Dephasing by electron-electron interactions in a ballistic Mach-Zehnder interferometer

We consider a ballistic Mach-Zehnder interferometer for electrons propagating chirally in one dimension (such as in an integer Quantum Hall effect edge channel). In such a system, dephasing occurs when the finite range of the interaction potential is taken into account. Using the tools of bosonization, we discuss the decay of coherence as a function of propagation distance and energy. We supplement the exact solution by a semiclassical approach that is physically transparent and is exact at high energies. In particular, we study in more detail the recently predicted universal power-law decay of the coherence at high energies, where the exponent does not depend on the interaction strength. In addition, we compare against Keldysh perturbation theory, which works well for small interaction strength at short propagation distances.


Introduction
The loss of quantum mechanical phase coherence by a fluctuating environment plays an essential role in many different branches of modern physics [1]. It governs the transition from the quantum to the classical world [2,3] and occurs as an unavoidable consequence of any measurement process [4]. It introduces the dephasing time as the time-scale during which it is possible to observe quantum coherent dynamics in qubits and other quantum objects. Furthermore, the interference contrast in man-made interferometers is determined by decoherence. This includes setups for electrons in semiconductors or metals, for single photons, neutrons, neutral atoms and larger objects. In each of those examples various different sources of fluctuations contribute, e.g. thermal radiation, or the interaction with phonons and other particles.
The paradigmatic setup that is treated in much of the literature on decoherence is the following: A "small" quantum system with few degrees of freedom or even a finite-dimensional Hilbert space (atom, spin, qubit, single particle) interacts with an equilibrium environment that comprises an infinite number of degrees of freedom, i.e. a "bath". In the majority of cases, the model of the environment is restricted even further, to consist of a collection of harmonic oscillators (e.g. photon or phonon modes). Quantum dissipative systems of that kind already offer a rich phenomenology, including exponential or power-law decay of the coherence in time, as well as dissipative phase transitions at strong coupling [5,1].
However, there are several situations in which one is forced to go beyond that class of models. We mention three of the most important examples, where the nature of the environment has to be reconsidered. (i) The environment may be driven out of equilibrium (as in the interaction with electrical currents or a photon stream emanating from a laser), which is particularly important for measurement setups. As a consequence, extra fluctuations are introduced and there is no simple relation any more between the dissipative response and the fluctuations. (ii) The environment may be different from a bath of harmonic oscillators, such that the resulting fluctuations are not renormalization are observed. The main results of their study are that at low voltages and temperatures the interference contrast becomes perfect, while the suppression of contrast at increasing voltages and temperatures depends on the details of the interaction potential.
In our work, we will first review the general expression for the current that consists of two parts, a flux-independent term and the interference contribution that will be suppressed by interactioninduced decoherence (section 2). In contrast to [25], we formulate the answer in terms of the Green's function in energy-position space. This has the advantage of corresponding directly to the contribution of an electron at energy that travels a distance x inside one of the arms of the interferometer. Next, we review the model Hamiltonian and the solution by bosonization (section 3). We discuss the general features of the Green's function |G > ( , x)| that has been obtained by numerical evaluation of the exact bosonization expressions, and study the influence of the coupling constant. At low energies, the decay with propagation distance x is weak. It becomes faster when the energy rises above the characteristic energy that characterizes the finite range of the interaction. Finally, at large energies, the decay becomes independent of energy.
The latter asymptotic regime is then the subject of a more detailed analysis using a physically transparent semiclassical picture (section 4). This picture of a single electron interacting with the potential fluctuations produced by the other electrons has been exploited by us in a recent short paper [36] to discuss dephasing in an interacting chiral fermion system. It had been introduced earlier to deal with dephasing of ballistically propagating electrons in contact with an arbitrary quantum environment [17,21], and has also been suggested independently in the context of two interacting Luttinger liquids [30]. Here, we provide more details of the calculation and an extended discussion of the fluctuation spectrum that is seen by the moving electron in its frame of reference. One of the main results is that at high energies there is a "universal" power-law decay |G > ( , x)| ∝ 1/x of the electron's coherence, with an exponent independent of interaction strength. We also analyze the situation at finite temperature, where one has to discuss the transformation of the lab-frame temperature into an effective temperature in the co-moving frame of reference.
Although in principle, for this particular problem, the bosonization solution is fully sufficient, we conclude our analysis with a section on perturbation theory (section 5). This is done in anticipation of going to different setups where an exact solution is no longer possible. Even though we will only discuss the equilibrium Green's function of the interacting system, we employ the nonequilibrium (Keldysh) diagrammatic technique, to provide for a straightforward extension to situations where this is needed. We calculate the self-energy up to second order in the interaction. This includes a diagram that describes the decay by emission of a plasmonic excitation, which is however partially cancelled at low energies by an exchange-type diagram. We will show that at short propagation distances and for modest coupling strengths, the Keldysh result provides a good approximation to the exact solution for |G > ( , x)|, even though its structure in ( , k)-space is qualitatively different from the bosonization result.

The electronic Mach-Zehnder Interferometer
The electronic Mach-Zehnder interferometer is one of the simplest model systems where the interplay of quantum mechanical coherence and many-body effects can be studied, both in theory and experiment.
To set up the description, we first neglect interactions and imagine a single electron moving through the interferometer. The interferometer itself is described as two parallel one-dimensional channels in which electrons propagate into the same direction (see Fig.1a). At two tunnel contacts (i.e. quantum point contacts (QPC's) in the experimental realization), these channels are coupled by tunneling amplitudes t A and t B . Further below, we will assume these tunneling probabilities to be small perturbations, coupling lead 1 (left channel) and lead 2 (right channel). Furthermore, a magnetic flux is enclosed by the interferometer, which leads to an additional Aharonov-Bohm phase φ.
In the experiment the current I through the interferometer, i.e. the current between the two leads, measured at the output port, is the quantity of interest (see Fig.1). It contains two types of contributions: one flux-independent constant term and one interference term that depends oncos(φ). The contrast of the interference fringes observed in I(φ) = I 0 + I coh (φ) can be quantified via the so-called visibility where I max (I min ) are the maximum (minimum) current as a function of flux. This definition is chosen that the visibility is equal to one for perfect interference contrast. This can be used as a direct measure for the coherence of the system. The coherence can be destroyed by the influence of an external bath as well as by internal interactions like the Coulomb interaction between the electrons inside the interferometer.
Treating the interferometer as a many-body system yields expressions for the current through the interferometer, which are not as obvious in a physical sense as in the single particle picture. Therefore the goal of this section is to formulate the quantities of interest in the physically most intuitive and transparent way.

