Phase sensitivity of an $\operatorname{SU}(1,1)$ interferometer via product detection

EPJ Quantum Technology

Table 1 Phase sensitivities with various detections and QCRB of an $\operatorname{SU}(1,1)$ interferometer with different input states

Input states	Product	Intensity	Parity	Homodyne	QCRB
\|0〉⊗\|0〉	$1/\kappa ^{1/2}$	$1/\kappa ^{1/2}$	$1/\kappa ^{1/2}$	Not available [42]	$1/\kappa ^{1/2}$
\|α〉⊗\|0〉	$1/[\kappa (N_{a}+1)]^{1/2}$	$\Delta \phi _{\mathrm{onecoh}}^{\mathrm{I}}$^∗	$1/[\kappa (N_{a}+1)]^{1/2}$	$1/(\kappa N_{a})^{1/2}$	$1/[\kappa (2 N_{a}+1) +2N_{a}(N_{\mathrm{PA}}+2)]^{1/2}$
$\vert \alpha e^{i\eta _{1}}\rangle \otimes \vert \alpha e^{i\eta _{2}}\rangle $^†	$1/\{2N_{a}[(N_{\mathrm{PA}}+2)(\sqrt{\kappa }+1)+\kappa ]+\kappa \}^{1/2}$	$1/(2\kappa N_{a})^{1/2}$	Ref. [42]	$\simeq 1/[4\kappa N_{a}]^{1/2}$	$1/\{4N_{a}[(N_{\mathrm{PA}}+1)\sqrt{\kappa }+\kappa +1]+\kappa \}^{1/2}$
\|α〉⊗Ŝ(r)\|0〉	$1/[\kappa (N_{a}e^{2r}+\cosh r e^{r})]^{1/2}$	$\Delta \phi _{\mathrm{cohsqz}}^{\mathrm{I}}$^‡	$1/[\kappa (N_{a}e^{2r}+\cosh ^{2}r)]^{1/2}$	$1/[\kappa N_{a}e^{2r}]^{1/2}$	$1/[2N_{a}(N_{\mathrm{PA}}+2)+N_{\mathrm{PA}}^{2}\sinh ^{2}(2r)/2+\kappa (2N_{a}\cosh r e^{r}+\cosh ^{2} r)]^{1/2}$
Ŝ(r)\|0〉⊗Ŝ(r)\|0〉	$1/[\kappa (2N_{s}+1)]^{1/2}$	$\Delta \phi _{\mathrm{twosqz}}^{\mathrm{I}}$^§	$1/[\kappa (2N_{s}+1)]^{1/2}$	Not available	$1/[(1+N_{\mathrm{PA}})^{2}\times \cosh 4r -1]^{1/2}$

Input states	Product	Intensity	Parity	Homodyne	QCRB
\|0〉⊗\|0〉	\(1/\kappa ^{1/2}\)	\(1/\kappa ^{1/2}\)	\(1/\kappa ^{1/2}\)	Not available [42]	\(1/\kappa ^{1/2}\)
\|α〉⊗\|0〉	\(1/[\kappa (N_{a}+1)]^{1/2}\)	\(\Delta \phi _{\mathrm{onecoh}}^{\mathrm{I}}\)^∗	\(1/[\kappa (N_{a}+1)]^{1/2}\)	\(1/(\kappa N_{a})^{1/2}\)	\(1/[\kappa (2 N_{a}+1) +2N_{a}(N_{\mathrm{PA}}+2)]^{1/2}\)
\(\vert \alpha e^{i\eta _{1}}\rangle \otimes \vert \alpha e^{i\eta _{2}}\rangle \)^†	\(1/\{2N_{a}[(N_{\mathrm{PA}}+2)(\sqrt{\kappa }+1)+\kappa ]+\kappa \}^{1/2}\)	\(1/(2\kappa N_{a})^{1/2}\)	Ref. [42]	\(\simeq 1/[4\kappa N_{a}]^{1/2}\)	\(1/\{4N_{a}[(N_{\mathrm{PA}}+1)\sqrt{\kappa }+\kappa +1]+\kappa \}^{1/2}\)
\|α〉⊗Ŝ(r)\|0〉	\(1/[\kappa (N_{a}e^{2r}+\cosh r e^{r})]^{1/2}\)	\(\Delta \phi _{\mathrm{cohsqz}}^{\mathrm{I}}\)^‡	\(1/[\kappa (N_{a}e^{2r}+\cosh ^{2}r)]^{1/2}\)	\(1/[\kappa N_{a}e^{2r}]^{1/2}\)	\(1/[2N_{a}(N_{\mathrm{PA}}+2)+N_{\mathrm{PA}}^{2}\sinh ^{2}(2r)/2+\kappa (2N_{a}\cosh r e^{r}+\cosh ^{2} r)]^{1/2}\)
Ŝ(r)\|0〉⊗Ŝ(r)\|0〉	\(1/[\kappa (2N_{s}+1)]^{1/2}\)	\(\Delta \phi _{\mathrm{twosqz}}^{\mathrm{I}}\)^§	\(1/[\kappa (2N_{s}+1)]^{1/2}\)	Not available	\(1/[(1+N_{\mathrm{PA}})^{2}\times \cosh 4r -1]^{1/2}\)