# Common values of a class of linear recurrences

Attila Pethő  Department of Computer Science,
University of Debrecen,
H-4002 Debrecen, P.O. Box 400<EMAIL_ADDRESS>
###### Abstract.

Let $(a_{n}),(b_{n})$ be linear recursive sequences of integers with
characteristic polynomials $A(X),B(X)\in{\mathbb{Z}}[X]$ respectively. Assume
that $A(X)$ has a dominating and simple real root $\alpha$, while $B(X)$ has a
pair of conjugate complex dominating and simple roots $\beta,\bar{\beta}$.
Assume further that $\alpha/\beta$ and $\bar{\beta}/\beta$ are not roots of
unity and $\delta=\log|\alpha|/\log|\beta|\in{\mathbb{Q}}$. Then there are
effectively computable constants $c_{0},c_{1}>0$ such that the inequality

$|a_{n}-b_{m}|>|a_{n}|^{1-(c_{0}\log^{2}n)/n}$

holds for all $n,m\in{\mathbb{Z}}^{2}_{\geq 0}$ with $\max\\{n,m\\}>c_{1}$.

###### Key words and phrases:

linear recurrences

###### 2010 Mathematics Subject Classification:

11D61, 11B37, 11D75

##  1\. Introduction

Let $p_{0},\ldots,p_{k-1},g_{0},\ldots,g_{k-1}\in{\mathbb{Z}}$ and

(1.1) $g_{n+k}=p_{k-1}g_{n+k-1}+\dots+p_{0}g_{n}$

for all $n\geq 0$ then $(g_{n})$ is called linear recursive sequence with
characteristic polynomial $G(X)=X^{k}-p_{k-1}X^{k-1}-\dots-p_{0}$. To avoid
improper examples we assume $p_{0}$ and at least one of $g_{0},\dots,g_{k-1}$
non-zero. Denote $\gamma_{1},\dots,\gamma_{l}$ the distinct zeroes of $G(X)$
with multiplicities $m_{1},\dots,m_{l}$ respectively. Set
${\mathbb{K}}={\mathbb{Q}}(\gamma_{1},\dots,\gamma_{l})$. It is well known,
see e.g [12] Chapter C, that there exist polynomials
$G_{j}(X)\in{\mathbb{K}}[X]$ of degree at most $m_{j}-1,j=1,\dots,l$ such that

$g_{n}=G_{1}(n)\gamma_{1}^{n}+\dots+G_{l}(n)\gamma_{l}^{n}$

holds for $n\geq 0$.

The breakthrough works of A. Baker [1], [2] on lower bounds for linear forms
in logarithms of algebraic numbers made it possible, among others, the
systematic study of diophantine properties of linear recursive sequences.
Moreover the results obtained with the help of Baker’s theory are effective,
they serve computable upper bounds for the size of the solutions. You find an
excellent overview of earlier results in the book of Shorey and Tijdeman [12].

The subspace theorem of W.M. Schmidt [11] became an other strong tool in the
theory of linear recurrences, see e.g. J.-H. Evertse [5]. Usually the results
obtained with the derivations of the subspace theorem are more general, but
not effective.

One of the basic questions is the description of the common terms of
recurrences. This includes values appearing several times in a recurrence. As
recurrences are linear combinations of exponential functions, one expects that
they have usually only finitely many common terms, and, in the opposite case,
they have to be strongly related. This was proved ineffectively by J.-H.
Evertse [5] and M. Laurent [7]. Effective version is known only for the
equations $g_{n}=g_{m}$ provided the characteristic polynomial of $(g_{n})$
has at most three roots with the same absolute value and for $g_{n}=h_{m}$
when both recurrences have dominating simple and real roots. The book of T.N.
Shorey and R. Tijdeman [12] gives detailed overview on the related results.

Recently E.F. Bravo et al. [4], [3] established all solutions
$n,m\in{\mathbb{Z}}$ of the diophantine equation $T_{n}=T_{m},$ where
$(T_{n})$ denotes the tribonacci sequence, which is defined by the initial
terms $T_{-1}=T_{0}=0,T_{1}=1$ and by the recursion
$T_{n+3}=T_{n+2}+T_{n+1}+T_{n},n\geq-1$. The relation (1.1) allows to extend
$(g_{n})$ to negative indices, but that terms are usually not integers. If,
however, $p_{0}=\pm 1$ the terms with negative indices are integers too. This
holds to the tribonacci sequence, which negative branch is defined as
$T_{-n-3}=-T_{-n-2}-T_{-n-1}+T_{-n}$. The characteristic polynomial of
$(T_{n})$ is $T(X)=X^{3}-X^{2}-X-1$, which has a dominating real root. On the
other hand the characteristic polynomial of $(T_{-n})$ is
$X^{3}+X^{2}+X-1=-X^{3}T(1/X)$, whose conjugate complex roots are dominating,
i.e. the absolute values of them are larger than the absolute value of the
third root. In this circumstances the older methods are not applicable, the
authors needed a novel idea. As I mentioned Bravo et al. were interested to
find all common terms for a given recurrence, but after a closer look became
clear that the method is capable to prove effective finiteness result for a
wide class of recurrences. My original goal was to generalize the result of
E.F. Bravo et al. [4], [3] to common values of $k$-generalized Fibonacci
numbers, see e.g. [10]. Although I was unable to fulfill the original
expectation, the results are general enough to summarize in these notes.

To formulate our theorems we have to introduce some notation. Let
$(a_{n}),(b_{n})$ be linear recursive sequences of integers with
characteristic polynomials $A(X),B(X)\in{\mathbb{Z}}[X]$ respectively. Let
$\alpha_{1},\ldots,\alpha_{k}$ and $\beta_{1},\ldots,\beta_{l}$ be the
distinct roots of $A(X)$ as well as of $B(X)$ with multiplicities
$m_{1},\ldots,m_{k}$ and $n_{1},\ldots,n_{l}$ respectively. We assume that
$|\alpha_{1}|>|\alpha_{2}|\geq|\alpha_{3}|\geq\ldots\geq|\alpha_{k}|$; further
$\beta_{2}=\bar{\beta}_{1}$ and
$|\beta_{1}|=|\beta_{2}|>|\beta_{3}|\geq\ldots\geq|\beta_{l}|$.

In the sequel $c_{0},c_{1},\ldots$ denote effectively computable constants
depending only on the degrees and on the roots of the characteristic
polynomials of $(a_{n})$ and $(b_{n})$. The algebraic numbers $\alpha,\beta$
are called multiplicatively dependent if there are non-zero integers $a,b$
such that $\alpha^{a}\beta^{b}=1$.

######  Theorem 1.1.

Let $(a_{n})$ and $(b_{m})$ be as above. Assume that
$m_{1}=n_{1}=1,\alpha_{1}/\beta_{1}$ and $\beta_{2}/\beta_{1}$ are not roots
of unity and $\delta=\log|\alpha_{1}|/\log|\beta_{1}|\in{\mathbb{Q}}$, i.e.
$\alpha_{1}$ and $|\beta_{1}|$ are multiplicatively dependent. Then the
diophantine inequality

(1.2) $|a_{n}-b_{m}|>|a_{n}|^{1-(c_{0}\log^{2}n)/n}$

holds for all $n,m\in{\mathbb{Z}}^{2}_{\geq 0}$ with $\max\\{n,m\\}>c_{1}$,
where

$c_{0}>\frac{9c_{15}}{\log|\alpha_{1}|}$

and $c_{1},c_{15}$ denote effectively computable positive constants defined in
the proof.

An immediate consequence of our theorem is

######  Corollary 1.2.

Under the assumptions of Theorem 1.1 the equation

$a_{n}=b_{m}$

has only finitely many, effectively computatble solutions in
$n,m\in{\mathbb{Z}}^{2}_{\geq 0}$.

Using the theory of weighted $S$-unit equations, see e.g. [6] it is easy to
prove that if $\alpha_{1}/\beta_{1}$ and $\beta_{2}/\beta_{1}$ are not roots
of unity then the equation $a_{n}-b_{m}=K$ has for any given integer $K$ only
finitely many solutions in $(n.m)\in{\mathbb{Z}}_{\geq 0}^{2}$. We do not give
here the proof, but refer to the proof of Theorem 3.4 of [10].

The assumption $\alpha_{1}/\beta_{1}$ and $\beta_{2}/\beta_{1}$ are not roots
of unity is natural because, otherwise, $a_{n}=b_{m}$ may have infinitely many
solutions. Indeed, set $\alpha_{1}=2,\beta_{1}=2i,\beta_{2}=-2i$ and
$a_{n}=\alpha_{1}^{n},b_{m}=\beta_{1}^{m}+\beta_{2}^{m}$, then
$b_{4k}=2^{4k+1}=a_{4k+1}$ hold for all $k\geq 0$. Let now $(b_{m})$ be the
same and $(a_{n})$ any sequence satisfying our assumptions and with $a_{0}=0$.
As $b_{4k+1}=0$ for all $k\geq 0$ the equation $a_{n}=b_{m}$ has infinitely
many solutions.

