An observer says Meghan Markle "looked very chilled" as she supported Prince Harry at his polo match — also attended by Kate Middleton and her three children

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Meghan Markle is loving life as a new mom.

The Duchess of Sussex made her first public outing with 2-month-old Archie at husband Prince Harry and brother-in-law Prince William‘s charity polo match on Wednesday. It was a warm, sunny day, and Meghan mainly stayed under the shade of a tree with her son in her arms.

“She was doting on him, there’s no doubt about that,” an onlooker tells PEOPLE. “She was kissing and stroking him and bouncing him up and down.”

Despite the excitement of the polo match, Archie mainly dozed for his first big outing. Even when he did open his eyes, he was quiet. (Meghan described her son as “really calm” just two days after his birth, and it seems his demeanor hasn’t changed much!)

Meghan, Duchess of Sussex and Archie Harrison Mountbatten-Windsor attend The King Power Royal Charity Polo Day at Billingbear Polo Club on July 10, 2019 in Wokingham, England
Credit: Samir Hussein/WireImage
Mandatory Credit: Photo by Tim Rooke/Shutterstock (10331673az) Meghan Duchess of Sussex, Archie Mountbatten-Windsor, Catherine Duchess of Cambridge King Power Royal Charity Polo Day, Billingbear Polo Club, Wokingham, UK - 10 Jul 2019
Archie and Meghan Markle
| Credit: Tim Rooke/Shutterstock
Catherine Duchess of Cambridge, Meghan Duchess of Sussex, Archie Mountbatten-Windsor
Kate Middleton, Archie and Meghan Markle
| Credit: Tim Rooke/Shutterstock

Prince Harry, 34, joined his wife and son following the polo match. The trio walked to the parking lot and shared a sweet moment when both parents smiled at their baby boy, who was now bundled up in a blanket.

Photos of the family of three “showed real life,” the source adds.

Meghan Duchess of Sussex Archie Montbatten-Windsor Prince Harry
Meghan Markle, Archie and Prince Harry
| Credit: James Whatling/MEGA

Meghan and Harry are focused on giving their son a peaceful start.

“They are basking in the glow and enjoying each day,” a royal source tells PEOPLE in this week’s cover story. “As most new parents are, they’re just so excited. There’s something new every day. They’re just really happy.”

Prince Harry with Meghan Duchess of Sussex and Archie Harrison Mountbatten-Windsor
Meghan Markle, Archie and Prince Harry
| Credit: Tim Rooke/Shutterstock
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Even though the new mom, 37, is a member of the British royal family now, she hasn’t let go of her native California roots. She sported aviator style sunglasses with gold frames with a flowing khaki green dress by Lisa Marie Fernandez — choosing to forego the outfit’s belted bow option. Meghan let her long hair down, contributing to her free spirited vibe.

“She’s being mindful of what she puts in her body and on her skin and even her at-home cleaning supplies,” the friend added. “She’s hyperaware of what she’s using. She’s looking for natural things.”

Prince George, Meghan Duchess of Sussex, Archie Mountbatten-Windsor
Prince George, Archie and Meghan Markle
| Credit: Tim Rooke/Shutterstock
The Duke Of Cambridge And The Duke Of Sussex Take Part In The King Power Royal Charity Polo Day
Kate Middleton, Prince Louis, Meghan Markle and Archie
| Credit: Max Mumby/Indigo/Getty

William was also joined by wife Kate Middleton and their three kids — Prince George, Princess Charlotte and Prince Louis — at the King Power Royal Charity Polo Day on Wednesday. The match will help raise funds and awareness for charities supported by the royal brothers.

The special event marks the royal cousins’ first public playdate together.