Ultrashort pulses characterization by quantum state tomography

Stefan Lerch; André Stefanov

doi:10.1364/OE.24.030088

1. Introduction

The development of lasers that emit pulses shorter than the response times of any photodetectors stimulated the research field of pulse characterization. The goal of any characterization protocol is to describe how the spectral components of the electric field are changing in time. Apart from linear methods such as streak cameras [1], characterization techniques based on nonlinear processes have been developed. Those processes require broad spectral acceptance bandwidths, i.e. to be capable of resolving ultrashort features of the pulse. There exists a variety of techniques, such as frequency resolved optical gating (FROG) [2, 3], spectral phase interferometry by direct electric-field reconstruction (SPIDER) [4, 5], ptychography [6, 7] and multi-photon intrapulse interference phase scan (MIIPS) [8]. For many of those techniques, the pulses are reconstructed by iterative or deconvolution algorithms. Alternatively, there are tomographic reconstruction schemes allowing to directly determine the time-frequency distribution [9].

Many of those techniques assume the electric field to be fully coherent and do not account for the possibility of reduced coherence from pulse to pulse, or within a single pulse. In general, the chronocyclic Wigner function provides full information about an ensemble of partially coherent pulses [10–14]. To make a parallel with quantum mechanics, most quantum systems cannot be considered as closed and interact with the environment, and subsequently, undergo decoherence [15]. Therefore, a noisy quantum state cannot be described by a single pure state only but as a classical mixture of pure states: the density operator.

Following the analogy between quantum states and ultrashort pulses [12], we present here a pulse reconstruction scheme inspired by quantum mechanics which is based on linear inversion. Similarly to Quantum State Tomography (QST) for the reconstruction of quantum states [16], it allows to reconstruct the chronocyclic Wigner function of a pulse with full information about its coherence from a finite set of measurements. The paper is structured as follows. At first in the theory part, we recall the principle of QST using generalized Gell-Mann matrices. Then, we link the method to classical pulse reconstruction, and show how it can be applied using second-harmonic generation. We then describe the experimental setup and results. In the conclusion we give an outlook how to overcome the present limitations of the method.

2. Theory

2.1. Quantum state reconstruction

A pure quantum state |Ψ〉 belonging to a d-dimensional Hilbert space can be written as a superposition of orthonormal basis vectors $| Ψ 〉 = \sum_{j = 0}^{d - 1} c_{j} | u_{j} 〉$ with complex coefficients c_j. A more general description of a state is given by a statistical mixture of projections onto pure states |Ψ_n〉 with probabilities p_n. The corresponding density matrix

\hat{ρ} = \sum_{n} p_{n} | Ψ_{n} 〉 〈 Ψ_{n} |

can be diagonalized with eigenvalues λ_j and orthogonal eigenstates |Φ_j〉 such that

\hat{ρ} = \sum_{j} λ_{j} | Φ_{j} 〉 〈 Φ_{j} | .

The normalization sets

Tr {\hat{ρ}} = 1

, and the purity of the state is quantified by

γ = Tr {{\hat{ρ}}^{2}} = \sum_{j} λ_{j}^{2}

, where γ range from 1/d for a fully mixed state to 1 for a pure state.

The density matrix can be expanded into the unit operator 1_d and the SU(d) generators ${\hat{σ}}_{j}$ , also called generalized Gell-Mann matrices [17] such that

\hat{ρ} = \frac{1_{d}}{d} + \frac{1}{2} \sum_{j = 1}^{d^{2} - 1} r_{j} {\hat{σ}}_{j},

with

r_{j} = Tr {{\hat{σ}}_{j} \hat{ρ}} .

The goal of QST is to reconstruct the density matrix

\hat{ρ}

, i.e. to determine the real coefficients r_j. In general those coefficients cannot be measured directly but reconstructed by the expectation values of projective measurements. For this purpose, the Gell-Mann matrices are written as a superposition of projection measurements

{\hat{m}}_{k} = | m_{k} 〉 〈 m_{k} |

as

{\hat{σ}}_{j} = \sum_{k = 1}^{2 d^{2} - d} b_{j, k} {\hat{m}}_{k},

with

| m_{k} 〉 = \sum_{j = 0}^{d - 1} u_{k, j} | u_{j} 〉 .

