Transfer of temporal coherence in parametric down-conversion

TL;DR

This paper demonstrates that the temporal coherence of a partially coherent pump field in parametric down-conversion is fully transferred to the entangled photon pairs, impacting quantum communication applications.

Contribution

It provides a theoretical framework for understanding how pump coherence properties influence the entanglement and coherence of down-converted photon pairs.

Findings

Coherence functions factorize into pump and signal-idler components.

Explicit expressions derived for Gaussian Schell-model pump fields.

Concurrence of entangled states bounded by pump coherence.

Abstract

We show that in parametric down-conversion the coherence properties of a temporally partially coherent pump field get entirely transferred to the down-converted entangled two-photon field. Under the assumption that the frequency-bandwidth of the down-converted signal-idler photons is much larger than that of the pump, we derive the temporal coherence functions for the down-converted field, for both infinitely-fast and time-averaged detection schemes. We show that in each scheme the coherence function factorizes into two separate coherence functions with one of them carrying the entire statistical information of the pump field. In situations in which the pump is a Gaussian Schell-model field, we derive explicit expressions for the coherence functions. Finally, we show that the concurrence of time-energy-entangled two-qubit states is bounded by the degree of temporal coherence of the pump field. This study can have important implications for understanding how correlations of the pump field manifest as two-particle entanglement as well as for harnessing energy-time entanglement for long-distance quantum communication protocols.

Equations

$$\Delta \tau \equiv {\tau }_1-{\tau }_2$$

$$\Delta {\tau }^{\prime }\equiv {\tau }_1^{\prime }-{\tau }_2^{\prime }$$

$$\Delta \varphi \equiv {\varphi }_1-{\varphi }_2$$

$$\begin{gathered} |\psi {⟩}_1=A_1\iint d{\omega }_{s}d{\omega }_{i}V({\omega }_p){\Phi }_1({\omega }_s,{\omega }_i) \\ \times e^{i({\omega }_p{\tau }_{p1}+{\varphi }_{p1})}|{\omega }_s{⟩}_{{\omega }_{s1}}|{\omega }_i{⟩}_{{\omega }_{i1}}, \end{gathered}$$

$$\hat{\rho}=\big{\langle}|\psi\rangle\langle\psi|\big{\rangle}.$$

$${\hat{E}}_s^{(+)}(t)={\kappa }_{s1}{\hat{E}}_{s1}^{(+)}(t-{\tau }_{s1})+{\kappa }_{s2}{\hat{E}}_{s2}^{(+)}(t-{\tau }_{s2}),$$

$${\hat{E}}_i^{(+)}(t)={\kappa }_{i1}{\hat{E}}_{i1}^{(+)}(t-{\tau }_{i1})+{\kappa }_{i2}{\hat{E}}_{i2}^{(+)}(t-{\tau }_{i2}),$$

$${\hat{E}}_{s1}^{(+)}(t-{\tau }_{s1})=$$

$$e^{i{\varphi }_{s1}}{\int }_0^{\infty }$$

$$\begin{gathered} R_{si}^{(2)}(t_s,t_i)= \\ {\kappa }_1^2{R}^{(2)}(t_s,t_i,{\tau }_{s1},{\tau }_{i1})+{\kappa }_2^2{R}^{(2)}(t_s,t_i,{\tau }_{s2},{\tau }_{i2}) \\ +{\kappa }_1{\kappa }_2{\Gamma }^{(2)}(t_s,t_i,{\tau }_{s1},{\tau }_{i1},{\tau }_{s2},{\tau }_{i2})e^{-i\Delta \varphi }+c.c., \end{gathered}$$

$${\Gamma}^{(2)}(t_{s},t_{i},\tau_{s1},\tau_{i1},\tau_{s2},\tau_{i2})=\mathrm{Tr}\big{\{}\hat{\rho}\,\hat{E}_{s1}^{(-)}(t_{s}-\tau_{s1})\\ \times\hat{E}_{i1}^{(-)}(t_{i}-\tau_{i1})\hat{E}_{i2}^{(+)}(t_{i}-\tau_{i2})\hat{E}_{s2}^{(+)}(t_{s}-\tau_{s2})\big{\}},$$

$$R^{(2)}(t_s,t_i,{\tau }_{s1},{\tau }_{i1})={\Gamma }^{(2)}(t_s,t_i,{\tau }_{s1},{\tau }_{i1},{\tau }_{s1},{\tau }_{i1}).$$

$$\begin{gathered} {\Gamma }^{(2)}(t_s,t_i,{\tau }_{s1},{\tau }_{i1},{\tau }_{s2},{\tau }_{i2})= \\ \times {\Gamma }_p\left({\tau }_1-\frac{t_s+t_i}{2},{\tau }_2-\frac{t_s+t_i}{2}\right) \\ \times {\Gamma }_d\left({\tau }_1^{\prime }-\frac{t_s-t_i}{2},{\tau }_2^{\prime }-\frac{t_s-t_i}{2}\right), \end{gathered}$$