On the other hand the assumption $\alpha_{1}$ and $|\beta_{1}|$ are
multiplicatively dependent is not at al natural, moreover it is quite
restrictive, but without it we can prove only finiteness, but usually not
effective result. To highlight the reason consider the three appearances of
Baker type inequalities in our proof, which are

  1. (1)

the proof of a lower bound on $|b_{m}|$,

  2. (2)

the proof of a lower bound on $|n-\delta m|$,

  3. (3)

the proof of an upper bound on $n$.

The first two steps can be performed without any restrictions on $\delta$, but
it seems to be essential for Step 3.

Notice that Theorem 1.1 requires properties only on the characteristic
polynomials but not on the initial values of the sequences. Our second theorem
gives a sufficient condition for pairs of linear recurrences for which Theorem
1.1 is applicable. We show that the restriction
$\log|\alpha_{1}|/\log|\beta_{1}|\in{\mathbb{Q}}$ narrow considerably the
possible irreducible polynomials, but there are still infinitely many
examples. Multiplying such polynomials with suitable ones we get the
characteristic polynomials of linear recursive sequences for which the
assumptions of Theorem 1.1 hold. To avoid repetition of the phrase maximal
size of the roots of the polynomial $P(X)$ we denote it by $\overline{|P|}$.

######  Theorem 1.3.

With $a,b,p,q\in{\mathbb{Z}},p,q>0,\gcd(p,q)=1$ and $q$ even if $b$ is not a
square define $Q_{1}(X)=X^{2}+aX+b^{p}$ and $Q_{2}(X)=X^{3}+aX^{2}+bX+1$ such
that $Q_{2}(X)$ has one real root outside the unit circle and a pair of
conjugate complex roots. Assume that $P_{1}(X),P_{2}(X)\in{\mathbb{Z}}[X]$ and

  * (i)

either
$\bullet$
$a^{2}-4b^{p}<0,\;\overline{|P_{1}|}<|b^{q/2}|,\;\overline{|P_{2}|}<\overline{|Q_{1}|}$,
$\bullet$ $A(X)=(X\pm b^{q/2})P_{1}(X),\;B(X)=Q_{1}(X)P_{2}(X)$,

  * (ii)

or
$\bullet$
$\overline{|P_{1}|}<\overline{|Q_{2}|},\;\overline{|P_{2}|}<\overline{|X^{3}Q_{2}(1/X)|}$,
$\bullet$ $A(X)=Q_{2}(X)P_{1}(X),\;B(X)=X^{3}Q_{2}(1/X)P_{2}(X)$.

Let $(a_{n}),(b_{n})$ be non-zero linear recursive sequences with
characteristic polynomials $A(X)$ and $B(X)$ respectively. If $\alpha$ and
$\beta$ denote the dominating root of $A(X)$ and $B(X)$ respectively, then
they satisfy the assumptions of Theorem 1.1.

######  Remark 1.4.

The assumption $Q_{2}(X)$ has one real root and a pair of conjugate complex
roots means that the discriminant of $Q_{2}(X)$ is negative, i.e.
$-27+18ab+a^{2}b^{2}-4a^{3}-4b^{3}<0$.

Combining Theorems 1.1 and 1.2 we obtain a far reaching generalization of
Bravo et al. [4], [3].

######  Corollary 1.5.

Let $a,b\in{\mathbb{Z}}$ with $-27-18ab+a^{2}b^{2}-4a^{3}+4b^{3}<0$ Let
$f_{0},f_{1},f_{2}\in{\mathbb{Z}}$ not all zero and

$f_{n+3}=af_{n+2}+bf_{n+1}+f_{n},\;n\geq 0.$

Extend the sequence $(f_{n})$ toward negative indices by the recursion

$f_{-n}=-bf_{-n+1}-af_{-n+2}+f_{-n+3},\;n\geq 1.$

Then there are only finitely many effectively computable
$n,m\in{\mathbb{Z}},n\not=m$ with $f_{n}=f_{m}$.

##  2\. A Baker’s type inequality

An essential part of the proof of the main theorem depends on A. Baker’s
theory on linear forms in logarithms of algebraic numbers. Among the many
variants we use here the one due to Matveev, which is one of the sharpest
nowadays.

For an algebraic number $\eta$ with minimal polynomial

$f(X)=a_{0}(X-\eta^{(1)})\cdots(X-\eta^{(N)})\in{\mathbb{Z}}[X]$

with positive $a_{0}$, write $h(\eta)$ for its absolute logarithmic or Weil
height given by

$h(\eta)=\frac{1}{N}\left(\log
a_{0}+\sum_{j=1}^{N}\max\\{0,\log|\eta^{(j)}|\\}\right).$

We list important and well known properties of the function $h$ in the
following lemma.

######  Lemma 2.1.

Let $\gamma,\eta$ be algebraic numbers and $u\in{\mathbb{Q}}$. Then we have

  1. (1)

$h(\gamma\pm\eta)\leq h(\gamma)+h(\eta)+\log 2$

  2. (2)

$h(\gamma\eta^{\pm 1})\leq h(\gamma)+h(\eta)$

  3. (3)

$h(\gamma^{u})=|u|h(\gamma)$.

For the proof see e.g. Waldschmidt [13], Ch. 3.2.

Let $\mathbb{K}$ be an algebraic number field of degree $d_{\mathbb{K}}$ and
let $\eta_{1},\eta_{2},\ldots,\eta_{t}\in\mathbb{K}$ not $0$ or $1$, and
$b_{1},\ldots,b_{t}$ be nonzero integers. Put

$B=\max\\{|b_{1}|,\ldots,|b_{t}|,3\\}\qquad{\rm
and}\qquad\Gamma=\prod_{i=1}^{t}\eta_{i}^{b_{i}}-1.$

Let $A_{1},\ldots,A_{t}$ be positive integers such that

$A_{j}\geq\max\\{d_{\mathbb{K}}h(\eta_{j}),|\log\eta_{j}|,0.16\\},\quad{\text{\rm
for}}\quad j=1,\ldots t.$

Under the circumstances above, Matveev [8] proved

######  Lemma 2.2.

If $\Gamma\neq 0$, then

$\log|\Gamma|>-3\cdot 30^{t+4}(t+1)^{5.5}d_{\mathbb{K}}^{2}(1+\log
d_{\mathbb{K}})(1+\log tB)A_{1}A_{2}\cdots A_{t}.$

##  3\. Proof of Theorem 1.1

Under the assumptions there are polynomials
$A_{1},\ldots,A_{k}\in{\mathbb{Q}}(\alpha_{1},\ldots,\alpha_{k})[X]$,
$B_{1},\ldots,B_{l}\in{\mathbb{Q}}(\beta_{1},\ldots,\beta_{l})[X]$ of degrees
$m_{1}-1,\ldots,m_{k}-1$ as well as $n_{1}-1,\ldots,n_{l}-1$ respectively such
that

$a_{n}=A_{1}(n)\alpha_{1}^{n}+\ldots+A_{k}(n)\alpha_{k}^{n},\quad
b_{n}=B_{1}(n)\beta_{1}^{n}+\ldots+B_{l}(n)\beta_{l}^{n}$

hold for all $n\geq 0$. As $m_{1}=n_{1}=1$ and $\alpha_{1}$ and $\beta_{1}$ is
dominating among the roots of $A(X),B(X)$ respectively we have
$|\alpha_{1}|,|\beta_{1}|>1$, $\alpha_{1}$ is real $n_{2}=n_{1}=1$,
$\beta_{1}=\bar{\beta_{1}}$ and $A_{1},B_{1}$ and $B_{2}$ are constants.
Putting

$\delta_{a}=\frac{\log|\alpha_{2}|}{2\log|\alpha_{1}|}\quad\mbox{and}\quad\delta_{b}=\frac{\log|\beta_{3}|}{2\log|\beta_{1}|}$

then $\delta_{a},\delta_{b}<1$ and it is an easy exercise to prove

(3.1) $\displaystyle|a_{n}-A_{1}\alpha_{1}^{n}|$ $\displaystyle<$
$\displaystyle c_{2}|\alpha_{1}|^{\delta_{a}n},$ (3.2) $\displaystyle
c_{3}|\alpha_{1}|^{n}$ $\displaystyle<$
$\displaystyle|a_{n}|<c_{4}|\alpha_{1}|^{n}$

and

(3.3)
$|b_{m}-(B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m})|<c_{5}|\beta_{1}|^{\delta_{b}m}$

with suitable positive effective constants $c_{2},c_{3},c_{4},c_{5}$, provided
$n$ and $m$ are large enough.

