1. Introduction

Music can be described in various physical ways. For one there is the intensity of the sonic wave as a function of time, but not containing direct information on the involved pitches. If, on the other hand, one represents the piece only by its frequency content, the temporal succession of the involved tones is not obvious. Regular sheet music is a combination of both, and the music is written down in terms of notes with a certain duration. This joint time-frequency (jtf) representation is a very convenient way to represent and also to analyze music. Likewise, jtf representations are useful in quantum-mechanical and optical problems.

The use of femtosecond laser pulses as a tool for time-resolved ultrafast spectroscopy or as means to manipulate chemical reactions with tailored pulses has been a tremendous success. The problem of how to represent these ultrashort laser pulses is still open to discussion. Every optical property of the pulse, as well as every physical effect of the pulse when it interacts with matter, can be understood in both the time and frequency domains. Time and frequency correspond to two representations of the same physical process. The equivalence of the representations implies that no information is either gained or lost in going from one representation to the other. This equivalence of information content is widely understood, at least implicitly, when the two representations are related by Fourier transform. It manifests itself in the equivalence of the position and the momentum representation in quantum mechanics. In the case of discrete sampling it manifests itself in the fact that the Fast Fourier transform algorithm has the same number of points in time and frequency and is invertible.

It has become popular in recent years to use the analog of the P (Wigner) [1] and Q (Husimi-Kano) [2] representations to characterize shaped femtosecond laser pulses [3–13]. The joint visualization of pulses in the time and frequency domain should be useful in unraveling the mechanism by which the pulses interact with matter. For example, a pump-dump sequence, composed of two Gaussian pulses with different central carrier frequencies, shows up clearly as two Gaussian pulses in time-frequency, with an offset in their centers both in time and frequency. Similarly, up- or down-chirped pulses show up clearly via the slope of the pulse envelope in the jtf representation [14].

In the context of quantum mechanics, the Q and P phase-space representations are known to be fully equivalent to any of the usual representations, in the sense that they provide a complete and consistent formulation of quantum mechanics [15,16]. The consequence of the equivalence of these representations is that the information content in these representations can be neither greater than nor less than in the position or momentum representation. This is not completely obvious, since the Q and P representations are two-dimensional, while the position and momentum representations are one-dimensional; yet it must be true, provided that the states being described correspond to “pure states”. In the context of representing shaped ultrashort laser pulses in a jtf space, analogous conclusions must apply: the two-dimensional ω-t phase portrait can contain no more and no less information than the one-dimensional ω or t signals (we are always dealing with the equivalent of pure states).

When one discusses the equivalence of information content in the Q and P representations, one implicitly thinks about continuous functions. However, the representation of experimental pulses that are sampled on a grid in time or frequency is only discrete. If N points are given in time or frequency, one might naively expect that N ² points will be required in a jtf representation to achieve the same sampling density. However, the principle of conservation of information suggests that we should be able to use just √N sample points in time and √N in frequency, giving a total of N points in the jtf lattice, without any loss of information. One of the main points of this paper is to show that this is indeed the case, both analytically and numerically. Our analytical results are based on properties of the von Neumann lattice [17], a discrete version of the Husimi representation in which a complex Gaussian basis function (coherent state) is located at each lattice point in phase space [18, 19]. While the requisite formal properties of the von Neumann lattice were established many years ago, to the best of our knowledge an explicit comparison with the information content of the discrete Fourier representation and the demonstration of the equivalence has not been noted previously in the literature. The additional insight into the structure of a femtosecond laser pulse by being able to use a jtf picture allows one to develop a more mechanistic way of understanding the interaction of a given laser pulse with matter. The action of a given frequency at a given time becomes apparent.

In Section 2 we introduce the von Neumann representation as a new discrete jtf representation for ultrashort laser pulses. The information content of different representations is discussed in Section 3. A few examples of transformations into the von Neumann representation and back to the spectral domain are discussed in Section 4. Remarks on the interpretation of the von Neumann representation in conjunction with a comparison of different pulse representations are given in Section 5.

