Which is correct: "of order of nanokelvins" or "of order nanokelvins"?

Solution 1:

I'll add an answer, because there seems to be something of a US-UK divide here.

in/of the order of idiom [British]:

around or about (a specified number) : approximately

• The government has spent in the order of ten million dollars on the project.
• He receives something of the order of 100 e-mails a day.

[Merriam-Webster]

on the order of phrase [North American]

1 approximately

• It's a small fee, on the order of $10.
• Private economists at Goldman Sachs expect the deficits will be on the order of $300 to $375 billion.
• Even if economic output shrinks by 2 % this quarter, productivity would still grow on the order of 2 %.
• Fahrenheit 9/11 has sold something on the order of 13 million tickets.
• The whole operation appears to have cost on the order of $500,000.
• Her appearance fee was on the order of 10,000 yuan or more.
• The percentage of people who are university graduates in this country is still on the order of perhaps 25 percent of the adult population.

[Lexico]

However, the example OP gives is not exactly the same. A unit not a measure phrase (eg 'mile/s' not '3 miles') occurs here. We would not say 'approximately dollars' or 'in the neighbourhood of metres'. Here, the meaning is 'in the range normally addressed by the nanokelvin unit'. While results hereabouts are few, there are reputable-looking internet examples of all the following:

• temperatures in the order of millikelvins

• temperatures of the order of millikelvins

• temperatures on the order of millikelvins.

(as are

• distances in the order of parsecs
• distances of the order of parsecs
• distances on the order of parsecs);

I'd say all three are acceptable.

Solution 2:

I would coldly rephrase this on the order of nanokelvins.

Here is a an order-of-magnitude description of time.

As of May 2010, the smallest time interval uncertainty in direct measurements is on the order of 12 attoseconds... Source