Effect of dispersion on indistinguishability between single-photon wave-packets

Yun-Ru Fan; Chen-Zhi Yuan; Rui-Ming Zhang; Si Shen; Peng Wu; He-Qing Wang; Hao Li; Guang-Wei Deng; Hai-Zhi Song; Li-Xing You; Zhen Wang; You Wang; Guang-Can Guo; Qiang Zhou

doi:10.1364/PRJ.421180

Abstract

With propagating through a dispersive medium, the temporal–spectral profile of optical pulses should be inevitably modified. Although such dispersion effect has been well studied in classical optics, its effect on a single-photon wave-packet has not yet been entirely revealed. In this paper, we investigate the effect of dispersion on indistinguishability between single-photon wave-packets through the Hong–Ou–Mandel (HOM) interference. By dispersively manipulating two weak coherent single-photon wave-packets which are prepared by attenuating mode-locked laser pulses before interfering with each other, we observe that the difference of the second-order dispersion between two optical paths of the HOM interferometer can be mapped to the interference curve, indicating that (i) with the same amount of dispersion effect in both paths, the HOM interference curve must be only determined by the intrinsic indistinguishability between the wave-packets, i.e., dispersion cancellation due to the indistinguishability between Feynman paths; and (ii) unbalanced dispersion effect in two paths cannot be canceled and will broaden the interference curve thus providing a way to measure the second-order dispersion coefficient. Our results suggest a more comprehensive understanding of the single-photon wave-packet and pave ways to explore further applications of the HOM interference.

Similar to procedures for analyzing its classical counterpart, i.e., the electromagnetic wave, properties of single photons have been investigated through interference [1, 2]. Exciting demonstrations, which certify the genuine quantum nature of single photons, have been realized [3 - 5]. One important genuine quantum nature is the indistinguishability between single-photon wave-packets, which has been characterized through Hong-Ou-Mandel (HOM) interference in different degrees of freedom of single-photon wave-packets, such as in spatial mode [6], temporal mode [7], polarization mode [8], spectral mode [9 - 11], and orbital angular momentum mode [12]. Moreover, the HOM interference is also applied to quantum communications with dispersive quantum channels, such as quantum teleportation [13, 14] and measurement device independent quantum key distribution [15 - 17]. Generally, the dispersion distortion should take place after a single-photon wave-packet propagating through dispersive environment. For instance, the dispersion-induced temporal-mode broadening has been investigated with single-photon wave-packets in Ref. [18]. More importantly, the dispersion effect would also change the indistinguishability between single-photon wave-packets in the spectral-temporal profile, which has not yet been revealed so far.

In this work, we investigate the effect of dispersion on indistinguishability between single-photon wave-packets via the HOM interference and implement two proof-of-principle demonstrations with attenuated coherent single-photon wave-packets which are prepared by attenuating mode-locked laser pulses. Our analyses show that the difference of dispersion effect between the propagation paths of the two single-photon wave-packets can be mapped into the HOM interference curve. In one case, the mode-locked laser pulses propagate through a dispersion module, and then are separated into two parts which are attenuated to single-photon levels before sending them to a balanced HOM interferometer. Our results show that the width of the HOM interference curve is independent with the dispersion from the dispersion module, i.e., dispersion cancellation [19 - 21] due to the indistinguishability between Feynman paths [22, 23], thus restoring the original temporal width of the wave-packets. In another case, two attenuated coherent single-photon wave-packets are sent into an unbalanced HOM interferometer, i.e., different dispersion between optical interference paths. The unbalanced dispersion cannot be canceled and will modify the HOM interference curve. By measuring the change of the interference curve, we can obtain the second-order dispersion coefficient of the unbalanced dispersion, thus providing a new method for measuring it.

T_{0}

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Figure 1.Conceptual illustration of the HOM interference to reveal the dispersion effect on the indistinguishability between single-photon wave-packets. (a) HOM interferometer with dispersive manipulation modules. Two identical single-photon wave-packets are manipulated with dispersion modules along two optical paths, i.e., path A and path B, and then are sent into an HOM interferometer; (b) HOM interference curves without the second-order dispersion along two paths; (c) with the same second-order dispersion along two paths, i.e., balanced HOM interferometer; (d) with unbalanced second-order dispersions along two paths. To obtain the HOM interference curve, the travel time of the wave-packet in path A is fixed, and that in path B is varied and the time axis is in reference to the center of the pulse in path A. To guide eyes, envelopes of three sub-wave-packets are depicted with solid and dashed lines in red, green, and blue, respectively, and the black envelope covering the three sub-wave-packets is used to illustrate the widths of the wave-packets.

