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Xie Duan, Chen Haifeng. Phase precision of Mach–Zehnder interferometer in PM2.5 air pollution. Chinese Physics B, 2018, 27(7): 070304  
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Phase precision of Mach–Zehnder interferometer in PM2.5 air pollution
Xie Duan, Chen Haifeng
School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

 

† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant No. 61306131) and the Science Foundation of Shaanxi Provincial Department of Education, China (Grant No. 14JK1682).

Abstract

This paper theoretically explores the effect of PM2.5 air pollution on the phase precision of a Mach–Zehnder interferometer. With the increasing of PM2.5 concentration, phase precision for inputs of coherent state & vacuum state and inputs of coherent state & squeezed vacuum state will gradually decrease and be lower than the standard quantum limit. When the value of relative humidity is increasing, the precision of two input cases is decreasing much faster. We also find that the precision for inputs of coherent state & squeezed state is better than that of coherent state & vacuum state when PM2.5 concentration is lower. As PM2.5 concentration increases, the precision for inputs of coherent state & squeezed state decreases faster, and then the two precisions tend to be the same while the concentration is higher.

PACS: 03.67.-a;03.65.Ta
Keyword:Mach-Zehnder interferometer;PM2.5;phase precision
1. Introduction

Optical interferometry is becoming increasingly important in modern physics. In a classical optical system, the phase precision can in principle be proportional to , commonly referred to as the standard quantum limit (SQL) or the shot-noise limit. Compared with its classical counterpart, quantum-enhanced interferometry is used to estimate phases beyond the shot-noise (or standard quantum) limit.[1–3] Quantum-enhanced interferometry was initiated by Caves, who first suggested how to reach a sub-shot-noise sensitivity by using coherent and squeezed-vacuum light as the input of a Mach–Zehnder interferometer.[4] Using Caves’ scheme, the phase sensitivity is bounded by Δφ∼1/N3/4.[5] Subsequently, some experiments proved that the precision could exceed the standard limit by using Caves’ scheme.[6–8] In addition to squeezed vacuum states, some studies found that by using other non-classical quantum states, such as a spin squeezing state,[9,10] a NOON state,[11,12] and an entangled coherent state,[13,14] the phase precision could achieve the Heisenberg limit.

With the development of industry and the growth of populations, air pollution is becoming more and more serious. In July 1997, the United States Environmental Protection Agency (EPA) promulgated new primary and secondary national ambient air quality standards for PM2.5 (particulate matter less than 2.5 μm). The diameter of PM2.5 particles are very small and have long residence time in the atmosphere. In free space, photon loss comes from the light scattering or absorption by the abundant fine particles of atmospheric aerosol, which will reduce the visibility.[15,16] The ability of light scattering is strongest when the diameter of fine particles is close to the light wavelength.[17] In quantum communication, the light wavelength is about 0.8 μm. So PM2.5 particles are mainly responsible for photon attenuation. In optical interferometry, photonic loss is the main obstacle to the practical implementation of quantum enhanced protocols, which will reduce the estimated precision and degrade the performance of non-classical quantum states.[18–20] However, research into the relationship between PM2.5 air pollution and the phase precision of optical interferometry has not yet been started. In this paper, we will study the effect of PM2.5 air pollution on the phase precision of the Mach–Zehnder interferometer.

We focus on the phase precision of the two input cases under the influence of PM2.5 air pollution. In the first input case, one input port of the Mach–Zehnder interferometer is not excited, that is to say, the port is fed with a vacuum state. The other input port is fed with an important classical state, a coherent state. In the other input case, one input state is a coherent state, and the other input state is a non-classical state, a squeezed vacuum state. This kind of strategy is relatively easy to realize in current technology. Other non-classical states, such as the NOON state and spin squeezing state with a definite photon number, are notoriously hard to be realized. For example, researchers can only prepare the NOON state with a photon number N of no more than 10 to this day.[21,22] So we are not concerned with those non-classical states in this paper.

