Evidence of Superfluidity in a Dipolar Supersolid Through Non-Classical

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Evidence of superfluidity in a supersolid under rotation Giulio Biagioni Dipartimento di Fisica e Astronomia and LENS, Università di Firenze, and CNR-INO, sezione di Pisa

106° SIF National Congress, 14-18 September 2020 Supersolid state of matter

• Classical solid Crystalline order

Macroscopic consequence: rigidity of the crystal

• Superfluid Coherent wave function

Macroscopic consequences: metastibility of supercurrents, reduced moment of inertia

• Supersolid Both kinds of order Macroscopic consequences: ?

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 2 Our dipolar supersolid

휃 = 휋/2 162 푉푖푛푡(풓) = 푉푐표푛푡 풓 + 푉푑푖푝(풓) 퐷푖푝표푙푎푟 푎 Dy 휀 = = 푑푑 푑푑 퐶표푛푡푎푐푡 푎 휇 = 10휇 퐵 Isotropic and Repulsive repulsive 휃 = 0 Attractive

Confinement Vertical trapping + anisotropy clustering effect !

Superfluid Normal solid (Bose-Einstein condensate) Supersolid (Droplet crystal)

휀푑푑 Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 3 Our dipolar supersolid

휀푑푑 lattice mode

8 10

Superfluid 1

5 0 0 9

3 k

. y 5 k x Supersolid

) 9

( 7

G 6

( 2

)

B 5

Droplet crystal

88 2

9 7

. 5 휀푑푑 5 8 28 48 100 t (ms) 1.25 superfluid mode

Variance of

) 2

d 1.00 uniform random

r ( Evidence for double symmetry breaking

e phase

c 0.75

i r

L. Tanzi et al., Phys. Rev. Lett.,

a L. Tanzi et al., Nature, 574, 774 (2020) V

0.50

e 122, 130405 (2019)

a h

P 0.25 Cluster supersolid with ~3 − 4 droplets and ~104 atoms per droplet 0.00 15 30 45 60 75 Time (ms) Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 4 Related works

G. Natale et al., Phys. Rev. Lett. 123, 050402 (2019) Innsbruck M. Guo et al., Nature 574, 386 (2019) Stuttgart

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 5 Superfluids under rotation

푖휑(푟) • Macroscopic wavefunction Ψ0 푟 = Ψ0(푟) 푒

ℏ 푣Ԧ = ∇휑 ⇒ ∇ ∧ 푣Ԧ=0 ⇒ ׯ 푣Ԧ ∙ 푑푙Ԧ = 0 ⇒ classical irrotational hydrodynamics • 푚

• Reduced moment of inertia ⇒ 퐼 = 1 − 푓푠 퐼푐 (in a cylindrical geometry)

መ 푣Ԧ = 휔푅휃 • Normally, 푓푠 → 1 for 푇 → 0 ⇒ 퐼 = 0 z 휔 • No angular momentum is acquired < 퐿 >= 퐼휔 from the container when 휔 → 0

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 6 Superfluids under rotation

Helium-4 (1967)

Hess-Fairbank Helium-3 (1982)

related Meissner effect in superconductors

BECs (2000)

Giulio Biagioni Exploring the supersolid state of matter 7 Leggett’s argument: can a solid be superfluid?

푖휑(푟) Also supersolids are described by a macroscopic wavefunction Ψ0 푟 = Ψ0(푟) 푒

The density modulation appears in the form of the superfluid fraction:

−1 (퐼 = 1 − 푓푠 퐼푐 with 푓푠 ≤ ׬푑푥/휌ҧ(푥

so that 0 < 퐼 < 퐼푐 at 푇 = 0 NCRI (Non Classical Rotational Inertia)

퐼 = 퐼푐 0 < 퐼 < 퐼푐 퐼 = 0

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 8 The scissors mode

How to probe the moment of inertia of our small and non-homogenous system?

We employ a collective mode of the trapped condensate: the scissors mode

In the small-angle limit:

휃 푡 = 휗0 sin(휔푠푐푡)

Experiments with non-dipolar BECs: 푥 2 2 O.M. Maragò et al., Phys. Rev. Lett. 휔푠푐 = 휔푥 + 휔푦 84, 2056-2019 (2000)

푦 푧

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 9 The scissors mode

Useful link with the moment of inertia

2 2 휔푥 + 휔푦 퐼 = 퐼푐훼 훽 2 휔푠푐

2 2 2 2 2 2 2 2 ۄ Τ〈 푥 + 푦ۄ 훼 = 휔푦 − 휔푥 ൗ 휔푥 + 휔푦 훽 = 〈푥 − 푦 Trap deformation Cloud deformation

D. Guéry-Odelin and S. Stringari, Phys. Rev. Lett. 83, 4452-4455 (1999)

Analogy with torsional oscillators 퐾 퐼 = 2 휔표푠푐

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 10 Scissors oscillation

BEC

휃

휃′ 휃′

휃′

Supersolid We excite the scissors mode .. then we extract the angle Single-frequency oscillation in both the changing temporary the at different times with a 2D BEC and the supersolid regime relative power between fit ODT2 and ODT3..

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 11 Scissors frequencies and moment of inertia

2 2 휔푥 + 휔푦 퐼 = 퐼푐훼 훽 2 휔푠푐

퐼퐵퐸퐶 < 퐼푠푠 < 퐼푐

Reduced moment of inertia ⇒ proof of the superfluid behavior under rotation

L. Tanzi et al., (2019) Preprint:

arXiv:1912.01910. Accepted for

publication in Science.

Theory, including 훽, by the Trento group: S. M. Roccuzzo et al., Phys. Rev. Lett., 124:045702 (2020)

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 12 Superfluid fraction

2 2 2 Our definition of superfluid 1−훼훽 휔푥+휔푦 /휔푠푐 퐼 = 1 − 푓 퐼 + 푓 훽2퐼 ⇒ 푓 = fraction for an anisotropic system: 푠 푐 푠 푐 푠 1−훽2 numerical simulations

푓푠~1 (cluster supersolid)

Comparison with the 1D Leggett’s −1 model: 푓푠 ≤ ׬푑 푥Τ휌ҧ (푥) (white droplets triangles, density profile from the superfluidity

Trento group) 휇 휌 Our experiment: 0.88±0.14 Leggett formula Leggett: 0.25 Droplets superfluidity: 0.50 휀푑푑 = 1.50 휇 Helium experiments: ~0.01 −4 Leggett: 10 휀푑푑 = 1.42

15/09/2020 Evidence of superfluidity in a supersolid under rotation 13 Conclusions and outlook

We have demonstrated the superfluid properties of the dipolar supersolid under rotation through a measurement of its moment of inertia.

Next goals:

• Realizing larger, 2D supersolids to detect a sub-unity superfluid fraction, study vortices and measure a finite shear modulus

• Josephson effect without a barrier

• Other types of supersolids with cold atoms (e.g. dipoles in 2D stabilized by 3-body forces, or lattice supersolids)

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 14 The team

Current team: Luca Tanzi Giovanni Modugno Giulio Biagioni Nicolò Antolini Andrea Fioretti Carlo Gabbanini

Former members: Eleonora Lucioni Francesca Famà Jacopo Catani Julian Maloberti

Theory by: Russell Bisset, Luis Santos (Hannover); Alessio Recati, Sandro Stringari (Trento); Michele Modugno (Bilbao); Luca Pezzè, Augusto Smerzi (Firenze); Maria Luisa Chiofalo (Pisa), Adriano Angelone (Trieste) …

Giulio Biagioni Evidence of superfluidity in a supersolid under rotation 15