1. INTRODUCTION

Frequency chirping is an efficient control technique to provide dynamical systems with a wide range of possible energies to be absorbed within a single interaction [1,2]. Here the increase (or decrease) of the instantaneous frequency in time is commonly employed in many sonar, laser, or radar applications. A very relevant application for strong-field physics is the chirped pulse amplification (CPA), which is a technique for amplifying an ultrashort laser pulse up to the petawatt level with the pulse being stretched out temporally and spectrally prior to amplification [3,4].

If frequency-chirped laser fields interact with classical or quantum mechanical systems, they can play a dual dynamical role with regard to the efficiency of energy absorption and the degree of excitation of the system. On the one hand, by having a wider energy spectrum than their corresponding (unchirped) monochromatic counterparts, they can more effectively excite those nonlinear systems whose dynamical response is associated with multiple resonances. On the other hand, as each moment in time can be characterized by a specifically chosen instantaneous value of the frequency, it also provides a very time-sensitive control tool for the excitation process during one interaction pulse. As we will show in this work, there are some systems for which the final amount of absorbed energy after the interaction is solely based on the first process. We name this a (phase-insensitive) “incoherent” chirping mechanism, as the final degree of excitation depends here solely on the field’s energy spectrum $|{F}(\omega ){|^2}$ associated with the force ${F}({t})$. Alternatively, there are also systems whose dynamical response depends on the temporal details, which we would call a “coherent” chirping mechanism, as the final dynamical response is sensitive to the phases [5] of the (complex) Fourier transform ${ F }(\omega )$. As ${ F }({t})$ and its temporally time-reversed force ${ F }({T} - {t})$ share the same energy spectrum $|{ F }(\omega ){|^2}$, the possibility of a different dynamical response to these two excitation fields can be used to differentiate between both mechanisms. For a general system it is certainly not clear a priori which of the two mechanisms is the more dominant one.

This work was motivated in part by recent studies that examined the electron-positron pair-creation process from the vacuum under suitable chirped external oscillatory electric fields [6–11]. It was predicted that chirping can increase the yield of created electron-positron pairs after the interaction by several orders in magnitude. However, the physical mechanism of such a spectacular enhancement is still an open question. This is an important research area of fundamental importance, as the possibility to convert light directly into matter is one of the most striking predictions of quantum electrodynamics (QED) [12–16], but its direct experimental verification is still lacking. Also, due to recent significant progress in the development of high-powered laser systems, the optimization of electromagnetic field configurations to maximize the total number of created electron-positron pairs [17–24] is of obvious experimental interest.

This work is structured as follows. In Section 2 we illustrate the difference between coherent and incoherent chirping mechanisms for classical oscillators and quantum mechanical multi-level atoms. In Section 3 we prove that the temporally excited vacuum decay process is phase insensitive and therefore does not permit a coherent chirping, but a space-time-dependent excitation force permits also the coherent chirping decay process. We summarize this work together with a brief outlook on future challenges in Section 4.

2. EXAMPLES FOR COHERENT AND INCOHERENT CHIRPING MECHANISMS

Using the concrete examples of two classical mechanical and two quantum mechanical model systems, we will illustrate the coherent and incoherent chirping mechanisms. We will see that incoherent chirping is the only possible excitation mechanism for the harmonic oscillator and the two-level system, while the quartic oscillator as well as a general three-level system permit coherent chirping. The three-level system in itself is interesting, as here the existence of special symmetries (with respect to energy level spacings and dipole oscillator strengths) can also permit incoherent chirping mechanisms.

A. Time-Dependent Chirped Force Field and Its Frequency Spectrum

In all of the dynamical systems discussed in this work, the external time-dependent electric force field is characterized by four parameters. They are the electric field’s amplitude ${{ F }_0}$, the initial frequency ${\omega _i}$, the chirping rate ${ b }$, and the total pulse duration ${T}$. Neglecting for simplicity any temporal turn-on or turn-off durations, it is given by ${ F }({t}) = {{ F }_0}\;{\sin}[({\omega _i} + { b t})\;{t}]$ $\theta ({T-t})\, \theta(t)$, where $\theta$ denotes the unit step function. Here and below we use atomic units (${ m } = { e } = \hbar = {1}\;{\rm a}.{\rm u}.$) such that the frequency $\omega $ is numerically identical to the energy $\hbar \omega $.

The spectral content of ${ F }({t})$ is naturally given by its Fourier decomposition, defined here as ${ F }(\omega ) \equiv {({2}\pi )^{ - 1/2}}\;\int \; {\rm d} t { \exp}[ - { i }\;\omega \;{t}]\;{ F }({t})$. In the absence of any chirping (${ b } = {0}$), the energy spectrum, defined as $|{ F }(\omega ){|^2}$, is peaked at $\omega = {\omega _i}$ with a width that is proportional to ${1/T}$. As ${ b }$ is increased from zero, the distribution widens to larger energies $\omega $, covering the effective range from about $ - { 1/}({2}\;{T}) + {\omega _i}$ to ${ 1/}({2}\;{T})\;{ + }\;({\omega _i} + 2{ b }\;{T})$. As ${ F }({t})$ is real, the real and imaginary parts of the complex Fourier amplitudes have the symmetries ${\rm Re}[{ F }( - \omega )] = { \rm Re}[{ F }[(\omega )]\;$ and ${\rm Im}[{ F }( - \omega )] = - { \rm Im}[{ F }(\omega )]$, such that the energy spectrum is even in $\omega $: $|{ F }[ - \omega ]{|^2} = |{ F }[\omega ]{|^2}$. Therefore, we can focus on positive energies only.

The Fourier amplitudes of the corresponding time-reversed electric field, given by ${{ F }_{ R }}({t}) \equiv { F }({T} - {t})$, can naturally be related to that of the original field, i.e., ${{ F }_{ R }}(\omega ) = {\exp}[ - { i }\;\omega \;{T}]\;{ F }{(\omega )^*}$, such that they have identical energy spectra: $|{{ F }_{ R }}(\omega ){|^2} = |{ F }(\omega ){|^2}$.

Due the positive chirping (${ b } \gt {0}$), ${ F }({t})$ offers the system at earlier time smaller frequencies, whereas ${{ F }_{ R }}({t})$ starts to be more oscillatory. In order to get a quantitative estimate, the specific numerical parameters (${{ F}_0} = {{2}^{1/2}}\;{\rm a}.{\rm u}.$, ${\omega _i} = {1}.{4}\;{\rm a}.{\rm u}.$, ${ b } = {0}.{065}\;{\rm a}.{\rm u}.$, and ${T} = {20}\;{\rm a}.{\rm u}.$) are used for all four interactions discussed in the sections below. We have graphed ${ F }({t})$, ${{ F }_{ R }}({t})$, and $|{ F }(\omega ){|^2}$ in Fig. 1.

