Universal formalism of Fano resonance

The phenomenon of Fano resonance is ubiquitous in a large variety of wave scattering systems, where the resonance profile is typically asymmetric. Whether the parameter characterizing the asymmetry should be complex or real is an issue of great experimental interest. Using coherent quantum transport as a paradigm and taking into account of the collective contribution from all available scattering channels, we derive a universal formula for the Fano-resonance profile. We show that our formula bridges naturally the traditional Fano formulas with complex and real asymmetry parameters, indicating that the two types of formulas are fundamentally equivalent (except for an offset). The connection also reveals a clear footprint for the conductance resonance during a dephasing process. Therefore, the emergence of complex asymmetric parameter when fitting with experimental data needs to be properly interpreted. Furthermore, we have provided a theory for the width of the resonance, which relates explicitly the width to the degree of localization of the close-by eigenstates and the corresponding coupling matrices or the self-energies caused by the leads. Our work not only resolves the issue about the nature of the asymmetry parameter, but also provides deeper physical insights into the origin of Fano resonance. Since the only assumption in our treatment is that the transport can be described by the Green’s function formalism, our results are also valid for broad disciplines including scattering problems of electromagnetic waves, acoustics, and seismology.


I. INTRODUCTION
A fundamental phenomenon associated with quantum or wave scattering dynamics is Fano resonance. 1,2A typical scattering system consists of incoming channels, a scatterer or a conductor, and outgoing channels.The scatterer, when isolated, can be regarded as a closed system with a discrete spectrum of intrinsic energy levels.When the energy of the incoming particle or wave matches an energy level, a resonant behavior can arise in some experimentally measurable quantities, such as the conductance in a quantum-transport system.The resonance profile is typically asymmetric, and can in general be expressed as (ε + q) 2 /(ε 2 + 1), where ε is the normalized energy deviation from the center of the resonance, and q is the parameter characterizing the degree of asymmetry of the resonance.The asymmetry parameter q is of great experimental importance as it determines how the experimental data can be fitted by the Fano profile.
The profile was first derived by Fano 1 in the study of inelastic scattering of electrons off the helium atom and auto-ionization, although the phenomenon was predicted earlier in elastic neutron scattering. 3Being a general wave interference phenomenon, Fano resonance has been found in many contexts in physics, such as photon-ionization, 4 Raman scattering, 5,6 photon-absorption in quantumwell structure, 7,8 scanning microscopy tunneling in the presence of impurity, [9][10][11] transport through approximate expression of the width of the Fano resonance with a clear physical picture, which relates the strong localized scar/pointer [38][39][40][41][42] states to sharp conductance fluctuations.Concluding remarks are then presented in Sec.IV.

A. General scheme for quantum transport
Without loss of generality, we consider a two-terminal quantum-dot system, which can be formulated as follows. 43The system is divided into three parts: left lead, conductor (scatterer) of arbitrary shape, and right lead, where the semi-infinite left and right multi-mode electronic waveguides are assumed to be uniform, connected only to the conductor (or the scattering region), as shown in Fig. 1.The conductor is chosen to include all the irregular components in the scattering system such as geometrical shape, arbitrary electric or magnetic potential, and/or disorder, etc.The conductor can be a single molecule, a quantum dot, or any other small-scale structure through which electronic waves pass coherently.The tight-binding Hamiltonian matrix can then be written as where H C is a finite-size square matrix of dimension N C × N C , N C is the number of discrete points in the conductor, and H L, R are the Hamiltonians of the left and right leads, respectively.The various couplings between the conductor and leads are given by the matrices H LC , H CL , H CR , and H RC .
The effect of the semi-infinite uniform leads can be treated by using non-Hermitian self-energy terms, Σ L , Σ R , for the left and right leads, respectively, which are determined by where G L, R are Green's functions for the left and the right leads.][46] The retarded Green's function for the conductor is given by where Σ = Σ L + Σ R is the total self-energy from both leads.The coupling matrices Γ L (E) and Γ R (E) are the difference between the retarded and the advanced self-energy: FIG. 1.A general scheme for transport through a conductor or a scatterer connected with two semi-infinite leads.The leads are assumed to be uniform.The conductor contains any irregular component of the scattering system.For example, it can be a scatterer with a nonuniform shape, or with a whole that mimics the Aharonv-Bohm interferometer, or with irregular electric or magnetic potential or disorders, or a single molecule.
which characterize the coupling between the conductor and the leads, and are Hermitian: The quantum transmission T(E), as a function of the Fermi energy, is given by 43 The classic Landauer's formula 47 can then be used to calculate the conductance: where T(E) is the transmission of the conductor and To be concrete, we focus on the low-temperature conductance, or equivalently, the transmission T.