Current
The observable of interest in the present setup is the current through the interferometer (see Fig.1a)due to a finite bias voltage between the two leads, i.e. µ 1 − µ 2 = 0. Dealing with the electron-electron interaction exactly using the bosonization technique has the disadvantage that we have to treat the tunneling between the channels in perturbation theory. In the following we define the quantum point contacts A and B to be at the positions x A j = 0 and x B j = x j , respectively (where j = 1, 2 is the channel index). Then the tunneling Hamiltonian is given bŷ The current into channel 1 is (q e < 0):Î Thus the Heisenberg equation of motion yieldŝ Now we change to the interaction picture with respect toĤ 0 (whereĤ 0 denotes the interacting electron Hamiltonian in the absence of tunneling, which we define in section 3), settingÂ H0 (t) ≡ e iĤ0tÂ e −iĤ0t . We are interested in the steady-state current through the interferometer, which we obtain as a Kubo-type expression, in linear response with respect to the tunneling Hamiltonian, at arbitrary bias voltages: a b Figure 1: (a) Scheme of the interferometer setup. The two channels 1 and 2 of length x 1,2 and the corresponding chemical potentials µ 1,2 are indicated. The electrons can tunnel at QPCs A and B, with tunnel amplitudes t A and t B . By tuning the magnetic flux Φ through the interferometer, one observes an interference pattern I(φ). The solid orange lines denote the current through the interferometer.
(b) A single electron propagating at high energies feels a fluctuating quantum potentialV (t), due to the interaction with the density fluctuations in the sea of other electrons. We show a density plot of the potential, which is produced by the electronic density fluctuations (plasmons) in the channel. The plasmons are moving with a renormalized velocityv (see main text) while the highenergy electron moves with the bare Fermi velocity v F . It picks up a random phase, which leads to dephasing. This is the picture underlying the semiclassical approach in Section 4.

Evaluation of the current
Starting from Eq. (5) and plugging in the apropriate definitions an explicit expression for the current can be found. It will be convenient to express the current in terms of the unperturbed Green's functions of the chiral electron liquids in the two channels. These Green's functions will therefore be the primary object of our subsequent discussions.

Green's functions
In particular, we will analyze the particle-and hole-propagators where we omit the channel index for brevity, unless needed for clarity. The Fourier transforms are obtained as In addition, we will need the spectral density A(k, ω), where The energy-dependent tunneling density of states follows as

Flux-independent part of the current
The flux-independent part of the current is found usingEq. (2), (4) and (5) which we rewrite by going to the frequency domain where ¡ (dω) . . . = ¡ dω/2π . . .. Using the identities where f j (ω) denotes the Fermi function. We can reformulate the expression as a function of the tunneling density of states ν(ω), which finally yields the most intuitive form describing the sum of tunneling currents at two point-like locations: In particular, these expressions show that the flux-independent part of the current only depends on the tunneling density of states. It is independent of the length of the interferometer arms. This is to be expected, as that part of the current is insensitive to the electrons' coherence, and therefore the decay of coherence as a function of propagation distance will not enter here.

Coherent part
The Mach-Zehnder setup is intended to investigate the coherence of the electron system and therefore the most interesting quantity is the coherent part of the current, which we define to be the flux-dependent contribution. Again, using Eq. (2), (4) and (5) it yields : At T = 0, in a situation where the particle current flows from channel 2 to 1, only the first term (and its complex conjugate) contributes.
It might be helpful to see how the structure of this term G > 1 G < 2 can be understood in an intuitive, if slightly imprecise, way, that also relates to our subsequent semiclassical discussion. When the full beam in channel 2 impinges onto the first beam-splitter A, we obtain a superposition between two many-particle states: With an amplitude near unity, nothing happens (no tunneling takes place), and we denote this state as |0 . There is a small chance (of amplitude t A ) for a particle to tunnel through A into channel 1, leaving behind a hole in channel 2. As time passes, the second part acquires an amplitude (relative to the first) that is given by the product of propagation amplitudes for the electron (ψ 1 ) and the hole (ψ * 2 ), resulting in: Including the action of the second beam splitter B, and the Aharonov-Bohm phase, the total probability to detect an extra electron in the output port (channel 1) is therefore which gives rise to the interference term Averaging the amplitudes over phase fluctuations induced by the interaction, we arrive at the propagators, replacing ψ 1 by the particle propagator G > 1 , and ψ * 2 by the hole propagator G < 2 . The full analysis keeps track of energy conservation Thus, in the many-body picture, the observation of an interference term in the current is seen to depend both on the passage of an electron through channel 2 (G > 2 ) as well as the coherent propagation of the corresponding hole, of the same energy ω, in channel 1 (G < 1 ). This issue has been discussed before, both for the Mach-Zehnder interferometer and weak localization [17,21,10,8].

