The rule regarding a tenant moving out is that he must return the unit to the landlord in the same condition he found it, minus reasonable wear and tear. Therefore, he is not technically required to have the carpets professionally cleaned, but he can not leave it in a condition that exceeds reasonable wear and tear. If I were you, I would present him with the receipt from when you had the carpets professionally cleaned when he moved in, and inform him that the law requires him to leave the premises in the exact same condition as he found it minus reasonable wear and tear.
Under Civil Code 1950(b)(3), a landlord may deduct from the tenant's security deposit, the reasonable cost of:
"The cleaning of the premises upon termination of the tenancy necessary to return the unit to the same level of cleanliness it was in at the inception of the tenancy." (The amendments to this paragraph enacted by the act adding this sentence shall apply only to tenancies for which the tenant's right to occupy begins after January 1, 2003..)
Under Civil Code 1950.5(g), the landlord must provide an itemized statement of deductions indicating the amount and purpose of each deduction with copies of any invoices for outside work and/or materials. If the landlord or landlords employees did the work, then there should be documentation of the work performed, time spent and hourly rate charged.
You must refund the balance of the tenant's security deposit and provide itemized receipts within 21 days.