Probing decoherence through Fano resonances
Andreas Bärnthaler, Stefan Rotter, Florian Libisch, Joachim Burgdörfer, Stefan Gehler, Ulrich Kuhl, Hans-Jürgen Stöckmann
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Probing decoherence through Fano resonances
Andreas B¨arnthaler, Stefan Rotter, ∗ Florian Libisch, and Joachim Burgd¨orfer
Institute for Theoretical Physics, Vienna University of Technology, A–1040 Vienna, Austria, EU
Stefan Gehler, Ulrich Kuhl, and Hans-J¨urgen St¨ockmann
Fachbereich Physik, Philipps-Universit¨at Marburg, Renthof 5, D-35032 Marburg, Germany, EU (Dated: August 30, 2017)We investigate the effect of decoherence on Fano resonances in wave transmission through resonantscattering structures. We show that the Fano asymmetry parameter q follows, as a function of thestrength of decoherence, trajectories in the complex plane that reveal detailed information on theunderlying decoherence process. Dissipation and unitary dephasing give rise to manifestly differenttrajectories. Our predictions are successfully tested against microwave experiments using metalcavities with different absorption coefficients and against previously published data on transportthrough quantum dots. These results open up new possibilities for studying the effect of decoherencein a wide array of physical systems where Fano resonances are present. PACS numbers: 73.23.-b,03.65.Yz,42.25.Bs
One of the central issues of current research in quan-tum mechanics is decoherence [1], i.e., the loss of coher-ence induced in a system by the interaction with its en-vironment. Studying the ubiquituous effects of decoher-ence is not only of fundamental interest for the under-standing of the quantum–to–classical crossover, but isthe key to the realization of operating quantum informa-tion devices which rely on long coherence times [2, 3].To this end, decohering processes need to be controlledand suppressed. In practice, however, enumeration andidentification of sources of decoherence is already a chal-lenging task on its own (see, e.g., [4, 5]). This is, in part,due to the fact that different decoherence channels aredifficult to distinguish from one another since their influ-ence on the observables of interest is often very similar.Decoherence in quantum systems is, typically, de-scribed within the framework of an open quantum sys-tem approach by a quantum master equation of, e.g., theLindblad form [6]. In this framework the reduced den-sity operator, ρ , of the open system with Hamiltonian H S interacting with the environment through Lindbladoperators L j evolves as, i ˙ ρ = [ H S , ρ ] + i X j L j ρL † j − (cid:16) L † j L j ρ + ρL † j L j (cid:17) . (1)The system-environment interactions allow for a deco-hering, yet unitary evolution of the system within theMarkov approximation. In the special case that only thelast term of the coupling in Eq. (1) ( ∼ L † j L j ρ + ρL † j L j ) ispresent, interaction with the environment is purely dis-sipative. The counter-term ( ∼ L j ρL † j ) acts as sourceand preserves the unitarity of the evolution. Charac-terizing the system-environment interaction for a givenphysical realization is one of the major challenges of de-coherence theory. In this letter we show that Fano reso-nances, specifically the asymmetry parameter q , allow to dephasing (b)Im( q ) Re( q )0 q x dissipation (a)Im( q ) Re( q )0 q FIG. 1: Dependence of the complex Fano asymmetry param-eter q on the decoherence strength χ . In the case of a uni-formly dissipative system (a) an increase of χ shifts q ( χ ) inthe complex plane along a straight line away from a real value q while for a uniformly dephasing system (b) the trajectory q ( χ ) forms a circular arc . This arc is centered at x on thereal axis thus featuring a vertical tangent at q . disentangle different decoherence mechanisms present inresonant scattering devices such as quantum dots [5, 7–10] (for a review see [11]). The q parameter follows, as afunction of the decoherence strength, trajectories in thecomplex plane that are specific to the underlying envi-ronmental coupling. In particular, dissipation and deco-herent dephasing can be distinguished from each other.Fano line