Visibility
In the Mach-Zehnder setup, the so called visibility is used as a measure of the coherence of the system. There are different definitions (experimentally, often the differential visibility is employed). However we will define the visibility in terms of the total current, as The bias voltage is defined as µ 1 − µ 2 = q e V and we set V > 0. We will also focus on zero temperature, as this seems to be the most interesting case. Now the visibility can be written in a compact form (here shown for T = 0): Note that the channel indices of the Green's functions (GFs) are omitted, as in this formula the GFs are defined with respect to a fixed density (and potential u j = 0) and all the explicit dependence on the bias voltage is shifted to the GF arguments. Motivated by the structure of Eq. 15 in the following we will focus the attention on analyzing the function G > j ( , x) in three different ways. First we will apply the bosonization technique, i.e. we will include the intrinsic interaction in all orders. The main disadvantage of the bosonized picture is that we are looking at single particles tunneling between channels, while we are phrasing the description in terms of collective, bosonic excitations which prohibit a simple physical picture of the process of dephasing. Therefore, as an alternative point of view, we will discuss a transparent semiclassical model for electrons propagating high above the Fermi energy, subject to the fluctuations produced by the rest of the electrons. Finally, we will complement this analysis by studying the system in Keldysh perturbation theory for the fermions, up to second order in the coupling strength.

Solution by bosonization
In this section we recall how to obtain the Green's functions needed here via bosonization [37,38]. Experienced readers may skip the section, and can refer to it later regarding the notation.

Hamiltonian and formal solution
We start from interacting right-moving chiral fermions in the two channels j = 1, 2, employing a linearised dispersion relation (we set = 1): where v F denotes the Fermi velocity and u j is a constant which fixes the chemical potential of the channel. The particle operators for the chiral electrons arê where L describes the size of the artificial normalization volume (with L → ∞ in the end). We also introduce the density fluctuations within each channel: such that the Fourier components are given bŷ withρ −q,j =ρ † q,j . The average densityρ j enters µ j = u j + 2πv Fρj . As we take the two interferometer channels to be spatially separated, we only have to take care of intrachannel interactions. Transforming the interaction Hamiltonian into momentum space yieldŝ where are the Fourier components of the interaction potential. We construct bosonic operators from the Fourier components of the density in the standard way. As we are only dealing with chiral electrons, we only need to consider q > 0 in the following: These operators fulfill As usual, the main advantage of bosonization consists in being able to write the kinetic part of the Hamiltonian as a quadratic form in boson operators: The interaction part of the Hamiltonian reads: Thus, the Hamiltonian is already in diagonal form, with the plasmonic dispersion relation of an interacting chiral 1D electron system: For the following discussions, we introduce the dimensionless coupling constant α = U (q→0) 2πv F , where α ∈] − 1, ∞[. The renormalized plasmon velocity at small wavenumbers isv = v F (1 + α). Negative values of the coupling constant are related to attractive interations, positive values to repulsion (at small wavenumbers). For α → −1 the plasmon velocity tends to zero,v → 0. For α < −1 the system is unstable, i.e. formally ω(q) < 0 for q > 0.
The final step is to express the single-particle operators using bosonic fields [38,37] The ultraviolet cutoff length a is sent to zero at the end of the calculation. The Klein operatorF j annihilates a fermion in a spatially homogeneous way, with the following commutation relations: The explicit time dependence of the Klein operators is obtained by using the Heisenberg equation of motion, i.e. dF j /dt = −i[F j ,Ĥ] = −iµ jFj ⇒F j (t) = e −iµj tF j (0). In the end taking into account the relation k F,j = 2π LN j we arrive at

Green's function from bosonization
Now we are able to evaluate the Green's function defined above explicitly using the bosonized single particle operatorsΨ. The calculation is done quickly using the fact, that the Hamiltonian in terms of the bosonic operators is quadratic, i.e. the fieldΦ[b,b † ] can be treated like a Gaussian (quantum) variable, resulting in and analogously for G < . By factoring off the non-interacting Green's function G >/< 0 , we can write where G with β ≡ 1 T and k B ≡ 1. All the effects of the interaction now are included in the exponent where we have to subtract the non-interacting contribution:

Discussion: Green's function in space and time
Here we discuss the Green's function as a function of space and time. A more detailed discussion can be found in [25]. The absolute value of the Green's function G > (x, t) is shown in Fig.2, at zero temperature (to which we restrict our discussion).In the following all the numerical evaluations are performed using a generic smooth interaction potential U q = U 0 e −(|q|/qc) s . We note that all the qualitative results are valid for potentials which are finite at zero momentum (U 0 = 0) and which are cut off beyond some momentum scale. Those assumptions are not restrictive and for example are fulfilled for a Coulomb potential with screening in a quasi one-dimensional channel of finite width.
In Fig.2, we observe as the main feature that the Green's function splits into two parts during its propagation. One of those propagates with the bare Fermi velocity v F and represents the unperturbed Green's function, i.e. the high energy part. For increasing time its weight decreases, i.e. the amplitude of the bare electron to arrive at x without being scattered decreases. The other peak represents the low energy part, stemming from energies below − µ ∼ v F q c . It moves with the renormalized velocityv.
We can obtain this structure of G > (x, t) from a crude approximation. Namely for T = 0 in a first approximation we can cut the momentum integral due to the fact that for q q c the integrand vanishes, i.e.
The integrals are known and yield Therefore the structure of the Green's function is given by displaying both the δ peak at x = v F t and the broadened peak at x =vt, whose width is set by q −1 c . In Fig.2 one can observe the fairly good agreement between the full result and this first approximation.  . In the high-energy limit, the semiclassical solution is valid for arbitrary coupling strength. However, the energies for which the description is valid become larger for increasing coupling strength. In (b) this limit is beyond the presented energy interval. Therefore here we do not show the semiclassical solution.