2. Phase precision of two input cases in noiseless case

In this section, we will calculate the precision of two input cases without photon loss. As shown in Fig. 1, BS1 and BS2 are two 50/50 beam splitters in the Mach–Zehnder interferometer. Two incoming light beams are injected from both ports of BS1 and then recombined by a second beam splitter BS2. The phase difference between two arms is ϕ. We assume that the two beams acquire −π/2 and π/2 phase when passing through BS1 and BS2, respectively.

Fig. 1. Schematic diagram of the Mach–Zehnder interferometer.

Let |in⟩ be the initial state entering the interferometer and |out⟩ be the final state leaving the interferometer. Then the relationship between two states is[23]

where Jx and Jz denote x and y components of the total angular momentum J, and . The photon number difference between the two out ports is n1 −n2 = 2Jz. The average of the Jz operator is
It was found that
Hence the variance of Jz is
where
The phase precision can be calculated through
If the two input ports of the Mach–Zehnder interferometer are fed with a coherent state |α⟩ and a vacuum state |0⟩, respectively, in other words, the input state is |in⟩ = |α⟩ |0⟩, we find Thus the precision is
Since the mean photon number of input states is N = |α|2, we acquire the last term of Eq. (7). When φ = π/2 or 3π/2, the optimal precision is achieved, which is

The input state |in⟩ = |α⟩ |r⟩ represents a coherent state entering one port and a squeezed vacuum state entering the other. Thus The phase precision is

When φ = π/2 or 3π/2, the phase precision is expressed as
The optimal result can be obtained numerically. When the photon number of squeezed vacuum states is smaller than mean photon number N and approaches , then[5]
where the mean photon number is given by N = |α|2 + sinh2r. Mixing a coherent state with a squeezed state provides a precision higher than the standard quantum limit.

Figure 2 indicates the precision of two input cases. When a small amount of the squeezed vacuum state is mixed with the coherent state, the precision is better than the precision for inputs of coherent state & vacuum state.

Fig. 2. (color online) Phase precision of the two input cases in noiseless case.
3. Quantum state in the presence of noise

That PM2.5 fine particles scatter or absorb photons can be regarded as that the quantum system suffers from unwanted interactions with the environment. These unwanted interactions show up as noise in quantum information processing systems. Supposing the input ports and the output ports are in a clean indoor environment, PM2.5 particles always exist in the free space during the transmission. So we focus on the precision influenced by photon loss in the stage of transmission. Figure 3 indicates where photon loss may appear (gray area).

Fig. 3. Possible positions where PM2.5 fine particles can appear during the transmission.

Mathematically, the interaction of a quantum system with its environment can be expressed as some set of {Ek}.[24] The output state due to the interaction can be described as

Most quantum states are not single qubit states. However, these states possess symmetry under particle exchange, so
where P is an arbitrary permutation of N indices. Taking into account the symmetry property, an l-mode Fock state |n⟩ = |n1⟩ |n2⟩...|nl⟩ with N = n1 + n2 + ··· + nl indistinguishable particles can be expressed as the particle description[25,26]
where l1, l2,..., lnl indicates nl indices (or particles) in mode l and is the summation over all permutations of the indices (or particles) inside the brackets. Because any quantum state can be constructed by a Fock state |{n}⟩, we can translate all quantum states into a particle description.

In the particle description of quantum state ρin with definite N photon number, the photons it brings can be treated as distinguishable particles. The Kraus operators can act independently on each particle. So the initial state ρin under the influence of quantum noise is written as

where Eki denotes the Kraus operator Ek for the i-th particle (photon).

Considering the quantum noise, figure 4 indicates the process of the initial state ρin traveling through a Mach–Zehnder interferometer. The transformation means the i-th photon of the initial state ρin traveling through BS1, denotes the phase shift, represents the i-th photon traveling through BS2, and ξ is the quantum noise acting on the photon during the transmission. Symbols and are the Pauli x and the Pauli z matrices acting on the i-th qubit. The combined operation of all photons is a tensor product

where , , and . So
Because and , the tensor product of the phase shift operation can be written as
and the transformations of BS1 and BS2 are
Inserting Eqs. (19)–(21) into Eq. (18) and getting rid of the operators {Ek} in Eq. (18), we find Eq. (18) to be similar to Eq. (1) in the noiseless case.