In contrast, it is a consequence of the deep theory of A. Baker that if
$\beta_{1}/\beta_{2}$ is not a root of unity then

(3.4) $|\beta_{1}|^{m-c_{6}\log
m}<|B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}|<c_{7}|\beta_{1}|^{m},$

see Corollary 3.7 of [12]. The last two inequalities imply

(3.5) $|\beta_{1}|^{m-2c_{6}\log m}<|b_{m}|<2c_{7}|\beta_{1}|^{m}.$

provided $m$ is large enough. We derive only the lower bound, because the
proof of the upper bound is straight forward

$|b_{m}|>|B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}|-c_{8}|\beta_{3}|^{m}>|\beta_{1}|^{m-c_{6}\log
m}-c_{8}|\beta_{1}|^{\delta_{b}m}>|\beta_{1}|^{m-2c_{6}\log m}.$

Contrary to the statement of Theorem 1.1 we assume that there exist infinitely
many $(n,m)\in{\mathbb{Z}}_{\geq 0}^{2}$ for which (1.2) does not holds. We
show that the choice of any such solutions with $n$ large enough leads to a
contradiction. In the sequel let $(n,m)\in{\mathbb{Z}}_{\geq 0}^{2}$ be a
solution of

$||a_{n}|-|b_{m}||\leq|a_{n}|^{1-(c_{0}\log^{2}n)/n}$

and $n$ is large enough. As $||a_{n}|-|b_{m}||=|a_{n}-b_{m}|,$ if $a_{n}$ and
$b_{m}$ have the same sign and by $|-a_{n}-b_{m}|$, we have to deal with the
two cases separately. In the sequel we are dealing with the first case because
replacing $a_{n}$ by $-a_{n}$ in the argument below one gets easily the proof
of the second case. Thus we are to do with the inequality

(3.6) $|a_{n}-b_{m}|\leq|a_{n}|^{1-(c_{0}\log^{2}n)/n}.$

We also assume $a_{n}\geq b_{m}$. The case $a_{n}<b_{m}$ can be handled
similarly.

Instead of $a_{n}-b_{m}$, which may have many terms we will study
$A_{n,m}=A_{1}\alpha_{1}^{n}-(B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m})$, which
has only three summands, thus much more easy to treat. Moreover $|A_{n,m}|$ is
not far from $a_{n}-b_{m}$. Indeed

$|A_{n,m}-(a_{n}-b_{m})|\leq|a_{n}-A_{1}\alpha_{1}^{n}|+|b_{m}-(B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m})|<c_{2}|\alpha_{1}|^{\delta_{a}n}+c_{5}|\beta_{1}|^{\delta_{a}m}.$

Using (3.1) and (3.3) we obtain

$\displaystyle|A_{n,m}|$ $\displaystyle\leq$
$\displaystyle|a_{n}-b_{m}|+|a_{n}-A_{1}\alpha_{1}^{n}|+|b_{m}-(B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m})|$
$\displaystyle\leq$
$\displaystyle|a_{n}|^{1-c_{0}(\log^{2}n)/n}+c_{4}|\alpha_{1}|^{\delta_{a}n}+c_{5}|\beta_{1}|^{\delta_{b}m}$
$\displaystyle\leq$
$\displaystyle(c_{4}+c_{5})\max\\{|\beta_{1}|^{\delta_{b}m},|\alpha_{1}|^{\delta_{a}n}\\}+c_{3}^{1/2}|\alpha_{1}|^{n-c_{0}\log^{2}n}.$

We distinguish two cases depending on the maximum.

Case I. $|\alpha_{1}|^{\delta_{a}n}\geq|\beta_{1}|^{\delta_{b}m}$. The first
summand on the right hand side of (3) is now at most
$(c_{4}+c_{5})|\alpha_{1}|^{\delta_{a}n}$. Dividing by $|\alpha_{1}|^{n}$ and
taking into account that $\delta_{a}<1$ we get

$\left|\frac{A_{n,m}}{\alpha_{1}^{n}}\right|<(c_{4}+c_{5})|\alpha_{1}|^{(\delta_{a}-1)n}+c_{3}^{1/2}|\alpha_{1}|^{-c_{0}\log^{2}n}<2c_{3}^{1/2}|\alpha_{1}|^{-c_{0}\log^{2}n}<|A_{1}|/2,$

whenever $n$ is large enough. This implies

(3.8)
$|A_{1}||\alpha_{1}|^{n}/2<|B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}|<3|A_{1}||\alpha_{1}|^{n}/2.$

A direct consequence of this inequality and (3.4) is that

$|A_{1}||\alpha_{1}|^{n}/2<c_{7}|\beta_{1}|^{m},$

thus

$n-\delta m<c_{8}\quad\mbox{with}\quad c_{8}=\frac{\log
c_{7}-\log(|A_{1}|/2)}{\log|\alpha_{1}|}.$

On the other hand he inequalities (3.4) and (3.8) imply,

$3|A_{1}||\alpha_{1}|^{n}/2>|\beta_{1}|^{m-c_{6}\log m},$

thus

$n-\delta m>-c_{6}\delta\log m-c_{9}\quad\mbox{with}\quad
c_{9}=\frac{\log(3|A_{1}|/2)}{\log|\alpha_{1}|}.$

Case II. $|\alpha_{1}|^{\delta_{a}n}<|\beta_{1}|^{\delta_{b}m}$. The first
summand of the right hand side of (3) is now at most
$(c_{4}+c_{5})|\beta_{1}|^{\delta_{b}m}$. Our assumption ($a_{n}\geq b_{m}$)
together with the previous estimations implies

$|\alpha_{1}|^{n}>\frac{1}{2|A_{1}|}|a_{n}|\geq\frac{1}{2|A_{1}|}|b_{m}|>\frac{1}{2|A_{1}|}|\beta_{1}|^{m-2c_{6}\log
m}.$

Dividing (3) by $|\alpha_{1}|^{n}$ we get

$\displaystyle\left|A_{1}-\frac{B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}}{\alpha_{1}^{n}}\right|$
$\displaystyle<$
$\displaystyle(c_{4}+c_{5})\frac{|\beta_{1}|^{\delta_{b}m}}{|\alpha_{1}|^{n}}+c_{3}^{1/2}|\alpha_{1}|^{-c_{0}\log^{2}n}$
$\displaystyle<$ $\displaystyle
2(c_{4}+c_{5})|A_{1}||\beta_{1}|^{\delta_{b}m-m+2c_{6}\log
m}+c_{3}^{1/2}|\alpha_{1}|^{-c_{0}\log^{2}n}$ $\displaystyle<$
$\displaystyle|A_{1}|/2,$

provided $m$ (thus $n$) is large enough. From here on we can repeat the
argument of Case I and get the same lower and upper estimates for $n-\delta
m$.

For later use we summarize them in the inequality

(3.10) $-c_{10}\log m<n-\delta m<c_{8},$

where $c_{10}=c_{4}\delta+c_{9}$ and $c_{8}$ are positive constants. Its
simpler, but weaker form is,

(3.11) $\delta m/2<n<2\delta m,$

which is sometimes easier to apply and is valid if
$m>\max\\{c_{8}/(2\delta),(2c_{10}/\delta)^{2}\\}$. Plainly, if $m$ does not
satisfy the last inequality, but $(n,m)$ is a solution of (3.6) $n$ is
effectively bounded too.

In Case I we proved $|A_{n,m}|<2c_{3}^{1/2}|\alpha_{1}|^{n-c_{0}\log^{2}n}$.
Now we are in the position to prove the same bound for Case II too. Indeed, we
can estimate the exponent of the first term of (3)

$\delta_{b}m-m+2c_{6}\log
m<\frac{\delta_{b}-1}{2\delta}n+2c_{6}\log(2n/\delta)<-c_{11}n$

with an effective, positive constant $c_{11}$, which is our claim.

So far we proved that if $(n,m)\in{\mathbb{Z}}^{2}$ is a solution of (3.6)
such that $n$ is large enough then

(3.12) $|A_{n,m}|<2c_{3}^{1/2}|\alpha_{1}|^{n-c_{0}\log^{2}n}$

and (3.10) holds.

In the next step we prove a lower bound for $|A_{n,m}|$. To achieve that write
$\beta_{1}=rz$ with $r\in{\mathbb{R}}_{>0}$ and $z\in{\mathbb{C}},|z|=1$, and
we see that $\beta_{2}=r/z$ and
$B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}=r^{m}(B_{1}z^{m}+B_{2}z^{-m})$.
Remember that $r=|\beta_{1}|=|\alpha_{1}|^{\delta}$. Substituting this into
(3), dividing by $|B_{1}r^{m}|$ and using (3.10) and (3.12) we get

$\displaystyle\left|z^{m}-\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}+\frac{B_{2}}{B_{1}}z^{-m}\right|$
$\displaystyle\leq$
$\displaystyle\frac{2c_{3}^{1/2}}{|B_{1}|}\frac{|\alpha_{1}|^{n-c_{0}\log^{2}n}}{r^{m}}$
$\displaystyle=$
$\displaystyle\frac{2c_{3}^{1/2}}{|B_{1}|}|\alpha_{1}|^{n-\delta
m-c_{0}\log^{2}n}$ $\displaystyle<$ $\displaystyle
c_{12}|\alpha_{1}|^{-c_{0}\log^{2}n},$

with $c_{12}=2|\alpha_{1}|^{c_{8}}c_{3}^{1/2}/|B_{1}|$ .

