1. Introduction

Quantum information physics is attractive not only from a scientific but also from application point of view, and three kinds of applied fields have been defined, i.e., cryptography [1], communication [2], and computation [3,4]. Although these applications have tremendous capabilities beyond classical physics, they require ideal quantum states. Because quantum states are easily destroyed through interactions with circumstances, it is generally difficult to install the quantum information technologies in realistic application fields. We should therefore find an optimal solution to satisfying realistic conditions and to exploiting quantum information technologies. Tomaru et al. has proposed using antisqueezed light to optical communications as one of these solutions [5]. The squeezed component of a squeezed state is easily pulled back to a shot-noise level through loss. The antisqueezed component of the squeezed state, on the other hand, is not pulled back to this level until the light is completely lost. The antisqueezed component is loss tolerant, and similarly amplification tolerant. The proposed idea was to use an elliptic figure of fluctuations in phase space to achieve secure optical communications, e.g., using Yuen et al.’s protocol [6]. The proposal of Tomaru et al. did not question whether or not the squeezed component was squeezed less than the shot-noise level. Therefore, they used the term “antisqueezed light” instead of squeezed light.

The method of using the Kerr effect of optical fibers is useful for generating antisqueezed light at wavelength region of 1.55 µm. Several kinds of squeezed states and squeezing methods have been proposed [7–9]. The ratio of antisqueezed fluctuations to the amplitude of signal light is an important parameter in the phase-encoded secure optical communications because the large ratio makes eavesdropping difficult. In this sense, the method of using a Sagnac interferometer that produces a quasi-squeezed vacuum is appropriate [7]. Squeezing strength increases with the product of optical power and fiber length. Although a solid-state laser or fiber laser is ordinarily used as an optical source in squeezing experiments, a laser diode (LD) is better if we consider installing an antisqueezed light source in optical-communication systems. However, we must modify the originally proposed configuration of the Sagnac interferometer to a low-power version because the power of LD light is low. We must first use a long optical fiber, e.g., 5 km, to compensate for the low optical power. A reflection-type configuration is better than the conventional loop configuration for such a long fiber to improve the stability of the interferometer against environmental fluctuations [10,11]. Moreover, if possible, it is better not to use an expensive polarization-maintaining fiber. The antisqueezed light that is reported here is generated through a reflection-type interferometer consisting of a standard single-mode fiber (SMF), using LD light.

2. Experimental configuration

Figure 1 shows the block diagram for the improved experimental configuration. The reflection-type interferometer is composed of a polarizing beam splitter PBS3-to-Faraday mirror (FM). Amplified LD light with an erbium-doped fiber amplifier (EDFA) transmits all optical components to PBS3 and is divided 50:50 into orthogonal polarization components at PBS3 (This is a balanced case.). PBS4 combines the two orthogonal components with delay. This delay ensures these two components propagate fibers independently. The optical fiber for antisqueezing is a standard SMF. A polarization controller (PC) between PBS4 and SMF compensates for the polarization-mode dispersion of SMF. FM rotates each polarization component by 90°. The two returned orthogonal components are combined to one pulse with one polarization, which is output to port 2 of PBS2, and is used as local light when noise is measured. The squeezed vacuum is generated at PBS3 with the polarization orthogonal to the combined pump light [11]. The squeezed vacuum transmits PBS2 and is reflected at PBS1. If the two polarization components at PBS3 are not fully combined, imbalanced light displaces the squeezed vacuum. The displacement is problematic but rather preferable because the displaced squeezed vacuum is used as carrier light.

The laser diode was directly modulated with a sinusoidal wave of 2 GHz. The chirp of the output pulse was compensated for with dispersion-compensating fibers (DCF) (40 ps/nm). A DC (10 mA) bias applied to the LD (NEL NLK5C5EBKA) and the DCF length were determined to yield minimum pulse width. Although higher AC power gives narrower pulse, the power was maintained at 12 dBm to prevent the LD from being destroyed. The output pulse was characterized by Second-Harmonic Generation-Frequency Resolved Optical Gating (SHG-FROG) [12].