Fig. 4. Process of each photons in noisy Mach–Zehnder interferometer during the transmission.

The average value of an observable O can be expressed by

where
If the observable O = Jz, we can derive the phase precision of definite N photon number states in a noisy interferometer.

In Eqs. (8) and (12), we find the precision scale as 1/N1/2 or 1/N3/4, where N denotes the mean photon number for the input of coherent state & vacuum state or that of coherent state & squeezed vacuum state. Obviously, the phase bound is related with mean photon number. While calculating the phase precision, we consider those indefinite photon number states with mean photon number N as a quantum state with a definite photon number N. Thus, the above-mentioned calculation for definite photon number state can apply to indefinite photon number states, such as inputs of coherent state & vacuum state, or to inputs of coherent state & squeezed vacuum state.

4. Effects of PM2.5 on the phase precision

The IMPROVE project established empirical formulas[27] to evaluate the contribution of bulk PM2.5 chemical species to the total light extinction

where bext is the light extinction coefficient, expressed in inverse megameters (Mm−1). Species concentrations shown in brackets are in microgram per meter cubed (μg/m3). OMC, LAC, SOIL, and CM represent organic carbon mass, light-absorbing carbon, fine soil, and coarse mass. 10 is the extinction from gas molecules. f(RH) is the water growth factor as a function of relative humidity (RH). Most aerosol species, such as gas molecules, (NH4)2SO4, and NH4NO3, can contribute to light scattering. While other species, such as LAC, can cause light absorption. Some research has found that CM has a very small contribution to bext.[28,29] So the equation (24) can be rewritten as
where P1 to P4 indicate the proportion of (NH4)2SO4 and NH4NO3, OMC, LAC, and SOIL in the total concentration of PM2.5.

We can evaluate the average value of Pi from data samples from November 2013 to November 2014 in Hangzhou, China.[30] The Pi values are shown in Table 1. The value f(RH) varies with RH, as shown in Table 2.[31]

Table 1.

The average proportion of each species in the total concentration of PM2.5.

.
Table 2.

Statistical summary of mean f(RH) values in selected relative humidity ranges.

.

Due to light scattering or absorption by the abundant fine particles in PM2.5, there is photon loss when the quantum state is transmitted in free space. The attenuation of light propagation through free space according to Beer’s law can be described as[32]

where I0 is the initial light intensity and I is light intensity after it passes through the distance L. Influenced by PM2.5 air pollution, the probability η of losing a photon is
Amplitude damping describes the phenomenon of energy dissipation. The operation elements are
E1 operation changes state |1⟩ into state |0⟩ with probability η, corresponding to the physical process of losing a photon to the environment. E0 leaves |1⟩ unchanged with probability 1−η, which means that the photon is not lost.

For the Jz observable, there is

so
whereIis the identity matrix. Using Eq. (23) and Eqs. (28)–(31), we can conclude that
If the two input ports are fed into a coherent state and a vacuum state,
When ϕ = π/2 or 3π/2, the optimal phase precision is
Similarly, if the input states are |in⟩ = |α⟩ |r⟩, we obtain
Obviously, the numerator is the smallest when φ = π/2 or 3π/2, and equation (36) becomes

Substituting Eq. (27) into Eq. (35) and Eq. (37), we can find the relationship between phase precision and PM2.5 air pollution.

5. Comparison and discussion

As shown in Fig. 5, the precisions of two input cases in the presence of PM2.5 get worse as the PM2.5 concentration gets bigger. When the PM2.5 concentration is close to zero, the photon loss mainly comes from the scattering of gas molecules. Because the scattering effect of gas molecules is relatively weak, the phase precision is close to the precision of the non-loss case. With PM2.5 concentration continuously increasing, the fine particles, such as organic carbon mass, light-absorbing carbon, fine soil, and coarse mass increase. Due to the light scattering or absorbing from the abundant fine particles, the precision gets worse as PM2.5 concentration increases. As seen in Eq. (26), long distance transmission will accelerate light attenuation. The precision via longer transmission distance L = 1 km is worse than precision through transmission distance L = 0.1 km.

