† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11474095, 11654005, and 11234003) and the National Key Research and Development Program of China (Grant No. 2016YFA0302000).
We theoretically study the quantum Fisher information (QFI) of the SU(1,1) interferometer with phase shifts in two arms by coherent ⊗ squeezed vacuum state input, and give the comparison with the result of phase shift only in one arm. Different from the traditional Mach–Zehnder interferometer, the QFI of single-arm case for an SU(1,1) interferometer can be slightly higher or lower than that of two-arm case, which depends on the intensities of the two arms of the interferometer. For coherent ⊗ squeezed vacuum state input with a fixed mean photon number, the optimal sensitivity is achieved with a squeezed vacuum input in one mode and the vacuum input in the other.
Quantum enhanced metrology which has received a lot of attention in recent years is the use of quantum measurement techniques to obtain higher statistical precision than purely classical approaches.[1–16] Mach–Zehnder interferometer (MZI) and its variants were used as a generic model to realize highly precise estimation of phase. In order to achieve the ultimate lower bounds,[17,18] much work has been devoted to finding the methods to improve the sensitivity of phase estimation, such as (i) using the nonclassical input states (quantum resources): squeezed states[3,19,20] and NOON states,[21,22] (ii) using the new detection methods: homodyne detection[23,24] and parity detection,[25–28] and (iii) using the nonlinear processes: amplitude amplification[29] and phase magnification.[13] Here we focus on the nonlinear amplitude amplification process to improve the sensitivity. In 1986, Yurke et al.[29] introduced a new type of interferometer where two nonlinear beam splitters (NBSs) take the place of two linear beam splitters (BSs) in the traditional MZI. It is also called the SU(1,1) interferometer because it is described by the SU(1,1) group, as opposed to the traditional SU(2) MZI for BS. The detailed quantum statistics of the twomode SU(1,1) interferometer was studied by Leonhardt.[30] The SU(1,1) phase states were also studied theoretically in quantum measurements for phase-shift estimation.[31,32] Furthermore, the SU(1,1)-type interferometers have been reported by different groups using different systems in theory and experiment, such as all optical arms,[33–36] all atomic arms,[37–39] atom-light hybrid arms,[40–45] light-circuit quantum electrodynamics system hybrid arms,[46] and all mechanical modes arms.[47] These SU(1,1)-type interferometers provide different methods for basic measurement.
At present, many researchers are focusing on how to measure the phase sensitivities, where several detection schemes have been presented.[23,28,36] In general, it is difficult to optimize all the detection schemes to obtain the optimal estimation protocol. However, the quantum Fisher information (QFI)[4,5] characterizes the maximum amount of information that can be extracted from quantum experiments about an unknown parameter (e.g., phase shift ϕ) using the best (and ideal) measurement device. Therefore, the lower bounds in quantum metrology can be obtained by using the method of the QFI. In recent years, many efforts were made to obtain the QFI of different measurement systems.[48–66] For the SU(1,1) interferometers with phase shift only in one arm, the QFI with coherent states input was studied by Sparaciari et al.,[60,62] and the QFI with coherent ⊗ squeezed vacuum state input was presented by some of us.[28] Nevertheless in some measurement schemes, phase shifts in two arms are required to measure. For example, the phase sensitivity of phase shifts in two arms for the SU(1,1) interferometer with coherent states input was experimentally studied by Linnemann et al.[37] Jarzyna et al. studied the QFIs of phase shifts in the two-arm case for a MZI, and presented the relationship with the result of phase shift in the single-arm case.[52] Since phase shift in the single arm is not simply equivalent to the phase shifts in two arms where one phase shift of them is 0, the QFIs of phase shifts in two arms for an SU(1,1) interferometer needed to be researched. In this paper, we study the QFI of SU(1,1) interferometer of phase shifts in two arms with two coherent states input and coherent ⊗ squeezed vacuum state input, and give the comparison with the result of phase shift only in one arm. These results should be useful for some phase measurement processes.
The QFI
In an SU(1,1) interferometer, the NBSs take the place of the BSs in the traditional MZI shown in Fig.
Secondly, we describe the phase shifts process. Different from the BS, the NBS involves three light fields where the pump field is classical and with a classical reference phase. The uncertainty of classical pump field
Now, we give the QFIs with different input states under the condition of phase shifts in two arms. From Eq. (
Next, we consider a coherent light combined with a squeezed vacuum light as the input ⎸Ψin〉 =⎸α〉a ⊗ ⎸0,ς〉b (
So far, we have given the QFI of SU(1,1) interferometer where the phase shifts in the two arms, and they as well as the QFIs with phase shift in the one arm case are summarized in the Table
Whatever the measurement chosen, the QCRB can give the lower bound for the phase measurement[4,5,48–50]
For coherent ⊗ squeezed vacuum state input, η is equal to sinh2r/Nin (Nin = Nα+ sinh2r), where the parameter η can be used to label the squeezing fraction of the mean photon number. When η = 0 or η = 1, the input state is only a coherent state ⎸α〉a or only a squeezed vacuum state ⎸0,ζ〉b. When 0 < η < 1, the input state is a coherent ⊗ squeezed vacuum state. For coherent ⊗ squeezed vacuum state input case, only the squeezed vacuum light as input and without the coherent state, the phase sensitivity is the highest shown in Fig.
For the lossy interferometers, the pure states evolve into the mixed states and the QFI will be reduced. However, the QFI of pure state puts an upper bound on that of mixed state. Here, we focus on the maximal QFI of the SU(1,1) interferometer, and then we ignore the losses in the interferometer.
The analytical expressions of QFI for an SU(1,1) interferometer with two coherent states and coherent ⊗ squeezed vacuum state inputs have been derived. For single-arm case, the QCRBs of phase shift in upper arm and in lower arm are slightly different because the intensities in two interferometric arms are asymmetric. The phase sensitivities of phase shifts between the single-arm case and two-arm case are also compared. The QCRB of single-arm case can be slightly higher or lower than that of two-arm case, which depends on the intensities of the two arms of the interferometer. For coherent state ⊗ squeezed vacuum state input with a definite input number of photons, the optimal condition to obtain the highest phase sensitivity is a squeezed vacuum in one mode and the vacuum state in the other mode.
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