 

Fig. 1. Block diagram of experimental configuration. FR means a Faraday rotator.

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Fig. 2. Typical SSB noise traces measured at 10 MHz under conditions of 5-km fiber length and 17.6-dBm input power. The traces were averaged ten times. (a) Signal. (b, c) Local light noise detected at one of PDs. (d) Standard quantum limit. (e) Noise floor. Resolution bandwidth was 300 kHz and sweep time was 1 s.

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Fig. 3. Dependence of noise on frequency, measured with 14.4 GHz bandwidth balanced PDs. The signs (a, b, c, d, e) have similar meanings to those in Fig. 2.

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Table 1. Characteristics of antisqueezed light.

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3. Experimental results

The noise of generated antisqueezed light was measured by balanced homodyne detectors with a spectrum analyzer [11]. Figure 2 shows typical single-side band (SSB) noise traces obtained at 10 MHz, where the signal light phase was scanned by a piezo actuator. The length of SMF was 5 km, and the input power to it was 17.6 dBm (14.6 dBm for each polarization). The full width at half maximum (FWHM) of autocorrelation (FROG trace) of the input pulse was 17 ps. The FWHM of the retrieved pulse was 16 ps, and the retrieved spectral width was 0.18 nm. Although third-order dispersion remained, the linear chirp was compensated for. The soliton order N of the input pulse is estimated to be 1.9 through the relation, N ²=γP₀τ ²/β ₂, assuming a sec h²(t/τ) pulse shape. Peak power P ₀ is estimated, using average input power P _ave and repetition frequency f _r, by the relation, P ₀=P_ave/2τf_r. The nonlinear coefficient γ is estimated, using nonlinear refractive index n ₂, fiber diameter d of effective cross section A _eff, and wavelength λ, by γ=8n ₂/λd[13]. Here, the assumed values are n ₂=2.66×10^-20 m²/W, d=10.5 µm, λ=1.55 µm, and β₂=2.3×10^-26 s²/m. Trace (a) in Fig. 2 gives signal light noise. The maximum noise that gives antisqueezed noise is 41.5 dB, and the minimum noise is 23.6 dB, normalized to the standard quantum limit (SQL), which is the noise level measured without the signal light. The tiny imbalances in the beam splitter (BS) and photodiodes (PDs) were compensated for with an aperture, which was adjusted to obtain a minimum noise trace for SQL. Traces (b) and (c) give local light noise, which was measured with only local light being input to one of PDs. Noise data measured at different conditions are summarized at Table 1.

Figure 3 shows the dependence of noise on frequency measured with 14.4-GHz bandwidth balanced PDs under the same conditions as in Fig. 2. Although SQL is hidden in the noise floor because of low trans-impedance gain, antisqueezing is assured over a 10-GHz bandwidth.

Squeezing parameter r is estimated by sinh r=γP₀z/2, where z is the fiber length [14]. Antisqueezed noise is calculated to be 10log e^2r=18.9 dB for the experimental conditions in Fig. 2. When fiber propagation is described by the nonlinear Schrödinger equation, which is linearized on quantum parts, a circular fluctuation in phase space is transformed into an elliptic shape [14]. In this understanding, the squeezed state output from the fiber interferometer is in a purely vacuum-transformed state, originating in an input vacuum from port 2 of PBS2, even if the pump pulse is a noisy mixed state. The reference level for 18.9 dB should therefore be SQL. However, 18.9 dB is much less than the measured antisqueezed noise. This means that extra noise enters the squeezed vacuum, which is also apparent from large measured minimum noise. The experimentally obtained noise is the result of elliptic transformation in phase space including the extra inflow noise.