$$\begin{gathered} {\Gamma }_p\left({\tau }_1-\frac{t_s+t_i}{2},{\tau }_2-\frac{t_s+t_i}{2}\right) \\ =e^{-i{\omega }_{p0}\Delta \tau }\iint d{\bar{\omega }}_p^{\prime }d{\bar{\omega }}_p^{\prime \prime }⟨V^{\ast }({\bar{\omega }}_p^{\prime }+{\omega }_{p0})V({\bar{\omega }}_p^{\prime \prime }+{\omega }_{p0})⟩ \\ \times \exp \left[-i{\bar{\omega }}_p^{\prime }\left({\tau }_1-\frac{t_s+t_i}{2}\right)\right]\exp \left[i{\bar{\omega }}_p^{\prime \prime }\left({\tau }_2-\frac{t_s+t_i}{2}\right)\right], \end{gathered}$$

$$\begin{gathered} {\Gamma }_d\left({\tau }_1^{\prime }-\frac{t_s-t_i}{2},{\tau }_2^{\prime }-\frac{t_s-t_i}{2}\right) \\ =e^{-i{\omega }_{d0}\Delta {\tau }^{\prime }}\iint d{\bar{\omega }}_d^{\prime }d{\bar{\omega }}_d^{\prime \prime }⟨g_1^{\ast }({\bar{\omega }}_d^{\prime })g_2({\bar{\omega }}_d^{\prime \prime })⟩ \\ \times \exp \left[-i{\bar{\omega }}_d^{\prime }\left({\tau }_1^{\prime }-\frac{t_s-t_i}{2}\right)\right]\exp \left[i{\bar{\omega }}_d^{\prime \prime }\left({\tau }_2^{\prime }-\frac{t_s-t_i}{2}\right)\right], \end{gathered}$$

$$g_1(\omega )={\Phi }_1\left({\omega }_{s0}+\frac{\omega }{2},{\omega }_{i0}-\frac{\omega }{2}\right)f_{s1}\left(\frac{\omega }{2}\right)f_{i1}\left(-\frac{\omega }{2}\right),$$

$${\bar{\Gamma }}^{(2)}$$

$$={⟨{\Gamma }_p\left({\tau }_1-\frac{t_s+t_i}{2},{\tau }_2-\frac{t_s+t_i}{2}\right)⟩}_{\frac{t_s+t_i}{2}}$$

$$\times {⟨{\Gamma }_d\left({\tau }_1^{\prime }-\frac{t_s-t_i}{2},{\tau }_2^{\prime }-\frac{t_s-t_i}{2}\right)⟩}_{\frac{t_s-t_i}{2}},$$

$${\bar{\Gamma }}^{(2)}$$

$$=\sqrt{{\bar{I}}_1{\bar{I}}_2}\sqrt{{\bar{G}}_1{\bar{G}}_2}{\bar{\gamma }}_p\left(\Delta \tau \right){\bar{\gamma }}_d\left(\Delta {\tau }^{\prime }\right)$$

$$={\bar{R}}^{(2)}{\bar{\gamma }}_p\left(\Delta \tau \right){\bar{\gamma }}_d\left(\Delta {\tau }^{\prime }\right).$$

$$\begin{gathered} {\bar{R}}_{si}^{(2)}={\kappa }_1^2{\bar{R}}^{(2)}+{\kappa }_2^2{\bar{R}}^{(2)}+{\kappa }_1{\kappa }_2{\bar{R}}^{(2)}{\bar{\gamma }}_p\left(\Delta \tau \right){\bar{\gamma }}_d\left(\Delta {\tau }^{\prime }\right) \\ \times e^{i({\omega }_{p0}\Delta \tau +{\omega }_{d0}\Delta {\tau }^{\prime }+\Delta \varphi )}+c.c. \end{gathered}$$

$$\big{\langle}V^{*}(\omega^{\prime\prime}+\omega_{0})V(\omega^{\prime}+\omega_{0})\big{\rangle}=\\ A\exp\left[-\frac{({\omega^{\prime}}^{2}+{\omega^{\prime\prime}}^{2})}{4\left(\Delta\omega_{p0}\right)^{2}}\right]\exp\left[-\frac{\left(\omega^{\prime}-\omega^{\prime\prime}\right)^{2}}{2\left(\Delta\omega_{c}\right)^{2}}\right].,$$

$${\Gamma }_{\mathrm{GS}}(t_1,t_2)=\sqrt{I(t_1)I(t_2)}{\gamma }_{\mathrm{GS}}(\Delta t),$$

$$I(t_{1(2)})$$

$$\mathrm{and}{\gamma }_{\mathrm{GS}}(\Delta t)$$

$${\Gamma }^{(2)}(t_s,t_i,{\tau }_1,{\tau }_2)={\Gamma }_p\left({\tau }_1-\frac{t_s+t_i}{2},{\tau }_2-\frac{t_s+t_i}{2}\right)$$