shapes in transport result from the interfer-ence between different channels in the transmission am-plitude [12], t ( k ) = z r Γ / k − k res + i Γ / t d , (2)with t d the amplitude of a (smoothly varying) direct (orbackground) channel and z r the strength, k res the po-sition, and Γ the width of the resonant channel. Thetransmission probability in the vicinity of the resonance, | t ( k ) | takes on the form of a Fano profile, | t ( k ) | = | t d | | ε + q | ε , (3)in terms of the reduced wavenumber (or energy) ε =( k − k res ) / (Γ / q determinesthe shape of the Fano resonance. In the limit q → ±∞ ,the symmetric Breit-Wigner shape is recovered while for q → q is strictly real [13, 14]. When TRSis broken, Eq. (3) still holds, but q may take on complexvalues [15]. The generalization of the Fano q parameteris therefore ideally suited as a sensitive probe of TRS-breaking processes. An Aharonov-Bohm ring exposedto a TRS-breaking magnetic field was recently shown toexhibit q parameters performing periodic oscillations inthe complex plane [8]. Decoherence, being a prime ex-ample for breaking TRS, should leave distinct signaturesin the behavior of q as well [8–10, 15–17]. In the presentletter we provide experimental and theoretical evidencethat the complex q -trajectory reveals details on the un-derlying decoherence process that are characteristicallydifferent for dissipation and irreversible dephasing.We first consider ballistic transport through a scat-tering cavity in the presence of uniform dissipation, thestrength of which is independent of the wavelength orposition inside the cavity [Fig. 1(a)]. For the correspond-ing open quantum system this corresponds to the reduc-tion of the coupling to the environment [Eq. (1)] to sinkterms ( ∼ L † j L j ρ + ρL † j L j ) only. Physical realizations ofpure dissipation in quantum dots include electron-holerecombination and currents leaking into the substrate.For classical wave scattering [18] the presence of dissipa-tion in a resonant device shifts the resonance positions k res → k res = k res − ∆ k − iκ . Since, in relative terms, thebroadening of the resonance width κ usually dominantesover ∆ k , Eq. (2) is modified as follows, t ( k ) ≈ z r Γ / k − k res + i Γ / iκ + t d , (4)with a broadened resonance width (Γ + 2 κ ).As convenient measure for the strength of decoherencewe use the ratio χ = 2 κ/ (Γ + 2 κ ), with the limitingcases χ = 0 in the absence of decoherence and χ = 1for dissipation-dominated broadening. With Eq. (4) thegeneralized q parameter in Eq. (3) now becomes, q ( χ ) = q + χ ( i − q ) , (5)where q is the real q parameter in the absence of dissi-pation. For increasing dissipation strength the complexFano q parameter follows a straight line trajectory in the k[1/mm] (a)(b)(c)(d) d i ss i p a ti v e d i ss i p a ti v ec oh e r e n t steelbrasscoppernumericalexperiment T ( k ) T ( k ) T ( k ) FIG. 2: (Color online) (a) Rectangular microwave cavitiesof identical dimensions but out of different materials. Theleft two cavities are shown with a shutter at the opening asused for tuning the Fano parameter. (b)-(d) Wavenumberdependence of transmission T ( k ) through a cavity. In (b)the fully coherent limit is shown. (c) and (d) display thetransmission with the dissipative decoherence as present inthe steel cavity at room temperature. Theoretical MRGM(c) and experimental data (d) show excellent agreement onan absolute scale. complex plane [see Fig. 1(a)] which, for large dissipationstrength ( χ → q = i . The linearform of q ( χ ) follows from the assumption entering Eq. (4)that only the resonant but not the direct amplitude ( t d )is affected by decoherence—an assumption which gener-ally holds well for resonant scattering devices (includingmicrowave cavities).The analytical dependence of q on the decoherencestrength has previously been studied in the case of de-phasing [Fig. 1(b)] as the main source of decoherence [15].Here, in contrast to the dissipative case, the flux in thesystem is conserved even at finite coherence lengths. Asimple realization of flux-conserving decoherent dephas-ing for ballistic transport is given by the B¨uttiker dephas-ing probe [19]: By attaching a