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# Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect

M. Mazzanti  Van der Waals-Zeeman Institute, Institute of Physics, University
of Amsterdam, 1098 XH Amsterdam, Netherlands  R. Gerritsma  Van der Waals-
Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH
Amsterdam, Netherlands  QuSoft, Science Park 123, 1098 XG Amsterdam, the
Netherlands  R. J. C. Spreeuw  Van der Waals-Zeeman Institute, Institute of
Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  QuSoft,
Science Park 123, 1098 XG Amsterdam, the Netherlands  A. Safavi-Naini  QuSoft,
Science Park 123, 1098 XG Amsterdam, the Netherlands  Institute for
Theoretical Physics, Institute of Physics, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, the Netherlands

###### Abstract

We consider the implementation of quantum logic gates in trapped ions using
tightly focused optical tweezers. Strong polarization gradients near the
tweezer focus lead to qubit-state dependent forces on the ion. We show that
these may be used to implement quantum logic gates on pairs of ion qubits in a
crystal. The qubit-state dependent forces generated by this effect live on the
plane perpendicular to the direction of propagation of the laser beams opening
new ways of coupling to motional modes of an ion crystal. The proposed gate
does not require ground state cooling of the ions and does not rely on the
Lamb-Dicke approximation, although the waist of the tightly focused beam needs
to be comparable with its wavelength in order to achieve the needed field
curvature. Furthermore, the gate can be performed on both ground state and
magnetic field insensitive clock state qubits without the need for counter-
propagating laser fields. This simplifies the setup and eliminates errors due
to phase instabilities between the gate laser beams. Finally, we show that
imperfections in the gate execution, in particular pointing errors $<30$ nm in
the tweezers reduce the gate fidelity from $\mathcal{F}\gtrsim 0.99998$ to
$\gtrsim 0.999$.

Trapped ions are one of the most mature platforms for the implementation of
quantum computing and quantum logic gates have been implemented with very high
fidelity in these systems [1, 2]. Usually, the quantum logic gates in trapped
ions rely on state-dependent forces applied to the ions by laser fields or
magnetic fields. The exchange of motional quanta between the ions then leads
to effective qubit-qubit interactions. Several recent works have explored how
the use of state-of-the-art optical tweezer technology can benefit the trapped
ion quantum computer. Optical tweezers can be used to confine atoms very
strongly by inducing a dipole in them and find application in neutral atomic
quantum simulators, in which tweezers are used to levitate individual atoms
[3, 4, 5, 6, 7]. In trapped ions, tweezers may be used to tune the soundwave
spectrum in the ion crystal and thereby to program the interactions between
the qubits [8, 9, 10]. Furthermore, in a recent work [11] we have proposed
combining state-dependent optical tweezers with oscillating electric fields to
build a universal trapped ion quantum computer with extremely long-ranged
interactions between the qubits.

In this work, we consider another scenario, in which we make use of the strong
polarization gradients that occur in optical tweezers. We note that strong
gradients in optical potentials have been previously investigated to implement
two-qubit gates without the need for ground-state cooling [12, 13, 14].
However, our approach utilizes the state-dependent displacement of the tweezer
potential due to polarization gradients [15, 16, 17]. We propose to use this
optical analogue of the Magnus effect to implement quantum logic gates in
trapped ions.

Figure 1: Schematic representation of the two-qubit gate. a) We apply tweezers
propagating along the $-y$ direction on the two ions forming the gate. The
tweezer intensity can be decomposed into three polarization components. b)
Simplified level scheme of an alkaline-earth like ion without nuclear spin
showing the encoding of the qubit in its Zeeman ground states. The two
polarization components of the tweezer couple to different states in the
$P_{1/2}$ manifold with detuning $\Delta$. This causes the minima of the
tweezer potentials to be shifted by an amount $\pm\lambdabar$ depending on the
qubit state. Bottom : main polarization components for a Gaussian and
Laguerre-Gaussian ($l=1$, $n=0$) tightly focused tweezer.

_Setup –_ We consider linearly $x$-polarized, Gaussian tweezers, pointing in
the $-y$ direction and tightly focused at two qubits between which we wish to
implement a quantum logic gate. The quantum computing platform here considered
is a linear crystal of $N$ alkali-like trapped ions of mass $m$. In the focal
plane the ions experience a strong polarization gradient along the $x$
direction, such that the polarization is linear ($x$) in the center and
circular $(\sigma^{\pm})_{z}$ in the wings of the Gaussian. A direct
calculation [18] decomposing the field in the focal plane into its circular
components $(\sigma^{\pm})_{z}$ (and $\pi_{z}$) shows that, to a good
approximation, the circular components are near-Gaussian distributions,
displaced in opposite directions along the $x$ axis. We depict this setup in
Fig. 1. Note that the circular components rotate in the $xy$ plane, i.e. a
plane containing the $\mathbf{k}$ vector of the light. As shown in Fig. 1, the
$(\sigma^{\pm})_{z}$ component is displaced by an amount
$\pm\lambdabar\equiv\pm\lambda/2\pi$, with $\lambda$ the tweezer wavelength.
As the total field is the superposition of two displaced Gaussians, its
intensity is slightly elongated along $x$. Hollow tweezers (Gaussian-Laguerre)
can be used instead of Gaussian ones. This will provide the needed field
curvature while keeping near-zero intensity at the center of the beam,
drastically reducing the probability of off-resonant scattering that might
limit the gate fidelity.

For simplicity, we first consider ions without nuclear spin, such as 40Ca+,
88Sr+, 138Ba+ and 174Yb+. The qubits are encoded in the electronic ground
states ${}^{2}S_{1/2}$ and $|0\rangle=|j=1/2,m_{j}=1/2\rangle$ and
$|1\rangle=|j=1/2,m_{j}=-1/2\rangle$ with $j$ the total electronic angular
momentum and $m_{j}$ its projection on the quantization axis. The magnetic
field lies along the $z$-direction and the tweezers are polarized along the
$x$-direction, such that the ions experience linearly polarized laser light.
The direction along the $x$-axis is the long direction of the ion trap, with
trap frequency $\omega_{x}$. The motion of the ions along the $x$-direction
can be described by collective modes of harmonic motion with frequencies
$\omega_{m}$ and mode vectors $b_{i,m}$, with $m$ labeling the mode and $i$
the ion [19].

