1. Introduction

The phase is one of important and manageable information carried by photons which can be used as qubits in quantum communication and computation, thus the ability to control the optical phase at the single-photon level will be a benefit to quantum information science [1–4,6–8]. In recent decades an unique phenomenon of electromagnetically induced transparency (EIT) with the property of extremely high dispersion has been extensively studied and applied in many interesting subjects [9, 10]. Several studies based on EIT have proposed to efficiently enhance photon-photon interaction even at the single-photon level, such as photon switching [11] and cross-phase modulation [12, 13]. However, owing to the narrow EIT transparency window, the width of a light pulse propagating through an EIT medium is usually selected as long as possible to avoid additional loss and achieve steady-state transmission. Typically, the width of a light pulse for EIT-based experiments is about a few µs [14–16]. The power of a 2.5-µs single-photon pulse is 100 fW, where the laser wavelength is 780 nm. Hence, the ultralow-light-level phase measurement in the EIT-based system is an important issue that allows us to determine the phase properties of light pulses at the single-photon level for realizing practical applications in quantum information processing, such as quantum phase gates [17–21].

A simple method called the beat-note interferometer, which can be used to measure the phase evolution from the head to tail of a light pulse, was proposed and demonstrated by Chen et al. [22]. The authors claimed that a phase shift of a Gaussian pulse with a peak power of 400 pW was observed. They also suggested that the sensitivity of this method can be improved by using a high-gain detector. In this paper, we adopt this suggestion and show an experimental demonstration of utilizing a beat-note interferometer to simultaneously measure the phase and amplitude variations of light pulses after propagating through an EIT medium at femtowatt-light levels. The experimental results also show that the transmission of a light pulse with an energy at the single-photon level can be detected by the beat-note interferometer without needing a single photon counter module. In the presented scheme, the sensitivity of the interferometer for the phase measurement was enhanced at least 5 orders compared with the previous study. It is a significant improvement when applying this method to the single-photon-level phase measurement in an EIT-based system or a highly dispersive medium. Moreover, we observe that the measured phase noise approaches the shot-noise level arising from the fluctuations of detected photons.

 

Fig. 1. (a) Three-level Λ-scheme used for EIT in ⁸⁷Rb D ₂ line. (b) A schematic diagram of the experimental setup. BS, beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; POL, polarizer; SM Fiber, single-mode fiber; AOM, acousto-optic modulator; PD, photo detector; PMT, photomultiplier tube. The 1/e diameters of probe and coupling beams are around 1 and 5 mm, respectively.

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2. Experimental details

We experimentally study the EIT effect in the laser-cooled ⁸⁷Rb atoms produced by a vapor-cell magneto-optical trap (MOT). Around 3 × 10⁹ atoms are trapped in the MOT, as measured by the optical-pumping method. Figure 1(a) shows the relevant energy levels of ⁸⁷Rb D ₂ transitions that is used for the Λ-scheme EIT experiment. This scheme consists of the two levels ∣F = 1〉 ≡ ∣1〉 and ∣F = 2〉 ≡ ∣2〉 of the 5S _1/2 ground states, and the ∣F′ = 2〉 ≡ ∣3〉 level of the 5P _3/2 excited state. A strong coupling field with frequency ω_c and a weak probe field with frequency ω_p are employed to drive ∣2〉 ↔ ∣3〉 and ∣1〉 ↔ ∣3〉 transitions, respectively. These two fields, which are circularly polarized with σ+ polarization, form a three-level Λ-type configuration of EIT system.

The response of the atomic ensemble experienced by the probe field can be described in terms of the probe-field linear susceptibility χ (ω_p) [10]. The transmission and phase shift of the probe field induced by the EIT medium are defined as T = exp[−Imχ (ω_p)kL] and $∆{\phi }_{\mathrm{EIT}}=\frac{1}{2}\mathrm{Re}\chi \left({\omega }_p\right)\mathrm{kL}$ , respectively, where k is the wave number of the probe field and L is the optical path length of the medium. They are given by

where nσ ₁₃ L is the optical density for the probe transition (n is the atomic density, and σ ₁₃ is the atomic cross section of the ∣1〉 ↔ ∣3〉 transition), and Ω_c is the Rabi frequency of the coupling transition. The one-photon (probe) detuning is defined as Δ_p = ω_p − ω ₃₁, where ω ₃₁ is the frequency of the ∣1〉 ↔ ∣3〉 transition, and the two-photon detuning is defined as δ = Δ_p − Δ_c, where Δ_c is the detuning of coupling field (see Fig. 1(a)). γ ₂₁ is the decoherence rate between the two ground states, and γ ₃₁ = Γ + γ _3deph is the total coherence decay rate out of the excited state ∣3〉, where Γ = 2π × 6 MHz is the spontaneous decay rate of the excited state, and γ _3deph is the dephasing rate of the state ∣3〉 arising from the linewidth of laser fields and elastic collisions between atoms.