$$=\sqrt{I\left({\tau }_1-\frac{t_s+t_i}{2}\right)I\left({\tau }_2-\frac{t_s+t_i}{2}\right)}{\gamma }_p(\Delta \tau ),$$

$$I\left({\tau }_1-\frac{t_s+t_i}{2}\right)=\frac{2\pi \Delta {\omega }_{p0}A}{T}\exp \left[-\frac{{\left({\tau }_1-\frac{t_s+t_i}{2}\right)}^2}{2T^2}\right],\mathrm{etc},$$

$$\mathrm{and}{\gamma }_p(\Delta \tau )=\exp \left[-\frac{\Delta {\tau }^2}{2{\tau }_{\mathrm{coh}}^2}\right].$$

$${\bar{\Gamma }}^{(2)}={\bar{\Gamma }}_p({\tau }_1,{\tau }_2)$$

$${\bar{\gamma }}_p(\Delta \tau )=\exp \left[-\frac{\Delta {\tau }^2}{2{\bar{\tau }}_{\mathrm{coh}}^2}\right],$$

$$\rho =\left(\begin{array}{ccccccc}a& & 0& & 0& & c\\ 0& & 0& & 0& & 0\\ 0& & 0& & 0& & 0\\ c^{\ast }& & 0& & 0& & b\end{array}\right)$$

$$c=\eta {\kappa }_1{\kappa }_2{\bar{R}}^{(2)}{\bar{\gamma }}_p(\Delta \tau ){\bar{\gamma }}_d(\Delta {\tau }^{\prime })e^{i({\omega }_{p0}\Delta \tau +{\omega }_{d0}\Delta {\tau }^{\prime }+\Delta \varphi )}.$$

$$C({\rho }_{2qubit})=2|c|=\frac{2{\kappa }_1{\kappa }_2{\bar{R}}^{(2)}}{{\kappa }_1^2{\bar{R}}^{(2)}+{\kappa }_2^2{\bar{R}}^{(2)}}{\bar{\gamma }}_p(\Delta \tau ){\bar{\gamma }}_d(\Delta {\tau }^{\prime }).$$

Full text

I Introduction

Coherence and entanglement are intimately related concepts. The recent attempts at developing a resource-based theory of coherence also reveal such relations vogel2014pra ; baumgratz2014prl ; girolami2014prl ; chitambar2016prl . One of the physical processes in which the relations between coherence and entanglement can be systematically explored is parametric down-conversion (PDC)—a nonlinear optical process in which a pump photon interacts with a nonlinear crystal to produce a pair of entangled photons, termed as signal and idler burnham1970prl . Using the PDC photons, coherence and entanglement effects have been observed in several degrees of freedom including polarization brendel1995pra , time-energy hong1987prl ; zou1991prl ; herzog1994prl ; pittman1996prl ; jha2008pra ; franson1989prl ; brendel1991prl ; jha2008prl ; brendel1999prl ; thew2002pra , position and momentum fonseca1999pra ; neves2007pra ; jha2010pra , and orbital angular momentum (OAM) nagali2009natphot ; jha2010prl ; pires2010prl ; jha2011pra .

There have been several studies on how coherence and entanglement properties of the down-converted field are affected by different PDC setting and pump field parameters hong1985pra ; rubin1996pra ; monken1998pra ; joobeur1996pra ; ribeiro1997pra ; fonseca1999pra ; saleh2005prl . However, regarding how the intrinsic correlations of the pump field get transferred to manifest as two-photon coherence and entanglement, there have been efforts mostly in the polarization and spatial degrees of freedom monken1998pra ; walborn2004pra ; jha2010pra ; kulkarni2016pra . In the spatial degree of freedom, a very general spatially partially coherent field was considered and it was shown that the spatial coherence properties of the pump field get entirely transferred to that of the down-converted two-photon field jha2010pra . However, in the temporal degree of freedom, the effects due to the temporal correlations of the pump field have only been studied in two limiting situations: one, in which the constituent frequency components are completely correlated (fully-coherent pulsed field) grice1997pra ; keller1997pra ; brendel1999prl ; tittel2000prl ; mikhailova2008pra ; inagaki2013optexp and the other, in which the constituent frequency components are completely uncorrelated (continuous-wave field) hong1987prl ; zou1991prl ; herzog1994prl ; pittman1996prl ; jha2008pra ; franson1989prl ; brendel1991prl ; jha2008prl ; ou1989pra ; rubin1994pra ; milonni1996pra ; kwon2009optexp ; nasr2008prl ; okano2015scirep ; tanaka2012optexp . In this article, we study the coherence transfer in PDC for a general temporally partially coherent pump field and explicitly quantify this correlation transfer for the special case of a partially coherent Gaussian Schell-model field paakkonen2002optcomm , in which the correlations between the constituent frequency components have a Gaussian distribution.

