New Jersey Statutes Title 53. State Police 53 § 2-25

a. In any investigation, interview, or other proceeding conducted under oath by the division or any duly authorized officer, employee, or agent thereof, if a person refuses to answer a question or produce evidence of any other kind on the ground that the person may be incriminated thereby, and notwithstanding the refusal, an order is made upon 24 hours' prior written notice to the Attorney General of the State of New Jersey, and to the appropriate district attorney or prosecutor having an official interest therein, by the Superintendent of the division or the superintendent's designee, that the person answer the question or produce the evidence, the person shall comply with the order. If the person complies with the order, and if, but for this section, would have been privileged to withhold the answer given or the evidence produced by the person, then immunity shall be conferred upon the person, as provided for herein. Immunity shall not be conferred upon any person except in accordance with the provisions of this section. If, after compliance with the provisions of this section, a person is ordered to answer a question or produce evidence of any other kind and complies with the order, and it is thereafter determined that the Attorney General or appropriate district attorney or prosecutor having an official interest therein was not notified, that failure or neglect shall not deprive that person of any immunity otherwise properly conferred upon the person. But the person may nevertheless be prosecuted or subjected to penalty or forfeiture for any perjury or contempt committed in answering, or failing to answer, or in producing or failing to produce evidence, in accordance with the order, and any answer given or evidence produced shall be admissible against the person upon any criminal proceeding concerning such perjury or contempt.

b. If a person, in obedience to a subpoena directing the person to attend and testify, is in this State or comes into this State from the State of New York, the person shall not, while in this State pursuant to such subpoena, be subject to arrest or the service of process, civil or criminal, in connection with matters which arose before the person's entrance into this State under the subpoena.