On the Time Dependence of Adiabatic Particle Number

# On the Time Dependence of Adiabatic Particle Number

Robert Dabrowski and Gerald V. Dunne Department of Physics, University of Connecticut, Storrs CT 06269-3046, USA
###### Abstract

We consider quantum field theoretic systems subject to a time-dependent perturbation, and discuss the question of defining a time dependent particle number not just at asymptotic early and late times, but also during the perturbation. Naïvely, this is not a well-defined notion for such a non-equilibrium process, as the particle number at intermediate times depends on a basis choice of reference states with respect to which particles and anti-particles are defined, even though the final late-time particle number is independent of this basis choice. The basis choice is associated with a particular truncation of the adiabatic expansion. The adiabatic expansion is divergent, and we show that if this divergent expansion is truncated at its optimal order, a universal time dependence is obtained, confirming a general result of Dingle and Berry. This optimally truncated particle number provides a clear picture of quantum interference effects for perturbations with non-trivial temporal sub-structure. We illustrate these results using several equivalent definitions of adiabatic particle number: the Bogoliubov, Riccati, Spectral Function and Schrödinger picture approaches. In each approach, the particle number may be expressed in terms of the tiny deviations between the exact and adiabatic solutions of the Ermakov-Milne equation for the associated time-dependent oscillators.

###### pacs:
12.20.Ds, 11.15.Tk, 03.65.Sq, 11.15.Kc

## I Introduction

The stimulated production of particles from the quantum vacuum is a remarkable feature of quantum field theory that can occur when the vacuum is subjected to an external perturbation, such as gauge or gravitational curvature. Notable examples include the Schwinger effect from applying an external electric field to the quantum electrodynamic (QED) vacuum he (); sch (); greiner (); dunne (), Friedmann-Robertson-Walker (FRW) cosmologies parker68 (); zeldovich (); Parker:1974qw (); mukhanov (); hu (), de Sitter space times Birrell:1982ix (); mott (); bousso (); gds2010 (); Polyakov:2007mm (); and (), Hawking Radiation due to blackholes and gravitational horizon effects gibbhawk (); vacha (); ford (); Boyanovsky:1996rw (); Brout:1995rd (); padma (), and Unruh Radiation seen by an accelerating observer unruh (); Schutzhold:2006gj (). This particle production paradigm plays an important role in the physics of non-equilibrium processes in heavy-ion collisions gelisvenu (); kharzeevtuchin (); Gelis:2010nm (), astrophysical phenomena ruff (), and the search for nonlinear and non-perturbative effects in ultra-intense laser systems mourou (); mattias (); dunne-eli (); DiPiazza:2011tq (). There are also close technical analogues with driven two-level systems, relevant for atomic and condensed matter processes oka (); Nation (), such as Landau-Zener-Stückelberg transitions lzs (), the dynamical Casimir effect and its analogues Dodonov:2010zza (); nori (), Ramsey processes and tunnel junctions AkkermansDunne:2012 (); reulet ().

Particle production involves evolution of a quantum system from an initial (free) equilibrium configuration to a new final (free) equilibrium configuration through an intervening non-equilibrium evolution due to a perturbing background. Quantifying the final asymptotic particle number involves relating the final equilibrium configuration to the initial one. This is a comparison of well-defined asymptotic vacua where the identification of positive (particles) and negative (anti-particles) energy states is unambiguous and exact. On the other hand, a quantitative description of particle production at all times, not just at asymptotically early and late times, requires a well-defined notion of time-dependent particle number also at intermediate times. This is a challenging conceptual and computational problem, especially if one wants to include also back-reaction effects and the full non-equilibrium dynamics. In this paper we discuss in detail one significant aspect of this problem: the role of the truncation of the adiabatic expansion in the conventional definition of time-dependent particle number.

