Field theoretical approach to quantum transmission in time-dependent potentials
We develop a field theoretical approach based on the temporary basis description as a tool to investigate the transmission properties of a time-driven quantum device. It employs a perturbative scheme for the calculation of the transmission of a monochromatic beam of particles through the time-dependent set-up. The main advantage of the proposed treatment is that it permits the use of the particle picture for the calculation of the scattering matrix and the transmission coefficient. Therefore the elementary physical processes contributing to the transmission can be identified and interpreted in a transparent way. We apply the method to the simple but prototype problem of transmission through an one-dimensional oscillating delta potential and we demonstrate how it enables a deep understanding of the underlying physical processes.
The transmission of quantum particles through a time-dependent potential has been the subject of extensive studies in the last three decades (see (1) and references therein). The main goal in these studies is to classify and understand the mechanisms of quantum tunnelling in such a potential. Typical examples of processes involving transmission through time dependent devices are the tunnelling of a test particle from a metastable state (2) using instanton techniques, the scattering off a dissipative environment (3) and the tunneling through a time-modulated barrier (4). Particularly in (4) it was shown that at low modulation frequencies the traversing particle sees a static barrier and at high frequencies the particle tunnels through the time-averaged potential. It was also demonstrated that inelastic processes can occur, where the tunneling particle gains or loses energy quanta from the modulation field. In the same work a fundamental question was raised concerning the definition of the traversal time through such a fluctuating device, an issue which is still under debate (5). Recently novel perspectives for the study of driven quantum systems emerged. In particular it has been realized that there is the possibility to exploit phenomena occurring in these systems for the development of novel technology such as the design of quantum pumps, i.e. devices capable to create quantum directed transport through periodic external driving (6), or the control of quantum devices for information processing (7).
One of the simplest examples for studying the transmission in a time-dependent environment is the tunnelling through a delta-barrier with a harmonically oscillating coupling which has been considered by several authors (8); (9); (10). Although the device is very simple, the presence of the time-dependent coupling leads to interesting phenomena, observable in the transmission coefficient, such as Fano resonances, threshold enhancements related to sideband modes and their interplay (9). More complicated setups, involving several oscillating delta barriers, have been investigated in the context of quantum pumps (11). In this case dephasing effects may also occur, influencing the appearance of resonances (12). In (13) an infinite periodic chain of delta-barriers with harmonically oscillating strength has been studied demonstrating the modification of the conductance zones through the time-dependence.
If the driving of the potential barrier is periodic the standard approach used to solve the quantum dynamics is based on Floquet theory (14) which is very well suited for the straightforward calculation of the transmission properties, discriminating between the contributions from elastic and inelastic channels. In the Floquet treatment the considered problem is first rendered time-independent and then solved using standard numerical techniques. This represents certainly an advantage from the computational point of view. However, it is conceptually a handicap since it does not allow to isolate contributions or identify mechanisms leading to a specific dynamical behaviour that allow for a better understanding of the underlying physics.
Path integral methods, on the other hand side, have been commonly used for the study of the quantum dynamics in time-dependent potentials (15) but they show usually a slow convergence. Thus, despite of a few exceptional cases where an analytical solution can be obtained (16), an efficient general purpose path integral treatment of time-dependent potentials is still lacking. Progress in this direction has recently been achieved by introducing a rapidly converging scheme in the framework of a high-order short-time expansion of transition amplitudes in time-dependent potentials (17). This approach holds for a general time-dependence of the potential. The disadvantage of this approach is that it can only be applied to smooth potentials. This is a relevant point in view of the fact that previous works have mostly been employing either the or a rectangular barrier for the study of the time-dependent systems.
In the present work we develop a field theoretical approach to the investigation of the quantum dynamics in time-dependent potentials. The method is suitable for studying any kind of potentials (also non-smooth) as well as external driving (periodic or not). In addition it allows for a classification and decomposition of the quantum dynamics in terms of fundamental processes which represent the skeleton of the quantum evolution in these systems. Particularly we reveal the role of virtual “multi-photon” processes which determine to a large extent details of the resonant structures in the dependence of the transmission coefficient on the incoming energy. Within our treatment these sub-processes can be clearly distinguished from the real “multi-photon” exchange which lead to inelastic transmission (4). The power of our method is demonstrated using the example of the oscillating delta barrier which has been extensively studied in the literature. We show that it is possible to calculate and understand transmission properties which have not been accessible so far. An example is the transmission zero associated with the Fano resonance characterizing the quantum dynamics in this system. The virtual “multi-photon” processes, mentioned above, are relevant for the determination of its location.
