His attorney, San Diego-based Antonio Yoon, said he had just been retained by Barak's family, adding he needs more time to prepare for the case.
Based on that, Benitez also continued Barak's detention hearing until June 7.
The families of both Perry and Barak were in court as both men were led into the court. They were shackled and dressed in county jail court overalls.
Perry family members declined to speak to a reporter.
A sister of Barak's said her brother is a good man not capable of the crime for which he has been charged.
Following the hearing Yoon had no comment on the case. As he drove away, he did say he hasn't had a chance to review any of the documents in the case yet.
Levine commented to members of the media who had gathered outside the courtroom after the day's proceedings.
Without speaking in detail, he said, "The facts will show he is a good police officer, but he has been described as somewhat naive and gullible."
Levine said "someone took advantage of those character traits."
He did not elaborate on that statement.
According to the complaint against Perry and Barak, the two men each have been charged with one count of importation of a controlled substance.
The complaint states both men were in a car that was attempting to enter the United States at the downtown Calexico Port of Entry on May 24.
A U.S. Customs inspector at a primary checkpoint booth recognized Perry and cleared the car to pass.
However, at that time Customs started a "block blitz" program in which a group of vehicles is directed into the secondary inspection area for a random check.
The car reportedly driven by Barak was one of those directed into the secondary inspection area.
Barak, according to reports, did not comply. Instead he raced through the port into the city. Customs inspectors put down a spike trip before he left the port, which punctured at least two of the tires.
The car came to a stop blocks away from the port on Cesar Chavez Boulevard.
A U.S. Border Patrol agent responded to the vehicle and saw a man running from the area. The agent stopped the man, who identified himself as Perry.
When the agent asked for identification Perry showed his credentials as a police officer for Calexico.
The agent let him go. During a press conference last week, officials said the agent thought at the time that Perry was involved in the investigation.
Authorities continued their search and arrested Barak. The keys to the car were found in his pocket, according to the complaint.
Both he and the car were returned to the port, where a search of the vehicle uncovered more than 800 pounds of cocaine in the trunk.
Late that night Perry, who had returned to Mexico, reportedly voluntarily came back to the United States through the port. He then was taken into custody.
Calexico police officials said an officer used a cell phone to communicate with Perry and talked him into turning himself into authorities.
Both Barak and Perry will remain in custody without bail until the June 7 detention hearing, when Benitez will decide whether to set a bail amount for each man.
Outside the court room, Levine said his client is not a flight risk and should have a bail amount set.
"I don't think there is any concern he will flee," Levine said, adding if his client were guilty he had the opportunity to avoid arrest by remaining in Mexicali.
Instead, he said, his client turned himself over to prove his innocence.
Assistant U.S. Attorney Pete Lewis argued in court Perry is a flight risk.
He alleged, based on the complaint, Perry fled through the port in the car in which the cocaine was found, ran from the car and went into Mexico.
Lewis also said in court he is unsure what circumstances led Perry to turn himself over to authorities.
Staff Writer Darren Simon can be reached at 337-4082.