In the United States, the general way to challenge violations of your constitutional rights is a civil action under 42 U.S.C. § 1983. This broad statute allows anyone injured by such a violation to obtain damages and an injunction against future conduct.
Police officers are entitled to qualified immunity from suit; a plaintiff must show that the officers' conduct violated "clearly established statutory or constitutional rights of which a reasonable person would have known." Messerschmidt v. Millender, 132 S. Ct. 1235, 1244 (2012). In other words, the search must not only have been unconstitutional; it must have been clearly so, under established law. This is a heavy burden.
Suing the police department involves some other hurdles; generally, one would claim that police policy was unconstitutional or that the department negligently hired or supervised the officers.
It's possible to prevail on facts similar to your hypothetical, however, so long as the search was clearly unconstitutional. I wasn't able to find any cases involving just an unlawful search; the closest I found was Frunz v. City of Tacoma, 468 F.3d 1141 (9th Cir. 2006), which also involved a seizure:
The facts are remarkable. Plaintiff, Susan Frunz, and her two guests
were in Frunz’s home in Tacoma, Washington, when police surrounded the
house, broke down the back door and entered. The police had no warrant
and had not announced their presence. Frunz first became aware of them
when an officer accosted her in the kitchen and pointed his gun,
bringing the barrel within two inches of her forehead. The police
ordered or slammed the occupants to the floor and cuffed their hands
behind their backs—Frunz for about an hour, until she proved to their
satisfaction that she owned the house, at which time they said “never
mind” and left.
As the officers doubtless knew, physical entry into
the home is the “chief evil against which the wording of the Fourth
Amendment is directed.” United States v. United States District Court,
407 U.S. 297, 313 (1972); see also Murdock v. Stout, 54 F.3d 1437,
1440 (9th Cir. 1995) (“[P]rotection of individuals from unreasonable
government intrusion into their houses remains at the very core of the
Fourth Amendment.”). To safeguard the home, we normally require a
warrant before the police may enter. “The right of privacy was deemed
too precious to entrust to the discretion of those whose job is the
detection of crime and the arrest of criminals . . . . And so the
Constitution requires a magistrate to pass on the desires of the
police before they violate the privacy of the home.” McDonald v.
United States, 335 U.S. 451, 455-56 (1948); see also Groh v. Ramirez,
540 U.S. 551, 560 (2004). What extraordinary circumstances justified
sundering the privacy and protection of Frunz’s home without a
Id. at 1142–43. Read the rest of the opinion; it's short, understandable, and relevant to this question. Spoiler alert: the circumstances were not sufficiently extraordinary.
(By the way, the Ninth Circuit later imposed sanctions on the defendants for raising a frivolous appeal.)