u_{13}^6 u_{23}^4 + \\
2 b_3^2 u_3^6 u_{012}^2 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^2 u_{123}^6 u_{13}^6 \
u_{23}^4 + b_3^2 u_3^6 u_{012}^2 u_{0123}^2 u_{013}^6 u_{02}^2 u_{023}^2 u_{03}^4 u_{123}^6 \
u_{13}^6 u_{23}^4 - \\
b_3^2 u_3^6 u_{012}^6 u_{0123}^6 u_{013}^2 u_{02}^2 u_{023}^6 u_{03}^4 u_{123}^6 u_{13}^6 \
u_{23}^4 - 2 b_3^2 u_3^6 u_{012}^6 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^6 u_{123}^6 \
u_{13}^6 u_{23}^4 + \\
b_3^2 u_3^6 u_{012}^4 u_{0123}^6 u_{013}^4 u_{02}^4 u_{023}^4 u_{03}^4 u_{123}^4 u_{13}^8 \
u_{23}^4 - b_3^2 u_3^6 u_{012}^8 u_{0123}^6 u_{013}^4 u_{02}^4 u_{023}^8 u_{03}^4 u_{123}^4 \
u_{13}^8 u_{23}^4 - \\
b_3^7 u_3 u_{012}^6 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^3 u_{13} u_{23}^5 +
b_3^7 u_3 u_{012}^2 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^3 u_{13} u_{23}^5 \
- \\2 b_3^5 u_3^3 u_{012}^4 u_{0123}^5 u_{013}^3 u_{023}^3 u_{03}^3 u_{123}^5 u_{13} u_{23}^5 +
3 b_3^5 u_3^3 u_{0123}^5 u_{013}^3 u_{023}^7 u_{03}^3 u_{123}^5 u_{13} u_{23}^5 - \\
b_3^5 u_3^3 u_{012}^4 u_{0123}^7 u_{013}^5 u_{023}^5 u_{03}^5 u_{123}^5 u_{13} u_{23}^5 -
2 b_3^5 u_3^3 u_{012}^6 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03} u_{123}^3 u_{13}^3 \
u_{23}^5 - \\b_3^5 u_3^3 u_{012}^6 u_{0123} u_{013}^5 u_{02}^2 u_{023} u_{03}^3 u_{123}^3 u_{13}^3 \
u_{23}^5 + 2 b_3^5 u_3^3 u_{012}^2 u_{0123}^5 u_{013} u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^3 \
u_{13}^3 u_{23}^5 - \\
4 b_3^5 u_3^3 u_{012}^6 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^3 u_{13}^3 \
u_{23}^5 + 4 b_3^5 u_3^3 u_{012}^2 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^3 \
u_{13}^3 u_{23}^5 - \\
2 b_3^5 u_3^3 u_{012}^6 u_{0123}^3 u_{013}^7 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^3 u_{13}^3 \
u_{23}^5 + b_3^5 u_3^3 u_{012}^2 u_{0123}^7 u_{013}^3 u_{02}^2 u_{023}^7 u_{03}^5 u_{123}^3 \
u_{13}^3 u_{23}^5 + \\
2 b_3^5 u_3^3 u_{012}^2 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^7 u_{123}^3 u_{13}^3 \
u_{23}^5 - b_3^3 u_3^5 u_{012}^4 u_{0123}^5 u_{013} u_{023}^3 u_{03} u_{123}^5 u_{13}^3 u_{23}^5 - \\
2 b_3^3 u_3^5 u_{012}^4 u_{0123}^3 u_{013}^3 u_{023} u_{03}^3 u_{123}^5 u_{13}^3 u_{23}^5 +
b_3^7 u_3 u_{012}^4 u_{0123}^3 u_{013}^3 u_{02}^4 u_{023} u_{03}^3 u_{123}^5 u_{13}^3 u_{23}^5 \
+\\ 9 b_3^3 u_3^5 u_{0123}^3 u_{013}^3 u_{023}^5 u_{03}^3 u_{123}^5 u_{13}^3 u_{23}^5 -
2 b_3^3 u_3^5 u_{012}^4 u_{0123}^7 u_{013}^3 u_{023}^5 u_{03}^3 u_{123}^5 u_{13}^3 u_{23}^5 \
- \\4 b_3^3 u_3^5 u_{012}^4 u_{0123}^5 u_{013}^5 u_{023}^3 u_{03}^5 u_{123}^5 u_{13}^3 u_{23}^5 -
3 b_3^7 u_3 u_{012}^8 u_{0123} u_{013}^5 u_{02}^4 u_{023}^3 u_{03}^5 u_{123}^5 u_{13}^3 \
u_{23}^5 + \\2 b_3^7 u_3 u_{012}^4 u_{0123}^5 u_{013}^5 u_{02}^4 u_{023}^3 u_{03}^5 u_{123}^5 \
u_{13}^3 u_{23}^5 +
b_3^5 u_3^3 u_{012}^2 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03} u_{123}^7 u_{13}^3 \
u_{23}^5 +\\ 2 b_3^5 u_3^3 u_{012}^2 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^7 \
u_{13}^3 u_{23}^5 -
2 b_3^5 u_3^3 u_{012}^6 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^7 u_{13}^3 \
u_{23}^5 - \\b_3^5 u_3^3 u_{012}^6 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^7 u_{123}^7 \
u_{13}^3 u_{23}^5 -
2 b_3^3 u_3^5 u_{012}^6 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03} u_{123}^3 u_{13}^5 \
u_{23}^5 + \\2 b_3^3 u_3^5 u_{012}^2 u_{0123}^3 u_{013} u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^3 \
u_{13}^5 u_{23}^5 -
4 b_3^3 u_3^5 u_{012}^6 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^3 u_{13}^5 \
u_{23}^5 - \\b_3^3 u_3^5 u_{012}^6 u_{0123}^7 u_{013}^5 u_{02}^2 u_{023}^7 u_{03}^3 u_{123}^3 \
u_{13}^5 u_{23}^5 +
b_3^3 u_3^5 u_{012}^2 u_{0123} u_{013}^3 u_{02}^2 u_{023} u_{03}^5 u_{123}^3 u_{13}^5 u_{23}^5 \
+\\ 4 b_3^3 u_3^5 u_{012}^2 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^3 u_{13}^5 \
u_{23}^5 - 2 b_3^3 u_3^5 u_{012}^6 u_{0123}^5 u_{013}^7 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^3 \
u_{13}^5 u_{23}^5 + \\
2 b_3^3 u_3^5 u_{012}^2 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^7 u_{123}^3 u_{13}^5 \
u_{23}^5 + 3 b_3 u_3^7 u_{0123} u_{013}^3 u_{023}^3 u_{03}^3 u_{123}^5 u_{13}^5 u_{23}^5 - \\
2 b_3 u_3^7 u_{012}^4 u_{0123}^5 u_{013}^3 u_{023}^3 u_{03}^3 u_{123}^5 u_{13}^5 u_{23}^5 +
4 b_3^5 u_3^3 u_{012}^4 u_{0123}^5 u_{013}^3 u_{02}^4 u_{023}^3 u_{03}^3 u_{123}^5 u_{13}^5 \
u_{23}^5 - \\b_3 u_3^7 u_{012}^4 u_{0123}^3 u_{013}^5 u_{023} u_{03}^5 u_{123}^5 u_{13}^5 u_{23}^5 +
2 b_3^5 u_3^3 u_{012}^4 u_{0123}^3 u_{013}^5 u_{02}^4 u_{023} u_{03}^5 u_{123}^5 u_{13}^5 \
u_{23}^5 -\\ 9 b_3^5 u_3^3 u_{012}^8 u_{0123}^3 u_{013}^5 u_{02}^4 u_{023}^5 u_{03}^5 u_{123}^5 \
u_{13}^5 u_{23}^5 +
2 b_3^5 u_3^3 u_{012}^4 u_{0123}^7 u_{013}^5 u_{02}^4 u_{023}^5 u_{03}^5 u_{123}^5 u_{13}^5 \
u_{23}^5 +\\ b_3^5 u_3^3 u_{012}^4 u_{0123}^5 u_{013}^7 u_{02}^4 u_{023}^3 u_{03}^7 u_{123}^5 \
u_{13}^5 u_{23}^5 +
b_3^3 u_3^5 u_{012}^2 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03} u_{123}^7 u_{13}^5 \
u_{23}^5 +\\ 2 b_3^3 u_3^5 u_{012}^2 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^7 \
u_{13}^5 u_{23}^5 -
2 b_3^3 u_3^5 u_{012}^6 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^7 u_{13}^5 \
u_{23}^5 -\\ b_3^3 u_3^5 u_{012}^6 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^7 u_{123}^7 \
