The phrase "at the home" is typically used in a context that says whose home is meant. For example:
The Vigil for the Deceased: this is a service of prayers, songs and homilies either at the home of the deceased or in church, before the day of the funeral. (Christian funerals, BBC, 2009; emphasis mine)
The phrase "at the home" can also be used in contexts where the inhabitants of the place have just been mentioned. For example:
A Florida man shot dead his pregnant wife after allegedly waking up and mistaking her for a prowler. (...) When police arrived at the home in Stuart, 100 miles north of Miami, the woman was on the floor with a severe head wound. (Baby born after wife shot dead in Florida, The Times, 7 November 2020; emphasis mine)
(If you do a web search for "at the home", you will find countless examples where "at the home" means "at the home where the incident happened".)
The phrase "at the home" can also be used to refer to an institution or a nursing home. For example:
My elderly sister (86) was admitted to Springfield in February 2020 after five weeks in hospital followed by a couple of weeks of unsuccessful home care. (...) From the moment she arrived at the home, she was given constant care and attention by all staff, be it by the home's manager, right through to the nursing and care staff. (Quoted from a review on the page Springfield Nursing Home on Carehome.co.uk, 23 March 2020; emphasis mine)
So when you write "spring has arrived at the home", it needs to be clear from the preceding context whose home is meant (because of the definite article "the"), otherwise the phrase may have strange connotations.
However, if you leave out the definite article and write "spring has arrived at home" or "spring has arrived home", there may be ambiguity about whose home is meant: one of the character's or the personified Spring's home? One might avoid this by writing "our home" (assuming that the poetic narrator and Ghalib share the same home). If "home" is not required by the source, one may also write "Spring has come" or "Spring has arrived", which avoids all the ambiguities mentioned above.