Quantum-enhanced super-sensitivity of Mach–Zehnder interferometer using squeezed Kerr state

In the science and technology of metrology, the central task is to perform accurate measurements of certain parameters.¹ By exploiting the peculiar properties of quantum mechanics,^1–4 quantum metrology deals with the precision measurements of such parameters.⁵ For precise measurement of certain parameters, which are not measurable directly via conventional techniques, phase estimation^1–4 via optical interferometers plays an important role. In order to implement the phase measurement scheme, usually, SU(2) or SU(1,1) based interferometers⁶ are used. SU(2)-type interferometers, such as Michelson interferometer (MI) and Mach–Zehnder interferometer (MZI), are based on passive type beam splitters, while SU(1,1)-type interferometers are based on active elements, e.g., optical parametric amplifiers (OPAs), in place of the beam splitters.^6–9 

Theoretically and experimentally, it is found that the performance of the interferometer maximally depends on the input light sources.⁵ The performances of the interferometer, which depend maximally on the input light, in the ascending order would be thermal lights, coherent lights, and, maximally, nonclassical lights.^5,10

Nonclassical lights are a class of lights that are only understood by the quantum mechanical theories,¹¹ e.g., single photon state,¹² squeezed states,¹³ twin Fock states,¹⁴ Schrödinger’s cat states,^15–17 and N00N states.^18–22 The well-known combination of coherent and squeezed vacuum as the input states^23–25 became a famous choice for their good performance in the low as well as high-power range^26–28 of interferometry and, also, due to its very recent application in the gravitational wave detection.^29–33 Since the seminal work by Caves²³ four decades ago, squeezing-assisted optical interferometry^34,35 has become a centerpiece of theoretical^28,36–38 and experimental⁴ quantum metrology. A MZI injected with an intense light in the coherent state (which is an eigenstate of the annihilation operator, denoted as |β⟩) at one input port and the squeezed-vacuum state at the other input port can attain the phase sensitivity $Δϕ=e−r/n̄$⁠, where r(≥0) is the squeezing parameter, $n̄$ is the total average number of photons inside the interferometer, and ϕ is the relative phase shift between the two arms of the interferometer.²³ This scheme can beat the shot-noise limit (SNL), $ΔϕSNL=1/n̄$⁠, by an amount depending on the squeezing parameter r. To date, squeezing factors³⁹ of more than 10 dB have been observed in several experiments.^40–42 Interestingly, for a similar intensity, a MZI can achieve Heisenberg limit (HL), $ΔϕHL=1/n̄$⁠, for the coherent and squeezed-vacuum states at the inputs.⁴³ 

In order to achieve the HL, we must have squeezing in such amount that $e−r≈1/n̄$⁠. Experimentally, it is very tough for higher values of photons.^40–42 In 2000, Dowling group proposed a new type of state known as the N00N state¹⁸ by which one can easily attain the HL. However, for higher photon numbers, generation of N00N state is very much challenging.^44,45 So, this has led to open a new area of research having a significant amount of work in the optimization and generation of the nonclassical light.³⁵ In order to generate the nonclassical light, special types of nonlinear materials⁴⁶ and techniques⁴⁷ are being used. For example, parametric processes in second-order χ⁽²⁾ media generate squeezing and entanglement.⁴⁸ On the other hand, the Kerr effect occurring in third-order nonlinear χ⁽³⁾ media is being used to perceive quantum nondemolition measurements^49,50 and to generate quantum superpositions^15,51 as well as squeezing⁵² and entanglement.^53,54 The most common methods of generating nonclassical light are parametric downconversion (PDC),⁴⁸ four-wave mixing,^55,56 and the Kerr effect.⁵⁷ Unlike the former two approaches, squeezing via the Kerr interaction^58,59 is inherently phase-matched, which allows for flexibility in the wavelength of the probe light. These features meant that the utilization of the Kerr effect is a robust and flexible approach. The Kerr interaction requires high optical powers to reach sufficient nonlinearity, and this is commonly achieved by using ultrashort pulses.^60,61 However, this requires careful control of the pulses, since dispersion can act to spread out the pulse and, therefore, reduce the nonlinearity. Control of pulse spreading may be achieved by generating optical solitons, where the nonlinearity and dispersion are perfectly balanced.^58,62 For Kerr squeezing, the possibility of using materials such as optical fiber lends significant flexibility and it does not require a cavity⁵⁸ to enhance the strength of interaction as well as simplify the experimental requirements. For high nonlinearities and good transparency, chalcogenide- and tellurite-like glasses^63–65 are good candidates for optical fiber fabrications.

