We demonstrate second-order phase transition in optical parametric oscillators corresponding to a transition from degenerate to non-degenerate oscillation at a critical cavity detuning which can be utilized for enhanced sensing, and information processing

Fejer, Martin

Jahani, Saman

Langrock, Carsten

Marandi, Alireza

Roy, Arkadev

English

Caltech Authors - Main

FTh4A.5.pdf CLEO 2020 © OSA 2020
Observation of second-order spectral phase
transition in optical parametric oscillator
Arkadev Roy1, Saman Jahani1, Carsten Langrock2, Martin Fejer2, Alireza Marandi1
1 Electrical Engineering Department, California Institute of Technology, Pasadena, California, 91125,USA
2Edward L. Ginzton Laboratory, Stanford University, Stanford, California, 94305, USA
marandi@caltech.edu
Abstract: We demonstrate second-order phase transition in optical parametric oscillators
corresponding to a transition from degenerate to non-degenerate oscillation at a critical
cavity detuning which can be utilized for enhanced sensing, and information processing.
Driven dissipative systems in non-equilibrium steady states exhibit a wide range of intriguing phenomena
including self-organization, pattern formation, emergent phase and dynamical phase transition [1], and
exceptional point arising from PT-symmetry breaking [2]. Driven nonlinear optical resonators provide a fertile
ground to explore non-equilibrium physics [3], however, such systems are not extensively studied in the spectral
domain. Here, we show that an optical parametric oscillator (OPO) near degeneracy can undergo a dynamical
phase transition from its degenerate (ordered state) to non-degenerate (disordered) state by cavity detuning. We
define and analyze the variation of an order parameter as a function of cavity detuning, and experimentally and
numerically show a rapid (square-root) spectral splitting as a function of the detuning.
We consider a doubly-resonant OPO, where the signal and the idler are resonant in a lossy dispersive cavity,
with a pump at twice the frequency of the mean of the signal and idler frequencies [4]. We observe a second-
order spectral phase transition at a critical cavity detuning corresponding to a transition from degenerate to non-
degenerate optical parametric oscillation. This novel nonlinear phenomena of spectral self-organization adopted
by the system in the pursuit of minimizing the threshold belongs to the class of dynamical phase transitions and
can have myriad applications.
In the CW-driven, high-finesse limit, the OPO is governed by the mean-field evolution equation:
∂a
∂ξ
= (−α + iΔφ)a+ga∗ − iβ2
2
∂ 2a
∂ t2
−
[
ε2
2u2
∫ Lu
0
(Lu− τ)a(t− τ)2dτ
]
a∗, (1)
where, a describes the signal envelope under the slowly varying envelope approximation limit. Here ξ , t refers
to the slow time and the fast time, respectively. α , Δφ , β2, and g denote the loss, detuning, second-order group
velocity dispersion (GVD), and the phase-sensitive parametric gain, respectively. g in the CW-limit is expressed
as εbL, where b is the pump amplitude. L refers to the cavity round trip length where the nonlinear interaction
is encountered, ε includes the strength of the nonlinear interaction and u is the walk-off parameter. The last term
to the right of the equation is responsible for the gain saturation. Each of these terms are normalized by suitable
normalization factors.
We assume a perturbation in the signal around the half-harmonic frequency to be of the form: a = a+eiδωt +
a−e−iδωt . Linear stability analysis to determine the most unstable longitudinal mode gives:
d
dξ
[
a+
a∗−
]
=
[
−α + iΔφ + i β22 (δω)2 g
g∗ −α− iΔφ − i β22 (δω)2
][
a+
a∗−
]
(2a)
λ± =−1±
√
|g|2− (Δφ +β2(δω)2)2 (2b)
By analyzing the eigenvalue (growth rate)(Eq(2b)) of the linear stability matrix we arrive at two scenarios. When
sgn(Δφ) = sgn(β2), we find that the most unstable frequency of oscillation is δω = 0, and the corresponding
threshold (i.e. when λ+ = 0) is |g|th =
√
α2 +(Δφ)2, leaving the OPO in the degenerate phase. When sgn(Δφ) =
−sgn(β2), the most unstable frequency of oscillation is given by δω = ±| 2Δφβ2 |
1
2 , and the associated threshold is
|g|th = α , leaving the OPO in the non-degenerate phase. This proves the existence of the spectral phase transition.
The spectral splitting
(
δω =±| 2Δφβ2 |
1
2
)
in the non-degenerate phase shows square-root behavior with detuning.
This is akin to what is exhibited by exceptional points arising in the context of non-Hermitian degeneracies and
has been utilized for enhanced sensitivity [5]. The square-root behaviour around the critical point results in a large
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(a) (b) (c) (d)
Fig 1. Second-order spectral phase transition in OPO. a) Schematic of the experimental setup. b) Order parameter
as a function of detuning; continuous order parameter but discontinuous derivative atypical of a second order
phase transition is plotted. The inset shows the gain curve as a function of cavity detuning. The gain curve has its
maximum at the critical point (CP). c) Spectrum as a function of detuning: Experimental results (top) capturing
the second-order critical point, numerical simulation result (bottom) (dotted line plots the theoretically expected
spectral splitting). It closely follows the square-root behaviour in the non-degenerate regime. d) Emergence of
the carrier-envelope-offset frequencies of the signal and idler around the critical point. Measured RF spectrum
captured using a fast detector as a result of interfering the OPO output and a local oscillator.
sensitivity when biased around the critical point. Thus our work can pave the way for the development of critical/
transition edge sensors leveraging the phenomenon of spectral phase transition.
Figure 1a shows the experimental setup which comprises a periodically poled lithium niobate (PPLN)
waveguide which provides parametric gain around 1550 nm using a pump around 775 nm [6]. The resonator is
formed using polarization maintaining fiber and a free-space portion, which can provide precise detuning (Δφ )
using a PZT. The OPO is pumped by ∼ 15-ps pulses ( 775 nm) at 250 MHz repetition rate. Figure 1b shows the
analytical and numerical results of the OPO behavior around the spectral phase transition point, and Fig. 1c,d
show the experimental and numerical results of the spectral behaviors of the OPO.
We have shown that OPOs around degeneracy can undergo a spectral phase transition, which we have
demonstrated experimentally and verified numerically. Our results shed light on spectral behaviors of a new
operation regime of nonlinear photonic resonators which can potentially enable new opportunities for sensing
and information processing. The square-root behavior can be a resource for enhanced sensing, similar to
exceptional points [5], and the dynamics around the critical point may be utilized for information processing
similar to optical Ising machines [7]. Extension of this study to below the oscillation threshold can lead to
opportunities for quantum information processing. Such a spectral phase transition can also happen in Kerr
OPOs. Recent progress in lithium niobate based integrated devices will enable development of sensors based on
phase transition on chip [8].
This work is supported by ARO Grant No. W911NF-18-1-0285 and NSF Grant No. 1846273.
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3. M. Vaupel, A. Maitre, and C. Fabre, Physical Review Letters 83 (25) , (1999).
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Khajavikhan, Nature 548 (7666) , (2017).
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Observation of second-order spectral phase transition in optical parametric oscillator

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Observation of second-order spectral phase transition in optical parametric oscillator

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