u_{13}^5 u_{23}^5 -
b_3 u_3^7 u_{012}^6 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^3 u_{13}^7 \
u_{23}^5 + \\b_3 u_3^7 u_{012}^2 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^3 u_{13}^7 \
u_{23}^5 + b_3^3 u_3^5 u_{012}^4 u_{0123}^7 u_{013}^3 u_{02}^4 u_{023}^5 u_{03}^3 u_{123}^5 \
u_{13}^7 u_{23}^5 + \\
2 b_3^3 u_3^5 u_{012}^4 u_{0123}^5 u_{013}^5 u_{02}^4 u_{023}^3 u_{03}^5 u_{123}^5 u_{13}^7 \
u_{23}^5 - 3 b_3^3 u_3^5 u_{012}^8 u_{0123}^5 u_{013}^5 u_{02}^4 u_{023}^7 u_{03}^5 u_{123}^5 \
u_{13}^7 u_{23}^5 - \\
b_3^6 u_3^2 u_{012}^6 u_{0123}^2 u_{013}^4 u_{02}^2 u_{023}^2 u_{03}^2 u_{123}^4 u_{13}^2 \
u_{23}^6 + 2 b_3^6 u_3^2 u_{012}^2 u_{0123}^4 u_{013}^2 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^4 \
u_{13}^2 u_{23}^6 - \\
2 b_3^6 u_3^2 u_{012}^6 u_{0123}^4 u_{013}^6 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^4 u_{13}^2 \
u_{23}^6 + b_3^6 u_3^2 u_{012}^2 u_{0123}^6 u_{013}^4 u_{02}^2 u_{023}^6 u_{03}^6 u_{123}^4 \
u_{13}^2 u_{23}^6 - \\
b_3^4 u_3^4 u_{012}^4 u_{0123}^4 u_{013}^2 u_{023}^2 u_{03}^2 u_{123}^6 u_{13}^2 u_{23}^6 +
3 b_3^4 u_3^4 u_{0123}^4 u_{013}^2 u_{023}^6 u_{03}^2 u_{123}^6 u_{13}^2 u_{23}^6 - \\
2 b_3^4 u_3^4 u_{012}^4 u_{0123}^6 u_{013}^4 u_{023}^4 u_{03}^4 u_{123}^6 u_{13}^2 u_{23}^6 +
b_3^4 u_3^4 u_{012}^2 u_{0123}^4 u_{02}^2 u_{023}^4 u_{03}^2 u_{123}^4 u_{13}^4 u_{23}^6 - \\
4 b_3^4 u_3^4 u_{012}^6 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^2 u_{123}^4 u_{13}^4 \
u_{23}^6 + 2 b_3^4 u_3^4 u_{012}^2 u_{0123}^2 u_{013}^2 u_{02}^2 u_{023}^2 u_{03}^4 u_{123}^4 \
u_{13}^4 u_{23}^6 - \\
2 b_3^4 u_3^4 u_{012}^6 u_{0123}^2 u_{013}^6 u_{02}^2 u_{023}^2 u_{03}^4 u_{123}^4 u_{13}^4 \
u_{23}^6 + 2 b_3^4 u_3^4 u_{012}^2 u_{0123}^6 u_{013}^2 u_{02}^2 u_{023}^6 u_{03}^4 u_{123}^4 \
u_{13}^4 u_{23}^6 - \\
2 b_3^4 u_3^4 u_{012}^6 u_{0123}^6 u_{013}^6 u_{02}^2 u_{023}^6 u_{03}^4 u_{123}^4 u_{13}^4 \
u_{23}^6 + 4 b_3^4 u_3^4 u_{012}^2 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^6 u_{123}^4 \
u_{13}^4 u_{23}^6 - \\
b_3^4 u_3^4 u_{012}^6 u_{0123}^4 u_{013}^8 u_{02}^2 u_{023}^4 u_{03}^6 u_{123}^4 u_{13}^4 \
u_{23}^6 + 3 b_3^2 u_3^6 u_{0123}^2 u_{013}^2 u_{023}^4 u_{03}^2 u_{123}^6 u_{13}^4 u_{23}^6 - \\
b_3^2 u_3^6 u_{012}^4 u_{0123}^6 u_{013}^2 u_{023}^4 u_{03}^2 u_{123}^6 u_{13}^4 u_{23}^6 -
2 b_3^2 u_3^6 u_{012}^4 u_{0123}^4 u_{013}^4 u_{023}^2 u_{03}^4 u_{123}^6 u_{13}^4 u_{23}^6 \
+ \\2 b_3^6 u_3^2 u_{012}^4 u_{0123}^4 u_{013}^4 u_{02}^4 u_{023}^2 u_{03}^4 u_{123}^6 u_{13}^4 \
u_{23}^6 - 3 b_3^6 u_3^2 u_{012}^8 u_{0123}^2 u_{013}^6 u_{02}^4 u_{023}^4 u_{03}^6 u_{123}^6 \
u_{13}^4 u_{23}^6 + \\
b_3^6 u_3^2 u_{012}^4 u_{0123}^6 u_{013}^6 u_{02}^4 u_{023}^4 u_{03}^6 u_{123}^6 u_{13}^4 \
u_{23}^6 + b_3^4 u_3^4 u_{012}^2 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^2 u_{123}^8 \
u_{13}^4 u_{23}^6 - \\
b_3^4 u_3^4 u_{012}^6 u_{0123}^4 u_{013}^4 u_{02}^2 u_{023}^4 u_{03}^6 u_{123}^8 u_{13}^4 \
u_{23}^6 - b_3^2 u_3^6 u_{012}^6 u_{0123}^6 u_{013}^4 u_{02}^2 u_{023}^6 u_{03}^2 u_{123}^4 \
u_{13}^6 u_{23}^6 + \\
2 b_3^2 u_3^6 u_{012}^2 u_{0123}^4 u_{013}^2 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^4 u_{13}^6 \
u_{23}^6 - 2 b_3^2 u_3^6 u_{012}^6 u_{0123}^4 u_{013}^6 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^4 \
u_{13}^6 u_{23}^6 + \\
b_3^2 u_3^6 u_{012}^2 u_{0123}^2 u_{013}^4 u_{02}^2 u_{023}^2 u_{03}^6 u_{123}^4 u_{13}^6 \
u_{23}^6 + 2 b_3^4 u_3^4 u_{012}^4 u_{0123}^6 u_{013}^4 u_{02}^4 u_{023}^4 u_{03}^4 u_{123}^6 \
u_{13}^6 u_{23}^6 + \\
b_3^4 u_3^4 u_{012}^4 u_{0123}^4 u_{013}^6 u_{02}^4 u_{023}^2 u_{03}^6 u_{123}^6 u_{13}^6 \
u_{23}^6 - 3 b_3^4 u_3^4 u_{012}^8 u_{0123}^4 u_{013}^6 u_{02}^4 u_{023}^6 u_{03}^6 u_{123}^6 \
u_{13}^6 u_{23}^6 + \\
b_3^5 u_3^3 u_{012}^2 u_{0123}^3 u_{013} u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^5 u_{13}^3 \
u_{23}^7 - 2 b_3^5 u_3^3 u_{012}^6 u_{0123}^3 u_{013}^5 u_{02}^2 u_{023}^3 u_{03}^3 u_{123}^5 \
u_{13}^3 u_{23}^7 + \\
2 b_3^5 u_3^3 u_{012}^2 u_{0123}^5 u_{013}^3 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^5 u_{13}^3 \
u_{23}^7 - b_3^5 u_3^3 u_{012}^6 u_{0123}^5 u_{013}^7 u_{02}^2 u_{023}^5 u_{03}^5 u_{123}^5 \
u_{13}^3 u_{23}^7 + \\b_3^3 u_3^5 u_{0123}^3 u_{013} u_{023}^5 u_{03} u_{123}^7 u_{13}^3 u_{23}^7 -
b_3^3 u_3^5 u_{012}^4 u_{0123}^5 u_{013}^3 u_{023}^3 u_{03}^3 u_{123}^7 u_{13}^3 u_{23}^7 + \\
b_3^3 u_3^5 u_{012}^2 u_{0123}^5 u_{013} u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^5 u_{13}^5 \
u_{23}^7 - 2 b_3^3 u_3^5 u_{012}^6 u_{0123}^5 u_{013}^5 u_{02}^2 u_{023}^5 u_{03}^3 u_{123}^5 \
u_{13}^5 u_{23}^7 + \\
2 b_3^3 u_3^5 u_{012}^2 u_{0123}^3 u_{013}^3 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^5 u_{13}^5 \
u_{23}^7 - b_3^3 u_3^5 u_{012}^6 u_{0123}^3 u_{013}^7 u_{02}^2 u_{023}^3 u_{03}^5 u_{123}^5 \
u_{13}^5 u_{23}^7 + \\
b_3^5 u_3^3 u_{012}^4 u_{0123}^5 u_{013}^5 u_{02}^4 u_{023}^3 u_{03}^5 u_{123}^7 u_{13}^5 \
u_{23}^7 - b_3^5 u_3^3 u_{012}^8 u_{0123}^3 u_{013}^7 u_{02}^4 u_{023}^5 u_{03}^7 u_{123}^7 \
u_{13}^5 u_{23}^7 + \\
b_3^4 u_3^4 u_{012}^2 u_{0123}^4 u_{013}^2 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^6 u_{13}^4 \
u_{23}^8 - b_3^4 u_3^4 u_{012}^6 u_{0123}^4 u_{013}^6 u_{02}^2 u_{023}^4 u_{03}^4 u_{123}^6 \
u_{13}^4 u_{23}^8) = 0.
\end{multline}