The next argumentation was used for the tribonacci numbers by Bravo et al. [4,
3] (especially the proof of Lemma 1 in [3]). The zeroes of the quadratic
polynomial $X^{2}-\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}X+\frac{B_{2}}{B_{1}}$
are

$2\lambda_{1}=\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}+\sqrt{\left(\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}\right)^{2}-4\frac{B_{2}}{B_{1}}},\quad
2\lambda_{2}=\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}-\sqrt{\left(\frac{A_{1}\alpha_{1}^{n}}{B_{1}r^{m}}\right)^{2}-4\frac{B_{2}}{B_{1}}},$

thus

$|z^{m}-\lambda_{1}||z^{m}-\lambda_{2}|\leq
c_{12}|\alpha_{1}|^{-c_{0}\log^{2}n}.$

We may assume without loss of generality
$|z^{m}-\lambda_{1}|\leq|z^{m}-\lambda_{2}|$, hence

$|z^{m}-\lambda_{1}|\leq\sqrt{c_{12}}|\alpha_{1}|^{-c_{0}(\log^{2}n)/2},$

which implies

$|\lambda_{1}|\geq|z|^{m}-\sqrt{c_{12}}|\alpha_{1}|^{-c_{0}(\log^{2}n)/2}\geq
1/2,$

provided $n$ is large enough. Thus

(3.14) $|z^{m}/\lambda_{1}-1|\leq
2\sqrt{c_{12}}|\alpha_{1}|^{-c_{0}(\log^{2}n)/2}.$

Set $\Gamma=z^{m}/\lambda_{1}-1$. If $\Gamma=0$ then
$A_{1}\alpha^{n}=B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m}$. For an algebraic
integer $\eta$ denote $P(\eta)$ the largest prime lying below the prime ideal
divisors of $(\eta)$. On one hand $P(A_{1}\alpha^{n})$ is uniformly bounded,
but as $\beta_{1}/\beta_{2}$ is not a root of unity
$P(B_{1}\beta_{1}^{m}+B_{2}\beta_{2}^{m})$ can be arbitrarily large by Pethő
[9]. Thus $\Gamma\not=0$ and we can apply Lemma 2.2 with he actual parameters:

${\mathbb{K}}={\mathbb{Q}}(z,\lambda_{1}),t=2,\eta_{1}=z,\eta_{2}=\lambda_{1},b_{1}=m,b_{2}=1.$

As $z=r/\beta_{2}=\sqrt{\beta_{1}/\beta_{2}}$, the field ${\mathbb{Q}}(z)$ is
included in a quadratic extension of $Q(\beta_{1},\ldots,\beta_{l})$.
Similarly ${\mathbb{Q}}(\lambda_{1})$ is a subfield of a quadratic extension
of ${\mathbb{Q}}(\alpha_{1},\ldots,\alpha_{k},\beta_{1},\ldots,\beta_{l})$,
thus $d_{{\mathbb{K}}}\leq 4k!l!$.

In the preceding argumentation we assumed $n$ large enough several times.
Thus, by (3.11) $m$ is large enough too, and we may assume $B=m$. Plainly
$h(\eta_{1})=h(z)\leq 2h(\beta_{1})$. The estimation of $h(\eta_{2})$ is much
more complicated and seems to be possible only if
$\delta=\log|\beta_{1}|/\log|\alpha_{1}|$ is rational, which is assumed in the
theorem. Indeed, in this case $u=n-m\delta$ is rational too, and as
$\frac{\alpha_{1}^{n}}{r^{m}}=\pm|\alpha_{1}|^{u}$, we have

$h\left(\frac{A_{1}}{B_{1}}\frac{\alpha_{1}^{n}}{r^{m}}\right)\leq|u|h(\alpha_{1})+h\left(\frac{A_{1}}{B_{1}}\right)$

by Lemma 2.1 (3). Applying several times Lemma 2.1 we finally obtain

$h(\lambda_{1})\leq 2|u|h(\alpha_{1})+c_{13}$

with a computable constant $c_{13}$. Finally taking into account (3.10) we get

$h(\eta_{2})=h(\lambda_{1})\leq c_{14}\log m+c_{13}$

with $c_{14}=2h(\alpha_{1})c_{10}$. Setting
$A_{1}=8k!l!h(\beta_{1}),A_{2}=4k!l!c_{14}\log m+4k!l!c_{13}$ the application
of Lemma 2.2 yields

$\log|\Gamma|>-c_{15}\log^{2}m-c_{16},$

where

$c_{15}=3^{6.5}\cdot 2^{11}\cdot 30^{6}(k!l!)^{4}\log(k!l!)h(\beta_{1})c_{14}$

and $c_{16}=c_{15}c_{13}/c_{14}$.

Comparing the lower bound for $\log|\Gamma|$ with the upper bound for
$|\Gamma|$, i.e. with (3.14) we obtain

$c_{0}(\log|\alpha_{1}|)(\log^{2}n)/2-\log(2\sqrt{c_{11}})<c_{15}\log^{2}m+c_{16}.$

Using (3.10) we get

$c_{0}<\frac{2c_{14}}{\log|\alpha_{1}|}\left(1+\frac{\log(2/\delta)}{\log
n}\right)^{2}+\frac{2(c_{15}+\log(2\sqrt{c_{11}})}{\log|\alpha_{1}|}\frac{1}{\log^{2}n}$

after some simple manipulation. If

$n>\max\left\\{\frac{2}{\delta},\exp\sqrt{\frac{c_{16}+\log(2\sqrt{c_{12}})}{c_{15}}}\right\\}$

then

$c_{0}<\frac{9c_{15}}{\log|\alpha_{1}|},$

but this contradicts our assumption.

##  4\. Proof of Theorem 1.3

Assume that $A(X),B(X)\in{\mathbb{Z}}[X]$ satisfy the assumptions of Theorem
1.1. Let $P_{1}(X),P_{2}(X)\in{\mathbb{Z}}[X]$ be such that
$\overline{|P_{1}|}<\overline{|A|}$ and $\overline{|P_{2}|}<\overline{|B|}$
then $A_{1}X),B_{1}(X)$ with $A_{1}(X)=A(X)P_{1}(X),B_{1}(X)=B(X)P_{2}(X)$
satisfy obviously the assumptions of Theorem 1.1 too. Thus it is enough to
characterize the irreducible polynomials among them.

Let $A(X),B(X)\in{\mathbb{Z}}[X]$ be irreducible, and denote $\alpha$ the real
root of $A(X)$ with maximal size, while $\beta$ the non-real root of $B(X)$
with maximal size. In Theorem 1.1 a crucial (and unnatural) assumption is that
$|\alpha|$ and $|\beta|$ are multiplicatively dependent. This means that there
are coprime, non-zero integers $p,q$ such that
$\pm\alpha^{p}=(\beta\bar{\beta})^{q/2}$ or, equivalently

(4.1) $\alpha^{2p}=(\beta\bar{\beta})^{q}.$

As $\alpha$ is the dominant real root of $A(X)$ and $\beta$ one of the
dominant complex roots of $B(X)$ both have to lie outside the unit circle and
the last equation is only possible if $p$ and $q$ have same sign, but then we
may assume that both are positive. Denote ${\mathbb{K}}$ the normal closure of
${\mathbb{Q}}(\alpha,\beta,\bar{\beta})$ and $G$ its Galois group. If
$\sigma\in G$ then $\sigma(\alpha)$ is a root of $A(X)$, thus
$|\sigma(\alpha)|<|\alpha|$. Because of the same reason
$|\sigma(\beta)|<|\beta|$ for all $\sigma\in G$ except when
$\sigma(\beta)=\bar{\beta}$.

Let $\sigma\in G$ not the identity be such that $\sigma(\alpha)=\alpha$. If
$\sigma(\beta)\not=\bar{\beta}$ then

$(\beta\bar{\beta})^{q}=\alpha^{2p}=\sigma(\alpha^{2p})=\left(\sigma(\beta)\sigma(\bar{\beta})\right)^{q},$

but this is impossible because $|\sigma(\beta)|<|\beta|$ and
$|\sigma(\bar{\beta})|\leq|\bar{\beta}|$. Thus either $\alpha\in{\mathbb{Z}}$
and $\deg G=2$ or $B(X)$ has apart $\bar{\beta}$ a third root, say $\gamma$.