fictitious voltage probe,the coherent scattering paths from source to drain areaccompanied by incoherent paths via the voltage probewhich randomize the phase information. To convert thedissipative voltage probe into a flux-conserving dephasingprobe, the potential of the probe is chosen such that theflux leaving through the probe is incoherently injectedback into the cavity. This corresponds to the presenceof both sink and source terms in the Liouvillian opera-tor [Eq. (1)] with an infinite number ( j = 1 , . . . , ∞ ) ofcoupling terms with random phases. Such an incoherentreinjection of flux is fully accounted for by an additionalBreit-Wigner shaped term in the Fano profile, | t ( k ) | = | t d | " { ε ′ + Re[ q ( χ )] } ε ′ + Im[ q ( χ )] ε ′ , (6)with the reduced wavenumber now rescaled to the in-creased resonance width ε ′ = ( k − k res ) / (Γ / κ ) as wellas Re[ q ( χ )] = q (1 − χ ) and Im[ q ( χ )] = χ (cid:2) q (1 − χ )]. By eliminating χ from these expressions, one finds { Re[ q ( χ )] − x } + Im[ q ( χ )] = r , (7)where x = [ q − /q ] / r = 1 + x are indepen-dent of χ . Thus q ( χ ) describes a circle in the complexplane centered at ( x ,
0) on the real axis and convergingto q ( χ →
1) = i [see Fig. 1(b)]. We find circular trajec-tories in the complex q -plane also for a different (moregeneral) scenario of dephasing modeled by gradually sup-pressing the interference term between the direct and theresonant transmission in Eq. (2) as χ → q ( χ →
1) may differ from i . We conclude that for the same Fano resonance as de-termined by the q ( χ →
0) = q limit, the complex gener-alization of q evolves along different trajectories for finite χ for purely dissipative (on a straight line) and dephasing(on a circular arc) decoherence. Even for small χ where q is close to the real axis characteristic differences ap-pear: the dephasing trajectory has a tangent parallel tothe imaginary axis while the dissipative trajectory takesoff at an angle arctan(1 /q ) relative to the x -axis. Thisfinding suggests that by following the q ( χ ) trajectory fora given Fano resonance, the underlying decoherence pro-cess can be unambiguously identified.An ideal system for the controlled experimental verifi-cation of the above theoretical results are microwave cav-ities which have been successfully employed in the past asanalog simulators of a wide variety of quantum transportphenomena [20]. As was shown recently, well-separatedFano resonances can be measured with high accuracy insuch systems [17]. The transport of microwaves into andout of the cavity can be controlled via shutters at bothends, which in turn determine the q values of resonances.Due to the finite conductivity of the cavity and the re-sulting dissipation of flux in the cavity walls, decoher-ence is naturally present. Furthermore, we can controlthe degree of dissipation by cooling the cavities to lowertemperatures or by fabricating cavities with identical ge-ometry out of different materials. To a good degree ofapproximation, the power loss can be assumed to be uni-form and mode-independent [as in Eq. (4)].For the experiment we used rectangular microwavecavities (length L = 176 mm, width D = 39 mm) [see I m ( q ) (a) Re(q) Re(q)numericaldata Ref. [7]dissipationcopperbrasssteel dephasing I m ( q ) (b) ~ ~ ~ ~ FIG. 3: (Color online) (a) Complex Fano q parameter ex-tracted from microwave experiments on copper, brass andsteel cavities. For each material, measurements were per-formed at room temperature (filled symbols) and liquid ni-trogen temperature (open symbols). (b) Fano parameter cor-responding to the theoretical MRGM data for a dephasingcavity (diamond symbols) and the experimental data fromRef. [7] (asterisk symbols). Data points for the same res-onance at different decoherence strengths are connected bycolored lines and rescaled, q → ˜ q , to reach ˜ q = 1 (˜ q = i ) inthe fully coherent (incoherent) limit. The straight [circular]black curve in (a) [(b)] displays the theoretical prediction fol-lowing Eq. (5) [Eq. (7)] for the case of uniform dissipation[dephasing]. Fig. 2(a)] made out of copper, brass and steel withdifferent conductivities σ [at room temperature: σ =54 .