At intermediate times, when the system is out of equilibrium, it is less clear how to distinguish between positive and negative energy states. The standard approach parker68 (); mukhanov (); hu (); brezin (); popov (); gavrilov (); KME:1998 (); Habib:1999cs (); kimpage (); winitzki (); Kim:2011jw (); Gelis:2015kya (); Zahn:2015awa () involves using the adiabatic expansion to specify a reference basis set of approximate states, under the assumption of a slowly varying perturbation. Then a time-dependent particle number is defined by the projection of the evolving system onto these approximate states. With this procedure, the final particle number at asymptotically late time is independent of the basis choice. However, the particle number at intermediate times has a significant dependence on the basis choice, often varying over several orders of magnitude before settling down to its final basis-independent late-time value DabrowskiDunne:2014 (). At first sight, this basis dependence would seem to immediately invalidate any attempt to define a physically sensible intermediate-time particle number. In particular, since the adiabatic expansion is a divergent expansion, we expect that its truncation should be performed at its optimal order, which is not fixed at a particular order but depends on the physical parameters of the perturbation. But here we can invoke a remarkable universality result due to Dingle and Berry. Dingle found that the large-order behavior of the divergent adiabatic expansion has a universal form, providing accurate estimates of its behavior under optimal truncation dingle (). Berry BerryAsymptotic () applied Borel summation to find a generic smoothing of the associated Stokes phenomenon [i.e., particle production dumludunne ()], leading to a universal time evolution. We have previously applied these technical results to the physical phenomena of particle production in time dependent electric fields and in de Sitter space time DabrowskiDunne:2014 (). Here we present a systematic analysis of the influence of the choice of order of truncation of the adiabatic expansion, which corresponds directly to the non-uniqueness of specifying the approximate adiabatic reference states.

This surprising universality suggests a natural definition of time-dependent adiabatic particle number at all times, corresponding to an optimal adiabatic approximation of the time evolution. This raises interesting questions regarding the physical nature of such a definition of particle number, some aspects of which have begun to be tested experimentally in analogous non-relativistic quantum systems lim-berry (); expberry (); demirice (); BB (); delCampo (); jarz-shortcut (); CDexp1 (); CDexp2 (). We will address these questions in the quantum field theory context in future wrok.

In this paper, we examine the truncation of the adiabatic expansion using several common (and equivalent) formulations of particle production: the Bogoliubov brezin (); popov (); KME:1998 (); and (), Riccati dumludunne (), Spectral Function FukushimaHataya:2014 (); Fukushima:2014 () and Schrödinger vacha (); padma () approaches. The analysis also extends straightforwardly to the quantum kinetic approach KME:1998 (); Habib:1999cs (); Rau:1995ea (); schmidt (); Huet:2014mta (), and the Dirac-Heisenberg-Wigner approach with time-dependent background fields Hebenstreit:2010vz (); Hebenstreit:2010cc (). For definiteness we study the Schwinger effect in scalar QED (sQED) with spatially homogeneous but time-dependent electric fields, but the basic results apply to a wide variety of quantum systems, as mentioned above. In Section II we review the relation between the Klein-Gordon equation and the Ermakov-Milne steen (); erm (); milne (); pinney () equation, associated with the exact solution to the quantum harmonic oscillator with time-dependent frequency husimi (); LewisInvariant1 (); DittrichReuter (). The projection of the adiabatic states onto the exact solution of the Ermakov-Milne equation leads to an analytic expression for the time-dependent adiabatic particle number, which clearly illustrates the basis dependence and simplifies its evaluation. The four approaches to time-dependent particle production yield precisely the same form, demonstrating that basis dependence is a universal feature of the adiabatic particle number at intermediate times. In Section III we examine the influence of different truncations of the adiabatic expansion. This also yields a new perspective: the adiabatic approximation of time-dependent particle production is completely characterized by the exponentially small deviations from the exact Ermakov-Milne solution. Section IV is devoted to a brief discussion of the results.