We choose the detuning between the tweezers and the D1 transition to be large
enough to avoid photon scattering, but much smaller than the spin-orbit
coupling splitting of the ${}^{2}P$ state. In this way, we can neglect
coupling to the $P_{3/2}$ state. In what follows we will show that this
requirement can be satisfied experimentally. Close to the center of the
tweezer, strong polarization gradients appear and as a result, the two qubit
states experience slightly different tweezer potentials. In particular, as we
show in Fig. 1(a), the optical Magnus effect causes each qubit state to
experience a tweezer potential that is offset from the apparent center of the
tweezer by $\sim\lambdabar$ [16]. Hence, we may approximate the tweezer
potential as :

$\displaystyle\hat{U}(x)$
$\displaystyle=-U_{0}\exp\left(-2(\hat{x}+\hat{\sigma}_{z}\lambdabar)^{2}/w_{0}^{2}\right)$
(1)
$\displaystyle\approx-\tilde{U}_{0}+\frac{1}{2}m\omega_{\text{tw}}^{2}\hat{x}^{2}+gx\hat{\sigma_{z}}$
(2)

with
$\omega_{\text{tw}}=\sqrt{4\tilde{U}_{0}(w_{0}^{2}-4\lambdabar^{2})/(mw_{0}^{4})}$,
$g=4\tilde{U}_{0}\lambdabar/w_{0}^{2}$, and
$\tilde{U}_{0}=U_{0}\exp(-2\lambdabar^{2}/w_{0}^{2})\approx U_{0}$. Here
$U_{0}$ is the tweezer potential in the center and the beam waist is $w_{0}$.
Our approximation replaces the tweezer potential with a harmonic potential and
is valid for $w_{0}\gg l_{m}$, with $l_{m}=\sqrt{\hbar/2m\omega_{m}}$. The
last term in $U(x)$ is the result of the spin-dependent force $g$ coupling the
internal state of the qubit, $\hat{\sigma}_{z}$, to its motion $\hat{x}$.
Thus, the optical Magnus effect allows us to straightforwardly implement a
quantum gate.

_Tweezer Hamiltonian –_ In the interaction picture with respect to
$\hat{H}_{0}=\hbar\omega_{m}\hat{a}_{m}^{\dagger}\hat{a}_{m}$ the tweezer
Hamiltonian on ions $i$ and $j$ is:

$\hat{H}_{1}=A(t)\left(\frac{1}{2}m\omega_{\rm
tw}^{2}\left(\hat{x}^{2}_{i}+\hat{x}_{j}^{2}\right)+g\left(\hat{\sigma}_{z}^{(i)}\hat{x}_{i}+\hat{\sigma}_{z}^{(j)}\hat{x}_{j}\right)\right).$
(3)

Here,
$\hat{x}_{i}=\sum_{m}l_{m}b_{im}\left(\hat{a}_{m}e^{-i\omega_{m}t}+\hat{a}_{m}^{\dagger}e^{i\omega_{m}t}\right)$
is the position operator of ion $i$ in the interaction picture, with
$\hat{a}_{m}^{\dagger}$ the creation operator for the mode $m$, and $0\leq
A(t)\leq 1$ specifies the time-dependence of the tweezer intensity. The qubit-
state independent terms in $\hat{H}_{1}$ do not alter the dynamics of the
quantum gate. We ignore these terms and arrive at:

$\hat{H}_{2}=A(t)g\left(\hat{x}_{i}\hat{\sigma}_{z}^{(i)}+\hat{x}_{j}\hat{\sigma}_{z}^{(j)}\right),$
(4)

which takes the form of a spin-phonon coupling Hamiltonian reminiscent of the
Mølmer-Sørenson scheme for phonon-mediated quantum gates in trapped ions [20].
However, at this stage we still have various choices available for $A(t)$,
depending on which type of quantum gate we would like to implement. For
instance, pulsed $A(t)$ could be used to perform fast gates. Here, we choose
$A(t)$ to obtain a geometric phase gate. For this, we set $2A(t)=1-\cos(\nu
t+\phi)$ where $\phi=0$ assures a smooth ramp of the tweezer intensity and
$\nu=\omega_{\text{c}}+\delta$ with the subscript $c$ denoting the center-of-
mass (c.o.m.) mode for which $\omega_{\text{c}}=\omega_{x}$ and
$b_{i,\text{c}}=1/\sqrt{N}$. We write the operators $\hat{x}_{i}$ and
$\hat{x}_{j}$ in terms of $\hat{a}_{c}$ and $\hat{a}_{c}^{\dagger}$ and
perform the rotating wave approximation to arrive at:

$\hat{H}_{3}=\frac{gl_{\text{c}}}{4\sqrt{N}}\left(\hat{a}_{\text{c}}e^{i\delta
t}+\hat{a}^{\dagger}_{\text{c}}e^{-i\delta
t}\right)\left(\hat{\sigma}_{z}^{(i)}+\hat{\sigma}_{z}^{(j)}\right).$ (5)

To derive the qubit-qubit interactions forming the geometric phase gate, we
perform a unitary transformation
$\hat{U}_{1}=e^{-i\delta\hat{a}^{\dagger}_{\text{c}}\hat{a}_{\text{c}}t}$ to
eliminate the time dependence, followed by a Lang-Firsov [21] transformation,
$\hat{U}_{2}=\exp\left(\hat{\alpha}\left(\hat{a}^{\dagger}_{\text{c}}-\hat{a}_{\text{c}}\right)\right)$
with
$\hat{\alpha}=-\frac{\tilde{g}}{\delta}\left(\hat{\sigma}_{z}^{(i)}+\hat{\sigma}_{z}^{(j)}\right)$.
Disregarding qubit-independent terms, we obtain

$H_{\text{eff}}=\frac{2\tilde{g}^{2}}{\hbar\delta}\hat{\sigma}_{z}^{(i)}\hat{\sigma}_{z}^{(j)},$
(6)

with $\tilde{g}=gl_{\text{c}}/(4\sqrt{N})=\tilde{\eta}\tilde{U}_{0}$, with the
proportionality factor $\tilde{\eta}=\lambdabar
l_{\text{c}}/(\sqrt{N}w_{0}^{2})$. This Hamiltonian generates qubit-qubit
interactions that can be used to implement a geometric phase gate by setting
the gate time $\tau=2\pi/\delta$ and
$\frac{\tilde{g}^{2}\tau}{\hbar^{2}\delta}=\pi/4$.