 

Fig. 1. Temporal profiles of the chirped electric force fields ${ F }({t})$ and ${{ F }_{ R }}({t})$ used to excite the classical oscillators and quantum systems discussed in Section 2. We also show their energy spectrum $|{ F }(\omega ){|^2} = |{{ F }_{ R }}(\omega ){|^2}$. In all of our numerical illustrations, we use the atomic unit system and ${ F }({t}) = {{2}^{1/2}}\;{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]\;{\rm a}.{\rm u}$.

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We see that the instantaneous value of the frequency, ${\omega _i} = {1}.{4}\;{\rm a}.{\rm u}.$, nearly doubles (increases by 93%) to its final instantaneous value, ${\omega _i} + 2{ b }\;{T} = 4\;{\rm a}.{\rm u}.$ In other words, we have a very significant amount of chirping here. Due to the spectral broadening associated with the relatively short interaction time [${ F }({t})$ has only nine maxima during its duration ${T}$], we see that the effective energy spectrum $|{ F }(\omega ){|^2}$ contains also much smaller and larger energies than the range from $ - { 1/}({2}\;{T})\;{ + }\;{\omega _i}$ to ${ 1/}({2}\;{T})\;{ + }\;({\omega _i} + 2{ b }\;{T})$. The short interaction time ${T}$ manifests itself also in the complicated multi-peaked structures with its two major peaks at $\omega = {2}\;{\rm a}.{\rm u}.$ and $\omega = {3}.{3}\;{\rm a}.{\rm u}$.

B. Classical Harmonic Oscillator

Naturally, the free particle is the least sensitive system to any temporal characteristics of the external force ${ F }({t})$, as its final kinetic energy after the interaction, ${ e }({T}) \equiv {({{d}{x}}/{{ d}{t}})^2}({T})/{2}$, is simply given by the global quantity ${ e }({T}) = {[\int_0^{T} \; {{\rm d}t}\;{ F }({t})]^2}/{2}$. This time integral is obviously identical for our pair of forces, ${ F }({t})$ and ${{ F }_{ R }}({t}) \equiv { F }({T} - {t})$, making any attempt at coherent chirping unsuccessful, as the final energy absorbed under the time-reversed excitation [denoted by ${{ e }_{ R }}({T})]$ is equal to ${ e }({T})$. The equality ${ e }({T}) = {{ e }_{ R }}({T})$ is independent of the particle’s initial velocity. The expression for ${ e }({T})$ also reflects the well-known fact that a free particle cannot pick up any final drift velocity from those force fields ${ F }({t})$ whose “area” $\int_{0}^{T}{{\rm d}t\, F(t) }$ vanishes.

The next complicated system is the harmonic oscillator, whose elongation ${x}({t})$ is given by ${{d}^2}{ x}/{{{d}t}^2} = - \;\Omega^2\;{x} + { F }({t})$. For the initial conditions ${x}({0})\;{= }\;{dx/dt}({0})\;{= }\;{0}$, it has the usual analytical Green’s function solution given by ${x}({T}) = \int_0^{\:T} \; {\rm d}\tau {\sin}[\Omega ({T} - \tau )]/\Omega \;{ F }(\tau )$ and ${dx}/{dt}\;({T}) = \int_0^{\:T} \; {\rm d}\tau \;{\cos}[\Omega ({T} - \tau )]\;{ F }(\tau )$. Therefore, the analytical form of the final energy is given by ${ e }({T}) \equiv {({dx}/{dt})^2}/{2}\;{ + }\;{\Omega ^2}\;{{x}^2}/{2}\;{= }$ ${[\int _0^{\:T}\!{{\rm d}t}{\cos}[\Omega ({T} - {t})]{ F }({t})]^2}/{2}{ + }{[\int _0^{\:T}\!{{\rm d}t}{\sin}[\Omega ({T} {-} {t})]{ F }({t})]^2}/{2}$. Expressing the products of the integrals as a double integral and using the trigonometric relationship ${\sin}(\alpha ){\sin}(\beta ) + {\cos}(\alpha ){\cos}(\beta ) = {\cos}(\alpha - \beta )$, the energy reduces to ${ e }({T}) = \int_0^{\:T} \int_0^{\:T} \; {{{\rm d}t}_1}\;{{{\rm d}t}_2}\;{\cos}[\Omega ({{t}_1} - {{t}_2})]{ F }({{t}_1})\;{ F }({{t}_2})/{2}$. This expression is equal to the real part of $\int_0^{\:T} \; \int_0^{\:T} \; {{{\rm d}t}_1}\;{{{\rm d}t}_2}{ \exp}[{ i }\;\Omega ({{t}_1} - {{t}_2})]{ F }({{t}_1})\;{ F }({{t}_2})/{2}$, such that we obtain ${ e }({T}) = |\;\int_0^T {{{\rm dt}}} { \exp}[{ i }\;\Omega \;{t}]{ F }({t})\;{|^2}/{2}$. If we substitute the new variable ${t} \equiv {T}\; - \;\tau $ we obtain immediately ${{ e }_{ R }}({T}) = { e }({T})$. This proves analytically that the harmonic oscillator also is insensitive to the specific timing of the chirped signal. In other words, only the incoherent chirping mechanism is capable of exciting this system.

For those force functions ${ F }({t})$ that vanish outside the time interval between 0 and ${T}$, the final absorbed energy is equal to the absolute value square of the Fourier amplitude of ${ F }({t})$ evaluated at the oscillator’s natural frequency $\Omega $. Quite interestingly, this suggests that only those force fields that have a non-vanishing Fourier component at the specific frequency $\Omega $ can contribute to the final absorbed energy at time ${T}$.

In Fig. 2 we illustrate the temporal growth pattern for the two energies ${ e }({t})$ and ${{ e }_{ R }}({t})$. While the corresponding time dependence of the elongations ${x}({t})$ and ${{x}_{ R }}({t})$ obtained for general signals ${ F }({t})$ and ${ F }({T} - {t})$ are entirely different at any time ${t}$ (including the final time ${T}$), the final energies ${ e }({T})$ and ${{ e }_{ R }}({T})$ do take identical values. This is fully consistent with our analytical derivation above. However, the equality ${ e }({T}) = {{ e }_{ R }}({T})$ is generic to only those oscillators, for which the initial conditions for both the position and the momentum are zero.