2. Von Neumann representation

In this section we introduce the von Neumann transformation, which allows one to convert a discrete complex-valued signal from the time or the frequency domain into a two-dimensional joint time-frequency grid (von Neumann plane) and vice versa. Without loss of generality we consider the case that the signal ε˜ (ω) is defined in the frequency domain ranging from ω_min to ω_max = ω_min Ω. If the signal is given in the time domain, it can always be first converted to the frequency domain by application of the discrete Fourier transform. The discrete signal is sampled at N points ω_min, ω_min + δω,…, ω_min + (N - 1) δω = ω_min + Ω. The corresponding signal ε(t) in the time domain ranges from t = -T/2 to t = T/2 where the timespan T of the signal due to the Nyquist sampling theorem is proportional to the inverse of the sampling step in frequency δω through the relation T = π/δt. The sampling step in time δt is consequently obtained by the analogous equation Ω = 2π/δt (Fig. 1).

 

Fig. 1. Definition of the von Neumann parameters. A discrete signal in the frequency or in the time domain can be mapped one-to-one on the von Neumann joint time-frequency grid. This grid covers the complete spectral and temporal ranges, Ω and T, defined by the Fourier relation, but the spectral and temporal resolutions (δt and δω in time and frequency domain and Δt and Δω on the von Neumann grid) are different such that the total number of sample points, N, remains the same in all three cases. The FWHM of the Gaussian basis function of the von Neumann representation σ_t and σ_ω are illustrated with respect to the grid spacing by the circle.

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We use a basis set consisting of complex Gaussians|α_ωntm〉 on an evenly spaced lattice in phase space. This basis is called the von Neumann basis, since such a lattice was originally introduced by von Neumann [17]. Boon and Zak [18] described formal properties of the discrete lattice as a basis; Davis and Heller [19] used this basis for the description of molecular wave functions. The basis functions are defined to be

and span a two-dimensional grid covering the complete time and frequency range of the signal ε˜ (ω) (Fig. 1). Each basis function is centered at a point (ω_n,t_m) in the von Neumann plane. This is evident for ω_n in the given frequency representation (1) and becomes apparent for t_m after an inverse Fourier transform of the basis functions:

The points (ω_n,t_m) of the von Neumann representation are distributed uniformly over the complete frequency and time range Ω and T (Fig. 1). With Δω= Ω/k and Δt = T/k being the von Neumann step sizes in frequency and time this means ${\omega }_n={\omega }_{min}+\left(n-\frac{1}{2}\right)\mathrm{\Delta \omega },n=1\dots k$and $t_m=-\frac{T}{2}+\left(m-\frac{1}{2}\right)\Delta t,m=1\dots k$, m=1 … k. The number of points k is the same in time and frequency and is chosen such that the product yields k × k = N, which is the number of sample points of the original signal. This equivalence will be shown and justified in section 3. A quadratic signal sample number N can always be obtained via zero padding of the signal.

These complex Gaussians all have the same full width at half maximum σ_ωthat is determined by the parameter α as

Similarly, the width in time is ${\sigma }_t=\sqrt{\alpha 16\ln 2}$, obeying the Fourier relation between time and frequency:

This means that the parameter α defines the ratio between these respective widths,

Since the von Neumann plane is intended for a joint time-frequency representation the resolution should be equal in both domains to have the same amount of information distributed over time as well as frequency. This sets another constraint on the widths of the basis functions with respect to the full signal lengths T and Ω:

If we now combine Eqs (5) and (6), we obtain

Having determined the parameter α as a function of the spectral and temporal range, the signal, |ε〉 can now be represented in terms of the von Neumann basis set. As in quantum mechanics, a convenient way to transform to a new representation is to express the identity operator in the new basis. In the von Neumann basis set the identity operator takes the form

where

is the overlap matrix of the basis functions, which is non-diagonal since the von Neumann basis is nonorthogonal [20]. This resembles closely the identity operator that is known for the coherent states |α〉 with the difference that the overlap matrix has to be applied in the discrete case here. Its analytical form is obtained by inserting Eq. (1) into Eq. (9), rearranging the coefficients and evaluating the resulting Gaussian integral, which results in

Using the usual scalar product defined by〈a|b〉 = ∫a ^* (ω) b (ω)dω we can operate with Eq. (80 on the state |ε〉 to obtain

where the coefficients Q˜_ωntm are given by

We will call the coefficients Q͂_wntm the von Neumann representation.

