1. Introduction

Underwater communication is vital for undersea exploitation and modern communication. Conventional ways which employ acoustical technique for underwater communication have their drawbacks [1,2], including high path loss, narrow bandwidth, high bit error rate, among which unconditional security is more demanding due to commercial and secure interest. The question arises of whether quantum communication [3–7] can be achieved by employing seawater as a reliable channel.

Since the seminal work of Bennett and Brassard on transferring quantum states and cryptographic keys through 0.3-meter-long free-space air [8], quantum communication has become an ultimate approach for the goal of unconditional communication security. Massive experimental efforts have been made in channels of fiber [9–11] and free-space air [12–14], pushing the distance up to the order of 100 kilometers [15–19]. Although quantum key distribution and quantum teleportation have been achieved via optical fiber installed underneath Geneva Lake and the River Danube [11] respectively, experimental investigation in free-space seawater has never been done so far.

Fortunately, as existence of transmission window around 800nm in free-space air [14], there is a blue-green optical window at the wavelength regime of 400–500nm in free-space seawater [20], wherein photons experience less loss and therefore can penetrate deeper. Photonic polarization may be a desirable carrier of quantum bits since isotropic seawater induces very limited birefringent effect. In this letter, to verify feasibility of seawater quantum channel, we experimentally explore polarization preservation properties of single photon and quantum entanglement.

2. Experiment setup and results

As shown in Fig. 1, Our seawater samples were collected from the surface of costal sea in the zone between Dalian city and Zhangzi island, which conforms to Jerlov water type I [21]. Six sampling sites are several kilometers apart from one another. The distance between site I and VI is up to 36 kilometers. Investigations in a large scale and diverse waters would ensure generalizability of our experimental results. A schematic layout of the experimental setup is shown in Fig. 2, a 3.3-meter-long glass tube equipped with inlet and outlet is utilized as a testbed of quantum channel by filling different seawater sample as well as distilled water. A 405-nm semiconductor laser running at CW mode is fed into the channel and detected by power meter. Together with a reference detection for eliminating laser fluctuation, we obtain average attenuation coefficient 0.354 ± 0.007m⁻¹ in six seawater samples. The value is a few times higher than the result of 0.081 ± 0.009m⁻¹ obtained in distilled water [see Appendix.I] (Table 1).

 

Fig. 1 Location of the seawater samples. In order to reduce position-dependent uncertainty of seawater, we make experimental investigation in large area. The sites where we collected the samples locate at the north of the Yellow Sea, which is on the eastern coast of Liaodong peninsula, lying between Dalian city and Zhangzi island. The GPS coordinates provide specific position information of each site.

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Fig. 2 Experimental quantum state transfer of single photons in free-space seawater. a, Sketch of experimental system. The semiconductor laser can run in both continuous wave (CW) and pulsed modes. CW mode is adopted in the experiment of estimating loss of seawater samples, and pulsed mode is used to prepare single photon source with tunable attenuation. Quantum states of single photons are prepared in polarization by using a polarizing beam splitter followed by a half and a quarter wave plates, reversed order of these three elements in the output act as a state analyzer for quantum tomography. Polarization compensator consisting of two quarter- and one half-wave plates is utilized to compensate polarization rotation in fiber and other linear optical devices. b, 2D color chart of state (|H〉,|V〉, |D〉, |A〉,|R〉, |L〉,) fidelities through different channels including six seawater samples, distilled water and air (empty tube). Maximum likelihood estimation is used for keeping density matrix physical. Note that the start point of color bar is set at 98% to visualize the distinction better. c, Measured density matrices of six receiving states through seawater sample VI.

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Table 1. Measured Attenuation Coefficients, State Fidelities and Process Fidelities.

View Table

Besides the loss associated with maximum secure distance, it is more important to find a desirable degree of freedom of photon with which quantum states can be encoded and transferred with high fidelity, i.e. without significant degrade induced by seawater. As water is a uniform isotropic medium, it is very likely that seawater, though incorporating dissolved salts and microbes, does not lead to massive polarization rotation or depolarization of photon. Random scattering on water molecular and suspended particulate matter can introduce depolarization accompanying with transverse angle diffusion [22]. However, we can spatially filter the depolarized photons with small receiving angle defined by optical fiber. We develop a simple depolarizing model of single photon scattering to explain why polarization preservation and high-fidelity state transfer are feasible under strong photon scattering (see Appendix.II). In pure Rayleigh scattering regime, wide scattering angle and weight make photons survive in receiving angle with a very limited proportion, which means the dominant role of scattering is loss rather than depolarization. In the case of tight forward scattering, we found the polarization can be well preserved even under multiple scattering.

