# Conditional probabilities of measurements, quantum time and the Wigner’s
friend case

M. Trassinelli<EMAIL_ADDRESS>Institut des NanoSciences
de Paris, CNRS, Sorbonne Université, Campus Pierre et Marie Curie, 75005
Paris, France

###### Abstract

Considering a minimal number of assumptions and in the context of the timeless
formalism, conditional probabilities are derived for subsequent measurements
in the non-relativistic regime. Only unitary transformations are considered
with detection processes described by generalized measurements (POVM). One-
time conditional probabilities are unambiguously derived via the Gleason-Bush
theorem, including for puzzling cases like the Wigner’s friend scenario where
their form underlines the relativity aspect of measurements. No paradoxical
situations emerge and the roles of Wigner and Wigner can be seen by his friend
as being in a superposition.

##  I Introduction

The measurement problem is one of the most fundamental issues in Quantum
Mechanics on which the numerous interpretations and foundations differ. Von
Neumann was the first to try to formalize the problem of measurement assuming
that there are basically two types of evolution in quantum mechanics Von
Neumann (1955). The first is unitary evolution, while the other is the
collapse of the wave-function during the measurement from a superposition of
states to only one of the studied systems. Contrary to unitary evolution, the
second measurement process, formalized by the projection postulate, is
irreversible, non-causal and, of course, non-unitary; it is generally viewed
as the passage between the quantum and the classical realms. When such a
process is not postulated, different mechanisms can be evoked to justify it,
like the coupling to the environment (decoherence) Zurek (2003), the addition
of non-linearities in the evolution equation Ghirardi  _et al._ (1979) and
specific coupling to the gravitational fields Penrose (1996), among many other
approaches.

If an ideal Ockham’s razor would be applied to the different interpretations
of Quantum Mechanics, only those with a minimal number of hypotheses should be
considered (and that agree with experimental measurements of course). Instead
of postulating two type of natures, classical and quantum, the simplest
approach is to consider only the second one; quantum phenomena cannot be
explained by classical physics, whereas classical behavior can emerge from
quantum systems. With the same minimalist approach, only one evolutionary
process–the unitary processes–should be taken into account. If only unitary
transformations are considered, the measurement process has to be interpreted
then as a unitary interaction between the detector system and the studied
system. For this task, the standard approach is to describe the detection
process by a positive valued operator (effect) in the context of the Positive
Valued Operator Measure (POVM) framework Auletta  _et al._ (2009); Busch  _et
al._ (2016). In this context, non-ideal measurements can also be treated
including destructive detections.

By considering the measuring apparatus as a quantum system, as well as the
entire chain up to the observer’s neurons, the measurement theory becomes part
of the quantum theory of interacting compound systems. In this framework, the
correlations created by interactions between systems play a central role, like
in Relational Quantum Mechanics Rovelli (1996, 2018), in the original Everett
formulation Everett (1957) and its many-worlds theory derivation Dewitt and
Graham (2015), in the Consistent Histories Griffiths (2003); Omnès (1992) , in
the History Projection Operator formalism Isham (1994) and in QBism Caves  _et
al._ (2002). This simplicity has however a price. The entanglement between
observer and observable leads to different measurement results for
independent, i.e. non-interacting, observers. This relativity of what is
normally considered universal remains a big conceptual leap for beings living
in a world where quantum behaviors are barely visible. This discomfort
manifests itself in paradoxes like in the case of the Wigner’s friend scenario
Wigner (1995), where two observers can have contradictory descriptions of the
same system at the same moment.

Another thorny problem of the standard formulation of quantum mechanics is the
special role occupied by the time coordinate $t$. In the unitary evolution,
$t$ is the evolutionary parameter; in the Schrödinger equation, time
explicitly appears in the derivation with respect to it. In both cases, time
plays a special role in the description of a system’s evolution. Moreover, the
formal description of a system before and after a measurement (via a short
interaction) requires considering two distinct Hilbert spaces
$\mathcal{H}_{in}\otimes\mathcal{H}_{out}$ Isham (1994); Oreshkov  _et al._
(2012); Flatt  _et al._ (2017); Castro-Ruiz  _et al._ (2018). This
complexification and the central role of the time coordinate can be eliminated
at once by including the measurement of time itself in a more general
description of the studied interaction together with a clock system.
Originally designed to be compatible with general relativity, this formalism
allows for considering dynamic processes without time as parameter. This leads
to a simpler formulation of the measurement process acting in a unique Hilbert
space, where time is an observable among others resulting in a timeless
description of dynamics.

