Optical Communication with Weak Coherent Light Fields

Entanglement and superposition are foundations for the emerging field of quantum commu‐ nication and information processing. These two fundamental features of quantum mechanics have made quantum key distribution unconditionally secure (Scarani et al., 2009; Weedbrook et al., 2010) compared to communication based on classical key distribution. Currently, implementation of an optical quantum communication is mainly based on discrete and continuous quantum variables. They are usually generated through nonlinear interaction processes in χ(2) (Kwiat et al., 1995) and χ(3) (Lee et al., 2006,2009) media. Discrete-variable qubit based implementations using polarization (Liang et al., 2006, 2007; Chen et al. 2007, 2008; Sharping et al., 2006) and time-bin (Brendel et al., 1999; Tittel et al., 1998, 1999) entanglement have difficulty to obtain unconditional-ness, and also usually have low optical data-rate because of post-selection technique with low probability of success in a low efficient single photon detector at telecom-band (Liang et al., 2005, 2006, 2007). Continuous-variable imple‐ mentations using quadrature entanglement (Yonezawa et al., 2004; Bowen et al., 2003; Silberhorn et al., 2002) and polarization squeezing (Korolkova et al., 2002) can have high efficiency and high optical data-rate because of available high speed and efficient homodyne detection. However, the quality of quadrature entanglement is very sensitive to loss, which is imperfect for implementing any entanglement based quantum protocols over long distance. Continuous-variable protocols that do not rely on entanglement, for instance, coherent-state based quantum communication (Yuen, 2004; Corndorf et al., 2003; Barbosa et al., 2003; Grosshans et al., 2002, 2003; Qi et al., 2007; Wilde Qi et al., 2008), are perfect for long distance optical communication. Several experimental approaches were taken to resolve transmission loss for long distance optical communication by using coherent light source. Optical wave


Introduction
Entanglement and superposition are foundations for the emerging field of quantum communication and information processing. These two fundamental features of quantum mechanics have made quantum key distribution unconditionally secure (Scarani et al., 2009;Weedbrook et al., 2010) compared to communication based on classical key distribution. Currently, implementation of an optical quantum communication is mainly based on discrete and continuous quantum variables. They are usually generated through nonlinear interaction processes in χ (2) (Kwiat et al., 1995) and χ (3) (Lee et al., ,2009 media. Discrete-variable qubit based implementations using polarization (Liang et al., , 2007Chen et al. 2007Chen et al. , 2008; Sharping et al., 2006) and time-bin Tittel et al., 1998Tittel et al., , 1999) entanglement have difficulty to obtain unconditional-ness, and also usually have low optical data-rate because of post-selection technique with low probability of success in a low efficient single photon detector at telecom-band (Liang et al., 2005(Liang et al., , 2007. Continuous-variable implementations using quadrature entanglement (Yonezawa et al., 2004;Bowen et al., 2003;Silberhorn et al., 2002) and polarization squeezing (Korolkova et al., 2002) can have high efficiency and high optical data-rate because of available high speed and efficient homodyne detection. However, the quality of quadrature entanglement is very sensitive to loss, which is imperfect for implementing any entanglement based quantum protocols over long distance. Continuous-variable protocols that do not rely on entanglement, for instance, coherent-state based quantum communication (Yuen, 2004;Corndorf et al., 2003; Barbosa et al., 2003;Grosshans et al., 2002Grosshans et al., , 2003Qi et al., 2007;Wilde Qi et al., 2008), are perfect for long distance optical communication. Several experimental approaches were taken to resolve transmission loss for long distance optical communication by using coherent light source. Optical wave mechanical implementations (Lee et al., 2002(Lee et al., , 2004 of entanglement and superposition with coherent fields have been demonstrated.
We discuss and demonstrate a new type of optical communications based on weak coherent light fields in detail in this chapter.