The paper is organized as follows. In section 2 we present the main idea of the perturbative scheme used to describe the transmission properties of a time-dependent quantum device. In section 3 we present our method focusing on the case of periodic driving and using as a simple example the delta barrier with a harmonically oscillating strength. In section 4 we give the results for the transmission properties of the oscillating delta-barrier setup. In particular we demonstrate how one can systematically calculate and interpret details of the behaviour of the transmission coefficient as a function of the incoming energy within our approach. Section 5 contains our concluding remarks. Finally, extensive formulas and their derivation are provided in the appendices.
Ii Field theoretical perturbative approach for transmission in time-dependent potentials
We consider the transmission of a quantum particle through a localized time-dependent potential. Our specific analysis is performed assuming a one-dimensional setup. The proposed approach can however be easily generalized to higher dimensional cases. Initially, i.e. for , the wave function of the quantum particle for is a plane wave with energy and momentum directed from the left to the right (incoming quantum particle). We also assume that the wave function for and is a plane wave with energy and the same direction of the momentum (outgoing quantum particle). The amplitude for the scattering of the considered quantum particle by a specific local time-dependent potential is in general given in terms of the corresponding causal Green’s function as:
where and are the input and output momenta of the particle while and are the corresponding energies. In eq. (1) we have introduced the temporary basis with being a collective index for all the quantum numbers needed to fully identify each temporary eigenstate of the time-dependent Hamiltonian at time . In this basis the instantaneous Schrödinger equation reads:
The relevant quantity for determining the Green’s function are the geometrical phases given as:
The diagonal elements are the Berry phases. In the temporary basis the propagating kernel fulfils the Green’s equation:
where the infinite dimensional matrix contains only the non-diagonal geometrical phases:
while the Berry phases are included in the effective energies :
In operator notation the propagator can be written as:
The non-diagonal matrix in (7) renders the considered problem analytically intractable in its general form. To proceed with the calculation of the amplitude (1) it is necessary to develop a scheme allowing for the expansion of the propagator in terms of simpler calculable sub-processes. The main assumption in our approach is that there exists an ordering with respect to the magnitude of the transition amplitudes of the different dynamical processes taking place in the scattering off a time-dependent potential. The latter is implied by the energy difference between the incoming and the outgoing state as stated in eq.(3). The amplitudes of the elastic processes dominate while inelastic processes with small energy transfer are more probable than those with large energy transfer. This property, if valid and consistently applicable, suggests that the non-diagonal matrix can be treated as a perturbation and the following expansion of is possible:
In eq. (8) the expansion breaks down in the case of zero eigenvalues of the denominator requiring a special treatment. We will come back to this point later on. Using the expansion (8) we can write the amplitude (1) as follows:
where the term in the above sum involves insertions of the transition matrix while is the zeroth order term which does not contain . The first term in the sum on the r.h.s. of eq. (9) is:
and the higher order terms have a similar structure as implied by the expansion (8). The form (9) allows the use of a particle picture for the interpretation of the dynamics in scattering off a time-dependent potential having at the same time a simple diagrammatic interpretation: the amplitude is decomposed in a sum of sub-processes. Each sub-process is a sequence of two elementary processes: the particle propagation being in a particular eigenstate and the transition between two states of the temporary basis. The explicit form of depends on the applied potential and especially on the temporary basis. The power of the diagrammatic representation of is that it allows the calculation of desired properties (like the energy of a transmission zero or a local transmission maximum) by isolating the contributing dynamical processes. In general the spectrum of contains both a discrete as well as a continuous part. Therefore the allowed elementary processes can be classified as follows:
continuum-continuum (c/c) transitions
continuum-bound (c/b) and bound-continuum (b/c) transitions
bound-bound (b/b) transitions
propagation in a continuum state
propagation in the bound state
A typical sub-process contributing to is shown in Fig. 1. The curly line indicates propagation in a continuum state while the dashed line means propagation in a bound state. The full black circles indicate c/c transitions while the circles containing a cross indicate a c/b or b/c transition.
Iii The case of time-periodic potentials: a delta-barrier with oscillating strength
The calculation of the transmission properties in a time-dependent potential significantly simplifies if the driving is periodic. In this section we will demonstrate how the perturbative scheme introduced in the previous section works in practice by performing an analysis of the transmission properties of a monochromatic wave passing through a periodically varying potential. As a concrete example we consider the transmission through a delta barrier with oscillating strength in one dimension. As mentioned already in the introduction, our main purpose is to calculate and explain features of the transmission behaviour which are not easily accessible by other approaches like direct integration or Floquet theory. The Schrödinger equation of the considered problem reads:
with being the period of the oscillating -potential and the associated frequency. For fixed the problem becomes static and can be easily solved. Introducing the length scale and the energy scale the static version of eq.(11) can be written in dimensionless form as follows:
with , , and is the corresponding wave function. For a given time instant the spectrum of the Hamiltonian consists of continuum states and one bound state with energy . The later exists only during the time period for which . The associated complete and orthonormal temporary basis is:
as given in (18).