## Ii Adiabatic Particle Number

### ii.1 Field Mode Decomposition: Klein-Gordon and Ermakov-Milne Equations

We consider scalar QED for simplicity.111Apart from the opposite phase of interference effects, the physics is very similar to that of spinor QED, but it is notationally simpler. For a charged (complex) scalar field in a time-dependent and spatially homogeneous classical electric field, the scalar field can be separated into spatial Fourier modes, , so that the Klein-Gordon equation, , reduces to decoupled linear time-dependent oscillator equations:

 Klein-Gordon equation:¨fk(t)+ω2k(t)fk(t)=0 (1)

Here the effective time-dependent frequency is brezin (); popov (); KME:1998 ()

 ω2k(t)≡m2+k2⊥+(k∥−A∥(t))2 (2)

where and are the momenta of the produced particles along and transverse to the direction of the electric field, respectively. The magnitude of the electric field varies with time as . There is an analogous mode decomposition for particle production in cosmological and gravitational backgrounds parker68 (); and (); gds2010 (); vacha ().

We define quantized scalar field operators and momenta for each mode as

 ϕk(t) = fk(t)ak+f∗−k(t)b†−k (3) π†k(t) = ˙fk(t)ak+˙f∗−k(t)b†−k (4)

with (time independent) bosonic creation and annihilation operators to describe particles and anti-particles. Bosonic commutation relations impose the Wronskian condition on the mode functions :

 fk(t)˙f∗k(t)−˙fk(t)f∗k(t)=i (5)

Writing the complex mode function in terms of its real amplitude and phase ,

 fk(t)≡ξk(t)e−iλk(t) (6)

the Klein-Gordon equation (1) reduces to the Ermakov-Milne steen (); erm (); milne (); pinney () equation for the amplitude function :

 Ermakov-Milne equation:¨ξk(t)+ω2k(t)ξk(t)=14ξ3k(t) (7)

As usual, unitarity determines the time-dependent phase in terms of as:

 λk(t)=∫tdt′2ξ2k(t′). (8)

Note that with the definition (6), the Ermakov-Milne equations (7, 8) are completely equivalent to the original Klein-Gordon equation (1). Another equivalent way to express the time-evolution is achieved by defining the square of the amplitude function, , which satisfies a nonlinear second-order equation, and its corresponding linear third-order equation:

 Gel'fand-Dikii equation:2Gk¨Gk−˙G2k+4ω2k(t)G2k = 1(nonlinear form) (9) ...Gk+4ω2k(t)˙Gk+4ωk(t)˙ωk(t)Gk = 0(linear form) (10)

This is known as the Gel’fand-Dikii equation gelfand (), arising in the analysis of the resolvent Green’s function for Schrödinger operators, which can be written in terms of products of solutions to the Klein-Gordon equation (1). The resolvent approach has been used in the analysis of Schwinger effect Balantekin:1990aa (); dunnehall ().

The particle production problem consists of the following physical situation: at initial time the vacuum is defined with respect to the (time-independent) creation and annihilation operators in (3). Then as time evolves the vacuum is subjected to a time-dependent electric field, which turns off again as . At , after the electric field has been turned off, the production of particles from vacuum can be inferred from the fraction of negative frequency modes in the evolved mode functions. As is well known brezin (); popov (); dumludunne (), this can be expressed as an “over-the-barrier” quantum mechanical scattering problem, in the time domain, by interpreting the Klein-Gordon equation (1) as a Schrödinger-like equation

 −¨fk−(k∥−A∥(t))2fk=(m2+k2⊥)fk (11)

with physical “scattering” boundary conditions brezin (); popov (); gavrilov ():

 (12)

The scattering coefficients and defined at satisfy . So, we can evolve the mode oscillator equation (1) with initial conditions

 fk(t→−∞) ∼1√2ωk(−∞)e−iωk(−∞)t (13) ˙fk(t→−∞) ∼−i√ωk(−∞)2e−iωk(−∞)t (14)

or, equivalently the Ermakov-Milne equation (7) with initial conditions

 ξk(t→−∞) ∼1√2ωk(−∞) (15) ˙ξk(t→−∞) ∼0 (16)

A numerical advantage of the Ermakov-Milne equation is that the amplitude function typically varies more smoothly than the mode function [and recall from (8) that the phase is determined by ]. This is illustrated in Figure 1, for an explicit example of a single-pulse electric field, for which a well-known analytic exact solution is possible, as reviewed in the Appendix V. In this paper we primarily express particle number in terms of the amplitude function .

