Frequency-domain arbitrary linear transformations for photons
One or more optical resonators are coupled to an optical waveguide in sequence. Each of the resonators includes a corresponding modulator. A signal controller is configured to electrically drive each modulator with a corresponding composite electrical signal. Each composite electrical signal includes two or more frequency components of a frequency comb defined by the one or more resonators. The result of this configuration is that an input-output relation between an input of the waveguide and an output of the waveguide is a linear transformation defined by the composite electrical signals using frequencies of the frequency comb as a basis. Such linear transformations can be reciprocal or non-reciprocal, unitary or non-unitary.
1 . Apparatus comprising:
one or more resonators coupled to a waveguide in sequence;
wherein each of the one or more resonators includes a corresponding modulator;
wherein N f is a number of frequency components, wherein N f >1, wherein Ω R is a free spectral range of a frequency comb defined by the one or more resonators, wherein θ l is a phase of an l-th frequency component, and wherein A l is an amplitude of an l-th frequency component;
a signal controller configured to electrically drive each modulator with a corresponding composite electrical signal
∑
l
=
1
N
f
A
l
cos
(
l
Ω
R
t
+
θ
l
)
;
whereby an input-output relation between an input of the waveguide and an output of the waveguide is a linear transformation defined by the composite electrical signals using frequencies of the frequency comb as a basis.
2 . The apparatus of claim 1 , wherein the one or more resonators include one or more adjustable resonators having adjustable center frequencies.
3 . The apparatus of claim 2 , further comprising a closed-loop controller configured to adjust the adjustable center frequencies of the one or more adjustable resonators to lock the selected resonators to the frequency comb.
4 . The apparatus of claim 1 , wherein the one or more resonators include one or more band-limited resonators that act only on a predetermined set of frequencies of the frequency comb.
5 . The apparatus of claim 4 , wherein at least one band-limited resonator is coupled to an auxiliary resonator such that selected resonator modes of the at least one band-limited resonator are perturbed away from the frequency comb to provide band-limiting.
6 . The apparatus of claim 1 , wherein the composite electrical signals are derived from the linear transformation using automatic differentiation to expedite gradient-based inverse numerical design.
7 . The apparatus of claim 1 , wherein the linear transformation is unitary.
8 . The apparatus of claim 1 , wherein the linear transformation is non-unitary.
9 . The apparatus of claim 1 , wherein the linear transformation is reciprocal.
10 . The apparatus of claim 1 , wherein the linear transformation is non-reciprocal.