Green's function vs. position and energy
As shown above in Eq. (15) and (20), the current through the interferometer is determined by the propagators G >/< ( , x). Therefore, in the following our main focus will be on this function, which can be thought of as the amplitude for an electron of energy to propagate unperturbed over a distance x. The function is shown in Fig. 3, where we plot the numerical evaluation of the exact result obtained using the bosonization technique. This is done for two values of the coupling strength α and for different interaction potentials. There are some main features which can be observed by having a brief look at Fig. 3, while for a detailed discussion we refer to the following section.
(i) At x = 0, where |G > (x = 0, )| = ν( ) equals the tunneling density of states, there is a finite dip at low energies. This is a static interaction effect. For repulsive interactions it represents the suppression of the tunneling density by a factor v F /v, due to the interaction-induced increase of the velocityv. At high energies ( v F q c ), the non-interacting density of states is recovered. (ii) At any fixed energy , the Green's function decays with increasing propagation length x. The shape of the decay (as a function of x) becomes independent of energy for high energies. In contrast, the decay is suppressed for energies below ∼ v F q c , and there is no decay in the limit → 0. The decay of the GF is equivalent to dephasing (since in our model there are no interbranch interactions and correspondingly no vertex corrections). As a consequence, the absence of decay at zero energy will lead to perfect visibility at T = 0, V → 0.
(iii) At larger x, there are oscillations in the Green's function. These result from the doublepeak structure in the time-domain, with peaks at x = vt and x =vt. These lead to a beating term Therefore, the period of oscillations in the energy domain is determined by the difference between the bare and the renormalized velocity (see Fig.3), viz.:

Large coupling constants
In this section we want to discuss briefly the shape of the Green's function in terms of the coupling strength. We emphasize that, once the shape of the interaction potential is given, the only dimensionless parameter left is the coupling constant α = U 0 /(2πv F ). All the other parameters may be absorbed into a rescaling of the result.
In Fig. 4 we show |G > ( , x)| as a function of energy for various coupling strengths (different curves), both at x = 0 and at some finite propagation distance x = 0. For small coupling α > 0, we just observe the suppression of the tunneling density of states discussed above. Upon increasing the coupling strength, a series of rounded steps emerges, suppressing the tunneling density even further. The same features can be seen in the shape of the GF at finite x, though there they are superimposed by the decay (describing decoherence) and the oscillations as a function of energy (discussed in the preceding section). To identifiy the oscillations in energy which we observe even for small coupling strength in Fig.4c we divide the GF for x = 0 showed in Fig.4b by the tunnel density of states. As expected, those oscillations are robust against a change in the coupling strength. We have not found any simple analytical model to discuss the structures observed here. However, note that in Fig.3b one observes that the step structure is more pronounced for the Gaussian potential compared to the results for the exponential shape. That shows the strong influence of the shape of the interaction potential on the step structure.
We note that the previous discussion in the ( , k)-space (as opposed to ( , x)) had found nonanalytic structures for the case of a box-shape potential U q [34].

Semiclassical model of dephasing
Up to now we have applied the bosonization technique in order to get exact information about the decoherence the electron suffers while passing through the interferometer. However we actually do not know what is going on in more physical terms. First of all, we do not know in detail how to distinguish between the single electron we are considering while traveling through the interferometer and the bath electrons which are present at the same time. In a Fermi liquid there is no question about the nature of the single electron, i.e. in the vicinity of the Fermi edge it can be described as a quasiparticle. In contrast, in the bosonization approach everything is described in terms of collective bosonic excitations, i.e. density fluctuations. Therefore, as soon as the electron tunnels into the interacting system, there is no way to trace this special electron any longer. This fact leads to difficulties in understanding the decoherence intuitively, since in the moment of tunneling the coherent phase information is encoded into the bosonic degrees of freedom. Fortunately, it turns out that indeed it is possible to find simple physical pictures which are helpful in understanding the process of dephasing in more detail besides the mathematical solution.
For energies much higher than the Fermi energy (to be made more precise below), it turns out that it is possible to describe the interaction of a single propagating electron with all the other electrons, by viewing them as a bosonic quantum bath. To see this, in section 4 we apply an intuitive, semiclassical method which is able to reproduce exactly the results from the full bosonization formalism in the high energy limit.
In a recent work [36] we have already briefly reported on universal dephasing for high-energy electrons at long distances for T = 0, based on the semiclassical approach to be discussed in more detail in the following.

