# A short note on exponential sums, exceptional characters and primes in short
intervals

Runbo Li  The High School Affiliated to Renmin University of China, Beijing
100080, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

We focus on primes in short intervals $[x-x^{\alpha},x]$ under the assumption
of the existence of exceptional Dirichlet characters and get a small
improvement of a 2004 result of Friedlander and Iwaniec. By using our new
estimation of exponential sums, we extend the previous admissible range
$0.4937\leqslant\alpha\leqslant 1$ to $0.4923\leqslant\alpha\leqslant 1$.

###### Key words and phrases:

primes, short intervals, exponential sums, exceptional characters

###### 2020 Mathematics Subject Classification:

11M06, 11N05, 11N37

##  1\. Introduction

The famous Riemann Hypothesis is equivalent to the asymptotic formulas

$\psi(x):=\sum_{n\leqslant
x}\Lambda(n)=x+O\left(x^{\frac{1}{2}+\varepsilon}\right)\quad\operatorname{and}\quad\pi(x)=\operatorname{Li}(x)+O\left(x^{\frac{1}{2}+\varepsilon}\right),$
(1)

where $\Lambda(n)$ is the von Mangoldt function. These imply that

$\psi(x)-\psi(x-x^{\alpha})=x^{\alpha}+O\left(x^{\frac{1}{2}+\varepsilon}\right)\quad\operatorname{and}\quad\pi(x)-\pi(x-x^{\alpha})=\left(\operatorname{Li}(x)-\operatorname{Li}(x-x^{\alpha})\right)+O\left(x^{\frac{1}{2}+\varepsilon}\right)$
(2)

for $\frac{1}{2}+\varepsilon<\alpha\leqslant 1$. Clearly this shows that there
is always a prime number in the short interval
$[x-x^{\frac{1}{2}+\varepsilon},x]$. Unfortunately, we can’t prove (2)
unconditionally. Now the best unconditional result is due to Baker, Harman and
Pintz. In [1] they showed that $0.525\leqslant\alpha\leqslant 1$ is
admissible. But even we assume the Riemann Hypothesis, we can’t extend the
range $\frac{1}{2}+\varepsilon<\alpha\leqslant 1$ anymore.

In 2004, under the assumption of the existence of exceptional Dirichlet
characters, Friedlander and Iwaniec [3] first extended the range of $\alpha$
to some numbers below $\frac{1}{2}$. Actually they proved the following
theorem:

######  Theorem 1.1.

([[3], Theorem 1.1]). Let $\chi$ be the real primitive character of conductor
D. Let $x\geqslant D^{r}$ with $r=18289$ and
$\frac{39}{79}\leqslant\alpha\leqslant 1$. Then

$\psi(x)-\psi(x-x^{\alpha})=x^{\alpha}\left\\{1+O\left(L(1,\chi)(\log
x)^{r^{r}}\right)\right\\}$

and

$\pi(x)-\pi(x-x^{\alpha})=\left(\operatorname{Li}(x)-\operatorname{Li}(x-x^{\alpha})\right)\left\\{1+O\left(L(1,\chi)(\log
x)^{r^{r}}\right)\right\\},$

where $L(s,\chi)$ is the Dirichlet L-function.

In fact, their result strongly depends on a deep result involving product of
three Dirichlet $L$-series by themselves ([4], with only classical Van der
Corput method):

######  Theorem 1.2.

([[4], Theorem 4.2]). Let $\chi_{j}\left(\bmod D_{j}\right)$ denote primitive
characters and $D=D_{1}D_{2}D_{3}$. For any $x\geqslant 1$, we have

$\displaystyle\Delta:=$ $\displaystyle\sum_{n_{1}n_{2}n_{3}\leqslant
x}\chi_{1}\left(n_{1}\right)\chi_{2}\left(n_{2}\right)\chi_{3}\left(n_{3}\right)-\operatorname{Res}_{s=1}\left(L\left(s,\chi_{1}\right)L\left(s,\chi_{2}\right)L\left(s,\chi_{3}\right)\frac{x^{s}}{s}\right)$
$\displaystyle\ll$ $\displaystyle
D^{\frac{38}{75}}x^{\frac{37}{75}+\varepsilon}.$

In 2017, based on his previous work on higher order derivative tests for
exponential sums [7], Nowak [8] [9] combined the classic Van der Corput method
with Huxley’s ”Discrete Hardy–Littlewood method” (see [5]) and obtained the
following two estimations of $\Delta$:

######  Theorem 1.3.

([[8], Theorem 1], [[9], Theorem 3]). Put
$D_{\max}:=\max\left(D_{1},D_{2},D_{3}\right)$, then for any $x\geqslant 1$,
we have

$\Delta\ll
x^{\varepsilon}\left(D^{\frac{1153}{5073}}D_{\max}^{\frac{474}{1691}}x^{\frac{2498}{5073}}+D^{\frac{14077}{81168}}D_{\max}^{\frac{54661}{162336}}x^{\frac{26507}{54112}}+D^{\frac{38017}{144981}}D_{\max}^{\frac{3986}{16109}}x^{\frac{71090}{144981}}+D^{\frac{856679}{2404602}}D_{\max}^{\frac{1281}{6764}}x^{\frac{2185055}{4809204}}\right)$
(A)

and

$\displaystyle\Delta\ll$ $\displaystyle
x^{\varepsilon}\left(D^{\frac{1153}{5073}}D_{\max}^{\frac{474}{1691}}x^{\frac{2498}{5073}}+D^{\frac{3043}{12387}}D_{\max}^{\frac{1086}{4129}}x^{\frac{6086}{12387}}+D^{\frac{2719}{13362}}D_{\max}^{\frac{2719}{8908}}x^{\frac{13129}{26724}}+D^{\frac{1}{4}}D_{\max}^{\frac{1}{2}}x^{\frac{1}{4}}\right.$
$\displaystyle\left.+D^{\frac{2}{7}}D_{\max}^{\frac{2}{7}}x^{\frac{3}{7}}+D^{\frac{1}{3}}x^{\frac{1}{3}}+D^{\frac{1}{6}}D_{\max}^{\frac{1}{2}}x^{\frac{1}{3}}+D^{\frac{328}{2719}}D_{\max}^{\frac{1}{2}}x^{\frac{2063}{5438}}+D^{\frac{205}{1422}}D_{\max}^{\frac{1}{2}}x^{\frac{253}{711}}\right).$
(B)

Obviously, both (A) and (B) in Theorem 1.3 imply the same bound $\Delta\ll
D^{\frac{2575}{5073}}x^{\frac{2498}{5073}+\varepsilon}$. In this paper, we
further combine Nowak’s work with Bourgain’s new bound (see [2]) and get some
improvement on Theorem 1.3. We will give a detailed proof of one of our
results which is the following one:

######  Theorem 1.4.

Put $D_{\max}:=\max\left(D_{1},D_{2},D_{3}\right)$, then for any $x\geqslant
1$, we have

$\displaystyle\Delta\ll$ $\displaystyle
x^{\varepsilon}\left(D^{\frac{118}{519}}D_{\max}^{\frac{97}{346}}x^{\frac{511}{1038}}+D^{\frac{121}{692}}D_{\max}^{\frac{467}{1384}}x^{\frac{675}{1384}}+D^{\frac{56039}{213309}}D_{\max}^{\frac{69941}{284412}}x^{\frac{419257}{853236}}+D^{\frac{17936}{50343}}D_{\max}^{\frac{131}{692}}x^{\frac{91507}{201372}}\right)$
$\displaystyle\ll$ $\displaystyle
D^{\frac{527}{1038}}x^{\frac{511}{1038}+\varepsilon}.$

Then by the similar arguments as in [3], we can get the following result on
primes in short intervals:

######  Theorem 1.5.

Let $\chi$ be the real primitive character of conductor D. Let $x\geqslant
D^{r}$ with $r=433433$ and $0.4923\leqslant\alpha\leqslant 1$. Then

$\psi(x)-\psi(x-x^{\alpha})=x^{\alpha}\left\\{1+O\left(L(1,\chi)(\log
x)^{r^{r}}\right)\right\\}.$

and

$\pi(x)-\pi(x-x^{\alpha})=\left(\operatorname{Li}(x)-\operatorname{Li}(x-x^{\alpha})\right)\left\\{1+O\left(L(1,\chi)(\log
x)^{r^{r}}\right)\right\\}.$

######  remark.

We have $0.4936<\frac{39}{77}<0.4937$ and $0.4924<\frac{2500}{5077}<0.4925$.

Compare our Theorem 1.5 with Theorem 1.1, we can find that our result is non-
trivial if

$L(1,\chi)\ll(\log x)^{-433433^{433433}-1}$ (3)

or

$L(1,\chi)\ll(\log D)^{-433433^{433433}-1}$ (4)

if $D$ is a positive power of $x$. Unfortunately, Zhang [10] posited that
there is no $L$-function with

$L(1,\chi)\ll(\log D)^{-2022}.$ (5)

##  2\. Higher order derivative tests for single exponential sums

######  Lemma 2.1.

For $r\geqslant 4$ a fixed integer, and positive real parameters $M\geqslant
1$ and $T$, suppose that $F$ is a real function on some compact interval
$I^{*}$ of length $M$, with $r+1$ continuous derivatives satisfying throughout

$F^{(j)}\asymp TM^{-j}\quad\operatorname{for}\quad j=r-2,r-1,r.$

Then, for every interval $I\subseteq I^{*}$,

$E:=\sum_{m\in I}\mathrm{e}^{2\pi\mathrm{i}F(m)}\ll
M^{a_{r}}T^{b_{r}}+M^{\xi_{r}}T^{\eta_{r}}+M^{\alpha_{r}}+M^{\gamma_{r}}T^{-\delta_{r}}$

where
$a_{4}=\frac{1}{2},b_{4}=\frac{13}{84}+\varepsilon,\xi_{4}=0,\eta_{4}=\frac{31}{84}+\varepsilon,\alpha_{4}=\frac{334}{411},\gamma_{4}=1,\delta_{4}=\frac{1}{2}$,
and for every $j>4$,

$\begin{gathered}a_{j}=\frac{a_{j-1}+b_{j-1}+1}{2\left(b_{j-1}+1\right)},\quad
b_{j}=\frac{b_{j-1}}{2\left(b_{j-1}+1\right)},\\\
\xi_{j}=\frac{\left(\xi_{j-1}+1\right)\left(b_{j-1}+1\right)-a_{j-1}\eta_{j-1}}{2\left(b_{j-1}+1\right)},\quad\eta_{j}=\frac{\eta_{j-1}}{2\left(b_{j-1}+1\right)},\quad\alpha_{j}=\frac{\alpha_{j-1}+1}{2},\\\
\gamma_{j}=\frac{a_{j-1}\delta_{j-1}+\left(b_{j-1}+1\right)\left(\gamma_{j-1}+1\right)}{2\left(b_{j-1}+1\right)},\quad\delta_{j}=\frac{\delta_{j-1}}{2\left(b_{j-1}+1\right)}.\end{gathered}$