However, there is a problem in identifying the Kerr squeezing in direct detection experiments in the resulting fields. The nonlinear index of refraction of the medium modifies the phases of the number states in the initial coherent state, but photon statistics of the field remain Poissonian.⁶⁶ This produces a squeezing ellipse in phase space that is oriented in an oblique direction, neither in the direction of the phase nor in the direction of the amplitude quadrature.⁶⁷ Consequently, it has been difficult to identify Kerr squeezing. A method to observe Kerr squeezing in fibers was proposed by Kitagawa and Yamamoto.⁶⁸ They used an asymmetric Sagnac loop to displace the squeezing ellipse in phase space so that the short axis lined up with the amplitude quadrature.^59,69 Another method was proposed by Gerry and Grobe.⁶⁶ They applied the Kerr state to a two-photon parametric process, such as degenerate parametric downconversion or degenerate four-wave mixing. The resulting radiation is termed “squeezed Kerr states (SKS),” since the unitary evolution operator in the interaction picture for these processes is of the form of a squeeze operator.⁶⁶ It is shown that the effect of the Kerr medium on photon statistics becomes readily apparent by the application of the squeeze operator to a Kerr state. Statistics of SKS are sub-Poissonian as well as super-Poissonian, and its quadrature squeezing is improved. Furthermore, higher-order nonclassical properties of SKS have been studied by Mishra.⁷⁰ 

So, based on the nonclassical properties of the SKS,^66,70 we are motivated to study the improvement in phase sensitivity by using the SKS at the input of MZI. We will also compare our results with previous studies and results. To the best of our knowledge, no such study has been done to date.

The total phase shift in MZI arms is ϕ = ϕ_us + ϕ_es, where ϕ_us is the phase change due to unknown sources and ϕ_es is the phase change due to controllable experimental setup. However, ϕ_us ≪ ϕ_es, so we can write ϕ ≈ ϕ_es, and for simplicity, throughout the paper, we can ignore the suffix of ϕ_es and denote it as ϕ. Therefore, for precise observation, one must adjust the phase difference between the arms of the MZI nearest to ϕ. We employ the single intensity detection (SID), intensity difference detection (IDD), and homodyne detection (HD) schemes,^26,37 and we calculate the optimal solutions of phase sensitivity for all these detection schemes. In this paper, we are focused on the single parameter case, i.e., the phase change only in one arm of the interferometer. So, with the help of the single parameter quantum Fisher information (QFI) and its associated quantum Cramér–Rao bound (QCRB),^28,71,72 we analyze the better performance in our MZI setup.

This paper is organized as follows: In Sec. II, we discuss the basics of interferometry by using different types of detection schemes and briefly discuss the SKS. Section III describes the phase sensitivity of MZI with SKS and coherent states as the inputs of MZI under the lossless condition. Section IV describes the phase sensitivity of MZI with SKS and coherent states as the inputs of MZI under lossy conditions. In Sec. V, we conclude our results.

Here, we will discuss the basics of phase estimation and parameter estimation with MZI and the generation of SKS. Here, we will see the operator transformations under the MZI action and the method of standard error propagation formula to calculate the phase sensitivity of the interferometer for different detection schemes. We will discuss the lower bound in phase sensitivity by considering the single parameter quantum Fisher information (QFI) and the corresponding quantum Cramér–Rao bound (QCRB). Furthermore, we will review the interaction of coherent light with Kerr media followed by the squeezer.