Semiclassical approach to the Green's function
Electrons at high energies propagate at the unperturbed speed v F , as can be observed from the corresponding limiting behaviour of the plasmonic dispersion relation. The decoherence, i.e. the Green's function, can be described in a transparent semiclassical framework, that becomes exact in the limit of high energies. We will confirm this later by comparing against the full bosonization solution. Within this semiclassical picture, one thinks of the electron propagating through the channel, while accumulating an additional phase due to the interaction with the bath formed by all the other electrons. To model the effective, bosonic bath acting on the single electron, we make use of the plasmonic dispersion relation which was derived using the full bosonization technique (Eq. 33). As the electron we consider is flying high above the Fermi sea, we can neglect the backaction of the electron onto the bath. In this picture the electron only experiences the intrinsic fluctuations of the bath. The potential acting on such a single high-energy electron is obtained by convoluting the density fluctuations with the interaction potential: Note that this definition implies, that the effective potential fluctuationsV (t) experienced by the single electron are just the fluctuations of the bath evaluated at the classical electron position x = v F t at time t. This is why we call this model "semiclassical". If we were dealing with a classical fluctuating potential V (t), the electron would simply pick up a random phase ϕ(t) = − ¡ t 0 dt V (t ). In that case the non-interacting Green's function would have to be multiplied by a factor e iϕ(t) to obtain the correct Green's function. However, if the quantum nature of the bath becomes important one has take care of the non-commutativity of the operatorV (t) at different times. This can be done by introducing a time-ordering symbol: The time t = x/v F in Eq. (46) is determined by the propagation length. This is actually identical to the decay of coherence of a single level whose energy fluctuates. In various contexts, this is known as the "independent boson model" [39,40], or the case of "pure dephasing" in a (longitudinally coupled) spin-boson model [1]. We note that the same kind of approach to dephasing of ballistically propagating electrons has been introduced previously, both for a situation with a general quantum bath [17,21,10], as well as for two coupled Luttinger liquids [30].
Furthermore we note that the decay is independent of energy . This is because the propagation speed is energy-independent, and the distance to the Fermi edge becomes unimportant at high energies as well. Qualitatively, we have seen this feature before in our discussion of the full bosonization solution.
In summary, the decay of coherence, described by F (t), is completely determined by the fluctuation spectrum VV ω = ¡ dt e iωt V (t)V (0) of the potential seen by the electron in the moving frame. To proceed further we express the time-ordered correlator TV (t 1 )V (t 2 ) as a sum of commutator and anti-commutator part: The real part of F (t) and therefore the decay of the Green's function depends on the symmetrised part of the correlator. This part is formally similar to the correlator of classical noise, though it also contains the zero-point fluctuations of the plasmon field: In addition, a phase −ImF (t) shows up in the exponent. It is due to the commutator ofV , and thus it represents a purely quantum mechanical contribution. In terms of the Fourier transform of the spectrum, this yields From Eq. (45), we obtain for the potential spectrum in the co-moving frame The argument ω + v F q indicates that we are dealing with the Galileo-transformed spectrum of the density fluctuations. As a result, the spectrum of the density fluctuations gets tilted compared to the original dispersion relation (see Fig. 5). Making use of the plasmonic dispersion ω(q), which we obtained from the bosonization method, the density-density correlator yields We obtain The symmetrized correlator can be written as Plugging in these correlators in Eq. (48) and Eq. (49), finally yields F (t): Indeed, up to an additional energy renormalization − is identical to the one stemming from the bosonization technique for x = v F t [compare to Eqs. (40) and (41)]. This constant energy shift had been incorporated into a redefinition of the chemical potential, as noted in the section on bosonization.
To be precise, Eqs. (55,56) show that the decoherence of electrons moving along the trajectory x = v F t can be described exactly in terms of the semiclassical approach (which is based on the assumption of high-energetic electrons). In order to prove the statement that the decoherence at large energies can be understood within the semiclassical description, one has to confirm the initial assumption that high energetic electrons actually move with the bare Fermi velocity v F . Already, in section 3.3 we observed that the Green's function G > (x, t = x v F ) determines the high-energy regime of the Fourier transform G > (x, ). This followed from the analysis of the Green's function G > (x, t). Beside a broad peak moving withv, one observes a sharp peak in time, moving along the trajectory x = v F t, which is obviously responsible for the contributions to G > ( , x) at large energies. In addition, we confirm the assumption numerically. As we are not able to perform the Fourier transformation analytically, instead Fig. 5 shows the numerical evaluation of the function e −F (x) and the direct Fourier transform of the Green's function G > (x, t) resulting from the bosonization in the limit → ∞. Obviously, those turn out to be identical. Therefore we conclude that the semiclassical approach is becoming exact in the limit of high-energy electrons. We emphasize that, apart from the transparent interpretation of the decoherence of high energetic electrons via the semiclassical method, even the observation that the Fourier transform of the Green's function at high energies turns out to be identical to [G > (x, t)/G > 0 (x, t)]G > 0 ( , x) with x = v F t is non-trivial and serves as a starting point for the analytical discussion of the decoherence in this regime.
Having established the connection between the semiclassical approach and the full bosonization solution, we now turn to the properties of the symmetrized potential spectrum. For this, first we focus on the zero temperature case T = 0 (see Fig. 5). At high frequencies, we obtain a singularity {V ,V } T =0 ω ∝ 1/ ω max − |ω| at the cutoff frequency ω max = max(ω(q) − v F q), which is the maximum frequency in the Galilei-transformed plasmon dispersion relation. Such a maximum frequency arises due to the momentum cut-off in the dispersion relation (which results from the finite range of the interactions). Due to this smooth momentum cut-off, the velocity of plasmons in the limit of large momenta is identical to the Fermi velocity v F of the electron. By transforming the potential to the moving frame, it gets tilted (see Fig.5). As the velocity of the plasmons in the limit of large momenta is identical to the velocity of the moving frame, the effective dispersion relation shows a maximum. The singularity in the spectrum arises due to the fact that ω(q) ≈ ω max +ω (q −q * ) 2 /2 in the vicinity of q * , where ω(q * ) = ω max . Note that an interaction potential U q with a nonmonotonous decay in q may give rise to several such singularities, corresponding to the local maxima of ω(q) − v F q.
At low frequencies ω v F q c , the spectrum increases linearly in ω, corresponding to "Ohmic" noise, which is ubiquitous in various other physical contexts [1]. For interaction potentials that are smooth in real space (i.e. where all the moments of |U q | are finite), we find that the leading low-frequency behaviour is determined solely by the contribution to Eq. (50) stemming from small q. The result is (here for α > 0): Most remarkably, the dimensionless prefactor (the slope) of the noise spectrum turns out to be completely independent of the coupling strength α, which drops out. This is in contrast to the typically studied non-chiral Luttinger liquids, where an Ohmic spectrum has been found with an interaction-dependent prefactor [30]. As a direct consequence, the electron's Green's function shows a universal power-law decay at long distances, as we will discuss in more detail in the next section.