###### Proof.

The proof is very similar to that of [[7], Theorem 1]. The only difference is
that we use the bound $E\ll M^{\frac{1}{2}}T^{\frac{13}{84}+\varepsilon}$ in
the range $T^{\frac{17}{42}+\varepsilon}\ll M\ll T^{\frac{1}{2}}$ and the
bound $E\ll M^{\frac{1}{2}}T^{\frac{32}{205}+\varepsilon}\ll
M^{\frac{333}{410}+\varepsilon}\ll M^{\frac{334}{411}}$ in the range
$T^{\frac{1}{2}}\ll M\ll T^{\frac{181}{328}+\varepsilon}$. ∎

######  Lemma 2.2.

For positive real parameters $M\geqslant 1$ and $T$, suppose that $F$ is a
real function on some compact interval $I^{*}$ of length $M$, with 6
continuous derivatives satisfying throughout

$F^{(j)}\asymp TM^{-j}\quad\operatorname{for}\quad j=3,4,5.$

Then, for every interval $I\subseteq I^{*}$,

$E\ll
M^{\frac{139}{194}}T^{\frac{13}{194}+\varepsilon}+M^{\frac{163}{388}}T^{\frac{31}{194}+\varepsilon}+M^{\frac{745}{822}}+M^{\frac{215}{194}}T^{-\frac{21}{97}}.$

###### Proof.

This is the case $r=5$ of our Lemma 2.1. ∎

##  3\. A problem considered by Friedlander, Iwaniec and Nowak

We follow in all essentials the argument of [4], taking a bit more care and
using our Lemma 2.2. Writing $e(w):=\mathrm{e}^{2\pi\mathrm{i}w}$ as usual, we
start from the estimate

$\Delta\ll\left|D^{\frac{1}{6}}x^{\frac{1}{3}}\sum_{n_{1}n_{2}n_{3}\leqslant
N}\frac{\bar{\chi}_{1}\left(n_{1}\right)\bar{\chi}_{2}\left(n_{2}\right)\bar{\chi}_{3}\left(n_{3}\right)}{\left(n_{1}n_{2}n_{3}\right)^{\frac{2}{3}}}e\left(\pm
3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\right)\right|+x^{\varepsilon}\left(\frac{Dx^{2}}{N}\right)^{\frac{1}{3}}$
(6)

where the sign $\pm$ is chosen so that the modulus involved becomes maximal.
This result is immediate from [[4], Proposition 3.2]. The exponential sum here
can be split up into $O\left(\log^{3}x\right)$ subsums

$S\left(N_{1},N_{2},N_{3}\right):=\sum_{\begin{subarray}{c}n_{j}\sim N_{j}\\\
j=1,2,3\end{subarray}}\frac{\bar{\chi}_{1}\left(n_{1}\right)\bar{\chi}_{2}\left(n_{2}\right)\bar{\chi}_{3}\left(n_{3}\right)}{\left(n_{1}n_{2}n_{3}\right)^{\frac{2}{3}}}e\left(\pm
3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\right)$

with $N_{1}N_{2}N_{3}\leqslant N$ throughout. Without loss of generality we
assume that $N_{1}\leqslant N_{2}\leqslant N_{3}$, and put $P=N_{1}N_{2}N_{3}$
for short. Then obviously

$S\left(N_{1},N_{2},N_{3}\right)\ll
P^{-\frac{2}{3}}\sum_{\begin{subarray}{c}n_{j}\sim N_{j}\\\
j=1,2\end{subarray}}\left|\sum_{n_{3}\sim
N_{3}}\bar{\chi}_{3}\left(n_{3}\right)e\left(\pm
3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\right)\right|.$ (7)

Here integration by parts has been used with respect to $n_{3}$, and trivial
estimation with respect to $n_{1},n_{2}$. For each fixed pair
$\left(n_{1},n_{2}\right)$, the range for $n_{3}$ is chosen in such a way that
the absolute value on the right hand side becomes maximal.

The next step is to express the character $\bar{\chi}_{3}$ by means of the
Gauss sums $G\left(m,\bar{\chi}_{3}\right)$ :

$\bar{\chi}_{3}\left(n_{3}\right)=\frac{1}{D_{3}}\sum_{m=1}^{D_{3}}G\left(m,\bar{\chi}_{3}\right)e\left(-\frac{mn_{3}}{D_{3}}\right),\quad
G\left(m,\bar{\chi}_{3}\right):=\sum_{k=1}^{D_{3}}\bar{\chi}_{3}(k)e\left(\frac{km}{D_{3}}\right)$

Since $\left|G\left(m,\bar{\chi}_{3}\right)\right|\leqslant
D_{3}^{\frac{1}{2}}\leqslant D_{\max}^{\frac{1}{2}}$, it follows that for each
fixed $\left(n_{1},n_{2}\right)$, with $n_{1}\sim N_{1}$, $n_{2}\sim N_{2}$,

$\sum_{n_{3}\sim N_{3}}\bar{\chi}_{3}\left(n_{3}\right)e\left(\pm
3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\right)\ll
D_{\max}^{\frac{1}{2}}\max_{m=1,\ldots,D_{3}}\left|\sum_{n_{3}\sim
N_{3}}e\left(3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\mp\frac{mn_{3}}{D_{3}}\right)\right|.$

The last exponential sums will now be bounded by Lemma 2.2. The conditions of
Lemma 2.2 are satisfied with

$M=N_{3},\quad T=\left(\frac{Px}{D}\right)^{\frac{1}{3}}.$

Therefore, by Lemma 2.2 we have

$\displaystyle\sum_{n_{3}\sim
N_{3}}\bar{\chi}_{3}\left(n_{3}\right)e\left(3\left(\frac{n_{1}n_{2}n_{3}x}{D}\right)^{\frac{1}{3}}\right)$
$\displaystyle\ll$ $\displaystyle
D_{\max}^{\frac{1}{2}}\left(N_{3}^{\frac{139}{194}}\left(\left(\frac{Px}{D}\right)^{\frac{1}{3}}\right)^{\frac{13}{194}+\varepsilon}+N_{3}^{\frac{163}{388}}\left(\left(\frac{Px}{D}\right)^{\frac{1}{3}}\right)^{\frac{31}{194}+\varepsilon}+N_{3}^{\frac{745}{822}}+N_{3}^{\frac{215}{194}}\left(\left(\frac{Px}{D}\right)^{\frac{1}{3}}\right)^{-\frac{21}{97}}\right).$
(8)

By (7)–(8) we get that

$\displaystyle D^{\frac{1}{6}}x^{\frac{1}{3}}S\left(N_{1},N_{2},N_{3}\right)$
$\displaystyle\ll$ $\displaystyle
D^{\frac{1}{6}}x^{\frac{1}{3}}P^{\frac{1}{3}}D_{\max}^{\frac{1}{2}}\left(D^{-\frac{13}{582}}N_{3}^{-\frac{55}{194}}P^{\frac{13}{582}}x^{\frac{13}{582}}+D^{-\frac{31}{582}}N_{3}^{-\frac{225}{388}}P^{\frac{31}{582}}x^{\frac{31}{582}}+N_{3}^{-\frac{77}{822}}+D^{\frac{7}{97}}N_{3}^{\frac{21}{194}}P^{-\frac{7}{97}}x^{-\frac{7}{97}}\right)$
$\displaystyle\ll$ $\displaystyle
D_{\max}^{\frac{1}{2}}\left(D^{\frac{14}{97}}N_{3}^{-\frac{55}{194}}P^{\frac{69}{194}}x^{\frac{69}{194}}+D^{\frac{11}{97}}N_{3}^{-\frac{225}{388}}P^{\frac{75}{194}}x^{\frac{75}{194}}+D^{\frac{1}{6}}N_{3}^{-\frac{77}{822}}P^{\frac{1}{3}}x^{\frac{1}{3}}+D^{\frac{139}{582}}N_{3}^{\frac{21}{194}}P^{\frac{76}{291}}x^{\frac{76}{291}}\right).$
(9)

By the trivial facts $P^{\frac{1}{3}}\leqslant N_{3}\leqslant N$ to get rid of
$N_{3}$ and $P\ll N$, we have

$\displaystyle\Delta\ll$
$\displaystyle\left(\frac{Dx^{2}}{N}\right)^{\frac{1}{3}}+D_{\max}^{\frac{1}{2}}\left(D^{\frac{14}{97}}N^{\frac{76}{291}}x^{\frac{69}{194}}+D^{\frac{11}{97}}N^{\frac{75}{388}}x^{\frac{75}{194}}+D^{\frac{1}{6}}N^{\frac{745}{2466}}x^{\frac{1}{3}}+D^{\frac{139}{582}}N^{\frac{215}{582}}x^{\frac{76}{291}}\right).$
(10)

Finally, by choosing

$N=D^{\frac{55}{173}}D_{\max}^{-\frac{291}{346}}x^{\frac{181}{346}},$

we obtain the bound

$\displaystyle\Delta\ll$ $\displaystyle
x^{\varepsilon}\left(D^{\frac{118}{519}}D_{\max}^{\frac{97}{346}}x^{\frac{511}{1038}}+D^{\frac{121}{692}}D_{\max}^{\frac{467}{1384}}x^{\frac{675}{1384}}+D^{\frac{56039}{213309}}D_{\max}^{\frac{69941}{284412}}x^{\frac{419257}{853236}}+D^{\frac{17936}{50343}}D_{\max}^{\frac{131}{692}}x^{\frac{91507}{201372}}\right)$
$\displaystyle\ll$ $\displaystyle
D^{\frac{527}{1038}}x^{\frac{511}{1038}+\varepsilon},$ (11)

which is our Theorem 1.4, improves Nowak’s Theorem 1.3(A) (we have
$0.4924<\frac{2498}{5073}<0.4925$ and $0.4922<\frac{511}{1038}<0.4923$).