If $\alpha\in{\mathbb{Z}}$ then the only algebraic conjugate of $\beta$ is
$\bar{\beta}$, thus $\beta\bar{\beta}\in{\mathbb{Z}}$, which has to be a
$p$-th power, say $b^{p}$. Hence $\alpha^{2}=b^{q}$ and $\alpha=\pm b^{q/2}$.
If $b$ is a square then $q$ is arbitrary, otherwise it must be even. Hence the
defining polynomial of $\beta$ has the shape $X^{2}+aX+b^{p}$ such that
$a^{2}-4b^{p}<0$, further the defining polynomial of $\alpha$ is $X\pm
b^{q/2}$. This proves that if $A$ has an integer root then (i) is not only
sufficient, but also necessary that the pair $A(X),B(X)$ fulfill the
requirements of Theorem 1.1.

We now turn to the second case, but now we can prove only sufficiency. Denote
$\alpha$ the real root and $\gamma,\overline{\gamma}$ the pair of conjugate
complex roots of $Q_{2}(X)$. We have $\alpha\gamma\overline{\gamma}=1$, thus
$|\gamma|<1$ because by assumption $|\alpha|>1$. Further the last equation
means $|\alpha||\gamma|^{2}=1$ too, i.e $|\alpha|$ and $|\gamma|$ are
multiplicatively dependent. The roots of $X^{3}Q_{2}(1/X)$ are
$1/\alpha,1/\gamma,1/\overline{\gamma}$. With $\beta=1/\gamma$ we have that
$\alpha$ and $\beta$ are multiplicatively dependent.

It remains to prove that neither $\alpha/\beta$ nor $\beta/\overline{\beta}$
is a root of unity. In contrary assume that $\alpha/\beta=\omega$ is a root of
unity. Taking conjugate and multiplying the resulting equation with the first
one we get

$\frac{\alpha}{\beta}\frac{\alpha}{\overline{\beta}}=\omega\overline{\omega}=1.$

We also have $\alpha/(\beta\overline{\beta})=1$, which together with the last
equation implies $\alpha^{3}=1$, i.e. $\alpha=1$, which is absurd.

Assume finally that $\beta/\overline{\beta}=\omega$ is a root of unity. Then,
especially $\omega\in{\mathbb{K}}={\mathbb{Q}}(\beta,\overline{\beta})$. We
also have $\alpha\in{\mathbb{K}}$ because $\alpha+\beta+\overline{\beta}=a$,
thus the field ${\mathbb{K}}$ is normal. There is an automorphism $\sigma$ of
${\mathbb{K}}$, with $\sigma(\beta)=\alpha$. It maps $\overline{\beta})$ to
itself or to $\beta$ and $\omega$ to one of its conjugates, which is also root
of unity. In both cases $\alpha/\beta$ is a root of unity, which is by the
last paragraph absurd. Hence the assumptions of Theorem 1.1 fulfill. $\Box$

##  5\. Proof of Corllary 1.5

The characteristic polynomial of $(f_{n})$ is $F(X)=X^{3}-aX^{2}-bX-1$, while
that of $(f_{-n})$ is $F_{-}(X)=X^{3}+bX^{2}+aX-1=-X^{3}F(\frac{1}{X})$, hence
the roots of $F(X)$ and $F_{-}(X)$ are reciprocal. We have further
$F(X)=-((-X)^{3}+a(-X)^{2}-b(-X)+1)$. The polynomial in bracket has the same
shape as $Q_{2}(X)$, but with $-b$ instead of $b$. The assumption
$-27-18ab+a^{2}b^{2}-4a^{3}+4b^{3}<0$ means that its discriminant is negative,
i.e. $F(X)$ has one real and a pair of conjugate complex roots. Denote by
$\alpha$ its real root. Because $F(0)=-1$ and $F(x)\to\infty$ with $x$,
$\alpha$ has to be positive. Interchanging the positive and negative
directions we may assume $\alpha>1$. Denote the other roots of $F(X)$ by
$1/\beta,1/\overline{\beta}$. They are in absolute value less than one. The
roots of $F_{-}(X)$ are $\beta,\overline{\beta},1/\alpha$. With these
notations we have

(5.1)
$f_{n}=a\alpha^{n}+b\beta^{n}+\overline{b}\overline{\beta}^{n},\;n\in{\mathbb{Z}}$

with some non-zero $a,b\in{\mathbb{C}}$. To prove the corollary we have to
distinguish three cases.

Case 1, $n,m\geq 0$. A simple, elementary argument, which is left to the
reader, is enough to settle it.

Case 2, $n,m<0$. Now $f_{-n}$ is still exponentially growing, but also
oscillating. To prove the corollary we have to invoke the theory of linear
forms in logarithms of algebraic numbers. More precisely Theorem 3.5 of [12]
with the choice
$x_{n}=y_{n}=b\beta^{n}+\overline{b}\overline{\beta}^{n},A=B=1$ implies the
statement.

Case 3, $n\cdot m<0$. We may assume $n>0,m<0$ without loss of generality.
Setting $m=-k,k>0$ our equation becomes

$f_{n}=f_{-k}.$

The characteristic polynomial of $(f_{n})$ is $F(X)$ and that of $(f_{-n})$ is
$F_{-}(X)$. By the remarks at the beginning of the proof setting
$Q_{2}(X)=-F(-X),P_{1}(X)=P_{2}(X)=1$ the assumptions of Theorem 1.3 hold,
i.e. $(f_{n})$ and $(f_{-n})$ are such sequences for which Corollary 1.2 is
applicable, thus our equation has only finitely many effectively computable
solutions in this case too..

## References

  * [1] A. Baker, Linear forms in the logarithms of algebraic numbers, Mathematika, 13 (1966), 204-216.
  * [2] A. Baker, Transcendental Number Theory, Cambridge Univ. Press, 1975.
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  * [5] J.-H. Evertse, On sums of $S$-units and linear recurrences, Compositio Math. 53 (1984), 225-244.
  * [6] J.-H. Evertse and K. Győry, Discriminant equations in Diophantine number theory, New Mathematical Monographs, 32, Cambridge University Press, Cambridge, 2017, xviii+457,
  * [7] M. Laurent, Équations exponentielles polynômes et suites récurrentes linéaires, Astérisque No. 147 - 148 (1987), 121 - 139, 343 - 344.
  * [8] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125–180. (English translation in Izv. Math. 64 (2000), 1217–1269.
  * [9] A. Pethő, On the greatest prime factor and divisibility properties of linear recursive sequences, Indag. Math. (N.S.) 1 (1990), no. 1, 85 – 93.
  * [10] A. Pethő, On the $k$-generalized Fibonacci numbers with negative indices, Publ. Math. Debrecen, to appear
  * [11] W.M. Schmidt, Diophantine Approximations, Lecture Notes in Mathemmatics 785, Springer Verlag, 1980.
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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Quantum and Classical Two-photon Interference of Single Photons with
Ultralong Coherence Time

Manman Wang  Beijing Academy of Quantum Information Sciences, Beijing 100193,
China.  Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China.  University of Chinese Academy of Sciences, Beijing 101408, China.
Yanfeng Li  Beijing Academy of Quantum Information Sciences, Beijing 100193,
China.  Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China.  University of Chinese Academy of Sciences, Beijing 101408, China.
Hanqing Liu  Institute of Semiconductors, Chinese Academy of Sciences, Beijing
100083, China.  College of Materials Science and Opto-Electronic Technology,
University of Chinese Academy of Sciences, Beijing 101408, China.  Haiqiao Ni
Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China.  College of Materials Science and Opto-Electronic Technology,
University of Chinese Academy of Sciences, Beijing 101408, China.  Zhichuan
Niu  Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China.  College of Materials Science and Opto-Electronic Technology,
University of Chinese Academy of Sciences, Beijing 101408, China.  Xiaogang
Wei  Beijing Academy of Quantum Information Sciences, Beijing 100193, China.
Renfu Yang  Beijing Academy of Quantum Information Sciences, Beijing 100193,
China.  Chengyong Hu<EMAIL_ADDRESS>Beijing Academy of Quantum Information
Sciences, Beijing 100193, China.

###### Abstract

Two-photon interference (TPI) is a fundamental phenomenon in quantum optics
and plays a crucial role in quantum information science and technology. TPI is
commonly considered as quantum interference with an upper bound of $100\%$ for
both the TPI visibility and the beat visibility in contrast to its classical
counterpart with a maximum visibility of $50\%$. However, this is not always
the case. Here we report a simultaneous observation of quantum and classical
TPI of single photons with ultralong coherence time which is longer than the
photon correlation time by five orders of magnitude. We observe a TPI
visibility of $94.3\%\pm 0.2\%$ but a beat visibility of $50\%$. Besides an
anti-bunching central dip due to single-photon statistics, we observe two
bunching side peaks in cross-correlation curves for indistinguishable photons.
Using either classical wave superposition theory or quantum field approach, we
derive the same expressions for the cross-correlation functions which
reproduce and explain the experiments well. We conclude that quantum TPI with
a stream of single photons is equivalent to classical TPI, both of which are
the fourth-order interference arising from the second-order interference
occurring on the time scale of photon coherence time.