Universal dephasing for high-energy electrons
We insert (57) into the long-distance limit of Eq. (48): where the choice of ω c ω max is arbitrary (but related to the constant), and |G > ( → ∞, x)|v F = e −Re[F (x)] . As a consequence of the logarithmic contribution to the exponent, the leading asymptotic behaviour of the Green's function is a power-law decay with an interaction-independent exponent 1: We recall the fact that Eq. (59) refers to the Green's function in energy-coordinate space, as a function of propagation distance x at large energies. Having shown previously that the current and thus the visibility only depend on this function (see Eqs. (15,20)), we may denote it the 'coherence' of the electron. Therefore, Eq. (59) shows that the coherence itself displays a universal powerlaw decay. Please observe that in the absence of interactions |G > ( , x)| would be constant, so the decay indeed implies interaction-induced decoherence. The surprising feature is the independence  Figure 6: Coherence of an electron propagating at high energies in an interacting chiral system, as a function of propagation distance for various values of interaction strength α: 2πα = 0.1 (solid black line), 2πα = 1.0 (solid green line) and 2πα = 5.0 (solid red and blue). The potential is taken as U q = U 0 e −|q/qc| and temperature is zero except for the red line where T /q c v F = 0.005. The non-interacting case would give v F |G > ( , x)| ≡ 1. The long-distance decay is universally given by ∝ 1/x, independent of interaction strength. Note that for decreasing coupling strength the asymptotic power-law decay sets in for increasingly larger propagation distances. At finite temperatures, this power-law decay turns into an exponential decay for large x with a decay rate Γ ϕ depending on interaction strength (inset). of the exponent from the strength of the interactions (while the prefactor, not displayed here, would indeed depend on the details of the potential).
In writing down Eq. (58), we have neglected the contributions of large momenta in Eq. (50). These will lead to a subleading correction to the power-law, which we discuss below. The third term in Eq. (58) is responsible for oscillations in the coherence |G > ( , x)|, on top of the decay (see below). Fig. 6 shows the decay of G > ( → ∞, x) for different coupling constants α.
In order to understand how this generic result for the asymptotic decay is compatible with the non-interacting limit (α = 0, where |G > ( , x)| is constant), we have to discuss the range of validity of the asymptotic behaviour. As the linear slope in the effective spectrum V ,V ω>0 applies only at |ω| ω max , we must certainly require ω max x/v F 1.
Since ω max vanishes linearly with α, the limiting regime is reached at ever larger values of x when the interaction strength is reduced. This shift of the range of validity is shown in Fig. 6.
The oscillatory modulation is due to the square root singularity at ω → ω max in V ,V ω>0 .
Its amplitude depends on the interaction strength |α| but vanishes at long distances (see appendix A.1): where C is a numerical constant, which solely depends on the explicit form of the potential. We now discuss the deviations from the leading low-frequency behaviour of V ,V ω . Taking into account the contributions resulting from large q in Eq. (50), for smooth potentials like U q = . In real space this expression translates into a correction 1 s ln(ln(|α|q c x)) to the decay function . At finite temperature T = 0, the long-time limit is given by an exponential decay |G > (x, )| ∝ exp[−Γ ϕ x/v F ], with a decay rate This follows from the long-time limit of Eq. (48) together with Eq. (54): Using the identity lim a→∞ sin 2 (ax) ax 2 = πδ(x), we get Re[F (x)] = πT |1 + α −1 | −1 x, from which we obtain the aforementioned decay rate Γ ϕ . In Fig. 6 the decay rate is shown as a function of the coupling α. For small α, this rate vanishes as Γ ϕ = πT |α|, i.e. it is non-analytic in U 0 ∝ α. Such dephasing rates proportional to T have also been found in non-chiral Luttinger liquids [31,30,32]. At large repulsive coupling, U 0 → +∞, we have the universal result Γ ϕ → πT . For attractive interaction, Γ ϕ diverges upon approaching the instability at α → −1, wherev → 0 and where the resulting low-frequency modes are thermally strongly excited.
We note that this behaviour is somewhat surprising when compared to other problems of dephasing. When considering pure dephasing of a two-level system by an Ohmic bath, a power-law decay t −γ at T = 0, with an exponent γ set by the coupling, automatically implies an exponential decay at a rate Γ ϕ = πγT at finite temperatures. This follows from the fluctuation-dissipation theorem (FDT) which turns the T = 0 Ohmic spectrum into a white-noise spectrum with a weight proportional to T .
However, in the present case, we have to take into account the Galileo transformation, which turns the laboratory-frame temperature T into an effective temperature T eff in the frame moving along with the particle at speed v F . In order to establish the FDT for the effective potential fluctuations in the electron frame of reference we therefore have to define the effective temperature T eff by demanding for ω ↓ 0 In the low-frequency limit we get V ,V ω = 2πω · coth ω 1 + α −1 /(2T ) . Together with Eq. (52), this yields the effective temperature T eff ≡ T 1 + α −1 −1 . In the moving frame the frequencies are reduced by q c v F and therefore the effective temperature is also smaller.Only for large repulsive interactions (v v F ), the transformation does not matter. Therefore, in this limit T eff = T and the universal, coupling-independent power-law for T = 0 turns into a universal decay rate at finite temperatures.
Finally, turn briefly to the question of observing these features in experiments. We note that the universal power-law decay of the Green's function should be observable in principle in the Mach-Zehnder interferometer setup, as it directly translates into a decay of the visibility itself. To obtain a numerical estimate, we assume a screened Coulomb potential with screening length q −1 c ∼ 10 −7 m and a finite channel width b ∼ 10 −7 m. Following [25] the edge state velocity can be assumed to be: v F ∼ (10 4 − 10 5 ) m s . Then the dimensionless coupling constant α is of the order 1. To reach the high energy limit, the applied bias voltage V has to fulfill: q e V q c v F . For v F = 10 4 m s , it turns out that one has to apply V ∼ 10 2 µV which is in the range of the bias voltages typically applied in experiments (in [12] V ∼ 10 1 µV ). We mention that the long-distance limit should be reached for x v F /ω max ∼ 10 −1 µm, which is shorter than the typical interferometer arm length (e.g, in [12] x 1,2 ∼ 10 1 µm). Thus there is some hope that the high-energy as well as the long-distance limit is accessible in the experiment. However, we note that the magnitude of the Green's function G > ( , x) gets suppressed strongly when reaching the long-distance limit, i.e. x v F /ω max . For α = 1, the magnitude of |G > ( , x)| yields: |G > ( , x)| ∼ 10 −2 (see Fig.6). With help of Eq. (20) this translates into a visibility of the order v I ∼ 10 −4 (for this see also section 6). Therefore, the direct measurement of the power-law dependence of the visibility for large bias voltages seems to be a challenging task, unless one finds a way to optimize further the relevant parameters.

