Chlorophyll enters an excited state when the right frequency of light interacts with it. Photovoltaics operate on a similar principle, though it would be a stretch to say they were designed "in the image" of chlorophyll.
Photovoltaics harness sunlight more cheaply and efficiently than manufactured chlorophyll. Once the sunlight is harnessed, you can do whatever you want with it such as sequestering carbon dioxide.
The only advantage chlorophyll-based sunlight harvesting has over photovoltaics is that plants are easy to grow. Manufacturing chlorophyll in a laboratory would buy the inefficiency of plants at the price of inorganic manufacturing. It's the worst of both worlds.
On the other hand, chlorophyll is used as a method of carbon sequestration. This is called "growing forests".
We have. See Daniel Norcera's work on the artificial leaf. Last I heard, it was plagued with difficulties scaling up the prototype.
Humans have imitated nature since prehistoric times. Aircraft, submarines and even cars are designed in the image of wild animals.
Huh? This is so misleading that I have trouble reading beyond it. These things are designed and tested for functionality in environments that _also_ have animals (wild or not), and many of the same forces that designs cater to, also shape the evolution of said animals. For a lot of aspects, studying the solutions that have evolved over millions of generations is a great illumination and often a jumpstart to designs that work well. But they're not "designed in the image of", they're "designed with some of the functionality of".
Directly speaking, there are zero animals that can safely and cheaply carry people in the air, under water, or on roads. We had to invent new things for these purposes. What new things are you imagining that require artificial photosynthesis that aren't already performed well by natural photosynthesis?
Today, we can synthesize chlorophyll, and it's reasonable to think we could easily fabricate a leaf to use it to photosynthesize O2 and carbohydrates from CO2. We're a lot further from packaging the technology into a self-replicating tiny starter unit ("seed"). But it doesn't matter, as we can _directly_ use plants for this purpose.
It will be interesting to see whether, as we do discover new uses for photosynthesis (and other biological transformations of chemicals), we start from the ground up with purely fabricated solutions, or whether we find it better to engineer plants and animals that do the job.
Humans have imitated nature since prehistoric times. Aircraft, submarines and even cars are designed in the image of wild animals. So why haven't we imitated plants yet?
The reason this question is significant is due to the increasing discordance between the existence of humanity and nature. Try as we may, the coronavirus pandemic clearly showed us just how much the national economy of many countries depends on the sale and consumption of non-renewable resources like petroleum.
Since we can neither reduce the consumption of such resources nor kill off the majority of the human population, we should try our best to transform this planet to sustain the ever-increasing human population.
The first step for such an endeavour is unavoidably the imitation of photosynthesis. It harnesses the inexhaustible supply of solar energy to convert carbon dioxide into glucose and oxygen, both of them extremely important for humans. An artificial contraption will have many advantages over its natural counterpart as it can run 24/7 and be less susceptible to external factors like disease or parasites. The productivity of such a unit can be scaled infinitely once the prototypes have been optimised.
We already have the necessary tools to create such technology. We know the composition of chlorophyll. We are sufficiently experienced in creating super thin structure and even microscopic circuits. Even though I do not have a degree in Botany or Chemistry, I believe that with sufficient stimulus, human researchers can develop such a technology rapidly.
So the only question remains is: Why haven't the developed countries of this planet invested in such a project already and what will it take for the scientific community to focus on this topic?