Here, we will discuss the phase sensitivity of the MZI using the coherent state, |α⟩, and SKS, |ψ_SK⟩, as the inputs in the 1st and 2nd input ports, respectively (Fig. 1). The phase sensitivity, i.e., Δϕ, for different detection schemes are functions of six different variables (ϕ, α, β, θ, γ, and r). In order to optimize the parameters, we divide our results into different subsections by considering the special cases.

Note that, throughout the calculations, without loss of generality, we take α = |α|, β = |β|, and θ = π, since, analytically, we found that θ = π gives better results in comparison with other values of θ in all the cases.

The central task of our work is to find the effect of Kerr nonlinearity (in terms of γ) and Kerr nonlinearity with squeezing parameter (r) on the Δϕ of the MZI for three different detection schemes. So, in each case, we will try to see the variation of Δϕ with γ. To be more clear, we divide our discussions into two parts: (i) |ψ⟩_in = |0⟩₁ ⊗ |ψ_SK⟩₂; (ii)|ψ⟩_in = |α⟩₁ ⊗ |ψ_SK⟩₂.

From Eq. (34), we find that for r = 0, Δϕ_hd depends on γ, as one can see in Fig. 3. So, for the HD scheme with γ ≠ 0, the optimum value of ϕ varies with |β|. We analytically found that for a wide range of values of |β|(∼1–100), the optimum value of ϕ is approximately 7π/4. So, it is interesting to mention that with the optimum value of ϕ = 7π/4, Δϕ_hd beats SNL for some non-zero values of γ, keeping the first input port as vacuum (Fig. 3). This is in agreement with the recent study of Takeoka et al.³⁸ that a system working with a vacuum state in one port and a nonclassical state on the other port can beat the SNL if only one of the arms of the MZI has an unknown phase shift (i.e., single parameter estimation case) and the detector uses any external phase reference and power resource during the detection process. In our work, we are taking the Kerr state as a nonclassical state⁶⁷ and we are not ignoring the global phase factor as is obvious in Eqs. (2) and (3). If we ignore the global phase factor, the phase sensitivity never beats the SNL; we can see this in Fig. 4. This means that the global phase factor acts as an external phase source for the HD scheme and beating of the SNL, in Fig. 3, is not the violation of the “no-go theorem.”^23,38

We note that for the case of the HD scheme, the normalized phase sensitivity [(Δϕ)/SNL] cannot reach up to 1 for γ = 0 (Fig. 3). This is because, here, we plot the graph at ϕ = 7π/4, while the optimal value of ϕ is π/2 for the case of γ = 0. Similar cases arise for Figs. 5 and 6.

Case (ii): The r ≠ 0 case, i.e., |ψ⟩_in = |0⟩₁ ⊗ |ψ_SK⟩₂. From Eqs. (33) and (B4), we find that Δϕ_sid and Δϕ_idd are dependent on γ. This means that the Kerr medium plays a role in the variation of phase sensitivity in the case of the SID and IDD schemes when r ≠ 0, but it should be noted that still, phase sensitivity only saturates the SNL (Figs. 5 and 6). In order to visualize the effect of γ on Δϕ, we consider two values (lower and higher energies) of |β| = 5 and 100 and see the variation of Δϕ/Δϕ_SNL with γ for different values of the squeezing parameter r (plots are shown in Figs. 5 and 6). In Figs. 5 and 6, we can see that phase sensitivity for SID and IDD saturates the SNL for all values of γ. Meanwhile, a significant enhancement in phase sensitivity occurs for the HD scheme. We also plot Δϕ_QCRB/Δϕ_SNL and Δϕ_HL/Δϕ_SNL, and we can see that for some values of γ, phase sensitivity for the HD scheme saturates QCRB and approaches HL. As we can see, phase sensitivity is enhanced with an increase in r, so to enhance the precision, one can use the higher squeezing for a better phase sensitivity. It is important to mention here that the current record for the squeezing factor (15.3 dB or r = 1.7) is reported in Ref. 41.