### ii.2 Bogoliubov Transformation and Adiabatic Particle Number

In processes that involve a time-dependent background field, a unique separation into positive and negative energy states with which to identify particles and anti-particles is only possible at asymptotic times brezin (); popov (), when the electric field is turned off. This is the same as the non-uniqueness of defining left- and right-moving modes inside an inhomogeneous dielectric medium budden (); berry-mount ().

To proceed, we define a time-dependent adiabatic particle number in the presence of a slowly varying time-dependent background, with respect to a particular set of reference mode functions defined as

 ~fk(t)≡1√2Wke−i∫tWk(t)t→−∞−−−−→1√2ωk(−∞)e−iωk(−∞)t (17)

Clearly there is an infinite number of such reference mode functions, all having the same initial asymptotic behavior. The problem is to define a physically suitable set of mode functions for use at intermediate times.

Insisting that , as defined in (17), be a solution to the Klein-Gordon equation (1), the function is related to the effective frequency by the well-known Schwarzian derivative form:

 W2k(t)=ω2k(t)−⎡⎣¨Wk(t)2Wk(t)−34(˙Wk(t)Wk(t))2⎤⎦ (18)

This can be solved by a systematic adiabatic expansion in which the leading order is the standard leading WKB solution to the mode oscillator equation (1) of the form BerryAsymptotic (); DabrowskiDunne:2014 (). Higher order terms are analyzed in detail in section III.

The Bogoliubov Transformation is a linear canonical transformation that defines a set of time-dependent creation and annihilation operators, and , from the original time-independent operators, and , defined at the initial time in (3, 4) popov (). They are related by

 (~ak(t)~b†−k(t))=(αk(t)β∗k(t)βk(t)α∗k(t))(akb†−k) (19)

where unitarity requires for scalar fields, for all . As a result of the Bogoliubov transformation, the equivalent decomposition of the scalar field operator in terms of these reference mode functions is

 ϕk(t)=~fk(t)~ak(t)+~f∗k(t)~b†−k(t) (20)

This can also be interpreted as a linear transformation between the exact mode functions and the reference adiabatic mode functions , as

 fk(t)=αk(t)~fk(t)+βk(t)~f∗k(t) (21)

We also need to specify the transformation of the scalar field momentum operator :

 π†k(t)=Qk(t)~fk(t)~ak(t)+Q∗k(t)~f∗k(t)~b†−k(t) (22)

with a corresponding decomposition of the first derivative:

 ˙fk(t)=Qk(t)αk(t)~fk(t)+Q∗k(t)βk(t)~f∗k(t) (23)

Here is defined as

 Qk(t)=−iWk(t)+Vk(t). (24)

The inclusion of the real time-dependent function , specified later, in the decompositions (21) and (23) represents the most general decomposition of the exact solution that is consistent with unitarity (the preservation of the bosonic commutation relations, or equivalently the Wronskian condition (5)). The freedom in the choice of and encodes the arbitrariness of specifying positive and negative energy states at intermediate times. We will see later that a ‘natural’ choice is , coming from the derivative of the factor in the definition of the reference mode functions (17).

The scattering coefficients in (12) are realized as the Bogoliubov coefficients evaluated at asymptotically late time, after the perturbation has turned off: and . The time-dependent adiabatic particle number, for each mode , is defined as the expectation value of the time-dependent number operator with respect to the asymptotic vacuum state. Assuming no particles are initially present, the time-dependent adiabatic particle number is

 ~Nk(t)≡⟨~a†k(t)~ak(t)⟩=|βk(t)|2. (25)

This reduces the problem to the direct evaluation of the time evolution of the Bogoliubov transformation parameters and . The decompositions (21) and (23) are exact provided they satisfy the mode oscillator equation (1), which implies the following evolution equations for the Bogoliubov transformation parameters and :

