Yes, both you and your one-year-old can visit. Being married is not a requirement for visitation at the Metropolitan Detention Center in Los Angeles.
For you, the process involves submitting a visitation application to the facility and completing a background check. MDC Los Angeles is a federal facility operated by the Bureau of Prisons, and the BOP requires all prospective visitors to be approved before visiting. Once approved your name goes on the inmate's approved visitor list and you can schedule visits according to the facility's visitation schedule.
For your baby, children can visit federal facilities and a one-year-old is welcome. Minor children typically visit under the supervision of an approved adult visitor, meaning as long as you are approved the baby comes with you. The baby does not need their own separate application, but you should confirm the specific policy at MDC LA when you submit your application since federal facilities can have specific procedures around visiting with minors.
Before your first visit, confirm the current visitation schedule for MDC Los Angeles. Federal facilities designate specific visiting days and hours, and availability can vary by the inmate's housing unit. Contact the facility directly or check the BOP website at bop.gov for current visiting information for that specific location.
Also confirm with your baby's father that he has submitted your name to be added to his approved visitor list on his end. Both sides of the process need to be complete before a visit can happen.