Starting from a minimalistic description of nature using only quantum systems
and unitary interactions, this article focuses on the formulation and
properties of the conditional probability function in the quantum time
formalism. Since the early formulations from Page and Wootters in 1983 Page
and Wootters (1983), several works have discussed the conditional probability.
This work takes advantage of the best of both the timeless and POVM formalisms
to provide a coherent description of different measurements.

Differently from previous works Hellmann  _et al._ (2007); Gambini  _et al._
(2009); Giovannetti  _et al._ (2015); Castro-Ruiz  _et al._ (2020); Höhn  _et
al._ (2021); Baumann  _et al._ (2021), here the conditional probability for
subsequent measurements is derived starting from first principles. More
precisely, the most general measurement operator is built for the total
Hilbert space that includes the clock system itself. By applying the Gleason-
Bush theorem, the associate probability is derived. As it will be show, in
this way the probability expression is univocally defined. In particular for
Wigner’s friend measurement scenarios, the associated probability expressions
can be simply and unambiguously formulated.

In the following section, the basic notion of the quantum time/timeless
formalism is introduced, including detector systems. In section III, we derive
the corresponding probability expressions and in Sec. IV we apply the
developed tools to study the case of the Wigner’s friend. Section V will be
our conclusions.

##  II Non-relativistic quantum time (timeless evolution) formalism

Similarly to Refs. Hellmann  _et al._ (2007); Giovannetti  _et al._ (2015);
Baumann  _et al._ (2021), the Wheeler-DeWitt equation is taken as the starting
point for describing the dynamics of the global state $\ket{\Psi}$ in the
space-time continuum Rovelli (2007):

$\hat{H}\ket{\Psi}\rangle=0.$ (1)

$\hat{H}$ is the total Hamiltonian acting in the kinematic Hilbert space
$\mathcal{K}$ where space-time coordinates are measured. The notation with the
double ket $\ket{\cdot}\rangle$ indicates the inclusion of a clock system in
$\ket{\Psi}\rangle$ from which the time coordinate is measured. In the
following paragraphs, we implicitly assume $c=\hbar=1$. Solutions of the above
equation are also solutions of

$\ket{\Psi}\rangle=\int d\alpha
e^{-i\alpha\hat{H}}\ket{\Psi}\rangle=\mathbb{P}\ket{\Psi}\rangle.$ (2)

The operator $\mathbb{P}$ can be considered as a “projector” from
$\mathcal{K}$ to the dynamical Hilbert space $\mathcal{H}$, the Hilbert space
of the dynamical solutions Rovelli (2007).

In the non-relativistic approximation where the perfect clock system $C$ is
considered non-interacting with the system of interest $S$, the kinematic
Hilbert space can be decomposed into
$\mathcal{K}=\mathcal{H}_{T}\otimes\mathcal{H}_{S}$ where $\mathcal{H}_{T}$
and $\mathcal{H}_{S}$ are the Hilbert sub-spaces of the clock and the studied
system, respectively. The corresponding Hamiltonian is

$\hat{H}=\hat{p}_{T}\otimes\mathbb{1}_{S}+\mathbb{1}_{T}\otimes\hat{H}_{S}$
(3)

where $\hat{H}_{T}=\hat{p}_{T}$ consists of the conjugate operator of the time
operator $\hat{T}$, with $[\hat{T},\hat{p}_{T}]=i$. $\\{\ket{t}_{T}\\}$ are
the corresponding time states that form the base of $\mathcal{H}_{T}$, with
$\hat{T}\ket{t}_{T}=t\ket{t}_{T}$ and
${}_{T}\\!\braket{t}{t^{\prime}}_{T}=\delta(t-t^{\prime})$. A more general
case where the clock can interact with the studied system or with another
clock is treated in Refs. Ruiz  _et al._ (2017); Smith and Ahmadi (2019). The
case of relative or non-ideal clocks is discussed in Refs. Loveridge and
Miyadera (2019); Rossi and Soares-Pinto (2020).

The total wave-function can be then decomposed into
$\ket{\Psi}\rangle=\ket{t}_{T}\otimes\ket{\psi(t)}_{S}$ where
$\ket{\psi(t)}_{S}=_{T}\\!\braket{t}{\Psi}\rangle$ is the wave-function
obtained by the condition of measuring the time $t$ in the clock.
Consequently, we can also write

$\ket{\Psi}\rangle=\int dt\ket{t}_{T}\otimes\ket{\psi(t)}_{S}.$ (4)

