"It is right to be excited - not just necessarily about the future, but about what Emma Raducanu is doing right here, right now."

She did not look overawed, or overly nervous. A few steadying breaths, a moment to sort through her bag, and then she jogged out to serve.

Just over an hour later, she was receiving a standing ovation after blitzing past home favourite Shelby Rogers to reach her maiden Grand Slam quarter-final, prompting British player Naomi Broady to say this is the time to be excited.

And she is still only 18 years old.

"I was watching my name and face on the big boards at the back of the court and I couldn't believe it was me," Raducanu told BBC tennis correspondent Russell Fuller.

"Playing an American at her home Slam, I could still hear my name being chanted. I was super happy and grateful for the support I received."

For Broady, who knows her well, Raducanu's success is no surprise.

"Emma and I were hoping to play some doubles together over the grass-court season and I said to her, if you ever have any questions, feel free to ask them," she said on BBC Radio 5 Live.

"She asked me if she should play mixed doubles alongside women's doubles and singles at Wimbledon. This was before the wildcards were announced, so she didn't know if she was getting one or would be playing any of those events.

"Her thoughts were: hold on a minute, if I'm doing really well in singles and I'm into the second week, I might be too tired with the mixed doubles.

"It just shows the foresight and the belief that, even before she knew the wildcard or the draw, she could get through to the second week.

"She is already so wise. She's got that self-belief before she's even gone into her first Grand Slam and it's why I'm not surprised or shocked with what she is doing out there."

It is not just that Raducanu has kept winning in New York. She has won in emphatic fashion.

In her seven matches from qualifying to the quarter-finals, she has not dropped a set. She has lost just 15 games - the fewest by any player in the first four rounds since Serena Williams in 2013.

Her composure has attracted the most attention. She did not look overawed on Ashe, or when she went 2-0 down early in the first. She reset, kept going for her shots, and reeled off six straight games to take the opener.

Born in Canada, her parents moved to London when she was two, and she started playing tennis three years later.

After turning professional in 2018, she impressed those in the know in British tennis, with Anne Keothavong calling her up to the Fed Cup team in 2020.

She was originally given a wildcard for Wimbledon qualifying but was upgraded after her winning the final of the Nottingham ITF event in the weeks before.

A few weeks after her Wimbledon run, she got her A-level results - an A* in Maths and an A in Economics - before travelling to the US with new coach Andrew Richardson.

Wanting to draw on the experience gained at Wimbledon, she decided to play as much as she could. A first-round loss in the WTA event in California was followed by a quarter-final run in an ITF tournament in Pennsylvania.

A week later she was in Chicago for a Challenger event, reaching the final, where she lost to world number 77 Clara Tauson in three sets.

By the time she arrived in New York, she had plenty of matches under her belt and, crucially, experience of the heat and humidity that players face at Flushing Meadows.

From there, she raced through qualifying and is now only the third qualifier in the Open era to reach the US Open last eight.

"It definitely feels surreal. I didn't expect to be here at all," Raducanu said.

"I feel very confident in my game and with each one I am growing and learning new things."

While no-one wants to overhype Raducanu, it is difficult not to get excited about the way she has gone about her tennis so far.

She combines punchy shot-making - as demonstrated in her emphatic win over Sara Sorribes Tormo - with an ability to suss out an opponent's strengths and adjust her tactics accordingly.

She has been described as a breath of fresh air - easy-going in front of the cameras but with a strong sense of determination on court. She is also, crucially, fun to watch.

"It wouldn't surprise me if she becomes British number one by this time next year, even perhaps by the grass-court season," Broady said.

"Her level has just shot through the roof since she's been out in the US."

Raducanu is among a group of teenagers who have broken through in New York. Canada's Leylah Fernandez, who turned 19 on Monday, ousted defending champion Naomi Osaka in the third round, and followed that with victory over three-time Grand Slam champion Angelique Kerber.

In the men's draw, 18-year-old Carlos Alcaraz stunned third seed Stefanos Tsitsipas in round three and won another gruelling five-setter on Sunday to advance to the quarters.