M. Zong, M.A,F.H. van den Broek and H.J. Bakker

Spatio-Temporal Correlation of the Zero-Photon Elec­tromagnetic Field

Abstract

We propose a new imaging technique based on parametric generation to investigate the cor­rel at ion properties of the zero-photon electromagnetic field. In a first measurement this technique is used to investigate the time correlation of this field. In the near future, the technique will be used to study the spatio-temporal correlation of the zero-photon electro­magnetic field.

1 Introduction

Atoms in an excited state can decay to the ground state by spontaneous emission of a photon. In spite of its name, the rate of the spontaneous emission process is not only determined by the excited atom itself but also by the properties of the zero-point (zero-photon) electromagnetic field. For instance, the rate of spontaneous emission can be modified by changing the mode density ofthe zero-point field at the spontaneous-emission frequency with a cavity. [1,2,3,4]. Clearly, excited atoms can thus be used as a probe of the zero-point electromagnetic field. However, a major disadvantage of using an atom is that the field can only be probed at the resonant emission frequencies. The zero-point electromagnetic field can only be studied over a wide frequency range with a non-resonant process in which the properties of the light at the quantum-noise level are retained. Such a process is parametric generation.

Parametrie generation is a second-order nonlinear optica! proces in which the photons of an intense pump pulse are split into two photons of smaller energy (signal and idler) in a nonlinear crystal under the condition of energy conservation : wp = W. + Wi, with p, s and i denoting pump, signa! and idler respectively [5]. This process has been widely used for the generation of intense mid-infrared pulses [6, 7, 8).

Interestingly, although parametric generation can be used to generate high-intensity pulses, the process cannot be described using a classical description of the electromagnetic fields and the nonlinear optical interaction. In sueb a classical description there will only be a conversion from pump light to signal and idler if either the signalor the idler is already present at the beginning of the process to stimulate the conversion. However, it is important to realize that the absence of a classical signa! and idler field only implies that the expectation value of these fields will be zero. Due to the positive zero-point energy of every mode of the electromagnetic field, there will be a certain spreadjuncertainty in the electric fields of signal and idler. This uncertainty is referred to as quantum fluctuations of the zero-point field and seeds the parametric-generation process.

Zong, Van den Broek and Bakker 93

In the parametrie generation process, the quantum fluctuations of the zero-photon elec­tromagnetic field are amplified and result in a (stochastic) phase modul at ion of the signal and idler fields . Due to the phase-sensitive nature of the parametric-generation process the phase modulations of the amplified signal and idler will be complimentary [9]. If the signal and idler fields overlap in time within their coherence time, the phase modulations cancel and the sum-frequency spectrum consists of a narrow peak (with the width of the pump-pulse spectrum) even though the individual spectra of signal and idler can be extremely broad. The occurence of a narrow peak in the sum-frequency spectrum can also be explained with a quantum-mechanical description of parametrie generation. The frequencies of signal and idler photons that originate from a common pump photon will be anticorrelated since the sum of their frequencies must be equal to the pump frequency. Hence, if signal and idler are recombined without any delay between the two, there is a large chance that the photons recombine with their twin brothers leading to a narrow peak in the sum-frequency spectrum (twin-correlation peak) [10].

2 Imaging the spatio-temporal correlation of the zero-photon elec­tromagnetic field

In previous twin-correlation experiments the signal and idler fields always originated from the same spatial region of the crystal. Hence, up to now there exists no information on the correlation of the quantum fluctuations at different positions. Here we present an imaging technique by which this spatio-temporal correlation of the zero-photon electromagnetic field can be investigated in detail.

A schematic picture of the experiment al set-up is presented in Fig. 1. A short nonlinear crystal is used for parametric generation. In this process the crystal is illuminated with a st rong pump beam (Nd:YAG) that is relatively homogeneous both longitudinally (having a pulse duration of 35 picoseconds) and transversally (having a beam diameter of ::::::,7 mm). In the illuminated volume of the crystal, the quantum fluctuations of the zero-photon field will be amplified leading to the generation of signal and idler. We intend to investigate the time correlation of two signal fields that started at different positions rl and r2 in the crystal. These two positions are imaged using a microscope objective in combination with two pinholes. The distance rl - r2 between the imaged regions of the crystal can easily be varied by changing the distance ra - rb between the two pinholes.

The imaged signal fields are parametrically amplified in two different chains of LiNb03

crystals. These crystals allow the amplification of mid-infrared pulses tunable between 1.4 and 4.5 {tm to an energy of a few hundred {tJ per pulse. The parametrie amplification of the signal field at rb williead to the generation of an idler field. This idler field is combined with the parametrically amplified signal field at ra in a short nonlinear crystal to generate the sum frequency. In the experiment the sum-frequency spectrum is measured as a function of time delay and distance between the imaged regions of the crystal. When signal and idler have time overlap, the sum-frequency spectrum will contain astrong twin-correlation peak [10]. Mathematically, the twin-correlation peak can be described as follows. The sum-frequency

94 Spatio-Temporal Correlation of the Zero-Photon Electromagnetic Field

SpectTo­meter

Figure 1: Schematic picture of the experimental set-up for measuring the correlation of the signal fields generated in different directions. The abbreviations have the following meaning: OPG : optical parametric generation j OPA: optical parametric amplificationj LNB : LiNb03j SFG : sum­frequency generationj OMA: optical multichannel analyzer.

electric field ês,(t) is given by :

ês,(t,7) '" ê.(Ta, t)ê,(Tb, t + 7), (1 )

with ês(Ta, t) the signal field coming from position Ta and ê,(Tb, t + 7) the idler field coming from position Tb with a delay 7 with respect to the signal.