The paper is organized as follows. In Sec. II, we consider a general temporally partially coherent pump field and show that its temporal coherence properties get entirely transferred to the down-converted two-photon field. We work out the two-photon temporal coherence functions for both infinitely-fast and time-averaged detection schemes and show that in each scheme the coherence function factorizes into two separate coherence functions with one of them carrying the entire statistical information of the pump field. In Sec. III, we show that the entanglement of time-energy entangled two-qubit states is bounded by the degree of temporal coherence of the partially coherent pump field. We present our conclusions in Sec. IV.

II Tranfer of Temporal coherence in PDC

II.1 Detection with infinitely fast detectors

We follow the formalism worked out in Ref. jha2008pra and represent a general two-alternative two-photon interference of the PDC photons by the two-photon path diagrams shown in Fig. 1. The pump is a general temporally partially coherent field. Alternatives 1 and 2 are the two pathways by which a pump photon is down-converted and the down-converted signal and idler photons are detected in coincidence at single-photon detectors $D_{s}$ and $D_{i}$, respectively. There are six independent time parameters in this setting. The subscripts $p$, $s$ and $i$ denote the pump, signal and idler respectively. We adopt the convention that a signal photon is the one that arrives at detector $D_{s}$ while an idler photon is the one that arrives at detector $D_{i}$. The symbol $\tau$ denotes the traversal time of a photon while $\phi$ denotes the phase, other than the dynamical phase, accumulated by a photon. Thus, $\tau_{s1}$ denotes the traversal time of the signal photon in alternative $1$, etc. The various signal, idler and pump quantities are used to define the following parameters:

[TABLE]

The parameters defined above are identical to those defined in Ref. jha2008pra , except for $\tau^{\prime}_{1},\tau^{\prime}_{2}$ and $\Delta\tau^{\prime}$, which have been scaled down by a factor of 2. It is found that this rescaling imparts the equations in this paper a neat and symmetric form.

The two-photon state $|\psi\rangle_{1}$ produced in alternative $1$ in the weak-downconversion limit is given by ou1989pra ; wang1991pra ; grice1997pra :

[TABLE]

where $V(\omega_{p})$ is the random, spectral amplitude of the pump field at frequency $\omega_{p}$ and $\Phi_{1}(\omega_{s},\omega_{i})$ is the phase-matching function in alternative 1. The two-photon state $|\psi\rangle_{2}$ in alternative $2$ can be similarly defined. The complete two-photon state $|\psi\rangle$ at the detectors is the sum of the two-photon states in alternatives $1$ and $2$ and can be written as $|\psi\rangle=|\psi\rangle_{1}+|\psi\rangle_{2}$. The corresponding density matrix $\rho$ of the state at the detectors is therefore:

[TABLE]

Here $\langle\cdots\rangle$ represents an ensemble average over infinitely many realizations of the two-photon state.

We now denote the positive frequency parts of the electric fields at detectors $D_{s}$ and $D_{i}$ by $\hat{E}^{(+)}_{s}(t)$ and $\hat{E}^{(+)}_{i}(t)$, respectively, and write them as

[TABLE]

where $\kappa_{s1(2)}$ and $\kappa_{i1(2)}$ are scalar amplitudes and where

[TABLE]

is the positive frequency parts of the electric field at detector $D_{s}$ in alternative $1$, etc. The function $f_{s1}(\omega-\omega_{s0})$ is the amplitude transmission function of the filter $F_{s}$ placed at detector $D_{s}$, etc. The filters $F_{s}$ and $F_{i}$ are centered at frequencies $\omega_{s0}$ and $\omega_{i0}$, respectively, and we assume the phase-matching condition $\omega_{p0}=\omega_{s0}+\omega_{i0}$, where $\omega_{p0}$ is the central frequency of the pump field $V(\omega_{p})$. The coincidence count rate $R_{si}^{(2)}(t_{s},t_{i})$ of the two detectors is the probability per (unit time)2 that a signal photon is detected at time $t_{s}$ and the corresponding idler photon is detected at time $t_{i}$, and it is given by $R^{(2)}_{si}(t_{s},t_{i})=\mathrm{Tr}\{\hat{\rho}\,\hat{E}^{(-)}_{s}(t_{s})\,\hat{E}^{(-)}_{i}(t_{i})\hat{E}^{(+)}_{i}(t_{i})\,\hat{E}^{(+)}_{s}(t_{s})\}$ glauber1963pr . Using the definitions and expressions of Eqs. (1)-(II.1), we evaluate $R^{(2)}_{si}(t_{s},t_{i})$ to be

[TABLE]