The perturbative calculation of the scattering amplitude (eq. (1)) requires the summation of all the contributing sub-processes in increasing order. According to the discussion in the previous section the order of a term in the perturbation series is determined by the number of transitions (c/c, b/c or c/b) and can be diagrammatically presented by the series of graphs shown up to second order in Fig. 2. Since there is only one bound state in the temporary spectrum there are no possible transitions between bound states.
To first order only processes with a single transition, necessarily of c/c
type, participate. These may be elastic or inelastic. In
the inelastic case the final energy is given as:
where takes the values
under the restriction that
remains positive. This is a consequence of the periodic driving
(4) valid for all orders of the proposed
perturbation expansion (real “multi-photon” exchange).
Before going on with the explicit calculation of the transmission
amplitude (1) it is useful to present the matrix elements
for the building blocks of the contributing sub-processes.
The transition amplitude from one continuum state to another has the form:
with . The superscript is omitted in since the r.h.s. of eq. (14) is independent of these signs. As expected the diagonal term diverges and needs to be regularized. We adopt the usual box regularization in order to define a regularized Berry phase:
where with being the delta function regularized on a finite volume . With this choice in eq. (14) is finite fulfilling . After including also the off diagonal terms we get the general, regularized expression, for :
Continuum-bound state transitions
Denoting with the unique bound state of the potential (existing only for with ) we can express the c/b transition as:
Obviously the b/c transition amplitude is the complex conjugate of .
The zero-order approximation of the scattering amplitude (1) is given as:
In eq.(18) is the propagator in the bound state while is the propagator in a continuum state. The causal form of the latter reads:
The bound-state propagator is, in the interval , the solution of the Green’s equation:
The causal solution of eq.(20) is:
while the solution obeying the periodic boundary condition reads:
In obtaining the -matrix amplitude one has to perform the limits , . For simplicity we will assume here . Thus a consistent treatment requires such that . In this case the zero-order contribution to the scattering amplitude is trivial containing no transitions (free transmission). We will use in the following the notation for the classification of the contribution of the various sub-processes to the -matrix. means that the considered sub-process contains in total transitions. From these transitions are of c/c type while are of b/c or c/b type. Using this notation we write:
The first order term, containing a single c/c transition, becomes:
Obviously there are no first order processes involving a single b/c or c/b transition since the outgoing and incoming states belong necessarily to the continuum spectrum (positive energy). Eq. (24) can be rewritten as:
Since the renormalized c/c transition obeys:
the term is not included in the sum of eq. (25). The calculation of the amplitude , introduced to describe the -matrix contribution of the c/c transitions, is straightforward and leads to the expression:
with . The various contributing to the sum in eq. (25) correspond to the usual higher Floquet modes. As it can be seen from eq. (27) the amplitude for the inelastic continuum-continuum transitions decays exponentially with for (where :
in accordance with the fast convergence of the Floquet sum observed (but not explained) in the analysis of the dynamics of the oscillating delta-barrier in the literature. It must be noticed here, that, for , although the approximation (28) does not hold, the perturbative expansion is still valid. The reason is that, also in this case, the contribution of the diagrams with increasing number of transitions decreases. Since the dimensionless coupling is given by the original coupling multiplied by a factor proportional to our treatment is necessarily non-adiabatic and becomes exact either in the weak coupling or in the rapid oscillations (very large oscillation frequency ) limit. One important issue to be noticed here is that the expansion of the -matrix in terms of the number of transitions, as described above, allows the decomposition of the transmission process into elementary sub-processes giving a consistent meaning to our approximation procedure. The emerging perturbative scheme can also be understood in terms of an expansion in powers of the ratio of the coupling over magnitude of the incoming wave vector . As can be directly confirmed from eqs. (16) and (17), the c/c transitions are of order while c/b or b/c transitions are of order . The magnitude of the various diagrammatic contributions to the -matrix is then quantified by the leading power of , a power that increases as the number of transitions increases.