Figure 2: We calculate the gate fidelity for a ground state cooled ion
$\bar{n}_{\text{c}},\ \bar{n}_{\text{s}}=0$ (blue), sub-Doppler cooled thermal
state with $\bar{n}_{\text{c}}=0.62,\ \bar{n}_{\text{s}}=0.23$ (orange) and
$\bar{n}_{\text{c}}=15,\ \bar{n}_{\text{s}}=0.23$ (red, using in this case a
Fock cutoff $n_{c}\leq 120$, $n_{s}\leq 10$). (a) Process fidelity of the two-
qubit Magnus gate for different gate times. (b) Effects of misalignment
$\epsilon$ (orange) and intensity noise $\Lambda_{1/\tau}$ (blue) on the gate
fidelity. The size of each intensity noise data point represents the standard
deviation of 20 simulation where we generated a random Gaussian noise with
$\sigma=\Lambda_{1/\tau}$ on each of the two pulses. This implies a noise on
the laser intensity at frequency $1/\tau$ that can not be removed by the spin-
echo sequence.

_Characterization of the gate –_ We analyse the gate dynamics by performing
numerical simulation of the full dynamics generated by the Hamiltonian
$\hat{H}_{\mathrm{sim}}=\hat{H}_{0}+\hat{U}\left(x_{i}\right)+\hat{U}\left(x_{j}\right)$
for a two dimensional ion crystal where the tweezers potentials
$\hat{U}\left(x_{i;j}\right)$ on ions $i$ and $j$ have been expanded up to
fourth-order including spin-independent terms. We use realistic experimental
parameters: $\sim$ 156 $\mu$W of tweezer laser power focused to a waist of
$w_{0}\sim$ 0.5 $\mu$m and tuned 15 THz to the red from the
${}^{2}S_{1/2}\rightarrow\,^{2}P_{1/2}$ transition in 174Yb+
($\lambda=369.5\,$nm). This results in $\tilde{U}_{0}/h\sim 1.6$ MHz,
$\tilde{g}/h=2.1$ kHz$/\sqrt{N}$, and setting $\delta=2\pi\times 12.2$
kHz$/\sqrt{N}$ the gate time for the geometric phase gate is
$\tau=170\sqrt{N}\mu$s. With a calculated qubit-state independent tweezer
potential of $\omega_{\text{tw}}\sim 2\pi\times 37$ kHz, the center-of-mass
mode frequency ($\omega_{c}/2\pi\sim 1$ MHz) is shifted by $\sim
2\omega_{\text{tw}}^{2}/\omega_{c}N\sim 2\pi\times 710/N$ Hz. This shift can
easily be taken into account by correcting $\delta$ accordingly. In these
estimates, we neglected the contribution from other dipole allowed
transitions, that are detuned by $\sim 66$ THz (the relatively weak
${}^{2}S_{1/2}\rightarrow\,^{3}[3/2]_{3/2}$ transition) and 115 THz (the
strong D2 line) or more.

We consider the gate unitary with a spin-echo sequence given by
$U(0,\tau)=X^{\otimes 2}U(\tau/2,\tau)X^{\otimes 2}U(0,\tau/2)$, where
$X^{\otimes 2}$ is a qubit flip on both qubits. This spin echo sequence is
needed in order to remove local rotations on the qubits states and possible
timing errors. We calculate the unitary time evolution operator $U(0,\tau)$
for a system of two ions with their motional c.o.m. and _stretch_ modes and
truncate their respective Hilbert spaces to $n_{\text{c}}\leq 18$ and
$n_{\text{s}}\leq 10$. In figure 2 we show the process fidelity of the gate
assuming the ions are in their motional ground state ($\bar{n}=0$) as a
function of gate time.

The gate fidelity of $\mathcal{F}=$ 0.999988 with
$n_{\text{c}}=n_{\text{s}}=0$ rivals the current standard approaches.
Moreover, the performance of our gate is robust to the thermal occupation of
the motional modes. We characterize the gate performance in presence of
thermal phonons using the average gate fidelity [22, 18] and find that it
depends weakly on the motional state of the two ions. In fact, using
$\bar{n}_{\text{c}}=0.62$, $\bar{n}_{\text{s}}=0.23$, the fidelity is almost
unaltered at $\mathcal{F}_{\rm th}=$ 0.999989.

One of main experimental challenges is perfect tweezer alignment. We have
studied the resilience of the gate against misalignment of the tweezer in the
$x$-direction, which we denote by $\epsilon$. In the presence of misalignment,
$\tilde{U}_{0}\to
U_{0}\exp^{-2\left(\epsilon+\hat{\sigma}_{z}\lambdabar\right)^{2}/\omega_{0}^{2}}$.
Thus, the misalignment has two effects: (i) it changes the tweezer potential
at the center of the tweezer and therefore the phase accumulation in the phase
gate, and (ii) it shifts the potential in a qubit-state-dependent way. The
second contribution is corrected to lowest order by a spin-echo sequence.
Figure 2(b) shows the infidelity as a function of $\epsilon$. Here we assume
that the tweezers are misaligned on both ions in the same way which seems the
experimentally most likely case. The unitary $U(0,\tau)$ leads to phase space
trajectories for $\langle x(t)\rangle$ and $\langle p_{x}(t)\rangle$
associated with the c.o.m. motion[18]. As expected, we find approximately
circular phase-space orbits for the even parity states $\ket{00}$, $\ket{11}$,
and very little motion for the odd parity ones. We see that every state
combination leads to ion motion, but the difference in motion still leads to a
high fidelity of $\gtrsim$ 0.999 as shown in Figure 2(b).