Furthermore, in Fig. 3 and also in Figs. 5 and 6, we can see that the maximum phase sensitivity depends on |β| and on γ in case of the HD scheme. As |β| changes, the corresponding optimal value of γ for which we get the maximum phase sensitivity also changes. So, in order to explore the effect of |β| and γ on Δϕ_hd, we plot a graph between |β| and γ and show the variation in phase sensitivity via color change in the graph (Fig. 7). Figure 7 gives two important results from an experimental point of view: (i) an enhancement in Δϕ_hd (phase sensitivity) with an increase in the value of |β|, and (ii) for higher values of |β|, a decrease in the optimal value of γ. The decrease in the value of γ means that the interaction time of light with the Kerr medium decreases [Eq. (19)]. Thus, we can say that the Kerr medium plays an important role in the enhancement of the phase sensitivity with the HD scheme.

For the case of SKS and coherent state as the inputs of the MZI, i.e., |ψ⟩_in = |α⟩₁ ⊗ |ψ_SK⟩₂, the expressions of the corresponding phase sensitivity associated with the SID, IDD, and HD schemes are given in Eqs. (27)–(29), respectively.

Since Δϕ is dependent on the input number of photons, N = g₁ + g₂, and N depends on the parameters |α|, |β|, γ, θ, and r, we can figure out the role of these parameters on Δϕ by looking the variation of N with these parameters. We have seen in previous cases that the phase sensitivity is better for θ = π in comparison with the other values of θ and, so, here we consider θ = π. Now, we see the variation in N with γ for different values of r in three different cases by considering low and high energy limits at the input, viz., (i) |α| = 3 and |β| = 2, (ii) |α| = 100 and |β| = 2, and (iii) |α| = 100 and |β| = 100, as shown in Fig. 8. We find that N is independent of γ in the case of r = 0, but for the case r ≠ 0, a rapid growth in N is seen as γ is increasing. This is because the nonlinear index of refraction of the medium modifies the phases of the number states in the initial coherent state, but photon statistics of the field remain Poissonian.⁶⁶ The effect of the Kerr medium on photon statistics becomes readily apparent by application of the squeeze operator.⁶⁶ As we can see that, in Fig. 8, in the case of squeezed coherent state, |ψ_S⟩ (r ≠ 0 and γ = 0), the photon number decreases, while for γ ≠ 0, it varies rapidly. That is, variation in the input number of photons, N, is dependent on the interaction of photons in the Kerr medium.

As a particular case, let us consider two cases: (i) r = 0, i.e., |ψ⟩_in = |α⟩₁ ⊗ |ψ_K⟩₂ (when the second input is the Kerr state), and (ii) r ≠ 0, i.e., |ψ⟩_in = |α⟩₁ ⊗ |ψ_SK⟩₂ (when the second input is the squeezed Kerr state).

It is obvious from Eqs. (27)–(29) that for r = 0, Δϕ still depends on |α|, |β|, γ, and ϕ. In the case of |α| = 0, the calculation of the optimal value of ϕ for different detection cases was straightforward, but, here, it is relatively hard. So, keeping in mind the lower and high energy inputs, we consider four cases: (i) |α| = 3 and |β| = 2, (ii) |α| = 50 and |β| = 2, (iii) |α| = 3 and |β| = 50, and (iv) |α| = 50 and |β| = 50, and plot the ϕ vs γ graph as shown in Figs. 9–12, respectively, where color variation shows Δϕ/Δϕ_SNL. We can see that the phase sensitivity is better in the IDD scheme as compared to the SID and HD schemes having optimal phase 3π/4 or 7π/4 in case (i), as shown in Fig. 9. In cases (ii)–(iv), all three detection schemes exhibit a better phase sensitivity with different optimal phases, as shown in Figs. 10–12, respectively. Multiple color regions in the phase sensitivity pattern in cases (iii) and (iv) are because of the fluctuation in N, as previously we saw in Fig. 8(c).