The explicit form of projections

{\hat{m}}_{k}

and their relation to

{\hat{σ}}_{j}

are given in the appendix. The measurement of

{\hat{m}}_{k}

on many copies of identically prepared states, results in a signal

s_{k} \propto Tr {{\hat{m}}_{k} \hat{ρ}} .

Inserting Eq. (5) into Eq. (4) and (3) and using Eq.(7), it is possible to reconstruct the density matrix from the measured signals

\hat{ρ} = \frac{1_{d}}{d} + \frac{1}{2 N} \sum_{j = 1}^{d^{2} - 1} {\hat{σ}}_{j} \sum_{k = 1}^{2 d^{2} - d} b_{j, k} s_{k} .

The normalization factor

N = \sum_{k = 1}^{2 d^{2} - d} Tr {{\hat{m}}_{k}} s_{k}

ensures

Tr {\hat{ρ}} = 1

.

Such direct reconstruction is computationally easy to perform, once the projection measurements are defined. However, because of the noise inherent in all measurements (both systematic and statistical), the measured signals differ from the expected ones. As a consequence the reconstructed matrix is obviously not faithful but affected by this noise. Without additional constrains on the matrix, the direct reconstruction leads with high probability to a non-physical state, i.e. with negative eigenvalues. A way to guarantee positivity is to find within the set of positive matrices the one which predicts the results the closest to the measured ones. Unfortunately such optimization is computationally intensive and also leads to systematic errors [18]. In general, evaluating the error on the density matrix is a complex task, rigorously defined only recently [19]. Therefore, in the following and in order to focus on the method, we only use direct reconstruction and we do not evaluate the error on the reconstructed states.

2.2. Classical pulse reconstruction

In order to apply the methods of QST to the reconstruction of classical electromagnetic fields, we consider the complexed valued function Ψ(Ω) which, for the moment, we assume to be the envelop of the pulse to reconstruct ℰ₀(Ω). If the function has a compact support and is square integrable, it can be decomposed on a set of orthonormal functions P_j (Ω) as Ψ(Ω) = ∑_j c_j P_j (Ω), with complex coefficients give by a scalar product

c_{j} = 〈 P_{j} | Ψ 〉 = \int_{- a}^{b} d Ω Ψ (Ω) P_{j}^{*} (Ω),

with the orthonormality condition

δ_{i, j} = \int_{- a}^{b} d Ω P_{i} (Ω) P_{j}^{*} (Ω)

. As the space of those functions, together with the scalar product, forms an Hilbert space, we can immediately apply the results from the preceding paragraph about reconstruction of quantum states. Once a basis is chosen the representation of Ψ(Ω) is equivalent to the one of a quantum state |Ψ〉, or in the Dirac notation Ψ(Ω) = 〈Ω|Ψ〉.

In the following we extend this equivalence for incoherent fields for which the description by a unique function Ψ(Ω) is not sufficient. Analogically to the quantum mechanical case of Eq. (2), partially coherent pulses can be represented by an ensemble of coherent pulses Ψ_n(Ω) with respective occurrence probabilities p_n. This ensemble is usually described by the two-frequency correlation function defined as

C (Ω_{a}, Ω_{b}) = 〈 ℰ_{0} (Ω_{a}) ℰ_{0}^{*} (Ω_{b}) 〉 .

From the definition of the average we have, up to a normalization constant,

C (Ω_{a}, Ω_{b}) = \sum_{n} p_{n} Ψ_{n}^{*} (Ω_{b}) Ψ_{n} (Ω_{a}) .

The two-frequency correlation function has therefore a similar expression to the density matrix of quantum states previously introduced. Indeed, using Eq. (1) as the definition of the density matrix, and we have

{\hat{ρ}}_{p u l s e} (Ω_{a}, Ω_{b}) = 〈 Ω_{a} | {\hat{ρ}}_{p u l s e} | Ω_{b} 〉

= \sum_{n} p_{n} 〈 Ω_{a} | Ψ_{n} 〉 〈 Ψ_{n} | Ω_{b} 〉

= C (Ω_{a}, Ω_{b}) .

