Alice Eve is fishing in her pocket for her iPhone to find me a picture. She's just been to Russia on a promotional tour for what is easily the biggest film of her career to date – J J Abrams' sci-fi sequel Star Trek Into Darkness – and the blonde Brit wants to show me a gift she was given, a set of specially commissioned Russian dolls featuring the images of Kirk, Spock and other Star Trek characters. "Isn't that genius?" she beams, proudly showing off memorabilia that would probably fetch a fortune at a Trekkie convention.
With this in mind, studio executives at Paramount will be relieved to know the 30-year-old actress didn't steal anything for the bridge of the Starship Enterprise – "I'm not a klepto!" she giggles. Not even a uniform? "The fabric is very expensive," she trills. "They're almost artefacts." Today, sporting a quite stunning mint-coloured mother-of-pearl frock coat – decorated with a swarm of bees – she simply seems delighted to be cast in one of the summer's most anticipated blockbusters. "It's a very special moment in my career trajectory."
Eve plays Dr Carol Marcus, a character who last appeared (as played by Bibi Besch) in 1982's Star Trek II: The Wrath of Khan. Then she was a molecular biologist and mother to Captain Kirk's only son. Here, joining the Starship Enterprise as a weapons specialist, there is a slight frisson with Chris Pine's Kirk. But her mind is on higher things, with the crew fighting Benedict Cumberbatch's villain. "She is a genuine superbrain," says Eve. "I just took it that she was as clever if not the cleverest person on board."
There may be a slightly gratuitous shot of Eve in her bra – even Abrams can't resist the sexy scientist cliché – but the actress admirably convinces as a brain-box. She did, after all, read English at Oxford. "I'm not intimidated by academia. I pursued it quite heavily myself… and I went all the way with education. I suspect if I hadn't gone all the way myself, then I wouldn't have the – pun intended – weaponry within myself to go and tackle something like that. But I felt confident in that department, that I could pretend to be as clever as her."
Confidence is something Eve radiates. Until Star Trek Into Darkness at least, Hollywood may have tapped her up for a rather uninspiring selection of roles – the love interest in romcom She's Out of My League; Charlotte's "boobtastic" nanny in Sex and the City 2; Edgar Allen Poe's damselin distress in The Raven and a young Emma Thompson with Sixties bouffant hair in Men in Black 3. But there's far more to Eve than that. Not many would have the guts to walk into a room with Tom Stoppard and Trevor Nunn, and leave with a part in their production of the former's play Rock'n'Roll – "a good moment," she says.
The daughter of actors Trevor Eve (of Shoestring fame) and Sharon Maughan, Eve has almost circumvented the traditional "costume drama" route taken by most British actresses. Her screen debut may have been a brief appearance in Richard Eyre's film Stage Beauty, but there's barely a petticoat or lace glove on her CV. Rather, echoing her childhood – when her family moved to Los Angeles for her to try and crack the US market – she made a deliberate play and left for America.
So what has it taken to break into Hollywood? "Dedication and being sure of your dream and following it with steadfastness, with an absolute focus," she says. "I was clear about what I wanted to do and the world I wanted to be a part of. I love films and I love the way they make them in Hollywood and I wanted to be a part of that, so I pursued it." It helped, she says, that she'd partly grown up there, that she understood the American way of life – which was hers until she returned with her family to England when she was 13.
While her accent is pure Home Counties, "I was an American girl," she admits. "I had to come back here and learn to be English. I spoke with an American accent and I liked American things. And when my parents moved back, I had to learn what English girls spoke like and what English girls liked." The oldest of three, she returned to study at Bedales, which – while playing Olivia in Twelfth Night – was where she caught the acting bug. "Then it's like a drug and you're addicted."
Having worked with J J Abrams, it's only going to get worse. The director behind Lost and Super 8 has just committed to taking on the next Star Wars episode, though Eve stops short of throwing her hat in the ring for a role. "I wouldn't be greedy. I think it's been a great pleasure to work with JJ at all." Still, you can bet she wouldn't turn down the chance to brandish a lightsaber if Abrams came asking.
While she's attached to the planned Entourage movie spin-off – she played the British girl who captured the heart of fictional Hollywood star Vincent Chase in the TV show – she's not all about blockbusters. She's just completed Neil LaBute's Some Velvet Morning, a two-hander with Stanley Tucci about sexual politics, which she dubs "a real-life horror film without any blood". Then there's Cold Comes the Night, in which she plays a widowed single mother who turns the motel she owns into a brothel. "It took me about three months to come back from [playing] her."
She tells me that acting sometimes can sap your soul. "You can be left with less of yourself." And it certainly hasn't left much room for love. After dating actors Rafe Spall and Rufus Sewell, last year, she called time on her romance with Oxford-educated poet Adam O'Riordan. Yet she doesn't seem panic-stricken; not even about turning 30. "I was ready," she says. "I think if you're not ready, then it might be difficult. But I was really ready. My twenties were quite complex but I'm really happy." The way her career is going, she has every right to be.
'Star Trek Into Darkness' is out now
*This article appears in tomorrow's print edition of Radar Magazine