An unexpected delay in a standard quantum optical process generates pairs of photons

29 Nov 2024, 15:43 by David Appell

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(a) Experimental setup. BP: bandpass filter, LP: longpass filter, HWP: half-wave plate, PBS: polarizing beam splitter, 2-SNSPD: two-element superconducting nanowire single photon detector. BBO crystal is used as a linear optical element to increase the group delay between the signal and idler photons produced by the ppKTP. Detection time of the photons is related to their frequency due to the dispersive fiber. (b) Group delay T measured as a function of parametric gain ε. Horizontal dashed line is an analytic model limited to the first-order Magnus term Ωˆ 1, in which case the expected group delay is simply the group velocity walk-off β0 = L(v _s⁻¹ − v_i⁻¹ )/2. Dotted-dashed line is T = β0 − 24τε² / √ 3π(12 + ε ² ) based on Magnus expansion up to Ωˆ 3. Shaded black curve is the numerical model NeedALight [68], where shaded region represents 95% confidence interval. (c) Example of a spectrally-resolved coincidence histogram N(ω1, ω2) measured when ε = 1.407(3). (d) Fourier transform of N(ω1, ω2). Color bar shows number of events per bin collected in five minutes. Credit: Guillaume Thekkadath et al

Since it was first demonstrated in the 1960s, spontaneous parametric down-conversion (SPDC) has been at the center of many quantum optics experiments that test the fundamental laws of physics in quantum mechanics, and in applications like quantum simulation, quantum cryptography, and quantum metrology.

SPDC is the spontaneous splitting of a photon into two photons after it passes through a nonlinear object like certain crystals. The process is nonlinear and instantaneous, and the two output photons (called the signal photon and idler photon) satisfy conservation of energy and momentum compared to the input photon (the pump photon). SPDC is often used with a specially designed crystal to create pairs of entangled photons.

A research team from Canada has discovered that there is a delay between the detection of the two output photons, one that depends on the intensity of the incoming light that impacts the crystal. They call this a "gain-induced group delay."

Their study, published in the journal Physical Review Letters, uses theory and numerical simulations, and then data from their own experiment. The delay means applications whose design requires precisely timed photons, such as quantum sensors and quantum computers, could be affected.

The time delay (also called the group delay) was discovered theoretically, by studying SPDC using so-called perturbation theory, a standard technique in physics in which complicated mathematical-like energy operators are simplified by only including the leading terms of their expansion, akin to a Taylor series expansion of a function. (Feynman diagram calculations rely heavily on perturbation expansions.) This makes the equations much easier to calculate with.

Here, each term in the expansion represents a different, increasingly complicated type of SPDC photon scattering. The lowest-order is basic—a pump photon scatters into a pair of lower energy photons. The next-leading term is when the two produced photons scatter off one another, with a total of three photons produced as output.

It's assumed both the signal and idler fields are initially in a vacuum state, so this term comes to zero. The third-order term describes a scattering process in which two pump photons each produce pairs of down-converted photons followed by the up-conversion of two of the latter photons.

Each term is proportional to the power of the interaction strength, so the successive terms get smaller and smaller. Complicated quantum mechanical equations are thus used to calculate an expression for the group delay.

They then developed a model of the SPDC perturbation processes to numerically calculate time delays.

To test their theoretical analysis, the group used interferometry. A rapidly pulsed laser whose power varied from 0 to 60 milliwatts generated 180 femtosecond long pulses (that's 0.18 trillionths of a second) at wavelengths centered on 779 nanometers, with the wavelengths of the pulse being 5.37 nm at full width at half maximum.

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This light is in the very near infrared compared to human vision. Using a specially designed 2-mm long nonlinear crystal of potassium titanyl phosphate (KTP), they produced a collinear photon pair with both wavelengths centered on 1,558 nm (near-infrared light with frequencies of 192 terahertz each).

Their polarizations were perpendicular to one another, and they separated the signal and idler using a polarizing beam splitter and sent the photons through an interferometer. The measured time delay was 150 nanoseconds.

"We demonstrated that photon pairs generated by spontaneous parametric down-conversion exhibit a gain-induced group delay," the researchers, led by lead author Guillaume Thekkadath of the National Research Council of Canada in Ottawa, wrote. (The six other co-authors are in Canada as well.)

Their results show that the joint amplitude of the output light is not simply that of the two photons overlapping. Such setups are "...becoming increasingly relevant for applications like photonic quantum computing, Gaussian boson sampling, interferometry and quantum frequency conversion," they write.

They say such delays are straightforward to compensate for in bulk optics—solid instruments like mirrors, lenses, prisms and windows and crystals, but not thin films. However, "the delay may pose complications when designing quantum interference circuits integrated into chips."

Although their study examined SPDC starting with ultrashort laser pulses as the pump photons, they write that it also applies to spontaneous four-wave mixing—interactions between photons of two or three wavelengths to produce photons of one or two new wavelengths—because they have an identical quantum mechanical description.

This is often used in integrated circuits or optical fibers. They note that the observed group delay will also affect sources pumped by longer pulses or even continuous-wave light.

More information: Guillaume Thekkadath et al, Gain-Induced Group Delay in Spontaneous Parametric Down-Conversion, Physical Review Letters (2024). DOI: 10.1103/PhysRevLett.133.203601. On arXiv: DOI: 10.48550/arxiv.2405.07909

Journal information: Physical Review Letters , arXiv