With λ_j the eigenvectors of the density matrix, and using

Φ_{j} (Ω) = \sum_{k} v_{j}^{(k)} P_{k} (Ω)

, the decomposition of the eigenfunctions onto the orthonormal basis functions, we find

{\hat{ρ}}_{p u l s e} (Ω_{a}, Ω_{b}) = \sum_{k, l} ρ_{k, l} P_{k}^{*} (Ω_{a}) P_{l} (Ω_{b}),

with

ρ_{k, l} = \sum_{j} λ_{j} ν_{j}^{(k) *} ν_{j}^{(l)} .

If the density matrix has only one non-zero eigenvalue λ_j, it describes a perfectly coherent pulse. However, in case of statistical mixture, the off-diagonal entries of the density matrix contain the full information about coherences. Althought the density matrix representation is complete, in ultrafast optics, a usual way to represent an ensemble of laser pulses is the chronocyclic Wigner function [10–13]

W (t, Ω) = \int \frac{d Ω^{'}}{2 π} C (Ω + \frac{Ω^{'}}{2}, Ω - \frac{Ω^{'}}{2}) e^{- i Ω^{'} t} .

This representation is convenient because the marginal distributions result in the temporal |f(t)|² = ∫ dΩ W(t, Ω) and spectral |f(Ω)|² = ∫ dt W(t, Ω) profiles of the pulse. Quantitatively, a degree of coherence is given by [4, 13]

μ = \frac{\int d Ω \int d Ω^{'} {| C (Ω + \frac{Ω^{'}}{2}, Ω - \frac{Ω^{'}}{2}) |}^{2}}{{[\int d Ω C (Ω, Ω)]}^{2}}, 0 \leq μ \leq 1 .

Interestingly, using the normalization

{\int d Ω \hat{ρ}}_{p u l s e} (Ω, Ω) = 1

, it is easy to see that µ is nothing else than the purity of the density matrix γ

μ = \int d Ω \int d Ω^{'} {| {\hat{ρ}}_{p u l s e} (Ω + \frac{Ω^{'}}{2}, Ω - \frac{Ω^{'}}{2}) |}^{2}

= \int d Ω_{a} \int d Ω_{b} {| {\hat{ρ}}_{p u l s e} (Ω_{a}, Ω_{b}) |}^{2}

= Tr {{\hat{ρ}}^{2}} .

The Wigner function and the density matrix are related by

W (t, Ω) = \int \frac{d Ω^{'}}{2 π} {\hat{ρ}}_{p u l s e} (Ω + \frac{Ω^{'}}{2}, Ω - \frac{Ω^{'}}{2}) e^{- i Ω^{'} t},

or in term of the eigendecomposition of Eq. (16),

W (t, Ω) = \int \frac{d Ω^{'}}{2 π} \sum_{j = 0}^{d - 1} λ_{j} [\sum_{k = 0}^{d - 1} ν_{j}^{(k)} P_{k} (Ω + \frac{Ω^{'}}{2})] [\sum_{l = 0}^{d - 1} ν_{j}^{(l)} P_{l} (Ω - \frac{Ω^{'}}{2})] e^{- i Ω^{'} t} .

The problem of measuring the Wigner function is therefore equivalent to the one of determining the density matrix. This later one can be tackled by performing the adequate set of measurements as described in 2.1. For that purpose we define a set of functions as for Eq. (6)

M_{k} (Ω) = \sum_{j} u_{k, j} P_{j} (Ω),

and of signals

S_{k} = {| \int_{- a}^{b} d Ω Ψ (Ω) M_{k}^{*} (Ω) |}^{2} .

The signals are given by an overlap integral (or scalar product) that projects the function Ψ(Ω) onto M_k (Ω). As a consequence, when truncating the sum in Eq. (24) to a finite number of summands d, the signals S_k are formally equivalent to Eq. (7). We also assign to each basis vector a polynomial function |u_j〉 ↔ P_j (Ω). The density matrix and its coefficients ρ_k,l can then be reconstructed from the signals using Eq. (8). The resulting matrix is then diagonalized and the corresponding Wigner function is computed from Eq. (23). The key benefit of QST is thus the ability to directly reconstruct the coherence properties of an ensemble of pulses by a finite set of projective measurements.