{| ψ_{0} ⟩}_{A, B} = | α_{A, B} (t) ⟩,

P^{'} (τ)

1.12 \pm 0.01 ps

Figure 2.Experimental setup. The setup consists of attenuated mode-locked laser pulses, HOM interferometer, and photon detection with a dispersion module. VOA, variable optical attenuator; SMC, single-mode fiber coupler; PBS, polarization beam splitter; PMC, polarization-maintaining fiber coupler; PC, polarization controller; SNSPD, superconducting nanowire single photon detector; TDC, time to digital convertor.

1.31 \pm 0.01 ps

Figure 3.HOM interference curves (a) without a dispersion module, and (b) with 50 km long fiber as the dispersion module at the output of the mode-locked laser, respectively. The blue dots are experimental results. The solid purple lines are Gaussian fitting curves obtained via Monte Carlo method with 1000-time random sampling around each measured data assumed as Poissonian distribution.

17.0 ps / (km \cdot nm)

0.93 \pm 0.01 ps

3.76 \pm 0.08 ps

Figure 4.HOM interference curves with a 80 m long single-mode fiber inserted in one path, which corresponds to 16 periods of mode-locked laser pulses.

0.5 ps / (km \cdot nm)

0.5 ps / (km \cdot nm)

Note added: A similar dispersion effect on single-photon wave-packets has recently been reported for single photons generated from the spontaneous parametric downconversion process [44].

Acknowledgment. The authors thank Professor Zhe-Yu Jeff Ou and Professor Xiaoying Li for useful discussions, and Professor Wei Zhang for offering us a mode-locked femtosecond laser.

The paths A and B in Fig.?1(a) are defined as spatial modes A and B, respectively. Assuming the single-photon wave-packets in spatial modes A and B are both in coherent state, we can present their initial quantum state as [24]

{| ψ_{0} ?}_{A, B} = | α_{A, B} (t) ?,

(A1)

satisfying

a_{A, B} (t^{'}) | α_{A, B} (t) ? = α_{A, B} (t^{'}) | α_{A, B} (t) ?,

(A2)

and

\int d t {| α_{A, B} (t) |}^{2} = {| α_{A 0, B 0} |}^{2},

(A3)

where

a_{A, B} (t^{'})

is the annihilation operator of field in modes A or B, and

{| α_{A 0, B 0} |}^{2}

is the averaged photon number in each wave-packet. After passing through the dispersive media in the propagation pathways, the coherent state evolves from Eq.?(A1) into

{| ψ_{1} ?}_{A, B} = | α_{A 1, B 1} (t) ?,

(A4)

where

α_{A 1, B 1} (t) = {(2 π)}^{? 1 / 2} \int d Ω_{A, B} α_{A, B} (Ω_{A, B}) e^{? i (ω_{0} + Ω_{A, B}) t + i β_{A, B} (Ω_{A, B}) L_{A, B}},

(A5)

with

α_{A, B} (Ω_{A, B}) = {(2 π)}^{? 1 / 2} \int d t α_{A, B} (t) e^{i (ω_{0} + Ω_{A, B}) t}

and the central angular frequency of the two wave-packets both assumed to be

ω_{0}

. In Eq.?(A5),

L_{A, B}

and

β_{A, B}

are the length of the dispersive medium on path A or B, and the corresponding propagation constant at angular frequency of

ω_{0} + Ω_{A, B}

. Before being incident on the beam splitter in Fig.?1(a), the wave-packet in mode B is delayed by

τ

and its quantum state turns into

{| ψ_{2} ?}_{B} = {| α_{B 1} (t + τ) ?}_{B} .

(A6)

Introducing spatial modes C and D corresponding to the two output ports of the beam splitter, we can obtain the annihilation operators

a_{C, D} (t)

of fields in the two modes via the transforms

a_{C} (t {) = 2}^{? 1 / 2} [a_{A} (t) + a_{B} (t)],

(A7)

a_{D} (t {) = 2}^{? 1 / 2} [a_{A} (t) ? a_{B} (t)] .