Note that the inverse of the overlap matrix exists since strictly speaking S _(n,m),(i,j) has no zero eigenvalues. In practice, S may have a number of very small eigenvalues which should be eliminated through a contraction process before inverting [19]. Since the Q˜ are complex numbers they can be described by an amplitude and a phase whose meaning will be explained in section 5. Furthermore, in the same section we will elucidate the relation between the Husimi Q function and the von Neumann coefficients. Equation (12) together with Eq. (1), and Eq. (13) together with Eq. (2), allow for the direct transformation of a signal from either of the two Fourier domains (frequency or time) to the von Neumann representation and back again via Eq.(11).

3. Information content

In the discrete Fourier transformation, the frequency coverage Ω (defined above) is related to the sampling spacing δ_t in time by

Similarly, for the time span T and the sampling spacing δω in frequency we have

Thus, grid spacing in the time domain corresponds to grid range in the frequency domain, and vice versa (Fig. 1). The volume in phase space covered by the Fourier representation is calculated as the product of the ranges in time and frequency, V = T × Ω. With the number of points in either time or frequency domain given by N = T/δt = Ω/δω, the total phase space volume is then

This simple result has the following appealing interpretation: the total volume in phase space is proportional to the number of grid points N, where the phase space volume per grid point is 2π. In the phase space of position and momentum which is well known from quantum mechanics an analogous consideration yields Planck’s constant h as volume per grid point [21,22].

Now consider the von Neumann lattice [17]. We assume, as above, a quadratic grid in phase space, with the same number of points in time and in frequency, k = T/Δt = Ω/Δω. Here Δt and Δωare the step sizes in time and frequency, respectively (Fig. 1). As shown by Perelomov [23], the infinite von Neumann basis set for phase space cells with volume ΔpΔq = 2πh¯ is complete but not overcomplete, hence in our case, choosing a spacing

should be complete but not overcomplete. (Technically, the infinite von Neumann lattice is overcomplete by one basis function [23]. Although in our application the von Neumann lattice is finite, the neglect of basis functions outside of the phase space region of interest can be expected to have only a minor effect on the representation of functions localized in the interior.) The total number of points in phase space is given by

where in the second step we have used Eq. (17). Equation (18) indicates that the number of sampling points can be kept constant in jtf space although we now have a representation that is two-dimensional and has spectral and temporal information at the same time. As a consequence, the grid spacing in the von Neumann lattice in each coordinate is much coarser than in either of the Fourier representations, namely $\Delta t=\frac{N}{k}\mathrm{\delta t}=\mathrm{k\delta t}$ and $\mathrm{\Delta \omega }=\frac{N}{k}\mathrm{\delta \omega }=\mathrm{k\delta \omega }$ (Fig. 1), and yet contains the same information.

4. Examples

In this section we discuss numerical examples using Eqs. (11) and (12) for the transformation of a signal from the frequency domain to the von Neumann representation and back (Fig. 2). The transformation was implemented in C++, replacing the integral in Eq. (12) by a sum with step size δω over all signal sampling points from ω_min to ω_max. The overlap matrix S _{(n,m),(i, j)} was calculated with Eq. (10) and then inverted numerically. In order to prove that the direct inversion of the von Neumann transformation is possible, the original signal in frequency domain (black lines in Fig. 2) is shown together with the signal obtained after back transformation from the von Neumann representation (red circles in Fig. 2).

The first example is an ultrashort laser pulse with a Gaussian spectrum centered at 800 nm (Fig. 2(a), black line) that supports a pulse duration of 70 fs. A quadratic spectral phase of 30000 fs² is applied (Fig. 2(b), black line) which causes an elongation of the pulse to about 1000 fs. The spectral field is defined on 121 points and the von Neumann grid accordingly on √121 = 11 points along both the frequency and the time axis. Despite the rather low resolution the pulse duration and the tilt which is typical for chirped pulses are clearly visible in the von Neumann intensity (Fig. 2(c)). The von Neumann phase is not shown in this figure. After inversion of the von Neumann transformation yielding again spectral intensity (Fig. 2(a), red circles) and phase (Fig. 2(b), red circles), the original signal is recovered to an excellent approximation with only minor deviations. This shows that the signal information is indeed conserved by the von Neumann transformation.