B. Fast-slow expansion
To analyze the Fano resonance profile, the scales of variations (e.g., fast or slow with respect to the energy variation) of the various quantities in Eq. ( 5) are the key.Indeed, the self-energy Σ varies slowly with the energy, so do the coupling matrices Γ L, R .That is, in the energy scale where the transmission exhibits rapid, nearly abrupt oscillations, e.g., through a Fano resonance, the value of the self-energy matrix elements can be regarded as approximately constant, as shown in Fig. 2. The change in the transmission must then come from the energy-dependence of the Green's function G(E).Our idea to probe into the Fano resonance profile is then to consider a small energy range about such a resonance and decompose the Green's function G(E) into two components: where G 0 (E) and G 1 (E) are the slow and fast components, respectively.The transmission can then FIG. 2. For a graphene quantum dot, a demonstration of the slow variation of the self-energy through a Fano resonance, where the solid curve is transmission versus E with sharp resonances, the dotted and dash-dotted curves denote the real and imaginary parts of an arbitrary element of the self-energy matrix, respectively.The spikes in the self-energy indicate the energy value at which the number of the transmitting modes in the leads increases.The inset shows a zoom-in with a few transmission resonances.It can be seen that in any small energy interval about a single resonance, the self-energy can be regarded as a constant.
be expressed as where Γ L , Γ R , and G 0 (E) are treated as constant matrices.Among the four terms in Eq. ( 7), only T 00 varies slowly with the energy.Unlike the standard perturbation theory, here G 1 (E) is the fast changing component of the Green's function, but it is not necessarily small.Thus the remaining three terms T 01 , T 10 , and T 11 exhibit approximately the same behaviors in terms of their magnitudes and rates of change with the energy.Consequently, T 11 cannot be regarded as a second-order term.Note that T 01 and T 10 are not independent of each other: where we have used the relation Γ † L, R = Γ L, R and the invariant property of trace under cyclic permutations.Similarly, we have T † 11 = T 11 so that T 11 is real.

C. Construction of G 1
The Green's function can be expressed in matrix form as 43 , where the summation is over all the eigenstates, Ψ α and Φ α are the right and left eigenvectors associated with the eigenvalue ε α of the generalized Hamiltonian matrix In general, the self-energy is not Hermitian and, hence, Ψ α and Φ α are not identical but form a bi-orthogonal set: Φ † α Ψ β = δ α, β .The eigenvalues ε α are generally complex: ε α = E α − iγ α , where the imaginary part originates from the self-energy and characterizes the lifetime of the corresponding state before it tunnels into the leads.
Consider a small energy interval that contains a transmission resonance, in which G(E) can be decomposed into a slowly varying and a fast changing components: G(E) = G 0 (E) + G 1 (E).Let E 0 denote the center of the resonance.In the small energy interval about E 0 , we can identify a small set Ω 0 of eigenstates and write , and the summation over all the other eigenstates can be denoted as G 0 (E).G 1 (E) is the fast component because, for E around E 0 , only those states whose energies are close to E 0 will contribute to the variation of G(E).Note that although the identification of Ω 0 can be somewhat arbitrary, practically, when an eigenstate is well separated from others and the corresponding eigenvalue has a small imaginary part, i.e., γ α < E α+1 − E α and γ α < E α − E α−1 , it will result in a transmission resonance by itself with energy scale γ α , as shown in Fig. 3.In this case, Ω 0 can be chosen to have only one eigenstate α.This will be our focus in the following derivation of the resonance profile.
Situation can also arise where the eigenenergies of a small number of eigenstates are close to each other and have large γ α values, i.e., γ α is larger than the spacing between the real parts of adjacent eigenvalues.In this case, the transmission resonances from these eigenstates are not separable, and the resulting resonance profile is the mixture of the isolated profiles.