Eq. (6) is the interference law for the two-photon field. The first and the second terms are the coincidence count rates in alternatives 1 and 2, respectively. The interference term ${\Gamma}^{(2)}(t_{s},t_{i},\tau_{s1},\tau_{i1},\tau_{s2},\tau_{i2})$ appears when both the alternatives are present, and it will be referred to as the two-photon cross-correlation function of the down-converted field. We now make the assumption that the spectral width $\Delta\omega_{p0}$ of the pump field is much smaller than the central frequency $\omega_{p0}$ and the spectral widths of the phase-matching functions and filter functions. As a result, the phase-matching and filter functions can be taken to be approximately constant in the frequency range $\left(\omega_{p0}-\Delta\omega_{p0}/2,\omega_{p0}+\Delta\omega_{p0}/2\right)$. This assumption remains valid for most PDC experiments employing continuous wave pump field hong1987prl ; zou1991prl ; herzog1994prl ; pittman1996prl ; jha2008pra ; franson1989prl ; brendel1991prl ; jha2008prl ; kwon2013optexp and pulsed pump field grice1997pra ; keller1997pra ; brendel1999prl ; tittel2000prl ; inagaki2013optexp and may only be invalid for experiments employing ultrashort pulsed pump fields marcikic2002pra ; marcikic2003nature ; marcikic2004prl . We use the relations $\omega_{p}=\omega_{s}+\omega_{i}$, $\omega_{d}=\omega_{s}-\omega_{i}$, $\omega_{p0}=\omega_{s0}+\omega_{i0}$, $\omega_{d0}=\omega_{s0}-\omega_{i0}$ and define the integration variables $\bar{\omega}_{p}=\omega_{p}-\omega_{p0}$ and $\bar{\omega}_{d}=\omega_{d}-\omega_{d0}$. Using Eqs. (1)-(6), we obtain after a long but straightforward calculation:

[TABLE]

where,

[TABLE]

and,

[TABLE]

with

[TABLE]

etc. The ensemble average $\left\langle\,V^{*}(\bar{\omega}^{\prime}_{p}+\omega_{p0})V(\bar{\omega}^{\prime\prime}_{p}+\omega_{p0})\,\right\rangle$ is the cross-spectral density function of the pump field. It is at once clear from Eq. (8) that the coherence function ${\Gamma}_{p}\left(\tau_{1}-\frac{t_{s}+t_{i}}{2},\tau_{2}-\frac{t_{s}+t_{i}}{2}\right)$ and the cross-spectral density function $\left\langle\,V^{*}(\bar{\omega}^{\prime}_{p}+\omega_{p0})V(\bar{\omega}^{\prime\prime}_{p}+\omega_{p0})\,\right\rangle$ are connected through the generalized Wiener-Khintchine relation mandel&wolf with parameters $\tau_{1}-(t_{s}+t_{i})/2$ and $\tau_{2}-(t_{s}+t_{i})/2$. So, in terms of the two-photon time parameters $\tau_{1}-(t_{s}+t_{i})/2$ and $\tau_{2}-(t_{s}+t_{i})/2$, the coherence function ${\Gamma}_{p}\left(\tau_{1}-\frac{t_{s}+t_{i}}{2},\tau_{2}-\frac{t_{s}+t_{i}}{2}\right)$ has the same functional form as that of the cross-correlation function of the pump field. The function $\left\langle g_{1}^{*}(\bar{\omega}^{\prime}_{d})\,g_{2}(\bar{\omega}^{\prime\prime}_{d})\right\rangle$ is also in the form of a cross-spectral density function, and as is clear from Eq. (9), it forms a generalized Wiener-Khintchine relation with the coherence function ${\Gamma}_{d}\left(\tau^{\prime}_{1}-\frac{t_{s}-t_{i}}{2},\tau^{\prime}_{2}-\frac{t_{s}-t_{i}}{2}\right)$. Therefore, the function ${\Gamma}_{d}\left(\tau^{\prime}_{1}-\frac{t_{s}-t_{i}}{2},\tau^{\prime}_{2}-\frac{t_{s}-t_{i}}{2}\right)$ not only carries all the information about the phase-matching conditions and the crystal parameters but also carries information about any statistical randomness that the down-converted photons go through jha2010pra2 . It is interesting to note that any statistical randomness encountered by the photons after the down-conversion affects only ${\Gamma}_{d}\left(\tau^{\prime}_{1}-\frac{t_{s}-t_{i}}{2},\tau^{\prime}_{2}-\frac{t_{s}-t_{i}}{2}\right)$ and has no effect on ${\Gamma}_{p}\left(\tau_{1}-\frac{t_{s}+t_{i}}{2},\tau_{2}-\frac{t_{s}+t_{i}}{2}\right)$. This fact can potentially be used for encoding information in the pump’s coherence function and decoding if from the down-converted photons even after the down-converted photons have passed through turbulent media.