General remarks
The semiclassical method we introduced in the preceding section provides us with an intuitive picture for the case of electrons flying high above the Fermi sea. To complete the analysis, in the present section we employ perturbation theory to discuss the behaviour at short propagation distances and weak coupling for all energies. This includes the low energy regime, where the influence of the Fermi edge becomes important. We will apply Keldysh (i.e. nonequilibrium) perturbation theory up to second order in the interaction strength α. The main outcome of the perturbation theory is that the tunnel density of states is affected by renormalization effects while the decay of the GF in the close vicinity of the Fermi edge is suppressed. We will find that the suppression of decoherence is brought about by a cancellation between two second order diagrams near the Fermi edge. During this subsection we fix the notation, following the review [41]. Using the Keldysh time leads to an additional matrix structure of the GF which reflects the fact that one has to differentiate between points in time which lie on the backward or the forward branch of the Keldysh contour. Beyond this additional structure, all the known Feynman rules remain exactly the same. After performing the rotation in Keldysh space [41], the representation of the matrix GF G and the related matrix self-energy is given by where we introduce the Keldysh GF, . First we will derive an expression for the retarded self-energy Σ R ( .k), which will be used to calculate G R that can be related to the single particle propagator G > in equilibrium. Starting from the matrix Dyson equation G( , k) = G 0 ( , k) + G 0 ( , k) · Σ( , k) · G( , k), one finds that the retarded Green's function only depends on the retarded self-energy: In the following we calculate the diagrams up to second order for a linearized dispersion relation, but for finite temperature and for an arbitrary interaction potential. In the end we compare the results of the perturbation theory with the results of the bosonization technique. The relevant processes are shown in Fig.7. There are two second order diagrams, which can be identified as the interaction with a plasmonic excitation and a corresponding diagram containing an additional exchange process (that can be viewed as a vertex correction diagram). The crucial point is that the vertex correction counteracts the plasmonic processes in the vicinity of the Fermi edge, leading to a suppression of the decay of the GF.

Evaluation of the diagrams
The starting point of the calculation is the evaluation of the the unperturbed electronic propagator matrix G 0 . In addition to the usual retarded and advanced GFs of free electrons, G In contrast to the advanced and retarded GFs, the Keldysh propagator contains information about the electronic spectrum as well as about the occupation of those states. Therefore at this point one could introduce arbitrary non-equilibrium states which is the main advantage of working on the Keldysh contour. However, as we are describing channels which are only weakly tunnel-coupled to each other, we will calculate equilibrium Green's functions.
As in second order we explicitly include interactions with free plasmons, we also derive their propagators here. To this end we identify the bosonic field with the potentialV ( . This is identical to the potential we introduced in the semiclassical description [although in the latter case we specialized to x ≡ v F t]. The bosonic propagators are defined as

respectively. A straightforward calculation yields for the retarded and advanced propagators
As we assume the system to be in equilibrium, we use the FDT to obtain the plasmonic Keldysh propagator: Now we can proceed calculating the various contributions to the self-energy up to O(α 2 ). Considering all the possible Feynman diagrams, we are left with the first order Hartree-Fock diagrams and in second order with the plasmon diagram and the vertex correction (see Fig.7). Thus we can express the self-energy as Σ R (2) = Σ Hartree + Σ Fock + Σ Plasmon + Σ Vertex . Those can be evaluated according to the rules given in [41]. We use the equal-time interaction propagators U R/A (q) ≡ U q and set U K ≡ 0, as usual.
First-order contributions: Hartree-Fock diagrams The Hartree diagram yields a global energy renormalization ∆E Hartree = U (q = 0)ρ. In the framework of the Luttinger model the electron density diverges as there is no lower boundary of the electron spectrum. However, formally one can include the energy shift into the definition of the chemical potential (see subsection 3.1). In the following we omit the Hartree contribution.
The self-energy contribution due to the Fock diagrams is which for zero temperature yields Σ Fig.8b].
The resulting k-dependent energy shift describes, in particular, the renormalization of the eletron velocity near the Fermi edge. This also affects the tunneling density of states, leading to a suppression (for repulsive interactions, α > 0) or enhancement (α < 0). This can be seen in Fig.3. Again, the constant shift can be incorporated in the definition of the chemical potential.
Second-order contributions: Plasmonic excitations and vertex correction The electron's coherence decays by interacting with the plasmons, i.e. the density fluctuations of the other electrons. The plasmon diagram (see Fig. 7) represents one of the contributions describing this physics. It yields where the second term contains the Fermi function, which introduces the effects of the Fermi edge on the coherence. Inserting the propagators, the contribution can be written in a compact form Thus, at T = 0 this contribution vanishes for k → 0. We note in passing that the structure "coth + tanh" generically occurs in discussions of dephasing, where it describes both the strength of the thermal fluctuations and the influence of the Fermi function, i.e. the physics of Pauli blocking [42,17,21,10,8,9]. In the limit of high energies, the result of Eq. (70) can be rewritten in terms of the potential fluctuations at the particle position, as discussed in the preceding section. Specifically, we have lim k→∞ Σ R P,T ≡0 = G R 0 · V (x = 0, t = 0) 2 . For a plot of the function Σ R P,T =0 , see Fig.8c. Finally we derive the vertex correction, mentioned above, which after a rather lengthy calculation yields (see appendix B) This expression simplifies for The contribution from the vertex correction as well as the total second order correction to the self-energy σ R P+V ≡ Σ R P + Σ R V are shown in Fig.8. The crucial feature is that up to second order in momentum k the plasmon diagram and the vertex correction cancel exactly against each other, whereas for high momenta [k q c ] only the plasmon contribution remains while the vertex correction tends to zero. In summary, when calculating up to second order in the coupling, we already see that the dephasing is suppressed in the vicinity of the Fermi edge. As the comparison with the exact bosonization solution shows, this conclusion holds true qualitatively to all orders. We can the preceding expressions to evaluate the retarded Green's function G R and from this the propagator G > . The final results has evaluated numerically. The results are illustrated in Fig.8. Expanding the Green's function for small propagation distance , yields: which coincides with the expanded exact result. The good agreement of the bosonization result and the Keldysh perturbation theory for small |α| is shown in Fig. 8, as well as in Fig. 5. However, for large x the perturbation theory fails. For a detailed study of chiral interacting electrons of the spectrum in ( − k)-space, starting from the bosonization result, we refer the reader to [34].

