing openness beyond a certain value of ξ (cid:38) .
2. The latter result is surprisingbecause it is opposite to that observed in systems with preserved T invariance ( ξ = 0). We come tothe conclusion that the effect of T -invariance violation on the elastic enhancement then dominatesover the openness, which is crucial for experiments which rely on enhanced backscattering, since,generally, a decrease of the openness is unfeasible. Motivated by these experimental results we,furthermore, performed theoretical investigations based on random matrix theory which confirmour findings. I. INTRODUCTION
The features of the classical dynamics of a closedHamiltonian system are reflected in the spectral fluctua-tion properties of the corresponding quantum system [1–3]. For a chaotic dynamics they are predicted to coincidewith those of random matrices from the Gaussian or-thogonal ensemble (GOE) if the system is time-reversal( T ) invariant. This was confirmed in numerous exper-imental and numerical studies of nuclear systems [4–7], and of various other systems [8–18]. We report onexperiments with flat microwave resonators referred toas microwave billiards [19–23] emulating quantum bil-liards [24–26]. Systems with violated time-reversal ( T )invariance are described by the Gaussian unitary ensem-ble (GUE), as observed, e.g., in atoms in a constantexternal field [27], in quantum dots [28, 29], in Ryd-berg excitons [30] in copper oxide crystals, nuclear re-actions [31, 32], microwave networks [33–35] and in mi-crowave billiards [22, 36, 37]. A random matrix theory(RMT) description was also developed for partially vi-olated T invariance [29, 38–41] and applied recently toexperimental data obtained with a superconducting mi-crowave billiard [37].Similar observations concerning the descriptiveness byRMT were also made for quantum chaotic scattering ∗ email: [email protected] † email: [email protected] systems. In fact, RMT was originally introduced inthe field of nuclear physics [9]. Nuclear-reaction ex-periments yield cross sections of which the fluctuationshave been investigated thoroughly for the T -invariantcase and compared to RMT predictions for quantumscattering systems [42, 43] and for other many-bodysystems [44–46]. The case of T -invariance violation(TIV) was considered in Ref. [31] for nuclear spectra,in Refs. [47–51] for compound-nuclear reactions, and inRefs. [29, 52, 53] for other devices. Analytical expressionshave been derived within RMT for the scattering ( S )-matrix autocorrelation function for preserved [54] andpartially violated T invariance [55, 56] and verified ex-perimentally with microwave billiards. This is possiblebecause the scattering formalism describing them [57]coincides with that of compound nuclear reactions [58]and both the modulus and phase of S -matrix elementsare accessible, whereas in compound nuclear reactionsonly cross sections, that is, the modulus, can be deter-mined. Furthermore, large data sets may be obtained forsystems with preserved, partially or completely violated T invariance. The analogy has been used in numerousexperiments [34, 55, 56, 59–64] for the investigation ofstatistical properties of the S matrix using as indicatorfor TIV that the principles of reciprocity, S ab = S ba , andof detailed balance, | S ab | = | S ba | , no longer hold.Another statistical measure of quantum chaotic scat-tering, which is of particular importance, e.g., in nuclearphysics because it can be determined from cross-sectionmeasurements and does not depend on the mean reso- a r X i v : . [ phy s i c s . c l a ss - ph ] S e p nance spacing, is the elastic enhancement factor (EEF)as a measure for the enhancement of elastic scatteringprocesses, that is, scattering in backward direction orback to the initial scattering channel over inelastic onesto other directions or scattering channels. Such an en-hancement was observed in compound-nucleus cross sec-tions [65, 66] and, actually, is an universal wave phe-nomen [52, 67–71]. The elastic enhancement factor wasproposed as a tool to characterize a scattering process byMoldauer [72] and serves as a probe of quantum chaos innuclear physics [54, 73, 74] and in other fields [56, 75–77].The nuclear cross section σ ab provides a measure forthe probability of a nuclear-reaction process involving anincoming particle in scattering channel b scattered, e.g.,at a nucleus, thus forming a compound nucleus and even-tually decaying into a residual nucleus and a particle inscattering channel a . Its energy average is expressed inthe framework of the Hauser-Feshbach theory [78, 79] interms of the S matrix elements S ab ( ν ; η, γ, ξ )) = (cid:104) S ab (cid:105) + S fl ab ( ν ; η, γ, ξ ), σ fl ab = σ ab −(cid:104) σ ab (cid:105) = | S fl ab | ≡ C ab (0; η, γ, ξ ).Here, ν denotes the