D. The universal formula
For a single separated resonance, G 1 (E) is given by: and G 0 (E) can be regarded as a constant matrix and approximated by its value at There is a well separated resonance profile about E 0 , which occurs at ε α .
can be expressed as The coupling matrices vary slowly with the energy and can be regarded as constant matrices evaluated at E 0 as well: With the separation of the various quantities into slow and fast components, we can express the transmission T(E) as where T 00 is a constant and T sum (E) gives the approximation of transmission curve T(E) about a single, isolated resonance.Substituting the expression of G 1 (E) into the above, we have where T 01 (E 0 ), T 10 (E 0 ), and T 11 (E 0 ) can be evaluated numerically.Figure 4 shows the energy variations of different terms in the expansion expression (9).We observe that the numerical value of T 11 (E) is comparable with those of T 01 (E) and T 10 (E), and T 01 (E) = T 10 (E) * , where * denotes complex conjugate.Substituting these expressions into T sum , we obtain Without loss of generality, we set E 0 = E α , leading to we have Letting E 0 = E α and ε = (E − E α )/γ α , we finally arrive at Validation of Eq. ( 10) is illustrated in Fig. 5, where the numerically obtained three isolated resonances from a graphene quantum dot are shown, together with the respective theoretical profiles.Excellent agreement is observed for energy around the resonance.
To compare with the Fano formula, we rewrite Eq. ( 10) as where the second term is the standard Fano resonance profile 1 and the third term is a symmetric bias.Our Eq. ( 10) thus represents a more general formulation of the Fano-resonance profile, and it is consistent with previous results, e.g., the Fano resonance profiles of conductance from the scattering-matrix elements. 28,29In the original formula (ε + q r ) 2 /(ε 2 + 1), the asymmetry parameter q r is proportional to the ratio of the transmission amplitudes for the resonant and non-resonant channels, 1 and it is regarded as a real parameter.To see why q can take on complex values, as asserted in most existing works (e.g., those on conductance fluctuations 28 ), we consider the case q 2 r < 2, let q i =  2 − q 2 r , define q = q r + iq i , and rewrite Eq. (11) as In previous works, q r and q i are related to each other in a complicated way [see, for example, Eqs. (4-12) in Ref. 28].For q r < √ 2, our formula Eq. ( 11) is equivalent to Eq. ( 12) in Ref. 28, where the first complicated term in their Eq.( 12) is replaced by 2, i.e., q 2 i = 2 − q 2 r .This is consistent with Fig. 4 of Ref. 28, which shows damped oscillations of q r versus the magnetic flux with the maximum value of 1.41.This relation was also observed in previous experimental studies 15,16 of transport through a quantum dot in an Aharonov-Bohm interferometer, where it is found that |q| is generally independent of the magnetic field strength and traces out a circle in the complex q plane, although |q| can be larger than 2 and the center is not at the origin [Fig.6(b-c) in Ref. 16].The former is understandable as q is normalized by ∆T, which can be quite different depending on the specific fitting procedure employed.However, given that ∆T is fixed and determined as in our formula, |q| 2 should be 2. Thus, our derivation allows us to verify that in the normalized Fano resonance profile, the asymmetry parameter can in general be complex, regardless of the time reversal symmetry, 29 and |q| = √ 2. This is a universal value, independent of any details of the transport or scattering process, insofar as it can be treated by the non-equilibrium Green's function formalism.Thus we expect our formula Eq. ( 10) to be valid for general coherent transport through quantum dots, small scale organic crystals, single molecules, and other small-scale structures.