Summary of the Keldysh perturbation theory
To summarize, the perturbation theory shows that there are two different energy regimes: In general, the GF decays as a function of propagation distance due to the interaction with the density fluctuations. This is particularly pronounced at high energies v F q c , where we have also shown that the Keldysh result and the semiclassical (or bosonization) approach coincide at short distances. For low energies v F q c , the decay is suppressed. On the other hand, at low energies the Fermi velocity is renormalized due to virtual processes, leading to a modification of the tunneling density of states. It is important to note that these two different energy regimes only emerge since we are dealing with interaction potentials of finite range. The curve for x ≡ 0 is equal to the flux-independent current I coh (x ≡ 0, V ) = I 0 (V ) (red line), which implies that the interference contrast is perfect at zero armlength. b) The visibility v I as a function of bias voltage V for various arm-lengths x 1 = x 2 = x. The red line denotes the semiclassical calculation. The small deviations from the bosonization result vanish completly for larger V . The plot is done for U q = U 0 e −(q/qc) 2 with U0 v F = 2πα = 3.

Visibility and Current
As mentioned in the beginning, the results for the GF G > ( , x) we worked out in the foregoing section can be applied directly to the evaluation of the current and the visibility (Fig.9). Fig.9a shows the current through the interferometer as a function of voltage for different arm lengths. Here we restrict the considerations to the symmetric case, i.e. x 1 = x 2 . For x = 0 the coherent part of the current obviously is identical the flux-independent part, which implies a perfect visibility (at T = V = 0). The suppression of the current at small voltages is due to the velocity renormalization which lowers the tunnel density (for repulsive interactions). However, as the change in the tunnel density influences the classical and the coherent part in the same way, it does not show up in the visibility at all, i.e. v I (V, x = 0) ≡ 1 (see Fig.9b).
The behaviour of the coherence of the GF for high energies discussed above (using the semiclassical approach) can be transferred directly to the discussion of the visibility. Therefore, in the limit of high voltages V and T = 0 the visibility is determined by the factor |G > ( → ∞, x)| 2 . This follows from the fact that for higher voltages the contribution of the high-energy electrons becomes dominant. It also implies that the visibility at high voltages becomes voltage-independent. Correspondingly, the exponential decay at finite temperature is transferred to the visibility v I as well. For x = 0 the dephasing reduces the coherent (flux-dependent) part of the current which leads to a decrease of the visibility. At small voltages |q e V | q c v F the visibility decays only very slowly with increasing interferometer length (see the discussion of the Green's function). In the limit V → 0 the visibility is approaching unity v I → 1 , which is consistent with the fact that in equilibrium and at zero temperature there is no dephasing. In contrast to the Green's function itself, which shows oscillations as a function of , the visibility does not show pronounced oscillations as a function of V .

Conclusions
In the present paper, we have studied dephasing by electron-electron interactions in a ballistic interferometer. We have considered the case of electrons moving inside ballistic, one-dimensional, chiral channels, such as edge channels in the integer quantum Hall effect. The interference contrast, in the limit of low transmission at the beam splitters, can be expressed via the Green's function of the interacting system. We have studied the decay of the electron's coherence as a function of propagation distance, employing three different approaches: The exact bosonization solution, a semiclassical approach that becomes exact for high energies, and diagrammatic nonequilibrium (Keldysh) perturbation theory. Our most important physical result is that at high energies the decay of coherence at T = 0 becomes a power-law with a universal exponent, independent of interaction strength. We have also shown that second-order perturbation theory compares well with the exact solution at modest coupling strength and short distances. This may be important for potential applications to more elaborate setups that cannot be treated exactly any more.

A Semiclassical approach
A.1 Amplitude of the oscillations in G > ( , x) as a function of x for T = 0 The third term in Eq . (58) is responsible for the oscillations in G > ( → ∞, x), while the main contribution stems from the square-root singularity at ω = |ω max |. This contribution yields: From Eq. (54) we obtain the ω-dependence of the correlator in the vicinity of the singularity where ω(q * ) = max(|ω q − v F q|), q 0 ≡ q * + |ωmax|−ω |ξ| Evaluating the integral in the limit x → ∞ gives where C denotes a numerical prefactor which depends only on the form of the interaction potential, not on the interaction strength α.
A.2 Sub-leading corrections for smooth potentials U q at T = 0 The subleading corrections result from the contributions of large q in Eq. (54) (here ω > 0). We start with the expression The sub-leading correction to the low-frequency behaviour of {V ,V } ω for a smooth potential U q = U 0 e −|q/qc| s is given by where q sub fulfills: 2πω |U0|qc = q sub qc e −|q sub /qc| s . Therefore, as q q c for ω small enough we obtain for q sub q sub = q c ln q c |U 0 | 2πω Finally, the subleading corrections yields for ω |α|v F q c .

A.3 Sub-leading correction in time-domain for T = 0
The subleading correction to the decay function F (x) results from the expression F (sub) = With Eq. (80) (omitting an additional constant) it yields where for reasons of brevity we set |α|v F q c ≡ ω * . As we can choose the cutoff frequency ω c such that ω c /ω * < 1, we can assume 0 < ω < 1. This enables the substitution ω = e −x :

B Keldysh perturbation theory: Vertex correction
The vertex correction to the retarded self-energy has the structure (compare to the review of [41]):