##  I INTRODUCTION

When two identical photons are simultaneously incident on a 50/50 beam
splitter from two input ports, they will always bind together and leave the
beam splitter from the same output port. This phenomenon is called two-photon
interference (TPI) - a fundamental effect in quantum optics which was
discovered in 1987 by Hong, Ou, and Mandel (HOM) [1]. Since its discovery, TPI
has been considered as quantum interference with no analogue in classical
physics [2, 3], and widely applied in quantum information science and
technology, e.g., measuring photon’s bandwidth and timing[1], testing the
degree of photon indistinguishability [4, 5], Bell-state analysis/measurement
[6] and entanglement swapping/generation [7] for quantum communications and
networks [8, 9, 10, 11], quantum information processing [12, 13, 14] and
quantum metrology [15, 16].

Recently there has been great interest in the fourth-order interference
between two weak lasers (it is also called the HOM interference, or classical
TPI by some authors) for practical implementations of the measurement-device-
independent quantum key distribution (MDI-QKD) [17, 18, 19] which uses TPI to
post-select entangled states as a time-reversed version of entanglement-based
QKD [20]. The HOM interference between two lasers or classical light is
usually interpreted as classical interference with a visibility less than
$50\%$ [21, 22] in contrast to TPI with a maximum visibility of $100\%$ when
two photons are indistinguishable [1, 2, 5]. This visibility difference
usually serves as a criterion to distinguish between quantum TPI and classical
TPI. It is well known that all photons in laser light are identical, thus the
interference visibility with an upper bound of $50\%$ underestimates the
degree of photon indistinguishability of a laser. Therefore it is natural to
ask: What’s the relationship between the quantum TPI of two photons and the
HOM classical interference of two lasers? And how to characterize the genuine
photon indistinguishability of a laser?

In this work, we answer the above questions by performing time-resolved TPI
experiments in an asymmetric Mach-Zehnder interferometer (AMZI) using single
photons with ultralong coherence time. In most TPI experiments reported so
far, single photons have short coherence time limited by the single-photon
correlation time or twice the emitter’s radiative lifetime. As a result, TPI
and the HOM classical interference mix up with the single-photon statistics
and it is hard to distinguish between them. In this work we coherently convert
laser light into single photons using a single quantum dot (QD) coupling to a
doubled-sided optical microcavity in the Purcell regime. Such single photons
inherit the incident laser’s ultralong coherence ($>10\ \mu$s) which is five
orders of magnitude longer than the photon correlation time, so allow us to
discriminate quantum and classical effects in TPI and related interference
beat. We observe a TPI visibility of $94.3\%\pm 0.2\%$ but a beat visibility
of $50\%$, indicating the coexistence of quantum and classical TPI. Besides an
anti-bunching central dip due to single-photon statistics, we observe two
bunching side peaks in cross-correlation curves for indistinguishable photons.
We find these intricate phenomena are linked to each other. Using either
classical wave superposition theory or quantum field theory, we derive the
same expressions for the cross-correlation functions which reproduce and
explain the experiments well. Based on these results, we conclude that quantum
TPI with a stream of single photons is equivalent to classical TPI, both of
which are the fourth-order interference arising from the second-order
interference occurring on the time scale of photon coherence time.

Figure 1: (color online) Experimental setup for TPI measurement using a
modified AMZI where two cascaded AOM frequency shifters are placed in one arm
to make tunable frequency shifts. Either fibre or free-space delays are used
to vary the time difference $\Delta t$ between two AMZI arms. Time-resolved
correlation measurements are performed with an 8-channel time-correlated
single photon counter with a time jitter of $3\ $ps. Single photons with
ultralong coherence time are generated by a single QD in an optical
microcavity placed inside a closed-cycle cryostat. BS, $\mathrm{BS_{A}}$ and
$\mathrm{BS_{B}}$: beam splitters, FS: AOM frequency shifter, LP1 and LP2:
linear polarizers, HWP: half-wave plate, SNSPD1 and SNSPD2: superconducting
nano-wire single-photon detectors with a time resolution of $20\ $ps.

##  II Generation of single photons with ultralong coherence time

We designed and fabricated a pillar microcavity containing a single self-
assembled In(Ga)As QD resonantly coupling to the fundamental cavity mode with
the cooperativity parameter $C=2g^{2}/(\kappa\gamma_{\perp})\gg 1$ and
critical photon number $n_{0}=\gamma_{\perp}\gamma_{\parallel}/(4g^{2})\ll 1$,
where $g$ is the QD-cavity interaction strength, $\kappa$ is the cavity photon
decay rate, $\gamma_{\parallel}$ is the QD spontaneous emission rate into
leaky modes, $\gamma_{\perp}=\gamma_{\parallel}/2+\gamma^{*}$ is the QD
polarization decay rate, and $\gamma^{*}$ is the QD pure dephasing rate. Such
design allows the incident laser light interacts with the QD
deterministically, cavity-enhanced coherent scattering and strong nonlinearity
at the single-QD and single-photon level. The cavity is defined by two mirrors
made up of 18 and 30 pairs of GaAs/Al0.9Ga0.1As distributed Bragg reflectors
(DBRs), respectively. The two DBR mirrors are made asymmetric in the realistic
devices such that the leakage rate from the top mirror can balance the total
leakage rates from the bottom mirror, cavity side and background absorption
resided in materials. This cavity structure mimics a double-sided cavity with
zero reflectivity at the center of the fundamental cavity mode. The details
for sample growth and device fabrication can be found in our recent work [24].
The cavity quantum electrodynamics(CQED) parameters for the sample used in
this work are $g/2\pi=4.7\ $GHz, $\kappa/2\pi=36.8\ $GHz,
$\gamma_{\parallel}/2\pi=0.35\ $GHz and $\gamma^{*}/2\pi\simeq 0\ $GHz. So the
cooperativity parameter is $C=6.9$ and the critical photon number is
$n_{0}=6.9\times 10^{-4}$.

Figure 2: (color online) Generation and characterization of single photons
with ultralong coherence time ($>10\ \mu$s). (a) Reflection spectra measured
by scanning the laser’s wavelength. The QD-cavity coupled system (i.e., hot
cavity) behaves like a cold or empty cavity at high laser powers when the QD
gets saturated by excitation. (b) Measured second-order autocorrelation
$\mathrm{g}^{(2)}(\tau)$ of reflected light with the laser frequency fixed at
the QD resonance. $\mathrm{g}^{(2)}(0)=0.030\pm 0.002$ is achieved. (c)
Measured degree of first-order coherence $|\mathrm{g}^{(1)}(\tau)|$ versus
time delay at different driving powers with a Michelson interferometer. The
solid curves are calculations using the master equation[24]. We choose a low
driving power with $S=0.01$ for TPI experiments in this work. (d) Cross-
correlation measurements for cross polarizations (blue dash) and parallel
polarizations (red solid) at $\Delta t=2.1\ $ns and $\Omega/2\pi=0\ $kHz. The
experiment time is more than half an hour. (e) Calculated cross-correlation
functions for cross polarizations (blue dash) and parallel polarizations (red
solid) at $\Delta t=2.1\ $ns and $\Omega/2\pi=0$. (f) TPI visibility versus
the time delay $\tau$ at $\Delta t=2.1$ ns and $\Omega/2\pi=0$. The visibility
peak with $V_{HOM}(0)=94.3\%\pm 0.2\%$ is accompanied by a $50\%$ background.

The sample was placed inside a closed-cycle cryostat (see Fig. 1) and the QD
transition was in resonance with the cavity mode by temperature tuning. A cw
tunable Ti:saphhire laser with the linewidth $<100\ $kHz was used to drive the
cavity. We monitor the intensity or correlation of the reflected light. Fig.
2(a) presents the coherent reflection spectra measured by scanning the laser’s
wavelength. We get nearly zero reflectivity ($R=0.89\%$) at the center
frequency of cavity mode at higher laser powers when the QD saturates. The QD
transition induces a sharp reflection peak ($R=46.6\%$) inside the cavity-mode
resonance [Fig. 2(a), red solid line] at lower laser powers.

Fixing the laser frequency on the QD resonance, we measure the second-order
autocorrelation function of reflected light and achieve
$\mathrm{g}^{(2)}(0)=0.030\pm 0.002$ [see Fig. 2(b)]. High single-photon
purity is observed at low driving fields with the saturation parameter $S<1$.
The reflected light consists of a superposition of the driving field and the
cavity output field. The driving laser field shows Poissonian statistics while
the cavity output field exhibits super-bunching [24] due to photon-induced
tunneling [25, 26] and multi-photon scattering [27], so the common picture
that a single QD can only scatter (or absorb and emit) single photons cannot
be applied at low driving fields. We identify that fully destructive
interference between the driving field and the cavity output field erases the
two-photon probability amplitude in reflected light and converts the driving
laser light into single photons [24], affirming the interference picture on
antibunching in resonance fluorescence proposed 40 years ago [28].