The equation (<ref>) can be written as quadratic equation  (<ref>) with respect to
$ x=s_{02} $  with coefficients $A$ (<ref>) , $B$ (<ref>)  , $C$ (<ref>)  .

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††thanks: Present address: IonQ Inc, 4505 Campus Dr, College Park, Maryland
20740, USA††thanks: E-mail<EMAIL_ADDRESS>
# Two-mode squeezing and SU(1,1) interferometry with trapped ions

J. Metzner1, A. Quinn1, S. Brudney1 I.D. Moore1 S.C. Burd2, D.J. Wineland 1
D.T.C. Allcock1 1Department of Physics, University of Oregon, Eugene, OR, USA
2Department of Physics, Stanford University, Stanford, CA, USA

(August 28, 2024)

###### Abstract

We experimentally implement circuits of one and two mode operations on two
motional modes of a single trapped ion. This is achieved by implementing the
required displacement, squeezing, two-mode squeezing, and beamsplitter
operations using oscillating electric potentials applied to the trap
electrodes. The resulting electric fields drive the modes resonantly or
parametrically without the need for optical forces. As a demonstration, we
implement SU(2) and SU(1,1) interferometers with phase sensitivities near the
Cramér-Rao bound. We report a maximum sensitivity of a SU(2) interferometer
within 0.67(5) dB of the standard quantum limit (SQL) as well as a single and
two-mode SU(1,1) sensitivity of 5.9(2) dB and 4.5(2) dB below the SQL
respectively.