In the second example the same spectrum is used (Fig. 2(d), black line) but now with piece-wise linear spectral phases that correspond to temporal shifts of the different spectral parts proportional to the slope of the phase (Fig. 2(e), black line). The “red” part (from 2.27 fs^-1 to 2.34 fs^-1) of the spectrum is shifted to earlier times by -1000 fs, the “blue” part (from 2.34 fs^-1 to 2.44 fs^-1) to later times by about 400 fs. This effect is clearly visible from the two distinct points at (ω ₁ ≈ 2.32 fs^-1,t ₁ ≈ -1000 fs) and ((ω ₂ ≈ 2.36 fs^-1,t ₂ ≈ 400 fs) in the corresponding von Neumann intensity plot (Fig. 2(d)). The good agreement between the original signal (black lines in Figs 2(d) and 2(e)) and the signal obtained by transforming the von Neumann representation back to the Fourier representation via Eq. (11) (red circles in Figs 2(d) and 2(e)) confirms again the conservation of information content and the numerical stability of the inversion procedure.

 

Fig. 2. Two examples of the von Neumann transformation. The first example is a chirped laser pulse with a quadratic spectral phase. Its spectral intensity (a) and phase (b) is sampled at 121 frequency points and shown as black lines. The second example is an ultrashort laser pulse with the first half of the spectrum shifted forward and the second half backward in time. The corresponding spectral intensity (d) and phase (e) are again plotted in black. These electric fields were transformed onto a von Neumann grid of 11 × 11 points and subsequently transformed back to the frequency domain. The von Neumann intensity of the chirped pulse (c) shows a characteristic shape and the von Neumann intensity of the double pulse (f) displays two subpulses of different central frequency. The reconstructed spectral intensity and phase is added in (a), (b), (c) and (d) as red circles. Apart from small discrepancies, the reconstruction quality is very good and the original signal is recovered.

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A different example is shown in Fig. 3. In contrast to Fig. 2, here we start with a pulse defined in the von Neumann representation that is transformed to the Fourier domain using Eq. (11) and subsequently reconstructed in the jtf representation via Eq. (12). The original von Neumann representation shown in intensity |Q˜|² (Fig. 3(a)) and phase (Fig. 3(b)) corresponds to an arrangement of three “subpulses” that have a spectral separation of 0.035 fs^-1 and a temporal separation of 1400 fs. The Fourier representations, obtained by applying Eq. (11), are displayed in Figs 3(c) (spectral) and 3(d) (temporal). The spectral intensity (Fig. 3(c), red line) is centered around two frequency points ω ₁ = 2.33 fs^-1 and ω ₂ = 2.38 fs^-1. Additionally, the interference at ω ₂ provides evidence of the second, temporally shifted, subpulse arriving at this frequency. The temporal intensity (Fig. 3(d), red line) shows a corresponding behavior with two intensity peaks at t ₁ = -400 fs and t ₂ = 800 fs and an interference occurring at the latter position showing the presence of the spectrally shifted subpulse at t ₂. The spectral phase (Fig. 3(c), dashed) as well as temporal phase (Fig. 3(d), dashed) contain further information about the structure of the electric field, however it is difficult to interpret this information since the phases of all three subpulses are involved. Note that even though the Fourier representation contains all the information on the signal, the interpretation of the signal structure is strongly simplified in the von Neumann representation. Finally, the signal was transformed from the Fourier domain back to the von Neumann representation in order to check the reconstruction quality, and the result is shown in Figs 3(e) (von Neumann intensity) and 3(f) (von Neumann phase). Comparing the original (Figs 3(a) and 3(b)) and reconstructed (Figs 3(e) and 3(f)) von Neumann representations, perfect agreement is found, thus proving once more the complete and direct invertibility of the von Neumann transformation and the conservation of information content despite the different spectral-temporal resolutions.