We prepare single photons by driving the semiconductor laser to 2-ns pulse with a repetition rate of 50MHz and strongly attenuating it to 0.3∼0.6 photon per pulse typically adopted in decoy state protocols [23–25]. We encode single photons with six initial input states |ψ_in〉 in polarization |H〉, |V〉, $|D〉=1/\sqrt{2}\left(|H〉+|V〉\right)$, $|A〉=1/\sqrt{2}\left(|H〉-|V〉\right)$, $|R〉=1/\sqrt{2}\left(|H〉+i|V〉\right)$, $|L〉=1/\sqrt{2}\left(|H〉-i|V〉\right)$ respectively, where H and V represent horizontal and vertical polarization. After going through seawater, we project each output state on these three sets of orthogonal basis [Fig. 2(a)]. The background induced by ambient light scattering is filtered out carefully in spatial, spectrum and time domain. We employ quantum state tomography [26] method to reconstruct the density matrices of output states with the counts of avalanche photodiode detectors APD1 and APD2. To eliminate the error from differential loss and detection efficiency, we characterize the ratio of two orthorgonal projection measurements and incorporate the value into our density matrix construction (see Appendix.III). By making coincidence with synchronization signal of laser pulse from photodiode, we suppress noisy counts including background count and dark count of avalanche photodiode detectors from 500cps to 150cps, when the coincidence window of FPGA is 3.5ns. Figure 2(b) shows fidelities of receiving states for six initial states in six seawater samples, distilled water and air (empty tube), see their average value in Table 1. We can see that fidelities are above 98% for all initial states and seawater samples. As an example, Figure 2(c) shows density matrices of six receiving states through seawater sample VI.

To reveal physical processes of seawater quantum channel, we employ quantum process tomography [27], which can provide complete information of the channel changing arbitrary input state into another state. Any quantum process can be written as

In Fig. 3(a), polarization entangled photon-pair source at 810nm is generated by a blue laser beam (power 11mW, wavelength 405nm) pumping a quasi-phase matched periodically-poled KTiOPO₄(PPKTP) crystal with type II spontaneous parametric down conversion [30]. The source is prepared as the singlet state

 

Fig. 3 Experimental entanglement distribution in free-space seawater. a, Schematic of experimental setup. Polarization entangled photon-pair source is produced by a blue laser pumped PPKTP crystal (25-mm long) in a Sagnac ring interferometer. One photon is measured locally, the other is distributed through the channels and analyzed at the output. b, c, polarization correlation properties observed in air (b) and sample VI (c). Four curves in each chart are obtained by projecting one photon at polarization angles |H〉,|V〉, |D〉, |A〉,respectively and scan the other one.

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By filling seawater sample VI into the glass tube, we have observed a change of polarization correlation from 95.28 ± 0.14% to 94.99 ± 0.42% in H/V basis, from 94.15 ± 0.16% to 94.6 ± 0.44% in D/A basis [see Fig. 3(b) and 3(c)], indicating no clear decrease even under 30dB additional loss at 810nm wavelength. State tomography method is used to construct the density matrix of the entangled state under two conditions, i.e. air and sample VI [see Fig. 4]. We have obtained a fidelity 0.9946 from the distance of these two density matrices by $F_S={\{tr[\sqrt{\sqrt{{\rho }_{seawater}}{\rho }_{air}\sqrt{{\rho }_{seawater}}}]\}}^2$.

 

Fig. 4 Diagrammatic representation of reconstructed real (Re) and imaginary (Im) components of polarization entangled state. Density matrix is obtained by linear state tomography under two conditions: empty tube (a) and sample VI (b). The states in both cases appear to be highly entangled in polarization, which demonstrate entanglement can well preserved in seawater. This complements the evidence of high process fidelities provided by Fig. 2 and Table 1.

