Yes, your sentence is correct.
How can a single photon be detected?
The indefinite article "a" tells me you want to know how to detect a photon, any photon, not a specific one. If you wanted to detect a specific photon, you would have used the definite article "the."
The word "single" tells me you want to know how to detect ...
I will now say something because I have to say something.
What you have here is an elliptical sentence. That omission/ellipsis is quite common in speech and informal writing.
This is what the Chicago Manual of Style Guide (17th ed.) says
A grammatical ellipsis (sometimes called an omission) occurs when part of a clause is left understood and the reader ...
To repeat what the other answers have already said, yes, your example sentence is perfectly grammatical and idiomatic. For that matter, so are all of these sentences as well:
How can one single photon be detected?
How can the single photon [emitted by some specific source] be detected?
How can single photons be detected?
How can two single ...
In English, this type of construction is called a compound noun: it is used to describe a specific type of something. The final noun is the general thing, and any nouns in front of it (yes, there can be more than one) specify exactly what kind of thing it is.
As an example, a can is a noun, an opener is a noun, and we can put these two nouns together to form ...
Sometimes words are left out to avoid repetition. In your example the verb "speak" is omitted after the subject "fools" to avoid repetition because the verb 'speak' is already mentioned after "wise men".
"Wise men speak because they have something to say; fools speak because they have to say something."
To avoid the repetition of the verb "speak", we can ...
I'd call it stylistic (or poetic / literary) inversion. Familiar to many in contexts like One swallow does not a summer make, it may have been more common in natural speech a few centuries ago (I've no idea), but it's definitely "non-standard" today.
DON'T COPY THIS STYLE - unless you're really good at English, and wish to be "whimsical"!
BUT... note that ...
You are correct that the subject and verb do not agree. It should be: a new approach... is in order.
The subject of the clause is "a new approach," and the conjugated form of the verb to be is "is." The author likely got confused by the plural object "capabilities" directly preceding the verb.
The meaning of the two sentences is essentially the same in this context. However, the second version is awkward because it's never (or almost never) used.
Nonetheless, the second sentence can be rephrased in the same way as the first—you just need to add a word. In fact, if you do that, it makes it natural.
The cocktail is easy to make with this bottle.→ ...
The phrase as soon as possible is so common that I do not think people would see it as harsh. They do not think of it as meaning immediately but more as soon as you have time. It is so common that people usually write ASAP in an informal context.
Right after would, I suspect, be fine in US English but it sounds ugly to me.
After you have finished your tasks ...
Most of your interpretations are correct, but...
Dr. Fox had described Elizabeth as having 'an intellect of quite a high order' - that is, as being fairly intelligent. At the time she had wept from disappointment because he had not felt able to say that she was very intelligent (had an intellect of a high order). Now, she thinks it probably was meant as a ...
In some cultures, the order of children is important- for example in Bali, the first child is usually called Wayan, the second Made, the third Nyoman... etc... though they do not distinguish between genders, so Wayan, Made and Komang could be boys or girls.
In the UK, boys and girls are given different names according to gender, but the gender is not ...
Agreeing with @Peter
"They waste their time thinking that they would make up the loss in
This could be construed to mean, they waste their time by spending it thinking. In other words they are sitting around actively "thinking that they would make up for the loss". That would be a big waste of time.
"They waste their time,...
The manner in which Haldi was given the lift (an assisted journey) by Smiley was that Haldi rode Smiley. Constructs such as "He rides a horse" or "she rode a bicycle" are in common usage and this example with a giraffe is no different. It is clear, unambiguous and correct.
English speakers generally don't use the modals will or would inside temporal or conditional clauses, even where there is clear future meaning:
When I go (not when I will go)
until she comes (not until she will come)
If you see him (not if you will see him)
and in the past
Until they grew up (not until they would grow up).
There are exceptions, but ...
It is not idiomatic English - unless you are talking about cooked apple as a substance.
What kinds of pie filling do you like?
Apple is good, but I like cherry best.
When speaking about the natural, uncooked fruit, we would always say
Apples/cherries/oranges are good.
 Your shirt is of the same colour as John's.
 Your shirt is the same colour as John's.
Yes,  and  have the same meaning, but NPs cannot be adjectival, as I explained to you in comments.
In  the NP "the same colour" is complement of the prep "of", and the whole PP "of the same colour ..." is complement of "be".
In  the NP "the same ...
It originally (as written by Tryon Edwards) had a comma after "fools". You can make sense of it as shorthand for "fools talk".
"Wise men talk because they have something to say; fools, because they would like to say something."
This can be found in the misattributed section of Wikiquote's Plato page. Beware of Churchillian Drift.
Also, you generally don'...
"It wonders me" is used in the Pennsylvania Dutch-English dialect. The usage comes from the Pennsylvania German verb wunnere being reflexive, so that it takes a personal pronoun. If you have heard the expression "it wonders me" in English, it is possible that you were hearing someone of Pennsylvania Dutch or other Germanic background.
When a verb acts as an adjective, it is called a participle adjective. Sometimes verbs or verbal phrases in English can act nouns. These are called gerunds.
You can usually tell the meaning from the context of the sentence.
The cook prepared a four-star meal. [Prepared is the verb]
The meal was prepared yesterday. [The verb is "was prepared"...
Where did he disappear?
Answer: He disappeared in the woods.
Where did you disappear to? [meaning: where did you go? I didn't see you]
Answer: I went out jogging
disappear to is used in colloquial spoken language to mean: go, when it seems that a person has "disappeared". Indeed, they have disappeared from the view of the speaker.
Your eyes look like Daddy's would be acceptable. In my experience, the most idiomatic way to say it would be You've got your daddy's eyes, (obviously meaning you have inherited the appearance of your eyes from your father.)
I think "use nothing" is perfectly grammatical; as a programmer myself, I think it is more clear to say something different, perhaps "Don't use anything".
(I've always felt that multiple return statements out of a subprogram are a bad idea, but that's not an English issue.)
There is the word “invaluable”, which literally means you can’t value something.
A common idiom people use is to say you “can’t put a price on” something. For example:
Though surrogate mothers are well paid, you really can’t put a price on the risks involved in pregnancy and delivery.
A “teat” refers to a non-human mammary. In humans, we usually refer to a “breast”.
There is an expression about babies “clinging to their mother’s breast”, which refers to how babies naturally cling to their mothers for the nourishment and comfort.
Referring to humans clinging to a “teat” is a metaphor describing dependency on something non-human. “Fat” and “...
I see why you think the direction is redundant - the ceiling must be "up", so why mention it?
It could be argued that one could shoot at a diagonal angle and still hit the ceiling. If you want to paint the picture that the gun was perpendicular to the ceiling, that direction "up" helps do that.
If you wanted to omit the direction, it ...
In English it is possible to use a noun as an adjective
Here is an internet article about this:
As you know, a noun is a person, place or thing, and an adjective is a word that describes a noun ...
Sometimes we use a noun to describe another noun. In that case, the first noun "acts as" an adjective ... The "noun as adjective" always
"Formulated" can be better understood as a participle, and not past tense. In regular (weak) verbs the participle and past tense have the same form, so this can be confusing, but consider:
John appeared driven to achieve his goals.
You can see in this sentence that the particple "driven" is used and not the past tense "drove".