Combine 1/2 cup granulated sugar, 1/2 cup water, and 3 rosemary sprigs in a small cream enameled saucepan. Set over medium heat.

Bring the mixture to a gentle simmer, stirring now and then, until the sugar disappears completely and the liquid runs clear with only a faint green tint, about 3 minutes.

Take the pan off the heat, cover, and let the rosemary steep in the hot syrup for 15 minutes: it will smell distinctly piney and the color will deepen slightly.

Strain the syrup through a fine mesh strainer into a small glass jar, pressing the rosemary sprigs lightly to release the last of the oil, then discard the sprigs.

Let the syrup cool at room temperature for 20 minutes, then refrigerate for at least 30 minutes until fully cold: a warm syrup will melt the ice too fast.

Fill a tall clear glass with ice cubes until it is about three quarters full.

Pour 1/2 cup chilled pomegranate juice, 1 tablespoon fresh lime juice, and 1 tablespoon of the cooled rosemary syrup over the ice.

Top the glass slowly with 1/2 cup chilled sparkling water, pouring down the inside of the glass so the bubbles stay lively rather than flattening on contact.

Give the drink one gentle stir with a long spoon, just enough to blend the layers without knocking out the fizz, then taste: it should land tart before sweet.

Drop in a tablespoon of pomegranate seeds and tuck a small rosemary sprig against the rim. Serve immediately while the ice is still full and the bubbles are high.