Quantum information process usually involves quantum bits (qubits) that are independent of each other. Quantum operations are performed on them to make them entangled to each other. If photons are used as qubits, nonlinear optical process is usually needed to make them entangled. However, recent protocol with linear optical elements makes linear quantum optical computation possible [1]. Crucial in this protocol is the Hong-Ou-Mandel (HOM) interference effect [2] that allows two independent photons to interfere and generate entanglement between them after some projections such as coincidence measurement. As a matter of fact, creation of entanglement between independent photons was already realized in the schemes such as quantum state teleportation [3] and entanglement swapping [4].

So far, qubits are produced in the form of heralded single-photon state [5] from parametric process such as parametric down-conversion (PDC) and four-wave mixing (FWM) [3, 4, 6, 7, 8, 9, 10, 11]. These processes generate nonclassical two-photon states entangled in a variety of degrees of freedom. Recently, interest arises in obtaining a single-photon state from a weak classical source of light such as a weak coherent state. Two-photon interference was demonstrated with a heralded single-photon state and a weak coherent state which provides another source of independent photons [12, 13, 14].

Investigation of quantum interference of two photons between quantum state and classical state started as early as in 1988 [15], right after the famous HOM two-photon interference experiment [2]. It was predicted [15] that when the classical field is much weaker than the single-photon state, the visibility of the two-photon interference can approach 100%. The secret behind this is the visibility of two-photon interference involving single-photon state is independent of the ratio between the intensities of the two fields, unlike interference between two classical fields [16]. In 1991, Tan et al [17] made the first proposal of using two-photon interference between a single-photon state and a classical coherent state to demonstrate nonlocality in phase correlation. Later, Rarity and Tapster [18] proposed a scheme for creating entangled three-photon GHZ state with a coherent state and two-photon state from PDC. Recently, Pittman and Franson [13] first realized the scheme proposed in Ref.[17] and demonstrated two-photon interference between a heralded single-photon state and an independent coherent state.

The reason that a classical state can provide a source as a single-photon is the projection measurement that is realized in photon coincidence measurement. When the light source is very weak, the dominating term in the state describing any field, classical or quantum, is the vacuum state. But vacuum can’t contribute to the photon detection. So the next dominating term is the non-vanishing lowest photon number state. In most cases such as a coherent state and a thermal state, it is a single-photon state. As a matter of fact, a weak thermal state was initially used as a single-photon source for the BB84 protocol of quantum cryptography [19]. However, two-photon interference cannot be realized with two classical sources because the two-photon events in any one of the sources alone are of the same order as the two-photon events from two such sources respectively. The two-photon events in any one of the sources alone do not participate in two-photon interference and will raise the baseline and reduce the visibility to less than 50%. From the theory in Ref.[15], we need at least one source to have less or zero probability of two-photon events in order to achieve a visibility larger than 50%. A single-photon source is an ideal such source.

In this paper, we report an experiment of two-photon HOM type interference between a heralded single-photon state and a classical source, which are generated from two independent dispersion-shifted fibers (DSF) via FWM. This classical source is of thermal nature and yet we observed a visibility of (82±11)%, which is obviously larger than the classical limit of 50% in two-photon interference.

Our experiment setup is shown in figure 1. The correlated signal and idler photon pairs are generated by FWM process in a DSF. Two independent sources of photon pairs are obtained by passing the pulsed pump through a 60/40 coupler and then pumping DSF1 and DSF2, respectively. Pump pulses of about 4 ps duration and central wavelength λ_p=1538.9 nm arrive at 40 MHz rate. DSF1 and DSF2, both with a length of about 300 m and zero dispersion wavelength λ ₀=1538±2 nm, are loosely wrapped into coils and merged in liquid nitrogen (77 K) to suppress the Raman scattering. To adjust the pump power of each fiber source independently, an attenuator is placed between the output of the 60/40 coupler and DSF1.

 

Fig. 1. The experimental setup. FPC, fiber polarization controller; PBS, polarization beam splitter; F, filter; Att, attenuator; SPD, single photon detector.

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For each DSF-based source, which is described in more detail in Ref.[20], signal and idler photon-pairs at wavelengths of 1546.9nm and 1530.9 nm, respectively, are produced by pumping DSFs with the pulsed pump. Signal1 (signal2) and idler1 (idler2) photons co-polarized with the pump are selected by properly adjusting the fiber polarization controller FPC1 (FPC2) placed in front of the polarization beam splitter PBS1 (PBS2). To reliably detect the signal and idler photons, an isolation between the pump and signal/idler photons in excess of 100 dB is required, because of the low efficiency of spontaneous FWM in DSF. We achieve this by passing the output of DSF1 (DSF2) through a dual-band filter F composed by cascading a double grating filter with array-waveguide gratings or a tunable filter [20]. The bandwidth of the pump, signal and idler photons are about 1 nm, 0.33 nm, and 0.7 nm respectively.

The generated signal and idler photons are counted by single photon detectors (SPDs, id200 and PLI-AGD-SC-Rx) operated in the gated Geiger mode. The rate of the 2.5 ns gate is about 600 KHz, which is 1/64 of the repetition rate of the pump pulses, and the dead time of the gate is set to be 10 µs. The quantum efficiency of SPD1, SPD2, and SPD3 is 7%, 10%, and 20%, respectively. The total detection efficiency for signal1 (signal2) and idler2 is 0.6% (0.7%) and 3%, respectively, when the efficiencies of the other transmission components, such as the DSF, PBS, and filter F, are included.