energy of the incoming particles ina nuclear reaction or the microwave frequency in a mi-crowave billiard, and C ab ( ε ; η, γ, ξ ) is the S -matrix au-tocorrelation function. Both S ab and C ab depend on theopenness, that is, on the number M of open channels andthe strength of their coupling to the environment givenin terms of the parameter η and the absorption γ , and onthe size of TIV quantified by a parameter ξ . The EEFcan be expressed in terms of C ab (0; η, γ, ξ ), F M ( η, γ, ξ ) = (cid:112) C aa (0; η, γ, ξ ) C bb (0; η, γ, ξ ) (cid:112) C ab (0; η, γ, ξ ) C ba (0; η, γ, ξ ) (1)In the sequel we will suppress the dependence of S ab ( ν ) = S ab ( ν ; η, γ, ξ ) and C ab ( (cid:15) ) = C ab ( (cid:15) ; η, γ, ξ ) on η, γ and ξ like is commonly done.Analytical results are obtained for the EEF by insert-ing those for C ab ( (cid:15) ) [54, 55] interpolating between pre-served ( β = 1 , ξ = 0) and completely violated ( β =2 , ξ (cid:39) T invariance [56]. The limiting values, attainedfor well isolated resonances, where the resonance widthΓ is small compared to the average resonance spacing d ,and for strongly overlapping ones are known, F ( β ) M ( η, γ ) → (cid:26) /β for Γ /d (cid:28) /β for Γ /d (cid:29) . (2)Accordingly, a value of the EEF below 2 indicatesTIV [56]. For the case of partial TIV the features of F ( β ) M ( η, γ, ξ ) as function of ξ and M are not yet well un-derstood. The objective of the present article is to fillthis gap by performing thorough experimental and RMTstudies of F M ( η, γ, ξ ) in the ( η, ξ ) plane.Properties of the EEF have been investigated exper-imentally in microwave networks [80] and in microwaveresonators [56, 81–84] with two attached antennas, thatis, M = 2 open channels, of similar size of the couplingto the interior states, which is quantified by the trans-mission coefficients T c = 1 − |(cid:104) S cc (cid:105)| (cid:39) T, c = 1 ,
2. Weak coupling corresponds to T (cid:39) T = 1 [54, 73]. Recently, we investigated the EEF ina microwave billiard [84] as function of the openness byvarying M and thus η = M T [74] while keeping the ab-sorption fixed. In the present article we report on thefirst experimental study of the EEF for increasing M in the presence of T violation. Such experiments are ofparticular relevance for nuclear physics and, generally,experiments relying on enhanced backscattering, becausethere typically the number of open channels can be large.In Sec. II we introduce the experimental setup and thenpresent experimental results in Sec. III. Then we explainhow we determined the experimental parameters, i.e., theopenness η , absorption γ and size of TIV ξ on the basisof analytical RMT results and then finally discuss the ex-perimental and RMT results for the enhancement factorin the ( η, ξ ) plane. II. EXPERIMENTAL SETUP
We used the same microwave billiard as in our previousstudies [84]. A schematic top view of the cavity is shownin Fig. 1. It has the shape of a quarter bow-tie billiardwith area A = 1828 . and perimeter L = 202 . h = 1 . ν max = c/ d (cid:39) .
49 GHz with c the speed oflight in vacuum. Below ν max only transverse-magneticmodes are excited so that the Helmholtz equation de-scribing the microwave billiard is scalar and mathemat-ically identical to the Schr¨odinger equation of the quan-tum billiard of corresponding shape. The inner surfaceof the cavity is covered with a 20 µ m layer of silver toreduce internal absorption. The top lid of the cavityhas 9 randomly distributed holes of same size markedfrom 1 to 9 in Fig. 1. The sub-unitary two-port S ma-trix S ab , a (cid:54) = b, a, b ∈ { , . . . , } was measured yielding C ab ( (cid:15) ) and the associated EEF. For this, wire antennas oflength 5.8 mm and pin diameter 0.9 mm are attached tothe holes a, b and connected to an Agilent E8364B VectorNetwork Analyzer (VNA) with flexible microwave cables.The additional open channels are realized by successivelyattaching to the other holes according to their numberingantennas of the same size but shunted with 50 Ω loads.Since identical antennas are used, the associated trans-mission coefficients take similar values, so that η = M T .The amplitudes of the resonances in the spectra | S ab ( ν ) | depend on the size of the electric field at the positions ofthe emitting and receiving antennas. Since the resonatorhas the shape of a chaotic billiard, and thus the aver-age electric field intensity is distributed uniformly overthe whole billiard area, the EEF does not depend on thechoice of positions of the measuring antennas. Therefore,we will present results only for the measurements wherewe chose antenna positions at a = 1 and b = 2.All measurements are performed in the frequencyrange ν ∈ [6 ,
12] GHz. To realize an ensemble of 100 mi-
