TOKYO — North Korea’s declaration at the end of the Winter Olympics of its willingness to start a dialogue with the United States offered a sliver of optimism that the political pageantry of the Games would lead to more substantial results.

A day after the closing ceremony, President Trump responded that the United States, too, was interested in talking.

“We want to talk also,” Mr. Trump said in remarks to American governors at the White House on Monday. But he quickly added: “only under the right conditions.”

“Otherwise,” he said, “we’re not talking.”

But Mr. Trump’s hint that talks might be possible came just hours before word emerged of a potential complication to any peace efforts: the looming departure of Joseph Y. Yun, one of the State Department’s most knowledgeable and experienced diplomats on North Korea. Mr. Yun abruptly announced his plan to retire by the end of the week, a departure that could undermine any chances of talks taking place, much less progress being made on curbing North Korea’s nuclear programs.

Mr. Yun, the top American envoy on North Korea, helped negotiate the release of Otto Warmbier, the American college student who was imprisoned by North Korea and died days after returning home in a coma last year. He has been a strong advocate for a diplomatic — rather than military — resolution to the North Korean nuclear crisis.

It was not clear why Mr. Yun had decided to retire so suddenly. He did not immediately return a request for comment. Heather Nauert, spokeswoman for the State Department, said in a statement that Mr. Yun was retiring “for personal reasons” and that Rex W. Tillerson, the secretary of state, “reluctantly accepted his decision.”

Analysts were taken aback.

“It’s definitely sad news,” said Andrei Lankov, a North Korea expert at Kookmin University in Seoul, South Korea. Mr. Yun “is very much in favor of compromise and negotiations, and it seems that his voice is not going to be heard.”

For the last year and a half, tensions have been building on the Korean Peninsula as both Mr. Trump and Kim Jong-un, North Korea’s leader, have ratcheted up bellicose attacks that seemed to push the United States and the North closer to a confrontation.

The respite of the Olympics — to which North Korea sent 22 athletes and an entourage of dignitaries, cheerleaders and musical performers — along with a break in North Korean missile and nuclear tests since last November raised hopes that tensions were finally easing.

But no sooner had the North and the United States declared their willingness to talk to each other — something they have in fact done before— than it became clear that the two sides remained stubbornly far apart.

“The North Koreans have always said they would be happy to talk to the United States, and in fact they are eager to come and talk to us — as one nuclear weapons state to another,” said Ralph Cossa, president of the Pacific Forum at the Center for Strategic and International Studies. “And the U.S. is willing to talk to the North Koreans if they are prepared to put nuclear weapons on the table. So both sides are willing to talk, but not about the same thing.”

While leading the United States delegation to the Olympics in Pyeongchang, South Korea, earlier this month, Vice President Mike Pence planned to secretly meet with a high-level delegation from North Korea, but the North Koreans canceled at the last minute, according to the State Department.

“This is the real challenge with North Korea,” said Mr. Cossa. “If you try to confront them, they get their backs up and feel they have to be more confrontational back, to show they are not afraid.”

”But if you make an overture, they see this as a weakness they have to exploit,” he added. “And if you offer them the moon and the stars, they say ‘O.K., we want the sun also.’”

The United Nations Security Council has imposed increasingly strict sanctions on North Korea over its weapons program, but policing violations can be difficult. On Tuesday, Japan’s Foreign Ministry announced that a Japanese military plane on Saturday had detected a North Korean and a Maldavian-flagged ship conducting what Japan judged to be ship-to-ship transfers banned by the Security Council.

Many analysts say that the North Korean leadership will never agree to talks if they have to promise to give up their nuclear weapons to get the dialogue started.

“I’m very suspicious that there are conditions under which North Korea will denuclearize peacefully,” said Bridget Coggins, an associate professor of political science at University of California at Santa Barbara.

Given that North Korea believes that nuclear weapons protect the country from an attack by the United States, “I don’t see there being a lot of breakthroughs,” she said. “There would never be a security guarantee that would be sufficient enough for denuclearization to happen.”

To get any kind of dialogue going, Ms. Coggins said the United States might have to accept a freeze in missile or nuclear tests instead of demanding that the North agree to give up its arsenal altogether.

“It seems like there have been a lot of opportunities to win points and move the ball forward in terms of pulling back from crisis if that is in fact what the United States wants,” said Ms. Coggins. “And those opportunities haven’t been seized.”

Mr. Trump did not specify what he meant by the “right conditions” for talks, but Sarah Huckabee Sanders, the White House press secretary, added context in remarks to reporters on Monday.

“Let us be completely clear,” she said. “Denuclearization must be the result of any dialogue with North Korea. Until then, the United States and the world must continue to make it known that North Korea’s nuclear and missile programs are a dead end.”

Mr. Yun had previously hinted that while getting North Korea to give up its weapons program was the ultimate goal of the United States, talks might begin on the basis of a freeze in the North’s nuclear and missile testing.

“North Korea stopping missile tests and nuclear tests would be a great first step,” Mr. Yun said last month at a conference in Tokyo.

Analysts said one of the obstacles to talks was that the Trump administration continuously sends inconsistent signals.

“The administration isn’t reading from one book on North Korea policy,” said Ankit Panda, a senior editor at the Diplomat, a foreign affairs magazine, and a regional security analyst. “We’ll hear different officials hint at different approaches.”

With the United States and South Korea likely to resume joint military exercises soon, Mr. Panda warned of “a spiral of U.S. and South Korean exercises being responded to by new provocations from the North.”

There is more promise of talks continuing between North and South Korea. On Tuesday in Seoul, Kim Yong-chol, the chief North Korean delegate, had breakfast with Unification Minister Cho Myoung-gyon of South Korea and Suh Hoon, director of the South’s National Intelligence Service. The two sides agreed that the North and South would continue to work to improve ties and help ensure peace on the peninsula.

Ms. Coggins said that such talks were not likely to lead to a change in the North’s nuclear program, but they could lead to promising moves, like reunions of families that have been divided since the Korean War.

“These are smaller things that are impactful and meaningful, in not just a sentimental way but to kind of bridge the social divide between North and South in a way that is productive going forward,” said Ms. Coggins. She said such talks might even help bring about the release of three Americans still detained in North Korea.

Analysts said that if South Korea’s president, Moon Jae-in, accepts an invitation to visit Kim Jong-un in Pyongyang, Mr. Moon might broker talks between the United States and the North.

“The scope and space for diplomacy is wider today than it was,” said Scott Snyder, director of United States-Korea policy at the Council on Foreign Relations.