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We further evaluate the quality of output entangled state via a clear violation of the Clauser-Horne-Shimony-Holt (CHSH) -type Bell’s inequality [31], with regard to classical limit S=2 and perfect entangled state $S=2\sqrt{2}~2.828$. We achieved S = 2.6936 ± 0.0074 in air and S = 2.6695 ± 0.0203 in seawater sample, violating CHSH-type Bell’s inequality with 93 and 33 standard deviations respectively. Again, there is no distinctive difference in the two conditions, though confidence drops as the result of less coincidence events collected in seawater due to loss. These results indicate polarization entangled state can survive well in seawater channel.

3. Discussion

Our results have verified free-space seawater desirable as quantum channel enabling high-fidelity distribution of single photon and entanglement. It encourages us to look into a longer achievable communication distance. The seawater farther from the coastline, which contains less suspended particulate matter and therefore has much lower attenuation, may promise longer transmission distance for photons. According to data reported [21, 32], the loss to photons in blue-green window can be as low as 0.018m⁻¹. An achievable communication distance 885 meters can be derived if we apply attainable threshold of quantum communication against loss of 70dB [7]. Geometric effect and underwater turbulence on optical beam may introduce more loss. In spite of being shorter than the achievable distance in fiber and air channel, there have been extensive applications with hundreds of meters of seawater quantum channel. For example, an encrypted control command can be sent from an aircraft vehicle to objects underwater, even they are close to the surface of the water.

However, further experimental tests in the scenario of longer channel and field condition are necessary, where new techniques have to be developed to solve emerging problems. The aspects including the wavefront degradation, the beam wandering and the photon statistics may induce loss of single photon rather than the error of photon polarization. In future field test, These aspects associated with adaptive optics must be thoroughly investigated to minimize the channel loss. For example, we have observed slow beam wandering in our comparatively static ambient condition, see Appendix.VI. It implies that APT (acquiring, pointing and tracking), a technique of active locking of beam pointing via feedback [16], has to be developed according to real dynamic environment of open sea.

Quantum repeater [28] combining entanglement swapping [33] and quantum memory [34] in blue-green window, though technically more challenging, can efficiently extend the achievable distance. A scenario we have conceived is that, in a quantum communication network consisting of many non-stationary platforms, each platform communicate with neighbouring one in a moderate distance meanwhile can serve as quantum repeater node connecting two or more neighbouring platforms. Another scenario we have thought of is that, a ship can connect one platform in free-space seawater and one in free-space air, where different wavelength windows of light can be exploited respectively, serving as a quantum repeater node as well as sea-air interface. As its elemental ingredients, short-wavelength quantum entanglement has been developed in semiconductor [35], and short-wavelength quantum memory may be achieved by virtue of frequency conversion in quantum regime [36–38].

4. Conclusion

In summary, we have experimentally demonstrated the distribution of polarization qubits and entangled photons over seawater channel. While implemented in a short distance, the obtained high process fidelities indicate that the seawater associated with suspended particulate matter introduces very limited depolarization and disentanglement, which verify the feasibility of quantum communication and quantum cryptography in free-space seawater. Future explorations include field experiment in open sea, blue-green band quantum repeater and sea-air quantum communication interface.

Appendix

I. Loss budget and measurement

Loss caused by seawater could not be measured directly because of additional loss of optical devices and possible fluctuation of the laser power. We use a reference beam to remove the influence. A combination of two quarter waveplates and one half waveplate is employed to compensate the polarization rotation induced by single mode fiber, as well as to introduce a tunable but fixed splitting ratio between reference and signal beam on the polarizing beam splitter. The splitting ratio is constant for measurement on all samples. Real loss induced by seawater can be derived by subtract the value of empty tube out of full tube. Differential reflection at glass-seawater and glass-air interface must be taken into account, which can be figured out by Fresnel formula.

(3)$$\tilde{t}=\frac{2n_{1}cos{i}_1}{n_{1}cos{i}_1+n_{2}cos{i}_2}$$

(4)$$T=\frac{n_2}{n_1}{t}^2$$

where i₁ (i₂) is incident angle (transmission angle). Refractive index of medium can influence the transmission of light. n_air = 1, n_water = 1.34 (T_c = 20°C, λ = 405nm), n_BK7 = 1.5302(λ = 405nm).