 (26)

where

 δk =12iWk(ω2k−W2k+(˙Vk+V2k)) (27) Δk =˙Wk2Wk+Vk (28)

Note that vanishes with the choice . The numerical evaluation of this coupled differential equation completely determines the time evolution of and with respect to the basis . The time evolution of the adiabatic particle number is obtained by the modulus squared of the time evolution of the Bogoliubov coefficient following (25), solved using the initial conditions and , consistent with the scattering scenario in (12) and the assumption of no particles being initially present. The evolution equations (26) are dependent on the choice made for the basis functions and , which influences the time evolution of the adiabatic particle number at intermediate times but does not affect its final asymptotic value at future infinity, DabrowskiDunne:2014 (). This is because the final value is determined by the global information of the Stokes phenomenon dumludunne ().

The time evolution of the coefficients and can also be expressed directly through the time evolution of the amplitude function . Solving the linear equations (21) and (23) we find

 αk(t) =i~f∗k(t)(˙fk(t)−Q∗k(t)fk(t)) (29) βk(t) =−i~fk(t)(˙fk(t)−Qk(t)fk(t)), (30)

Furthermore, from (6) and its time-dependent phase (8), we find the identity

 ˙fkfk=˙ξkξk−i2ξ2k, (31)

Thus, the Bogoliubov coefficients may be rewritten in the uncoupled form as

 αk(t) (32) βk(t) =−ξk√2Wk[(12ξ2k−Wk)+i(˙ξkξk−Vk)]exp[−i∫t(12ξ2k+Wk)] (33)

This expresses the time evolution of the Bogoliubov coefficients as a comparison between the time evolution of the amplitude function, , obtained by solving the Ermakov-Milne equation (7), and the reference mode basis . The Adiabatic Particle Number then follows:

 |αk(t)|2 =ξ2k2Wk⎡⎣(12ξ2k+Wk)2+(˙ξkξk−Vk)2⎤⎦ (34) ~Nk(t)=|βk(t)|2 =ξ2k2Wk⎡⎣(12ξ2k−Wk)2+(˙ξkξk−Vk)2⎤⎦ (35)

It is straightforward to confirm that unitarity is preserved: .

The expression (35) for the time-dependent particle number is one of the primary results of this paper. It emphasizes clearly the dependence of the adiabatic particle number on the basis choice of reference mode functions . It is not enough to know the time evolution of : one must also compare it to the reference functions. With the choice , the expression for the adiabatic particle number simplifies further to a direct comparison between and :

 ~Nk(t) =ξ2k2Wk⎡⎣(12ξ2k−Wk)2+(˙ξkξk+˙Wk2Wk)2⎤⎦ (36)

In subsequent sub-sections we show how exactly the same expression arises in other different but equivalent, methods for defining and computing the adiabatic particle number. Then in Section III we show how in the adiabatic expansion the expression (36) can be viewed as a measure of the tiny deviations between the exact solution of the Ermakov-Milne equation and various orders of the adiabatic approximation for .

### ii.3 Riccati Approach to Adiabatic Particle Number

The time evolution of the Bogoliubov coefficients can be re-expressed in Riccati form by defining the ratio popov (); dumludunne ()

 Rk(t)≡βk(t)αk(t) (37)

which can be viewed as a local (in time) reflection amplitude for this Schrödinger-like equation (11) popov (); brezin (). Using the unitarity condition, , the time-dependent adiabatic particle can be rewritten as

 ~Nk(t)=|Rk(t)|21−|Rk(t)|2 (38)

In the semi-classical limit in which is the dominant scale (as is relevant in QED), this over-the-barrier scattering problem has an exponentially small reflection probability, which implies that the adiabatic particle number is well approximated by .