E. Connection to the Fano formula with complex and real asymmetric parameters
We now demonstrate a surprising consequence of Eq. ( 10), bridging of the Fano formulas with complex and real asymmetry parameter.In particular, we can show that the standard Fano formula with complex q parameter is equivalent to Eq. (10), and using another transformation, Eq. ( 10) is equivalent to the standard Fano formula with real q parameter.The indication here is that the Fano formulas with complex and real q parameters are connected through Eq. (10).Thus in an experimental fitting both real and complex q values are meaningful, and they give essentially the same physics. 49he Fano formula with complex asymmetric parameter has the form where |t d | 2 is the direct transmission without the presence of a scattering region, q ′ is the complex asymmetric parameter.To see its relation with Eq. ( 10), we can write Expanding both sides and letting the coefficients of different ε terms be equal, we obtain where the parameters of Eq. ( 10) can be expressed by the parameters for Fano formula with complex asymmetric parameters, and vice versa.That is, the two are equivalent.Moreover, from the relation FIG. 6.For a single resonance given by Eq. ( 10), (a) T 00 = 0.3, ∆T = 0.3, q r = 1.2, the resulting parameters for Eq. ( 13) are 2, the resulting parameters for Eq. ( 13) are we see that |q ′ | = √ 1 + ∆T/T 00 .When T 00 (or |t d | 2 ) is large, since ∆T is on the order of one, q ′ r is close to zero while q ′ i is close to 1.This may be misleading because for q ′ i ≫ q ′ r , it may appear that the resonance is almost symmetric as discussed in previous studies, 35 but this may not be the case.Figure 6 shows the resonance profile for two cases, one with T 00 = 0.3 (a) and another with T 00 = 10.3 (b), with ∆T and q r being fixed.One can see that, since only T 00 is different, the resonance profile only has a shift for the two cases, and it has a strong asymmetric form.However, the resulting (q ′ r , q ′ i ) for Eq. ( 13) are (1.2, 0.748) for (a) and (0.035, 1.014) for (b), indicating an asymmetric form for the former and a symmetric form for the latter. 35Thus, although the two cases has the same resonant form, one may not be able to see that from the numerical values of (q ′ r , q ′ i ) as they change drastically.When there is only one transmitting mode in the leads, the maximum transmission is 1.In this case, when time reversal symmetry is present, the asymmetric parameter q ′ can be chosen to be real 35 (q ′ i = 0).To distinguish with the above discussions, we use the symbol q for the real asymmetric parameter.This leads to 1 + ∆T/T 00 − q 2 r (∆T/T 00 ) 2 = 0, or q 2 r = (T 00 /∆T) 2 + T 00 /∆T.
Since q r = Im(T 01 (E 0 ))/∆T, this yields The relations between the parameters are r − 1), q r = q/(q 2 − 1),  (a) Contour plot of q r on the complex asymmetry parameter q ′ plane.Note that q r diverges on the unit circle |q ′ | = 1.For better visualization, for |q r | > 1, it is normalized to sign(q r )[1+log 10 |q r |].(b) q r versus the real asymmetry parameter q, where it has two branches, |q| > 1 (solid blue curves) and |q| < 1 (dashed brown curve), corresponding to |q ′ | > 1 and |q ′ | < 1 in panel (a), respectively.The asterisks indicate the calculated values of q r and q from the corresponding q ′ , i.e., asterisks in (c).The sequence numbers (1-4) indicate the correspondence to the change complex asymmetry parameter q ′ in (c).(c) Transmission (shifted) versus normalized energy demonstrating Fano profiles with a complex asymmetry parameter q ′ (asterisk) circulating one cycle, where |q ′ | = √ 2, thus q r = q ′ r . or, To view the relation more directly, we can rewrite the above equations as follows.With q ′ = q ′ r + iq ′ i and q r = q ′ r /(|q ′ | 2 − 1) (Fig. 7(a)), we have, where the left-hand side has the form of Eq. ( 10), and there is no constraint on q ′ .With q r = q/(q 2 − 1) or q = 1/(2q r ) ±  1/(4q 2 r ) + 1 (Fig. 7(b)), we have Note that the offset on the right-hand side can be moved to the left side.The relation between q and q r is shown in Fig. 7(b).It has two separate branches, i.e., for |q| > 1 and |q| < 1, where the latter flips the shape as the factor q 2 − 1 is negative.When the complex parameter q ′ = q ′ r + iq ′ i is given, q r can be uniquely determined.However, when q r is given, q ′ r and q ′ i cannot be determined uniquely, but trace out a circle: different q ′ on the circle may lead to distinct Fano profiles, they will collapse to a single curve determined by q r by applying a proper shift and scaling.In Ref. 30, it is found that such a parameter q ′ corresponds to a dephasing process [Eq.(7) in Ref. 30].This implies that, although dephasing breaks TRS and the resonance profile may look quite different for different values of q ′ , it leaves clear footprints that all the profiles during a dephasing process can be tracked back to a single profile by simple rescaling and shifting.In situations where |q ′ | is a constant, q ′ r is proportional to q r , and the change of sign in q r indicates a sign change in q ′ r as well.Thus the sign change in q ′ r can simply be revealed from the incline of the resonance profile. 32Figure 7(c) shows a series of Fano profiles with complex asymmetry parameter q ′ and |q ′ | = √ 2 (so q r = q ′ r ).The calculated values of q r and q are plotted on the q r (q) curves of Fig. 7(b).The corresponding changes in q and q r , when q ′ circulates the cycle, are indicated by the sequence numbers.A similar discussion linking an expression similar to Eq. ( 10) to the real q-parameter Fano formula was provided by Shore in 1967. 50he above argument indicates that, in the experimental analysis of conductance or transmission resonance, the standard Fano formula with either real or complex q parameter can be chosen, provided that an offset can be applied.An alternative method is to fit with our formula Eq. (10).A general conclusion is that, regardless of whether TRS is present or broken, the transmission resonance profile can always be fitted by the Fano formula with real q parameter, 49 although in certain cases the employment of complex q parameter may be more convenient, especially in applications such as probing decoherence. 30

