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Broadening the bandwidth of entangled photons: a step towards the generation of extremely short biphotons

We demonstrate a technique that allows to fully control the bandwidth of entangled photons independently of the frequency band of interest and of the nonlinear crystal. We show that this technique allows to generate nearly transform-limited biphotons with almost one octave of bandwidth (hundreds of THz) which corresponds to correlation times of just a few femtoseconds. The presented method becomes an enabling tool for attosecond entangled-photons quantum optics. The technique can also be used to generate paired photons with a very high degree of entanglement.

The development of methods for the generation of entangled photon pairs (biphotons) with a specific bandwidth has been of great interest in recent years. Narrow bandwidth of frequency correlations is important for the design of efficient atom-photon interfaces [1] in quantum networks [2], for long distance quantum communications [3], or to enable direct measurements of temporal correlations with current photodetectors [4,5].
On the other hand, some applications such as quantum optical coherence tomography [6] and nonlinear microscopy [7] require wide bandwidths. Wide bandwidths are a requisite for the generation of biphotons with very short correlation times [8] and when high fluxes of biphotons are desired [9]. A bandwidth of hundreds of THz can generate biphotons with a few femtoseconds of correlation time. These short temporal biphotons are of particular interest in the fields of quantum metrology [10] and for some protocols for timing and positioning measurements [11]. The narrow temporal correlation embedded in the biphoton can be transmitted over large distances thanks to the strong correlations of the entangled photons that allow to remotely compensate for chromatic dispersion [12].
Spontaneous parametric down conversion (SPDC) is the most convenient source for the generation of entangled photon pairs. When an intense laser beam illuminates a nonlinear material, with a certain small probability a pump photon might split into two lower energy photons. In analogy with a classical pulse, the temporal correlation width of the biphoton is determined by its spectral bandwidth and spectral phase [13]. The bandwidth of the SPDC pairs is determined by the type of phase-matching (type-I, type-II), the length and the dispersive properties of the nonlinear material, the geometry of the SPDC configuration (collinear or non-collinear) and by the spectral and spatial characteristics of the pump beam. By changing any of these parameters it is possible to modify the bandwidth and therefore the temporal correlations of the SPDC photons. For example, by choosing a particular material (PPLN) and pumping it at a specific wavelength (λ = 1885 nm), an ultra broad bandwidth of ∼ 1080 nm has been recently reported [14]. However, from this experiment no conclusion can be drawn about the biphoton's temporal length because the spectral correlations were not measured. In contrast, the group of Silberberg achieved to produce a bandwidth of 100 nm that yielded a biphoton with 23 fs of correlation time [7].
An alternative approach to modify the SPDC bandwidth is to use chirped quasi-phase matched crystals [15]. A frequency correlation bandwidth of ∼ 300 nm at a central wavelength of 812 nm has been achieved. Nevertheless, when this technique is used to obtain narrow temporal biphotons, compensation of the spectral phase is required as the biphotons are not transform limited.
In this Letter, we demonstrate experimentally a technique to increase the SPDC bandwidth in order to generate ultrashort near transform-limited biphotons. It employs angular dispersion (pulse-front tilt) to modify the effective group velocity and group velocity dispersion of the interacting waves. Differently from other approaches, the method is not based on the choice nor on the engineering of the nonlinear material and it works in any frequency band and in any nonlinear crystal of interest. That offers two advantages: First, the wavelength can be chosen in the region where single-photon detectors exhibit high detection efficiency; second, advantage can be taken of materials with high nonlinear coefficient that naturally do not provide the desired bandwidth.
Angular dispersion is a powerful enabling tool in many different areas of optics. It can be used to introduce negative group velocity dispersion in a beam propagating in free space [16], it is a key element in many techniques for pulse compression, it enables the generation of femtosecond second harmonic waves [17,18], and it has made possible the observation of temporal solitons [19], where the natural dispersion of the material would have made such observation impossible. In quantum optics, angu- lar dispersion enables to tailor the frequency correlations of entangled photons and allows to generate frequencyuncorrelated and frequency-correlated pairs of photons [20], quantum states that are not easily produced in most experimental configurations currently used [21,22].