Seawater water is a complex system, whose refractive index is related to its salinity, temperature and the incident light wavelength [39], which can be written as

(5)$$n_{seawater}=a_0+\left(a_1+a_2{T}_C+a_3{T}_C^2\right)S+a_4{T}_C^2+\left(a_5+a_{6}S+a_{7}T\right){\lambda }^{-1}+a_8{\lambda }^{-2}+a_9{\lambda }^{-3}$$

where empirical coefficients a₀ ~ a₉ are a₀ = 1.31405, a₁ = 1.779 × 10⁻⁴, a₂ = −1.05 × 10⁻⁶, a₃ = 1.6 × 10⁻⁸, a₄ = −2.02 × 10⁻⁶, a₅ = 15.868, a₆ = 0.01155, a₇ = − 0.00423, a₈ = −4328, a₉ = 1.1455 × 10⁶. Thus we can get the transmission rate T_air = 83. 57% and T_seawater = 90.535% under the conditions that the tube is empty and full, respectively.

Assuming that the intensity of reference beam is a₁(b₁) and the intensity of output signal beam is a₂(b₂), corresponding to the condition empty tube (full tube), we can obtain attenuation coefficient by

(6)$$\alpha =\frac{log\frac{b_2}{\frac{a_2{b}_1}{a_1}\times \frac{90.535\%}{83.57\%}}}{-3.3}$$

The attenuation coefficient of Jerlov type I coastal water is α = 0.35667m⁻¹ at λ = 425nm, which is close to the result we obtain in our experiment α = 0.35365 ± 0.0068m⁻¹ at λ = 405nm.

II. Scattering induced depolarization of single photon

To investigate how single photon scattering process affects the fidelity of the polarized photon, we employ the vector radiative transfer theory [22] and Monte Carlo method to establish a simple depolarizing model. Figure 5 schematically illustrate polarization-dependent photon scattering in seawater, where θ is the scattering angle and Φ is the azimuth angle between the scattering plane and the reference plane. In our model, we use a four-component Stokes column vector to describe the polarization of a photon.

(7)$$S=\left[\begin{array}{c}I\\ Q\\ U\\ V\end{array}\right]$$

where I is the total radiance, Q and U represent the linear polarization and V represents the state of circular polarization.

 

Fig. 5 A sketch of polarization-dependent photon scattering. The arrow represents the scattering path of one step, where θ is the scattering angle and Φ is the azimuth angle between the scattering plane and the reference plane. Rayleigh scattering is approximately dominant in pure clean seawater and the curve gives the phase function.

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In multiple scattering process, each step can occur between two different scattering planes, thus a rotation matrix is needed to rotate the current Stokes vector to next scattering plane. The rotation matrix R(Φ) is defined as

(8)$$R\left(\varphi \right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& cos2\varphi & sin2\varphi & 0\\ 0& -sin2\varphi & cos2\varphi & 0\\ 0& 0& 0& 1\end{array}\right]$$

The scattering process in each step can be described by a Mueller matrix with 4 × 4 elements. According to ref [40], Mueller matrix for seawater is not identical but similar to Rayleigh scattering, with the expression of

(9)$$M\left(\theta \right)=\left[\begin{array}{cccc}1& -\frac{si{n}^2\theta }{1+co{s}^2\theta }& 0& 0\\ -\frac{si{n}^2\theta }{1+co{s}^2\theta }& 1& 0& 0\\ 0& 0& \frac{2cos\theta }{1+co{s}^2\theta }& 0\\ 0& 0& 0& \frac{2cos\theta }{1+co{s}^2\theta }\end{array}\right]$$

In each step, we record the relative coordinates of the scattered photon and calculate the final receiving position specified by polar and azimuthal angles. A weight function given by Rayleigh scattering phase function can be incorporated to determine the probabilities of different scattering angle in our Monte Carlo procedure. After n times of step, the final Stokes vector becomes

(10)$$S_n=M\left({\theta }_n\right)R\left({\varphi }_n\right)\cdots M\left({\theta }_1\right)R\left({\varphi }_1\right)S_0$$

Figure 6 shows the fidelity of the scattered photon under different receiving polar and azimuthal angles after n = 1 scattering for different initial polarization states of horizontal [Fig. 6(a)], diagonal [Fig. 6(b)] and right circular [Fig. 6(c)]. A relatively large receiving angle (20mrad) in practical experimental scenario has been considered, only 0.0063% of total photons can be received while remain a high fidelity up to 99.99%. We then further calculate the n = 2 and n = 3 cases. The receiving probability of each case is 0.0068% and 0.0067% respectively. However, the fidelity after each step drops to 59.53% and 51.68%, which is close to the noise background given by unpolarized light. It should be noticed that the low-fidelity events contribute very minor to the total fidelities due to extremely small proportion. It means that, in pure Rayleigh scattering regime, the dominant role of scattering is loss rather than depolarization.