††preprint: APS/123-QED

Quantum-mechanical harmonic oscillators, described by infinite dimensional,
non-commuting observables, are currently being utilized for quantum sensing
[1, 2] and studies of continuous variable quantum computing (CVQC) [3]. The
quantum-mechanical harmonic oscillator is realized in trapped-ions, nano-
mechanical systems [4] and has an analog with microwave photons in
superconducting circuits [5] and optical photons in photonic systems [6].
Recent work, using motional states of trapped-ions has shown the ability to
detect small displacements [7, 8] and encode logical qubits [9, 10]. The use
of two-mode squeezed states could provide a quantum advantage [11](larger
quantum signal to noise ratio with the same number of resources) for sensing
and significant work has been done to generate these states in other platforms
[12, 13, 14]. Two-mode squeezed states have already proven useful for quantum
sensing using SU(1,1) interferometers [15, 16, 17, 18] in photonics, but can
be difficult to generate efficiently and suffer from photon loss.

In the following, we demonstrate different classes of interferometers using
the motional modes of a single trapped atomic ion. With a demonstration of
these interferometers, we show the ability to concatenate several operations
on the oscillator states of two modes in a phase coherent manner. Performing
these operations with digitally programmable frequency sources for relative
phase control simplifies significant calibration challenges when developing
more complicated circuits.

Trapped ions are well suited for such demonstrations due to the harmonic
nature of the trap and the potential to achieve relatively long mechanical
oscillator coherence times [19]. Reconstructing the motional states of these
oscillators is achieved by coupling them to the ion’s internal states via
motional sideband transitions and measuring the internal state populations
[20]. In the limit of a small number of motional quanta these transitions can
approximate a phonon counting type measurement. This enables a homodyne type
detection scheme which is used to quantify the sensitivity we achieve with the
different classes of interferometers demonstrated below.

Yurke _et al._ [21] proposed schemes to reach the Heisenberg limit [22] using
different classes of interferometers. Here, we work within the framework of
the Cramér-Rao bound [23, 24] which sets the ultimate limit for the resolution
of a sensing scheme. This bound is described by a parameter variance, instead
of the Heisenberg limit which is set by the number of resources used, i.e.
photon or phonon number [24], making it more suitable to the limits of the
experiments reported here. The required interactions for the interferometers
(described below) are colloquially called displacing, beam splitting, single-
mode squeezing, and two-mode squeezing. For trapped ions, these interactions
can be implemented with laser beams [25] or in a laser-free method using
additional potentials applied to the trap electrodes [26, 27]. Performance of
these interferometers can, in practice, be limited only by the motional
heating of the ion resulting from fluctuating patch potentials on the trap
electrodes [28, 29]. Here, we demonstrate an SU(2) interferometer (Mach-
Zender), which has previously been demonstrated in [25, 30, 31], (coherent
input states are used here in contrast to these works), as well as single-mode
and two-mode SU(1,1)111The Lie-group labelling is used to refer to the groups
that contain the operators that generate these states. interferometers. Each
of the interferometers described here is sensitive to different phase shifts,
which are single-mode phase $\phi_{a}$ (single-mode SU(1,1)), difference phase
$\phi_{a}-\phi_{b}$ (SU(2)), or sum phase $\phi_{a}+\phi_{b}$ (two-mode
SU(1,1)), where the $a,b$ labels refer to different modes. We show performance
close to the Cramér-Rao bounds for these interferometers.

To demonstrate these different interferometers we use a single trapped 40Ca+
ion as the mechanical oscillator. The ion is confined in a macroscopic, room-
temperature, linear Paul trap with an ion-electrode spacing of $\simeq$ 0.75
mm. Experiments are performed using the two ‘radial’ modes of the ion (normal
to the trap axis) with frequencies $\omega_{a}\approx 2\pi\times 1.80$ MHz and
$\omega_{b}\approx 2\pi\times 1.83$ MHz, and oscillator energy eigenstates
denoted by $\ket{n_{a},n_{b}}$. The mode splitting can be adjusted by changing
a DC potential on the RF electrodes [32] which also rotates the mode angle and
can be used to vary the achievable coupling rate of the interactions described
below. We use qubit states $\ket{\uparrow}\equiv\ket{m_{L}=+5/2}$ and
$\ket{\downarrow}\equiv\ket{m_{L}=+3/2}$ within the metastable $D_{5/2}$
manifold which has a lifetime of $\simeq$ 1.1 seconds [33], where $m_{L}$ is
the total angular momentum projection along the direction of the quantization
magnetic field of 0.498 G. The qubit transition frequency is
$\omega_{0}\approx 2\pi\times 2.63$ MHz. This is a magnetic field sensitive
qubit with a coherence time of $\simeq$ 1 ms, while stable trapping potentials
provide motional coherence times over 20 ms. The qubit is manipulated using an
orthogonal pair of laser beams red detuned from the transition frequency
between $P_{3/2}$ and $D_{5/2}$ manifolds by 44 THz to induce coherent
stimulated-Raman transitions between the two qubit states. In each experiment
the qubit is first prepared in the $\ket{\uparrow}\ket{0,0}$ state with high
fidelity, using optical pumping, electromagnetically-induced-transparency
(EIT) cooling [34] implemented with a pair of 397 nm beams on the
$S_{1/2}\rightarrow P_{1/2}$ transition and resolved-sideband laser cooling
using the qubit described above. Mode temperatures correspond to
$\bar{n}_{a,b}<0.1$. Qubit readout is performed by “deshelving” population in
the $\ket{\downarrow}$ state followed by applying a laser beam resonant with
the $S_{1/2}\leftrightarrow P_{1/2}$ cycling transition and detecting state-
dependent ion fluorescence [35]. Motional state analysis is performed by
setting the Raman beams difference frequency to $\omega_{0}\pm\omega_{m}$,
where $m$ denotes the desired mode for readout. Applying these sideband
interactions maps the motion onto the qubit states and produces Rabi
oscillations with multiple frequency components having amplitudes that depend
on the Fock state populations [20]. This interaction is generated by applying
the same pair of Raman beams that drive the qubit, but detuned above (red
sideband) or below (blue sideband), the qubit frequency. In the Lamb-Dicke
limit [20], the red sideband interaction drives transitions between
$\ket{\uparrow}\ket{n}$ and $\ket{\downarrow}\ket{n-1}$ with Rabi frequencies
proportional to $\sqrt{n}$ and can be used to verify preparation or return to
the motional ground state. The blue sideband interaction drives transitions
between $\ket{\uparrow}\ket{n}$ and $\ket{\downarrow}\ket{n+1}$ with Rabi
frequencies proportional to $\sqrt{n+1}$ and is used to fully characterize the
generated state populations [20].