Correlation functions of two weak light fields
Two orthogonal light fields are used to implement correlation function between two distant observers. In the Stapp's approach (Grib et al., 1999;Peres, 1995) for two distant observers A and B, when analyzer A is oriented along the polarization angle θ 1 , the transmitted |θ 1 // and reflected |θ 1 ⊥ polarization vectors of the light are given by, where the H and V are the horizontal and vertical axes. Analyzer A is a combination of half wave plate (HWP) and a polarization beam splitter (PBS) for projecting the linear polarization of the incoming photon. The operator associated with analyzer A can be represented by The operator A 1 has eigenvalues of ±1, such that, Depending on the photon is transmitted or rejected by the analyzer. Similarly, the analyzer B oriented at θ 2 can be defined as operator B 2 , Theory and Practice of Cryptography and Network Security Protocols and Technologies Operator A 1 (B 2 ) with eigenvalues of ±1 can be measured by using the balanced detection scheme as shown in Fig. 1. Two detectors are placed at the two output ports of a cube polarization beam splitter. Their output currents are subtracted from each other. The arrangement of this detection scheme can be used for measuring operator A 1 of Eq.(4) and B 2 of Eq. (7) that is the subtraction between the projection of the transmitted signal D // and the projection of the reflected signal D ⊥ .
Let's consider a beam of photons incidents on the PBS, if one photon goes through the PBS, it will produce non-zero signal at detector D // and zero signal at detector D ⊥ . Then, the subtraction yields positive signal as of D // − D ⊥ ≥ 0. If a photon is reflected from the PBS, it will go to the detector D ⊥ and produce non-zero signal at detector D ⊥ and zero signal at detector D // . Then, the subtraction yields negative signal as of D // − D ⊥ ≤ 0. For a certain amount of time, the subtraction records the random positive and negative spikes corresponding to the eigenvalues of +1 and -1 of operator A 1 , respectively, as shown in the inset of Fig. 1. If the incoming photons are in the superposition of |θ 1 // and |θ 1 ⊥ , the detection scheme A records a series of discrete random values, +1 and -1. Then, the mean value of A 1 is zero, that is A 1 = 0. Similarly, we can apply the same detection scheme for measuring operator B 2 and obtain B 2 = 0. The expectation value of the product A 1 B 2 or the mean value of the product signals of A 1 and B 2 will produce correlation functions, as given by,

Beam of photons
As shown in Eq.

Balanced homodyne detector
Balanced homodyne detector is utilised as the detection scheme for the weak coherent light fields for optical communication.
It consists of a 50/50 beam splitter, two photo detectors, a local oscillator field and a transimpendance amplifier. Superposed local oscillator field and weak light field will be detected by photodiodes D 1 and D 2 , lead to the generation of the photocurrent I 1 andI 2 . The photodiodes are connected together in such a way that the output equal to the I 1 minusI 2 as shown in Fig. 3. The balanced detector has two input ports. The signal field and local oscillator field optically mixed at the beam splitter. The local oscillator field is a large amplitude lightwave with the same frequency as the signal and having a well-defined phase with respect to the signal field. Generally, local oscillator field can be obtained from the same laser source as the signal field. The emerging output fields ε 1 and ε 2 are the superposition of signal and local oscillator fields.
The output fields ε 1 and ε 2 are given as, where ε LO and ε s are the amplitude of the signal and local oscillator field respectively.
Photocurrents that produced by the output fields ε 1 and ε 2 are given as Hence, the output of the balanced homodyne detector will be given as, 1 2 2 .
Since the signal and local oscillator fields are derived from the same laser source with relative phase φ. By considering only the real part of the signal and local oscillator fields, it can be described as, Where Aε s and Aε LO are the amplitude for signal and local oscillator fields, ω is optical frequency, φ is relative phase between the fields. Hence the output of the balanced homodyne detector is given by, 1 2 {cos( ) cos(2 )}.
The second term in the Eq.(15) is the fast varying term beyond the detection of the of the photo detector. Therefore, the output of the balanced homodyne detector is phase dependence, which is given by, 1 2 cos( ).
One of the main features of the balanced homodyne detector is the high signal to noise ratio compared to a single detector. For example, classical intensity fluctuations of the laser would affect the measurement of a single detector. Contrary, any changes in intensity will be canceled by the subtraction of the photocurrent with an ideal balanced homodyne detector.
However, due to the Poissonian statistics of the coherent light and random splitting process in the 50/50 beam splitter, fluctuations in intensity cannot be completely removed. Therefore even with the presence of only local oscillator field, the balanced homodyne detector will have a shot noise level above the electronics noise level as depicted in Fig.4, limiting the signal to noise ratio.