A. The expression of the width
In Eq. ( 10), the energy is normalized by γ α , thus γ α characterizes the width of the transmission resonance.In the following we shall develop a perturbation theory to calculate γ α .
Regarding the conductor as a closed system, the Hamiltonian matrix H C is Hermitian with a set of real eigenenergies and eigenfunctions: where N is the number of points in the discretized conductor.The eigenstates {ψ 0α |α = 1, . . ., N } are orthogonal and complete, thus they form a basis in the ℜ N space for the discrete spacial configuration of the wavefunctions.Recall that Since the self-energy matrix Σ has only nonzero elements in the subblock of boundary points connecting with the leads, for most of the eigenstates it can be treated as a perturbation.Thus for a given eigenstate α, we have where ∆ α , γ α , δ r and δ i are small quantities, E 0α − ∆ α = E α , and ψ αr (i) are the normalized perturbations for the real (imaginary) part of the eigenfunction ψ α0 .Substituting Eqs. ( 20) and ( 21) back into Eq.( 18) yields Neglecting the second-order terms on both sides and using Eq. ( 17), we get Projecting both sides of the above equation to state α, i.e., ⟨ψ 0α |•⟩, we obtain This gives the first-order approximation of the shift in the eigenenergy caused by the coupling with the leads.More explicitly, we can write [51][52][53] Since Since the self-energy characterizes the coupling between the conductor and the leads, γ α can be regarded as the tunneling rate of the state Ψ α , the eigenstate of the non-Hermitian Hamiltonian H C + Σ.We see that γ α is determined by the imaginary part of the self-energy Σ (or the coupling matrix Γ) and the corresponding eigenfunction ψ 0α of the isolated conductor.Since Σ (Γ) only has nonzero elements at the boundary points of the conductor connecting with the leads, only the values of ψ 0α on the same set of points, i.e., the boundary points, contribute to γ α .Therefore localized states, typically with small ψ 0α values at the boundary points, have small γ α thus generate sharp resonances.
Figure 8 shows, for a small graphene quantum dot, the correspondence between the transmission fluctuation and the calculated values of ε α .The self-energy Σ is evaluated at E 0 = 0.563t (somewhat arbitrary), and the values of ε α are then calculated as the eigenvalues of the non-Hermitian Hamiltonian H C + Σ.The positions of the resonances agree well with those from E α .In principle, the plots are FIG.8.For a graphene quantum dot, (a) transmission T versus energy E, (b) the real and imaginary parts of the eigenenergy ε α .The circles denote the eigenenergy values calculated directly from the non-Hermitian Hamiltonian of the entire open system: [H C + Σ]Ψ α = ε α Ψ α , where E 0 = 0.563t is used for calculating the self-energy.The crosses represent results from our first-order perturbation theory.The blue solid curve shows the line shape obtained by a further perturbation analysis of the self-energy.The dotted vertical lines are for eye guidance.
only accurate for the eigenenergies where E α is close to E 0 because Σ and hence ε α are all dependent upon the energy E. But we find that, even if E α is far from E 0 , we still obtain reasonable agreement, which can be attributed to the slow variation of the self-energy Σ with the energy.The blue solid curve in Fig. 8(b) is derived from a further approximation of Σ (Eq.( 32)).

