'I've seen my dog dreaming," says Jim Jarmusch over lunch in New York on a snowy December day. His voice is sedate, but excitement pops in his eyes. Other animals have imaginations, too, he thinks. "Once I left a mop outside the window of my apartment, and I saw a sparrow examining it for several days. It kept coming back, and then it started biting through to take away some strands to build a nest. It was thinking, you know?" Jarmusch does a sparrow voice, which sounds identical to his usual voice: "Man, I think this might work …"

Speaking to Jim Jarmusch, it turns out, isn't so different from watching one of his films. His work, like his conversation, doesn't cohere into stories so much as constellations, networks of seemingly isolated ideas which achieve a greater meaning arranged together just so. As a man, he's immediately identifiable: the Lee Marvin face, that shock of white hair that looks like Andy Warhol touched up with a Tesla coil.

As a director, too, there are recurring elements: a minimalist aesthetic, laconic but lovable characters (often played by musicians), a cool compositional remove that invites humour without sacrificing sincerity. These are films that believe everything is connected; theirs is a cinema of culture in conversation with itself. A young Japanese couple obsessed with Elvis. William Blake reborn into the American west. Instruments that resonate with every note that's been played on them, the world bound together by cab rides and cups of coffee. "Each one of us is a set of shifting molecules, spinning in ecstasy," says one character in The Limits of Control. "In the future, worn-out things will be made new again by reconfiguring their molecules."

Only Lovers Left Alive is a film about the urgency of that recycling process, a snickering genre tale that shacks up with a pair of exhausted paramours desperate to become new yet frustrated that they can't grow old. Jarmusch has been trying to make the movie for seven years, and whenever a bump in the road had him ready to abandon the project, Tilda Swinton would insist: "That's good news, it means that now is not the time. It will happen when it needs to happen." Now that the vampire film has become petrified by its own popularity, Only Lovers Left Alive may be arriving just in time. Every generation is convinced that they're living at the end of the world, and not a single one of them has yet to be proved right.

I'd happily argue Only Lovers Left Alive is Jarmusch's best film, but it might be more helpful to say it's his most fluent. The leads are Eve (Swinton) in Tangier, an ancient city forever on the cusp of rebirth, and Adam (Tom Hiddleston), in Detroit, contemporary America's most famous icon of decay. Both are exotic in their own way. She Skypes him on an iPhone. He answers on a rotary relic that he's rigged up through a tube television. They're vampires, and they're in love.

They live apart because they can, because it doesn't deprive them of time together. "If you live that long, separation for a year might feel like a weekend," says Jarmusch, his voice a spacey drawl. "It's not an obligation, it's an emotional connection." It's one so strong that Adam, a natural romantic who sees poetry in science, intimates that his relationship with Eve is an example of Einstein's theory of entanglement: "When you separate an entwined particle, and you move both parts away from the other, even on opposite ends of the universe if you alter or affect one, the other will be identically altered or affected."

In Detroit, Adam grows despondent about the stale state of human culture. In Tangier, Eve packs her favourite books into a small metal suitcase and arranges a series of night flights to the Motor City in order to see her immortal beloved, reserving her tickets under the name Fibonacci. "All entities in the universe are spherical, round or spiral," says Jarmusch. Circles are so crucial to the film that his script was originally threaded with quotes from Rumi, a dervish dancer, about waterwheels and turning. "It seemed a bit pretentious," he says.

It's hard not to see the theatrically suicidal Adam as Jarmusch in disguise, the director's neuroses in almost human form. For one thing, both of them love Swinton. "It's everything about her," says Jarmusch, eyes lost over my shoulder. "It's her physicality, the way that she moves … like a vestigial predator, like a wolf."

There's certainly a feral element to Eve's appearance; her character comes off as a Nobel laureate raised by wild animals. For Jarmusch, though, it's her clear eyes that are most compelling. "She has an ability to prioritise what's really important in life. Once I was listening to her, I think we were at lunch with Patti Smith, and I thought: 'Oh boy, if all culture breaks down, I'm following them. They're my leaders, the women are the way to go.' One of the great moments in my life," he continues, "was when we were shooting The Limits of Control, and we finished a take and I said: 'Oh Tilda, that was so beautiful, will you marry me?" And she replied: 'Oh darling, we already are.' I could have died."

Adam's problem, of course, is that he can't. Or he doesn't really want to. Like his creator, he's not suicidal, simply tormented by nausea at the sense that culture has run its course. Convinced that humans – whom he refers to as zombies – are rotting the world, he's the Platonic ideal of a hipster; how can you think anything is cool when you've lived for enough centuries to know that coolness is false? There's jaded, and then there's dismissing your old pals as "Shelley, Byron, and those French arseholes I used to hang around with … I don't have any heroes," he scoffs. "I'm sick of it – these zombies, what they've done to the world, their fear of their own imaginations."

Adam lives like a hermit, creating ambient drone music in his decrepit house on the edge of town (Jarmusch himself wrote the songs, performed by his band SQÜRL). Having insisted the music never leaves his house, Adam is livid to learn that Eve's younger sister, Eva (Mia Wasikowska), played one of his tracks in an LA club. He can recite the theory of entanglement verbatim, but struggles to embrace it. He thinks he can go it alone, but through Eve he's inextricably tied up in all things.

What Adam learns, and what Jarmusch understands, is that there's no upside to stepping out of the circle. Survival is an instinct, and for some it's the only option. Artists need to steal, and vampires need to feed. What Adam perceives as entropy, Eve recognises as hunger. Does Jarmusch desire immortality? "I wouldn't mind living to be maybe 300 years old … but eternally? Oh man, there's something about the cycle of life that's very important, and to have that removed would be a burden."

So Adam, it seems, isn't Jarmusch's proxy so much as his pale shadow. Unlike Adam, Jarmusch never stops looking for new heroes. It might seem a throwaway gag when Eve drives by the childhood home of a local Detroit legend and exclaims, without a hint of sarcasm, "I love Jack White!" In fact, the praise of a 3,000-year-old vampire is the ultimate artistic validation. "I believe her," says Jarmusch. "I despise hierarchical evaluation of culture. I go nuts when you say 'crime fiction is not an academically valid literature, or pop music vs classical music or whatever.'"

Auteur theory is, unsurprisingly, anathema. "I put 'A film by' as a protection of my rights, but I don't really believe it. It's important for me to have a final cut, and I do for every film. So I'm in the editing room every day, I'm the navigator of the ship, but I'm not the captain, I can't do it without everyone's equally valuable input. For me it's phases where I'm very solitary, writing, and then I'm preparing, getting the money, and then I'm with the crew and on a ship and it's amazing and exhausting and exhilarating, and then I'm alone with the editor again … I've said it before, it's like seduction, wild sex, and then pregnancy in the editing room. That's how it feels for me."

I tell Jarmusch that I always likened the process to preparing a meal. I see pre-production as listing the ingredients, production as shopping for them, and the pivotal step of post-production as the actual cooking. Jarmusch thinks this over for a moment, his eyes falling back to his empty plate. He stands, abruptly, and extends a big hand beneath a bigger smile: "Cooking is good too, but I prefer sex."