 

Fig. 3. Example for the reconstruction of von Neumann representations. A von Neumann representation was defined on a 11 × 11 von Neumann grid. It is shown on the left side in intensity (a) and phase (b). In order to test the quality of the von Neumann transformation it was transformed both to the frequency (c) and the time domain (d) representation using 121 sample points with red lines indicating the intensity and black dashed lines the corresponding phases. The von Neumann picture after transforming back from the spectral domain to the von Neumann grid is displayed as intensity (e) and phase (f). Since the phase for positions with negligible intensity has no meaning, we used “phase blanking” for simplification of the plot, i.e. setting all those von Neumann phase values to zero for which the corresponding intensities are less than 5% of the maximal intensity. The agreement between the original and the reconstructed von Neumann representation is excellent.

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5. Interpretation and comparison

5.1. The von Neumann intensity and the Husimi representation

In quantum mechanics the Husimi Q function [2,24] at a point (p ₀, q ₀) in phase space is defined as the absolute value squared of the overlap between the wave function |ψ〉 that is considered and the coherent state |α _p0q0〉 centered at that point [20],

Here |α _p0q0〉 is defined as

where q is the position and p is the momentum variable. Replacing |α_p0q0〉 by the von Neumann basis function |α _p0q0〉 and |ψ〉 by |η〉 [5] and inserting Eq. (1) we obtain the Husimi representation of an electric light field |η〉,

Comparing this to Eq. (12) shows the close relationship between the von Neumann and the Husimi representations. If the overlap matrix S _{(n,m), (i, j)} were diagonal and the grid spacing taken to zero, the absolute magnitude squared of the von Neumann representation would be identical to the Husimi representation of the corresponding signal. In other words, the von Neumann intensity would be identical with the Husimi representation.

It has been shown that a fully equivalent formulation of quantum mechanics in terms of the Husimi function is possible [15]. However, note that the Husimi representation involves a continuous basis while the von Neumann basis is discrete. Formally, the Husimi representation has the remarkable property that due to the overcompleteness of the basis, the diagonal elements contain all the information about the off-diagonal elements (in the position-momentum representation 〈q ₀ p ₀|Â|p ₀ q ₀〉 contains all the information of 〈q ₀ p ₀|Â|p′₀ q′₀〉 [23,25]. This means that in principle the amplitude and phase of the wavefunction can be reconstructed just from the absolute value squared of the wavefunction in a continuous vastly overcomplete basis. In practice, however, recovering the phase information from the Husimi distribution is ill-conditioned, as may be seen by a close reading of e.g. section 3 of [25], especially Eqs 2.13, 3.4 and 3.15.

In contrast, the von Neumann representation Q˜_ωntm contains the full phase information allowing for a direct inversion of the transformation back to either time domain or frequency domain. The overlap matrix not being diagonal leads to an additional summation with respect to overlap matrix elements in Eq. (12). Nevertheless, the interpretation of the von Neumann representation is just as intuitive as the interpretation of the Husimi representation and in fact contains equivalent information.

5.2. The von Neumann phase and the electric field representation

Here we investigate the interpretation of the von Neumann phase Φ_ωn,tm, with Q˜_ωn,tm = Q˜_ωn,tm| exp (iψ_ωn,tm) in more detail by comparing it with the phase of the electric field signal ψ_η(ω_n),

 

Fig. 4. The relation between the von Neumann phase and the phase of the electric field. A double pulse in the von Neumann plane is shown in intensity (a) and phase (b). The two parts of the double pulse have the same von Neumann intensity, which leads to equally intense pulses in the frequency and time domains. The von Neumann phase at these points is set to 0.75 π rad at ω ₁ = 2.33 fs^-1and to -0.25 π rad at ω ₂ =2.38 fs^-1. The corresponding signal in frequency domain (c) is represented as intensity (red line) and phase (black dashed line). The arrows connecting (b) and (c) show that the signal phase is identical with the von Neumann phase at positions ω ₁ and ω ₂.

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or analogously with ψ_ε (t_m). The nature of this relation becomes clear when writing the spectral field as

where we have used Eqs. (1) and (11).