##  4\. A new application: Primes in Short Intervals

In his articles [7]–[9], Nowak considered several problems concerning products
of $L$-series and pointed that his method is more powerful in the case of
exponential sums depending on several parameters. For exponential sums
depending on one parameter only, multiple sum estimations are often more
powerful. (see [[9], Concluding Remark]). Clearly our new estimation can
improve these results. Now we follow Friedlander and Iwaniec directly and show
that our Theorem 1.4 can be used to extend the range of $\alpha$. We first
introduce some arithmetic functions:

$\lambda(n):=\sum_{ad=n}\chi(d),\quad\nu(m):=\sum_{ad=m}\mu(a)\mu(d)\chi(d),\quad\rho(n):=\sum_{lm=n}\lambda(m),$

and a close relative of $\lambda(d)$, namely

$\lambda^{\prime}(d):=\sum_{kl=d}\chi(k)\log l$

which has the following properties:

$\lambda^{\prime}(d)=\sum_{bc=d}\lambda(b)\Lambda(c),\quad\Lambda(n)=\sum_{dm=n}\lambda^{\prime}(d)\nu(m),\quad
0\leqslant\lambda^{\prime}(d)\leqslant\tau(d)\log d.$

In the classical divisor problem, one need to get an asymptotic formula for
the functions:

$\mathcal{D}\left(x;\lambda\right):=\sum_{n\leqslant
x}\lambda(n),\quad\mathcal{D}\left(x;\lambda^{\prime}\right):=\sum_{n\leqslant
x}\lambda^{\prime}(n)$

and we also need the asymptotic formula for

$\mathcal{D}\left(x;\rho\right):=\sum_{n\leqslant x}\rho(n).$

######  Lemma 4.1.

For $x\geqslant 1$ we have

$\displaystyle\mathcal{D}\left(x;\lambda\right)=$ $\displaystyle
L(1,\chi)x+O\left(D^{\frac{1}{3}}x^{\frac{1}{3}+\varepsilon}\right),$ (12)
$\displaystyle\mathcal{D}\left(x;\lambda^{\prime}\right)=$ $\displaystyle
L(1,\chi)x\log
x+\left(L^{\prime}(1,\chi)-L(1,\chi)\right)x+O\left(D^{\frac{1}{3}}x^{\frac{1}{3}+\varepsilon}\right),$
(13) $\displaystyle\mathcal{D}\left(x;\rho\right)=$ $\displaystyle
L(1,\chi)x\log
x+\left(L^{\prime}(1,\chi)+(2\gamma-1)L(1,\chi)\right)x+O\left(D^{\frac{527}{1038}}x^{\frac{511}{1038}+\varepsilon}\right),$
(14)

where $\gamma$ is the Euler constant and the implied constant depends only on
$\varepsilon$.

###### Proof.

These can be proved by similar arguments as in [4], and the only difference is
that we use our Theorem 1.4 instead of Theorem 1.2 in order to prove the
formula (14). ∎

Now we split the functions $\rho=\rho^{*}+\rho_{*}$,
$\Lambda=\Lambda^{*}+\Lambda_{*}$ and further define

$\rho^{*}(n):=\sum_{\begin{subarray}{c}lm=n\\\ m\leqslant
D^{2}\end{subarray}}\lambda(m),\quad\rho_{*}(n):=\sum_{\begin{subarray}{c}lm=n\\\
m>D^{2}\end{subarray}}\lambda(m),$
$\Lambda^{*}(n):=\sum_{\begin{subarray}{c}dm=n\\\ m\leqslant
D^{2}\end{subarray}}\lambda^{\prime}(d)\nu(m),\quad\Lambda_{*}(n):=\sum_{\begin{subarray}{c}dm=n\\\
m>D^{2}\end{subarray}}\lambda^{\prime}(d)\nu(m),$
$\psi^{*}(x):=\sum_{n\leqslant
x}\Lambda^{*}(n),\quad\psi_{*}(x):=\sum_{n\leqslant x}\Lambda_{*}(n).$

######  Lemma 4.2.

[[3], Corollary 4.2]. For $D^{\frac{5}{2}}x^{\frac{3}{8}}<y\leqslant x$ we
have

$\psi^{*}(x)-\psi^{*}(x-y)=y+O\left(L(1,\chi)y(\log x)^{9}\right),$ (15)

where the implied constant is absolute.

By Lemma 4.1 and similar arguments as in [3], for
$D^{2}x^{\frac{492293}{1000000}}<y\leqslant x$ we obtain the following
asymptotic formula

$\mathcal{D}\left(x;\rho_{*}\right)-\mathcal{D}\left(x-y;\rho_{*}\right)=L(1,\chi)y\\{\log
x+O(1)\\}.$ (16)

Now by (16) and

$\left|\psi_{*}(x)-\psi_{*}(x-y)\right|\leqslant(\log
x)\sum_{\delta\leqslant\Delta}(2\tau(\delta))^{A}\left[\mathcal{D}\left(x\delta^{-1};\rho_{*}\right)-\mathcal{D}\left((x-y)\delta^{-1};\rho_{*}\right)\right]$
(17)

with the bound

$\sum_{\delta\leqslant\Delta}\tau(\delta)^{A}\delta^{-1}\ll(\log\Delta)^{2^{A}+1}$
(18)

where $\Delta=x^{\frac{1}{r}},A=\frac{r\log r}{\log 2}$ and $r$ is a positive
integer, we have

######  Lemma 4.3.

For
$D^{\frac{5}{2}}x^{\frac{492293}{1000000}+\frac{507707}{1000000r}}<y\leqslant
x$ with $r\geqslant 1$, we have

$\psi_{*}(x)-\psi_{*}(x-y)=O\left(L(1,\chi)y(\log x)^{r^{r}+3}\right),$ (19)

where the implied constant depends only on $r$.

Adding (19) to (15) we find that

$\psi(x)-\psi(x-y)=O\left(L(1,\chi)y(\log x)^{r^{r}+3}\right),$ (20)

subject to the conditions of Lemma 4.3. We require $D\leqslant
x^{\frac{1}{r}}$ and then choose a number $\theta$ with
$\frac{492293}{1000000}<\theta<\frac{1}{2}$ satisfies

$\frac{5}{2r}+\frac{492293}{1000000}+\frac{507707}{1000000r}\leqslant\theta.$

By choosing $\theta=0.4923$ and $r=433433$, we complete the proof of Theorem
1.5. We remark that the range $0.4923\leqslant\alpha\leqslant 1$ is rather
near to the limit obtained by this method. In his preprint [6], Merikoski
mentioned that some sieve arguments can be used to this problem.

## References

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070009 2015 A. Vindigni M. Perfetti, Dipartimento di Chimica, Universitá di
Firenze, Italy 070009

Dy2Ti2O7 is at present the cleanest example of a spin-ice material. Previous
theoretical and experimental work on the first-order transition between the
kagome-ice and the fully polarized state has been taken as a validation for
the dipolar spin-ice model. Here we investigate in further depth this phase
transition using ac-susceptibility and dc-magnetization, and compare this
results with Monte-Carlo simulations and previous magnetization and specific
heat measurements. We find signatures of an intermediate state between the
kagome-ice and full polarization. This signatures are absent in current
theoretical models used to describe spin-ice materials.

# An intermediate state between the kagome-ice and the fully polarized state
in Dy2Ti2O7

S. A. Grigera  [inst1, inst2] R. A. Borzi E-mail<EMAIL_ADDRESS>[inst3] D.
G. Slobinsky  [inst1] A. S. Gibbs Now at: Departamento de Ingeniería Mecánica,
Facultad Regional La Plata, Universidad Tecnológica Nacional, 1900 La Plata,
Argentina. [inst1]
R. Higashinaka  [inst4] Y. Maeno  [inst5] T. S. Grigera[inst3]

(17 May 2015; 12 June 2015)

††volume: 7

99 inst1 School of Physics and Astronomy, University of St Andrews, North
Haugh, St Andrews KY16 9SS, UK inst2Instituto de Física de Líquidos y Sistemas
Biológicos, UNLP-CONICET, 1900 La Plata, Argentina inst3Instituto de
Investigaciones Fisicoquímicas Teóricas y Aplicadas UNLP-CONICET and
Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de
La Plata, 1900 La Plata, Argentina inst4 Graduate School of Science, Tokyo
Metropolitan University, Hachioji, Tokyo 192-0397, Japan. inst5Department of
Physics, Kyoto University, Kyoto 606-8502, Japan.

##  1 Introduction

Spin-ice materials are deceptively simple in their constitution: classical
Ising spins with nearest-neighbour ferromagnetic interactions forming a
pyrochlore lattice. This crystal structure can be thought as an alternating
stack of kagome and triangular lattices along the [111] direction. The spins
sit at the vertices of tetrahedra and can point either to their center or
towards the outside. The magnetic frustration can be seen at the level of a
single tetrahedron: the energy is minimized by having two spins pointing
inwards and two outwards. This is the _ice rule_ , which corresponds exactly
to the Pauling rules for protons in water ice; like the latter, it also leads
to zero-point entropy, a characteristic signature of spin-ice systems [1].

We have chosen to work on Dy2Ti2O7 as the cleanest example of a spin-ice
material. Its ground state properties can be well described by a model with
only an effective nearest neighbour exchange interaction $J_{\rm si}$ of
$\approx 1.1\,$K [2]. Within this framework, when one applies an external
magnetic field $H$ in [111] below 1 K, the polarization of the system will
happen in two steps. First, the spins in the triangular lattice that lie
parallel to [111] will orient along the magnetic field, removing _part_ of the
residual entropy but with no change in the configurational energy [3, 4]. When
the magnetic moment of this sublattice has saturated, the magnetization $M$
cannot be further increased without breaking the spin-ice rule. This leads to
a plateau as a function of field at $M=3.33\,\mu_{\rm B}$/Dy-ion,
characteristic of the _kagome ice_ state. At higher fields, the spins in the
kagome lattice are finally fully polarized, leading to a sudden but continuous
increase in $M$ towards its saturation. This behavior was predicted
theoretically and found in Monte Carlo simulations [5, 6]. In spite of this,
something different happens in real spin-ice materials.

Figure 1: Real (left) and Imaginary (right) parts of the ac-susceptibility as
a function of temperature and magnetic field, for temperatures between 50 mK
and 6000 mK and magnetic fields between -4 and 4 T. The oscillatory field was
of amplitude 0.05 Oe and at a frequency of 87Hz. The zero-field Schottky-type
anomaly corresponding to the onset of spin-ice correlations, and the peaks
corresponding to the critical point at $\pm 1$ T and $\approx$400 mK are
clearly seen.

Since the magnetic moment of the magnetic ions in spin-ice materials is quite
large – the one associated with Dy3+ ions in Dy2Ti2O7 is near 10$\mu_{B}$–,
long range dipolar interactions have to be considered [7]. These interactions
do not alter the zero field ground state [8], but have a big effect on its
excitations. In relation to this, the transition to the fully polarized state
—which is the main concern of this paper— experiments a qualitative change. T.
Sakakibara and collaborators [9] studied experimentally the magnetization with
$H//$ [111] to temperatures much smaller than $J_{\rm si}$. After a well
defined plateau at $\approx 3.33\,\mu_{\rm B}$/Dy-ion, they observed a very
sharp increase in the magnetization. The presence of hysteresis was a
convincing argument that the real system reaches the fully polarized state
through a metamagnetic first order phase change at the lowest temperatures.
The change in $M$ becomes continuous at the critical end-point $T_{c}=360\,\pm
20$mK and $\mu_{0}H_{c}\approx 0.93\,$T [9].