 

Fig. 6 Transmission fidelities under different scattering angle and initial polarization state. Angle dependence of single scattering process for initial polarization states of horizontal (a), diagonal (b) and right circular (c). Apparently, in pure Rayleigh scattering regime, wide scattering angle and weight make photons survive in receiving angle with a very limited proportion, which means the dominant role of scattering is loss rather than depolarization. (d) Fidelities under tight forward scattering of 100 times. The polarization can be well preserved within 20-mrad receiving angle.

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In scenario of tight forward scattering, we consider n = 100 times of step. In a receiving angle of 20mrad, the scattered photons can still preserve a high fidelity over 99.5% shown in Fig. 6(d). One should note that our modelling and simulation may explain why polarization preservation and high-fidelity state transfer are feasible under strong photon scattering qualitatively. A more sophisticated model is needed to quantitatively describe the process for future experiments.

III. Differential efficiency in quantum tomography

Quantum state tomography can experimentally determine density matrix of an unknown state with a set of projection measurements. Density matrix of unknown state of single photon can be represented by four parameters {S₀, S₁, S₂, S₃} as $\widehat{\rho }=\frac{1}{2}{\displaystyle {\sum }_{i=0}^3{S}_i{\sigma }_i}$. Due to normalization, S₀ = 1. The other parameters can be determined with projection measurements as S₀ = P_|H〉+|V〉, S₁ = P_{|D〉−|A〉}, S₂ = P_{|R〉−|L〉}, S₃ = P_{|H〉−|V〉} For example, probe state |H〉 is sent through the seawater sample and output state is projected on |H〉 〈H| (|H〉〈V|), which is recorded by coincidence counts between photodiode and APD1 (APD2) denoted by C_H1 (C_H2). Stokes parameter S₃ can be determined by C_H1 (C_H2) and C_V1 (C_V2). The differential ratio of projection measurements |H〉 〈H| and |V〉 〈V| is ${\eta }_{HV}=\sqrt{\frac{C_{H1}{C}_{V1}}{C_{H2}{C}_{V2}}}$. To eliminate differential loss and detection efficiency, we correct the counts by $S_3=\frac{C_{H1}-C_{H2\eta HV}}{C_{H1}+C_{H2\eta HV}}$. By using the same method, we can get ${\eta }_{DA}=\sqrt{\frac{C_{D1}{C}_{A1}}{C_{D2}{C}_{A2}}}$, $S_2=\frac{C_{D1}-C_{D2\eta DA}}{C_{D1}+C_{D2\eta DA}}$ and ${\eta }_{RL}=\sqrt{\frac{C_{R1}{C}_{L1}}{C_{R2}{C}_{L2}}}$, $S_1=\frac{C_{R1}-C_{R2\eta RL}}{C_{R1}+C_{R2\eta RL}}$.

VI. Beam wandering in seawater

In the experiment, single count of detector present slow and quasi-periodical variation. We attribute this to the beam wandering associated with decoupling to optical fiber. Ambient condition change including mechanical vibration, temperature fluctuation and varying salinity may result in variational and inhomogenous refractive index distribution along propagating direction of the beam. According to the data collected in sample VI, the counts of two detectors present nearly synchronous variation. Such beam wandering issue, though in a faster speed, also exists as a main problem to solve in free-space quantum communication. It suggests that the active feedback system adopted in free-space air should also be applied to free-space seawater.

Funding

National Key Research and Development Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (11374211, 11690033); Shanghai Municipal Education Commission (SMEC)(16SG09, 2017-01-07-00-02-E00049); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405); Open fund from HPCL (201511-01); National Young 1000 Talents Plan.

Acknowledgments

The authors thank J.-W. Pan, I. A. Walmsley, L.-J. Zhang and X.-F. Ma for helpful discussions.

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