22 m / (Ω mm ) for copper, σ = 12 .
20 m / (Ω mm ) forbrass, and σ = 1 .
37 m / (Ω mm ) for steel]. The cavi-ties were terminated by two metallic shutters each withopening width s = 8 . l = 200 mm and width d = 15 . T ( k ) = | t ( k ) | recorded for the steel cavity display well-separated Fanoresonances [see Fig. 2(d)]. For comparison we also showthe corresponding numerical results including dissipation[see Fig. 2(c)] and without dissipation [see Fig. 2(b)]. Thenumerical data was obtained with the modular recursiveGreen’s function method (MRGM) [21], where uniformmicrowave attenuation by dissipation following [18] wastaken into account. Even though the effect of dissipationis quite sizeable, we find excellent agreement between the-ory and experiment [see Figs. 2(c),(d)]. The small oscilla-tions in the experimental data not reproduced by the nu-merical calculations can be attributed to standing wavesinduced by the minimal reflection from the adapters (lessthan 1%). The influence on the Fano resonances is, how-ever, negligible as compared to the dominant decoher-ence process, i.e., the ohmic losses in the cavity walls.Accordingly, Eq. (5) predicts the q parameters of Fanoresonances to display linear decoherence trajectories inthe complex plane. To verify this prediction, we now ex-tract the complex Fano q parameter from resonances atdifferent decoherence strengths as determined by the dif-ferent cavity materials and their temperature dependence(on each cavity one measurement was performed at am-bient temperature and at liquid nitrogen cooling). Sincethe Fano resonance formulas Eqs. (2),(3) are strictly onlyvalid for a direct amplitude t d which is k -independent, weexclude resonances from our analysis for which this re-quirement is not satisfied. For resonances satisfying thisrequirement we extract the q parameters by selecting theminimum, maximum and one intermediate value as fit-ting points in the experimental transmission probabilities[as, e.g., in Fig. 2(d)]. Following each resonance for dif-ferent degrees of dissipation thus allows us to obtain thedesired decoherence trajectories in the complex q plane.To facilitate the comparison of the behavior of differentmembers of the ensemble of resonances we rescale all data[ q → ˜ q = Re( q ) q − + i Im( q )] onto a single “universal” tra-jectory which connects the points ˜ q = 1 and ˜ q = i . Thedata shown in Fig. 3(a) demonstrates that the expectedlinear behavior is, indeed, observed.Such a linear trajectory due to pure dissipation cannow be contrasted with the circular trajectory for flux-conserving dephasing. For open quantum systems thiscorresponds to the presence of an infinite number ofsource ( ∼ L j ρL † j ) and sink terms ( ∼ L † j L j ρ + ρL † j L j )in Eq. (1) as induced, e.g., by wave number indepen-dent electron-phonon coupling. In the classical electro-magnetic cavity such a flux-restoring incoherent sourceis difficult to realize. However, in our simulation wecan numerically reinject the dissipated power into therectangular cavity, leaving the scattering system oth-erwise unchanged. The dissipated flux to be symmet-rically reinjected is determined as the difference be-tween transmission plus reflection and the unitary limit,1 − | t ( k ) | − | r ( k ) | . The q values extracted from thenumerically determined Fano profiles are also mappedonto a “universal” circular arc with radius 1 centeredat x = 0. The data obtained for several Fano reso-nances closely follow the circular trajectory [Fig. 3(b)]confirming the dependence of the q ( χ ) trajectory on theunderlying decoherence mechanism.To test our predictions also for a true quantum scatter-ing system, we reanalyzed published experimental dataon Fano resonances in transport through resonant quan-tum dots [7]. For these conductance measurements in thetemperature range 100mK ≤ T ≤ q ( T ) in the complex plane to be very welldescribed by a circular arc (see Fig. 3b)—as predictedfor flux-preserving dephasing. Details of this analysis as well as a comparison between the experimental and thetheoretical Fano resonance curves are provided in the ap-pendix.In