The change in character of this transition —from a crossover to a
discontinuity when dipolar interactions are included— was later understood in
terms of the defects associated with the breaking of the ice rules, or
_monopoles_. The nearest neighbors model corresponds to the case of free non-
conserved monopoles sitting in a diamond lattice. Including dipolar
interactions implies turning on a Coulomb interaction between these charges,
allowing them to condense through a real first order transition [10].
Numerical simulations (including Ewald summations to take into account long
range interactions) proved this picture right, and provided an additional
validation to the dipolar model [10]. The $M{\rm vs.\ }H$ curves obtained in
these simulations are quite symmetrical around $H_{c}$. The jump in the
magnetization $\Delta M(T)$ when crossing the first order transition line
grows very abruptly with decreasing temperature: for $T$ only $\approx 10\%$
below $T_{c}$, $\Delta M(T)$ amounts to $\approx 90\%$ of the total change in
magnetization from the kagome ice to full saturation. In other words, almost
full order is achieved in the system for temperatures just below $T_{c}$ and a
magnetic field of 1 T.

Specific heat $C_{p}$ measurements confirmed the existence of a critical end-
point —a sharp peak is clearly seen very near the precise spot in field and
temperature specified by Sakakibara et al. [11]. However, the identification
of a single first-order line below $T_{c}$ is less clear. The $C_{p}(T)vs.H$
curves show peaks at the fields $H_{c}(T)$ identified in [9] as the first-
order line, albeit of much smaller amplitude than that at $T_{c}$.
Additionally, below 300 mK, a second peak at higher fields is discernible
[11]. Even at the lowest temperatures ($100\,$mK), magnetic fields above 2 T
are needed to coerce the specific heat down to 0. This suggests that, in spite
of the absence of thermal excitations, the system does not reach full
polarization immediately after the first order transition from the kagome ice,
and an intermediate state establishes between these two well-known phases.
This specific heat features were confirmed by $ac$-susceptibility measurements
on the same samples [12]. In all cases, the sample sat at a fixed platform
with respect to the magnetic field and therefore the alignment with respect to
the [111] direction was within a few degrees. An angular dependent study of
the magnetization with Sato and coworkers [19] showed these asymmetries, and
additional features in the polarization transition were seen at small angles
away from [111]. The implications of these results in the current
understanding and modeling of the spin-ice materials have not been considered.

In this paper, we study in detail this additional intermediate state, and show
that it cannot be explained by any of the models currently used to study spin-
ice materials. Working at small angles away from [111], we looked for a
magnetic signature by repeating the static magnetization measurements in
several samples. In addition, improving the sensitivity by three orders of
magnitude, we measured ac-susceptibility at different frequencies, which also
allowed us to do a characterisation of the dynamics of the observed
transitions. In order to gain further insight into this possible intermediate
state, we performed Monte Carlo simulations of the experimental situation
using the currently accepted models including Ewald summations and exchange
interactions up to the third nearest neighbor [13, 14].

##  2 Methods

For our work, we measured several Dy2Ti2O7 single crystals grown in Kyoto and
in St Andrews with the floating-zone method. Samples were oriented using Laue
diffraction and cut into $3\,$mm long prisms of square or octagonal section of
approximately $1\,$mm2, with the [111] direction along the long axis to reduce
demagnetising effects with the field in the vicinity of [111] ($\approx
5^{o}$). The experiments were performed in a dilution refrigerator in St
Andrews. Samples were thermally grounded to the mixing chamber through gold
wires attached into them with silver paint. For susceptibility, we used a
drive field of $3.3\cdot 10^{-5}\,$T r.m.s., and counter-wound pickup coils
each consisting of approximately 1000 turns of $12\,\mu$m diameter copper
wire. The filling factor of the sample in the pick up coil was of
approximately 90%. We measured using drive fields of frequencies varying from
approx. $10\,$Hz to $1.0\,$kHz. Low temperature transformers mounted on the
$1\,$K pot of the dilution refrigerator were used throughout to provide an
initial signal boost of approximately a factor of 100. The magnetization was
measured using a home-built capacitance Faraday magnetometer [15].

##  3 Results and Discussion

Figure 1 shows the real ($\Delta\chi^{\prime}$, left) and imaginary
($\Delta\chi^{\prime\prime}$, right) parts of the ac-susceptibility $\chi$ as
a function of temperature and magnetic field in the whole area of interest.
The excitation frequency in this case is $\omega=87\,$Hz; the main features we
describe in the following are qualitatively independent of $\omega$. At zero
field, there is a very noticeable peak in both $\Delta\chi^{\prime}$ and
$\Delta\chi^{\prime\prime}$ for $T\approx 2\,$K. This corresponds to the
Schottky-type anomaly associated with the onset of spin-ice correlations of
the system. The magnetic field axis spans from -4 to 4 T, and we can clearly
see in the real part two peaks (at positive and negative fields) corresponding
to the critical point at $\approx\pm 1\,$T and $\approx 400\,$mK. For
temperatures below $400\,$mK, we see a much smaller feature in
$\Delta\chi^{\prime\prime}$, which has a correspondence in
$\Delta\chi^{\prime\prime}$: a ridge with an amplitude that decreases as a
function of temperature. The magnitude of the latter is comparatively very
small. At low temperatures and for fields $0.3\,$T$<|\mu_{0}H|<0.9\,$T, and
$2\,$T $<|\mu_{0}H|$, the susceptibility is very low, in accordance to the
kagome ice plateau and the saturation in the magnetization, respectively.

We now concentrate on the real part of the susceptibility at temperatures
below $J_{\rm si}$. In Fig. 2, we can see a series of curves at fixed
temperatures (from 50 to $500\,$mK) and fields between $-3.5$ and $3.5\,$T.
The excitation field used was $0.05\,$Oe, and the frequency $87\,$Hz. The
curves have been offset by 30% for clarity. The field was swept from negative
to positive values. Before the kagome ice state is established, the low field
susceptibility ($|\mu_{0}H|<0.3\,$T) at temperatures below $600\,$mK is
strongly dependent on the magnetic field sweep rate and direction (increasing
or decreasing), both signs of out-of-equilibrium behavior. At higher magnetic
fields, we only observe a small difference in the height of the peaks at
around $\pm 1\,$T, depending on whether the transitions are swept upwards or
downwards in field. The position changes very little, and the shape of the
features is unaltered. As we lower the temperature, the peak at $\approx 1\,$T
decreases markedly in amplitude, but without a corresponding change in its
high field side shoulder. Below $400\,$mK, it eventually splits into two
distinct features. Their separation in the field axis ($\approx$ 0.1 T at 300
mK) is consistent with previous measurements for a similar sample orientation
with respect to [111] [19]. While the first set of peaks has a correlate in
the imaginary part of $\Delta\chi$ (not shown here), no feature is discernible
in $\Delta\chi^{\prime\prime}$ for the peaks at higher fields.

Figure 2: Low temperature real part of the ac-susceptibility as a function of
field at fixed temperatures as indicated in the plot. The excitation field was
0.05 Oe at a frequency of 87Hz. The curves are offset by 30% for clarity. As
temperature is lowered from 400 mK, the peak at approx. 1 T at 400mK rapidly
decreases in amplitude, and splits into two peaks at lower temperatures.

Figure 3: Phase diagram with field slightly tilted from [111] ($\theta\leq
5^{o}$). An intermediate phase is seen between the kagome-ice and fully
polarized regions. The black circle is the critical point as identified from a
peak in the real part of the ac-susceptibility, $\chi^{\prime}$. The dotted
white circles correspond to a small doubled peaks seen in $\chi^{\prime}$ with
a corresponding feature in the imaginary part $\chi^{\prime\prime}$, while the
white circles denote small peak in $\chi^{\prime}$ with no signature in
$\chi^{\prime\prime}$. The red circles are taken from peaks in the specific
heat ($C$) measurements of reference [11]. The main divergence of $C$ seen in
reference [11] and identified as a critical point coincides with the critical
point (black circle).

In Fig. 3, we have plotted the position of these peaks as a function of field
and temperature (white circles), and the position of the critical point (black
circle). We have taken the specific heat data from reference [11] and
determined the position of the peaks in $C$ vs. $H$ for different
temperatures. These are plotted in this same graphic as red symbols. The
coincidence between these two experiments of different quantities, on
different samples, laboratories and experimental setup is remarkable.

As mentioned before, this secondary peak at higher fields is absent in the
$dM/dH$ data presented on Ref. [9]. We measured the magnetization using a
Faraday balance on the same samples and under similar temperature and field
conditions than before [16]. The main body of Fig. [17] shows our $dM/dH$ as a
function of field, compared with curves of $\Delta\chi$ at $T=100\,$mK and
frequencies spanning two orders of magnitude (from $\omega\approx 10$ to
$1000\,$Hz). For clarity, we have multiplied $\Delta\chi$ by a factor of
twenty. The peak in $dM/dH$ is markedly asymmetric, with an extended tail in
the high field side but no additional feature is seen at high fields, in
coincidence with Sakakibara’s observations. On the other hand, the second peak
is clearly seen for low temperature ($T<400\,$mK) at all measured frequencies
in the ac-susceptibility. While these two experiments seem to be in mutual
contradiction, the issue can be easily explained in terms of the resolutions
of both techniques. The inset of Fig. 4 shows both sets of data on the same
scale; we can see that the high field shoulder on the $dM/dH$ peak directly
corresponds (in the limit of long measurement times or low frequencies) to the
second peak detected with ac-susceptibility.

Through this analysis, we can see that the experimental volume of data
concerning this transition seems to be compatible. Between $\approx 300mK$ and
the lowest temperatures (50 mK in Ref. [9]), only $\approx 60\%$ of the total
change in magnetization occurs when traversing the first order transition
line. The remaining 40% is delivered _gradually_ when the field is further
increased to values well above 1.5$H_{c}$, in a fashion that does not seem to
depend much on temperature (see Fig. 3 on Ref. [9]). This gradual (as opposed
to discontinuous) change is behind the asymmetric shape of the magnetization
curves, and the second set of peaks in $C_{p}$ and $\Delta\chi$.

Figure 4: $dM/dH$ (dotted line) and real part of ac-susceptibility measured at
different excitation frequencies, from top to bottom: 19, 37, 77, 136, 277,
561, and 1117 Hz, and for $T=100{\rm mK}$. For ease of comparison, the latter
have been multiplied by a factor 20, and normalized to the amplitude of second
peak (no imaginary part has been measured for this feature). The inset shows
both sets of data in the same scale.