Fig. 5. (color online) Phase precisions influenced by PM2.5 for inputs of (a) coherent state & vacuum state (N = 100) and (b) coherent state & squeezed state (N = 100).

Except for transmission distance and PM2.5 concentration, relative humidity can also impact on the phase precision. Some species, such as (NH4)2SO4 and NH4NO3, have the property of moisture absorption. With RH value increasing, the particle sizes of (NH4)2SO4 and NH4NO3 increase by aerosol hygroscopic growth. Some research has found that the hygroscopic growth speed of aerosol particles with size 0.01–1 μm is faster and will finally increase to size 0.5–1.5 μm.[33] In quantum communication, the photon wavelength is about 0.8 μm. As the diameters of most particles are near photon wavelength, the scattering ability of PM2.5 was enhanced. Figure 6 shows that the phase precision decreases faster when the relative humidity is higher.

Fig. 6. (color online) Phase precisions in the presence of different relative humidity for inputs of (a) coherent state & vacuum state (N = 100) and (b) coherent state & squeezed state (N = 100).

In Hangzhou 2014, the lowest, highest, and annual average relative humidities are 51%, 81%, and 68%, respectively. As seen in Fig. 7, the precision changing trends of the above three cases of relative humidity are the same. At first, when PM2.5 concentration is not big, the precisions for inputs of coherent state & squeezed state are better than those for inputs of coherent state & vacuum state. Finally, when the PM2.5 concentration is larger, the precisions of the two input cases tend to be the same. The optimal phase precision for inputs of coherent state & squeezed state is obtained when the coherent beam carries most of the photons, which means that |α|2 ≫ sinh2r. So the mean photon number N ≈ |α|2 ≈ |α|2 − sinh2r and

Even when the amount of squeezed photon is smaller, the squeezing strength r is still big. Thus the term e−2r can be neglected when η is not close to zero. If η → 1, . In this case, the approximate optimal phase precision approaches and is the same as the precision for inputs of coherent state & vacuum state. Because η will increase along with the increasing of PM2.5 concentration, the two precisions are the same when PM2.5 concentration is higher.

Fig. 7. (color online) Comparison of phase precisions of two input cases (N = 100) in the presence of three relative humidities (a) RH = 51%, (b) RH = 68%, and (c) RH = 81%.
6. Conclusion

We analyzed the phase precision of a Mach–Zehnder interferometer in the presence of PM2.5 air pollution. With the increasing of PM2.5 concentration, the phase precision of two input cases will steadily decrease and be less than the standard quantum limit. Compared with the precision of coherent state & vacuum state, the precision of coherent state & squeezed state decreases faster as PM2.5 concentration increases. The precisions of two input cases tend to be the same when the fine particles concentration approaches 300 μg/m3. Except for PM2.5 concentration, transmission distance and relative humidity can also impact on phase precision. Because of the scattering or absorbing from the abundant fine particles, the long transmission distance will attenuate the intensity of light. The longer the distance is, the lower the precision is. Due to ammonium salt having the property of moisture absorption, high relative humidity improves the scattering ability of PM2.5. The precisions of two input cases decrease much faster when the value of relative humidity is higher.

Phase precision for inputs of coherent state & squeezed state is better than that of the other input case when PM2.5 concentration is smaller than 200 μg/m3. Coherent state & squeezed state will lose its advantage in heavy air pollution. High relative humidity is harmful to the Mach–Zehnder interferometer. As shown in Fig. 6, phase precisions of two input cases get close to the same when PM2.5 concentration is not bigger than 50 μg/m3, which means that high relative humidity has little effect on phase precision when air quality is good. If we want to obtain high precision in free space, the atmospheric parameters need to be monitored and we should ensure the experiment is carried out in a dry and clean environment.

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