2.3. Pulse reconstruction by means of second-harmonic generation

The condition to be able to reconstruct the state of a classical pulse using methods from QST is the ability to measure a signal proportional to the integral of Eq. (25) for a given set of M_k (Ω). Second-harmonic generation (SHG), first observed in 1961 [20], has become a widely used tool in nonlinear optics. In the non-depleted pump approximation and in the limit of a thick detection crystal the measured signal is proportional to

S_{S H G} \propto {| \int_{- \infty}^{\infty} d Ω ℰ (Ω) ℰ (- Ω) |}^{2},

where ℰ(Ω) is the slowly varying envelope of a laser pulse [21]. In our experiment, an appropriate optical setup allows to shape the frequency spectrum of the initial pulse ℰ₀(Ω) with a complex transfer function M(Ω)

ℰ (Ω) = ℰ_{0} (Ω) M (Ω) .

Inserting Eq. (27) into Eq. (26) allows to identify Ψ(Ω) = ℰ₀(Ω)ℰ₀(−Ω) and M_k(Ω) = M(Ω)M(−Ω). Clearly Ψ(Ω) and M_k(Ω) are even functions. Whereas Ψ(Ω) can be completely characterized, the reconstruction of the initial pulse ℰ₀(Ω) is limited. When writing ℰ₀(Ω) = e(Ω) + o(Ω) in form of a even [e(Ω)] and odd [o(Ω)] part, one sees that Ψ(Ω) = e(Ω)² − o(Ω)², such that only pulses with a well defined parity can be fully reconstructed.

3. Experiment

We experimentally investigate different set of basis functions P_j(Ω), and phase or amplitude modulations of the pulses. The experimental setup is depicted in Fig. 1. A Verdi (Coherent, 5W power) pumps a Ti:Sa oscillator (KM Labs, Model MTS), generating 46 nm broad pulses centered around λ₀ = 800 nm at a repetition rate of 90 MHz. The pulse duration of the outgoing beam is minimized by a prism compressor and is coupled into the pulse shaper. A periscope (P) manipulates its polarization, to guarantee phase-matching in the detection crystal.

Fig. 1 Schematic of the experimental setup. A Ti:Sa oscillator is pumped by a Verdi laser. A mirror (M0) sends the beam through a prism compressor (N-SF11 equilateral dispersive prism P1 and P2, mirror M1). A periscope (P) manipulates the polarization and guarantees the phase-matching condition in the SHG crystal. The pulses are sent through a grating compressor consisting of two transmission grating (G1 and G2, from LightSmyth 1503.76 grooves/mm), two plane mirrors (M2 and M5), and two cylindrical mirrors (M3 and M4, 600 mm radius). At the symmetry plane an SLM allows to shape the spectrum. The pulses are focused by the lens L1 in a nonlinear crystal (PPKTP), and the generated SHG light is collimated (lens L2) and detected by a photodiode (PD). The remaining laser light is filtered by a shortpass and a bandpass filters (F).

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The pulse shaper allows to control the phase and amplitude of the frequency spectrum of the incoming pulse. It is basically a grating compressor, whose elements are aligned in a Z-geometry within a plane. The rotation angle at the cylindrical mirror is set to six degrees for the center wavelength. At the symmetry plane the dispersed spectrum passes two identical nematic liquid crystal arrays of a programmable SLM (Jenoptik SLM-S640d). Each array consists of 640 single pixels with 97 µm in width and a gap of 3 µm between two adjacent pixels. The SLM in combination with the polarization-dependent detection method allows to shape phase and amplitude independently. The grating, the curvature of the mirrors, and width of the SLM define the frequency interval that can be shaped ([750 nm, 850 nm]). At the SLM plane the spectral components are spatially separated and a spectral calibration assigns a frequency to each pixel. The spectral resolution (FWHM) of δλ = (0.135 ± 0.007) nm translates to a spatial resolution (FHWM) of δx = (247.8 ± 13.6) µm. Dividing the spatial resolution by the size of one pixel, we find (2.4 ± 0.1) pixel. Thus, there are d_max = (266.1 ± 14.6) spectral degrees of freedom on the SLM out of the possible 640 ones. The action of the SLM can be described as a frequency depending transfer function M(Ω) such that the outgoing pulse reads ℰ(Ω) = ℰ₀ (Ω)M(Ω). Finally, a lens focuses the pulse into a 12 mm long periodically poled EKTiOPO₄ (PPKTP) crystal. The poling period is 3.25 µm, optimized for type-0 SHG at 800 nm. The SHG signal is proportional to the signal S of Eq. (26).

