When I did biofeedback therapy a couple of years ago, heart rate variability was the main proxy they used for relaxation. The goal for session was to obtain a steady heart rate.
Counterintuitively, it's actually the other way around -- at rest, more variability in your heart rate (over a roughly 20 second period) is correlated with lower levels of stress. Deep, slow, regular breathing tends to increase HRV, since heart rate is linked to respiration -- your heart beats a little faster when you inhale, a little slower when you exhale. It's desirable to have big, regular swings in heart rate, synchronized with your respiration.
Thankfully, biofeedback protocols will typically hide these details, and just calculate a single metric for you to attempt to optimize.
I remember what the chart looks like now, and yes the goal was in fact to have your heart rate vary with breath.
When I started the session, my heart rate would jump around almost arbitrarily. By the end, it would fall into a steady rhythm of rising and falling as I breathed.