B. Recovery of Eq. (12) for anti-resonance and resonant transmission
When there is only one transmitting mode, a particular focus is anti-resonance and resonant transmission.Here we shall demonstrate that in such a case, our approach can also exhibit these features.
From Eq. ( 12), we can obtain the extreme value of transmission, as follows.First, dT/dε = 0 leads to Substituting this into Eq.( 12) and letting Q = 1 + 4q 2 r , we obtain the extreme value of transmission as Note that q r → 0 leads to symmetric peak or dip, while q r → ∞ leads to the asymmetric form.For small q r , i.e., q r ≪ 1 (Im(T 01 ) ≪ ∆T), we have When there is only one transmitting mode in the leads and the TRS is preserved, for T 00 = 1 and ∆T = −1, we have T e = 0 and 1, corresponding to an anti-resonance; while for T 00 = 0 and ∆T = 1, we have T e = 0 and 1, corresponding to a resonant transmission.
Next we show that the above situation, e.g., ∆T = 1 or −1 can arise in certain circumstances.Recall that ∆T = 2ReT 01 (E α )) + T 11 (E α ).We shall calculate ReT 01 and T 11 .For large dots and narrow leads, the self-energy Σ can be regarded as a small perturbation of H C .For highly localized states, we have Ψ α ≈ Φ α ≈ ψ 0α , yielding For systems with an inversion symmetry, Ψ α and Γ will also have the same symmetry, leading to For T 11 (E α ), we have where the energy is evaluated at E α and G 1 = Ψ α Φ † α / (iγ α ).We thus have This has been verified numerically.Similarly, we have In the case of resonance, typically γ β , γ α β , γ βα , γ α are small and are of the same order of magnitude.We thus have γ β ∼ γ α β ∼ γ βα ∼ γ α ∼ γ, and For a system with N discrete points, γ 2 /(E α − E β ) 2 can be much smaller than 1/N, thus we have ReT 01 ∼ 0 and ∆T ∼ T 11 ∼ 1.

C. Expansion of self-energy and a further simplified expression of the width
We can go one step further by expanding the self-energy.In general, the self-energy matrix Σ can be expressed as 43 where χ m, p characterizes the eigenfunction of mode m in lead p.Note that only the values of ψ 0α on the boundary points of the conductor, say, ψ 0α, p , contribute to γ α .Since the boundary of the conductor can be set arbitrarily without affecting the transmission, we can always choose a boundary slice of points to be the same as a slice of the lead.For a slice of n discrete points, since { χ m, p |m = 1, . . ., n} form a complete and orthogonal basis, we can express ψ 0α, p as Substituting Eqs. ( 25) and (26) into Eq.( 22), we have, Since χ † i, p χ j, p = δ i j , we get 51-53 or For a two-terminal quantum dot with reflection symmetry, the eigenfunction also has this symmetry, and its value on the boundary points satisfy ψ 0α, L = ±ψ 0α, R , corresponding to even or odd parity.The self-energy matrices and the eigenfunctions χ i for the two leads are identical, so the contributions of the first-order approximation from the two semi-infinite leads are equal.We have or Since k m a is determined by the Fermi energy via the dispersion relation, the value of γ α is mostly determined by c α m , which is the expanding coefficient of ψ 0α, L on the transverse eigenfunction χ m for the leads.To be specific, we can write The wave function ψ 0α is normalized, i.e., ψ † For dispersive or transmitting states where ψ 0α takes on similar values on all points, the values on the boundary points are of the order of 1/ √ N.However, a localized state will have a large value of ψ 0α, dot on a small subset of points, e.g., points on a particular classically stable orbit in the dot.This will then lead to small values of ψ 0α, L on the boundary points, resulting in small c α m and γ α .This localization effect can be so strong that the values of γ α are several orders-of-magnitude smaller, i.e., 10 −5 t versus 10 −2 t for the dispersive states (See Refs.51-53 and also Fig. 3 in the main text).
When the leads are narrow, they will have only one transverse mode in a relatively large energy interval.In this case, the summation (30) has only one term and becomes where k(E) can be obtained from the dispersion relation, and we have assumed that the dependence of c α m on α is weak and so can be treated as a constant.Figure 3 in the main text shows the results of Eq. (32) together with the numerical results.We find a good agreement.