The theoretical prediction for the transition between kagome-ice to fully
polarized state with field in [111] was of a single transition—the “dimer to
monomer” transition of refs. [18, 6]. A small additional perpendicular field –
present in the experiments at small angles away from [111]– induces order in
the dimers in the kagome-ice state, but does not change the prediction of a
single transition into the fully polarised “monomer” state [18]. This might
hold true when further interactions are added, such as dipolar or further
neighbor exchange interactions. In order to investigate this, we performed a
numerical check. We did extensive Monte Carlo simulations of the dipolar model
including Ewald summations to account for the dipolar long range interactions.
We also added exchange interactions up to third nearest neighbors (taking the
exchange constants and other parameters within the constraints given by refs.
[13, 14]). We explored a wide range of field angles around [111], but were
unable to detect a double feature in $C_{V}$ at low temperatures compatible
with the experimental observations. It is then worth stressing that the very
observation of a second feature –even when taking into account a possible
sample misalignment– asks for new ingredients in the Hamiltonians that are
regularly used to describe spin-ice materials.

Given these considerations, it is difficult to discuss on the nature of this
intermediate state. It is tempting to think of some sort of “charge” ordering
in the diamond lattice (2-in 2-out tetrahedra within a majority of 3-1 and
1-3), previous to the final Zn-blende arrangement, where only $\approx
40-50\%$ of the sites are occupied by single monopoles. Note that this does
not rule out the possibility of still storing some residual entropy, since
there are different spin configurations that generate the same charge within a
given tetrahedron. We have not found previous data of the evolution of the
entropy as a function of field at temperatures well below $T_{c}$. However,
the very asymmetric shape of the entropy at 350 mK obtained using the
magnetocaloric effect shows that at this temperature the system is already
experiencing a strong first order metamagnetic transition, as mentioned in
Ref. [20]. This work shows that a big fraction of the residual entropy stored
in the kagome planes remains in the system well above $H_{c}$ [20], suggesting
that the intermediate state is indeed a partially disordered one.

##  4 Conclusions

In conclusion, we observe an intermediate state between the kagome-ice and the
fully polarized state when the field is slightly tilted from the [111]
direction. The signature of a double step we find in ac-susceptibility and
magnetization measurements is also present in earlier calorimetric
measurements, and suggested by magnetocaloric effect experiments. This feature
cannot be captured by the models regularly used to describe spin-ice systems,
fact that asks for further model refinements. At present, this data stands as
a challenge for the development of a realistic theoretical model of spin-ice
materials.

###### Acknowledgements.

We thank Joseph Betouras, Andrew Green and Chris Hooley for useful
discussions. SAG would like to acknowledge financial support from the Royal
Society (UK), RAB and TSG from CONICET, UNLP and ANPCYT (Argentina).

## References

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# Off-axis dipole forces in optical tweezers by an optical analog of the
Magnus effect

Robert J.C. Spreeuw  Van der Waals-Zeeman Institute, Institute of Physics,
University of Amsterdam,
PO Box 94485, 1090 GL Amsterdam, The Netherlands

###### Abstract

It is shown that a circular dipole can deflect the focused laser beam that
induces it, and will experience a corresponding transverse force. Quantitative
expressions are derived for Gaussian and angular tophat beams, while the
effects vanish in the plane-wave limit. The phenomena are analogous to the
Magnus effect pushing a spinning ball onto a curved trajectory. The optical
case originates in the coupling of spin and orbital angular momentum of the
dipole and the light. In optical tweezers the force causes off-axis
displacement of the trapping position of an atom by a spin-dependent amount up
to $\lambda/2\pi$, set by the direction of a magnetic field. This suggests
direct methods to demonstrate and explore these effects, for instance to
induce spin-dependent motion.

A common practice in many branches of sports is to send a ball onto a curved
trajectory by giving it a spin. In this famous example of the Magnus effect
Magnus (1853) the spinning ball deflects the stream of air around it, and is
pushed sideways by the reaction force perpendicular to its forward velocity.
In analogy, we may ask if a rotating dipole in an atom may similarly deflect a
beam of light, and thereby be pushed by a force perpendicular to the light
beam. C.G. Darwin already remarked that for circular dipoles “…the wave front
of the emitted radiation faces not exactly away from the origin, but from a
point about a wave-length away from it.”Darwin (1932). A recent experiment
confirmed that an atomic circular dipole can indeed appear to be displaced
from its true location, due to the emitted spiral-shaped wavefront Araneda
_et al._ (2019). Circular dipoles provide perhaps the simplest example of the
intrinsic coupling of spin and orbital angular momentum (SAM and OAM,
respectively) in non-paraxial light fields Allen  _et al._ (1992); Bliokh and
Nori (2015); van Enk and Nienhuis (1994); Dorn  _et al._ (2003); Nieminen  _et
al._ (2008); Monteiro  _et al._ (2009); Bliokh  _et al._ (2010); Rodríguez-
Herrera  _et al._ (2010); Bliokh  _et al._ (2011); Angelsky  _et al._ (2012).
Such fields, in the form of tightly focused laser beams, are of central
importance in a rapidly growing range of experiments involving (arrays of)
optical tweezers Thompson  _et al._ (2013); Barredo  _et al._ (2016); Endres
_et al._ (2016); Bernien  _et al._ (2017); Cooper  _et al._ (2018); Norcia
_et al._ (2018); Pagano  _et al._ (2019); Jackson  _et al._ (2020); Anderegg
_et al._ (2019); Saskin  _et al._ (2019); Wang  _et al._ (2020). These are
developed as precise tools to hold and manipulate single atoms or molecules at
the quantum level, in creating platforms for quantum simulation and
computation, as well as for quantum sensing and atomic clocks Madjarov  _et
al._ (2019); Norcia  _et al._ (2019).

Figure 1: Optical analog of the Magnus effect. (a) A linearly polarized
($\mathbf{E}\parallel x$), focused laser induces a circular dipole ($xz$
plane) on a $j=0\rightarrow j^{\prime}=1$ ($\Delta m_{j}=1$) transition, with
a magnetic field $\mathbf{B}\parallel y$ setting the quantization axis. The
spiral wave scattered by the circular dipole interferes with the incident
wave, producing two effects: (b) The incident beam is deflected in the $xz$
plane, with corresponding reaction force on the atom, transverse to the beam.
The direction changes sign with the detuning from the atomic resonance. (c) In
an optical tweezer (‘red’ detuning, $\Delta<0$), the transverse force shifts
the trapping position away from the optical axis by an amount
$\lambdabar=\lambda/2\pi$. See Fig. 3 for a far off-resonance case.

Here we predict that an atomic circular dipole can deflect the centered
focused laser beam that induces it. Conversely, the atom will experience a
transverse force when on-axis 111See also Dooghin  _et al._ (1992) for a very
different kind of analogy of the Magnus effect.. An important consequence of
this force can be seen in the off-axis displacement of the trapping potential
created by an optical tweezer Wang  _et al._ (2020). Thus, rather than ‘seeing
an atom where it is not’ Araneda  _et al._ (2019); Schwartz and Dogariu
(2006); Arnoldus  _et al._ (2008), here we describe its counterpart of
‘trapping an atom where the focus is not’ Thompson  _et al._ (2013); Wang  _et
al._ (2020); Caldwell and Tarbutt (2020). A simple geometric argument based on
light scattering shows that the true displacement of the trapping potential is
in fact a direct consequence of the apparent displacement of an emitting
circular dipole Darwin (1932); Araneda  _et al._ (2019). Tweezer trap
displacements have previously been calculated numerically, for specific beam
shapes, in terms of vector and tensor light shifts Thompson  _et al._ (2013);
Caldwell and Tarbutt (2020). The geometric argument given here directly shows
that for a circular dipole the displacement is simply
$\pm\lambdabar=\pm\lambda/2\pi$, and is remarkably independent of many
parameters, including the laser detuning, trap frequency, and even the
detailed shape of the trap. A comparison of a Gaussian beam with an angular
tophat beam illustrates this. This profound insight provides the basis for
state-dependent manipulation of atomic motion within the tweezer.

We describe these effects in terms of interference between the focused
incident beam with the wave scattered by the circular dipole, see Fig. 1. In
the optical theorem, such interference is used to describe the attenuation of
light in terms of the forward scattering amplitude Jackson (1999). In
contrast, here we concentrate on beam deflection, as a consequence of the tilt
of the spiral wavefront with respect to the incident wave. Two simple atomic
level schemes serve as examples; (i) a $j=0\rightarrow j^{\prime}=1$ and (ii)
a $j=1\rightarrow j^{\prime}=0$ transition. Case (i) is conceptually simpler
and most suited to observe the Magnus-like deflection of a weak, near-resonant
probe beam. Case (ii) offers interesting extra opportunities in the usual far
off-resonance regime of optical tweezer experiments.

Starting with case (i), the $j=0\rightarrow j^{\prime}=1$ transition, we focus
a linearly polarized ($\mathbf{E}\parallel x$), monochromatic laser onto a
single atom placed in the origin, see Fig. 1. A magnetic field
$\mathbf{B}\parallel y$ defines the quantization axis and splits the excited
state into three $|j,m_{j}\rangle_{y}$ sublevels, separated by the Zeeman
shift $\sim\mu_{B}B/\hbar$, with $\mu_{B}$ the Bohr magneton Schlesser and
Weis (1992). We tune the laser close to the $\Delta m_{j}=+1$ transition, with
a detuning $\Delta=\omega_{L}-\omega_{0}$ small compared to the Zeeman shift,
so that the $\Delta m_{j}=0,-1$ transitions can be neglected (for example,
$\Delta/2\pi\sim 10\,$MHz and $\mu_{B}B/h\sim 100\,$MHz.) The emission by the
induced circular dipole has a spiral wavefront in the $xz$ plane, tilted with
respect to the forward $\mathbf{\hat{z}}$ direction of the incident beam.

We represent the light fields by their angular spectrum Mandel and Wolf
(1995); sup , using spherical $k$-space coordinates $(k,\theta,\phi)$. For
monochromatic light, with $k=\omega/c$ fixed, the incident field can be
written as $\frac{1}{2}\mathbf{E}_{\text{in}}(\Omega)e^{-i\omega t}+c.c.$,
with $\Omega=(\theta,\phi)$. The total field is the sum of the incident and
scattered waves. Writing only the positive frequency ($\sim e^{-i\omega t}$)
components, the total field reads

$\mathbf{E}(\Omega)=\mathbf{E}_{\text{in}}(\Omega)+\mathbf{E}_{\rm{sc}}(\Omega)$
(1)

with $\mathbf{E}_{\rm{sc}}(\Omega)$ the scattered wave.