At low driving fields with $S<1$, the cavity output intensity is much weaker
than the driving field, so the first-order coherence of converted single
photons is simply determined by the driving field. Fig. 2(c) presents the
degree of first-order coherence $\mathrm{g}^{(1)}(\tau)$ of reflected light
versus time delay measured with a Michelson interferometer at different
driving powers. There are two coherence times observed, $\tau_{c1}\simeq 115$
ps and $\tau_{c2}>24.5$ ns which is limited by the longest path delay of
interferometer. The short 115-ps coherence time which is twice the QD
radiative lifetime $57\ $ps stems from the incoherent cavity output field due
to quantum fluctuations, while the long coherence time comes from the driving
field with ultra-long coherence time ($>10\ \mu$s). The incoherent cavity
output reduces the degree of first-order coherence by its intensity fraction
in reflected light, however, this reduction becomes less significant at low
driving fields.

##  III Two-photon interference in AMZI

Next we perform TPI experiments in a modified AMZI (see Fig. 1) where two
cascaded acousto-optical modulation (AOM) frequency shifters are placed in one
arm to make tunable frequency shift. The device works at a low driving field
with the saturation parameter $S=0.01$ in the Heitler regime. The reflected
single photons have a high degree of first-order coherence with
$|\mathrm{g}^{(1)}(\tau)|>95\%$ [see Fig. 2(c)] for $\tau$ less than the
coherence time. Fig. 2(d) presents the measured cross-correlation functions
$\mathrm{g}^{(2)}_{\perp}(\tau)$ and $\mathrm{g}^{(2)}_{\parallel}(\tau)$
between two detectors in cross- and parallel-polarization configurations at
$\Delta t=2.1$ns and $\Omega/2\pi=0$.

In cross-polarization configuration, interference phenomena are not expected
and we observe three anti-bunching dips at $\tau=0,\pm\Delta t$ with a
correlation time of $115\ $ps which is twice the QD radiative lifetime. The
central dip and two side dips arise from the single-photon correlations with a
depth of $0.5$ and $0.25$, respectively.

In parallel-polarization configuration, besides the central dip at $\tau=0$,
we observe two bunching side peaks at $\tau=\pm\Delta t$ which have not been
reported before [30, 29]. The HOM (or TPI) visibility is usually defined as
$V_{HOM}(\tau)=[\mathrm{g}^{(2)}_{\perp}(\tau)-\mathrm{g}^{(2)}_{\parallel}(\tau)]/\mathrm{g}^{(2)}_{\perp}(\tau)$.
From it, we get the TPI visibility $V_{HOM}(0)=94.3\%\pm 0.2\%$ [Fig. 2(f)]
which is well above the $0.5$ classical limit, indicating quantum nature of
TPI. The TPI visibility peak sits on a $V_{HOM}(\tau)=0.5$ background which
also stems from TPI as discussed later. For this reason, we normalize
$\mathrm{g}^{(2)}_{\parallel}(\tau)$ to $0.5$ in Fig. 2(d).

The bunching side peaks persist with increasing the fibre delays up to $5\ $
km (i.e., $\Delta t<25\ \mu$s) and then turn to antibunching dips when the
fibre delays are longer than $5\ $ km (i.e., $\Delta t\geq 25\ \mu$s) as shown
in Fig. 3.

If keeping the fibre delay to $1\ $km (i.e., $\Delta t=5\ \mu$s) but varying
the AOM frequency shifts, we observe periodic changeover between bunching side
peaks and antibunching side dips. For example, the bunching side peaks for
$\Omega=0$ [see Fig. 3(b)] turn to anti-bunching side dips inside the
interference beat for $\Omega/2\pi=48.0\ $kHz [Fig. 4(a)], $\Omega/2\pi=101.6\
$kHz [Fig. 4(b)], $\Omega/2\pi=147.1\ $kHz [Fig. 4(c)], $\Omega/2\pi=246.0\
$kHz [Fig. 4(e)], but turn back to bunching side peaks for $\Omega/2\pi=194.7\
$kHz [Fig. 4(d)].

In order to understand the above results, we adopt a general wave
superposition theory (see the Supplemental Material [23]) to calculate the
cross-correlation functions for a general light field in the modified AMZI
(see Fig. 1) in cross-polarization configuration

$\mathrm{g}^{(2)}_{\perp}(\tau)=\frac{1}{N}\Biggl{\\{}(R_{A}^{2}+T_{A}^{2})R_{B}T_{B}\mathrm{g}^{(2)}(\tau)+R_{A}T_{A}R_{B}^{2}\mathrm{g}^{(2)}(\tau+\Delta
t)+R_{A}T_{A}T_{B}^{2}\mathrm{g}^{(2)}(\tau-\Delta t)\Biggl{\\}},$ (1)

and in parallel-polarization configuration

$\begin{split}\mathrm{g}^{(2)}_{\parallel}(\tau)=&\frac{1}{N}\Biggl{\\{}(R_{A}^{2}+T_{A}^{2})R_{B}T_{B}\mathrm{g}^{(2)}(\tau)+R_{A}T_{A}R_{B}^{2}\mathrm{g}^{(2)}(\tau+\Delta
t)+R_{A}T_{A}T_{B}^{2}\mathrm{g}^{(2)}(\tau-\Delta t)\\\
&-2R_{A}T_{A}R_{B}T_{B}V_{0}|\mathrm{g}^{(1)}(\tau)|^{2}\cos\Omega\tau\sqrt{\mathrm{g}^{(2)}(\tau-\Delta
t)\mathrm{g}^{(2)}(\tau+\Delta t)}\Biggl{\\}},\end{split}$ (2)

where $N=(R_{A}^{2}+T_{A}^{2})R_{B}T_{B}+(R_{B}^{2}+T_{B}^{2})R_{A}T_{A}$ is
the normalization factor. $R_{A.B}$ and $T_{A,B}$ are the reflection and
transmission intensity coefficients of the beam splitters $\mathrm{BS_{A}}$
and $\mathrm{BS_{B}}$ with nominal values $R_{A}=T_{A}=50\%$ and
$R_{B}=T_{B}=50\%$. The parameter $V_{0}$ is introduced to consider the mode
overlap on $\mathrm{BS_{B}}$ (see the Supplemental Material [23]).
$\mathrm{g}^{(1)}(\tau)$ and $\mathrm{g}^{(2)}(\tau)$ are the first- and
second-order auto-correlation functions of the input light. $\Omega/2\pi$ is
the AOM frequency shift. Note that Eqs. (1) and (2) are suitable for general
light including single photons, laser light and thermal light. In case of
single photons, Eqs. (1) and (2) are also obtained in framework of quantum
field theory (see the Supplemental Material [23]).

A coincident event involves two photons detected by each of two detectors. The
first term in Eqs. (1) and (2) comes from the coincidence events for two
photons traveling through the same AMZI arm. The second and third terms count
in the coincidence events for two photons going through different arms. The
fourth term in Eq. (2) is due to the fourth-order interference of two photons
which essentially originates from the second-order interference occurring on
the time scale of photon coherence time (see Supplemental Material [23]). This
is the only interference term in Eq. (2), so TPI and the HOM classical
interference are the same interference. In parallel-polarization
configuration, single-photon correlation contributes to the antibunching
central dip with a depth of $0.5$ and a small width of $115\ $ps, while TPI
(i.e., the HOM classical interference) leads to a broad background (i.e., the
HOM dip) with a depth of $0.5$ and a width of the photon coherence time ($>10\
\mu$s). That is why we normalize $\mathrm{g}^{(2)}_{\parallel}(\tau)$ to $0.5$
when $\Delta t\ll\tau_{c}$ [see Fig. 2(d) and Fig. 3]. For the same reason,
the TPI visibility peak is observed on a $V_{HOM}(\tau)=0.5$ background in
Fig. 2(f). Taking $R_{A}=T_{A}=50\%$, $R_{B}=T_{B}=50\%$,
$\mathrm{g}^{(2)}(0)=0.03$ and
$|\mathrm{g}^{(1)}(\tau)|=\exp{(-|\tau|/\tau_{c})}$ where $\tau_{c}$ is the
photon coherence time ($>10\ \mu$s), Eqs. (1) and (2) well reproduce the
measured $\mathrm{g}^{(2)}_{\perp}(\tau)$ and
$\mathrm{g}^{(2)}_{\parallel}(\tau)$ in Fig. 2(d) with the calculated results
plotted in Fig. 2(e).