IV. CONCLUDING REMARKS
To summarize, we have developed a framework based on coherent quantum transport to establish the universality of the Fano resonance profile.Technically, our approach is to decompose the non-equilibrium Green's function into a fast and a slow components, enabling us to derive a general formula for the resonance.In cases where the fast component of the Green's function is dominated by only one eigenstate of the conductor, the Fano resonance profile can be described by Eq. (10), which is equivalent to the Fano formula with either real or complex q parameter.We have provided simple mathematical transformations to connect the three forms.Thus, for a given resonance, depending on the choice of the free parameters, it can be fitted by any of the three formulas with distinctly different q values.Note that for certain cases where the consideration of time-reversal symmetry is important, the choice of Fano formula with complex q parameter is preferred. 15,16,30However, as we have demonstrated in this work, if a resonance can be fitted by Fano formula with complex q parameter, then it can also be fitted by Eq. ( 10) and consequently the Fano formula with a real q parameter.Therefore, a fitting to Fano formula with complex q parameter is not absolutely necessary for any single resonance, although a series of resonances may preferably be fitted by the formulas with a a set of complex q parameters, which may be particularly useful in probing phenomena such as decoherence. 30he width of the resonance, which also characterizes the lifetime of the eigenstate in the originally closed conductor, is mainly determined by the eigenfunction of the conductor, especially the values at the boundary, given that the self-energies vary slowly with the energy.For dispersive or transmitting states, the boundary values of the eigenfunction can be large, leading to a strong coupling between the conductor and the leads.Tunneling to the lead is then facilitated, resulting in a short lifetime and, consequently, in a large resonance width.The conductance curve will appear "smooth" with energy variation.In the opposite case, if a state is highly localized, i.e., on a classically stable orbit, then the eigenfunction assumes large values on this orbit, but can be extremely small anywhere else, including the boundary.The coupling of this state to the leads can be significantly weaker, leading to a long lifetime and a much smaller resonance width.In this case, the conductance curve will exhibit abrupt variations as represented by a sharp Fano resonance.][40][41][42] While we have treated only two terminal transport systems, Fano resonance in multi-terminal systems can be treated similarly, where Eq. ( 10) and the interpretation of the resonant width remain valid.Due to the ubiquity of Fano resonance and the only assumption in our derivation is that the transport can be described by the Green's function formalism (which can be expanded to include decoherence in a proper way 27,30 ), our formula and finding apply not only to quantum transport, but also to wave scattering phenomena arising from diverse fields such as electromagnetic waves, acoustics, and seismology.

FIG. 3 .
FIG.3.Identification of the eigenstate set Ω 0 : for the same graphene quantum dot in Fig.1in the main text, transmission versus energy in a small interval that contains two Fano resonances.The circles below the curve indicate the locations of ε α = E α − iγ α .There is a well separated resonance profile about E 0 , which occurs at ε α .

34 FIG. 4 .
FIG. 4. A demonstration of the energy variations of the different terms in the expansion expression (9) with results from direct numerical calculation.Since T 10 (E) = T 01 (E) † , it is only necessary to plot T 01 (E).

FIG. 5 .
FIG.5.Comparison of Fano resonance profiles predicted by Eq.(10) with from direct numerical computation: (a) the graphene quantum dot system (inset) and the transmission curve for energy in the range 0 ≤ E/t ≤ 1, where t is the nearest-neighbor hopping energy of the graphene lattice, (b-d) numerical (solid curves) and theoretical (dotted curves) profiles of three isolated resonances.