We define the radiant intensity,

$J(\Omega)=|\mathbf{E}(\Omega)|^{2}/2Z_{0}.$ (2)

with $Z_{0}=1/\epsilon_{0}c$, so that $J(\Omega)d\Omega$ is the power flowing
out of an infinitesimal solid angle $d\Omega=\sin\theta\,d\theta\,d\phi$
around

$\mathbf{u}_{\Omega}=(\sin\theta\,\cos\phi,\sin\theta\,\sin\phi,\cos\theta)$
(3)

Combining Eqs. (1) and (2), the total radiant intensity is the sum of three
terms,

$J(\Omega)=J_{\text{in}}(\Omega)+J_{\text{sc}}(\Omega)+J_{\text{if}}(\Omega)$
(4)

The interference term

$J_{\text{if}}(\Omega)=\frac{1}{2Z_{0}}\left[\mathbf{E}^{\ast}_{\text{in}}(\Omega)\cdot\mathbf{E}_{\text{sc}}^{\text{(coh)}}(\Omega)+c.c.\right]$
(5)

contains only the coherent component of the scattered field. An incoherent
component would contribute to $J_{\text{sc}}(\Omega)$ but not to
$J_{\text{if}}(\Omega)$. For simplicity we assume that the scattered field is
entirely coherent, essentially restricting ourselves to the low-saturation
limit sup .

The deflection of the light beam can be expressed as the change in average
wave vector $\langle\mathbf{k}\rangle-\langle\mathbf{k}\rangle_{\text{in}}$
between the total (incident plus scattered) and the incident wave, using

$\left\langle\mathbf{k}\right\rangle_{\text{in}}=k\,\frac{\int\mathbf{u}_{\Omega}\,J_{\text{in}}(\Omega)\,d\Omega}{\int
J_{\text{in}}(\Omega)\,d\Omega}=k\,\frac{\int\mathbf{u}_{\Omega}\,J_{\text{in}}(\Omega)\,d\Omega}{P_{\text{in}}}$
(6)

and similar for $\langle\mathbf{k}\rangle$, omitting the subscript. Assuming
(again for simplicity) that non-radiative decay is absent, we shall write
$P_{\text{in}}=P$ throughout.

The deflection is entirely determined by the interference term
$J_{\text{if}}(\Omega)$. The scattered light itself does not contribute, due
to the symmetry of the dipole radiation pattern,
$J_{\text{sc}}(\theta,\phi)=J_{\text{sc}}(\pi-\theta,\pi+\phi)$, so that
$\int\mathbf{u}_{\Omega}J_{\text{sc}}(\Omega)\,d\Omega=0$. For the deflection
we therefore have

$\delta\langle\mathbf{k}\rangle=\langle\mathbf{k}\rangle-\langle\mathbf{k}\rangle_{\text{in}}=\frac{k}{P}\,\int\mathbf{u}_{\Omega}\,J_{\text{if}}(\Omega)\,d\Omega$
(7)

and for the force on the atom, by momentum conservation,

$\mathbf{F}=-\frac{P}{\omega}\,\delta\langle\mathbf{k}\rangle=-\frac{1}{c}\,\int\mathbf{u}_{\Omega}\,J_{\text{if}}(\Omega)\,d\Omega$
(8)

While this expression includes the forward radiation pressure force, in the
cases of interest here the main force will be transverse to the optical axis,
$\mathbf{F}\approx F_{x}\mathbf{\hat{x}}$. Then (approximately)
$\delta\langle\mathbf{k}\rangle\perp\langle\mathbf{k}\rangle_{\text{in}}$ and
with $\langle\mathbf{k}\rangle_{\text{in}}\approx k\mathbf{u}_{z}$ the
deflection angle is

$|\delta\theta|\approx\frac{|\delta\langle\mathbf{k}\rangle|}{k}$ (9)

We will choose $\delta\theta>0$ if $F_{x}<0$.

Let us now introduce specific field patterns to calculate
$J_{\text{if}}(\Omega)$. We take the dipole to be circular,
$\mathbf{p}=pe^{i\alpha}\mathbf{u}_{+}$, with
$\mathbf{u}_{\pm}=(\mathbf{\hat{x}}\mp i\mathbf{\hat{z}})/\sqrt{2}$ denoting
spherical unit vectors, and $\alpha$ the phase of the $p_{x}$ component of the
dipole, relative to the local driving field. The field radiated by a coherent
dipole Jackson (1999), in angular coordinates, takes the form sup :

$\mathbf{E}_{\text{sc}}(\Omega)=\mathcal{E}_{\text{sc}}\,ie^{i\alpha}\left(\mathbf{u}_{\Omega}\times\mathbf{u}_{+}\right)\times\mathbf{u}_{\Omega}$
(10)

with corresponding $J_{\text{sc}}(\Omega)$ given by Eq. (2). Here
$\mathcal{E}_{\text{sc}}=p\,k^{2}/4\pi\epsilon_{0}>0$ is a real-valued
amplitude. Assuming the steady state of the optical Bloch equations for a two-
level system, $\cot\alpha=-\Delta/\gamma$, with $\Delta=\omega-\omega_{0}$ the
detuning from the $\Delta m_{j}=+1$ transition, and
$\gamma=\omega_{0}^{3}D^{2}/6\pi\epsilon_{0}\hbar c^{3}$ the half width of the
transition, with $D$ the transition dipole moment.

For comparison, we consider two different types of incident beams, Gaussian
(‘G’) and ‘angular tophat’ (‘$\Pi$’), where the latter approximates the output
of a uniformly illuminated focusing lens. The field for these two beams can be
written as

$\displaystyle\mathbf{E}^{\text{(G)}}_{\text{in}}(\Omega)$
$\displaystyle\approx$
$\displaystyle\mathcal{E}_{0}^{\text{(G)}}\,\exp[-\theta^{2}/w_{\theta}^{2}]\,\mathbf{u}_{x}(\Omega)$
(11) $\displaystyle\mathbf{E}^{(\Pi)}_{\text{in}}(\Omega)$ $\displaystyle=$
$\displaystyle\mathcal{E}_{0}^{(\Pi)}\,\Pi(\theta/2r_{\theta})\,\mathbf{u}_{x}(\Omega)$
(12)

with amplitudes $\mathcal{E}_{0}^{\text{(G)}},\mathcal{E}_{0}^{(\Pi)}>0$. The
Gaussian beam has an angular width $w_{\theta}$ which is related to the
minimum waist $w_{0}$ ($1/e^{2}$ spatial radius of intensity) as
$w_{\theta}w_{0}=\lambda/\pi$. For the angular tophat,
$\Pi(\theta/2r_{\theta})$ is the rectangular function with angular half width
$r_{\theta}$ and unit amplitude. Its spatial profile near the focus is the
familiar Airy disk pattern. Note that neither propagation phases nor the Gouy
phase are visible here, as the above expressions are in angular coordinates
sup .

The polarization vector $\mathbf{u}_{x}(\Omega)$ is transverse to
$\mathbf{u}_{\Omega}$; it is obtained by co-rotating $\mathbf{\hat{x}}$ when
rotating $\mathbf{\hat{z}}\rightarrow\mathbf{u}_{\Omega}$, i.e. rotating by
$\theta$ around an axis $\mathbf{\hat{z}}\times\mathbf{u}_{\Omega}$ Richards
and Wolf (1959); Rodríguez-Herrera  _et al._ (2010),

$\mathbf{u}_{x}(\Omega)=\begin{pmatrix}\cos\theta\cos^{2}\phi+\sin^{2}\phi\\\
(\cos\theta-1)\sin\phi\cos\phi\\\ -\sin\theta\cos\phi\end{pmatrix}$ (13)

When combining Eq. (10) with Eq. (11) or (12) in Eq. (5), the interference
term contains the amplitude product
$\mathcal{E}_{0}^{\text{(G)}}\mathcal{E}_{\text{sc}}$ or
$\mathcal{E}_{0}^{(\Pi)}\mathcal{E}_{\text{sc}}$. In the low-saturation limit,
the amplitude $\mathcal{E}_{\text{sc}}$ is proportional to
$\mathcal{E}_{0}^{\text{(G)}}$ or $\mathcal{E}_{0}^{(\Pi)}$. Their ratio can
be obtained by requiring energy conservation sup . Upon insertion of the
resulting ratios $\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{\text{(G)}}$ and
$\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{(\Pi)}$ into Eq. (5), the
interference term $J_{\text{if}}(\Omega)$ becomes proportional to the total
power; the deflection angle is then independent of power.

In Fig. 2 we show $J_{\text{in}}(\Omega)$ in the plane of the dipole
($\phi=0$), together with the total radiant intensity $J(\Omega)$. For the
Gaussian beam, the effect of $J_{\text{if}}(\Omega)$ is to shift the peak and
the average of the direction of propagation away from $\theta=0$. For the
angular tophat, the interference leads to an intensity gradient across the
angular width of the beam, whereas the edges stay at the same angle. In this
case the intensity gradient leads to a change in average beam direction.

Finally, the deflection angle is obtained by integration as in Eq. (7),

$\delta\theta\approx\frac{3}{4}\frac{\gamma\Delta}{\left(\gamma^{2}+\Delta^{2}\right)}\times\left\\{\begin{array}[]{ll}w_{\theta}^{4}&\quad\text{(Gauss)}\\\
r_{\theta}^{4}/4&\quad\text{(angular tophat)}\end{array}\right.$ (14)

and the reaction force as

$F_{x}\approx-\frac{P}{c}\delta\theta$ (15)

The results are given as the leading order in $w_{\theta}$ and $r_{\theta}$.
The deflection angle reaches maximal values of $\delta\theta=\pm
3w_{\theta}^{4}/8$ and $\pm 3r_{\theta}^{4}/32$, respectively, for
$\Delta=\pm\gamma$; it vanishes in the plane-wave limit,
$w_{\theta},r_{\theta}\rightarrow 0$. In this central result we recognize in
the detuning dependence that the force is essentially a dipole force Gordon
and Ashkin (1980), arising from polarization gradients near the focus of a
linearly polarized light beam Bliokh  _et al._ (2011); Dorn  _et al._ (2003);
Monteiro  _et al._ (2009); Nieminen  _et al._ (2008); Thompson  _et al._
(2013); Wang  _et al._ (2020); Caldwell and Tarbutt (2020).

Figure 2: Beam deflection: radiant intensities in the plane of the
$\mathbf{u}_{+}$ dipole, for (a) a Gaussian incident beam with
$w_{\theta}=0.6$, and (b) an angular tophat incident beam with
$r_{\theta}=0.6$. In both cases, the gray/dotted curve shows
$J_{\text{in}}(\theta,\phi=0)$ of the incident beam, normalized to 1 for
$\theta=0$; red/solid and blue/dashed curves show the outgoing, or total
$J(\theta,0)$, for $\Delta=-\gamma$ and $+\gamma$, respectively. For clarity,
we identify $(\theta,0)\equiv(-\theta,\pi)$. Curves remain the same upon
switching simultaneously the signs of the detuning and the spin of the dipole.