Practical demonstration of the optical communication with two weak light fields
A proof-of-principle experiment to demonstrate the correlations of two weak light fields as described in section 2 is shown in Fig.5. A continuous wave laser at telecom band wavelength (1534nm) is used to provide two orthogonal weak light fields. We use a 50/50 beam splitter to optically mix the vertically and horizontally polarized coherent light fields. The beam 1 from Theory and Practice of Cryptography and Network Security Protocols and Technologies the output port 1 of the beam splitter is a superposition of the vertically and horizontally polarized weak light fields, similarly for beam 2 from output port 2 of the beam splitter. The balanced homodyne detectors are made of two p-i-n photodiodes (EXT500) and the signal measured by the balanced homodyne detectors will be further amplified by a transimpedance amplifier. A quarter wave plate at 45° as part of measuring device is inserted at beams 1 and 2 to transform the linearly polarized states to circularly polarized states. By using a quarter wave plate transformation matrix, the field amplitudes V 1 , H 1 , V 2 and H 2 are transformed as, where the phase shift due to the beam splitter is included. For simplicity we use unit vector notation and drop the amplitude of field notation. Now, analyzer A in beam 1 will experience homogeneous superposition of left circularly polarized and right circularly polarized weak light fields. Similarly for analyzer B in beam 2. Analyzer  A(B) is placed before the balanced homodynes detector A(B) to project out the phase angle 1(2) as, The superposed field in beam 1 after the λ/4 wave plate and the analyzer can be expressed as, ( cos sin ) ( cos sin ) .
and similarly for the superposed field in beam 2, where ω is optical frequency, and φ is the relative phase of the two orthogonal weak light fields. Thus, the interference signals obtained by the photodetector D 1// in balanced homodyne detectors at beam 1 are given as, On the other hand, for photodetector D 2// , the reflected beat signal becomes 22b Then, the balanced detector A measures Similarly, the interference signals obtained by the photodetectors in balanced homodyne detector at beam 2 can be written as, The interference signals of Eq.(23) and Eq.(25) above for balanced detectors A and B are the measurements of operators A 1 and B 2 , respectively. The interference signal in detector A is anti-correlated to detector B because of the phase shift of the beam splitter. The interference signals contain information of the projection angles of the analyzers. The average of the interference signals is zero, that is, <A 1 > = 0 and <B 2 > = 0. To further discuss the significant of measuring the operator A 1 , the interference signals obtained in balanced detector A can be rewritten as, which is identical in structure with operator A 1 as in Eq.(4), that is The factor of 2 in Eq. (26) (2  ) cos(2( )) cos(2( )).
A B q j q j q q q q j µ -+ + µ ---+ + Then, the mean value of this multiplied signal is measured. We obtain one of the correlation functions C(θ 1 , θ 2 ) as described in section 2, where the second term in Eq.(26) is averaging to zero due to the slow varying relative phase φ of the two orthogonal weak light fields from 0 to 2π. We normalized the correlation function C(θ 1 , θ 2 ) with its maximum obtainable value that is, θ 1 = θ 2 . Thus, for the setting of the analyzers at θ 1 = θ 2 , the normalized correlation function C(θ 1 , θ 2 )= −1 shows that the two beams are anti-correlated. To generate other correlation functions, such as C(θ 1 , θ 2 ) ∝ − cos2(θ 1 + θ 2 ) the λ/4 wave plate at beam 2 is rotated at -45°, then the beat signal measured by balanced homodyne detector B 2 of Eq.(25) is given by Hence, obtaining the correlation function,

Correlation measurement of a stable field and a noise field
To verify the above analysis and measurement method for weak light fields, we present an experiment measurement of one stable coherent light field and one random noise phase modulated light field.
One stable coherent field is mixed with one noise field in a beam splitter. The experimental result has been recently published (Lee, 2009). Fig. 6(a) and (b) are the beat signals obtained at A and B, where the phase ϕ c (t) is modulated with random noise through an acousto-optic modulator. The product of the beat signal at A and B is shown in Fig. 6(c). The mean-value measurement produces the bipartite correlation -cos2(θ 1 -θ 2 ), which is still classical correlation. However, it is obvious that the information of θ 1 and θ 2 are protected by classical noise not quantum noise. Classical noise is not completely random compared to quantum noise as inherited by coherent state.
In the next section, two weak coherent light fields | α and | β are used for generating quantum correlation, where the quantum noise ϕ (t) = ϕ β -ϕ α provided by mean photon number fluctuation.