Figure 1:  (a) View of trap along the trap axis showing the electrical
connections:pink—RF quadrupole electrodes, trap and parametric drive,
purple—displacement drive and DC electrode, yellow— other DC electrodes and
relevant control electrodes. The green box represents the circuitry necessary
to combine the drives [32]. Mode $a$ and mode $b$ label the two different
‘radial’ mode vectors of a single ion with the angles from the horizontal
shown. (b) A 3D rendering of the trap showing direction of the 976 nm laser
used for motional state preparation and state characterization, and cyan ball
representing the 40Ca+ ion

Generation of the motional states is accomplished by applying time-varying
potentials to the trap electrodes (see Fig. 1) producing a unitary operation.
The displacement operation is produced by applying an oscillating potential,
at the mode frequency, to a single trap electrode (purple electrode Fig. 1).
Single-mode squeezing, two-mode squeezing and the beam splitter are
accomplished by applying an oscillating potential at twice the mode frequency
$(2\omega_{a})$, at the sum of the mode frequencies $(\omega_{a}+\omega_{b})$
or at the difference frequency $(\omega_{a}-\omega_{b})$ respectively (pink
electrodes Fig. 1). These modulated potentials yields a generalized squeezing
interaction Hamiltonian,

$\hat{H}=i\hbar\dfrac{g}{2}\big{[}\hat{a}_{i}\hat{a}_{j}\exp(i\delta
t)\exp(-i\theta)-\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\exp(-i\delta
t)\exp(i\theta)\big{]}$ (1)

where $g$ is the parametric coupling strength, which has a different value for
each operation, $\delta$ is the detuning from the interaction resonance
(either single mode squeezing (mode $a$ or $b$) or two-mode squeezing), and
$\theta$ is the phase of the parametric modulation. With the proper choice of
detuning this implements the unitary squeezing operator with squeezing
parameter magnitude given by $r_{SMS}=gt$ and the two mode squeezing operator
with $r_{TMS}=2gt$. Here, we used a single-mode displacement coupling of
$g=2\pi\times 1.37$ kHz which gives a displacement parameter $\alpha=gt$. The
maximum single mode squeezing coupling is $g=2\pi\times 3.99$ kHz and a
maximum two-mode squeezing coupling of $g=2\pi\times 1.15$ kHz. Maximum
coupling refers to a coupling rate calculated from fits to the states with the
maximum voltage we used and the relevant pulse time for each operation. The
beam splitter coupling is $g=2\pi\times 0.66$ kHz calculated using calibration
data shown in [32].

In order to verify the proper performance of the beam splitter operation, we
use a calibration experiment described in [32]. Verification of the two-mode
squeezed state is more complicated given we are only able to readout a single
mode in a single shot of the experiment. Using the blue sideband readout
method, described above, is equivalent to tracing over a single mode and a
thermal state is measured [36]. This is analogous to measuring a single qubit
from a Bell state which produces a random qubit state. Additionally, we can
use a beam splitter operation after the two-mode squeezer to disentangle the
modes and produce single-mode squeezed states (Fig. 2). Performing both of
these measurements verifies we have produced the target two-mode squeezed
state and enables the use of these states for interferometric measurements.

Figure 2: (a) General quantum circuit diagram for characterizing the two-mode
squeezed state with blue sideband readout performed at one of two instances,
to verify we are producing the expected output (b) Measurement after the two-
mode squeezed state and before beam splitter produces a thermal state. (c)
applying the beam splitter operation after the two-mode squeezer and then
characterizing the state produces two single-mode squeezed states with
squeezing parameter $r$ equal to that of the two-mode squeezed state.

Implementing time reversal protocols is a method for generating these types of
interferometers and is a powerful tool for sensing applications [37]. These
protocols also enable us to verify that we are coherently generating the
desired states by time-reversing back to the initial state. The SU(2)
interferometer (Fig. 3c) is implemented by first displacing mode $a$ in our
experiment, which generates the desired initial coherent state. The beam
splitter is then used to generate a superposition of coherent and vacuum
states. The phase of the final beam-splitter is controlled by using a variable
delay, such that the accumulated phase is $(\omega_{a}-\omega_{b})t_{delay}$.
The final beam splitter will completely transfer the phonons to mode $b$ with
the correct choice of relative phase, and the red sideband readout will verify
we have returned to the ground state (Fig. 3c). We use a maximum displacement
of $\alpha\simeq\lvert 6\rvert$ to maintain a similar maximum $\bar{n}$ in the
single and two-mode squeezed states.

The single-mode (Fig. 3a) and two-mode (Fig. 3b) SU(1,1) interferometers are
implemented by using the parametric drive tuned to either twice the frequency
of mode $a$ or the sum frequency of mode $a$ and $b$ respectively. To avoid
increased decoherence, from long pulse durations, these pulses are run at a
fixed duration with the drive voltage being varied to control the size of the
state. A second squeezing pulse is then applied (Fig. 3b) with a varied phase
controlled by the driving electronics. We are limited in the amount of two-
mode squeezing we can generate due to our increased sensitivity to common-mode
phase fluctuations that this interferometer is sensitive to.

Figure 3:  (a) Single-mode SU(1,1) interferometer output for different amounts
of squeezing is shown by varying the relative phase of the second squeezing
pulse shown in the circuit diagram (inset). (b) repeat of (a) with two-mode
squeezing and two-mode SU(1,1) shown in the interferometer circuit inset.(c)
SU(2) interferometer output shown for various sizes of initial displacement
where relative phase is controlled with a variable delay shown in the SU(2)
circuit inset.