Using (37), the Bogoliubov coefficient evolution equations (26), with the basis (, become a Riccati equation:

 ˙Rk=(Δk−δk)e−2i∫tWk−2δkRk−(Δk+δk)e2i∫tWkR2k (39)

with and defined by equations (27, 28). This is straightforward to evaluate numerically with the initial conditions , and an initial phase of zero. It can also be solved semiclassically for , thereby yielding the final particle number , using complex turning points and the Stokes phenomenon dumludunne ().

Alternatively, using the forms calculated previously for and , equations (33), we obtain an analytic representation of the reflection probability as

 |Rk|2=(12ξ2k−Wk)2+(˙ξkξk−Vk)2(12ξ2k+Wk)2+(˙ξkξk−Vk)2 (40)

Expression (38) for the adiabatic particle number then yields

 ~Nk(t)=ξ2k2Wk⎡⎣(12ξ2k−Wk)2+(˙ξkξk−Vk)2⎤⎦ (41)

confirming the consistency with the Bogoliubov transformation expression (35).

### ii.4 Spectral Function Approach to Adiabatic Particle Number

Another physically interesting formalism to describe particle production at intermediate times is to define the time-dependent adiabatic particle number through the use of Spectral Functions Fukushima:2014 (); FukushimaHataya:2014 (), which are constructed in terms of correlation functions of the time-dependent creation and annihilation operators (19) used in (25). In this Section we show how the basis dependence arises in this formalism.

The Spectral Approach defines the adiabatic particle number through unequal time correlators of time-dependent creation and annihilation operators, in a limit that recovers the equal-time adiabatic particle number:

 ~Nk(t)=limt1,t2→t⟨~a†k(t1)~ak(t2)⟩ (42)

Using (19, 20), the time-dependent creation and annihilation operators can be written in terms of the decomposed field operators as

 ~ak(t) =i~f∗k(t)[∂0−Q∗k(t)]ϕk(t) (43) ~b†−k(t) =−i~fk(t)[∂0−Qk(t)]ϕk(t) (44)

which match smoothly to the initial creation and annihilation operators. Note the dependence on the choice of basis , through the function , defined in (24). We thus obtain

 ~Nk(t)=12Wk(t)limt1,t2→t([∂1−Qk(t1)][∂2−Q∗k(t2)])⟨ϕ†k(t1)ϕk(t2)⟩, (45)

where denotes a derivative with respect to time . This expression shows a clear separation between the computation of the correlation function , and the projection onto a set of reference modes, characterized by in (24). In Fukushima:2014 (); FukushimaHataya:2014 () a particular basis choice was made, and , corresponding to a leading-order adiabatic expansion and a particular phase choice via . (45) makes it clear that this is just one of an infinite set of possible choices, for which the final particle number at late asymptotic time is always the same, but for which the particle number at intermediate times can be very different.

Spatially homogeneous time-dependent external electric fields decouple the modes allowing the spectral functions, the Wigner transformed Pauli-Jordan function and Hadamard function , to be expressed as Fukushima:2014 (); FukushimaHataya:2014 ()

 Ak(t,k0) =1V∫dTeik0T⟨[ϕk(t+T2),ϕ†k(t−T2)]⟩ (46) Dk(t,k0) =1V∫dTeik0T⟨{ϕk(t+T2),ϕ†k(t−T2)}⟩ (47)

with the conjugate variable pair being the energy and the time separation . The spatial volume is denoted by .

The correlation function in (45) can be expressed through a linear combination of the inverse Wigner transformed functions (46, 47) as

 ⟨ϕ†k(t−T2)ϕk(t+T2)⟩ = V2∫dk02πe−ik0TWk(t,k0) (48) = (49)

where the total spectral function is defined as . Inserting this expression into (45), and taking the limit, yields an expression for the time-dependent adiabatic particle number in terms of the transformed correlation function as

 ~Nk(t)=V4Wk∫dk02π[14∂2t−Vk∂t+(Wk+k0)2+V2k]Wk(t,k0) (50)

This expression (50) is the natural extension of Fukushima’s result Fukushima:2014 (); FukushimaHataya:2014 (), which employed the leading adiabatic approximation choice of basis functions as and , to a general basis specified by and .