Correlation measurement of two weak light fields
By using the experiment setup as proposed in Fig.5, we are able to generate 4 types of bipartite correlation, given as To verify the analysis discussed in section 2, we perform systematic studies of the proposed experiment. We use a piezoelectric transducer (PZT) to modulate the phase of a weak light field. Then, all 4 types of correlation function were obtained by manipulation of experiment setup as discussed in previous section. We normalized the correlation function -cos 2(θ 1 -θ 2 ) with its maximum obtainable value, that is θ 1 =θ 2 . Fig.7 shows the normalized correlation function ±cos 2(θ 1 ± θ 2 ) as a function of the relative projection angle of the analyzer A and B. The blue line is the predicted theoretical value while the red circle with the error bar is the experimental data.
For each data point, we take ten measurements of the multiplied signal and obtain the average mean value. Each measurement was obtained by fix the projection angle of the analyzer A and rotates the projection angle of analyzer B. The error bar is mainly due to the electronic noises and temperature dependence of polarization optics.

Bit generation and measurement
After we established one of the bipartite correlation functions between observer A and B, bit generation and measurement for optical communications can be done by implementing bit correlations between them.
Lock-in-amplifier is used to measure the bit correlation of between observer A and B. Fig.8 depicts the experimental setup for bit measurement for observer A and B. To perform this measurement for the established correlation function of -cos 2(θ 1 − θ 2 ), we ramp the Piezoelectric transducer (PZT) at one of the weak light field to obtain one period of interference signal. An example of single period of interference signal measured at observer and reference signal for the lock-in amplifier is shown in Fig.9. For practical optical communication, phase locking of the two orthogonal weak light fields are required. We measure quadrature phases of orthogonal weak light fields with the step size of nπ/2 (n = integer) as shown in Fig. 10(a) (blue line). Using the same lock-in reference phase in the lockin amplifier, we measure the quadrature phases of weak coherent state at detector B as shown in Fig. 10(a) (dashed red line). We have observed the bits correlation between two parties for the shared correlation function of −cos 2(θ 1 − θ 2 ) as shown in Fig. 4(a), where the positive  (negative) quadrature signal is encoded as keys/bits '1' ('0'), respectively. By using the same lock-in reference phase, we observe bits correlations for the other three types of correlation functions −cos 2(θ 1 + θ 2 ), cos 2(θ 1 + θ 2 ), and cos 2(θ 1 − θ 2 ) as shown in Figs. 10(b), 10(c), and 10(d), respectively.
In real practice of long distance optical communication, we can establish one of the bit correlations for calibrating the lock-in reference phase at observer A and B. We further explore the feasibility of the scheme long distance optical communication for by performing bits correlations between two observers over a distance of 10 km through a transmission fiber. We couple one of the orthogonal weak light fields into 10 km of transmission fiber and a quarterwave plate and a half-wave plate are used at the output of the transmission fiber to compensate the birefringence. The correlation between two observers A and B are found to be preserved over the 10 km transmission fiber (Sua et al., 2011). We managed to establish four types of correlation functions and performed bits correlations for each shared correlation function between two observers.
In short, for our proposed weak coherent light fields optical communication scheme, information is encoded onto the superposition of the vertically and horizontally polarized weak light fields; decoding involves detection of the weak light fields by balanced homodyne detector and quadrature phases measurement by lock-in amplifier. For reliable measurement of the encoded signal, both phase and polarization of the weak light field must be stable.
Apparently, stability and accurate control of phase and polarization turned out to be the main challenge for the practical implementation of weak coherent light fields optical communication. The state of polarization of the light wave is not preserved in the typical transmission fiber. Dynamic control of the state of polarization of the light is critical to ensure the reliability the proposed optical communication scheme. Each dynamic polarization controller is bulky and expensive (Noe et al., 1999), severely limits the practicality of our scheme. Phase locking is another challenging obstacle as well. Phase locking is required between the two orthogonal weak light fields that used to implement the bit correlation between two observers. Without the phase locking, quadrature phases measurement performed by lock-in amplifier is meaningless. Therefore, optical phase-locked loop must be employed for the phase locking of two weak light fields. However, for high data rate optical communication, the delays allowed in the phased-locked loop are so small that phase locking becomes an enormous challenge (Barry et al., 1992;Kazovsky, 1986).

Conclusion
We have experimentally demonstrated a new type of optical communication protocol based on weak coherent light fields. Coherent bipartite quantum correlations of two distant observers are generated and used to implement keys (bits) correlation over a distance of 10 km. Our scheme can be used to provide security as a supplement to the existence decoy-state Bennett-Brassard 1984 protocol and the differential phase-shift quantum key distribution (DPS-QKD) protocol. The realization of intrinsic correlation of weak coherent light fields by using the measurement method is a first step toward linear-optics quantum computing with weak light fields and single-photon source.