We now address the question of how we can observe the deflection of a laser
beam, either directly or via the reaction force on the atom. As shown by Eq.
(14), the angle of deflection by a single atom is small compared to the
divergence angle, $|\delta\theta|\ll r_{\theta},w_{\theta}$. A direct
observation will thus require sufficiently high signal-to-noise ratio, similar
to what was achieved in the recent observation of apparent $\lambdabar$
displacement of an emitter Araneda  _et al._ (2019). With maximal signal
occurring near resonance ($\Delta=\pm\gamma$), where the photon scattering
rate is high, the best approach would be to hold the atom in an independent
trap, such as an ion trap or a tight optical tweezer. One can then look for
the deflection of a weak, near-resonant probe beam. A larger deflection angle
may be obtained if multiple atoms cooperate. For example, one may consider
dense clouds of sub-wavelength size, containing tens to hundreds of atoms,
that have been observed to show collective scattering properties Pellegrino
_et al._ (2014); Machluf  _et al._ (2019). Another possibility may be to use
elongated, (quasi-) one-dimensional samples with tight ($\lesssim\lambdabar$)
radial confinement, achievable, e.g., in optical lattices Moritz  _et al._
(2003); Paredes  _et al._ (2004); Kinoshita  _et al._ (2004) and on atom chips
Jacqmin  _et al._ (2011).

Figure 3: Optical tweezer operating on a $j=1\rightarrow j^{\prime}=0$
transition, leading to $\pm\lambdabar$ off-axis displacements for the
$(m_{j})_{y}=\mp 1$ sublevels (upper left). The four panels show, in clockwise
order, the effect of a rotation of the quantization axis ($\mathbf{B}$),
through a cycle $y\rightarrow z\rightarrow-y\rightarrow-z$. While the
$\mathbf{B}$-referenced $(m_{j})_{B}$ of an atom is conserved, the space-
referenced $(m_{j})_{y}$ is not. The locations of the $(m_{j})_{B}=\pm 1$
traps move up and down along the $x$ axis, in antiphase. If $\mathbf{B}$ is
rotated at the trap frequency, spin-dependent oscillatory motion in the
tweezer can be induced.

The second mode of observation, via the force on the atom, provides extra
opportunities to manipulate spin-dependent atomic motion in an optical
tweezer. To see this we consider case (ii): an optical tweezer trapping an
atom with a $j=1\rightarrow j^{\prime}=0$ transition. The $|m_{j}=\pm
1\rangle_{y}$ states now couple to the $(\sigma^{\mp})_{y}$ components of the
light field, and therefore experience opposite forces $F_{x}$. In this
configuration there is no need for a separate probe beam Wang  _et al._
(2020), the far off-resonance light ($\Delta/2\pi\sim 1-10\,$THz) of the
tweezer itself is sufficient. The photon scattering and associated heating
rates can thus be kept as low as in typical tweezer experiments. In this case
we assume that the Zeeman shift is large compared to the trap depth $U_{0}$
(for example $\mu_{B}B/h\sim 10\,$MHz, and $U_{0}/h\sim 1\,$MHz.) Looking at
the spiral wave of a $\mathbf{u}_{+}$ dipole shown in Fig. 1, we can readily
see that the relative tilt of the forward wavefronts will vanish if we
displace the atom by $\lambdabar$ in the $x$ direction. By thus aligning the
wave fronts, the transverse force should vanish. An atom in the
$|m_{j}=-1\rangle_{y}$ sublevel will therefore find an equilibrium position in
the tweezer at a displaced off-axis location $x_{\text{eq}}=\lambdabar$. By
the same reasoning, the $|m_{j}=+1\rangle_{y}$ sublevel will have the opposite
displacement, so that for the $j=1\rightarrow j^{\prime}=0$ transition:

$x_{\text{eq}}=-(m_{j})_{y}\lambdabar$ (16)

The tweezer thus traps the atom off axis, ‘where the focus is not’, in a spin-
dependent location. For the situation considered here the
$|m_{j}=0\rangle_{y}$ state would be untrapped, for a lack of $\pi$ component
in the laser polarization. This could be changed by rotating $\mathbf{B}$. In
particular, setting the angle between $\mathbf{E}_{\text{in}}$ and
$\mathbf{B}$ to $\arctan(\sqrt{2})$, the polarization components $\sigma^{-}$,
$\pi$, and $\sigma^{+}$ would become equal. At this ‘magic angle’ all three
spin components would be trapped with a Stern-Gerlach type separation Wang
_et al._ (2020); Stellmer  _et al._ (2011).

These simple geometric arguments are backed up by a calculation sup , that
shows that Eq. (14) for the beam deflection is multiplied by $1\mp kd$, for a
$\mathbf{u}_{\pm}$ dipole displaced by $d$ in the $x$ direction, to lowest
order in $d$. Thus the transverse force indeed vanishes for a transverse
displacement of $d=k^{-1}=\lambdabar$ in the $x$ direction. Remarkably, the
size of the displacement is independent of the detuning, the beam divergence
angle, the trap frequency, or even the precise shape of the beam (Gauss vs.
angular tophat). This profound insight follows from the geometric properties
of the scattering problem.

The off-axis trapping locations offer interesting opportunities to manipulate
the motion of atoms in the tweezer, see Fig. 3. Let us imagine an atom trapped
in the $|m_{j}=1\rangle_{y}$ state. As we slowly rotate the magnetic field in
the $yz$ plane, the orientation of the atom will adiabatically follow the
rotating quantization axis. After rotating the field $y\rightarrow
z\rightarrow-y$, the spin will have maintained its orientation relative to
$\mathbf{B}$, i.e. $|m_{j}=1\rangle_{B}\rightarrow|m_{j}=1\rangle_{B}$.
However, its orientation will have flipped in space,
$|m_{j}=1\rangle_{y}\rightarrow|m_{j}=-1\rangle_{y}$, since $\mathbf{B}$ has
changed direction. The space-referenced spin flip implies that the atom must
have moved to the other side of the optical axis. Thus, by rotating the
magnetic field in the $yz$ plane at a frequency $\omega_{B}$, we effectively
shake the trap back and forth:
$x_{\text{eq}}=-(m_{j})_{B}\lambdabar\cos\omega_{B}t$. The $m_{j}=\pm 1$
levels are shaken with opposite phase.

Shaking the trap at an amplitude $\lambdabar$ is equivalent to a harmonic
driving force $F_{x}=m\omega^{2}\lambdabar\cos\omega_{B}t$, with $\omega$ the
trap frequency. Resonant shaking, $\omega\approx\omega_{B}$, will induce an
oscillatory motion in the trap. For example, for a tweezer with a laser
wavelength of $\lambda\approx 0.8\,\mu$m, a Gaussian waist of 2$\,\mu$m,
holding an atom of mass $m=88u$ in a 20$\,\mu$K deep trap, the trap frequency
will be $\omega\approx 2\pi\times 7\,$kHz. In a simple driven harmonic
oscillator model only 3.5 drive cycles would impart enough energy to kick the
atom out of the trap, corresponding to a velocity of $\sim 6\,$cm/s. In
reality one would of course need to take anharmonicity into account. The point
here is that magnetic field modulation can easily induce oscillatory motion in
the trap which can then be detected either as trap loss, or by using time-of-
flight imaging methods. For the required magnetic field a few gauss should be
sufficient, to ensure that the Larmor frequency is large compared to the trap
frequency. Rotating the field at frequencies of $\sim 10\,$kHz is well
possible, being comparable to what is used in TOP traps Petrich  _et al._
(1995).

Many available atomic level systems should be suitable to display off-axis
tweezer trapping. For example, in 88Sr the transition
${}^{3}P_{2}\rightarrow{}^{3}S_{1}$ would provide a $j=2\rightarrow
j^{\prime}=1$ transition. The outer $(m_{j})_{y}=2\;(-2)$ state couples only
to the $\sigma^{-}\;(\sigma^{+})$ polarization component, so its spatial shift
will be $-\lambdabar\;(+\lambdabar)$. Using 87Rb one could operate a tweezer
red detuned to the $D_{1}$ line (795 nm), driving the two hyperfine lines
$F=2\rightarrow F^{\prime}=1,2$. Also in this case the outer state
$(m_{F})_{y}=2\;(-2)$ is displaced by $-\lambdabar\;(+\lambdabar)$, as long as
the detuning stays small compared to the fine structure splitting of the $D$
lines.

In summary, it is predicted that a circular dipole can deflect a focused laser
beam, similar to a spinning ball deflecting a stream of air in the Magnus
effect. The reaction force on the atom leads to spin-dependent, off-axis
displacement of atoms trapped in an optical tweezer. For a pure circular
dipole the displacement is $\pm\lambdabar$, independent of many trap
parameters. An external magnetic field can be used to induce spin-dependent
motion or to perform Stern-Gerlach type analysis of the spin states of the
atom in the tweezer.

I would like to thank N.J. van Druten, R. Gerritsma, J. Minar, and A. Urech
for stimulating and encouraging discussions as well as careful reading of the
manuscript. This work was supported by the Netherlands Organization for
Scientific Research (NWO).

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Supplemental Materials for:
Off-axis optical tweezers by an optical analog of the Magnus effect

Robert J.C. Spreeuw

Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam,
PO Box 94485, 1090 GL Amsterdam, The Netherlands

##  I Spatial vs. angular $k$-space coordinates

In this paper we express all fields by their angular spectrum
$\mathbf{E}(\Omega)=\mathbf{E}(\theta,\phi)$. This is usually defined for
fields propagating out into a half space $z\geq 0$ (see Ch. 3.2 in Mandel and
Wolf (1995)), as is clearly the case for the incident beams. The relationship
with the field in the plane $z=0$ is given by

$\mathbf{\tilde{E}}(x,y,0)=k^{2}\iint_{\theta\leq\pi/2}\mathbf{E}(\Omega)\sin\theta\,d\theta
d\phi$ (S1)

with $k=\omega/c$ the laser wave vector. For the Gaussian beam with angular
waist $w_{\theta}$ the above equation yields the familiar Gaussian beam cross
section, with minimum waist $w_{0}=\lambda/\pi w_{\theta}$, see also Ch. 5 in
Mandel and Wolf (1995). The angular tophat beam approximates the output of a
uniformly illuminated circular lens, and Eq. (S1) yields the resulting Airy
pattern in the focal plane $z=0$.