(A8)

The second-order correlation between fields in modes C and D can be obtained as

G_{C, D}^{(2)} (t, t + τ_{c}) = {}_{A}? ψ_{1} |_{B} ? ψ_{2} | a_{C}^{?} (t) a_{D}^{?} (t + τ_{c}) a_{D} (t + τ_{c}) a_{C} (t) {| ψ_{2} ?}_{B} {| ψ_{1} ?}_{A},

(A9)

where

τ_{c}

is the time delay introduced in calculating the second-order correlation function. Then, we can calculate the coincidence counts between the photons detected in modes C and D as

n_{CD} (τ) = η_{C} η_{D} \int_{T_{1}}^{T_{2}} d t \int_{? Δ τ_{c 0} / 2}^{Δ τ_{c 0} / 2} d τ_{c} G_{C, D}^{(2)} (t, t + τ_{c}),

(A10)

where

η_{C, D}

summarize the collection and detection efficiencies of the photons;

T_{1}

and

T_{2}

are the start and stop times of the coincidence counting measurement; and

Δ τ_{c 0}

is the width of the coincidence counting window. After substituting Eqs.?(A7)–(A9) into Eq.?(A10),

n_{CD} (τ)

comprises eight terms possible to be nonzero. We can list them as System.Xml.XmlElementSystem.Xml.XmlElement

At the last step in Eq.?(A12), we assume that

T_{2} ? T_{1}

and

Δ τ_{c 0}

are both remarkably larger than the temporal width of the dispersion-broadened wave-packets, and this is easily satisfied in our experiment. It is obvious that

n_{C D,1} (τ)

and

n_{C D,2} (τ)

originate from the second-order autocorrelation of the two wave-packets, respectively. In fact, in a Hanbury–Brown–Twiss experiment, the normalized second-order autocorrelation function of the two wave-packets is

g_{A, B}^{(2)} (τ_{c}) = \frac{\int_{T_{1}}^{T_{2}} d t \int_{τ_{c} ? Δ τ_{c 0} / 2}^{τ_{c} + Δ τ_{c 0} / 2} d τ_{c}^{'} {}_{A, B}? ψ_{1, 2} | a_{A}^{?} (t) a_{A}^{?} (t + τ_{c}^{'}) a_{A} (t + τ_{c}^{'}) a_{A} (t) | ψ_{1, 2} ?_{A, B}}{{[\int_{T_{1}}^{T_{2}} d t_{A, B} ? ψ_{1, 2} | a_{A}^{?} (t) a_{A} (t) | ψ_{1, 2} ?_{A, B}]}^{2}} .

(A13)

Thus, general forms of

n_{C D,1} (τ)

and

n_{C D,2} (τ)

are

n_{C D,1} (τ) = \frac{η_{C} η_{D}}{4} g_{A}^{(2)} (0) {[\int_{T_{1}}^{T_{2}} d t_{A} ? ψ_{1} | a_{A}^{?} (t) a_{A} (t) | ψ_{1} ?_{A}]}^{2},

(A14)

and

n_{C D,2} (τ) = \frac{η_{C} η_{D}}{4} g_{B}^{(2)} (0) {[\int_{T_{1}}^{T_{2}} d t_{B} ? ψ_{2} | a_{B}^{?} (t) a_{B} (t) | ψ_{2} ?_{B}]}^{2} \cdot

(A15)

Since we assume the single-photon wave-packets are in coherent state,

g_{A, B}^{(2)} (0) = 1

and therefore the results in Eq.?(A11) [Eq.?(A12)] and Eq.?(A14) [Eq.?(A15)] are consistent with each other. System.Xml.XmlElementSystem.Xml.XmlElement

It is obvious that the terms

n_{C D,3} (τ)

and

n_{C D,4} (τ)

oscillate quickly with variable

τ

, and their sum is

n_{C D,3} (τ) + n_{C D,4} (τ) = ? \frac{η_{C} η_{D}}{2} ? \cos (2 ω_{0} τ) Re {\int d Ω_{A} α_{B}^{*} (Ω_{A}) α_{A} (Ω_{A}) e^{? i [β_{B} (Ω_{A}) L_{B} ? β_{A} (Ω_{A}) L_{A}]}}^{2} .