Figure 3: Relevant energy levels of ${}^{171}\text{Yb}^{+}$ for implementing
the gate on hyperfine qubit splitted by $\omega_{q}$. The coupling can be
achieved using a pair of Raman beams detuned from the upper state
${}^{2}P_{1/2}$ by $\Delta$. In the brackets are the angular contributions to
the various dipole transition elements. Table 1: Main sources of gate errors.
We estimate $\gamma_{\mathrm{ph}}$ as the probability of a off-resonant
scattering in for 174Yb+ during the gate time ($\tau=240\,\mu$s) for a
Gaussian and Laguerre-Gaussian beams. Other typical sources of errors are
misalignment ($\epsilon$), tweezer intensity noise ($\Lambda_{1/\tau}$) and
timing ($\Delta\tau$). The values here reported are for laser parameters used
in our numerical simulations.

Error source |  | $\gamma_{\mathrm{ph}}$
---
Gaussian
| $\gamma_{\mathrm{ph}}$
---
Laguerre-Gaussian
| $\epsilon$
---
$30$ nm
| $\Lambda_{1/\tau}$
---
0.5%
| $\Delta\tau$
---
$\pm 5\,\mu\mathrm{s}$
$1-\mathcal{F}$ | $2\text{\times}{10}^{-3}$ | ${10}^{-6}$ | $1.3\text{\times}{10}^{-3}$ | $9.3\text{\times}{10}^{-5}$ | $2.7\text{\times}{10}^{-4}$

_Clock state case –_ While the calculation was performed for the electron spin
qubit states in ${}^{174}\text{Yb}^{+}$, it should also be possible to use the
hyperfine clock states $\ket{F=m_{F}=0}$ and $\ket{F=1,m_{F}=0}$ in
${}^{171}\text{Yb}^{+}$. This qubit is insensitive to magnetic field noise and
coherence times of up to an hour have been measured [23]. In this case, the
tweezers are formed by a bichromatic co-propagating laser field detuned by
$\Delta$ from the D1 transition at 369.5 nm with overall detuning
$\Delta\ll\omega_{\text{FS}}$, the fine structure splitting. We set the
frequency difference in the bichromatic tweezer to 12.6 GHz, corresponding to
the transition between the qubit states [24]. The tweezer laser then causes
Raman coupling between the qubit states via two distinct paths. In the first
path, the qubits are coupled via the state $|P_{1/2},F=1,m_{F}=-1\rangle$ due
to the $\sigma^{-}$ polarization component in the tweezer. In the other, the
qubits are coupled via the state $|P_{1/2},F=1,m_{F}=+1\rangle$ due to the
$\sigma^{+}$ component in the tweezer. We denote the Rabi frequencies of each
path as $\Omega^{\pm}_{1,2}(x)$. The corresponding Raman couplings of each
path interfere destructively in the center of the tweezer due to a relative
minus sign between $\Omega^{+}_{1}(x)$ and $\Omega^{+}_{2}(x)$ in their
Clebsch-Gordan coefficient,
$\propto(\Omega^{-}_{1}(0)\Omega^{-}_{2}(0)+\Omega^{+}_{1}(0)\Omega^{+}_{2}(0))/\Delta=0$.
However, the Magnus effect causes a strong position dependence of the relative
strength of both paths of magnitude

$\Omega_{\text{eff}}(x)=\frac{\Omega^{-}_{1}(x)\Omega^{-}_{2}(x)}{\Delta}+\frac{\Omega^{+}_{1}(x)\Omega^{+}_{2}(x)}{\Delta}\approx\frac{\Omega^{2}}{\Delta}\frac{4\lambdabar
x}{w_{0}^{2}},$ (7)

where we assumed $x\ll\lambdabar\ll w_{0}$ and
$|\Omega^{\pm}_{i}(0)|=\Omega/\sqrt{2}$ with $i=1,2$, such that both laser
frequencies have the same power. As a result, a qubit state-dependent force
appears as in Eq. (4), except that we must now replace
$\hat{\sigma}_{z}^{(i,j)}\rightarrow\hat{\sigma}_{x}^{(i,j)}$ and the gate
takes the form of the usual Mølmer-Sørensen interaction
$\propto\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}$ [20]. Amplitude
modulation via $A(t)$ allows again for resonant enhancement of the gate.

In addition to the Raman coupling, we obtain a tweezer potential (AC Stark
shift) for each qubit state of magnitude

$\delta^{|k\rangle}_{\text{AC}}(x)=\sum_{i=1,2}\sum_{j=+,-}\frac{|\Omega^{j}_{i}(x)|^{2}}{\Delta_{i,|k\rangle}}$
(8)

with $\Delta_{1,|0\rangle}=\Delta-\omega_{q}$, $\Delta_{2,|0\rangle}=\Delta$,
$\Delta_{1,|1\rangle}=\Delta$ and $\Delta_{2,|1\rangle}=\Delta+\omega_{q}$.
This causes an additional trapping potential
$\Phi(x)\approx\frac{1}{2}m\omega_{\text{tw}}^{2}x^{2}$ that is independent of
the qubit state as before, as well as a position-dependent differential Stark
shift
$\delta_{\text{AC}}(x)=\delta^{|1\rangle}_{\text{AC}}(x)-\delta^{|0\rangle}_{\text{AC}}(x)$.
In the limit $\omega_{q}\ll|\Delta|$,

$\displaystyle\delta_{\text{AC}}(x)$
$\displaystyle\approx-\frac{\omega_{q}}{\Delta^{2}}\sum_{i=1,2}\sum_{j=+,-}|\Omega^{j}_{i}(x)|^{2}$
(9) $\displaystyle=-\frac{\omega_{q}}{\Delta}\,\tilde{U}_{0}(x)$ (10)

This differential Stark shift is estimated to be small,
$\delta_{\text{AC}}/2\pi\approx 2.7\,$kHz for the numbers used in the
simulations, and can be compensated by a corresponding Raman detuning.