Let us consider collinear SPDC where the downconverted photons copropagate along the direction of the pump beam. The state of the downconverted photons at the output of the medium may be written as where Ω j (j = s, i) denote the signal and idler photons' angular frequency detunings from the central angular frequency ω 0 j , i.e, ω j = ω 0 j + Ω j , and q j are the corresponding transverse wavenumber vectors. The spectral and spatial properties of the down-converted photons are described by the joint spectrum [13] where E ω and E q are the spectral and transverse wavenumber distributions of the pump beam at the input face of the nonlinear crystal of length L, respectively.
To illustrate the principle behind the proposed method, let us consider the scheme shown in Fig. 1. In contrast to typical collinear SPDC configurations, the nonlinear crystal is placed between two optical elements that introduce angular dispersion, e.g., a pair of prisms or diffraction gratings. Angular dispersion ǫ causes the front of the pulse to be tilted by an angle ξ given by tan ξ = −λ 0 ǫ, where λ 0 is the central wavelength of the pulse and angular dispersion ǫ = m/ (d cos β 0 ), where m is the grating's diffraction order, d is the groove spacing and β 0 is the output diffraction angle.
The introduction of angular dispersion modifies the spatial distribution of the pump E q and the phasemismatch ∆k of the joint spectrum of Eq. 2. If the gratings introduce opposite angular dispersion in such a way that tan ξ s = − tan ξ p /α p and α p α s = 1, it is possible to modify ∆k only, and to effectively modify the group velocity and the group velocity dispersion of the pump, signal and idler photons. Here α j ≡ cos θ j / cos β j with θ j and β j being the incidence and diffraction angles of the pump beam at grating Gr1, respectively.
The effects of angular dispersion on the SPDC spectrum can be better understood by expanding ∆k in a Taylor series about the central frequencies ω 0 j . For a quasi-continuous-wave pump, the frequency detunings of the downconverted photons must be anticorrelated in order to satisfy energy conservation, i.e. Ω s = −Ω i , and ∆k becomes where N ′ j = N j + tan ξ p tan ρ j /c and g ′ j = g j − [tan ξ p /c] 2 /k 0 j play the role of effective inverse group velocity and effective group velocity dispersion, respectively. The inverse group velocity N j = (dk j /dω j ) ω 0 j and group velocity dispersion g j = d 2 k j /dω 2 j ω 0 j are modified in the presence of the pulse-front tilt ξ p and Poynting-vector walk-off ρ j . The new effective values of group velocity and group velocity dispersion depend on the angular dispersion experienced by the pump beam and allow us to control the SPDC bandwidth.
For example, let us consider type-II SPDC where the polarizations of the downconverted photons are mutually orthogonal. If tilt ξ p is chosen such that the effective group velocities are equal, N ′ s = N ′ i , the lowest non-zero term in phase mismatch ∆k is of the second order which results in an increase of the bandwidth. We obtain a type-II process where the dependence of the bandwidth on the length of the nonlinear crystal goes as 1/ √ L, instead of the typical 1/L [13]. The value of the pulse tilt that maximizes the bandwidth is On the other hand, in a type-I process the polarizations and the group velocities of the signal and idler photons are equal. The bandwidth is increased if the tilt is chosen such that g ′ s = g ′ i = 0. In such a case, the first non-zero term in ∆k is of the 4 th order and the dependence of the bandwidth on the length of the crystal goes as 1/L 1/4 . The tilt that maximizes the bandwidth is where k 0 j = k j ω 0 j , q s = q i = 0 . To demonstrate the feasibility of the proposed technique, an experiment was performed (see Fig. 1). A 2 mm thick, type-II BBO crystal flanked with two gratings Gr1 and Gr2 was pumped by the second harmonic (Radiantis Blue Stream, 40% efficiency in the picosecond regime) of a picosecond Ti:sapphire laser (Coherent Mira 900-P) tuned at 810 nm. The downconverted photons were separated by a polarizing beamsplitter and fed into monochromators M ono1 and M ono2 (Jobin Yvon Mi-croHR). Single-photon detectors (Perkin-Elmer SPCM-AQR-14-FC) measured single counts and coincidence electronics counted coincidence counts within a 3 ns window. The joint spectrum was obtained by measuring the number of coincidences while the two monochromators scanned the plane from 760 nm to 860 nm. Figure 2 shows the experimental results. Fig. 2(a) depicts the joint spectral density for tilt ξ p = 0 • when the gratings were removed. The typical frequency anti-correlation of the SPDC photons can be observed. Fig. 2(b) corresponds to the situation with tilt ξ max II = 38 • that maximizes the bandwidth. This was accomplished by taking grating Gr1 with a groove spacing d = 1/1200 mm and diffraction angle β p = 52 • . In order to satisfy tan ξ s = − tan ξ p /α p and α p α s = 1, grating Gr2 had d = 1/600 mm and β s = 18 • .