We thus find that the two-photon cross-correlation function factorizes into two separate coherence functions. The coherence function ${\Gamma}_{p}\left(\tau_{1}-\frac{t_{s}+t_{i}}{2},\tau_{2}-\frac{t_{s}+t_{i}}{2}\right)$ carries the entire statistical information of the pump field, and in this way the temporal correlation properties of the pump photon get entirely transferred to the down-converted photons. This result is the temporal analog of the effect described in Ref. jha2010pra in which it was shown that in PDC the spatial coherence properties of the pump field gets entirely transferred to the down-converted two-photon field. However, the present paper extends beyond just establishing this analogy. For example, in Ref. jha2010pra , the effect due to the phase-matching function was completely ignored, but in the present paper, we have included it through the coherence function ${\Gamma}_{d}\left(\tau^{\prime}_{1}-\frac{t_{s}-t_{i}}{2},\tau^{\prime}_{2}-\frac{t_{s}-t_{i}}{2}\right)$. Moreover, like most spatial-interference schemes, Ref. jha2010pra does not employ a detection scheme that involves space-averaging. However, most time-domain experiments employ time-averaged detection schemes. Therefore, in the present paper, we also work out how time-averaged detection schemes affect the temporal coherence transfer in PDC.

II.2 Time-averaged detection scheme

In most experiments, one does not measure the instantaneous coincidence rate $R_{si}^{(2)}(t_{s},t_{i})$ of Eq. (6). Instead, one measures the time-averaged coincidence count rate, averaged over the photon collection time $T_{\rm pc}$ and the coincidence time-window $T_{\rm ci}$. The time-averaged two-photon cross-correlation function ${\bar{\Gamma}}^{(2)}$ can be found by first expressing it as

[TABLE]

and then integrating it with respect to $(t_{s}+t_{i})/2$ over $T_{\rm pc}$ and with respect to $(t_{s}-t_{i})/2$ over $T_{\rm ci}$. In most experiments, the coincidence time-window $T_{\rm ci}$ spans a few nanoseconds, which is much longer than the inverse frequency-bandwidth of $g(\omega)$, typically of the order of picoseconds. The photon collection time $T_{\rm pc}$ is usually a few seconds and is much longer than the inverse frequency-bandwidth of the pump field $V(\omega_{p})$, typically of the order of microseconds. Thus we perform the above time-averaging in the limit $T_{pc},T_{ci}\to\infty$ to obtain

[TABLE]

Here ${\bar{I}}_{1}={\bar{\Gamma}}_{p}\left(\tau_{1},\tau_{1}\right)$, ${\bar{G}}_{1}={\bar{\Gamma}}_{d}\left(\tau^{\prime}_{1},\tau^{\prime}_{1}\right)$, $\bar{R}^{(2)}\equiv\sqrt{\bar{I}_{1}\bar{I}_{2}}\sqrt{\bar{G}_{1}\bar{G}_{2}}$, ${\bar{\gamma}}_{p}\left(\Delta\tau\right)={\bar{\Gamma}}_{p}\left(\tau_{1},\tau_{2}\right)/\sqrt{\bar{I}_{1}\bar{I}_{2}}$, and ${\bar{\gamma}}_{d}\left(\Delta\tau\right)={\bar{\Gamma}}_{d}\left(\tau^{\prime}_{1},\tau^{\prime}_{2}\right)/\sqrt{\bar{G}_{1}\bar{G}_{2}}$, etc. The function $\bar{{\gamma}}_{p}(\Delta\tau)$ satisfies $0\leq|\bar{{\gamma}}_{p}(\Delta\tau)|\leq 1$ and diminishes over a $\Delta\tau$-scale given by the inverse pump bandwidth $1/\Delta\omega_{p0}$. The function $\bar{{\gamma}}_{d}(\Delta\tau^{\prime})$ also satisfies $0\leq|\bar{{\gamma}}_{d}(\Delta\tau^{\prime})|\leq 1$ and diminishes over a $\Delta\tau^{\prime}$-scale given by the inverse frequency-bandwidth $1/\Delta\omega_{d0}$. The temporal widths of $\bar{\gamma}_{p}(\Delta\tau)$ and $\bar{\gamma}_{d}(\Delta\tau^{\prime})$ limit the ranges over which fringes could be observed as functions of $\Delta\tau^{\prime}$ and $\Delta\tau$, respectively, in a time-averaged two-photon interference experiment.

The coincidence count rate of Eq. (6) in the time-averaged scheme therefore becomes

[TABLE]

A similar expression was reported in Ref. jha2008pra , where various temporal two-photon interference effects have been described. The time averaged coherence function ${\bar{\gamma}}_{p}\left(\Delta\tau\right)$ has the same functional form as the time-averaged coherence function of the pump field. The time-averaged coherence function ${\bar{\gamma}}_{d}\left(\Delta\tau^{\prime}\right)$ depends on the phase-matching function and the crystal parameters, and its functional form shows up in the Hong-Ou-Mandel (HOM) hong1987prl and HOM-like effects pittman1996prl ; strekalov1998pra .

III The special case of a Gaussian Schell-model pump field

In the last section, we considered PDC with a very general non-stationary pump field and described how the temporal coherence properties of the pump field get transferred to the down-converted two-photon field. In this section, we consider the pump field to be a widely-studied class of non-stationary fields, namely, the Gaussian Schell-model field, also known as the non-stationary Gaussian pulsed fields paakkonen2002optcomm .