A schematic diagram of the experimental setup is shown in Fig. 1(b). The probe and coupling fields come from two diode lasers, respectively. The coupling laser is directly injection-locked by an external cavity diode laser (ECDL). One beam from the ECDL is sent through a 6.8 GHz electro-optic modulator (EOM, New Focus 4851). The probe laser is injection-locked by an intermediate laser seeded with the high-frequency sideband of the EOM output. The above arrangement can completely eliminate the influence of the carrier of the EOM output on the probe laser. The coupling and probe lasers, which are switched on or off by an acousto-optic modulator (AOM), propagate with an angle of 0.9°. A quarter-wave plate preceding the atomic sample converts the probe and coupling fields from linear into circular polarizations in the EIT experiment. The period of time sequence in the experiment is 10 ms. After all lasers and magnetic fields of the MOT are turned off and the coupling field is turned on for 100 µs, the 50-µs probe square pulse is switched on for measurement.

3. Beat-note interferometer

The beat-note interferometer is applied to simultaneously record the amplitude and phase variations of a probe pulse propagating through an EIT medium. The probe laser is first split into transmitted and reflected beams by a 50:50 beam splitter (BS1), as shown in Fig. 1(b). The transmitted beam passes the AOM to generate a first-order beam for the probe pulse and then recombines with the reflected beam (zeroth-order beam) on BS2 to form two beat-note signals. After leaving BS2, the two beat signals are sent to respective single-mode fibers to obtain the optimal spatial mode-matching. One beam is called the reference beat note, which is directly received by photo detector (PD1, New Focus 1801). The other beam, corresponding to the probe beat note is detected by PD2 (Hamamatsu H6780-20 and C9663) after interacting with atoms. Furthermore, the power of the probe laser is varied with polarizers and half-wave plates. The AOM driving frequency, ω_a, of the probe laser is 2π × 80 MHz, which is sufficiently large that the interaction between the zeroth-order beam and the EIT medium is negligible. Both reference and probe beat notes, which carries the beat frequency of ω_a, display the waveforms on the oscilloscope (OSC, Agilent MSO6034A).

where I _0r, I _1r, I _0p and I _1p are the intensities of zeroth- and first-order beams along the path of reference and probe beat notes, respectively. Δφ is the phase shift induced by the medium. φ_r and φ_p are phases that result from optical paths, components, or other factors. The difference between φ_r and φ_p is always fixed, thus we can directly measure the phase shift of a probe pulse by comparing the two beat signals using OSC. At the same time, the probe transmission (I _1p) can be obtained from the amplitude of probe beat notes according to Eq. (4). Throughout the experiment, only the phase shift within 200 ns of the end of the probe pulse is considered in order to obtain the steady-state results of EIT, and the power of the zeroth-order beam for the probe beat note is fixed at 2 nW. In addition, the measurement uncertainty is evaluated using 10 samples, with each sample averaged 4096 times using OSC.

 

Fig. 2. The beat-note amplitude (squares) and corresponding phase noise (circles) as a function of the peak power of probe pulse. Black and red dashed curves are fits of $f=a\sqrt{I_{1p}}+b$ and $g=\frac{p}{\sqrt{I_{1p}}}+q$ , respectively. $a=6.22\left[\frac{\mathrm{mV}}{\sqrt{\mathrm{pW}}}\right]$ , b = 0.88 mV, $p=0.016[\mathrm{rad}·\sqrt{\mathrm{pW}}]$ and q = 0.006 rad. The gain of PMT module is about 9 × 10⁷ V/W.

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Below we list two main differences between this work and the previous study [22]. (i) A high-gain photodetector (PD2) is adopted to measure the weak probe field. (ii) The experimental setup has been modified for the ultralow-light-level phase measurement (Fig. 1). For example, using the single-mode fiber in this experiment allows us to obtain the optimal spatial mode-matching between the zeroth- and first-order beams. Consequently, we can accurately and simultaneously determine the probe transmission from Eq. (4). These differences will enhance the sensitivity of the beat-note interferometer at least 5 orders of magnitude compared to the former investigation.

4. Experimental results and discussion

Figure 2 shows the measured beat-note amplitudes (squares) and the corresponding phase noise (circles) as a function of the peak power of the probe pulse in the absence of atoms. Squares are fitted by the function of $f=a\sqrt{I_{1p}}+b$ (black dashed lines), where $a=6.22\left[\frac{\mathrm{mV}}{\sqrt{\mathrm{pW}}}\right]$ indicates the gain for probe beat notes in the case of the zeroth-order beam of 2 nW, and b = 0.88 mV represents the noise level as the probe power is close to zero. The gain of the PMT module is about 9 × 10⁷ V/W in the experiment. We note that the measured beat-note amplitudes are only 80% of that estimated from the gain of the PMT module due to the finite frequency response of the PMT module.