3.1. Pulse preparation

In order to test the reconstruction scheme, well defined reference laser pulses are needed. The amplitude of the involved spectral components can be checked with a spectrometer, but the phase information is not accessible. A typical spectrum is depicted in Fig. 2(a). The generated pulses at the oscillator output are not transform-limited. The output mirror in the Ti:Sa oscillator introduces dispersion as well as the free propagation of 4.4 m through air (β = 10.1 fs²/m air). The prism compressor compensates for this dispersion. Its alignment is checked by using an autocorrelator. For different tip-tip distances of the prisms, the recorded autocorrelation functions ACF(τ) are fitted by Gaussian curves [see Fig. 2(b)]. The width σ_τ of the fits follows a quadratic dependency with the compressor tip-tip distance [see Fig. 2(c)], with a minimum of σ₀ = (30.7 ± 0.2) fs. The spectral width (FWHM) of ∆λ = 46 nm corresponds to a pulse width (FWHM) ∆τ =20.4 fs, whereas σ₀ translates into ∆τ = (36.1 ± 0.2). This discrepancy is explained most likely by the limited bandwidth of the autocorrelator that limits the measurable pulse width to 30 fs. However, the quadratic behaviour of the tip-tip scan validates the assumption that the pulses are nearly transform-limited after the prism compressor when operating at the minimal point. Higher order terms would lead to deviations, e.g. third-order dispersion would lead to a plateau in Fig. 2(c).

Fig. 2 (a) Spectrum at the output of the Ti:Sa oscillator, fiber coupled (fiber core 200 µm) and measured with the OSA (Yokogawa AQ6370). (b) Autocorrelation function ACF(τ): measurement (blue) with a Gaussian fit (red) at the tip-tip distance d = 209.6 mm. (c) Wisth of ACF(τ) as a function of the tip-tip distance of the prism compressor. The blue points indicate the measured widths, and the red curve is a fit of a quadratic function. (d) The folded geometry of the prism compressor leads to a spatially dispersed spectrum. Here, the dispersion in the vertical direction y is shown. The red line indicates the change of center wavelength.

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We operate the prism compressor in a folded geometry. This requires mirror M1 to be tilted such that the incoming and outgoing beam can be separated by the pickoff mirror M0. The beam offset should be as small as possible to suppress spatial dispersion. Here, spatial dispersion refers to a spatial dependency of the spectrum. Figure 2(d) shows the spatial dependence of the spectra after the periscope. The spectrum has been coupled into a 200 µm core fiber at different positions within the beam, and analyzed in an optical spectrum analyzer with 2 nm resolution. It illustrates that even with a minimized angle, the center wavelength changes (−1.8 ± 0.1) nm/mm in the vertical direction. In the horizontal direction the dispersion is only (0.8 ± 0.1) nm/mm. The spatial dispersion leads to different beam waists in x and y direction. They amount horizontally (2.2±0.2) mm and vertically (2.9±0.1) mm.

To summarize, the incoming pulse ℰ₀(Ω) can be considered as almost transform-limited, i.e. it is well described by its power spectrum. The spatial mode-mismatch is small, but can account for part of the discrepancies between expected and measured results. In order to guarantee measurements that are comparable and well controllable, we use the SLM not only to project onto functions M_k(Ω) but also to shape the incoming electric field ℰ₀(Ω) itself by a transfer function H(Ω). This means the transfer function reads $M (Ω) = \sqrt{M_{k} (Ω)} H (Ω)$ , where ℰ₀(Ω) → ℰ₀(Ω)H(Ω).

3.2. Transfer functions

As basis function P_j(Ω), we tested Legendre polynomials, Chebyshev polynomials, and frequency bin pairs. All bases are truncated to dimension d = 32 or d = 40 depending on the measurements. The first two bases are popular sets of orthogonal polynomials. The latter is a natural choice because of the pixelation of the SLM. The reconstruction quality depends on the number of basis functions needed to faithfully reconstruct a pulse, and on the signal intensity from each of the measurements. We observed that the best reconstructions were obtained with the Legendre polynomials, and we therefore only present those here. Legendre polynomials are given in the [−1, 1] interval by

P_{n} (x) = 2^{n} \sum_{k = 0}^{n} {(- 1)}^{k} x^{k} (\begin{matrix} n \\ k \end{matrix}) (\begin{matrix} \frac{n + k - 1}{2} \\ n \end{matrix})