The cross-spectral density function of a Gaussian Schell-model field is given by paakkonen2002optcomm

[TABLE]

where $\Delta\omega_{p0}$ is the frequency bandwidth of the field. The parameter $\Delta\omega_{c}$ is called the spectral correlation width and it quantifies the frequency-separation up to which different frequency components are phase-correlated. The limit $\Delta\omega_{c}\to 0$ corresponds to a continuous-wave, stationary field in which case the constituent frequency components are completely uncorrelated. The other limit $\Delta\omega_{c}\to\infty$ corresponds to a fully-coherent pulsed field in which case the constituent frequency components are perfectly phase-correlated. The corresponding temporal correlation function ${\Gamma}_{\rm GS}(t_{1},t_{2})$ can be calculated by using the generalized Wiener-Khintchine theorem paakkonen2002optcomm :

[TABLE]

with $\Delta t=t_{1}-t_{2}$ and where

[TABLE]

Here $\tau_{\rm coh}=(\Delta\omega_{c}/\Delta\omega_{p0})\left[1/(2\Delta\omega_{p0})^{2}+1/(\Delta\omega_{c})^{2}\right]^{1/2}$ is a measure of the coherence time of the field and $T=\left[1/(2\Delta\omega_{p0})^{2}+1/(\Delta\omega_{c})^{2}\right]^{1/2}$ is a measure of the temporal width of the non-stationary Gaussian pulse. The limit $\Delta\omega_{c}\to\infty$ yields $\tau_{\rm coh}\to\infty$ as expected for a fully-coherent field, and the other limit $\Delta\omega_{c}\to 0$ yields $\tau_{\rm coh}=1/\Delta\omega_{p0}$ as expected for a continuous-wave, stationary field.

Now, for conceptual clarity, we assume in this section that ${\Gamma}_{d}\left(\tau^{\prime}_{1}-\frac{t_{s}-t_{i}}{2},\tau^{\prime}_{2}-\frac{t_{s}-t_{i}}{2}\right)=1$ and take the pump field to be the Gaussian Schell-model field given by Eq. (13). Eq. (7) then becomes:

[TABLE]

where

[TABLE]

As expected from Eq. (7), we find that in terms of $\tau_{1}-\frac{t_{s}+t_{i}}{2}$ and $\tau_{2}-\frac{t_{s}+t_{i}}{2}$, the two-photon cross-correlation function in Eq. (15) assumes the same functional form as does the cross-correlation function in Eq. (14) in terms of $t_{1}$ and $t_{2}$. When integrated over $t$, Eq. (15) yields

[TABLE]

with $\bar{I}_{1}=\bar{I}_{2}=(2\pi)^{\frac{3}{2}}\Delta\omega_{p0}A$ and

[TABLE]

where $\bar{\tau}_{\rm coh}=1/\Delta\omega_{p0}$ is a measure of the coherence time. The time averaging washes out effects due to frequency correlations. Thus, only in the case of a stationary pump field $\bar{\gamma}_{p}(\Delta\tau)=\gamma_{p}(\Delta\tau)$ and $\bar{\tau}_{\rm coh}=\tau_{\rm coh}$.

IV Pump temporal coherence and two-qubit ENERGY-TIME ENTANGLEMENT

Two-qubit states are the necessary ingredients for many quantum information protocols ekert1991prl ; bennett1992prl ; bennett1993prl and have been realized by exploiting the entanglement of PDC photons in several degrees of freedom including polarization kwiat1995prl , time-energy franson1989prl ; brendel1999prl ; tittel2000prl ; thew2002pra ; ramelow2009arxiv ; kwon2009optexp ; kwon2013optexp , position-momentum neves2005prl ; neves2007pra ; rarity1990prl , and orbital angular momentum (OAM) vaziri2002prl ; langford2004prl ; leach2009optexp ; jha2010prl . There have been previous studies describing how correlations of the pump field in polarization and spatial degrees of freedom affect the entanglement of the generated two-qubit states. In the polarization degree of freedom it was shown kulkarni2016pra that the degree of polarization $P$ of the pump photon puts an upper bound of $(1+P)/2$ on the concurrence of the generated two-qubit state. In the spatial degree of freedom, effects of pump spatial coherence on the entanglement of the generated spatial two-qubit state have been worked out for two-qubit state that have only two non-zero diagonal elements, and for such states it has been shown that the concurrence is bounded by the degree of spatial coherence of the pump field jha2010pra . However, to the best of our knowledge, no such relation has so far been derived for the time-energy entangled two-qubit states.

