An American family survived a treacherous journey on rough seas if what authorities suspect is true: that they sailed from Florida to Cuba.

A Louisiana man who lost custody of his two young sons is accused of abducting them. The search is now focused on Cuba, where the man, together with his wife and the boys, may have fled, according to a Florida sheriff’s department.

Authorities believe the father, Joshua Michael Hakken of Louisiana, abducted his sons Chase, 2, and Cole, 4, early Wednesday after breaking into the Florida home of the children’s grandmother and tying her up.

A State Department official in Washington told CNN that the U.S. Interest Section in Havana “is aware of this case and is in contact with local authorities to get more information.” The official would not comment further, citing “privacy considerations.”

If the children are in Cuba, recovering them could be difficult, Rep. Ileana Ros-Lehtinen, R- Florida, told CNN.

“Unfortunately, these parents and these poor children, these innocent ones, will now be in a country where there are no laws, there is no redress, and that has been a refuge for fugitives and wanted criminals for many years,” she said.

In Havana, Lynn Roche, an official with the U.S. Interest Section, said section officials couldn’t discuss any case of an American who hasn’t signed a privacy act waiver.

And an official with the Cuban Foreign Ministry said Monday evening that his agency was unaware of the case and had not received any requests for help from U.S. officials.

The boys had been removed from the care of Hakken and his wife, Sharyn Patricia Hakken, last year, and on April 2, the couple’s parental rights were terminated in Louisiana, investigators say.

The abduction report triggered an Amber Alert for the children, but there were no reported sightings.

Joshua Hakken, described as an anti-government protester, was believed to be armed, according to the Hillsborough County sheriff’s office in Tampa, the site of last week’s alleged abduction. Investigators believe his wife was with him.

On Monday, the sheriff’s department released a statement saying it had “received information that the Hakken family had arrived in Cuba.”

The brief statement did not offer any specifics about the information but added, “We are currently working with the Federal Bureau of Investigation who is in contact with the U.S. State Department to verify this information.”

That followed a statement Friday from the same department saying it had obtained information that Joshua Hakken recently bought a 25-foot sailboat.

There is no extradition treaty between Cuba and the United States, although there have been recent cases in which Americans sought for crimes in the United States and discovered in Havana have been sent back by the Cubans.

The FBI currently estimates there are around 70 fugitives from U.S. justice in Cuba, which is one of the reasons Cuba remains on the U.S. list of countries that support state terrorism. Most of the American fugitives in Cuba have been there for decades and have ties to revolutionary movements or radical groups.

Last week, the police department in Slidell, Louisiana, issued its own statement offering background on the Hakkens and why the boys were taken from the parents last year.

In June of 2012, Slidell police responded to a disturbance report at a hotel where Joshua and Sharyn Hakken were staying with their sons, the police statement said.

“When police arrived, both Mr. and Mrs. Hakken were acting in a bizarre manner that alarmed officers. They were talking about ‘completing their ultimate journey’ and were traveling across the country to ‘take a journey to the Armageddon’,” the Slidell police statement said, adding, “Let it be noted that both of their children were present in the hotel room at the time.”

Because of the parents’ behavior and “the fact that narcotics and weapons were located inside of the hotel room,” the children were taken by child welfare officers, and Joshua Hakken was arrested on drug charges, the statement said.

“Approximately two weeks later, Slidell Police were notified that Mr. Hakken had shown up to the foster family home … with a firearm demanding the return of his children,” the Slidell police statement continued. “The foster parents called 911, and Mr. Hakken fled without his children. We have heard nothing until (Wednesday).”

At some point over the past few months, the children were sent to Tampa to live with their grandmother, Patricia Hauser, the mother of Sharyn Hakken.

“Somehow they found out where the kids were,” said Detective Larry McKinnon of the Hillsborough County sheriff’s office.

Sheriff’s investigators say Joshua Hakken entered Hauser’s home at 6:30 a.m. last Wednesday. She told police that he tied her up and fled with the children in her silver 2009 Toyota Camry. That vehicle was found later that day just a couple of blocks away from the home.

Those investigators told CNN they believe Joshua Hakken joined up with his wife, who was waiting in their pickup truck, and the family drove to a parking garage. A short time later, investigators said, Hakken is believed to have taken a sailboat out of a private slip in nearby Madeira Beach.

Surveillance images showed the boat sailing into the Gulf of Mexico about three and a half hours after the boys disappeared from their grandmother’s home, investigators said, adding that the photos showed adults and children onboard.

— CNN’s Patrick Oppmann in Havana, Rich Phillips and Kim Segal in Miami, Elise Labott and Emily Schultze in Washington, and Dave Alsup and Joe Sutton in Atlanta contributed to this report.