Photon scattering on the D1 transition can be estimated as
$\gamma_{\text{ph}}\sim\tilde{U}_{0}\Gamma/(\hbar\Delta)\sim 13$ s-1 with
$\Gamma=1.23\times 10^{8}$ s-1 in Yb+. This adverse effect may be reduced
significantly by employing hollow tweezers [25, 26, 11] at the expense of
added complexity. For a hollow beam with a waist $w_{0}=0.5\,\mu$m and $\sim
160\,\mu$W we obtain a reduction in scattering rate of $\sim 10^{-6}$ s-1. As
long as $\omega_{\text{tw}}\ll\Omega_{\text{rf}}$, the drive frequency of the
Paul trap, no parametric excitations can occur and micromotion of the ions is
not a problem. Other errors, such as due to intensity noise of the laser,
heating of the ions due to electric field noise and decoherence due to
magnetic field noise have the same effect as in other gate implementations.
Finally, we note that because the tweezers are far detuned from the closest
transitions, the exact overall frequency of the tweezer laser is irrelevant.

_Conclusions_ In conclusion, we have described a novel type of quantum phase
gate based on the optical Magnus effect using optical tweezers in a linear
chain of trapped ions. The main benefit is that the gate does not require
counter-propagating laser fields, greatly simplifying the setup and
eliminating errors due to phase instabilities between the gate laser beams.
Furthermore, the state-dependent force generated by the Magnus effect allows
to perform the gate by coupling to motional modes on the plane perpendicular
to the direction of propagation of the tweezers allowing novel experimental
implementations. The proposed gate does not require ground state cooling and
can perform a quantum logic gate on any pair of ion qubits by spatial
addressing. The expected gate fidelity rivals the state of the art also for
ions that are not cooled to the ground-state of motion.

## Acknowledgements

This work was supported by the Netherlands Organization for Scientific
Research (Grant Nos. 680.91.120 and 680.92.18.05, (R.G.). A.S.N is supported
by the Dutch Research Council (NWO/OCW), as part of the Quantum Software
Consortium programme (project number 024.003.037).

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Supplementary material for :
Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect

## Appendix I : Optical Magnus effect

A key characteristic of a tightly focused beam is the strong field curvature
near the focus. This not only affects the local intensity but also its
polarization structure. To calculate this, we take a superposition of plane
waves labeled by their wave vector in spherical coordinates,
$\mathbf{k}=(k,\theta,\phi)$. Taking $k=\omega/c$ as fixed we write

$\mathbf{E}(\mathbf{r})\propto\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\,\sin\theta\,\mathbf{u}_{x}(\theta,\phi)\,a(\theta,\phi)\,e^{i\mathbf{k}\cdot\mathbf{r}}$

with $\mathbf{u}_{x}(\theta,\phi)$ a polarization vector obtained by co-
rotating the $\mathbf{x}$ unit vector when $\mathbf{k}$ is rotated from
$\mathbf{z}$ to $(\theta,\phi)$, such that
$\mathbf{u}_{x}(\theta,\phi)\cdot\mathbf{k}=0$, see also Ref. [1]. In the
calculation we center the beam around $\theta=0$, and the focal plane is given
by $\mathbf{r}=(x,y,0)$. The shape of the beam is determined by the amplitude
function $a(\theta,\phi)$. For a Gaussian beam we set
$a(\theta,\phi)=\exp(-\theta^{2}/w_{\theta}^{2})$; for the lowest order
($l=1$) Laguerre-Gaussian (LG) beam we set
$a(\theta,\phi)=\theta\exp(i\phi-\theta^{2}/w_{\theta}^{2})$. After performing
the above integral we rotate the results for tweezers propagating along the
$-y$ direction. Finally, the circular field components $\sigma^{\pm}$ shown in
Fig. 1 of the main text are obtained as the projection onto unit vectors
$(\mathbf{x}\pm i\mathbf{y})/\sqrt{2}$. In Figure S-1, all three polarization
components for a Laguerre-Gaussian beam are shown. Note that the $\sigma^{-}$
and $\sigma^{+}$ components have similar intensity while the
$\pi$-polarization is suppressed by a factor $\sim 100$.

Figure S-1: Intensity of the polarization components for a LG beam calculated
at the focus. The $\pi$-polarization component has been enhanced by a factor
$100$ to make it visible. Here we set $w_{\theta}=0.6$

## Appendix II : Phase-space dynamics

We study the phase-space dynamics of the ions by simulating the time dependent
Hamiltonian using trotterization with time-steps of $10^{-4}\,\tau$. At each
time-step we evaluate the expectation value of the $\langle\hat{x}\rangle$ and
$\langle\hat{p}\rangle$ for the center of mass mode. As expected, we find
approximately circular phase-space orbits for the even parity states
$\ket{00}$, $\ket{11}$, and very little motion for the odd parity ones. In
Fig. S-2 it is possible to see the evolution in phase-space for all the four
spin states in case of perfectly aligned and slightly misaligned tweezers. As
described in the main text we simulate numerically the full Hamiltonian
defined as
$\hat{H}_{\mathrm{sim}}=\hat{H}_{0}+\hat{U}\left(x_{i}\right)+\hat{U}\left(x_{j}\right)$
where in case of misalignment $\epsilon$, $\hat{U}\left(x\right)$ reads as :

$\displaystyle U(x)\approx-
U_{0}\,e^{-2(\left(\hat{x}-\hat{\epsilon}\right)+\hat{\sigma}_{z}\lambdabar)^{2}/w_{0}^{2}}$
$\displaystyle\approx-\tilde{U}_{0}+4\tilde{U}_{0}\frac{\hat{\sigma}_{z}\lambdabar-\hat{\epsilon}}{w_{0}^{2}}\hat{x}$
$\displaystyle+\frac{1}{2}\tilde{U}_{0}\left(\frac{4\left(w_{0}^{2}-4\lambdabar^{2}\right)}{w_{0}^{4}}\right)\hat{x}^{2}-\frac{1}{2}\tilde{U}_{0}\left(\frac{16\left(\hat{\epsilon}^{2}-2\hat{\sigma}_{z}\hat{\epsilon}\lambdabar\right)}{w_{0}^{4}}\right)\hat{x}^{2}$
$\displaystyle-\left(8\tilde{U}_{0}\hat{\sigma}_{z}\lambdabar\frac{3w_{0}^{2}-4\left(3\hat{\epsilon}^{2}+\lambdabar^{2}\right)}{3w_{0}^{6}}\right)\hat{x}^{3}$
$\displaystyle+\left(8\tilde{U}_{0}\hat{\epsilon}\frac{3w_{0}^{2}-4\left(\hat{\epsilon}^{2}+3\lambdabar^{2}\right)}{3w_{0}^{6}}\right)\hat{x}^{3}$
$\displaystyle-\left(2\tilde{U}_{0}\hat{\sigma}_{z}\lambdabar\hat{\epsilon}\frac{-48w_{0}^{2}+64\left(\hat{\epsilon}^{2}+\lambdabar^{2}\right)}{3w_{0}^{8}}\right)\hat{x}^{4}$
$\displaystyle+\left(2\tilde{U}_{0}\frac{3w_{0}^{4}-24w_{0}^{2}\left(\hat{\epsilon}^{2}+\lambdabar^{2}\right)+16\left(\hat{\epsilon}^{4}+6\hat{\epsilon}^{2}\lambdabar^{2}+\lambdabar^{4}\right)}{3w_{0}^{8}}\right)\hat{x}^{4}.$