 

Fig. 2. Time dependence of the energy ${ e }({t}) = {({dx}/{\rm dt})^2}/{2}\;{ + }\;{\Omega ^2}{{x}^2}/{2}$ of a harmonic oscillator driven by the chirped force field ${ F }({t})$ and its time-reversed form ${ F_R}({t})$, presented in Fig. 1, with parameters $\Omega = {1}\;{\rm a.}{\rm u.}$ and ${ F }({t}) = {{2}^{1/2}}\;{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]\;{\rm a}.{\rm u}$. Here and in all of the following figures the energies ${ e }({t})$ and ${{ e }_{ R }}({t})$ are also in atomic units.

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The positively chirped force field ${ F }({t})$ excites the oscillator rather significantly at early times, when its instantaneous frequency (${\omega _i} = {1}.{4}\;{\rm a}.{\rm u}.$) is closer to the natural frequency $\Omega $, whose numerical value was chosen to be $\Omega = {1}\;{\rm a}.{\rm u}.$ As the chirped frequency moves further away from this natural frequency, the energy absorption varies and reduces in an oscillatory fashion. On the other hand, as the early instantaneous frequency for ${{ F }_{ R }}({t})$ is close to 4 a.u., it is far off from the resonance, and the early time absorption is negligible but increases to its final value ${{ e }_{ R }}({T}) = { e }({T})$ as time increases.

As an interesting side note, we should mention that even if the oscillator’s eigenfrequency $\Omega $ was chosen to be exactly the average of the largest and smallest instantaneous frequency of the two forces, i.e., $\Omega = (4\;{ + }\;{1}.{4}){/2}\;{\rm a}.{\rm u}.$, ${ e }({t})$ grows still significantly faster in time than ${{ e }_{ R }}({t})$ before they reach the same value ${ e }({T}) = {{ e }_{ R }}({T})$. This remarkable asymmetry suggests that the smallest and largest instantaneous frequencies of the chirped force field are dynamically not as relevant as the actual spectrum. We have shown in Fig. 1 that energy spectrum $|{ F }(\omega ){|^2}$ is peaked at $\omega = {3}.{3}\;{\rm a}.{\rm u}.$ and therefore contains frequencies around 4. We found that for $\Omega $ around 3.8 a.u., the time dependences of ${ e }({t})$ and ${{ e }_{ R }}({t})$ were qualitatively similar.

Note that if we include a damping mechanism that is linear in the velocity, then the system recovers its sensitivity to the specific timing of the force field, the chirping mechanism becomes more coherent, and we find ${ e }({T})\; \ne \;{{ e }_{ R }}({T})$.

C. Classical Quartic Oscillator

In order to increase the complexity of the classical mechanical system, we have also considered the dynamical response of the quartic oscillator [25–28] to both chirped fields ${ F }({t})$ and ${{ F }_{ R }}({t})$. Here we have to numerically solve the equation of motion given by ${{d}^2}{x}/{{dt}^2} = - \;{{x}^3} + { F }({t})$, where for a better comparison ${ F }({t})$ and ${{ F }_{ R }}({t})$ were chosen again to be identical to those discussed in Section 2.A. Here we note that due to the inherent nonlinearity, there are no known analytical and non-perturbative solutions available, even if the external force is simply monochromatic.

 

Fig. 3. Time dependence of the energy ${ e }({t}) = {({dx}/{dt})^2}/{2}\;{ + }\;{{x}^4}/{4}$ of a quartic oscillator driven by the chirped force field ${ F }({t})$ and its time-reversed form ${{ F }_{ R }}({t})$, as presented in Fig. 1, ${ F }({t}) = {{2}^{1/2}}$ ${\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$.

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In Fig. 3 we have converted the time-dependent position ${x}({t})$ and velocity ${dx}/{dt}({t})$ obtained from the equation of motion to the corresponding energy ${ e }({t}) \equiv {({dx}/{dt})^2}/{2}\;{ + }\;{{x}^4}/{4}$. We find that ${ F }({t})$ can excite this nonlinear oscillator significantly; for early times, the energy growth of ${ e }({t})$ suggests a quadratic scaling in time [28]. In fact, the specific numerical force parameters $({{ F }_0} = {{2}^{1/2}}{\rm a.u.}$, ${\omega _i} = {1}.{4}\;{\rm a}.{\rm u}.$, and ${ b } = {0}.{065}\;{\rm a}.{\rm u}.$) were actually chosen to trigger this rather resonance-like response. The key idea here is to attempt to loosely follow the oscillator’s own (elongation-dependent) natural frequency with the external force’s instantaneous frequency (${\omega _i}+{ 2b t}$) such that the force tries to remain in phase with the particle’s velocity at any time.

In order to illustrate that in this case the precise sequences of the offered frequencies are crucial, we have shown again by the dashed line the energy ${{ e }_{ R }}({t})$ absorbed from the time-reversed force. The different amounts of absorbed energies under both excitation modes are obvious and, in contrast to the linear oscillator, the final energies are quite different, i.e., ${ e }({T})\; \ne \;{{ e }_{ R }}({T})$.

D. Quantum Mechanical Two-Level Atom

The time evolution of the quantum state $|\Psi ({t})\rangle $ of a two-level system [29–31] is described by the two complex probability amplitudes ${{C}_1}({t})$ and ${{C}_2}({t})$ such that $\vert \Psi ({t})\rangle = {{C}_1}({t})$ $|{1}\rangle \;{ + }\;{{C}_2}({t})\;|{2}\rangle $. The Hamiltonian describing the energy transfer between the ground state $|{1}\rangle $ and the first excited state $|{2}\rangle $ is given (in atomic units) by

In order to examine if this two-level system is sensitive to the precise timing of the two force fields ${ F }({t})$ and ${{ F }_{ R }}({t})$, we have graphed in Fig. 4 the time dependence of the energy, defined as ${ e }({t})\; \equiv {\omega _1}|{{ C }_1}({t}){|^2} + {\omega _{2\:}}|{{ C }_2}({t}){|^2}$. We find that while ${ e }({t})$ and ${{ e }_{ R }}({t})$ are entirely different and very complicated functions of time, at the final time, however, both interactions have led apparently to an identical amount of energy absorption, i.e., ${ e }({T}) = {{ e }_{ R }}({T})$. In other words, similar to the classical harmonic oscillator, also the final amount of absorbed energy of this quantum mechanical system is completely phase insensitive to the precise timing of the external field. We have varied the parameters ${\omega _1},{\omega _2},{\omega _i}$, ${ b }$, and ${T}$ over a wide range and always found that ${ e }({T}) = {{ e }_{ R }}({T})$. This complete insensitivity to the temporal phase of the external force field clearly suggests that there cannot be any single unique chirping technique that can maximize the energy absorption of a two-level system, unless it has the symmetry ${ F }({t}) = { F }({T} - {t})$.

