Socially the woman has become a "kept woman". It would not be rape, at least in most jurisdictions, because the officer is using neither force nor threats of force. The duress is from her economic and general circumstance, and most jurisdictions would not consider that to be rape. At least technically, this is a form of prostitution, but a form very rarely prosecuted.
Maryland
maryland is a not untypical jurisdiction as US law goes in such matters. § 3-303 of the Naryland Criminal Code (Rape in the first degree) provides that:
(a) Prohibited. -- A person may not:
(a) (1)
(a) (1) (i) engage in vaginal intercourse with another by force, or the threat of force, without the consent of the other; or
(a) (1) (ii) engage in a sexual act with another by force, or the threat of force, without the consent of the other; and
(a) (2)
(a) (2) (i) employ or display a dangerous weapon, or a physical object that the victim reasonably believes is a dangerous weapon;
(a) (2) (ii) suffocate, strangle, disfigure, or inflict serious physical injury on the victim or another in the course of committing the crime;
(a) (2) (iii) threaten, or place the victim in fear, that the victim, or an individual known to the victim, imminently will be subject to death, suffocation, strangulation, disfigurement, serious physical injury, or kidnapping;
(a) (2) (iv) commit the crime while aided and abetted by another; or
(a) (2) (v) commit the crime in connection with a burglary in the first, second, or third degree.
Section 3-304 (Rape in the first degree) provides that:
(a) A person may not engage in vaginal intercourse with another:
(a) (1) by force, or the threat of force, without the consent of the other;
(a) (2) if the victim is a mentally defective individual, a mentally incapacitated individual, or a physically helpless individual, and the person performing the act knows or reasonably should know that the victim is a mentally defective individual, a mentally incapacitated individual, or a physically helpless individual; or
(a) (3) if the victim is under the age of 14 years, and the person performing the act is at least 4 years older than the victim.
Sections 3-306 thru 3-307 have similar provisions but they mention "sexual act" rather than "intercourse" and they do not define the crime as "rape" but as "sexual offense". But the emphasis on force or threat of force, and lack of consent is much the same.
So it seems that under Maryland law, at least, the situation described in the question would not be legally "rape" nor a "sexual offense".
Maryland Criminal code § 11-301 (2018) defines prostitution in subsection (c) which reads:
(c) Prostitution. -- "Prostitution" means the performance of a sexual act, sexual contact, or vaginal intercourse for hire.
Whether the provision of food, shelter, and protection by one person to another in return for sex would be considered "for hire" is arguable. In practice such arrangements, even if they also involve cash payment, are rarely if ever prosecuted.