Brings an output to ground perhaps?
An AA cell is pretty large.
I would use a logic level N channel mosfet, a couple of hearing aid batteries (or even one of those small 12V batteries used in garage remotes), a light dependant resistor and another high value resistor.
Connect the LDR and the resistor in series across the battery and connect the junction to the gate. The negative terminal connects to the source of the mosfet. The output is across the source and drain.
As you shine light on the LDR, the gate voltage increases and turns the MOSFET on.
Depending on the load you're switching (keep it small), the LDR, and the mosfet, the resistor might need to be between 100K and 1M. Maybe you could use a 1M trimpot here.
It won't turn on fast, but it should do what you require,
The maximum current will be determined by the resistor, for 100K it will be less than 120uA if you use a 12V battery
note that these batteries (MN21, A23, etc) will discharge in about 140 hours at 220 uA, so a battery would last about 10 to 12 days in fully illuminated conditions. If outdoors, I would guess you might get a month if you were lucky.