In the next order (two transitions) the c/b or b/c transitions are also possible. Typically, the corresponding sub-process is demonstrated by the fourth diagram on the right hand side shown in Fig. 2. The contribution of this term to the amplitude is given as:
This amplitude is non-zero only if the time variables are in the region . The needed bound-state propagator fulfils the periodic boundary condition: () and is given by eq. (22) where the allowed number of terms which must be summed up, depends on the difference . These facts complicate the calculation of the amplitude (29). We can considerably simplify things by replacing the time-dependent bound-state energy, in the framework of our approximation scheme, by its mean value over a period:
This approximation is a first order estimation, being justified in the case of very fast or very slow (static limit) oscillations of the potential which lead to an effective, time-independent, bound-state energy. Higher order corrections can be obtained by expanding the bound-state wave function around the effective coupling value . Following this scheme the b/c transition amplitude becomes in first order:
Introducing now the Fourier transformations:
we obtain for the amplitude :
where is given in eq. (27). It is straightforward to show that also the b/c transition amplitudes decay exponentially with the Floquet index when the applied perturbation scheme is valid, i.e.:
Using the expressions (32) we can calculate the contribution of the c/b or b/c transitions corresponding to the third diagram on the right hand side of Fig. 2 to the -matrix as follows:
and . The small parameter in the denominator of (37) is introduced in order to ensure convergence for and is equivalent to the demand that the continuum Green’s functions vanish in this limit. In the absence of the amplitude possesses a pole at . The occurrence of the pole is, from a mathematical point of view, a result of the perturbative expansion. In fact, the time-dependence of the potential produces an effective Hamiltonian containing an infinite series of geometrical phases (19). These terms are the origin of a non-vanishing imaginary part that naturally appears when higher order terms, involving virtual transitions between the continuum and the bound state (virtual “multi-photon” exchange), are summed up. As a consequence the denominator of the amplitude (37) never vanishes on the real axis of the energy. At the same time these higher order terms shift the energy by . This is equivalent with a change in the effective bound state energy of the form:
Including the higher order corrections in (37) we obtain a normalized redefinition of the continuum-bound-continuum (c/b/c) transition:
where is a normalization factor and , are the corrected values. In Appendix A we shall present the detailed calculation of the renormalized factors indicating the significance of the virtual “multi-photon” exchange processes. Here it suffices to note that in the limit we have which in turn means that, at least, (for , or ). On the other hand the behaviour of the corrected factors in the denominator of (39) depends only weakly on and (see eq. (51) and the discussion below this equation in Appendix A): , . As a consequence, the corrected values play an essential role only when the incoming energy is close to an integer value.
The expression (39) is finite and well-behaved for all the values of the incoming energy. However when the energy of the incoming particles differs from the effective bound-state energy by a positive integer the probability amplitude to arrive at the final state passing through the bound state has a sharp maximum. The impact of this maximum on the transmission properties will be discussed below. To complete the calculation of the -matrix with two transitions we have to calculate also the contribution of the second diagram on the right hand side of Fig. 2 containing two c/c transitions. After some straightforward steps we obtain:
and given by (27). Due to the integration over the regularizing imaginary part in the denominator of eq. (41) can be taken as zero. It can be further confirmed, both numerically and analytically (see Appendix B), that the imaginary part of the above amplitude
which is controlled by the pole at dominates over its real part
that avoids the pole ( denotes the principal value integration). In the case , the imaginary part is of order which, for , , is of order , while the real part is, at least, of order .
Already at this general stage of the analysis one can gain some insight into the expected behaviour of the transmission coefficient. Whenever a propagator is contained in a diagram contributing to the transmission amplitude we observe the appearance of denominators which are in fact associated with the factors occurring in the expansion (10). These denominators are mainly responsible for resonant structures in the transmission coefficient. The most important contributions come from diagrams involving propagators close to the incoming and outgoing states since in this case a pole structure in the complex energy plane emerges. If the propagators are bracketed between transitions which do not involve the incoming or the outgoing state then the pole structure is integrated out leading to a smoother contribution to the transmission coefficient. However, the effects of these denominators is also controlled by the transition amplitudes which occur in the numerator and play the role of residues in the final expression for the contribution of a given diagram. This interplay between propagation and transition (poles and residues) determines the overall behaviour of the transmission coefficient.
Iv Transmission properties
Having calculated the amplitudes of all sub-processes involving up to two transitions it is straightforward to calculate the associated -matrix amplitude up to this order. In general for scattering off a potential varying periodically with time the -matrix amplitude is given as:
where , are respectively the elastic and inelastic transmission amplitudes while , are the corresponding reflection amplitudes. They can be expressed in terms of the calculated sub-processes as follows:
The total transmission coefficient is obtained as:
Contribution of continuum-continuum versus continuum-bound state (or b/c) transitions
The decomposition (43) allows us to isolate the contribution to the transmission of the sub-processes involving c/b or b/c transitions and to compare with the corresponding contribution of sub-processes involving exclusively c/c transitions. This is not possible in any other approach. In addition, as in Floquet theory, we can also here discriminate between elastic and inelastic contributions to the transmission. Let us first concentrate on the dominating elastic transmission channel. According to eq. (43) the elastic part of the transmission coefficient is given as:
The discussion below eqs. (39) and (41) indicates that the dominant contributions in (44) are due to the and for all values of the incoming energy except for where the dominates. Thus it is suggestive to define the relative weight of contributions to the transmission coefficient involving transitions between the continuum and the bound state to contributions involving exclusively transitions between continuum states as follows:
In Fig. 3 we plot for illustration as a function of the incoming energy for two different values of the coupling . We see that the processes involving c/c transitions dominate for the complete energy regime except of a small region around the integer value .