Although the emission by a dipole is not confined to $z\geq 0$, the angular
representation of the radiation pattern of a dipole $\mathbf{p}$ in a
direction $\mathbf{u}_{\Omega}$ is well known to be given by Eq. (10) (main
text), see for example Ch. 9 in Jackson (1999). Only the radiating, or ‘far
field’, terms ($\sim r^{-1}$) are relevant in our case, because one can
evaluate the beam deflection at arbitrarily large distance of the dipole,
where the near fields ($\sim r^{-2},r^{-3}$) have become negligible.

In the plane ($\phi=0$) of a $\mathbf{u}_{\pm}$ dipole,

$\left(\mathbf{u}_{\Omega}\times\mathbf{u}_{\pm}\right)\times\mathbf{u}_{\Omega}=\frac{e^{\pm
i\theta}}{\sqrt{2}}\left(\cos\theta,0,-\sin\theta\right)$ (S2)

shows the spiral wave character in the prefactor $e^{\pm i\theta}$.

The factor $i$ in Eq. (10) (main text) is a crucial detail. It is a
consequence of expressing the spherical waves $e^{ikr}/r$ of the dipole field
as an angular spectrum of plane waves. The same factor $i$ can be recognized
in the Weyl representation of a diverging spherical wave Mandel and Wolf
(1995). In the case at hand, one can readily see that it also ensures that a
resonant beam is attenuated (absorbed) in the forward direction, due to
destructive interference of incident and scattered waves.

The phase factor $e^{i\alpha}$ in Eq. (10) (main text) follows from the steady
state of the optical Bloch equations Cohen-Tannoudji  _et al._ (1998). In a
two-level atom with states $e,g$, the induced dipole moment is given by the
off-diagonal density matrix element $\rho_{eg}$. If the atom is driven at
detuning $\Delta$ by a monochromatic field with (real) amplitude
$\mathcal{E}_{0}$ the steady state (for $s\ll 1$) is given by

$\rho_{eg}=\frac{i}{2}\frac{D\mathcal{E}_{0}/\hbar}{\gamma-i\Delta}$ (S3)

which has a complex argument $\alpha=\arg\rho_{eg}$ given by
$\cot\alpha=-\Delta/\gamma$. Here, since we choose an $x$ polarized incident
wave, $\alpha$ is the phase of the $p_{x}$ component of the dipole, relative
to the incident field.

##  II Low saturation limit

In the definition of the saturation parameter $s$ we include the detuning,
following Cohen-Tannoudji  _et al._ (1998),

$s=\frac{I/I_{0}}{1+\Delta^{2}/\gamma^{2}}$ (S4)

with $I$ the intensity and $I_{0}=2\pi hc\gamma/3\lambda^{3}$ the saturation
intensity. In the low-saturation limit, characterized by $s\ll 1$, the
scattered light is almost entirely coherent, with a small incoherent fraction
equal to $s/(1+s)$. In optical tweezer experiments, using far off-resonant
laser beams, typical values for $s$ are in the range $10^{-6}-10^{-8}$, so
that $s\ll 1$ is indeed well fulfilled and the incoherent scattering rate is
low.

##  III Field amplitudes

The peak amplitudes $\mathcal{E}_{0}^{\text{(G)}},\mathcal{E}_{0}^{(\Pi)}$ are
related to the total power in the incident beam by

$P=\int{J_{\text{in}}(\Omega)\;d\Omega}=\left\\{\begin{array}[]{ll}\approx\frac{(\mathcal{E}_{0}^{\text{(G)}})^{2}}{2Z_{0}}\times\frac{\pi
w_{\theta}^{2}}{2}\\\ \frac{(\mathcal{E}_{0}^{(\Pi)})^{2}}{2Z_{0}}\times
2\pi\left(1-\cos r_{\theta}\right)\end{array}\right.$ (S5)

for the Gaussian and angular tophat beam, respectively. The integrals were
performed using Mathematica software Mat . For the Gaussian, the equality is
only approximate, we give here the leading term of a power series in
$w_{\theta}$. The above expressions have been written as a product of the
forward ($\theta=0$) radiant intensity $\mathcal{E}_{0}^{2}/2Z_{0}$ and an
effective solid angle.

The average wave vector of the incident beams is shorter than the
corresponding value for a plane wave,

$\langle\mathbf{k}\rangle_{\text{in}}=k\mathbf{\hat{z}}\times\left\\{\begin{array}[]{ll}1-\frac{w_{\theta}^{2}}{4}+\mathcal{O}(w_{\theta}^{4})&\text{(Gauss)}\\\
\cos^{2}\left(\frac{r_{\theta}}{2}\right)&\text{(tophat)}\end{array}\right.$
(S6)

The amplitude ratios $\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{\text{(G)}}$
and $\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{(\Pi)}$ can be obtained from the
energy conservation condition

$\int\left[J_{\text{if}}(\Omega)+J_{\text{sc}}(\Omega)\right]\,d\Omega=0$ (S7)

The scattering term $J_{\text{sc}}(\Omega)>0$ would increase the outflowing
power, which must be cancelled by the interference term
$J_{\text{if}}(\Omega)$. As a result,

$\displaystyle\frac{\mathcal{E}_{\text{sc}}}{\mathcal{E}_{0}^{\text{(G)}}}$
$\displaystyle\approx$
$\displaystyle\frac{3\sin\alpha}{4\sqrt{2}}\,w_{\theta}^{2}$ (S8)
$\displaystyle\frac{\mathcal{E}_{\text{sc}}}{\mathcal{E}_{0}^{(\Pi)}}$
$\displaystyle=$
$\displaystyle\frac{3\sin\alpha}{4\sqrt{2}}\,\sin^{2}\left(\frac{r_{\theta}}{2}\right)(\cos
r_{\theta}+3)$ (S9)

where in the Gaussian case the leading order in $w_{\theta}$ is given.

With these ratios the interference terms in the radiant intensity, Eq. (5)
(main text) can be obtained as

$J_{\text{if}}(\Omega)=-\frac{\mathcal{E}_{\text{sc}}}{\sqrt{2}Z_{0}}\,f(\Omega,\Delta)\times\left\\{\begin{array}[]{ll}\sim\mathcal{E}_{0}^{\text{(G)}}e^{-\theta^{2}/w_{\theta}^{2}}\\\
\mathcal{E}_{0}^{(\Pi)}\,\Pi(\theta/2r_{\theta})\end{array}\right.$ (S10)

with

$f(\Omega,\Delta)=\frac{\gamma\left(\cos\theta\cos^{2}\phi+\sin^{2}\phi\right)-\Delta\sin\theta\cos\phi}{\sqrt{\gamma^{2}+\Delta^{2}}}$
(S11)

For the deflection, expressed as
$\delta\langle\mathbf{k}\rangle=\langle\mathbf{k}\rangle-\langle\mathbf{k}\rangle_{\text{in}}$
we evaluate the integral of Eq. (7) (main text) to obtain

$\delta\langle\mathbf{k}\rangle=\frac{3k}{4}\frac{\gamma\Delta}{\gamma^{2}+\Delta^{2}}\,\left(w_{\theta}^{4}+\mathcal{O}(w_{\theta}^{5}),0,-2\frac{\gamma}{\delta}w_{\theta}^{2}+\mathcal{O}(w_{\theta}^{4})\right)$
(S12)

for the Gaussian beam, and

$\delta\langle\mathbf{k}\rangle=\frac{3k}{16}\frac{\gamma\Delta}{\gamma^{2}+\Delta^{2}}\,\left(r_{\theta}^{4}+\mathcal{O}(r_{\theta}^{5}),0,-4\frac{\gamma}{\delta}r_{\theta}^{2}+\mathcal{O}(r_{\theta}^{4})\right)$
(S13)

for the angular tophat beam.

Note that in both cases the $y$ component is absent. To leading order, the
deflection angle is just given by

$\delta\theta\approx\frac{(\delta\langle\mathbf{k}\rangle)_{x}}{k}$ (S14)

which leads to Eq. (14) of the main text.

##  IV Calculation for a displaced atom

When the atom is located at a position $\mathbf{d}$ away from the origin, the
angular components of the scattered wave are phase shifted by an amount
$\exp({-ik\mathbf{u}_{\Omega}\cdot\mathbf{d}})$, so that the interference
term, Eq. (5) (main text), is modified to

$J_{\text{if}}(\Omega)=\frac{1}{2Z_{0}}\left[\mathbf{E}^{\ast}_{\text{in}}(\Omega)\cdot\mathbf{E}_{\text{sc}}^{\text{(coh)}}(\Omega)e^{-ik\mathbf{u}_{\Omega}\cdot\mathbf{d}}+c.c.\right]$
(S15)

For a displacement along $x$, we have $\mathbf{d}=d\mathbf{\hat{x}}$ so that

$k\mathbf{u}_{\Omega}\cdot\mathbf{d}=kd\,\mathbf{u}_{\Omega}\cdot\mathbf{\hat{x}}=kd\sin\theta\cos\phi$
(S16)

In the integrals $\int J_{\text{if}}(\Omega)\,d\Omega$ and
$\int\mathbf{u}_{\Omega}\,J_{\text{if}}(\Omega)\,d\Omega$, we develop the
integrand in a power series of $kd$, up to fourth order and integrate the
terms separately.

For the amplitude ratios
$\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{\text{(G)}}$ and
$\mathcal{E}_{\text{sc}}/\mathcal{E}_{0}^{(\Pi)}$ we find that their lowest
order ($\sim w_{\theta}^{2}$ and $\sim r_{\theta}^{2}$) is not affected by
$kd$. For the deflection angle the leading order in $w_{\theta},r_{\theta}$ is
still fourth order, and up to order $(kd)^{4}$ the angle is

$\delta\theta\approx\frac{3}{4}(1\mp
kd)\frac{\gamma\Delta}{\left(\gamma^{2}+\Delta^{2}\right)}\times\left\\{\begin{array}[]{ll}w_{\theta}^{4}&\quad\text{(Gauss)}\\\
r_{\theta}^{4}/4&\quad\text{(tophat)}\end{array}\right.$ (S17)

for a $\mathbf{u}_{\pm}$ dipole, respectively. This shows that the deflection
angle, and thus also the transverse force, vanishes if the atom is displaced
by an amount $d=\pm k^{-1}=\pm\lambdabar$.

## References

  * Mandel and Wolf (1995) L. Mandel and E. Wolf,  _Optical Coherence and Quantum Optics_ (Cambridge University Press, 1995) Ch. 3 and 5\.
  * Jackson (1999) J. D. Jackson,  _Classical Electrodynamics_, 3rd ed. (Wiley, New York, NY, 1999) Ch. 9 and 10.
  * Cohen-Tannoudji  _et al._ (1998) C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,  _Atom-Photon Interactions: Basic Process and Applications_ (Wiley-VCH, New York, 1998).
  * (4) Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2020).