(A18)

The oscillating period of the term

\cos (2 ω_{0} τ)

is smaller than 1?fs for the light field used in our experiment. In our experimental system, especially the one with optical fiber devices, the fluctuation of optical depth makes

τ

vary quickly with respect to the time scale of

T_{2} ? T_{1}

and therefore

n_{C D,3} (τ) + n_{C D,4} (τ)

in Eq.?(A18) turns to be zero. System.Xml.XmlElementSystem.Xml.XmlElement

At the last step in Eq.?(A12), the assumption that

T_{2} ? T_{1}

and

Δ τ_{c 0}

are remarkably larger than the temporal width of the dispersion-broadened wave-packets is used again. System.Xml.XmlElementSystem.Xml.XmlElement

After summing Eqs.?(A11), (A12), and (A16)–(A22) up, we can finally get

n (τ) = \frac{η_{C} η_{D} {| α_{A 0} |}^{2} {| α_{B 0} |}^{2}}{4} {\frac{{| α_{A 0} |}^{2}}{{| α_{B 0} |}^{2}} + \frac{{| α_{B 0} |}^{2}}{{| α_{A 0} |}^{2}} + 2 ? \frac{2}{{| α_{A 0} |}^{2} {| α_{B 0} |}^{2}} ? {| \int d Ω α_{A}^{*} (Ω) α_{B} (Ω) e^{? i [Ω τ + β_{A} (Ω) L_{A} ? β_{B} (Ω) L_{B}]} |}^{2}} \cdot

(A23)

Now, we introduce some assumptions for conveniently comparing Eq.?(A23) to the experiments. First, the averaged photon numbers in the two single-photon wave-packets are both 1, i.e.,?

{| α_{A 0} |}^{2} = {| α_{B 0} |}^{2} = 1

. For convenience,

α_{A 0}

and

α_{B 0}

are treated as 1. Second, the single-photon wave-packets before passing through the dispersive medium are indistinguishable, and their temporal profile is Gaussian, i.e.,?

α_{A, B} (t) = {(T_{0} \sqrt{π})}^{? 1 / 2} e^{? \frac{t^{2}}{2 T_{0}^{2}}} .

(A24)

Third, the effect of higher-order dispersions is negligible, and

β_{A, B} (Ω)

can be approximated by truncating the Taylor expansion to the second order, i.e.,?

β_{A, B} (Ω) \approx β_{A, B} (0) + β_{A, B}^{(1)} (0) Ω + β_{A, B}^{(2)} (0) Ω^{2},

(A25)

where

β_{A, B}^{(1)} (0)

and

β_{A, B}^{(2)} (0)

are the inverse group-velocity and group-velocity dispersion parameter of the dispersive path A (B), respectively.

Under the three assumptions above, Eq.?(A23) gets

n (τ) = η_{C} η_{D} {1 ? \frac{T_{0}^{2}}{\sqrt{4 T_{0}^{4} + α^{2}}} e^{? 2 {(τ ? δ τ)}^{2} / [4 T_{0}^{2} + {(α / T_{0})}^{2}]}},

(A26)

where

δ τ = β_{A}^{(1)} (0) L_{A} ? β_{B}^{(1)} (0) L_{B}

and

α = β_{A}^{(2)} (0) L_{A} ? β_{B}^{(2)} (0) L_{B}

. It is obvious that the coincidence count per trial

P (τ)

can be obtained by normalizing

n (τ)

in Eq.?(A26), i.e.,?

P (τ) = 1 ? \frac{T_{0}^{2}}{\sqrt{4 T_{0}^{4} + α^{2}}} e^{? 2 {(τ ? δ τ)}^{2} / [4 T_{0}^{2} + {(α / T_{0})}^{2}]} .

(A27)

In Eq.?(A27), the

P (τ)

versus

τ

forms the HOM interference curve and shows a dip centering at

τ = δ τ

. The FWHM of this dip is

d = 2 \sqrt{\ln 4} \sqrt{T_{0}^{2} + {(α / 2 T_{0})}^{2}} .