In the experiment, the heralded single-photon state is generated from the signal field of DSF2 by gating on the detection of the idler2 photons with SPD3. The thermal state is produced by DSF1 from only one of the correlated photons, i.e. signal1. This is so because it has been shown that the spontaneousFWM process, which is equivalent to a parametric amplifier, places each field of the two correlated photons in a thermal state [21]. To confirm this, we measure the normalized intensity correlation function g ⁽²⁾ for the individual signal photons and obtain g ⁽²⁾=2.00±0.03 [20]. This result shows that the signal photons produced by each of the fiber sources are of thermal nature and have high spatial and temporal coherence. The single photon state (heralded signal2 field) and the thermal state (signal1 field), both having a central wavelength of 1546.9nm and a FWHM of 0.33 nm, are carefully path matched and simultaneously fed into a fiber coupler acting as a 50/50 beam splitter from two input ports, respectively. Before coupling in the 50/50 beam splitter, thermal state in signal1 field is delayed by the reflector mirrors mounted on a translation stage. To ensure the two input fields of the 50/50 coupler have the identical polarization, the polarization of the thermal field is properly adjusted by FPC3. The two output ports of the fiber coupler are detected by SPD1 and SPD2, respectively.

 

Fig. 2. Three-fold coincidence rate versus the position of the translation stage when the average photon number per pulse of (signal1) thermal field is about (a) 0.26 and (b) 0.38. Three-fold background coincidences of 47 and 40 counts/40min are subtracted from the data in (a) and (b), respectively.

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The observation of the interference between the single photon state and the thermal state is achieved by measuring three-fold coincidences versus the position of the translation stage. During the measurement, the single counts, which are the outputs of SPD1, SPD2, and SPD3, and two-fold coincidence between arbitrary two SPDs are also recorded by the coincidence counting system. For an ideal single photon state, there is no need to modify the measured three-fold coincidences if the dark counts of the SPDs are negligible. But for the heralded single photon state, background counts caused by the multiphoton pairs of signal2 and idler2 fields should be subtracted. To reliably remove the background, single counts, two-fold and three-fold coincidence are measured when only the heralded single photon state is presented in the input of the 50/50 beam-splitter. The direct measurement of the background counts, i.e. the three-fold coincidences due to multiphoton pair events, should be consistent with the calculated results obtained by using the measured single counts and two-fold coincidences, and the experimental results agree with the prediction. Figure 2 shows the three-fold coincidences rate as a function of the position of the translation stage or the relative delay in the optical paths between the signal1 and signal2 fields, after subtraction of a background counts of 47/40min and 40/40min in Fig. 2(a) and (b), respectively. The production rate of photon pairs originated from DSF2 is about 0.15 pairs/pulse, and the average photon number of the signal1 (thermal) field produced by DSF1 is about 0.26 and 0.38 photons/pulse for the data in Fig. 2(a) and (b), respectively. The solid curves are the least-square fits to the data, which show a dip of (82±11)%and (67±11)% visibility are obtained in Fig. 2(a) and (b), respectively.

To understand the experimental results in Fig. 2, let us consider the density operator of the system as $\widehat{\rho }$_sys=$\widehat{\rho }$^th ₁⊗$\widehat{\rho }$ ^1p ₂, where $\widehat{\rho }$^th ₁ is the density operator for the thermal state and $\widehat{\rho }$ ^1p ₂=|1〉₂〈1| is for single-photon state, which is obtained by gating on the detection of idler2 photon. The field operators at SPD1 and SPD2, a _D1 and a _D2, are related to signal1 and signal2 as

where â ₁ and â ₂ are the field operators for the thermal state and single-photon state, respectively. The two-photon coincidence probability P ₂ is proportional to [15]

where n̅ is the average photon number of the thermal field, and I ₁₂=〈a ^† ₂ (t)a ^† ₁(t′)a ₂(t′)a ₁(t)〉/n̅ is the interference term, that leads to two-photon interference and depends on the overlap between signal1 and signal2 fields at the 50/50 beam splitter. When t=t′,I ₁₂=1 corresponding to the bottom of the dip in figure 2 but when t and t′ are very different, I ₁₂=0 corresponding to the wings in figure 2. From Eqs (1)–(2), we have $P_2=\frac{\overline{n}}{2}\left(\overline{n}+1-I_{12}\right)$ . So the visibility is simply $V=\frac{1}{1+\overline{n}}$ . It is clear that V approaches to 1 or 100% visibility if the average photon number of the thermal state n ̅≪1.

Experimentally, we can estimate n̄ from measured efficiencies of detectors and single counts and obtain n̄=0.26±0.035 and n̄=0.38±0.05, which leads to V_theory=0.79±0.03 and V_theory=0.72±0.03. These values are consistent with the results in Fig. 2.

In summary, we demonstrate a two-photon interference between a single-photon state and an independent thermal state with a high visibility. Because the experimental parameters of our DSF-based sources are carefully optimized [6, 20, 9], the observed visibility (82±11)% is obviously beyond the classical limit of 50%, while a recently reported HOM experiment with two heralded single photon states in the telecom band generated by optical fibers was only about 53% [10]. Comparing with the previous similar experiments reported in Ref.[12], in which the achieved visibility was much smaller than the calculated result due to mode mismatch, we obtain the quantum interference with a visibility consistent with theory expectation owning to the perfect mode match provided by the fiber sources [20, 11]. This may lead the fiber sources to have more applications in quantum information.