conclusion, we have demonstrated that Fanoresonances may serve as sensitive probes of decoherencein wave transport. We find that for increasing dephasingor dissipation strength the Fano asymmetry parameter q evolves on circular arcs or on straight lines in thecomplex plane. As confirmed by measurements onmicrowave cavities and on quantum dots, these char-acteristic signatures provide a means to determine notonly the degree but also the specific type of decoherencepresent in the experiment. It is hoped that the presentfindings will stimulate future experimental investigationsof the influence of decoherence on the Fano q -parameterin resonant quantum transport.We thank K. Kobayashi for very helpful discussionsand the Austrian FWF (P17359 and SFB016) as well asthe German DFG (FOR760) for support. Appendix with supplementary material
Complex q -trajectories: the case of dephasing Following previous analysis [15] based on the B¨uttikerdephasing probe model, we show in the main text of ourarticle that for this specific model the Fano q -parameterfollows a circular trajectory in the complex plane. It isnow instructive to inquire whether this result can also befound for a more general scenario of dephasing. One suchgeneric approach is the ensemble average over a randomphase φ between the resonant ( ∝ z r ) and the backgroundamplitude t d , t = t d + z r i + ε e iφ , where the reduced wavenumber ε = ( k − k res ) / (Γ / φ featuring a zero meanand a standard deviation σ the interference term in theensemble average of the total transmission h| t | i φ will begradually suppressed for increasing σ . This behavior canbe conveniently described by a prefactor (1 − χ ) ∈ [0 , χ ( σ ), h| t | i φ = | t d | + | z r | ε + (1 − χ ) 2Re (cid:18) t ∗ d z r i + ε (cid:19) = | t d | | z r /t d | + (1 − χ ) 2Re h z r t d ( − i + ε ) i + (1 + ε )1 + ε . The limit of complete dephasing ( χ →
1) correspondsto the incoherent addition of resonant and backgroundcontribution. With the real value of the Fano parameter q in the absence of decoherence given as q = i + z r /t d [15], we further obtain, h| t | i φ = | t d | [ ε + (1 − χ ) q ] − (1 − χ ) q + q + 2 χ ε . Comparing this expression with the general form of aFano resonance as in Eq. (6) reveals the evolution of thecomplex Fano parameter q ( χ ) as a function of the deco-herence strength χ ,Re[ q ( χ )] = (1 − χ ) q , Im[ q ( χ )] = 2 χ + q (2 χ − χ ) . Eliminating χ from the above two equations finally yields,Im[ q ( χ )] + { Re[ q ( χ )] + 1 /q } = 2 + 1 q + q , which describes a circle in the complex plane with radius r = p /q + q , centered at x = − /q on thereal axis. Note that in the limit of complete dephasing( χ →
1) the above trajectory q ( χ ) converges to a valueon the imaginary axis which, in contrast to the resultobtained with the B¨uttiker dephasing probe model [15],is not necessarily given by q = i .We thus arrive at the interesting conclusion that dif-ferent models of dephasing may yield a similar circularbehavior of q ( χ ) where, however, the radius of the cir-cular arc and its corresponding end point depend on thespecific dephasing scenario. Details on the employed fitting procedures
When extracting the complex value of q from a reso-nance profile, the imaginary part of q is highly sensitiveto the minimum value of the profile. General automatedfitting routines, however, do typically not account forthis specific dependence appropriately. To overcome thisdifficulty we used the minimum, maximum, and an in-termediate point of the resonance as interpolation pointsin our procedure. Furthermore, we took advantage ofthe knowledge that the minimum and maximum pointsare extreme values of the Fano resonance, resulting inaltogether five equations. Their solution yields the res-onance amplitude, the resonance position k res , the res-onance width Γ and the real and imaginary part of q .All the q -values in Fig. 3 corresponding to the microwaveexperiment and to the numerical dephasing data wereextracted in this way. Resonances with a non-uniformbackground were excluded from our analysis.As outlined in the main text, we also tested our pre-dictions against experimental data previously publishedin [7]. In that publication the resonant conductancethrough a quantum dot was studied as a function of tem-perature T (100mK ≤ T ≤ - - - - - H q L H q L - - - G H V L - - H G * h (cid:144) e L (b)(a) FIG. 4: (a) Voltage dependence of the conductance througha resonant quantum dot as measured in the experiment byZacharia et al. [7]. The black dots show the values extractedfrom [7] in steps of 100 mK between 100mK (lowest curve)and 800mK (highest curve). For better visibility the datasets with