To explain the bunching side peaks at $\tau=\pm\Delta t$, we rewrite Eq. (2)
as

$\begin{split}\mathrm{g}^{(2)}_{\parallel}(\tau)=&\frac{1}{N}\Biggl{\\{}(R_{A}^{2}+T_{A}^{2})R_{B}T_{B}\mathrm{g}^{(2)}(\tau)\\\
&+R_{A}T_{A}R_{B}\sqrt{\mathrm{g}^{(2)}(\tau+\Delta
t)}\Biggl{(}R_{B}\sqrt{\mathrm{g}^{(2)}(\tau+\Delta
t)}-T_{B}V_{0}|\mathrm{g}^{(1)}(\tau)|^{2}\cos\Omega\tau\sqrt{\mathrm{g}^{(2)}(\tau-\Delta
t)}\Biggr{)}\\\ &+R_{A}T_{A}T_{B}\sqrt{\mathrm{g}^{(2)}(\tau-\Delta
t)}\Biggl{(}T_{B}\sqrt{\mathrm{g}^{(2)}(\tau-\Delta
t)}-R_{B}V_{0}|\mathrm{g}^{(1)}(\tau)|^{2}\cos\Omega\tau\sqrt{\mathrm{g}^{(2)}(\tau+\Delta
t)}\Biggr{)}\Biggl{\\}}.\end{split}$ (3)

At first we discuss the $\Omega=0$ case. If
$T_{B}\sqrt{\mathrm{g}^{(2)}(0)}<R_{B}V_{0}|\mathrm{g}^{(1)}(\Delta t)|^{2}$
[refer to the third term of Eq.(3)], i.e.,

$\Delta
t<\frac{\tau_{c}}{2}\ln{\left[\frac{R_{B}V_{0}}{T_{B}\mathrm{g}^{(2)}(0)}\right]},$
(4)

a side peak at $\tau=\Delta t$ is expected, and otherwise we would observe a
side dip at $\tau=\Delta t$. Here we take $\mathrm{g}^{(2)}(\tau+\Delta t)=1$
at $\tau=\Delta t$ as $\Delta t$ is much larger than the photon correlation
time ($115\ $ps).

If $R_{B}\sqrt{\mathrm{g}^{(2)}(0)}<T_{B}V_{0}|\mathrm{g}^{(1)}(\Delta
t)|^{2}$ [refer to the second term of Eq.(3)], i.e.,

$\Delta
t<\frac{\tau_{c}}{2}\ln{\left[\frac{T_{B}V_{0}}{R_{B}\mathrm{g}^{(2)}(0)}\right]},$
(5)

a side peak at $\tau=-\Delta t$ is expected, and otherwise we would see a side
dip at $\tau=-\Delta t$. Here we take $\mathrm{g}^{(2)}(\tau-\Delta t)=1$ at
$\tau=-\Delta t$.

Figure 3: (color online) Cross-correlation measurements in parallel-
polarization configuration at different AMZI delays: (a) $0.12\ $km; (b) $1\
$km; (c) $2\ $km; (d) $8\ $km. The AOM frequency shift is set to zero. The
driving field is kept at $S=0.01$. The measurement time for each curve is
several hours. Figure 4: (color online) Cross-correlation measurements with
single photons in parallel-polarization configuration at different AOM
frequency shifts: (a) $\Omega/2\pi=48.0\ $kHz; (b) $\Omega/2\pi=101.6\ $kHz;
(c) $\Omega/2\pi=147.1\ $kHz; (d) $\Omega/2\pi=194.7\ $kHz; (e)
$\Omega/2\pi=246.0\ $kHz. The fibre length is fixed to $1\ $km, corresponding
to a time delay $\Delta t=5\ \mu$s. The driving field is kept at $S=0.01$.
Same measurement with the driving laser is plotted in (f) for comparison.

If taking $R_{B}=T_{B}=50\%$, $V_{0}=100\%$, and $\mathrm{g}^{(2)}(0)=0.03$,
the above inequalities (4) and (5) reduce to $\Delta t<1.8\tau_{c}$. Assuming
$\tau_{c}=10\ \mu$s, we would observe two bunching side peaks at
$\tau=\pm\Delta t$ if the AMZI fibre delays are less than $3.6\ $km (i.e.,
$\Delta t<18\ \mu$s), or two anti-bunching side dips if fibre delays are
longer than $3.6\ $km (i.e., $\Delta t>18\ \mu$s), in agreement with the
observations in Fig. 3 where the anti-bunching side peaks turn to dips when
the fibre length increases beyond $5\ $km. When $\Delta t\gg 1.8\tau_{c}$, the
interference effect cannot reach the side dips at $\tau=\pm\Delta t$ as
$\mathrm{g}^{(1)}(\Delta t)\approx 0$, so $\mathrm{g}^{(2)}_{\parallel}(\tau)$
around $\tau=\pm\Delta t$ is normalized to one in Fig. 3(d). The small
asymmetry between the two bunching side peaks is caused by the slight
difference between $R_{B}$ and $T_{B}$, or different detection efficiencies.
In previous work [29, 30], single photons generated by the QD spontaneous
emission have very short coherence time limited by twice the QD radiative
lifetime. As a result, $\Delta
t>\frac{\tau_{c}}{2}\ln{\left[V_{0}/\mathrm{g}^{(2)}(0)\right]}$ is always
met, so only antibunching side dips were observed in their experiments.

In the case of $\Omega\neq 0$, the interference term in Eqs. (2) or (3)
modulated by $\cos\Omega\tau$ generates an interference beat in
$\mathrm{g}^{(2)}_{\parallel}(\tau)$ curves (see Fig. 4). If
$\cos(\Omega\Delta t)=-1$ or $0$, we would always observe anti-bunching side
dips at $\tau=\pm\Delta t$ no matter what the fibre length is. This can be
easily seen from Eq. (3). If $\cos(\Omega\Delta t)=1$, whether side peaks or
dips are observed depends on the fibre length in the same way as the
$\Omega=0$ case. This explains the periodic changeover between side peaks and
dips at $\tau=\pm\Delta t$ when the AOM frequency shift is tuned (see Fig. 4).
Note that similar classical interference beat is also observed for the driving
laser light, but no sharp dips or peaks are observed [see Fig. 4(f)].

The beat visibility of $0.5$ in Figs. 4(a)-4(d) and the TPI visibility
background of $0.5$ in Fig. 2(f) both reflect the classical effect in TPI,
while the TPI visibility of $94.3\%\pm 0.2\%$ in Fig. 2(f) shows the quantum
effect in TPI. All these phenomena can be well explained by Eq. (2). Taking
$R_{A}=T_{A}=50\%$ and $R_{B}=T_{B}=50\%$, Eq.(2) yields
$V_{HOM}(0)=V_{0}/[1+\mathrm{g}^{(2)}(0)]$, specially $V_{HOM}(0)=V_{0}$ for a
stream of single photons with $\mathrm{g}^{(2)}(0)=0$, $V_{HOM}(0)=V_{0}/2$
for a stream of un-correlated photons (e.g., laser light) with
$\mathrm{g}^{(2)}(0)=1$ and $V_{HOM}(0)=V_{0}/3$ for thermal light with
$\mathrm{g}^{(2)}(0)=2$. Therefore, $V_{0}=V_{HOM}(0)[1+\mathrm{g}^{(2)}(0)]$
rather than $V_{HOM}(0)$ alone measure the true photon indistinguishability of
laser light, in accord with Mandel’s insight that indistinguishability equals
coherence [31].

##  IV DISCUSSION

To summarize, we have demonstrated a simultaneous observation of quantum and
classical two-photon interference of single photons with ultralong coherence
time which is five orders of magnitude longer than the photon correlation
time. We coherently convert laser light into single photons using a single QD
coupling to a double-sided optical microcavity in the Purcell regime. We
observe a TPI visibility of $94.3\%\pm 0.2\%$ but a beat visibility of $50\%$.
Besides the anti-bunching central dip due to single-photon statistics, we also
observe two bunching side peaks in cross-correlation curves for
indistinguishable photons due to second-order interference and intricate
photon correlations. We reproduce the experimental results using either
classical wave superposition theory or quantum field approach. We conclude
that quantum TPI with a stream of single photons is equivalent to classical
TPI, both of which are the fourth-order interference arising from the second-
order interference occurring on the time scale of photon coherence time. Our
work sheds new light on the nature of TPI.

Moreover, we point out that the technique to convert laser light into single
photons can be utilized to measure the genuine photon indistinguishability of
lasers, which is not accessible before. Inheriting the laser’s first-order
coherence time and robust photon indistinguishability, laser-converted single
photons [24] could be a key resource for interference-based quantum
information technologies, e.g., to establish high-quality TPI between remote
quantum memories [32] or to realize long-distance TPI [33], an important step
towards quantum internet [9, 10, 11].

###### Acknowledgements.

This work is supported by the Beijing Natural Science Foundation under the
grant IS23069. Z.C. Niu is grateful to the National Key Technology R$\&$D
Program of China under the grant 2018YFA0306101 for financial support.

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