with

$\displaystyle\tilde{U}_{0}=U_{0}e^{-2\left(\hat{\epsilon}+\hat{\sigma}_{z}\lambdabar\right)^{2}/w_{0}^{2}}$

A small tweezer misalignment $\epsilon$ gives rise to new spin-dependent terms
in the Hamiltonian that shift the trapping potential in a state dependent way.
In Fig.S-2 is shown how the dynamics is affected in the case where the
tweezers are misaligned by 30 nm.

Figure S-2: Center of mass mode phase-space dynamics for perfectly aligned
tweezer (left) and for $30$ nm misaligned ones (right). For the simulation we
used the same parameters as for $\tau/2=120\,\mu$s point in Figure 1(a) of the
main text.

## Appendix III : Gate fidelity

We characterize the gate by calculating the average process fidelity as
follows : [2]:

$\bar{F}(\hat{U}_{\text{id}},\hat{U}_{\hat{H}_{\mathrm{sim}}})=\frac{\sum_{j}\text{tr}\left[\hat{U}_{\text{id}}\hat{\sigma}_{j}^{\dagger}\hat{U}_{\text{id}}^{\dagger}\boldsymbol{\hat{\sigma}}_{j}(\hat{U}_{\hat{H}_{\mathrm{sim}}})\right]+d^{2}}{d^{2}\left(d+1\right)},$
(S-1)

where $\hat{U}_{\text{id}}$ is the unitary of an ideal geometric phase gate
and
$\boldsymbol{\hat{\sigma}}_{j}(\hat{U}_{\hat{H}_{\mathrm{sim}}})\equiv\text{tr}_{\text{FS}}(\hat{U}_{\hat{H}_{\mathrm{sim}}}\left[|n\rangle\langle
n|\bigotimes\hat{\sigma}_{j}\right]\hat{U}^{\dagger}_{\hat{H}_{\mathrm{sim}}})$
projects the unitary matrix generated by the time evolution of the Hamiltonian
used for the simulations $\hat{U}_{\hat{H}_{\mathrm{sim}}}$ on the Fock state
$|n\rangle$ and on a $d$-dimensional representation Pauli matrices.

## References

  * Spreeuw [2020] R. J. Spreeuw, Phys. Rev. Lett. 125, 233201 (2020).
  * Nielsen [2002] M. A. Nielsen, Physics Letters A 303, 249 (2002).


# Product measures without utilizing integrals

Hasan Gül111Middle East Technical University<EMAIL_ADDRESS>and Süleyman
Önal222Middle East Technical University<EMAIL_ADDRESS>
(13.05.2016)

Abstract: We show that product measures can be handled without resorting to
integrals.

MSC: 28A35
Keywords: Analysis; Classical measure theory; Measures and integrals in
product spaces

##  1 Introduction

In many textbooks on Real Analysis or Measure Theory, if not in all, product
measures on product spaces are handled by utilizing the monotone and dominated
convergence theorems which involve integrals with respect to some measures
(cf. [1, 2, 3, 4, 5]). Of course, we do not find this approach to be circular,
but feel a certain uneasiness.

The necessary foundation for studying a measure space $(X,\Sigma,\mu)$ is the
characterization of null sets with respect to $\mu$ and the establishment of
$\sigma$-additivity of it on the elements of the semiring ($\sigma$-algebra)
$\Sigma$ .

In this short note, we show that an introduction to product measure and laying
the aforementioned foundation without resorting to integrals is possible in a
very simple way.

##  2 Preliminaries

In this section, we recall some basic definitions.

Let $X$ be a nonempty set.

Definition 2.1. A collection $\Sigma$ of subsets of $X$ is called a semiring
if it satisfies the following properties:

1\. $\emptyset\in\Sigma$,

2\. If $A,B\in\Sigma$, then $A\cap B\in\Sigma$,

3\. For every $A,B\in\Sigma$, there exist $C_{1},...,C_{n}\in\Sigma$ such that
$A\setminus B=\cup_{i=1}^{n}C_{i}$ and $C_{i}\cap C_{j}=\emptyset$ if
$i\not=j$.

Definition 2.2. A nonempty collection $\Sigma$ of subsets of $X$ which is
closed under finite intersections and complementation is called an algebra of
sets. An algebra $\Sigma$ of subsets of $X$ is called a
$\mathbf{\sigma}$-algebra if every union of a countable collection of elements
of $\Sigma$ is again in $\Sigma$.

Definition 2.3. Let $\Sigma$ be a semiring of subsets of $X$. A set function
$\mu:\Sigma\rightarrow[0,\infty]$ is called a measure on $\Sigma$ if it
satisfies the following properties:

1\. $\mu(\emptyset)=0$ ,

2\. If $\\{A_{n}:n\in\mathbb{N}\\}$ is a disjoint sequence in $\Sigma$
satisfying $\cup_{n=1}^{\infty}A_{n}\in\Sigma$, then

$\mu(\cup_{n=1}^{\infty}A_{n})=\Sigma_{n=1}^{\infty}\mu(A_{n})$

holds.

Definition 2.4. A triplet $(X,\Sigma,\mu)$, where $\Sigma$ is a semiring of
subsets of $X$ and $\mu$ is a measure on $\Sigma$ is called a measure space .