The sum-frequency spectrum is measured as a function of the delay 7 . The power spec­trum I,,(w) is given by :

(2)

with (ês,(t, 7)ê;, (0, 7)) the time correlation function ofthe sum-frequency field. Substitution of equation (1) in this correlation function and using the fact that the idler field ê,(Tb, t) is the complex conjugate of the signal field ê.(Tb, t) gives :

(3)

This fourth-order correlation function can be investigated as a function of 7 and the positions Ta and Tb. In the twin-correlation experiments reported up to now, the idler has only been recombined with a signal coming from the same position Ta. If Ta = Tb, the fourth­order correlation function can be evaluated making use of the fact that the modulation of signal and idler are Gaussian processes:

(ês(Ta, t)ê;(Ta, t + 7)ê;(Ta, O)ê.(Ta, 7)) '" (e-(4In2)t2/T~e-(4In2)T2/T; + e-(4In2)T2N e-(4In2)t2N ),

(4)

Zong, Van den Broek and Bakker 95

1060 1065

WA VELENGTH (run)

(f) Delaysl.O pi

1060 1065 IO?O

WA VELENGTH (om) WA VELENGTH (nm)

Figure 2: Sum-frequency spectra at different time delays between signal and idler. The central wavelength of signal and idler is 1.8 ",m and 2.6 ",m, respectively.

with Tp the fuIl-width-at-half-maximum (FWHM) determined by the duration ofthe intensity profile of the signal pulse and Tc the correlation time of the signal field. In this equation it is assumed that the second-order correlation functions of the fields have a Gaussian dependence on the delay time T. The value of Tp is determined by the pulse duration of the Nd:YAG pulse used to pump the parametric-generation process and will in general be much larger than the value of the correlation time Tc . As aresuit, the sum-frequency spectrum wiIl consist of a narrow twin-correlation peak with the width of the pump-pulse spectrum (proportional to l/Tp ) superposed on a broad background with a width proportional to I/Tc.

3 Results and discussion

In Fig. 2 spectra of the sum-frequency light are shown obtained at different time delays between signal and idler in case the imaged regions are the same, Tl = T2 and T .. = Tb. The wavelengths of signal and idler are 1.8 and 2.6 p,m, respectively. The sum-frequency spectra clearly showastrong twin-correlation peak when signal and idler have time overlap. With increasing delay T the amplitude of the twin-correlation peak decreases. From this delay dependence the correlation time constant Tc can be determined.

In Fig. 3 the intensity of this twin-correlation peak is presented as a function of the

96 Spatio-Temporal Correlation of the Zero-Photon Electromagnetic Field

Width=I.O p. , ,,'/Gaussllt

·1.0 -0.5 0.0 0.5 1.0

Time delay (ps)

Figure 3: Intensity of the twin-correlation peak as a function of the time delay between signa! and idler.

time delay between signal and idler. From a Gaussian fit to the delay dependence we find that this peak has a full-width-at-half-maximum (FWHM) of 1.0 picosecond. This correlation time is inversely proportional to the phase-matching bandwidth of the parametric generation process. The phase-matching bandwidth is determined by the wavelength of signal and idler and increases with increasing wavelength of the signal and decreasing wavelength of the idler for type I phase matching. If the wavelengths of signal and idler are the same (degeneracy), the phase-matching bandwidth reaches its maximum value. The phase-matching bandwidth also depends on the dispersion and length of the nonlinear crystal used in the parametric generation process and decreases with increasing length of the nonlinear crystal.

The experimental results shown in Figs. 2 and 3 only give information on the time cor­rel at ion of the zero-point electromagnetic field, they do not give information on the spatio­temporal correlation, the length scale over which the time evolution of the field is correlated for a certain time. By measuring the amplitude of the twin-correlation peak as a function of the distance Tl - T2 the spatio-temporal correlation can be determined.

A spatio-temporal correlation of the zero-photon field can result from several effects. In the first place it can result from a structuring of the modes of the electromagnetic field. Such a mode structuring will occur in periodic or random media for which the length scale is on the order of the wavelength of light. Examples of such periodic media are Bragg reflectors and photonic crystals. A spatio-temporal correlation of the quantum fluctuations can also result from a nonlocality of the linear and nonlinear polarization in the nonlinear crystal used for parametric generation. The linear and nonlinear polarization will radiate and amplify the zero-photon signal and idler field. The non-Iocality of the polarization will lead to a phase relation of the fields at different points for a certain time and thus to a spatio-temporal correlation. It can be expected that the non-Iocality of the linear and second-order dielectric susceptibility will strongly dep end on the dielectric properties of the crystal. Especially for ferroelectric crystals like LiNb03 and LiTa03 long-range interactions are important so that the quantum fluctuations will be correlated over relatively large distances.

Zong, Van den Broek and Bakker 97

4 Conclusions

We investigated the time correlation of the zero-photon electromagnetic field using parametrie generation. In this process a signal and idler field are generated that contain the phase information of the quantum fluctuations of the zero-photon electromagnetic field. Sum­frequency generation of signal and idler gives rise to the so-called twin-correlation peak. The dependenee of the intensity of this peak on the delay between signal and idler gives information on the temporal correlation of the signal and idler fields. At a signal wavelength of 1.8 /-Lm (idler 2.6 /-Lm) we observe a Gaussian twin-correlation peak with a FWHM of 1.0 picosecond. In the near future we will use this technique to investigate the spatio-temporal correlation of the zero-photon electromagnetic field by combining signal and idler pulses that originate from different spatial regions.

References

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Author's address

FOM Institute AMOLF, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands.

98 Spatio-Temporal Correlation of the Zero-Photon Electromagnetic Field