 

Fig. 4. Time dependence of the total energy ${ e }({t}) \equiv {\omega _1}|{{ C }_1}({t}){|^2} + {\omega _2}$ $|{{ C }_2}({t}){|^2}$ of a two-level atom excited by the chirped force ${ F }({t}) = {{2}^{1/2\:}}{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$ and by its temporally reversed form ${{ F }_{ R }}({t})$. [The energies are ${\omega _1} = {0}\;{\rm a}.{\rm u}.$ and ${\omega _2} = {3}\;{\rm a}.{\rm u}.$; ${{d}_{12}} = {{d}_{21\:}} = {0}.{5}\;{\rm a}.{\rm u}.$]

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In order to prove analytically the corresponding phase insensitivity to ${ F }({t})$ and ${{ F }_{ R }}({t})$ for the classical free particle and the harmonic oscillator, we used the fact that there were analytical solutions to position ${x}({t})$ available. In the case of the two-level system, however, we do not have this advantage, as the set of Eq. (2) does not permit any closed-form analytical solutions, except for the trivial case when the external field is constant in time. Quite remarkably, however, it is nevertheless possible to prove in full generality for this system that ${ e }({T}) = {{ e }_{ R }}({T})$, despite the absence of any analytical solutions for any general ${ F }({t})$ or ${{ F }_{ R }}({t})$. We provide this proof in Appendix A.

So far we have examined the excitation of the atom being initially in the ground state, i.e., with vanishing polarization. However, if we assume that the atom is initially already excited $[|{{ C }_2}({t} = {0}){{\vert}^2}\; \ne \;{0}]$, such that we can choose a non-vanishing initial polarization, i.e., ${{ C }_1}({t} = {0})\;{= }\;{{2}^{ - 1/2}}$ and ${{ C }_2}({t} = {0})\;{= }\;{ i }\;{{2}^{ - 1/2}}$, then we actually find that ${ e }({T})\; \ne \;{{ e }_{ R }}({T})$.

E. Quantum Mechanical Three-Level Atom

In order to examine if the structure of a two-level system is too simple to permit for the coherent chirping mechanism, we have coupled a third level $\vert{3}\rangle $ with energy ${\omega _3}\;( \gt {\omega _2})$ to the first excited state $\vert{2}\rangle $. The state is therefore given by $|\Psi ({t})\rangle = {{ C }_1}({t})\;|{1}\rangle \;{ + }\;{{ C }_2}({t})\;|{2}\rangle \;{ + }\;{{ C }_3}({t})\;|{3}\rangle $. The Hamiltonian describing the energy transfer between the ground state $|{1}\rangle $ and the first and second excited states $|{2}\rangle $ and $|{3}\rangle $ is given (again in atomic units) by

In Fig. 5 we examine the special case of those atoms where the energy differences ${\omega _2} - {\omega _1}$ and ${\omega _3} - {\omega _2}$ are chosen to be equal to each other and also the oscillator strengths happen to match (${d_{12}} = {d_{23}}$). We see that also for this particular three-level atom the final amount of energy absorption is identical, i.e., ${ e }({T}) = {{ e }_{ R }}({T})$. However, once this perfect symmetry between the two transitions is broken, i.e., if we choose, for example, an atom with ${{d}_{12}}\; \ne \;{{d}_{23}}$ or ${\omega _2} - {\omega _1}\; \ne \;{\omega _3} - {\omega _2}$, we immediately obtain ${ e }({T})\; \ne \;{{ e }_{ R }}({T})$. This suggests that any general multi-level system (with more than two levels) should permit the coherent chirping mechanism, where the precise timing of the external laser field is crucial with regard to the final energy that has been absorbed.

 

Fig. 5. Time dependence of the energy ${\omega _1}|{{ C }_1}({t}){|^2} + \;{\omega _2}|{{ C }_2}({t}){|^2} + \;{\omega _3}|{{ C }_3}({t}){|^2}$ of a three-level atom when pumped by the chirped force ${ F }({t}) = {{2}^{1/2}}$ ${\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$ and by its temporally reversed force ${{ F }_{ R }}({t})$. Left figure: ${{d}_{12}} = {{d}_{23}} = {1}$ a.u. Right figure: ${{d}_{12}} = {1}\;{\rm a}.{\rm u}.$ but ${{d}_{23}} = {2}\;{\rm a}.{\rm u}$. The energies were ${\omega _1} = {0}\;{\rm a}.{\rm u}.$, ${\omega _2} = {3}\;{\rm a}.{\rm u}.$, and ${\omega _3} = {6}\;{\rm a}.{\rm u}$.

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In closing, we should mention that there are also other pairs of parameters with energy differences ${\omega _2} - {\omega _1}$ $\ne{\omega _3} - {\omega _2}$ and non-matching oscillator strengths $({{d}_{12}}\; \ne \;{{d}_{23}}$), for which we obtain ${ e }({T}) = {{ e }_{ R }}({T})$. However, the values of these particular pairs can only be constructed by trial and error based on repeated computer simulations. For example, if we assume that the time-dependent force ${ F }({t})$ is given by three piece-wise constant forces (${{ F }_1},\;{{ F }_2},$ and ${{ F }_3}$), as discussed more generally in Appendix A, the two propagators can be expressed as time-ordered products of the three individual operators ${{U}_1}$, ${{U}_2},$ and ${{U}_3}$ governing the evolution from ${t} = {0}$ to ${T}$. For the special case (${T} = {1}\;{\rm a}.{\rm u}.$, ${{ F }_1} = {1}\;{\rm a}.{\rm u}.$, ${{ F }_2} = \;\pi \;{\rm a}.{\rm u}.$, and ${{ F }_3} = {{7}^{1/2}}\;{\rm a}.{\rm u}.$) and the energies ${{E}_1} = {0}$, ${{E}_2} = {1}$, and ${{ E}_3} = {5}$ with ${{d}_{12}} = {1}$, we find that the particular set of dipoles ${{d}_{23}} = {0}.{3964},\;{0}.{7170},\;{1}.{351},\;{1}.{676},\;{2}.{079}\;{\ldots}\,{\rm a.u.}$ will lead to ${ e }({T}) = {{ e }_{ R }}({T})$.

To illustrate that the occurrence of the possible symmetry under time-reversed force signals can occur sometimes in an unexpected way, we complete this section with a brief preview of a four-level ladder system. As the deep underlying mechanisms for the occurrence of this symmetry are not yet fully understood, one has to rely on purely numerical explorations. If we choose for simplicity a level structure with equal energy differences (${ e }_{2}-{ e }_{1}={ e }_{3}-e_{2}={ e }_{4}-{ e }_{3}$) together with equal dipole moments (${d}_{12}={d}_{23}={d}_{34}$), then we consistently find that the final probability associated with the initial states is again invariant, i.e., $|{{ C }_1}({t}){|^2} = |{{ C }_{R,1}}({t}){|^2}$. This invariance is expected based on the initial state non-decay probability law as derived in Appendix B. In contrast, the probabilities for the next two excited states are not invariant. However, the probability for the highest energetic level quite unexpectedly reveals again the symmetry, i.e., $|{{ C }_4}({t}){|^2} = |{{ C }_{{ R },4}}({t}){|^2}$.