The rescaling of the energy axis from Ω ∈ [−Ω_max, Ω_max] to Ω′ ∈ [−1, 1] is performed by taking a linear map Ω → Ω′ = mΩ with m = 1/[Ω_max −k₀], $k_{0} \in ℝ$ . The reason is that, Legendre polynomials having a high ratio P_n(±1)/P_n(0) for large n, a low transmission has to be written on the SLM at Ω = 0 where we expect most of the spectral density. This would lead to small signals, but can be avoided by choosing the constant k₀ < 0 and normalizing the transmission curve to one. For the reconstruction, the signal is scaled accordingly. This procedure optimizes the signal to noise ratio and only introduce a small error due to the cutting of the polynomials at the edge of their domain.

The polynomials are labeled such that P_j(Ω) with even (odd) index j ∈ {1, 2, …, d − 1} correspond to an even (odd) function. Projections onto odd functions are not possible. Therefore, the reconstructed density matrix of dimension d × d has effectively only [d/2]² entries. The remaining entries are zero per definition. The protocol needs 2[d/2]² – [d/2]= 496 projection measurements (for d = 32). Each time, the SHG signal is recorded during 200 ms. Between two measurements, the acquisition is paused for 500 ms to guarantee stable orientation of the SLM’s liquid crystals. The total acquisition time is approximately 5.8 minutes. Figure 3(a) shows the set of data used for the reconstruction of the pulse with positive chirp shown in Fig. 4. As an example, the transfer function applied to perform the measurement number 50 is shown on Fig. 3(b).

Fig. 3 (a) Measured intensities of the upconversion signal for the 496 measurements needed for the pulse reconstruction of Fig. 4. (b) Example (for measurement number 50) of the amplitude (black continuous curve) and phase (red dashed curve) of the applied complex transfer function on the 640 pixels of the SLM. The asymmetry is due to the non-linear pixel to wavelength calibration.

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Fig. 4 Reconstruction results for different chirped pulses. From top to bottom: positive linear chirp (β = 500 rad/fs), negative linear chirp (β = −500 rad/fs), and an incoherent mixture of both. (a) Modulus and phase of each entry ${\hat{ρ}}_{i j}$ of the density matrix reconstructed using Eq. (8). (b) Plot of the eigenvalues λ_n of $\hat{ρ}$ and reconstructed spectra (normalized to one), using only the eigenvector corresponding to the largest eigenvalue λ_d−1. (c) Normalized Wigner function according to Eq. (23). The dotted line indicate the expected chirp of β = ±500 fs²; Marginal (blue) and expected (red) spectra from the Wigner function.

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4. Results and discussion

4.1. Reconstruction of different chirps

We first test the reconstruction protocol on purely phase modulated pulses. The SLM allows to shape the incoming pulse in a controllable manner without changing any other experimental parameters. This controllability allows to introduce well defined partial coherence. Combining two measurements of differently shaped pulses by taking the average of both signals, leads to an effective mixed ensemble. Here, we use pulses with positive or negative chirps β = ±500 fs² such that the transfer function is H(Ω) = exp {±iβΩ²}. In addition, we combine those two measurements by acquiring the signals sequentially for each pulse and adding them afterwards.

The measurement results are plotted in Fig. 4 for the three pulse configurations. The right column (a) shows the absolute values and the arguments of the density matrix entries computed from the data by using the QST reconstruction with Eq. (8). The second column (b) shows the eigenvalues of the density matrices and a plot of the eigenfunction corresponding to the largest eigenvalue. Finally, the last column (c) shows the Wigner functions reconstructed from the density matrices using Eq. (23), together with its marginal distribution |f(Ω)|².