In order to determine the sensitivities to small phase shifts that are
achievable with these different experiments, we calculate the quantum Fisher
information for a pure state [23]

$\mathcal{F}(\phi)=4(\Delta\hat{\mathcal{G}})^{2}$ (2)

where $(\Delta\hat{\mathcal{G}})^{2}$ is the variance in the initial state of
the probe (coherent state, squeezed state, two-mode squeezed state) and
$\hat{\mathcal{G}}=i[d\hat{U}^{\dagger}/d\phi]\hat{U}$. For phase sensing
$\hat{U}=e^{i\phi a^{\dagger}a}$ and therefore
$\hat{\mathcal{G}}=a^{\dagger}a$ and the Fisher information is proportional to
variance in the phonon number of the probe states. The phase sensitivity is
limited by the Cramér-Rao bound [23] given by

$\Delta\phi\geq\dfrac{1}{\sqrt{\mathcal{F(\phi)}}}.$ (3)

This means for the SU(2) interferometer we are limited at

$\Delta\phi=\dfrac{1}{\alpha}=\dfrac{1}{\sqrt{\expectationvalue{N}_{d}}}$ (4)

where $\expectationvalue{N}_{d}$ is the expected mean phonon number for the
displacement generated and this limit corresponds to the expected standard
quantum limit. The limit for the single-mode SU(1,1) interferometer is given
by [38]

$\Delta\phi=\dfrac{1}{\sqrt{8\expectationvalue{N}_{SMS}(\expectationvalue{N}_{SMS}+1)}}$
(5)

where $\expectationvalue{N}_{SMS}$ is the expected mean phonon number for the
input squeezed state. The two-mode SU(1,1) limit is given by [23]

$\Delta\phi=\dfrac{1}{\sqrt{\expectationvalue{N}_{TMS}(2+\expectationvalue{N}_{TMS})}}$
(6)

where $\expectationvalue{N}_{TMS}$ is the expected mean phonon number for the
input two-mode squeeze state.

The sensitivities achieved are calculated from the experimental data using the
Fisher information [23]

$F(\phi)=\sum_{j}\dfrac{1}{P_{j}(\phi)}\bigg{[}\dfrac{dP_{j}(\phi)}{d\phi}\bigg{]}^{2}$
(7)

where $P_{j}(\phi)$ is the probability of getting the experimental results $j$
if the value of the parameter is $\phi$. In our experiment with only two
possible outcomes, spin-up or spin-down, this simplifies to

$F(\phi)=\dfrac{1}{P_{\downarrow}(1-P_{\downarrow})}\bigg{[}\dfrac{\partial
P_{\downarrow}}{\partial\phi}\bigg{]}^{2}.$ (8)

The variance and slope of the signal are expressed analytically and are fit to
the phase signals that we generate (Fig. 3). These theoretical curves are
derived, see [32], without any adverse effects and represent the ideal
experimental output (Fig. 4). Systematic offsets in the phase fringes, seen in
the single-mode squeezing phase scans, from $\pi$ are due to an additional
pseudopotential [39] when the parametric drive is applied. This additional
potential accounts for a maximum shift of $\leavevmode\nobreak\ 10\,$kHz of
the mode frequencies. In the single mode SU(1,1) experiment the interaction
frequency is calibrated at the maximum coupling meaning as the coupling is
turned down, the mode frequencies decrease and larger phase offsets are
produced. This phase offset is calibrated away in the two-mode phase scans by
calibrating the mode frequencies at every parametric drive voltage. The
maximum phase sensitivity is extracted and plotted against the prediction from
the quantum Fisher information (Fig. 4). The results we achieve deviate from
the ideal limit due to motional and qubit decoherence as well as motional
heating. Initial mode temperatures also affects these results and contribute
to vertical offsets of the curves in Fig. 3. The ideal experimental output
deviates from the CR bound due to the choice of red side-band readout times,
where optimal pulse time varies with $\bar{n}$. Red side-band pulse times were
not optimized across all $\bar{n}$ due to the constraints on pulse times due
to off-resonant driving of other modes and increased pulse length causing
further decoherence.

Figure 4: Ideal experimental phase sensitivities ($\delta\phi$) for the SU(2)
(dashed green line), single mode SU(1,1) (dashed orange line) and two-mode
SU(1,1) dashed red line) are shown and can be compared to experimental results
shown as dots with error bars and Cramér-Rao (CR) bounds in solid lines.

We have demonstrated a wide class of trapped-ion motional state
interferometers using a ‘laser-free’ parametric drive achieving single and
two-mode SU(1,1) interferometer sensitivities of 5.9(2) dB and 4.5(2) dB below
the SQL, respectively and a SU(2) interferometer sensitivity within 0.67(5) dB
of the SQL. These results illustrate the measurement of phase shifts near the
Cramér-Rao bound for common-mode, relative, and single-mode phase
fluctuations. Enhanced sensitivity for measuring these motional phase
fluctuations could enable characterization and mitigation of motional
frequency noise which can benefit experiments involving motional states.
Beside providing a framework for implementing quantum-enhanced sensing
protocols, our methods provide the flexibility to concatenate multiple
motional operations into circuits for potential applications in CVQC.

During preparation of this manuscript we became aware of related work using
reservoir engineering to prepare two-mode squeezed states of trapped ions
[40].

###### Acknowledgements.

This research is supported by support from NSF through the Q-SEnSE Quantum
Leap Challenge Institute, Award #2016244, and the US Army Research Office
under award W911NF-20-1-0037. The data supporting the figures in this article
are available upon reasonable request from D.T.C.A.

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## supplementary materials

###  .1 Parametric Drive Electronics

We apply the voltages required to generate the quadrupole potential for
squeezing and beams splitter operations directly to the RF trapping
electrodes. This conveniently ensures the quadrupole is automatically well
centered on ions located on the axis of our linear trap. The trapping RF is
delivered to the trap electrodes by stepping up the 50 $\Omega$ RF source to
high voltage via a helical resonator acting as a transformer, as shown in Fig.
5. This RF drive is amplitude stabilized by the “Squareatron” [41] to improve
the motional coherence of the ion’s radial modes. The trapping RF and
beamsplitter RF are produced by a Rigol DG1022Z arbitrary waveform generator
(AWG) and the squeezing RF is produced by an Urukul DDS card [42], part of our
ARTIQ Sinara experimental control system [43]. The parametric drive tones are
applied directly to the ‘back’ of the helical resonator’s coil via bandpass
filters that ensure the two sources are isolated from each other. As these
tones are at frequencies well below the helical’s resonance they pass through
to the trap electrodes relatively unimpeded. Also connected to the back of the
resonator coil is the “squeeze box” containing a pair of LC circuits, which
provides three functions. Firstly, it provides a DC bias path to allow control
of the mode rotation and splitting. Secondly, it provides a low impedance path
to ground at the RF trapping frequency, ensuring the helical resonator works
correctly. Compared to a direct ground connection, the “squeeze box” decreases
the quality factor ($Q=f_{0}/\mathrm{FWHM}$) of the helical modestly, from 310
to 295, and shifts its resonant frequency of 14.5 MHz by approximately 6 kHz.
Thirdly, it provides an impedance match for the two parametric drive tones by
having the LC circuits in the box tuned to match the beamsplitter and squeeze
drive frequencies of 33 kHz and 3.6 MHz respectively. Electrical testing shows
we get voltages on the RF electrodes close to what we expect, as well as low
return loss at the squeezing drive frequency Fig. 6.