In Appendix B, we derive for a general ${N}$-level system interesting conservation laws for the final excitation under ${ F }({t})$ and ${{ F }_{ R }}({t})$. A major finding is the non-decay probability law, which states that the amplitude $\langle \Psi ({0}){\vert}\Psi ({T})\rangle $ is identical for ${ F }({t})$ and ${{ F }_{ R }}({t})$ if the underlying Hamiltonian matrix elements are real.

3. COHERENCE AND INCOHERENCE OF THE VACUUM DECAY PROCESS

While the largest possible energy that the multi-level quantum systems can absorb has an upper bound, we examine now a quantum field theoretical environment where the energy growth can—at least in principle—be unlimited. In this system, the amplitude of the external force field is extremely high such that it can break down the Dirac vacuum [32,33] and create electron-positron pairs. Here the absorbed energy manifests itself in the kinetic energy of the two created particles. We will show that if the external field is spatially uniform and varies only in time, then it is only possible to incoherently chirp the system. On the other hand, if the external force varies with both space and time, then it is possible to coherently chirp the system and to enhance the yield by providing the vacuum state with an optimized timing of the instantaneous frequencies of the field.

A. Electron-Positron Pair Creation Triggered by a Spatially Homogeneous Field

As generic to most theoretical approaches to this problem, we neglect the mutual attractive and repulsive Coulombic interactions between the created two fermions. For the sake of brevity, we refer the reader here to reviews [16,34] on various theoretical approaches and focus here immediately on the results.

In computational quantum field theory [35], the interaction of the vacuum with an external field given by ${\textbf{A}}({\textbf{r}},t)$ and ${V}({\textbf{r}},{\textbf{t}})$ is often modelled by the Hamiltonian ${H} = { C }$ ${\boldsymbol \alpha }$ $({\textbf{p}} - {\textbf{A}}/c) + {c^2}$ $\beta + {V}({\textbf{r}})$, which governs both the time evolution of the four spinor components of the electron-positron quantum field operator (via the Dirac equation [36]) and the dynamics of the fermion’s creation and annihilation operators (via the Heisenberg equation [37]). Here ${\boldsymbol \alpha } \equiv ({\alpha _1},{\alpha _2},\;{\alpha _3})$ and $\beta $ denote the set of the four ${4} \times {4}$ Dirac matrices. For simplicity, we restrict the dynamics to one spatial dimension and use again atomic units, where ${c} = {137}.{036}\;{\rm a}.{\rm u}.$ For those particular quantities that carry the units of energy, we express the numerical values in units of ${{c}^2}\;( = {1}.{88}\; \times \;{{10}^4}\;{\rm a}.{\rm u}.)$. This allows for a better comparison with the characteristic energy scale ${{2c}^2}$ given by the mass gap. Similarly, to permit a direct comparison with the electron’s Compton wavelength (${ 1/c}$), we express the lengths in units of ${ 1/c}$, and therefore the electric field strength in units of ${{c}^3}$.

Consistent with Dirac’s hole theory, the initial vacuum state is represented by the complete occupation of all negative ($\le-{{c}^2}$) energy eigenstates of ${H}$ (for ${\textbf{A}} = V = 0)$, denoted by $\vert {k;d} \rangle$ and characterized by their momentum ${k}$. The projection of the time-evolved states $\vert {k;d} {(t)}\rangle$ on all energy eigenstates with momentum ${p}$ and positive energy (denoted by $\vert{p};{u}\rangle $) represents the number of created particles, i.e., ${N}({t}) ={\Sigma _{{p}\:}}{\Sigma _{{k}}}\vert \langle {p};{u}\vert{k};{d}({t})\rangle\vert^{2}$. The energy carried by the electrons is therefore ${ e (t)}=\Sigma_{p}\Sigma_{k}\;e(p)\vert\langle p;u \vert k;d(t)\rangle \vert ^{2}$ with ${ e }({p}) \equiv {[{{c}^4} + {{c}^2}{{p}^2}]^{1/2}}$. To examine the phase sensitivity of the pair-creation process to chirped and reversely chirped force fields, we have modelled the external electric field by a Gaussian pulse

In Fig. 6 we compare the temporal growth of the energy ${ e }({t})$ of the created electrons obtained from ${ F }({t})$ with that computed from the time-reversed force field ${{ F }_{ R }}({t})$, denoted again by ${{ e }_{ R }}({t})$.

The average energy of the created electrons can be obtained by dividing the total energy ${ e }({T}) = {305}\;{{c}^2}$ by the final number of created electrons, which was ${N}({T}) = \;{228}$. For our parameters, we therefore obtain for the average electron an energy of ${305}\;{{c}^{2}}/{228}\;{= }\;{1}.{34}\;{{c}^2}$. This energy is close to $\omega /2$, which is fully consistent with a match of the main oscillating frequency of the external field (around $\omega \; = {2}.{{8\,\,c}^2}$) with the energy differences between the upper and lower energy continua.

While the temporal growth patterns of the two energies in Fig. 6 are entirely different, after the interaction they agree. This means that the temporally induced electron-pair-creation process is fully incoherent. Quite remarkably, despite the complicated structure of quantum field theory, this particular excitation process is therefore completely insensitive to the precise timing when the energy portions from the external field can be absorbed by the vacuum state. As we consider this one of the main findings of this work, we also have to provide an analytical proof for this. It is based on the separation of the contributions of intrinsically incoherent and coherent transitions to the final total particle yield.

 

Fig. 6. Growth of the energy of the created electrons during the interaction with a chirped external electric field. [The numerical box of length ${L} = {2}.{4}$ a.u. was discretized into ${{N}_x} = {512}$ spatial grid points. The temporal evolution was discretized into 5000 grid points. The electric field was given by ${ F }({t})\; =\; {{ F }_0}\;{\exp}[\; - \;{({t} \; - \; {{t}_1})^2}/({2}{\tau ^2})]\;{ \cos}[(\omega \;+ \;{ b }({t} \;-\; {{t}_1}))({t} \;-\; {{t}_1})]$, with ${ b } = {5}.{6} \times\;{{10}^6}\;{\rm a}.{\rm u}.$, $\omega = {2}.{8}\;{{ C }^2}$, ${\tau} = \;{5}.{325}\; \times \;{{10}^{ - 4}}\;{\rm a}.{\rm u}.$, ${{t}_{1\:}} =$ $\;{0}.{004}\;{\rm a}.{\rm u}.$, and ${{ F }_0} = {5}\;{{ C }^3}$.