Definition 2.5. Let $(X,\Sigma,\mu)$ be a measure space and $A\subseteq X$. If
$\mu(A)<\infty$, then $A$ is said to be of finite measure. If
$A=\bigcup_{n\in\mathbb{N}}A_{n}$, where each $A_{n}$ is of finite measure,
then $A$ is said to be of $\sigma$-finite measure.

Definition 2.6. A set function $\mu:P(X)\rightarrow[0,\infty]$ defined on the
power set $\mathcal{P}(X)$ of $X$ is called an outer measure if it satisfies
these properties:

l. $\mu(\emptyset)=0$ ,

2\. If $A\subseteq B$, then $\mu(A)\leq\mu(B)$,

3\. $\mu(\cup_{n=1}^{\infty}A_{n})\leq\Sigma_{n=1}^{\infty}\mu(A_{n})$ holds
for every sequence $(A_{n})_{n\in\mathbb{N}}$ of subsets of X.

Let $\mathcal{F}$ be a collection of subsets of $X$ containing the empty set.
Also, let $\mu:\mathcal{F}\rightarrow[0,\infty]$ be a set function such that
$\mu(\emptyset)=0$. For every subset $A$ of $X$ we define

$\mu^{*}(A)=\inf\\{\Sigma_{n=1}^{\infty}\mu(A_{n}):(A_{n})\text{ \rm is a
sequence of }\mathcal{F}\text{ \rm
with}A\subseteq\cup_{n=1}^{\infty}A_{n}\\}.$

If there is no sequence $(A_{n})_{n\in\mathbb{N}}$ of $\mathcal{F}$ such that
$A\subseteq\cup_{n=1}^{\infty}A_{n}$ , then we let $\mu^{*}(A)=\infty$.

The set function $\mu^{*}:P(X)\rightarrow[0,\infty]$ is an outer measure
(called the outer measure generated by the set function
$\mu:\mathcal{F}\rightarrow[0,\infty])$ satisfying $\mu^{*}(A)\leq\mu(A)$ for
each $A\in\mathcal{F}$.

Definition 2.7. Let $(X,\Sigma_{X},\mu_{X})$ and $(Y,\Sigma_{Y},\mu_{Y})$ be
measurable spaces. The subset

$\Sigma_{X}\otimes\Sigma_{Y}:=\\{A\times B:A\in\Sigma_{X},B\in\Sigma_{Y}\\}$

of $X\times Y$ is called the product semiring. The set function
$\mu_{X}\times\mu_{Y}:\Sigma_{X}\otimes\Sigma_{Y}\rightarrow[0,\infty]$
defined by

$\mu_{X}\times\mu_{Y}(A\times B)=\mu_{X}(A)\cdot\mu_{Y}(B)\ \ \text{\rm for
all }A\times B\in\Sigma_{X}\otimes\Sigma_{Y}.$

is a measure on $\Sigma_{X}\otimes\Sigma_{Y}$ and is called the product
measure of $\mu_{X}$ and $\mu_{Y}$ and is finitely additive.

##  3 Product measures

######  Theorem 1.

Let $(X,\Sigma_{X},\mu_{X})$ and $(Y,\Sigma_{Y},\mu_{Y})$ be measurable spaces
where $\Sigma_{X},\;\Sigma_{Y}$ are semi-rings ($\sigma-$algebras) and
$\mu_{X},\;\mu_{Y}$ are $\sigma-$ additive measures. Let $D\subseteq X\times
Y$ and suppose $D\subseteq\cup_{n\in\mathbb{N}}B_{n}\times C_{n}$ where
$\\{B_{n}\times C_{n}:n\in\mathbb{N}\\}$ are pairwise disjoint sequences with
$B_{n}\in\Sigma_{X}$ and $C_{n}\in\Sigma_{Y}$. Define $D^{>r}:=\\{x\in
X:\mu_{Y}^{*}(D^{x})>r\\}$ for $r\in\mathbb{R}^{+}$ and $D^{x}:=\\{y:(x,y)\in
D\\}$. Let $\mu_{X}^{*},\;\mu_{Y}^{*}$ be outer measures generated by
$\mu_{X}$ and $\mu_{Y}$, respectively, and suppose $\mu_{X}^{*}(D^{>r})>s$ for
some $s\in\mathbb{R}^{+},$ Then, there exists a finite subset
$F\subseteq\mathbb{N}$ such that

$rs<\underset{n\in F}{\Sigma}\mu_{X}(B_{n})\mu_{Y}(C_{n})$

.

###### Proof.

Let $x\in D^{>r}$ and define $N_{x}:=\\{n\in\mathbb{N}:x\in B_{n}\\}$. Then,
$D^{x}\subseteq\bigcup_{n\in N_{x}}C_{n}$. Now

$r<\mu_{Y}^{*}(D^{x})\leq\Sigma_{n\in N_{x}}\mu_{Y}^{*}(C_{n})\leq\Sigma_{n\in
N_{x}}\mu_{Y}(C_{n}).$

Let $M_{x}$ be a finite subset of $N_{x}\subseteq\mathbb{N}$ such that
$r<\Sigma_{n\in M_{x}}\mu_{Y}(C_{n})$. Note that $x\in B_{n}$ for all $n\in
M_{x}$. Thus, $D^{>r}\subseteq\bigcup_{x\in D^{>r}}(\underset{n\in
M_{x}}{\cap}B_{n})$ and the family $\\{\underset{n\in M_{x}}{\cap}B_{n}:x\in
D^{>r}\\}\subseteq\Sigma_{X}$ is countable since
$\\{M_{x}:M_{x}\subseteq\mathbb{N}\\}\subseteq\mathcal{P}_{<\aleph_{o}}(\mathbb{N}).$

We have

$s<\mu_{X}^{*}(D^{>r})\leq\mu_{X}^{*}(\bigcup_{x\in D^{>r}}(\underset{n\in
M_{x}}{\cap}B_{n})).$

Thus, there exist a finite subset $M$ of $D^{>r}$ with

$s<\mu_{X}^{*}(\underset{x\in M}{\bigcup}(\underset{n\in M_{x}}{\cap}B_{n}).$

Set $F:=\bigcup_{x\in M}M_{x}$. Since $\underset{x\in
M}{\bigcup}(\underset{n\in M_{x}}{\cap}B_{n}\times\underset{n\in
M_{x}}{\cup}C_{n})\subseteq\underset{n\in F}{\bigcup}(B_{n}\times C_{n})$ we
have
$rs<\underset{n\in F}{\Sigma}\mu_{X}(B_{n})\mu_{Y}(C_{n})$ by the finite
additivity of $\mu_{X}\times\mu_{Y}$ on the semiring
$\Sigma_{X}\otimes\Sigma_{Y}$.