T >
100 mK are each multiplied by a factor of √
10 from one temperature value to the next. The red curvesshow the analytical curves resulting from restricted parameterfits (see text) with the corresponding complex q -values beingdisplayed in (b) by the red asterisk symbols. The black curvein (b) shows a circular arc as prescribed for the dephasingmodel of decoherence. experimental data extracted directly from [7] are shownin Fig. 4(a) (black dots), right above. Unfortunately, forall the resonance curves provided in [7] the backgroundtransmission is very small, such that the correspondingFano resonance lineshapes in the conductance are verynear the symmetric Breit-Wigner limit (with | q | ≫ q -parameter as performed on the moreasymmetric Fano resonances (with | q | . | q | ≫ q ( χ )can be found that features good agreement with both theexperimental data and a circular form of q ( χ ). For thispurpose we carried out restricted parameter fits which,in line with [7], were performed on a logarithmic con-ductance scale with thermal broadening being includedseparately and the data for the fit being restricted to therange where the conductance is at least twice as large asthe measured conductance minimum (to reduce the in-fluence of neighboring peaks). The best curves which wefound in this way are shown in Fig. 4(a) (red lines) withthe corresponding q -values shown in Fig. 4(b) (red sym-bols) and in Fig. 3(b) (after rescaling to the unit circle).The overall very good agreement demonstrates that thearc-like behavior of q ( χ ) can very well describe the exper-imental data. To cross-check this result we also mappedthe measurement data on a linear q ( χ )-trajectory (notshown) as prescribed by the dissipation-dominated de-coherence in Eq. (5) and found much larger discrepan-cies. We hope future quantum transport experiments willmake more asymmetric Fano resonances (with | q | . ∗ Corresponding author, email: [email protected][1] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[2] D. Bouwmeester, A. Ekert, and A. Zeilinger, The physicsof quantum information (Springer, 2000).[3] J. J. Lin and J. P. Bird, J. Phys.: Condens. Matter ,R501 (2002).[4] R. Simmonds et al., Phys. Rev. Lett. , 077003 (2004). [5] B. Elattari and S. A. Gurvitz, Phys. Rev. Lett. , 2047(2000); J. Z. Bern´ad, A. Bodor, T. Geszti, and L. Di´osi,Phys. Rev. B , 073311 (2008).[6] G. Lindblad, Commun. Math. Phys.
119 (1976).[7] I. G. Zacharia et al., Phys. Rev. B , 155311 (2001).[8] K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye,Phys. Rev. Lett. , 256806 (2002); Phys. Rev. B ,235304 (2003).[9] A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C.Gossard, Phys. Rev. Lett. , 106803 (2004).[10] W. Gong and Y. Zheng and J. Wang and T. L¨u, Phys.Status Solidi B , 1175 (2008).[11] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar,arXiv:0902.3014.[12] U. Fano, Phys. Rev. , 1866 (1961).[13] Note that the q -parameter is complex for multi-channelscattering with TRS [14] (a case not considered here).[14] M. Mendoza et al., Phys. Rev. B , 155307 (2008).[15] A. A. Clerk, X. Waintal, and P. W. Brouwer, Phys. Rev.Lett. , 4636 (2001).[16] Z. Zhang and V. Chandrasekhar, Phys. Rev. B ,075421 (2006).[17] S. Rotter et al., Phys. Rev. E , 046208 (2004); PhysicaE , 325 (2005).[18] J. D. Jackson, Classical Electrodynamics (Wiley, 1998).[19] M. B¨uttiker, Phys. Rev. B , 3020 (1986); IBM J. Res.Dev. , 63 (1988).[20] H. J. St¨ockmann Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).[21] S. Rotter et al., Phys. Rev. B , 1950 (2000); ibid.68