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1. Incoherent and Coherent Contributions to the Final Particle Yield

The number of created electron-positron pairs was obtained from the double summation over all time-evolved negative (Dirac sea) states $\vert{k};{d(t)}\rangle$ and positive energy states $\vert{k};{u}\rangle$ as ${N}({t}) = {\Sigma _p}{\Sigma _k}\vert\langle{p};u \vert k;d(t)\rangle\vert^{2}$. As we will see below, it is advantageous to partition this expression into two contributions:

The first summation ${{ N}_{\rm inc}}({t})$ has an easy interpretation. Each term ($1-\vert\langle {k; d}\vert {k;d(t)\rangle\vert^{2}}$) is equal to the decay probability of the initial Dirac sea state $\vert {k;d\rangle}$. In other words, it is the probability that this Dirac sea state has developed a hole (meaning it becomes unoccupied). We remind the reader that the initial state conservation law (see the derivation in Appendix B) predicts $\langle \Psi ({0}){\vert}{U}\;\vert\Psi ({0})\rangle \;{= }\;\langle \Psi ({0})\vert\;{{U}_{ R }}\;\vert\Psi ({0})\rangle $, where ${U}$ and ${{U}_{ R }}$ are the unitary time-evolution propagators for the forces ${ F }({t})$ and ${{ F }_{ R }}({t})$. For $\vert\Psi ({0})\rangle \;{= }\;{\vert{k;d}}\rangle $, we immediately find that for any dynamics, $\langle {k;d} \vert {k;d}({t})\rangle $ is identically obtained from ${ F }({t})$ and from ${{ F }_{ R }}({T})$. Therefore, ${{ N}_{\rm inc}}({T})$ is the same for the vacuum decay triggered by each force.

We also see that the introduced nomenclature as “incoherent” for the portion ${{ N}_{\rm inc}}({T})$ is physically meaningful, as this quantity is indeed computed based on the mutually independent depletion of all initial states.

The depletion of the probability for $\vert{k;d}\rangle $ as calculated by ${{ N}_{\rm inc}}({t})$ alone, however, does not include the fact that other (also initially fully occupied) Dirac sea states $\vert{{k}^\prime };{d}\rangle $ can transfer part of their population to this particular state $\vert{k;d}\rangle $. As ${{ N}_{\rm inc}}({t})$ does not contain any non-diagonal terms, such as $\langle {k;d}\vert{{k}^\prime };{d}({t})\rangle $ for ${k} \ne {{k}^\prime }$, it does not take this transfer into account. Therefore, the expression for ${{ N}_{\rm inc}}({t})$ reflects only the sum of isolated decays. It therefore would overestimate the true number of final holes ${ N}({t})$ in the Dirac sea, i.e., the number of created electron-positron pairs.

In order to reduce the final (overestimated) number of created holes [based on ${{ N}_{\rm inc}}({t})$ alone], we have to also include those processes where other initially populated Dirac sea states $\vert{{k}^\prime };{d}\rangle $ repopulate the state $\vert{k;d}\rangle $. These processes reduce the number of holes [based on ${{ N}_{\rm inc}}({t})$], which is manifested by the fact that the second summation given by ${{ N}_{\rm coh}}({t}) \equiv - \;{\Sigma _{k\:}}{\Sigma _{k^\prime \ne k}}\;\vert\langle {{k}^\prime };{d} \vert {k;d}({t})\rangle \vert ^ 2$ is always negative. It therefore corrects the estimate for ${ N}({t})$ provided by ${{ N}_{\rm inc}}({t})$ downwards.

The rather different meanings of ${{ N}_{\rm inc}}({t})$ and ${{ N}_{\rm coh}}({t})$ can also be illustrated for the trivial case where the external fields ${ F }({t})$ and ${{ F }_{ R }}({t})$ are so weak that not a single electron-positron pair can be created. In this case, we have ${ N}({t}) = {0}$, and therefore ${{ N}_{\rm inc}}({t}) = - \;{{ N}_{\rm coh}}({t})$, as any decay (population decrease) of $\vert{ k;d}\rangle $ is being refilled by the population of other initial states $\vert{{k}^\prime };{d}\rangle $.

Let us return to the case of purely temporally induced pair-creation processes. In a notation where the external field is represented by a vector potential, the total canonical momentum is conserved such that all transitions’ matrix elements vanish: $\langle {{k}^\prime};{d\vert}{k;d}({t})\rangle = \langle {p;u\vert}{k;d}({t})\rangle {\vert}= {0}$, unless ${p} = {k}$ or ${k} ={{k}^\prime }$. This means that any initial Dirac sea state $\vert{k;d}\rangle $ can populate only a single state $\vert{p;u}\rangle $ with ${p} ={k}$. The overall pair-creation dynamics due to ${A}({t})$ are therefore equivalent to the dynamics of an infinite set of mutually independent two-level systems. As the summation in ${{N}_{\rm coh}}({t})\; \equiv - {\Sigma _{k}}{\Sigma _{k^{\prime} \ne k}}$ $\vert\langle {{k}^\prime };{d \vert}{k;d}({t})\rangle \vert ^ 2$ omits the only non-vanishing term ${k}= {{ k}^\prime }$, we have ${{ N}_{\rm coh}}({ t})= {0}$. In other words, we have proven here that a purely temporally induced vacuum decay can only be chirped incoherently, and we have ${N}({t}) ={{N}_{\rm inc}}({t})$. This means that ${F}({t})$ and ${ F}({T} - {t})$ lead to precisely the same number of created particle pairs.

2. Divergence of ${N_{\rm inc}}(t)$ and ${N_{\rm coh}}(t)$

We should note that the largest possible energies accessible in our numerical simulations are typically chosen to be sufficiently large such that the numerical value of the number of created electron-positron pairs ${N}({ t})$ is fully converged and therefore becomes independent of this energy cutoff. This convergence is possible as the coupling strength between the lower energy (Dirac sea) states and the states of the positive manifold, given by $\vert\langle {p;u \vert}{k;d}({t})\rangle \vert$, decreases rapidly with the momentum ${p}$.