The potential applied at the trap electrodes determines the parametric
coupling strength $g$ for the operations, as shown in Eq. 1. For measured
voltages on the trap RF electrode of 0.26 Vpp for the beam splitter and 0.58
Vpp for both single and two-mode squeezing we get couplings of
$g_{BS}=2\pi\crossproduct 0.93\,\mathrm{kHz}$, $g_{SMS}=2\pi\crossproduct
3.68\,\mathrm{kHz}$ and $g_{TMS}=2\pi\crossproduct 1.09\,\mathrm{kHz}$
respectively. These coupling values are calculated from theory using the above
voltages and agree reasonably with the values calculated from experiment shown
in the main text.

Figure 5: Electrical circuit diagram of the trapping RF, DC bias, and
parametric driving RF signals. The beam splitter RF is filtered by a Mini-
Circuits SLP-1.9+ MHz low-pass and a DC block BLK-18W-S+; the squeezing RF is
filtered by a Mini-Circuits SLP-5+ MHz low-pass filter and a custom made 5th
order elliptic 1.8 MHz high-pass filter. Figure 6: a) Voltage measured at the
ion trap vacuum feedthrough with beamsplitter and squeezing RF sources
outputting 0 dBm of power. Vertical dashed green and dash-dot red lines
indicate the beam splitter and squeezing frequencies, 33 kHz and 3.6 MHz
respectively. b) Return loss for the squeezing drive input showing good
impedance matching.

###  .2 Experimental output and Fisher information

To obtain an analytical expression for the output of these interferometers we
must determine the action of the red sideband on the final state and determine
the expectation values for the final qubit projection operator. We calculate
this below for the two-mode SU(1,1) case. The single-mode case can be treated
in a similar manner. We start by representing the two-mode squeezed state in
the Fock basis [44]

$\displaystyle\ket{TMS}=\dfrac{1}{\cosh{r}}\sum_{n=0}^{\infty}\tanh^{n}{r}\ket{n,n}$
$\displaystyle=$ (9)
$\displaystyle\sqrt{\dfrac{1}{1+\lambda}}\sum_{n=0}^{\infty}\left(\sqrt{\dfrac{\lambda}{1+\lambda}}\right)^{n}\ket{n,n}$

where $\lambda=\sinh{r}$. We need to determine
$\ket{out}=TMS(\phi)TMS\ket{0,0}$. For phase other than $\phi=\pi$ the output
is still a two-mode squeezed state with a different two-mode squeeze parameter
$r$, which we can express as a function of the phase.

$r(\phi)=\sinh^{-1}({\sinh{r_{0}}\cos{\phi/2}}),$ (10)

where $r_{0}$ is the input two-mode squeeze state magnitude. The red sideband
Hamiltonian is written as [20]

$H_{RSB}=\hbar\eta\dfrac{\Omega}{2}(\sigma^{+}a+\sigma^{-}a^{\dagger}).$ (11)

$\Omega$ is the qubit carrier Rabi frequency, $\eta$ is the Lamb-Dicke
parameter [20] for the given mode, and $\sigma^{+}$ and $\sigma^{-}$ are the
single qubit raising and lowering operators. The unitary operator for the
Hamiltonian can be written as

$U=e^{\dfrac{i\beta H_{RSB}t}{\hbar}}$ (12)

where $\beta=\eta\Omega t/2$. Expanding this unitary into an infinite series
gives

$\ket{n,n}\rightarrow\sum_{j=0}^{\infty}\dfrac{(i\beta)^{j}}{j!}(\sigma^{+}a+\sigma^{-}a^{\dagger})^{j}\ket{\downarrow,n,n}.$
(13)

Only terms having equal numbers of raising and lower operators for spin and
motion will have a final contribution to the projection, so we can re-write
this state

$U_{RSB}\ket{\downarrow,n,n}=\sum_{j=0}^{\infty}\dfrac{(i\beta)^{j}}{j!}(\sigma^{+}a\sigma^{-}a^{\dagger})^{j}\ket{\downarrow,n,n},$
(14)

where $\\{j\\}$ is only even integers. This leaves the final spin state
unaffected and only the number operator $a^{\dagger}a$ remains, leaving us

$\displaystyle
U_{RSB}\ket{\downarrow,n,n}=\dfrac{(i\beta)^{2j}}{(2j)!}n^{j}\ket{\downarrow,n,n}$
$\displaystyle=$ (15)
$\displaystyle\dfrac{(i\beta)^{2j}}{(2j)!}\sqrt{n}^{2j}\ket{\downarrow,n,n}=\cos{(\beta\sqrt{n})}\ket{\downarrow,n,n}.$

The expectation value of the spin projector is therefore

$\displaystyle\expectationvalue{\ket{\downarrow\otimes
1\kern-2.5pt\text{l}}\bra{\downarrow\otimes
1\kern-2.5pt\text{l}}}{\downarrow,n,n}$ $\displaystyle=$ (16)
$\displaystyle\dfrac{1}{1+\lambda}\sum_{n=0}^{\infty}\cos{(\sqrt{n}\eta\Omega\dfrac{t}{2})\bigg{(}\dfrac{\lambda}{1+\lambda}\bigg{)}^{n}}$

For the sake of simplicity we will call this expectation value
$\expectationvalue{\hat{X}}$ and we define the variance

$\Delta\hat{X}^{2}=\expectationvalue{\hat{X}^{2}}-\expectationvalue{\hat{X}}^{2}.$
(17)

For spin projection
$\expectationvalue{\hat{X}^{2}}=\expectationvalue{\hat{X}}$. We also need an
expression for the derivative of the expectation value as a function of the
phase;

$\begin{gathered}\dfrac{d\expectationvalue{\hat{X}}}{d\phi}=\dfrac{d\lambda}{d\phi}\dfrac{1}{1+\lambda}\\\
\bigg{(}\expectationvalue{\hat{X}}+\dfrac{1}{1+\lambda}\sum_{n=0}^{\infty}\cos^{2}{(\sqrt{n}\eta\dfrac{\Omega
t}{2})n\bigg{(}\dfrac{\lambda}{1+\lambda}\bigg{)}^{n-1}}\\\
\bigg{(}1-\dfrac{\lambda}{1+\lambda}\bigg{)}\bigg{)}\end{gathered}$ (18)

where $\dfrac{d\lambda}{d\phi}=-2\cosh^{2}{r}\sin{\phi}\sinh^{2}{r}$. The same
process is carried out and shown for the coherent state in the SU(2)
interferometer as these results differ significantly from the two-mode
SU(1,1). A coherent state is expressed in the Fock basis as