So, in conclusion, we find that, in enhancing the phase sensitivity of the interferometer, the coherent input with the Kerr state is more useful than the double coherent input, in approximately all the situations.

Here, we explore Eqs. (27)–(29) by considering r ≠ 0. For a better phase sensitivity, the optimal value of ϕ depends on the parameters |α|, |β|, γ, and r similar to Sec. III B 1. In order to explore the effect of γ on Δϕ, for the three detection schemes, we find a nearly optimal value of ϕ by using the analytical method. Analytically, we find that for the cases (i) |α| = 3, |β| = 2; (ii) |α| = 50, |β| = 2; (iii) |α| = 50, |β| = 50; and (iv) |α| = 3, |β| = 50, the optimal phases ϕ_opt for the SID scheme will be 9π/8, π/4, 9π/8, and 9π/8, respectively. However, all these four cases have ϕ_opt = π/2 for the IDD scheme and ϕ_opt = 0 for the HD scheme. Since we find that squeezing triggers the photon enhancement in the Kerr medium at several instances, as shown in Fig. 8, so here we choose the value of r = 1.5 for our convenience. So, we plot Δϕ/Δϕ_SNL vs γ by taking ϕ_opt for r = 1.5. Figures 13–16 show the phase sensitivity for the four cases (i)–(iv), respectively. We can see that the Kerr medium remarkably enhances the phase sensitivity. If we look at the performance of the three detection schemes, we find that the HD scheme is dominant in all four cases than the IDD scheme, which, in turn, is doing better than the SID scheme.

On comparison of the four cases (i)–(iv), we find that increased values of |α| and |β| enhance the phase sensitivity, but it is important to note that variation in |β| affects more than that for |α|. As we can see in Figs. 15 and 16, the increase in |α| is less effective for the larger values of |β|. On the other hand, the increase in |β| gets more effective even though |α| is large enough (Figs. 14 and 15).

Here, we see the effect of internal and external losses on the phase sensitivity of the interferometer and the behavior of the factor γ under realistic scenarios. So, for this purpose, we consider the cases of best performances under lossless conditions (Sec. III).

We can see that Eqs. (41)–(43) depict that internal (μ) and external (η) losses show an equal effect on the phase sensitivity in the case of the SID, IDD, and HD schemes, respectively. So, we can consider either internal or external loss and can see the variation in Δϕ with γ.

Now, we are considering the case in which the input state is |ψ⟩_in = |α⟩₁ ⊗ |ψ_SK⟩₂. The phase sensitivity for this case is given in Eqs. (41)–(43) for all three cases. Figures 23 and 24 show the variation of Δϕ/Δϕ_SNL with γ for r = 0 and r = 1.5, respectively. For the Kerr state, in Fig. 23, we can see that in all three cases, we can surpass the SNL in the lossy case (for $<30%$ loss) for some values of γ. If we take SKS, we can surpass the SNL even for more than 40% photon loss in the HD scheme (Fig. 24) and the IDD scheme gives a phase sensitivity below the SNL for more than 20% photon loss. For the SID scheme, even for lossless cases, we cannot beat the SNL.

We studied the phase super-sensitivity of a MZI using the SKS as one of the inputs. We studied the effect of the Kerr medium on the phase sensitivity of a MZI with squeezing (i.e., |ψ_SK⟩) and without squeezing state (i.e., |ψ_K⟩). We found several conditions under which the SKS gives a better sensitivity than other combinations of input states, viz., coherent plus squeezed vacuum state, coherent plus vacuum state, and coherent plus coherent state. We discuss our results in Secs. III and IV for lossless and lossy conditions, respectively. These sections are further divided into subsections based on the input combinations of the light.

To conclude the results found in Secs. III and IV, we have made Tables I and II, respectively. These tables contain the approximated best values of Δϕ/Δϕ_SNL for different cases from their respective plots for all three detection schemes, namely, SID, IDD, and HD, in order to conclude the results step by step.