4. Discussions

There are several causes for the extra noise. The first is propagation loss. The reason an output squeezed state is a squeezed vacuum is that two pulses divided at PBS3 have a correlation of quantum fluctuation-based accuracy, according to the beam splitter model of quantum mechanics. The correlation is weakened through propagation loss because uncorrelated vacuum noise flows in. Because the pump pulse has high noise as shown by traces (b) and (c) in Fig. 2, the output antisqueezed light has high noise according to the weakened correlation. For example, when the propagation length is 50 km, the loss is 10 dB, and the correlation is weakened to 1/10. In this case, the inflow noise is estimated to be 10 dB. The minimum noise for a 20-km (40 km for round trip) length is 25.2 to 28.8 dB in Table 1. This is larger than the estimated value, and the simple loss model does not give an adequate interpretation.

The second possibility is Brillouin scattering. This is backward scattering, and is homogeneously mixed into the backward-propagating pulses as noise. For example, we assume that -32 dBm Brillouin scattering power P _B for a 14.9-dBm input power and a 1-km fiber length is complete random noise, where whole backward scattering was regarded as Brillouin and was measured by replacing of FM with a terminator. The average photon number n mixed into a propagating pulse is estimated to be 2τP_B/hv, and the noise level is estimated to be 25.5 dB if fluctuation radius n ^1/2 is assumed. The measured minimum noise was 11.4 dB, which means that the scattered light has coherence between the two divided pulses, and the incoherent ratio is -14.1 dB. Similarly, if complete random noise is assumed for the case of 14.9-dBm input power and 30-km fiber length, a P _B of -23.5 dBm gives 34.0- dB noise, and its incoherent portion is 19.9 dB. This is much less than the measured minimum noise of 30.6 dB. Brillouin scattering also does not give an adequate interpretation.

The last possibility is forward scattering by Raman and guided acoustic-wave Brillouin scattering (GAWBS). Because we consider broadband noise and this experiment was done in a pulse scheme, GAWBS is neglected, compared with Raman scattering [15]. The Raman process considered here is intra-pulse stimulated Raman scattering. The scattering occurs pulse-by-pulse. Because phonons are not measured, the scattered optical state becomes a more mixed state, and one quantum fluctuation is superimposed per scattering. This picture weakens the correlation between two pulses, similar to loss. The Raman gain at low frequency is fitted to an equation, g(ν)=0.02ν+0.04ν ³, normalized to unity for peak gain, where ν are units of THz [16]. The peak gain is approximately 10^-13 m/W [17]. The Raman scattering probability is given by gPz/A_eff, where P is an appropriate optical power. We evaluate half of peak power for P, and again assume a sech² pulse shape. A spectral width of 0.18 nm is evaluated for ν as a rough estimate. For example, when input power is 11.9 dBm (one polarization component) and z=10 km, the scattering probability is 1.08×10^-3. Because one pulse consists of 6.04×10⁷ photons, the amount of scattering is 6.5×10⁴. This creates 48-dB noise. The actual conditions, i.e., propagation loss, polarization randomness, and pulse broadening decrease the estimated noise, and ν and P should be determined more accurately. Moreover, because intra-pulse Raman scattering conserves the photon number and produces phase noise, the weakened correlation is more remarkable in antisqueezed quadrature than in squeezed quadrature. The 48-dB noise, therefore, overestimates the effect of Raman scattering for the squeezed quadrature. However, the estimated noise is sufficiently high, compared with the measured minimum noise of 21.0 dB. This estimate therefore suggests that Raman scattering is dominant in increasing the noise transfer from the pump light to the squeezed vacuum.

The quality factors of antisqueezed light are summarized in Table 1. Ellipticity is one of most important parameters because applications of antisqueezed light use the elliptic figure of fluctuations in phase space. This is estimated from the measured maximum and minimum noise (dB_max and dB_min). The angle, 2φ, subtended by antisqueezed fluctuations at the origin of phase space is another important parameter because antisqueezed fluctuations make eavesdropping difficult. Signal light power is simply corrected by subtracting P _B, and its amplitude is estimated by n ^1/2 using the photon number. Half of the above angle is evaluated through the relation, $\tan \phi ={10}^{d{B}_{max}⁄20}⁄2n^{1⁄2}$. In addition to ellipticity and 2φ, the signal-to-noise ratio (S/N) that is determined by minimum noise is also important.

