March meeting C71.00338

1. Adiabatic and diabatic dynamical spin
squeezing
Zeyang Li1,2
, Boris Braverman1,2
, Enrique Mendez1,2
, and Vladan Vuleti´c1,2
1
Department of Physics, Massachusetts Institute of Technology; 2
Research Laboratory of
Electronics
Introduction
Deterministic generation of spin squeezing can be used for metrological
purpose [1]. To achieve high metrological gain, one has to fight against
non-unitarity [2] and technical noises due to imperfect control [3]. In this
work, we propose a method to overcome these difficulties which obtain
Heisenberg-like scaling squeezing and behave robustly against technical
limitations.
Concepts
One-Axis Twisting for Generating Spin Squeezing
N spin-1/2 systems with ˆSx,y,z = N
i=1 σi
x,y,z,
One-Axis Twisting (OAT) Hamiltonian H = χS2
z can generate spin
squeezing.
Kitagawa-Ueda parameter: ξ−2
≡ 4∆σ2
sqz/N,
scales as ξ2
∼ N−2/3
, limited by the finite curvature effect [4].
Two-Axis Twisting and the Heisenberg Limit
Two-Axis Twisting Hamiltonian H = χ(S2
z − S2
y )
generated squeezed states have certain squeezed axis
Heisenberg-like scaling of ξ−2
∼ N−1
.
However this method is not physically implemented yet.
The ultimate Heisenberg limit is σ2
HL = 1/4.
Unitarity
The cavity leakage and atomic spontaneous emission cause the contrast loss.
The squeezing light also generates unused measurement squeezing which
causes non-unitarity, quantified by Quantum Fisher Information.
Additional Rabi Drive and Adiabatic Squeezing
Time-Varying Controllable Hamiltonian
H = χ(t)S2
z + ∆(t)Sz − Ω(t)Sx
Three parameters are controlled by: cavity light strength, RF pulse detuning
and RF pulse amplitude.
Na¨ıve Adiabatic Approach
From g.s. of H(0) = −ΩSx to g.s. of H(t) = χS2
z
intermediate Hamiltonian H = −Ω(t)Sx + χ(t)S2
z
Exact Solution of Intermediate Hamiltonian Using
Holstein-Primakoff Approximation [5]
Sx =
N
2
− a†
a, Sy =
√
N
2i
(a†
− a), Sz =
√
N
2
(a†
+ a)
H ∼ α a†2
+ a2
+ β(a†
a)
applying transformation b ≡ (a + σa†
)/(
√
1 − σ2), where
σ =
+ 1 −
√
1 + 2
, =
Nχ(t)
2Ω(t)
,
H( ) = Ω
√
1 + 2 b†
b + const.
Dynamical Squeezing
Adiabatic is slow. For faster, keep a constant χ, ω,
∆Sz(t) = ∆Sz(0) cos2
(λΩt) +
∆Sy(0)
λ2
sin2
(λΩt), λ = 1 + 2 ∼
√
N.
Reference
[1] L. Pezz`e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein,
“Quantum metrology with nonclassical states of atomic ensembles,” Rev.
Mod. Phys., vol. 90, p. 035005, Sep 2018.
[2] B. Braverman, A. Kawasaki, and V. Vuleti´c, “Impact of non-unitary spin
squeezing on atomic clock performance,” New J. Phys., vol. 20, no. 10,
p. 103019, 2018.
[3] B. Braverman, A. Kawasaki, E. Pedrozo-Pe˜nafiel, S. Colombo, C. Shu, Z. Li,
E. Mendez, M. Yamoah, L. Salvi, D. Akamatsu, Y. Xiao, and V. Vuleti´c,
“Near-unitary spin squeezing in 171
Yb,” Phys. Rev. Lett., vol. 122, p. 223203,
Jun 2019.
[4] M. Kitagawa and M. Ueda, “Squeezed spin states,” Phys. Rev. A, vol. 47,
pp. 5138–5143, Jun 1993.
[5] T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain
magnetization of a ferromagnet,” Phys. Rev., vol. 58, pp. 1098–1113, Dec
1940.
States Illustration
Cavity Feedback Squeezing: decoherence and tiny rotation
=⇒
Squeezing with coherent Rabi: squeezed direction and robustness
=⇒
Extra broadening (Fisher Information) has less impact than feedback
squeezing.
No technical limitation on Sx rotation.
For clock purpose: integrate with Tycko π/2 pulse
Results (Numerical)
N = 400 (typical for our experiments), dynamical (fast) squeezing
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-15
-10
-5
0
t(ms)
Squeezing(dB)
Ω=2π×10 (Hz)
Ω=2π×27 (Hz)
Ω=2π×71 (Hz)
Ω=2π×188(Hz)
Ω=2π×500(Hz)
Maximum squeezing scaling
10 50 100
-15
-10
-5
0
Ω=2π×10
Ω=2π×20
Ω=2π×30
Ω=2π×40
Ω=2π×50
Ω=2π×100
Ω=2π×150
Ω=2π×200
where the maximum scales still with N−2/3
but for a certain Ω it can scale
with N until saturates.
Outlook
Relating to stability improvement [2].
Experimental Realization.
Exploring other quantum many-body systems.
bit.do/poster-PDP093 PDP-093/SKPP/III/2018