The measured data for the phase noise are fitted with the function of $g=\frac{p}{\sqrt{I_{1p}}}+q$ (red dashed lines), where $p=0.016[\mathrm{rad}·\sqrt{\mathrm{pW}}]$ and q = 0.006 rad. Here, q represents an unknown phase noise, which is not decreased to zero with increasing the light power. The fluctuation of optical paths from BS2 to two photodetectors, which are caused by mechanical vibrations, would result in phase noise for the measurement [22]. Nevertheless, the mechanical noise per unit length is ω_a/c or about 1.7 × 10⁻³ rad/mm in our experiment, where c is the speed of light. The unknown phase noise is much larger than the mechanical noise, hence it does not arise from the mechanical vibrations. The first term in function g exhibits the measured phase noise approaches the shot-noise level. The shot noise caused by the fluctuations of detected photons is predicted by the uncertainty principle ΔϕΔn ≥ 1, where Δϕ and Δn are the uncertainty in the optical phase and the photon number, respectively. For example, a probe pulse with a peak power of 100 pW contains 80 photons within the temporal length of 200 ns. Δn for the coherent-state light is equal to the square root of the mean photon number per light pulse. Therefore, the phase noise, Δϕ, is estimated to be about 0.001 rad among 10 samples, with each sample averaged 4096 times. Under the same conditions, the measured phase noise is about 0.002 rad, which is larger than the shot noise due to additional noise sources or other factors.

 

Fig. 3. (a) Observed EIT transmission versus probe laser detuning. The inset shows the EIT transmission window. (b) Measured phase shift of a light pulse induced by an EIT medium. Filled circles represent the peak power of light pulses below 10 fW. Red dashed lines in (a) and (b) correspond to theoretical curves calculated by Eqs. (1) and (2), respectively. The parameters for the theoretical curves in both figures are the same: nσ ₁₃ L = 16, γ ₂₁ = 0.002Γ, γ ₃₁ = 1.25Γ, Ω_c = 0.31Γ, and Δ_c = −0.7 MHz.

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Next, we measure the transmission and phase shift of a probe pulse after propagating through an EIT medium via beat-note interferometer. The input power of the probe pulse is set to 500 pW. Figure 3(a) shows the probe transmission as a function of probe detuning. The experimental data and theoretical predictions are plotted in circles and red dashed lines, respectively. The parameters for the theoretical calculation of Eq. (1) are determined from other experiments. Ω_c is 0.31Γ, as obtained from the separation of two absorption peaks in the EIT spectrum. γ ₂₁ is 0.002Γ, as estimated from the degree of transparency in the EIT spectrum. γ ₃₁ = 1.25Γ and Δ_c = −0.7 MHz are determined from the spectrum of one-photon absorption for the probe transition. Of note, nσ ₁₃ L is 16, as derived from the group delay time of slow light pulses [10], which means that under the condition of without EIT, a resonant probe pulse has a power loss of 16 or, equivalently, a transmission of e ⁻¹⁶. Figure 3(b) presents the synchronous observation of EIT phase shift of a probe pulse as a function of probe detuning. The red dashed line is the theoretical curve predicted by Eq. (2) with the same parameters in Fig. 3(a). Figures 3(a) and 3(b) show the good agreement between experimental data and theoretical predictions using the same parameters.

In particular, the filled circles shown in Fig. 3(b) represent the peak powers of the transmitted probe pulse below 10 fW. Here, the probe pulse has a large power loss due to its frequency is outside of the EIT transparency window, as indicated by Eq. (1). In the experiment, the phase shifts of light pulses at femtowatt-light levels were successfully measured, and the corresponding phase noise are between 0.1 ~ 0.3 rad which depend on the magnitude of the probe power. When the probe power is larger than 100 fW, the phase noise can be reduced to below 0.1 rad. In fact, the minimum light power we measured is few hundred attowatts, as shown with filled circles in Fig. 3(b). These data own the phase noise larger than 0.3 rad and a discrepancy compared to the theoretical curve. This deviation is attributed to the degradation of the signal-to-noise ratio (SNR). Although these data possess large phase noise, the measured values are still valid in virtue of the considerable phase shift.

 

Fig. 4. Measured phase noise versus number of average on oscilloscope. The red dashed line is a fitting curve based on equation h = α/√N + β, where α = 11.03 [rad · √N] and β = 0.009 rad.

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In order to further improve the phase noise at the femtowatt-light levels, the measurement time is increased using OSC. Figure 4 shows the phase noise dependence of the number of average, N, at Δ_p = −0.31 MHz [see Fig. 3(b)] corresponding to the transmitted probe power of 10 fW under EIT conditions. It is worth noting that a 50-µs probe pulse with a power of 10 fW contains only ~ 2 photons. The measured phase noise is decreased from 0.45 to 0.06 rad when the number of average is increased or, equivalently, SNR is enhanced. The experimental data are fitted by the function of h = α/√N + β, where β = 0.009 rad that is slightly larger the unknown phase noise of 0.006 rad (see Fig. 2) due to the instability of EIT system. It shows a well-known result that increasing N improves the SNR by √N.

5. Conclusion

In conclusion, we experimentally demonstrated the beat-note interferometer for directly and dynamically measuring the phase shift and transmission of a light pulse in an EIT system at the femtowatt-light levels. Also, we observe that the measured phase noise approaches the photon shot-noise level. The presented scheme not only reveals prospects for ultra-low-light phase measurement, but also has various applications in quantum and nonlinear optics. For instance, this apparatus can be used to study the single-photon-level cross-phase modulation [23] and its transient effects [24, 25], which will be a benefit to implement the quantum phase gates.