Comparing these two expressions (22) and (23) for |ε〉 shows that in the general case the relation between the von Neumann phase and the signal phase is not straightforward since a sum over the complete von Neumann plane is involved. However, when the signal consists of several well separated subpulses in time and frequency, the interpretation is strongly simplified.

As an example we consider a von Neumann grid with only two non-zero values, corresponding to two well-separated von Neumann points Q˜ω ₁ t ₁ and Q˜ω ₂ t ₂ . In Fig. 4(a) these two points have a von Neumann amplitude of |Q˜ω ₁ t ₁| = |Q˜ω ₂ t ₂| = 1 and are located at (ω ₁ = 2.33 fs^-1, t ₁ = 800 fs) and (ω ₂ = 2.38 fs^-1, t ₂ = -400 fs). The corresponding von Neumann phases ψ_ω1t1 = 0.75π and ψ_ω2t2 = -0.25π are shown in Fig. 4(b). With these example of two separated subpulses, Eq. (23) becomes

where the ${\mathit{Â}}_i\left(\omega \right)={\left(\frac{2\alpha }{\pi }\right)}^{\frac{1}{4}}\mid {\tilde{Q}}_{{\omega }_i{t}_i}\mid \exp \left[-\alpha {\left(\omega -{\omega }_i\right)}^2\right]\left(i=\mathrm{1,2}\right)$are real-valued amplitudes. Assuming furthermore  ₂ (ω ₁) ≈ 0 and  ₁ (ω ₂) ≈ 0, which means that exp [-α (ω ₁ - ω ₂]² is small, we see that ψ_ε (ω ₁) = ψ_ω2t2 and ψ_ε (ω ₂) = ψ_ω2t2. Thus the spectral signal phase at ω ₁ or ω ₂ is exactly equal to the phase values of the von Neumann grid at those two original points. This is shown in Fig. 4(c) where the frequency domain signal is shown in intensity (red line) and phase (dashed line). As predicted we have ψ_ε (ω ₁) = ψ_ω1t1 = 0.75π and ψ_ε (ω ₂) = ψ_ω2t2 = -0.25π (marked by circles). For other values ω ≠ ω ₁,ω ₂ the signal is given by Eq. (24). Therefore, the value of the signal phase ψ_ε (ω) is approximately [-t ₁ (ω -ω ₁) + ψ_ω1t1] for all frequencies ω with Â₁ (ω) > Â₂ (ω) and accordingly [-t ₂ (ω -ω ₂) + ψ_ω1t2] for all ω with Â₁ (ω) < Â₂ (ω) (Fig. 4(c) away from circled marks).

A similar interpretation of the von Neumann phase can be found in the time domain. Using Eq. (2) and reasoning by analogy we obtain

for the value of the signal phase in terms of the von Neumann phase at t_i = t ₁ or t_i = t ₂. This means that the temporal signal phase ψ_ε(t ₁) can in principle also be obtained directly from the corresponding von Neumann phase ψ_ω1t1; however the phase offset ω ₁ t ₁ must be taken into account. This is because the basis functions in frequency (Eq.(1)) and time (Eq.(2)) are not exactly symmetric due to the properties of the inverse Fourier transformation.

5.3. Representations for ultrashort laser pulses

We now compare different representations of ultrashort laser pulses that are used in the literature, with the von Neumann representation introduced here. The characteristic advantages and disadvantages are summarized in Table 1.

The most common representation is the description of the electric field as a complex quantity with phase and intensity either in the frequency or the time domain (Fig. 5(a)). In these cases, either the spectral or temporal features are visible with a resolution according to the number of

Table 1. Comparison between different representations for ultrashort laser pulses. The spectral, as well as the temporal representation, with and without SVEA of ultrashort laser pulses are compared with the most common jtf-representations (Wigner and Husimi). The von Neumann representation has favorable properties in all of the examined criteria.