A more quantitative comparison between the cases with and without pulse-front tilt is plotted in Fig. 3.
where Λ j is the detuning from the central wavelength, is associated with this antidiagonal. Without tilt, the FWHM bandwidth was measured to be ∆Λ − ∼ 7.5 nm (Fig. 3(c)). Applying a tilt ξ p = 38 • , the bandwidth broadened to ∆Λ − ∼ 52 nm   Fig. 3(d)). The sevenfold increase of the bandwidth is achieved without any modification of the nonlinear crystal or the working wavelength, only by introducing angular dispersion into the pump beam and the downconverted photons. It is this independence of the material properties and of the working wavelength which makes the method proposed here so promising for controlling the bandwidth of the SPDC photons.
The application of this method for the generation of entangled photons with very narrow temporal correlations can be seen in Fig 4. In general, temporal biphoton Ψ(t 1 , t 2 ) (t j is the clicking time of j -th detector) is given by the amplitude and phase of the joint spectrum Φ (Ω s , Ω i ). Fig. 4 shows the spectral density (a), spectral phase (b) and temporal shape (c) of a biphoton for the case of 2-mm thick Type-I BBO SPDC at 810 nm. Type-I phase matching was chosen because of its naturally broader spectrum. In the case without tilt, the spectral phase follows a quadratic dependence. In the case with tilt in the region where the spectral density varies, the spectral phase is almost constant due to its fourth-order dependence on the frequency. It is this fact that makes possible to generate nearly transform-limited biphotons. The mentioned increase of bandwidth translates into ultrashort biphotons with a temporal correlation of a few femtoseconds. This contrasts with other methods where the increase of the bandwidth is not directly accompanied by a decrease of the correlation time [15,23]. Pump beams with pulse-front tilt can also be employed to obtain a very high degree of entanglement. Up to now low values of the entropy of entanglement (E ∼ 1 − 2 ebits) of paired photons in the frequency continuum have been reported [24]. A higher degree of entanglement can be achieved using broadband SPDC photons. For entangled photons of the form Φ (Ω s , Ω i ) ∼ exp − (Ω s + Ω i ) 2 /B 2 p exp − (Ω s − Ω i ) 2 /B 2 c , the entropy of entanglement [25] depends on the ratio between the bandwidth of the pump beam B p (typically 2π 5 MHz) and the bandwidth of the biphoton B c . For a bandwidth of ∆λ s = 31 nm (B c ∼ 2π 16.4 THz) at 1064 nm [26], one has values of B c /B p ∼ 3.3 × 10 6 and E ∼ 21 ebits. The method presented here allows to reach values of ∆λ s > 500 nm (B c > 2π 420 THz), therefore allowing typical ratios greater than 8.4 × 10 7 and E > 26 ebits.
In conclusion, a scheme to broaden the bandwidth of the SPDC photons was experimentally demonstrated. The method is based on the introduction of angular dispersion (pulse-front tilt) into the pump and downconverted photons which allowed us to demonstrate a sevenfold increase of the original bandwidth without changing the nonlinear crystal or the frequency band. The potentiality of the proposed method to generate ultrashort nearly transformed-limited biphotons, and with a high degree of entanglement, has also been discussed. The described method can allow biphotons to enter the realm of attosecond quantum optics, since it allows the generation of entangled photons with correlation times below one femtosecond.