∎

######  Corollary 1.

Let $(X,\Sigma_{X},\mu_{X})$ and $(Y,\Sigma_{Y},\mu_{Y})$ be measurable spaces
where $\Sigma_{X},\;\Sigma_{Y}$ are semi-rings ($\sigma-$algebras) and
$\mu_{X}$ and $\mu_{Y}$ are $\sigma-$ additive measures. Then
$\mu_{X}\times\mu_{Y}$ is countably additive on $X\times Y$.

###### Proof.

Let $(X,\Sigma_{X},\mu_{X})$ and $(Y,\Sigma_{Y},\mu_{Y})$ be measurable spaces
where $\Sigma_{X}$ $,\Sigma_{Y}$ are semi-rings ($\sigma-$algebras) and
$\mu_{X},\;\mu_{Y}$ are $\sigma-$ additive measures. Let $D=B\times
C=\bigcup_{n\in\mathbb{N}}(B_{n}\times C_{n})$ with $B_{n},B\in\Sigma_{X}$ and
$C_{n},C\in\Sigma_{Y}$ for all $n\in\mathbb{N}$ and $B_{n}\times C_{n}$ are
pairwise disjoint for all $n\in\mathbb{N}$.

We may suppose, without loss of generality, $\mu(B)>0$ and also,$\mu(C)>0.$

Suppose $t<\mu_{X}(B)\cdot\mu_{Y}(C)$. We may find $r,s\in\mathbb{R}^{+}$ with
$0<r<\mu_{X}(B)$ and $0<s<\mu_{Y}(C)$ such that $t<r\cdot s$. Consider
$(B\times C)^{x}$ where $x\in B.$ Clearly,
$\mu^{*}((B\times C)^{x})=\mu(C)$ and $D^{>r}=B$. Thus, $r<\mu(C)$ and
$s<\mu^{*}(D^{>r}).$ By the theorem above, we may find a finite subset $F$ of
$\mathbb{N}$ such that

$t<\underset{n\in F}{\Sigma}\mu(A_{n})\cdot\mu(B_{n}).$

∎

######  Corollary 2.

Let $(X,\Sigma_{X},\mu_{X})$ and $(Y,\Sigma_{Y},\mu_{Y})$ be measurable spaces
where $\Sigma_{X},\;\Sigma_{Y}$ are semi-rings ($\sigma-$algebras) and
$\mu_{X},\;\mu_{Y}$ are $\sigma-$ additive measures. If $D\subseteq X\times Y$
and $(\mu_{X}\times\mu_{Y})^{*}(D)=0$ then $\mu_{Y}^{*}(D^{x})=0$ for
$\mu_{X}$-almost all $x\in X$. Conversely, if $D$ is
$(\mu_{X}\times\mu_{Y})^{*}$ measurable and $\mu_{Y}^{*}(D^{x})=0$ for
$\mu_{X}$-almost all $x$, then $(\mu_{X}\times\mu_{Y})^{*}(D)=0$, provided
$\mu_{X},\;\mu_{Y}$ are $\sigma$-finite measures.

###### Proof.

If $D\subseteq X\times Y$ and $(\mu_{X}\times\mu_{Y})^{*}(D)=0$ then
$\mu^{*}_{X}(D^{>r})=0$ for all $r>0$ by the theorem. Hence
$\mu_{X}^{*}(\bigcup_{n\in\mathbb{N}}D^{>\frac{1}{n}})=0$. Thus,
$\mu_{Y}^{*}(D^{x})=0$ for $\mu_{X}$-almost all $x\in X$.

Conversely, let $\mu_{X},\;\mu_{Y}$ be $\sigma$-finite measures. Suppose $D$
is $(\mu_{X}\times\mu_{Y})^{*}$ measurable and for $\mu_{X}$-almost all $x$,
$\mu_{Y}^{*}(D^{x})=0$. Then $(\mu_{X}\times\mu_{Y})^{*}(D)=0$ .

By the $\sigma$-additivity of measures, we may assume
$\mu_{X}(X),\;\mu_{Y}(Y)<\infty$. Suppose $\mu_{Y}(D^{x})=0$ for
$\mu_{X}$-almost all $x\in X$. Since $D^{c}=X\times Y\setminus D$, we have
$\mu_{Y}((D^{c})^{x})=\mu_{Y}(Y)$ for $\mu_{X}$-almost all $x\in X$. Thus,

$\mu_{X}^{*}(\\{x:\mu_{Y}((D^{c})^{x})=\mu_{Y}(Y)\\})=\mu_{X}(X)$

Hence, by the theorem,
$(\mu_{X}\times\mu_{Y})^{*}(D^{c})=\mu_{X}(X)\cdot\mu_{Y}(Y)$. Therefore,
$(\mu_{X}\times\mu_{Y})^{*}(D)=0$.

Note that both monotone and dominated convergence theorems themselves may be
proved with this approach, which is, using measure theory instead of
integration with a slightly different interpretation.

## References

  * [1] C. D. ALIPRANTIS and O. BURKINSHAW, Principles of Real Analysis, 3rd ed., Academic Press, New York, 1998.
  * [2] E. HEWITT and K. STROMBERG, Real and Abstract Analysis, Springer-Verlag, New York, 1965.
  * [3] S. LANG, Real and Functional Analysis, Springer-Verlag, New York, 1993.
  * [4] H. ROYDEN, Real Analysis, Macmillan, New York, 1988.
  * [5] W. RUDIN, Real and Complex Analysis, McGraw-Hill, New York, 1966.

∎