With this notation, considering the term
${}_{T}\\!\braket{t}{\hat{H}}{\Psi}\rangle$ with Eqs. (1) and (3), the
standard form of the Schrödinger equation $i\frac{\partial}{\partial
t}\ket{\psi(t)}_{S}=\hat{H}_{S}\ket{\psi(t)}_{S}$ is obtained. If
$\hat{H}_{S}$ does not depend on the time operator $\hat{T}$, a solution of
the equation is the unitary operator

$\hat{U}_{S}(t,t_{0})=e^{-i\hat{H}_{S}(t-t_{0})}.$ (5)

##  III Quantum measurement and probabilities

Similarly to the procedure described in Refs. Hellmann  _et al._ (2007);
Giovannetti  _et al._ (2015), we consider here a measurement at a time $t_{M}$
consisting of a unitary interaction of a negligible duration between the
system $S$ and a measuring system $M$. The total Hamiltonian describing the
system is

$\hat{H}=\hat{p}_{T}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{M}+\mathbb{1}_{T}\otimes\hat{H}_{S}\otimes\mathbb{1}_{M}\\\
+\hat{V}_{SM}\delta(\hat{T}-t_{M})+\mathbb{1}_{T}\otimes\mathbb{1}_{S}\otimes\hat{H}_{M}.$
(6)

$V_{SM}$ represents the interaction between $\mathcal{H}_{S}$ and
$\mathcal{H}_{M}$ at the time $t_{M}$, where $\hat{H}_{M}$ is the Hamiltonian
of the detector itself. With the detector space $\mathcal{H}_{M}$ we can
associate the entire sub-system chain from the detector to the observer’s
brain. The changes in $\mathcal{H}_{S}$ of a measurement with outcome $m$ is
given by the positive-valued operator $\hat{\Pi}^{m}$ linked to the detector
ancillary state $\ket{m}_{M}\in\mathcal{H}_{M}$. The interaction between the
detector and the system at time $t_{M}$ results in the unitary mapping

$\ket{\psi(t_{M})}_{S}\otimes\ket{r}_{M}\to\sum_{m}K_{M}^{m}\ket{\psi(t_{M})}_{S}\otimes\ket{m}_{M},$
(7)

where $\ket{r}_{M}$ is the “ready” detector state before the measurement, and
$\hat{K}_{M}^{m}$ are the different Kraus operators corresponding to the
outcomes $m$ with $\hat{\Pi}^{m}=(\hat{K}_{M}^{m})^{\dagger}\hat{K}_{M}^{m}$.
From Eq. (4), the total state $\ket{\Psi}\rangle$ can be written as

$\ket{\Psi}\rangle=\int_{-\infty}^{t_{M}}dt\ket{t}_{T}\otimes\ket{\psi(t)}_{S}\otimes\ket{r}_{M}\\\
+\int_{t_{M}}^{\infty}dt\ket{t}_{T}\otimes\sum_{i}\hat{U}_{S}(t,t_{M})\hat{K}_{M}^{i}\ket{\psi(t_{M})}_{S}\otimes\ket{i}_{M},$
(8)

where $\hat{U}_{S}$ is defined in Eq. (5).

We consider now the case of two subsequent measurements: one measurement $M$
on $S$ with output $m$ operated at a time $t_{M}$ and another $N$ with output
$n$ done at the time $t_{N}>t_{M}$ with no other interaction between the
different sub-systems. The corresponding Hamiltonian is

$\hat{H}=\hat{p}_{T}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{M}\otimes\mathbb{1}_{N}+\mathbb{1}_{T}\otimes\hat{H}_{S}\otimes\mathbb{1}_{M}\otimes\mathbb{1}_{N}\\\
+\hat{V}_{SM}\delta(\hat{T}-t_{M})\otimes\mathbb{1}_{N}+\hat{V}_{SN}\delta(\hat{T}-t_{N})\otimes\mathbb{1}_{M}\\\
+\mathbb{1}_{T}\otimes\mathbb{1}_{S}\otimes\hat{H}_{M}\otimes\mathbb{1}_{N}+\mathbb{1}_{T}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{M}\otimes\hat{H}_{N}$
(9)

and the associated wavefunction is Giovannetti  _et al._ (2015); Castro-Ruiz
_et al._ (2020),

$\ket{\Psi}\rangle=\int_{-\infty}^{t_{M}}dt\ket{t}_{T}\otimes\ket{\psi(t)}_{S}\otimes\ket{r}_{M}\otimes\ket{r}_{N}\\\
+\int_{t_{M}}^{t_{N}}dt\ket{t}_{T}\otimes\sum_{i}\hat{U}_{S}(t,t_{M})\hat{K}_{M}^{i}\ket{\psi(t_{M})}_{S}\otimes\ket{i}_{M}\otimes\ket{r}_{N}\\\
+\int_{t_{M}}^{\infty}dt\ket{t}_{T}\otimes\sum_{i,j}\hat{U}_{S}(t,t_{N})\hat{K}_{N}^{j}\hat{U}_{S}(t_{N},t_{M})\hat{K}_{M}^{i}\ket{\psi(t_{M})}_{S}\\\
\otimes\ket{i}_{M}\otimes\ket{j}_{N}.\\\ $ (10)

Note that this timeless representation of the total wave-function by a series
of superimpositions of system/detector entangled states is de facto equivalent
to the branched wavefunctions in Everett’s original paper Everett (1957),
where the central concepts turn around the relativity of states of measured
and measuring systems.