It is important to appreciate that the spectral function in (50) can be expressed directly in terms of the solutions to the Klein-Gordon equation or the Ermakov-Milne equation, without reference to the reference mode basis functions. Assuming no particles are initially present in the vacuum, the expectation value of the field operator commutator and anti-commutator are

 ⟨[ϕk(t+T2),ϕ†k(t−T2)]⟩ =fk(t+T2)f∗k(t−T2)−f∗k(t+T2)fk(t−T2) (51) ⟨{ϕk(t+T2),ϕ†k(t−T2)}⟩ =fk(t+T2)f∗k(t−T2)+f∗k(t+T2)fk(t−T2) (52)

Therefore, the spectral function assumes the form

 Wk(t,k0)=2V∫dTeik0Tfk(t−T2)f∗k(t+T2) (53)

Alternatively, this can be rewritten in terms of the amplitude function :

 Wk(t,k0)=2V∫dTeik0Tξk(t−T2)ξk(t+T2)exp[i∫t+T/2t−T/2dt′2ξ2k(t′)] (54)

Thus, the spectral function is determined without any knowledge of the basis functions and is exact provided that integration is performed over all possible values of the separation . The behavior of the spectral function (54) is shown in Figure 2 for the soluble case of a single-pulse electric field (see Appendix V), integrated over a finite range to , for various values of the cutoff . The two upper subplots and the lower left subplots in Figure 2 are plotted for the case when , in units with , with the upper left plot integrated with , the upper right plot integrated with , and the lower left plot integrated with . The lower right plot was plotted with the parameters used in Fukushima:2014 (), with and integration with . In each subplot of Figure 2, the dominant features of (54) are well approximated by the negative effective frequency , plotted with a blue-dashed line, which demonstrates that the spectral function represents the projection of the fundamental frequency on a plane spanned by time and the conjugate energy variable . Furthermore, we see that the oscillating features of the spectral function decrease as . Lastly, we compared the results obtained in Fukushima:2014 (), calculated by numerically evaluating the mode function and the subsequent integral in (54), with the exact solution to the mode-oscillator equation (see Appendix V), which indicates that the numerical approach suffers from sensitive numerical instabilities in the evaluation of (54) and the mode function .

We next show how the expression for the time-dependent adiabatic particle number that was previously derived in the Bogoliubov (35) and Riccati formalisms (41) is obtained in the Spectral Representation formalism. From equation (50), and using the spectral function (53), the expression is recovered by first re-writing the derivatives in terms of , and reorganizing the resulting terms via integration by parts to eliminate, apart from the exponential term , the dependence in the integrand. The integration produces a Dirac Delta function which, when integrated over , eliminates all integrations. Two terms appear: one corresponding directly to the adiabatic particle number, and the other to a surface boundary term. Recast in terms of using the identity (31), this lengthy but straightforward calculation leads to an expression for the time-dependent adiabatic particle number (50) as

 ~Nk(t)=ξ2k2Wk⎡⎣(12ξ2k−Wk)2+(˙ξkξk−Vk)2⎤⎦ (55)

noting that the total surface boundary term vanishes when . This agrees precisely with the Bogoliubov and Riccati expressions in (35). We see again that the adiabatic particle number is basis dependent at intermediate times, through the choice of the and functions. As before, is solved exactly without any knowledge of the basis functions, and the selected basis functions are inserted into the expression (45) to determine the adiabatic particle number with respect to that basis. In the spectral function approach this follows because the spectral function (54) is determined once and for all by the solution , and then the basis-dependent particle number is obtaind by the transform in (50).