There are two generic methods by which one makes a PDC-based time-energy entangled two-qubit state. In the first method, one uses a continuous-wave pump field, either single-mode franson1989prl or multi-mode kwon2009optexp ; kwon2013optexp . In the second method, one uses a pulsed pump field brendel1999prl ; tittel2000prl ; thew2002pra ; ramelow2009arxiv . In both these methods, a combination of post-selection strategies, such as selecting a faster coincidence detection-window, using arrival time of pump photon as a trigger etc., one makes sure that there are only two alternative pathways in which the signal and idler photons reach their respective detectors. The two alternative pathways form the two dimensional qubit space for the signal and idler photons. We represent by $|s1\rangle$ the state of the signal photon in alternative 1, etc. Therefore, the density matrix $\rho_{\rm 2qubit}$ of the two-qubit state can be written in the basis {$|s1\rangle|i1\rangle,|s1\rangle|i2\rangle,|s2\rangle|i1\rangle,|s2\rangle|i2\rangle$} as:

[TABLE]

where the diagonal terms $a$ and $b$ are the probabilities that the signal and idler photons are detected in states $|s1\rangle|i1\rangle$ and $|s2\rangle|i2\rangle$, respectively, and the off-diagonal term $c$ is a measure of coherence between states $|s1\rangle|i1\rangle$ and $|s2\rangle|i2\rangle$. In an experimental situation, the density matrix $\rho$ can be represented by the two alternative pathways of Fig. 1. Therefore, using Eq. (12), we write $a=\eta\kappa_{1}^{2}\bar{R}^{(2)}$ and $b=\eta\kappa_{2}^{2}\bar{R}^{(2)}$, where $\eta=1/[\kappa_{1}^{2}\bar{R}^{(2)}+\kappa_{2}^{2}\bar{R}^{(2)}]$ is the constant of proportionality. The off-diagonal term is given by

[TABLE]

The entanglement of $\rho_{\rm 2qubit}$, as quantified by Wootters’s concurrence $C(\rho_{\rm 2qubit})$ wootters1998prl , can be shown to be

[TABLE]

The pre-factor $2\kappa_{1}\kappa_{2}\bar{R}^{(2)}/(\kappa_{1}^{2}\bar{R}^{(2)}+\kappa_{2}^{2}\bar{R}^{(2)})$ is no greater than 1, and ${\bar{\gamma}}_{d}(\Delta\tau^{\prime})$ also satisfies $0\leq|{\bar{\gamma}}_{d}(\Delta\tau^{\prime})|\leq 1$. We therefore arrive at the relation: $C(\rho_{\rm 2qubit})\leq\bar{{\gamma}}_{p}(\Delta\tau)$. Therefore, we find that the concurrence $C(\rho_{\rm 2qubit})$ of the time-energy two-qubit state is bounded from above by the degree of coherence of the pump photon and thus that the temporal correlations of the pump field set an upper bound on the attainable concurrence for a two-qubit state of the form of Eq. (17). We note that in situations in which ${\bar{\gamma}}_{d}(\Delta\tau^{\prime})\approx 1$ and $\kappa_{1}=\kappa_{2}$, the maximum achievable concurrence for a pulsed field can be unity in principle and for a continuous-wave field it can be unity as long as $\Delta\tau$ is much smaller than the coherence time of the pump field. The above result is the temporal analog of the results obtained in the polarization kulkarni2016pra and spatial jha2010pra degrees of freedom. However, unlike in the spatial degree of freedom, which does not involve any space-averaged detection scheme, the results derived in this article show that even for the time-averaged detection schemes, the temporal correlation properties of the pump do directly decide the upper limit on entanglement that a time-energy entangled two-qubit state can achieve.

V CONCLUSIONS AND DISCUSSIONS

In conclusions, we have shown that in parametric down-conversion the coherence properties of a temporally partially coherent pump field get entirely transferred to the down-converted entangled two-photon field. Under the assumption that the frequency-bandwidth of the down-converted signal-idler photons is much larger than that of the pump, we have worked out the temporal coherence functions of the down-converted field for both infinitely-fast and time-averaged detection schemes. We have shown that in each scheme the coherence function factorizes into two separate coherence functions with one of them carrying the entire statistical information of the pump field. Taking the pump to be a Gaussian Schell-model field, we have derived explicit expressions for the coherence functions. Finally, we have shown that the concurrence of time-energy entangled two-qubit states is bounded by the degree of temporal coherence of the pump field. This result extends previously obtained results in the spatial jha2010pra and polarization kulkarni2016pra degrees of freedom to the temporal degree of freedom and can thus have important implications for understanding how correlations of the pump field in general manifest as two-particle entanglement. Our results can also be important for time-energy two-qubit based quantum communication applications. This is because it has been recognized that energy-time entangled two-qubit states are better than the polarization two-qubit states for long-distance quantum information marcikic2003nature ; marcikic2004prl , and our results show that the temporal coherence properties of the pump field can be used as a parameter for tailoring the two-qubit time-energy entanglement. Moreover, it is known that the purity of the individual photon states increases with the decrease in the entanglement of the two-photon state. Therefore, our work can also have implications for PDC-based heralded single photons sources obrien2007science ; cassemiro2010njp in the sense that the degree of purity of heralded photons can be tailored by controlling the coherence properties of the pump field.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support through an initiation grant no. IITK /PHY /20130008 from Indian Institute of Technology (IIT) Kanpur, India and through the research grant no. EMR/2015/001931 from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India.