How can we now associate a probability to the measurement outputs? Different
approaches have been developed in the past in the discussed context, as the
definition via transition amplitudes in Quantum Gravity Rovelli (2007);
Hellmann  _et al._ (2007) or postulating the validity of the Bohr rule
Giovannetti  _et al._ (2015). Our approach consists in considering the general
features of Hilbert spaces and requiring only basic properties of the
probability function $\wp$. For a Hilbert space with finite dimension
$\mathcal{H}$, the Gleason-Busch theorem Gleason (1957); Busch (2003)
demonstrates that the probability function is in fact univocally defined by
the trace rule

$\wp(a)=tr(\hat{\rho}\,\hat{\Pi}^{a}),$ (11)

where $\hat{\Pi}^{a}$ is a positive-valued operator in $\mathcal{H}$ and
$\hat{\rho}$ is the density matrix of a given state. This is actually the same
approach used by Pages and Wootters Page and Wootters (1983), where however
only projector operators are considered. It is natural to extend this
definition of probability to the infinite dimensional space $\mathcal{K}$.
Such an extension is in fact equivalent to the standard definition in Quantum
Gravity. In our case, a generic measurement with output $a$ must be
represented by an operator $\hat{\Pi}_{a}$ in the kinematic Hilbert space
$\mathcal{K}=\mathcal{H}_{T}\otimes\mathcal{H}_{S}\otimes\mathcal{H}_{M}\otimes\mathcal{H}_{N}$.
Because there is only one clock subspace $\mathcal{H}_{T}$, within the minimal
approach presented here, it is impossible to construct a two-time probability
for the event “$m$ at the time $t_{1}$ and $n$ at the time $t_{2}$”. On the
contrary, it is possible to consider the measurement “$m\land n\land t$”
$\equiv$ “m AND n AND t” corresponding to the “$m$ and $n$ results obtained at
the time $t$”, when the two measurements occur at the times $t_{M}$ and
$t_{N}$, respectively. Like in Refs. Giovannetti  _et al._ (2015); Hellmann
_et al._ (2007), the formulation of such a probability implies that the two
detector states $\ket{m}_{M}$ and $\ket{n}_{N}$ are stored in an internal
memory that is read at the time $t>t_{M},t_{N}$. It is important to note that,
in contrast to a previous work on subsequent measurements without timeless
notation Trassinelli (2020), here the operator “$\land$” $\equiv$ “AND” is
symmetric ($a\land b=b\land a$). The measurement order is in fact defined by
the Hamiltonian itself. We are not simply discussing $\wp(m\land n\land t)$,
but we are implicitly considering the probability $\wp(m\land n\land
t|M_{t_{M}}\land N_{t_{N}})$, where we explicitly indicate the dependence on
the prior information of the Hamiltonian structure, with the measurements $M$
at the time $t_{M}$ and $N$ at $t_{N}$. More complex scenarios with an
undefined measurement order in the Hamiltonian can be also considered Castro-
Ruiz  _et al._ (2020); Paunković and Vojinović (2020) but they are not
discussed here.

With respect to two-time probabilities, a naive form, based in wavefunction
two-time collapse, does not reproduce the correct propagator Kuchar (2011). An
alternative solution to this problem has been provided by Dolby Dolby (2004).
However, it has been demonstrated Hellmann  _et al._ (2007) that this solution
is in conflict with predictions of standard quantum mechanics. Recently, a new
proposition of two-time probability that resolves past criticisms has been
presented by Höhn and collaborators Höhn  _et al._ (2021). The operators
corresponding to the two measurements at the two times are built without
violating Eqs. (1) and (2), i.e. keeping the system in the physical Hilbert
space $\mathcal{H}$. In the present work, we take a simpler approach
considering measurement memories, such that the time coordinate is read only
once. In this way, we can easily define probability outputs from a generic
one-time operator $\hat{\Pi}_{a}=\hat{\Pi}^{m,n,t}_{MN}\in\mathcal{K}$ instead
of $\mathcal{H}$. Even if the total wave-function is not part of $\mathcal{H}$
after the measurement
($\ket{\Psi}\rangle\to\hat{K}_{a}\ket{\Psi}\rangle\notin\mathcal{H}$, where
$\hat{K}_{a}$ is the Kraus operator associated to
$\hat{\Pi}_{a}=\hat{K}_{a}\hat{K}_{a}^{\dagger}$), the probability $\wp(a)$ is
well defined by the Gleason-Bush theorem (Eq. (11)).