While in principle the summation ${N}({t})= {\Sigma _p}$ ${\Sigma _k}$ $\vert\langle {p ; u \vert}{ k;d}({ t})\rangle \vert ^ 2$ extends formally over infinitely many states, only the lowest momentum states $\vert{p;u}\rangle $ become populated. Similarly, only negative energy states $\vert{k;d}\rangle $ above a certain lower energy bound can decay into the upper positive energy continuum. We denote this subset of Dirac sea states as dynamically relevant states and use the notations $\vert{k;d}\rangle $ for these low momentum states. Similarly, the excitable upper continuum states are denoted by $\vert{ P;u}\rangle $. In other words, we can write the restricted sum as ${N}({t})={\Sigma _{P}}{\Sigma _K}\;\vert\langle { P;u\vert}{K;d}({t})\rangle \vert^2$ containing only a finite number of terms on a discretized momentum grid. The time evolution of the dynamically relevant and irrelevant states is given by

In contrast to the converged sum over all matrix elements $\vert\langle {p;u}\vert{k;d}({t})\rangle \vert^2$ required for ${ N}({t})$, each of the two summations ${{N}_{\rm inc}}({t})$ and ${{N}_{\rm coh}}({t})$ diverge with the energy cutoff. However, this does not constitute any problem as the large energy portions responsible for the divergence in ${{ N}_{\rm inc}}({t})$ and ${{N}_{\rm coh}}({t})$ mutually cancel out when being added up. This can be seen immediately if we insert Eq. (12) into the corresponding two summations. For the first summation ${{ N}_{\rm inc}}({t})$ we obtain $({\Sigma _{k}}{1}){ - }\;{\Sigma _{k}}\vert\langle { k;d}\;\vert{\Sigma _{k^{\prime}}}$ ${{D}_{k^{\prime}}}({k,t})$ $\vert{{k}^\prime };{d}\rangle \vert ^ 2$, which reduces to ${{N}_{\rm inc}}({t}) =({\Sigma _{k}}{1})- {1}$ if we use the orthogonality $\langle {k;d}\vert{{k}^\prime };{d}\rangle = {\delta _{k{^\prime}k}}$ and the normalization ${\Sigma _{k^{\prime}}}$ $\vert{{D}_{k^{\prime}}}({k},{t})\vert^2 ={1}$. Similarly, the expression for ${{N}_{\rm coh}}({t})$ becomes $ - {\Sigma _{k}}{\Sigma _{k^{\prime} \ne k}}$ $\vert\langle {{k}^\prime }{d\vert}{\Sigma _{k^{\prime\prime}}}$ ${{D}_{k^{\prime\prime}}}({k},{t})$ $\vert{{k}^{\prime \prime }};{d}\rangle \vert^ 2$, which simplifies to $ - ({\Sigma _{k^{\prime} \ne k}}{1})= -({\Sigma_{k^{\prime }}}{1})\;{+}{1}$. This means that the diverging terms in ${{N}_{\rm inc}}({t})$ and ${{N}_{\rm coh}}({t})$ are given by the same infinite summation $(\Sigma_{k}\,1)$ and therefore cancel out, and we consistently have ${{N}_{\rm inc}}({t}) + {{N}_{\rm coh}}({t})= {0}$ for the dynamically irrelevant Dirac states. This means we can therefore ignore the irrelevant states $\vert{k;d}\rangle $ in our analysis of pair creation, despite their diverging impact on ${{N}_{\rm inc}}({t})$ and ${{ N}_{\rm coh}}({t})$, respectively.

B. Electron-Positron Pair Creation Triggered by a Spatially Inhomogeneous Field

In this section will show that, in contrast to the fully incoherent temporally-only induced vacuum decay, the pair-creation process when triggered by a spatially inhomogeneous external force field can reveal coherent features. In other words, we will suggest that the original chirped force field and its time-reversed counterpart can lead to different final particle yields and therefore to different final energies ${e}({T})\; \ne \;{{e}_{R}}({T})$.

In order to model an external electric force field that, in addition to its temporal dependence has also a spatial variation, we have used the scalar potential in the product form ${V}({x},{t}) = {V}({x})\;{ F}({t},{x})$, where

In Fig. 7 we show the corresponding temporal growth of the energy that was absorbed by the vacuum state in this interaction. We see that—in direct contrast to Fig. 6 for a purely temporal excitation—in this case the final amount of energy after the interaction is different under ${V}({x},{ t})$ and ${V}({x},{T} - {t})$, i.e., ${e}({T})\; \ne \;{{e}_{R}}({T})$. This means that the precise timing of when which energy portion is absorbed by the system is important.

 

Fig. 7. Growth of the total energy of the created electrons during the interaction with a chirped external field. The numerical box of length ${L}= {2.4}$ a.u. was discretized into ${{ N}_x} = {512}$ spatial grid points. The temporal evolution was discretized into 5000 grid points. The scalar potential was given by ${ V}({x},{t})= {F}({t},{ x})$ ${V}({ x})$, where ${V}({ x}) = {{V}_0}\{{\tanh}[({ x} - {D})/{W}] - {\tanh}[({x} + {D})/{W}]\}/{2}$, with ${{V}_{0\:}} = {5}\;{{c}^2}$, ${W} = {0}.{ 5/c}$, ${D} = {0}.{6}\;{\rm a}.{\rm u}.$, and ${F}({t},{ x}) = {\rm exp}[\; - \;{({t} - {{ t}_1})^2}/({2}{\tau ^2})]{\cos}[(\omega + {b}({t} - {{t}_1}))({t} - {{t}_1})\; - \;{x}\omega /{ C }]$, with ${b} = {5}.{6} \times {{10}^6}\;{\rm a}.{\rm u}.$, $\omega = {2}.{8}$ ${{c}^2}$, $\tau = {5}.{325}\; \times \;{{10}^{ - 4}}\;{\rm a}.{\rm u}.$, and ${{ t}_1} = {0}.{004}\;{\rm a}.{\rm u}$.

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We have also varied the specific choice of our parameters ${W}$, ${D}$, $\tau $, ${ b}$, $\omega $, and ${{V}_0}$ to maximize the difference between ${e}({T})$ and ${{ e}_{R}}({T})$. While a systematic optimization of this six-dimensional parameter space is beyond the framework and purpose of this work, we have not been able to identify a better set of parameters. For example, if the amplitude ${{V}_0}$ is chosen to be smaller than the Schwinger limit, then the spatial decay mechanism would become less relevant, the temporal (incoherent) mechanism would become more dominant, and the difference between ${e}({ T})$ and ${{ e}_{ R}}({T})$ would decrease. The corresponding Keldysh parameter might also become relevant to govern some properties of the chirping mechanism. We should remark that, even independent of our main objective of comparing incoherent and coherent chirping, this process is complex, as the simultaneous interplay of temporally and spatially induced pair-creation mechanisms is presently not too well understood even for non-chirped fields.

