Are we Mindful in Jhana?

In the suttas, jhana is not a frozen stupor absorbed in a visual kasina, with body senses shut off. What sati and sampajano are doing in 3rd jhana, which explicitly state the jhana factors "sato and sampajano", is explained in AN 4.41, and MN 111. At the minimum, one is aware of the flux in changing sensations, perceptions, and verbal thoughts arising, persisting, fading.

I'll just give sutta references to most relevant passages having to do with what is called jhana;

This is what occurred to Sariputta through insight into mental qualities one after another:

"There was the case where Sariputta — quite secluded from sensuality, secluded from unskillful qualities — entered & remained in the first jhana: rapture & pleasure born of seclusion, accompanied by directed thought & evaluation. Whatever qualities there are in the first jhana — directed thought, evaluation, rapture, pleasure, singleness of mind, contact, feeling, perception, intention, consciousness,[2] desire, decision, persistence, mindfulness, equanimity, & attention — he ferreted them out one after another. Known to him they arose, known to him they remained, known to him they subsided. He discerned, 'So this is how these qualities, not having been, come into play. Having been, they vanish.' He remained unattracted & unrepelled with regard to those qualities, independent, detached, released, dissociated, with an awareness rid of barriers. He discerned that 'There is a further escape,' and pursuing it there really was for him. - mn111

Then, secluded from sensual pleasures, secluded from unwholesome states, I enter and dwell in the first jhāna, which consists of rapture and pleasure born of seclusion, accompanied by thought and examination. With the subsiding of thought and examination, I enter and dwell in the second jhāna, which has internal placidity and unification of mind and consists of rapture and pleasure born of concentration, without thought and examination. With the fading away as well of rapture, I dwell equanimous and, mindful and clearly comprehending, I experience pleasure with the body; I enter and dwell in the third jhāna of which the noble ones declare: ‘He is equanimous, mindful, one who dwells happily.’ With the abandoning of pleasure and pain, and with the previous passing away of joy and dejection, I enter and dwell in the fourth jhāna, neither painful nor pleasant, which has purification of mindfulness by equanimity.

“Then, brahmin, when I am in such a state, if I walk back and forth, on that occasion my walking back and forth is celestial. If I am standing, on that occasion my standing is celestial. If I am sitting, on that occasion my sitting is celestial. If I lie down, on that occasion this is my celestial high and luxurious bed. This is that celestial high and luxurious bed that at present I can gain at will, without trouble or difficulty.” - an3.63

“Well, sir, while meditating diligent, keen, and resolute, we perceive both light and vision of forms. But before long the light and the vision of forms vanish. We haven’t worked out the reason for that.”

“Well, you should work out the reason for that. Before my awakening—when I was still unawakened but intent on awakening—I too perceived both light and vision of forms. - mn128

Mahanama, at any time when a disciple of the noble ones is keeping in mind the Tathāgata, his mind is not overcome with passion, not overcome with hate, not overcome with delusion. His mind is straightened, based on the Tathāgata.

And when the mind is straightened, the disciple of the noble ones gets inspired by the goal, gets inspired by the dhamma, gains joy connected with the dhamma.

When he is joyful, rapture arises.

When the heart is rapturous, the body becomes tranquil. When the body is tranquil, he experiences happiness.

When he is happy, the mind becomes composed.

Mahanama, you should develop this recollection of the Buddha while walking, standing, sitting, or lying down. You should develop it while engaged in work and while living at home in a house full of children. - An11.12

Approaching them [monks], I will ask, ‘Venerable sirs, did the master [Buddha] ever come to Sāvatthī?’

If they tell me that he did come to Sāvatthī, then I will conclude that ‘Undoubtedly the master used a rains-bathing cloth of mine, or a meal for newcomers, or a meal for those going away, or a meal for the sick, or a meal for those tending the sick, or medicine for the sick, or gruel gifted every day.’

For me, recollecting that, joy will be born. When joyful, rapture will be born. When the heart is rapturous, the body becomes tranquil. When the body is tranquil, I will experience happiness. When I am happy, the mind will become composed

That will be the development of my (five) faculties, (five) strengths, and (seven) factors of enlightenment. - Visākhāvatthu (Vinaya)

Basically Jhana can & should be developed in many ways, not only with lights & visions as many unlearned people will have you believe.

Here on what is that 'mindfulness' in abhidhamma and sutta expression;

= Therein what is mindfulness-awakening-factor? Herein a monk is mindful, furnished with excellent mindfulness-penetration, he remembers, remembers constantly, what has long been done and long been said (concerning release). This is called mindfulness-awakening-factor.

That which is mindfulness, recollection, recall, mindfulness, remembrance, bearing (in mind), not losing, not confusing, mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness: this is called ‘mindfulness.’

The mindfulness which on that occasion is recollecting, calling back to mind; the mindfulness which is remembering, bearing in mind, the opposite of superficiality and of obliviousness; mindfulness as faculty, mindfulness as power, right mindfulness—this is the faculty of mindfulness that there then is. (Theravadin abhihamma)

The Blessed One said, “Suppose, monks, that a large crowd of people comes thronging together, saying, ‘The beauty queen! The beauty queen!’ And suppose that the beauty queen is highly accomplished at singing & dancing, so that an even greater crowd comes thronging, saying, ‘The beauty queen is singing! The beauty queen is dancing!’ Then a man comes along, desiring life & shrinking from death, desiring pleasure & abhorring pain. They say to him, ‘Now look here, mister. You must take this bowl filled to the brim with oil and carry it on your head in between the great crowd & the beauty queen. A man with a raised sword will follow right behind you, and wherever you spill even a drop of oil, right there will he cut off your head.’ Now what do you think, monks: Will that man, not paying attention to the bowl of oil, let himself get distracted outside?” “No, lord.” “I have given you this parable to convey a meaning. The meaning is this: The bowl filled to the brim with oil stands for mindfulness immersed in the body. Thus you should train yourselves: ‘We will develop mindfulness immersed in the body. We will pursue it, hand it the reins and take it as a basis, give it a grounding, steady it, consolidate it, and undertake it well.’ That is how you should train yourselves.”

Suppose, monk, that there were a royal frontier fortress with strong walls & ramparts and six gates. In it would be a wise, experienced, intelligent gatekeeper to keep out those he didn't know and to let in those he did. A swift pair of messengers, coming from the east, would say to the gatekeeper, 'Where, my good man, is the commander of this fortress?' He would say, 'There he is, sirs, sitting in the central square.' The swift pair of messengers, delivering their accurate report to the commander of the fortress, would then go back by the route by which they had come. Then a swift pair of messengers, coming from the west... the north... the south, would say to the gatekeeper, 'Where, my good man, is the commander of this fortress?' He would say, 'There he is, sirs, sitting in the central square.' The swift pair of messengers, delivering their accurate report to the commander of the fortress, would then go back by the route by which they had come.

"I have given you this simile, monk, to convey a message. The message is this: The fortress stands for this body — composed of four elements, born of mother & father, nourished with rice & barley gruel, subject to constant rubbing & abrasion, to breaking & falling apart. The six gates stand for the six internal sense media. The gatekeeper stands for mindfulness. The swift pair of messengers stands for tranquillity (samatha) and insight (vipassana). The commander of the fortress stands for consciousness. The central square stands for the four great elements: the earth-property, the liquid-property, the fire-property, & the wind-property. The accurate report stands for Unbinding (nibbana). The route by which they had come stands for the noble eightfold path: right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration." (Sutta)