View Table

 

Fig. 5. Shaped ultrashort laser pulse in different representations. This example is based on a Gaussian spectrum and piecewise linear spectral phase, creating “colored double pulses” with temporally shifted “red” and “blue” components. (a) Electric field (Fourier) representation in the frequency domain (upper panel) and in the time domain (lower panel) with intensity (red) and phase (dashed black). The two representations are equivalent and connected via Fourier transformation. (b) Real-valued and oscillating temporal electric field. (c) Wigner representation with positive and negative valued interference structures in the middle. (d) Non-negative Husimi representation obtained by smoothing the Wigner representation with a Gaussian. For these two jtf representations a resolution of 121 × 121 was applied in order to obtain time and frequency marginals with the same resolution as the electric field representations of 121 sample points. The von Neumann representation on a 11 × 11 grid with intensity (e) and phase (f) contains equal information as the other representations.

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sample points N. In order to obtain the conjugate representation a Fourier transformation has to be applied, i.e. spectral and temporal features are not visualized at the same time.

In the time domain the slowly-varying envelope approximation (SVEA) is normally used, factoring out the fast oscillating carrier frequency. But this is a good approximation only as long as the electric field intensity varies negligibly during one optical cycle. If this is no longer the case, as for very short pulses, the direct description in the time domain using the real-valued oscillating electric field at every time step is preferred (Fig. 5(b)). Of course, the number of sample points is normally higher than for the SVEA as the fast oscillations need to be resolved. If one assumes that in the SVEA one sample point per oscillation is used, yielding the total sampling number N from above, then according to the Nyquist limit at least twice this number is needed to resolve the oscillations in the real-valued representation. Of course this is only a lower bound, in general the required sampling will be much higher, i.e. N × M with M = 2.

The Wigner jtf representation [1] (Fig. 5(c)) is also quite common for ultrashort laser pulses. It is real-valued, but there can be negative regions as well. The probability to detect a photon with a given frequency ω at a certain time t can be obtained by integrating the Wigner distribution over a region of minimal uncertainty in time-frequency space around (ω,t). The marginals, i.e. the sum over one dimension, frequency or time, deliver the temporal or the spectral intensity of the signal, respectively. Interference terms in the two-dimensional representation often exist that cancel out in the marginals or in the integration procedure, such as seen also in Fig. 5(c) (middle region). However, these interferences make a direct interpretation difficult. The number of sampling points needed for an accurate characterization such that the interferences can be resolved is on the order of N ² where N is as before the number of sampling points needed in the Fourier representations.

The Husimi distribution [2] is another common jtf representation that can be obtained by smoothing the Wigner distribution with a Gaussian (Fig. 5(d)). In contrast to the Wigner distribution it does not have the above-mentioned problem of interference terms, since they cancel out in the smoothing process, but it lacks phase information between disjoint features in the jtf domain, and direct recovery of the time domain or frequency domain field is not possible. For example, with the double pulse of Fig. 5(d) it is unclear what the phase relation between the two features is. On the other hand there is a direct interpretation of the Husimi distribution as the probability for detecting a photon at a given time with a given frequency.

The von Neumann distribution (Figs 5(e) and (f)) has the interpretational advantages of the Husimi distribution as a jtf probability distribution without discarding the phase information. As shown above, the transformation from the Fourier representations to the von Neumann representation and back is straightforward. The number of sampling points necessary for an accurate description is the same as for the Fourier representations, and therefore only a fraction of the sampling points as compared to the other two-dimensional distributions is required. The advantage is most obvious when comparing Fig. 5(c) with Fig. 5 (e) and (f) and considering that they contain the same information.

6. Conclusion

We have introduced a new representation for ultrashort laser pulses: the von Neumann picture. It enables simultaneous recognition of temporal and spectral features of a given pulse shape, but in contrast to the Husimi representation there are straightforward and direct forward and backward transformations to the usual time-domain or frequency-domain fields. Another important characteristic of the von Neumann representation is the efficient coverage of the sampled jtf space with no more points than in the Fourier representation. This causes no loss of information even though the resolution along each of the two axes, frequency and time, is much coarser than in the usual Fourier representation. Because it employs an underlying basis of time-frequency Gaussians, we expect the von Neumann representation to provide an ideal interpretational tool as well an efficient way to implement genetic algorithms for quantum control, both in the laboratory and on the computer.