To examine the corresponding coherent and incoherent portions leading to the final number of created electron-positron pairs, we have also calculated ${{N}_{\rm inc}}({T})$ and ${{N}_{\rm coh}}({T})$ for the same dynamics shown in Fig. 7. Under the excitation by ${V}({x},{t})$, the final yield of ${N}({T}) ={36}.{246}$ is the result of the summation of ${{N}_{\rm inc}}({T}) = {193}.{528}$ and ${{N}_{\rm coh}}({T}) = - {157}.{282}$. This suggests that the average energy of the created electrons or positrons is about ${e}({T})/{N}({T})= {72}.{23}\;{{c}^2}/{36}.{246}\;{= }\;{1}.{99}\;{{c}^2}$. This amount is within the energy range of the Klein-energy window [37,38] associated with the spatial part ${V}({x})$ of the supercritical potential and usually given by ${{c}^2} \lt {\rm energy} \lt {{V}_0}\; - {{c}^2}$ (with ${{V}_0} = {{5c}^2}$). It is slightly larger than the most probable energy, which is for this potential $({{V}_0}\; - {2}\;{{c}^2})/{2}\;{=}\;{1}.{{5c}^2}$.

For the time-reversed excitation field ${{V}_{R}}({x},{t}) = {V}({x},{T} - {t})$, we obtained a slightly smaller final yield ${{N}_{R}}({T}) = {35}.{778}$. Here the decomposition into the incoherent and coherent portions yields ${{N}_{\rm inc}}({T}) = {193}.{528}$ and ${{N}_{\rm coh}}({T}) = - {157}.{75}$. Given the fact that the incoherent contributions for both processes are identical, ${{N}_{\rm inc}}({T}) = {{N}_{\rm inc}}({T}) = {193}.{528}$ is a nice numerical confirmation of the theoretical analysis derived in Section 3.A.1. Similarly, the difference between the final yields ${N}({T})$ can therefore clearly be associated with the different contributions of ${{N}_{\rm coh}}({T}) = - {157}.{282}$ for ${V}({x},{t})$ and ${{N}_{\rm coh}}({T}) = - {157}.{75}$ for ${{V}_{R}}({x},{t})$.

4. SUMMARY AND OPEN QUESTIONS

In this work we have examined the question of which dynamical systems are phase sensitive to the precise timing of an external force field. We have seen that for some physical systems entirely different dynamical pathways [associated with either ${F}({t})$ or its time-reversed counterpart ${{F}_{R}}({t})$] can lead to identical observables after the interaction. By comparing the dynamical responses to the two force fields ${F}({t})$ and ${F}({T} - {t})$, which have identical energy spectra, we could establish that the simple classical mechanical harmonic oscillator as well as a quantum mechanical two-level system are phase insensitive and can only be chirped incoherently. While the time evolutions caused by ${F}({t})$ and ${{F}_{R}}({t})$ are completely different for the elongation ${x}({t})$ and the velocity ${dx}/{dt}$ at all times ${t}$, the amount of final energy absorbed after the interaction becomes identical. This means that these systems are remarkably robust and not sensitive to the precise sequence of the temporal details of the force field. In contrast, we showed that a classical nonlinear oscillator can be coherently chirped. This means that the precise timing of the pulse is crucially important here. We also suggested that a quantum three-level system can also be coherently chirped, unless certain specific parameters (energies and dipole moments) destroy this phase sensitivity.

Another key finding of this work is the interesting impact of super-strong and time-dependent electric fields on the electron-positron pair-creation process. In the case of purely time-dependent fields (without any spatial inhomogeneity), we found that the final number of created particle pairs (and therefore their energy) does not depend at all on whether the field is positively or negatively chirped, suggesting that the vacuum decay process is completely incoherent. As a consequence, if one uses a specifically tailored sequence of pulses to optimize the final particle yield, offering the inversely timed sequence would always lead to the same final result. Quite interestingly, however, we found that a sufficiently large spatial inhomogeneity can lead to the possibility of coherent chirping, where the precise timing of the external field can lead to a different final response.

This also has some interesting implications about the temporal behavior of the particular force field that would lead to a global maximum for the final absorbed energy. If there exists a unique force field ${{F}_{\rm opt}}({t})$, then its time dependence necessarily has to satisfy the symmetry ${{F}_{\rm opt}}({t})= {{F}_{\rm opt}}({T} - {t})$. This symmetry was already observed numerically in prior studies [24] and can be very helpful for optimization studies, as this constraint significantly narrows down the space of possible candidates for ${{F}_{\rm opt}}({t})$.

We point out that while the effects of laser pulse chirping have been widely examined theoretically and experimentally in atomic and molecular optical physics and physical chemistry, it is presently unknown under which parameters a coherent chirping is possible and how this feature can be inhibited for specific characteristics of the systems. To identify the underlying physical mechanism for such an inhibition is certainly an interesting open question.

There are also interesting mathematical challenges that are motivated by this work. For example, we have proven analytically that for those theoretical models that can be described by a real Hamiltonian matrix, the initial state probability is identical for ${ F}({t})$ and its time-reversed counterpart ${{F}_{R}}({t})$, i.e., $\langle \Psi ({0}){\vert}{U}\vert{\Psi}({0})\rangle \;{= }\;\langle \Psi ({0}){\vert}\;{{U}_{R}}\;\vert\Psi ({0})\rangle $. It turns out that if the Hamiltonian matrix is complex, but tridiagonal, then the conservation law still holds, but the corresponding rigorous mathematical proofs are non-trivial but can be approached with symbolic computer algorithms for non-commutative algebras such as NCAlgebra based on Mathematica.

We consider it the main finding of this work to suggest that any efforts to construct an optimal external laser field that can optimize the final pair-creation yield cannot yield a single and unique optimum, as the corresponding time-reversed force field will always yield the same amount. Unique optima might therefore only exist if the vacuum decay is triggered by an external field that has a sufficient amount of spatial inhomogeneity. For example, the effects of additional magnetic fields on the pair-creation process have been widely studied.

We have used the force-free energy as a possible signature of the difference between coherent and incoherent chirping. However, there might be some other dynamical observables as well that could permit such a distinction. For example, maybe it is possible to construct other phase space variables (as a function of ${x}$ and ${p}$) for nonlinear oscillators that could reveal the force reversal symmetry at the final time. Another related question is if (in addition to the force-free energy) the harmonic oscillator can also permit other invariant quantities whose final values are invariant as well under the two excitations ${F}({t})$ and ${{ F}_{R}}({t})$.

In classical mechanics, dynamically invariant quantities can usually be associated with global symmetries. In our case, however, the observed invariance is local, i.e., concerning only the final value at time T. It is therefore a fascinating question to examine whether in the case of the “local” invariances introduced here one can also construct some Noether-like symmetry properties. Our analysis was restricted to comparing the final responses to ${F}({t})$ and ${F}({T} - {t})$. One could additionally analyze the effect of some other transformations on ${F}({t})$ that keep the energy spectrum $\vert{F}(\omega ){\vert^2}$ invariant. In this case, the number of “incoherent” systems should decrease. We are certainly only at the very beginning of our understanding.