### ii.5 Time Dependent Oscillator and Adiabatic Particle Number

Another common way to define adiabatic particle number is through the solution to the time-dependent oscillator problem, for each momentum mode popov (); DittrichReuter (); padma (); vacha (). We consider Schwinger vacuum pair production via the Schrödinger Picture time evolution of an infinite collection of time-dependent quantum harmonic oscillators, in the presence of a time-dependent background. The sQED hamiltonian becomes

 ^H(t)=∑k(12p2k+12ω2k(t)q2k) (56)

where labels each independent spatial momentum mode, and the field operators map to their quantum mechanical counterparts as and . The exact solution of the corresponding time-dependent Schrödinger equation can be written as DittrichReuter (); husimi (); LewisInvariant1 ()

 ψ(qk,t)=∑ncn,k(t)ψn(qk,t) (57)

where

 ψn(qk,t)=1√2nn!(12πξ2k(t))1/4e−12Ωk(t)q2kHn(qk√2ξk(t))e−i(n+12)λk(t) (58)

Here is the solution to the Ermakov-Milne equation (7), is defined by (8), and the time-dependent function in the Gaussian factor is defined as

 Ωk(t)=−i˙ξkξk+12ξ2k. (59)

These are normalized eigenfunctions of the exact invariant operator

 ^Ik(t)=q2k(˙ξ2k+14ξ2k)+ξ2kp2k−ξk˙ξk(pkqk+qkpk) (60)

satisfying

 ∂^Ik∂t+i[^H,^Ik]=0 (61)

and

 ^Ik(t)ψn(qk,t)=(n+12)ψn(qk,t) (62)

The function in (59) is directly related to the Riccati formalism of Section II.3, and the mode decomposition of the operator , the analog of the field (3), in the Heisenberg picture:

 iΩk(t)=˙ξkξk+i2ξ2k=˙f∗kf∗k=iWk(1−r∗k1+r∗k)+Vk (63)

Here, is again the solution to the Ermakov-Milne equation (7), is the solution to the Klein-Gordon equation (1), and the function is related to the reflection amplitude (37) by an extra phase:

 rk(t)=Rk(t)e2i∫tWk(t) (64)

Note that solving for in (63) in terms of leads directly to the analytical form (40) of the Riccati reflection probability.

We now define the adiabatic particle number by projecting these states onto a basis set of adiabatically evolving eigenstates of the time-dependent Hamiltonian. The most general expression for the adiabatically evolving eigenfunction , motivated by the assumption of a slowly varying potential given by , takes the form

 ζn(qk,t)=1√2nn!(Wk(t)π)1/4ei2Q∗k(t)q2kHn(√Wk(t)qk)e−i(n+12)∫tWk(t) (65)

where and are basis functions, with the function defined as in (24).

At asymptotic early and late times, these adiabatic eigenfunctions reduce to well-defined stationary harmonic oscillator eigenfunctions

 ζn(qk,t→±∞)∼1√2nn!(ωk(±∞)π)1/4e−12ωk(±∞)q2kHn(√ωk(±∞)qk)e−i(n+12)ωk(±∞)t (66)

A state initially prepared at a particular time can evolve to become a superposition of a variety of states at a later time . Assuming that the system is prepared in the ground state at , the probability amplitude of making a transition to the -th state is obtained by projecting the adiabatic eigenfunctions (66) onto the exact eigenfunction (58) for the ground state . The transition amplitude is

 Cn0,k(t) =∫∞−∞dqkζ∗n(qk,t)ψ0(qk,t)=(12ξ2kWk)1/4√2WkJk(2WkJk−1)n/2ei(n+1/2)∫tWk−iλk(t) (67)

where . Here . Recalling the form of (59) and (24), the function simplifies to

 Jk(t)=(12ξ2k+Wk)−i(˙ξkξk−Vk) (68)

Its modulus squared is related to the Bogoliubov coefficient and the Riccati reflection probability (40) as

 (69)

Using this result, the term in equation (67) simplifies to

 2WkJk−1=−(12ξ2k−Wk)−i(˙ξkξk−Vk)(12ξ2k+Wk)−i(˙ξkξk−Vk)=r∗k=R∗ke−2i∫tWk. (70)

Its magnitude is equal to the magnitude of the reflection amplitude . Thus the final form for the transition probability from the ground state to the -th state, can be expressed in terms of the reflection probability as

 |Cn0,k(t)|