FIG. 1. Schematic top view of the flat microwave res-onator with the shape of a quarter bow-tie billiard whichhas a chaotic classical dynamics. See Sec. II for a detaileddescription of the experimental setup. crowave billiards of varying shape, a metallic perturbermarked by a ’P’ in Fig. 1 with area 9 cm , perimeter21 cm is placed with its straight boundary part of length2 cm at the sidewall inside the cavity and moved step-wise along the wall with an external magnet [84]. Thesize of the steps of 2 cm is of the order of the wave lengthof the microwaves, which varies between 5 cm at 6 GHzand 2.5 cm at 12 GHz, and thus induces sufficiently largechanges in the spectra, as illustrated in Fig. 2, to attainstatistical independence of all realizations. In order to in-duce TIV, two cylindrical NiZn ferrites (manufactured bySAMWHA, South Korea) with diameter 33 mm, height6 mm and saturation magnetization 2600 Oe are insertedinto the cavity and magnetized by an external homoge-neous magnetic field of strength B (cid:39)
495 mT generatedby a pair of NdFeB magnets of type N42 with coercity11850 Oe placed above and below the cavity at the fer-rite positions marked by M and M in Fig. 1. Here,the positions of the ferrites were chosen such that largestpossible TIV is achieved. The magnetic field B induces amacroscopic magnetization in the ferrites which precessesaround B with the Larmor frequency ω o = γ G B , where γ G (cid:39) g eff ·
14 GHz/T and g eff (cid:39) . ν FR ≈ . S ( ν ) and S ( ν ) in the fre-quency range ν ∈ [8 ,
9] GHz, the principle of detailedbalance does not hold already well below ν FR . III. EXPERIMENTAL RESULTS
We used the cross-correlation coefficient C cross (0) = C cross ( ε = 0; η, γ, ξ ), C cross (0) = Re[ (cid:104) S fl12 ( ν ) S fl ∗ ( ν ) (cid:105) ] (cid:112) (cid:104)| ( S fl12 ( ν ) | (cid:105)(cid:104)| ( S fl21 ( ν ) | (cid:105) , (3) FIG. 2. Transmission spectra | S ( ν ) | (black full lines)and | S ( ν ) | (red dashed lines) for three consecutive po-sitions of the perturber in the microwave frequency range ν ∈ [8 ,
9] GHz. Violation of the principle of detailed balanceand, thus of TIV is clearly visible.FIG. 3. (a) Rescaled resonance width γ tot versus the mi-crowave frequency ν . The inset shows the average trans-mission coefficients. (b) Experimentally determined cross-correlation coefficient C cross (0) for M = 2 (red circles), 4(green squares) and 9 (blue triangles) open channels. (c) Sameas (b) for the strength ξ of TIV. as a measure for the size of TIV. It equals unity for T -invariant systems, and approaches zero with increas-ing size of TIV. We verified that the experimentalcross-correlation coefficient, average resonance width andtransmission coefficients are approximately constant ina frequency range of 1 GHz and accordingly evaluatedthe average of C cross (0) over the 100 cavity realizationsin 1 GHz windows. The result is shown in Fig. 3 (b).It exhibits a broad minimum in the frequency range ν ∈ [8 ,
9] GHz implying that strongest TIV is inducedby the magnetized ferrites at about half the value of ν FR . This may be attributed to the occurrence of modestrapped inside the ferrite [37]. Figure 3 (a) shows therescaled resonance widths γ tot = πd Γ. It results fromtwo contributions, namely the width Γ a due to absorp-tion of the electromagnetic waves in the walls of the cav-ity, ferrites and the metallic perturber and the escapewidth Γ esc originating from the additional open channelsdescribing the coupling of the internal modes to the con-tinuum. Absorption is accounted for by Λ (cid:29) T f (cid:28) T [73, 76]. Note, that choosing three differ-ent values for T f to account for the absorption propertiesof the cavity walls, the ferrites and the perturber whichare made from different materials, where the fractionsof fictitious channels are given by those of their perime-ters [61], yields similar results. The absorption strengthis related to Γ a according to the Weisskopf relation via γ = π Γ a d = Λ T f [85]. The openness η = M T [74] may beexpressed in terms of the Heisenberg time t H = πd andthe dwell time t W = esc which gives the time an incom-ing microwave spends inside the cavity before it escapesthrough one of the M open channels [76], η = t H /t W . Interms of the Weisskopf formula the escape width is givenby πd Γ esc = M T , so that γ tot = M T + Λ T f ≡ η + γ .The experimental EEF F M ( η, γ, ξ ) is obtained by av-eraging over the ensemble of 100 different cavity realiza-tions in 1 GHz windows. The result is shown in Fig. 4 (a)for M = 2 (red circles), M = 4 (green squares) and M = 9 (blue triangles) open channels. Here, the emptyand full symbols show the results for experiments withoutand with magnetized ferrite, respectively, and the errorbars indicate the standard deviation. The black dash-dotted line separates the cases of preserved and violated T -invariance. The value of F M ( η, γ, ξ ) is below two above6 GHz and it exhibits a pronounced minimum in the fre-quency interval [8,9] GHz. Furthermore, while for the T -invariant case the value of the enhancement decreaseswith increasing M as expected from Eq. (2), surprisinglythe opposite behavior is observed for the case of partialTIV. In order to confirm this behavior and for a betterunderstanding of its origin and of the occurrence of theminimum we performed studies based on RMT. IV. RANDOM MATRIX THEORY APPROACH