We first observe that the off-diagonal entries of the density matrices are more pronounced in the case of the positive and negative chirp compared to the incoherent mixture. This is a first sign of incoherence because off-diagonal elements ${\hat{ρ}}_{i j}$ are a measure for the coherence between the states |u〉_i and |u〉_j. For the reconstruction of the chirped pulses, we would expect a purely coherent pulse up to some noise. Indeed, we observe one more prominent eigenvalue. When combining both measurements, it is not possible to describe the ensemble by one single pulse. That is why two prominent eigenvalues are visible. This behaviour is clarified when looking at the reconstruction using only the eigenvector ν₃₁ corresponding to the largest eigenvalue λ₃₁. The artificially introduced quadratic phase is clearly present for the positive and negative chirps. On the contrary, the reconstruction with ν₃₁ for the incoherent mixture may suppose a positive chirp, but the Wigner function reveals a more complete picture of the ensemble. There, for coherent pulses, the positive and negative chirps are clearly visible and matches the expected ones, indicated by the dashed line. For incoherent mixture, the X-shaped Wigner function represents the two oppositely chirped pulses. The absence of components at the frequency-time points (Ω≠0 rad/fs, t = 0 fs) shows that they are incoherent. Quantitatively, the degree of coherence confirms the difference between the three reconstructions. The value for the incoherent mixture is significantly smaller (µ = 0.32) than for coherent pulses (µ = 0.50 for positive and µ = 0.67 for negative chirp).

The symmetry W(Ω, t) = W (−Ω, −t) is a consequence of reconstructing only even functions. The marginal spectrum |f(Ω)|² is compared to the expected one by assuming a transform-limited pulse. The measured transmitted power spectrum S(Ω) allows to calculate

e (Ω) = \frac{1}{2} [\sqrt{S (Ω)} + \sqrt{S (- Ω)}], o (Ω) = \frac{1}{2} [\sqrt{S (Ω)} - \sqrt{S (- Ω)}],

such that the expected marginal distribution is

{| f (Ω) |}^{2} = {| e {(Ω)}^{2} - o {(Ω)}^{2} |}^{2} = S (Ω) S (- Ω) .

For all three reconstructions, the measured marginal spectra match the expected chirps relatively well. The deviation and the non perfect coherence of the pulses could be attributed to some effective incoherences of the original pulses, due to the remaining spatial chirp, but also to imperfections of the measurement method. Those include the dark current of the power measurements and some possible stimulated SHG in the long crystal. A detailed treatment of the systematic and statistical uncertainties, together with their propagation into the Wigner function would require further investigation.

4.2. Reconstruction of amplitude modulated pulses

As a second example, we test the reconstruction of amplitude modulated pulses. We probe the coherence between two frequency regions of the pulse defined by a double Gaussian function. We apply therefore a transfer function such that the pulse spectrum reads

S (Ω) = \frac{1}{2} (e^{i \frac{{(Ω - Ω_{0})}^{2}}{σ^{2}}} + e^{i \frac{{(Ω + Ω_{0})}^{2}}{σ^{2}}}) .

The two frequency regions are centered around Ω₀ = 0.04 rad/fs and have a width that is defined by σ = 0.02 rad/fs. They are expected to arrive isochronously and to be coherent, i.e. in the Wigner function, we expect them centered at frequency-time point (Ω = ±0.04 rad/fs, t = 0 fs). The coherence should be manifested in some interference terms at Ω = 0 rad/fs.

This is indeed observed in the measurement results shown in Fig. 5. Here we expanded the reconstruction order to d = 40, in order to ensure more faithful results. Although, there is one prominent eigenvalue, the actual degree of coherence is rather low (µ = 0.42). This is due to the fact that the amplitude modulation attenuates many spectral components, and thus the signal to noise ratio decreases. The reconstruction with the eigenvector ν₃₉ shows to a quite good approximation the expected spectrum. The phase is fairly constant, and the two frequency parts peak at Ω = ±Ω₀. The Wigner function confirms this expectation. Both frequency parts are visible at Ω = ±0.04 rad/fs. The coherence between the frequency parts is clearly visible along the time axis at Ω = 0 rad/fs. The Wigner function alternates between positive and negative values which is typical for signal contribution due to interference. The marginal spectrum shows however negative contributions. This is non-physical and a consequence of constructing the Wigner function out of a non-physical density matrices with negative eigenvalues. Again, a more detailed treatment would be here required to precisely figure out the uncertainties on the density matrix.

Fig. 5 Reconstruction results with projections for amplitude modulated pulses. Two frequency parts are isolated by the transfer function as in Eq. (31). (a) Modulus and phase of each entry ${\hat{ρ}}_{i j}$ of the density matrix reconstructed using Eq. (8). (b) Plot of the eigenvalues λ_n of $\hat{ρ}$ and reconstructed spectra (normalized to one), using only the eigenvector corresponding to the largest eigenvalue λ_d₋₁. (c) Normalized Wigner function according to Eq. (23). The dotted line indicate the expected center frequencies at Ω = ±0.04 rad/fs.; Marginal (blue) and expected (red) spectra from the Wigner function.