$\ket{\alpha}=e^{-\absolutevalue{\alpha}^{2}/2}\sum_{n=0}^{\infty}\dfrac{\alpha^{2n}}{n!}\ket{n}.$
(19)

The red sideband produces a similar output,

$\displaystyle\expectationvalue{\ket{\downarrow\otimes
1\kern-2.5pt\text{l}}\bra{\downarrow\otimes
1\kern-2.5pt\text{l}}}{\downarrow,n}=\expectationvalue{\hat{X}}=$ (20)
$\displaystyle
e^{-\absolutevalue{\alpha}^{2}}\sum_{n=0}^{\infty}\dfrac{\alpha^{2n}}{n!}\cos^{2}{(\sqrt{n}\eta\Omega
t/2)}$

with $\alpha$ a function of the phase, $\alpha=\alpha_{0}\sin{\phi/2}$. The
variance can again be expressed in terms of this expectation value as in (17)
and the derivative of $\langle\hat{X}\rangle$ is given as

$\dfrac{d\expectationvalue{\hat{X}}}{d\phi}=e^{\absolutevalue{\alpha}^{2}}\sum_{n=0}^{\infty}\dfrac{\alpha^{2n}}{n!}\cos^{2}{(\sqrt{n}\eta\Omega
t/2)}(-\alpha^{2n+1}+n\alpha^{2n-1})$ (21)

###  .3 Beam splitter calibration

To produce the beam splitter operation described by the Hamiltonian [25]

$\hat{H}=i\hbar\dfrac{g}{2}\big{[}\hat{a}\hat{b}^{\dagger}\exp(-i\theta)-\hat{a}^{\dagger}\hat{b}\exp(i\theta)\big{]},$
(22)

we digitally trigger the drive AWG to output a pulse consisting of an integer
number of cycles (30 cycles in the runs in Fig. 7) at a fixed amplitude,
starting at a phase of zero. We cannot continuously adjust the length of these
pulses and therefore must calibrate the pulse amplitude to produce a 50/50
beam splitter pulse. Specifically, we select an amplitude that maximizes the
fringe contrast of our SU(2) interferometer (i.e. that allows a pair of 50/50
beam splitter pulses to fully transfer a displacement from one mode to
another). The starting phase of each pulse is fixed which means to adjust the
relative phase between beam splitter pulses in our SU(2) experiments we adjust
the delay time $t_{delay}$ between pulses, giving a phase offset of
$(\omega_{a}-\omega_{b})t_{delay}$, as shown in Fig. 7.

Figure 7: Starting with a coherent state in the low-frequency mode, the SU(2)
interferometer is performed with a variable delay between beam splitter
pulses, allowing coherent transfer of the coherent state between modes. The
amplitude of the 50/50 beam splitter pulses used can be calibrated by
maximizing contrast on mode $a$ fringes. Delay time is time passed between
beam splitter pulses.

###  .4 Squeezed state fitting

For performing motional state analysis we tune our Raman beams resonantly to a
blue motional sideband (BSB). This interaction ideally realizes the anti
Jaynes-Cummings (JC) Hamiltonian

$\hat{H}_{JC}=\hbar\Omega/2(\hat{\sigma}^{+}\hat{a}^{\dagger}+\hat{\sigma}^{-}\hat{a})$
(23)

where the qubit state transition operators
$\hat{\sigma}^{\pm}\equiv\dfrac{1}{2}(\hat{\sigma}_{x}\mp i\hat{\sigma}_{y})$
are constructed from the Pauli operators $\hat{\sigma}_{x}$ and
$\hat{\sigma}_{y}$ and $\hat{a}(\hat{a}^{\dagger})$ are the motional state
annihilation (creation) operators. This allows for an analytical solution for
the qubit state probabilities to be written[20]

$P_{\downarrow}(t)=\dfrac{1}{2}\Bigg{[}1+\sum_{n=0}^{\infty}P(n)\cos{(\Omega_{SB}\sqrt{n+1}t})\bigg{]}$
(24)

where $n$ is the oscillator Fock state, $t$ is the duration of the sideband
integration and $\Omega_{SB}=\eta\Omega_{carrier}$ is the sideband Rabi
frequency. This enables a general fit of the Fock state probabilities which
assumes no ideal model state. Due to the narrow frequency splitting of our
motional modes, the sideband interaction results in off-resonant driving of
our second mode which is separated in frequency from the first mode by only
$33$ kHz. The resulting interaction Hamiltonian is now more accurately written
as

$\hat{H}=\hbar\Omega/2(\hat{\sigma}^{+}\hat{a}^{\dagger}e^{i\delta_{1}t}+\hat{\sigma}^{-}\hat{a}e^{-i\delta_{1}t}+\hat{\sigma}^{+}\hat{b}^{\dagger}e^{i\delta_{2}t}+\hat{\sigma}^{-}\hat{b}e^{-i\delta_{2}t})$
(25)

where $\delta_{1}-\delta_{2}\approx 2\pi\times 33$ kHz. There is no known
analytical solution for the qubit probabilities under evolution of this
Hamiltonian therefore (24) cannot be used. In order to get a model free fit,
master equation simulations were performed using QuTiP [45] where each Fock
state probability in the inital state is a fit parameter while preserving the
trace. Estimates of fit uncertainties with this method proved the be a
challenge due to weak dependence on the off-resonant mode. Fitting was
performed starting with an ideal state and holding all Fock states except one
constant. The single Fock state was iterated through all Fock states for both
mode $a$ and $b$ which may produce an under estimate of the uncertainty. Data
where either mode $a$ or $b$ was driven resonantly was fit independently.
Figure 8 shows the results of the fit Fock state fits with the model state fit
to the probabilities for both modes independently and overlaid.

Figure 8: (a) Fits to the output of two-mode squeezing under interaction
described above. Blue bars indicate mode $a$ and red bars are mode $b$ with
associated error bars. Black line is a thermal state model fit to the Fock
state fits. (b) Fock state fits to output of the beam splitter after two-mode
squeezing. Blue data is mode $a$ and red is mode $b$ with associated error
bars. Black line is a squeezed state model fit to the Fock state fits.


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