A fiber length of 5 km (10 km for round trip) gives a maximum angle for 2φ in every input power although the S/N is minimal around that length, and 17.6-dBm input power gives a maximum angle although 2φ should ideally increase with input power. The dependence on input power suggests that complicated nonlinear effects break the ideal interference with the increase in input power. Ellipticity is maximum at a fiber length of 5–10 km, and increases with input power as a whole.

The fiber interferometer is balanced, including all nonlinear effects. The balance is adjusted with the λ/2 waveplate between PBS2 and PBS3. This adjustment becomes more delicate with the increase in input power. The optical system therefore prefers low input power if possible.

Judging from these comprehensive results, the optimized conditions for this system are a fiber length of 5 km and an input power of 17.6 dBm. When better S/N is required, the fiber length can be shortened or lengthened.

5. Application to secure optical communications

When secure optical communications are considered, determining the optimum S/N ratio is not simple, because higher S/N results in a lower bit-error rate (BER) for eavesdroppers. We do not discuss this problem here, but simply show the feasibility of antisqueezed light in secure optical communications. An eye diagram is a convenient demonstration to compare with ordinary optical communications, and a suitably high S/N condition was selected here. Figure 4 depicts the block diagram of the system used for eye pattern measurement. To obtain sufficiently displaced antisqueezed light, the splitting ratio at PBS3 in Fig. 1 was slightly imbalanced, and the relative phase between the two imbalanced components at PBS2 after a round trip was selected to be π. In this case, the reflected light at PBS2 is antisqueezed light, and the light reflected at PBS1 is local light. When the one-way SMF length was 10 km and input power was 14.9 dBm, 2δφ was 6.5°, ellipticity was 11.9, and S/N was 4.4×10⁴. The local light and antisqueezed light were combined at a 50:50 beam splitter with a 400-ps delay, and pseudo-randomly modulated with 0 or π at a lithium niobate (LN) phase modulator. These correspond to the transmitter side. At the receiver side, the local light and signal light were interfered using an asymmetric interferometer with a 400-ps delay (Optoplex DPSK demodulator), and detected by the 14.4-GHz balanced PDs. Output signals, which passed two low-pass filters with 1.866- and 2.5-GHz cutoff frequencies, were displayed on a sampling oscilloscope. When the Yuen et al.’s protocol is employed, eavesdroppers must detect signals without having information on the sender bases. This corresponds to detecting signals with inappropriate phases. Eye pattern (a) in Fig. 5 is an appropriate-phase case, whereas in pattern (b), the phase is 45° off. Eye pattern (b) not only closes more but also fluctuates more than pattern (a) because of the antisqueezed component. Eye pattern (c), which reflects a 90°- off phase, simply shows fluctuations. These fluctuations are purely the effect of antisqueezing. When the sender bases are randomly modulated according to the Yuen et al.’s protocol, datum points for eavesdroppers who do not know the sender bases are fully fluctuated. Figure 5 demonstrates that antisqueezed light additionally makes eavesdropping difficult in conjunction with cryptographic protocols, and a legitimate receiver who knows the sender bases can detect signals in a similar way to that used in ordinary optical communications.

 

Fig. 4. Block diagram for feasibility study on secure optical communications.

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Fig. 5. Eye patterns. Signal light and local light are interfered with appropriate phase in (a), with 45°-off phase in (b), and with 90°-off phase in (c).

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6. Summary

Antisqueezed light was generated through a reflection-type interferometer using an LD as an optical source. The fiber propagation of a 5-km round trip was required because of low-peak LD power even if that was amplified with an EDFA. However, the fiber interferometer worked without any problems. The fluctuations in output antisqueezed light were increased by noise transfer from the pump light mainly through Raman scattering. Thanks to the noise transfer, the antisqueezed fluctuation-subtended angle at the origin of phase space was large enough and this was 23° at maximum. The ellipticity of the obtained antisqueezed light was 7–9 in the balanced interferometer case.

The feasibility of secure optical communications was demonstrated with simple eye pattern measurements, using highly displaced antisqueezed light. The result suggests that practical and secure communications are possible.