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5. Conclusion and outlook

The proposed reconstruction protocol yields faithful results. Based on projection measurements, we are able to characterize an arbitrary ensemble of pulses, given that enough basis functions are used. The main advantage of the method is its ability of directly reconstructing a partially coherent ensemble from a set of measurements. The reconstruction protocol does not require iterative algorithms or deconvolution to retrieve the desired information. From an experimental point of view, although the alignment is more delicate compared to a simple FROG or GRENOUILLE [22] experiment, the setup does not contain any movable parts. The limitations of the method are the limited dynamic range because of the finite pixelation of the SLM and the requirement of precise characterization of the SHG process and of its added dispersion. In addition, the methods as implemented here, leads to the reconstruction of f (Ω) = ℰ₀(Ω)ℰ₀(−Ω) and not directly to ℰ₀(Ω). However by performing the SHG with a reference pulse in analogy to XFROG [23], it is possible extend the method to access the odd components of the pulse to be characterized. Finally, the analogy to quantum mechanics opens the possibility to study uncertainty propagation in the reconstruction by using recent theoretical tools developed in the context of quantum state tomography [18,19]. Such tools could be extended to rigorously define uncertainty for chronocyclic Wigner functions.

Annex: construction of the generalized Gell-Mann matrices

For the construction of the generalized Gell-Mann matrices we define the states

\begin{array}{l} | m_{j j^{'}}^{\pm} 〉 ≐ \frac{1}{\sqrt{2}} (| u_{j} 〉 \pm | u_{j^{'}} 〉), \\ | m_{j j^{'}}^{\pm i} 〉 ≐ \frac{1}{\sqrt{2}} (| u_{j} 〉 \pm i | u_{j^{'}} 〉), \\ | m_{j j} 〉 ≐ | u_{j} 〉 . \end{array}

Those projecting states replace the states in Eq. (6), i.e. $u_{k, j} \in {1, \pm 1 / \sqrt{2}, \pm i / \sqrt{2}}$ . Following this definition, we introduce the projection measurements ${\hat{m}}_{j j} = | m_{j j} 〉〈 m_{j j} |, {\hat{m}}_{j j^{'}}^{\pm} = | m_{j j^{'}}^{\pm} 〉〈 m_{j j^{'}}^{\pm} |$ and ${\hat{m}}_{j j^{'}}^{\pm i} = | m_{j j^{'}}^{\pm i} 〉〈 m_{j j^{'}}^{\pm i} |$ which replace the projection measurements ${\hat{m}}_{k}$ in Eq. (5). Thereby, the explicit decomposition of the Gell-Mann matrices in Eq. (5) is divided into three classes: the diagonal

{\hat{σ}}_{j^{2} - 1} = \sqrt{\frac{2}{j (j - 1)}} (\sum_{k = 1}^{j - 1} {\hat{m}}_{k k} - (j - 1) {\hat{m}}_{j j}),

the symmetric

{\hat{σ}}_{{(j - 1)}^{2} + 2 (j^{'} - 1)} = {\hat{m}}_{j j^{'}}^{+} - {\hat{m}}_{j j^{'}}^{-},

and the antisymmetric Gell-Mann matrices

{\hat{σ}}_{{(j - 1)}^{2} + 2 j^{'} - 1} = {\hat{m}}_{j j^{'}}^{+ i} - {\hat{m}}_{j j^{'}}^{- i},

with indices j ∈ {2, 3, …, d}, and j′ ∈ {1, 2, …, (j − 1)}.

Funding

This research was supported by the grant PP00P2_133596 of the Swiss National Science Foundation.

Acknowledgments

We thank Thomas Feurer for fruitful discussions.

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Ultrashort pulses characterization by quantum state tomography

Abstract

1. Introduction

2. Theory

2.1. Quantum state reconstruction

2.2. Classical pulse reconstruction

2.3. Pulse reconstruction by means of second-harmonic generation

3. Experiment

3.1. Pulse preparation

3.2. Transfer functions

4. Results and discussion

4.1. Reconstruction of different chirps

4.2. Reconstruction of amplitude modulated pulses

5. Conclusion and outlook

Annex: construction of the generalized Gell-Mann matrices

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (35)

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