(A28)

The visibility of the HOM-dip is given by

V = \frac{P_{\max} (τ) ? P_{\min} (τ)}{P_{\max} (τ)} = \frac{T_{0}^{2}}{\sqrt{4 T_{0}^{4} + α^{2}}},

(A29)

where

P {(τ)}_{\max}

P {(τ)}_{\min}

is the maximum or minimum of versus

τ

When the two wave-packets experience identical dispersion, i.e.,?

δ τ = 0

and

α = 0

, Eq.?(A16) degrades to

P (τ) = 1 ? \frac{1}{2} e^{? τ^{2} / (2 T_{0}^{2})} .

(A30)

Thus, the FWHM of the HOM-dip is

d = 2 \sqrt{\ln 4} T_{0} = \sqrt{2} T_{FWHM},

(A31)

where

T_{FWHM}

is the FWHM of the wave-packets, and the visibility equals 0.5. If we let

β_{A, B} (Ω) = 0

, i.e.,?there does not exist dispersion along the propagation pathways of the two wave-packets, we can get an expression of

P (τ)

identical to Eq.?(A30). This indicates that the HOM interference curve will not be affected by the balanced dispersion experienced by the two wave-packets. Therefore, we can obtain the original width of the single-photon wave-packet as

1 / \sqrt{2}

times the HOM interference curve even though the wave-packets have been broadened significantly by the dispersive manipulation. According to Eq.?(A28), when the dispersive manipulation in two paths is unbalanced, the difference of the second-order dispersion effect between two paths can accurately be mapped to the HOM interference curve by

α = T_{0} \sqrt{d^{2} / 2 ? \ln 2 ? 4 T_{0}^{2}},

(A32)

which can be utilized to measure the second-order dispersion effect of an unknown optical material.

In the experiment setup shown in Fig.?2, the pulsed light source is selected as a passively mode-locked fiber laser (PFL-200M, Alnair Labs) with the central wavelength of 1548.74?nm, pulse duration of

1.12 \pm 0.01 ?? ps

measured by the second-order autocorrelation method, repetition rate of 40?MHz, and 3?dB spectral bandwidth of 3.48?nm, as shown in Fig.?5(a). The laser pulses are attenuated to single-photon level with the mean photon number of 0.007 per wave-packet through a variable optical attenuator (VOA1, VOA2, and VOA3). To create two single-photon wave-packets, a 50:50 single-mode fiber coupler is used to divide the attenuated laser pulses into two parts propagating along two paths. Two wave-packets are injected into an HOM interferometer, which consists of the two variable optical attenuators (VOA2 and VOA3), two polarization beam splitters (PBS1 and PBS2), a 50:50 polarization-maintaining fiber coupler, and a fiber pigtailed variable optical delay line (Delay), as shown in Fig.?2. We can adjust the VOA2 and VOA3 in the two arms of the HOM interferometer to ensure that the mean photon numbers in the two paths are the same. The two PBS1 and PBS2 are used to ensure the polarization indistinguishability between the two wave-packets. A relative time delay between the two arms in the HOM interferometer is introduced by an optical delay line (MDL-002, General Photonics) with the accuracy of 10?fs and maximum time delay of 560?ps. Two output ports of the HOM interferometer are connected with two SNSPDs with detection efficiency of 68%. The electronic signals generated in photon detection events are input into a TDC (ID 900, ID Quantique) to obtain the coincidence counts. The HOM-dip can be observed in the HOM interference curve, i.e.,?coincidence counts versus the relative time delay between two arms of the HOM interferometer.

Figure 5.(a) Output of a mode-locked pulse laser, measured by a second-order autocorrelator. (b) Output of a mode-locked pulse laser after the dispersive manipulation, recorded by the single-photon detection and a TDC.

To demonstrate the ability of HOM interference in measuring the original width of laser pulses after dispersive manipulation, we insert a 50?km long fiber spool between the mode-locked laser and the VOA1 in Fig.?2 to introduce a remarkable dispersion before the laser pulses entering the HOM interferometer. The pulse width after such dispersive manipulation is directly measured through the single-photon detection and a TDC. The measured results are shown in Fig.?5(b). It shows that after propagating along a 50?km long fiber, the laser pulses have been broadened to

3.4 \pm 0.3 ?? ns

, including a time jitter of the SNSPD and TDC.