The input parameters of the RMT model are the trans-mission coefficients T = T a (cid:39) T b associated with anten-nas a and b , which are determined from the reflectionspectra, T c (cid:39) T of the remaining M − γ = Λ T f and the T -violation parame-ter ξ . The sizes of γ and ξ are determined by comparingthe distribution of the experimental reflection coefficients S , S and the cross-correlation coefficient to analyticaland numerical RMT results [55, 56, 75, 86]. Note, that inRef. [56] the absorption strength γ was determined fromthe resonance widths. This is not possible for the ex-periments presented in this article because it is too large(6 (cid:46) γ (cid:46)
15) due to the presence of the ferrites thatconsist of lossy material leading to a considerable degra-
FIG. 4. (a) Elastic enhancement factor deduced from thetwo-port scattering matrix ˆ S measured in 1 GHz windows for M = 2 (red circles), M = 4 (green squares), and M = 9(blue triangles) open channels, respectively. Empty and fullsymbols were obtained without [84] and with a magnetizedferrite inside the cavity. Each point is obtained by averagingover the 100 microwave billiard realizations. The error barsindicate the standard deviations. The black dash-dotted lineseparates the cases of preserved and violated T -invariance;see Eq. (2). (b) Same as (a) for the RMT results. dation of the quality factor, especially in the vicinity ofa ferromagnetic resonance and of trapped modes.The RMT results were obtained based on the S -matrixapproach [58] which was developed in the context of com-pound nuclear reactions and also applies to microwaveresonators [57],ˆ S ( ν ) = 1 − πi ˆ W † ( ν iπ ˆ W ˆ W † − ˆ H ) − ˆ W , (4)where ˆ S is ( M + Λ) dimensional and ˆ H denotes the N -dimensional Hamiltonian describing the closed mi-crowave billiard. We present results for the propertiesof the sub-unitary S -matrix with entries S ab , a, b = 1 , N × N -dimensional random matrices composed of real, sym-metric and anti-symmetric random matrices ˆ H ( S ) andˆ H ( A ) [55], respectively, H µν = H ( S ) µν + i πξ √ N H ( A ) µν , (5)interpolating between GOE for ξ = 0 and GUE for πξ/ √ N = 1, where GUE is attained already for ξ (cid:39) W accounts for the coupling of the N resonator modes to their environment through the M open and Λ fictitious channels [73, 76]. It is a ( M +Λ) × N dimensional matrix with real and Gaussian distributedentries W eµ of which the sum (cid:80) Nµ =1 W eµ W eµ = N v e , e = FIG. 5. Experimental distributions of the modulus of S for M = 2 (black histograms) and the corresponding RMTresults (red histograms) for different frequency ranges. Theparameter values are given in the panels. , . . . , M + Λ yields the transmission coefficients T e = π v e /d (1+ π v e /d ) [56].Figure 5 shows the experimental distributions of themodulus of S (black histogram) for a few examples.The red histograms show the RMT distributions best fit-ting the experimental ones. Figure 3 (a) shows the result-ing rescaled resonance widths γ tot = γ + η which indeed isconsiderably larger than in the experiments [84] withoutferrite. The largest absorption is γ (cid:39)
15 correspondingto strongest overlap of the resonances. Yet, the shape ofthe distributions of | S | in Fig. 3 shows that the limitof Ericson fluctuations, where a bivariate Gaussian dis-tribution is expected [87], is not yet reached. The exper-imental cross-correlation coefficients are shown in Fig. 3(b) and (c) exhibits the corresponding values for the TIVparameter ξ . These were determined by proceeding as inRef. [56], that is, we computed for each parameter set( η, γ ) the cross-correlation coefficient as function of ξ us-ing the analytical result of Ref. [55] and compared it withthe experimental ones to determine ξ as function of fre-quency. The left panel of Fig. 6 shows the analyticallydetermined cross-correlation coefficients in the frequencyrange ν ∈ [9 ,
10] GHz, the right one for M = 2 open chan-nels, where η and γ were chosen as in the experiments.Both reflect the features exhibited by ξ in Fig. 3 (c) inview of Fig. 3 (b).The cross-correlation parameter ξ has a pronouncedpeak in the frequency interval [8,9] GHz. There, thestrength of TIV is largest, ξ (cid:39) .
49. Above this inter-val its value still is comparatively large, ξ (cid:39) .
35. Ina given frequency range the size of ξ decreases with in-creasing M , that is, with openness η . Note, that thesize of TIV induced by the magnetized ferrite dependson the coupling of the spins in the ferrite precessingabout the external magnetic field to the rf magnetic-fieldcomponents of the microwaves, which in turn depends FIG. 6. Examples for the cross-correlation coefficients ob-tained from the analytical result [55], using the experimentalvalues of η and γ in a given frequency range ν ∈ [9 ,
10] GHz[left: M = 2 (black dash-dot line), M = 6 (red dashed line), M = 9 (orange dash-dash-dot line) and M = 10 (cyan fullline)] and for M = 2 [right: ν = 6 . ν = 7 . ν = 9 . ν = 10 . ν = 11 . on the electric field intensity in the vicinity of the fer-rite [37], and is largest at a ferromagnetic resonance orwhen microwave modes are trapped inside the ferrite andbetween the ferrite and top plate. Increasing the numberof open channels leads to an increasing loss of microwavepower and thus to a decrease of the electric-field intensitywhich explains the degression of ξ . To compute the EEF F M ( η, γ, ξ ) we used Eq. (1), that is, inserted the therebydetermined values for γ , ξ and η into the analytical re-sult for the autocorrelation function [55]. The results areshown in Fig. 4 (b). Empty symbols were obtained bysetting ξ = 0. The RMT results for ξ (cid:54) = 0 clearly repro-duce the course of the experimental ones for F M ( η, γ, ξ ).The pronounced dip exhibited by F M ( η, γ, ξ ) in the fre-quency range ν ∈ [8 ,
9] GHz coincides with that of largestachieved TIV, ξ (cid:39) . V. DISCUSSION AND CONCLUSIONS
We investigate the EEF F M ( η, γ, ξ ) for a fully chaoticquarter bow-tie microwave billiard with partial TIV. It isinduced by two magnetized ferrites and largest in the fre-quency range ν ∈ [8 ,
9] GHz which is below the frequencyof the ferromagnetic resonance at ν FR = 15 . M of open channels the EEF decreases with increasingsize of TIV, thus confirming the results for M = 2 ofRef. [56]. However, in distinction to the case of preserved T invariance it increases for fixed M with increasing ab-sorption γ = γ tot − M T as clearly visible, e.g., in the fre-quency range ν ∈ [9 ,
12] GHz where ξ is approximatelyconstant, and in a given frequency window with increas-ing M for ξ (cid:38) .
2. These observations are confirmedby RMT calculations. Figure 7 exhibits the resulting
FIG. 7. Three-dimensional plot of the computed EEF F M ( η, γ, ξ ) for M = 10 open channels versus η = MT with T ∈ [0 ,
1] and ξ ∈ [0 , γ = 10 were kept fixed.FIG. 8. Zoom of Fig. 7 into the region 0 . ≤ ξ ≤ . F M ( η, γ, ξ ). F M ( η, γ, ξ ) in the ( η, ξ ) parameter plane and Fig. 8 azoom into the region of 0 . ≤ ξ ≤ .
7. Here, M = 10and γ = 10 were kept fixed while T and ξ were var-ied. These computations show that for a fixed value of ηF M ( η, γ, ξ ) indeed decreases with increasing ξ . Further-more, it behaves like in the T invariant case for ξ (cid:46) . ξ ≥ . η until it reaches a minimum and thenincreases with η . The positions of the minima ( ξ ∗ , η ∗ ) of F M ( η, γ, ξ ) in the ( η, ξ ) plane are shown in Fig. 9. Theseresults are in accordance with our experimental findings. FIG. 9. Values of ( η, ξ ) of the minimum exhibited for ξ ≥ . F M ( η, γ, ξ ). Indeed, the experimental values of η are larger than η ∗ for ξ > .
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