It should be emphasized that there is no direct measurement on $S$ but only
via the ancillary detector states. The operator associated with this
measurement is
$\hat{\Pi}^{m,n,t}_{MN}=\ket{t}_{T}{{}_{T}}\\!\bra{t}\otimes\mathbb{1}_{S}\otimes\ket{m}_{M}{{}_{M}}\\!\bra{m}\otimes\ket{n}_{N}{{}_{N}}\\!\bra{n}$
that leads to

$\wp(m\land n\land
t)=tr(\hat{\rho}\,\hat{\Pi}^{m,n,t}_{MN})=||(_{T}\bra{t}\otimes_{M}\\!\bra{m}\otimes_{N}\\!\bra{n})\ket{\Psi}\rangle||^{2}\\\
=\begin{cases}||_{M}\\!\braket{m}{r}_{M}\
{}_{N}\\!\braket{n}{r}_{N}\ket{\psi(t_{M})}_{S}||^{2}&\text{ if }t<t_{M},\\\
||_{N}\\!\braket{n}{r}_{N}\hat{K}^{m}_{M}\ket{\psi(t_{M})}_{S}||^{2}&\text{ if
}t_{M}\leq t<t_{N},\\\
||\hat{K}^{n}_{N}\hat{U}_{S}(t_{N},t_{M})\hat{K}^{m}_{M}\ket{\psi(t_{M})}_{S}||^{2}&\text{
if }t\geq t_{N}.\\\ \end{cases}$ (12)

With two separate detections, we can now discuss the conditional probabilities
of the different outcomes. The conditional probabilities and $\wp(n|m\land t)$
can be calculated with the standard procedure from the Bayes’ rule and the
probabilities $\wp(n\land t)$ and $\wp(m\land t)$ by applying Eq. (11) with
the corresponding operators. We obtain in particular

$\wp(n|m\land t)=\frac{\wp(m\land n\land t)}{\wp(m\land
t)}=\frac{tr(\hat{\rho}\,\hat{\Pi}^{m,n,t}_{MN})}{tr(\hat{\rho}\,\hat{\Pi}^{m,t}_{M})}=\wp(n\land
t|m\land t),$ (13)

where
$\Pi^{m,t}_{M}=\ket{t}_{T}{{}_{T}}\\!\bra{t}\otimes\mathbb{1}_{S}\otimes\ket{m}_{M}{{}_{M}}\\!\bra{m}\otimes\mathbb{1}_{N}.$
For $\wp(m|n\land t)$ a similar equation can be derived. The validity of the
last equivalence in the above formula is justified by the presence of the time
measurement operator in both numerator and denominator. Note that $\wp(n\land
t|m),\wp(m\land t|n)$ probabilities could also be used to build conditional
forms. In this case, their evaluation is more complex due to the appearance of
$\wp(n)$ and $\wp(m)$ in the denominators. It requires in fact the
marginalization with respect the time operator basis $\\{\ket{t}_{T}\\}$ and
consequently its normalization. Such a normalization is discussed Ref.
Giovannetti  _et al._ (2015) but is not considered here. When only $\wp(m\land
n\land t)$ and $\wp(n\land t)$ are used to derive $\wp(m|n\land t)$, in fact
no problems of normalization are encountered.

##  IV Wigner’s friend scenario and conditional probability

###  IV.1 Formulae

With the notation and formulas presented above, we can now consider the case
of Wigner’s friend. This measurement scenario, introduced for the first time
by Wigner in 1961 Wigner (1995), is constituted by an observer $W$ (Wigner)
that observes another observer $F$ (his friend) who performs a quantum
measurement on a physical system $S$. The friend makes a measurement of the
system and records a well-defined result. During this measurement, and just
after, the system/friend ensemble is isolated from Wigner. For $W$, the
measurement of $F$ on $S$ becomes entangled and both are described by a
superposition of the different possible final states. Paradoxically, $W$ and
$F$ have at the same time a different description of $S$. In particular, when
quantum and classical worlds are both considered at the same time, with the
collapse of the wavefunction delimiting them, the collapse itself is not well
defined (especially when consciousness is considered as the collapse trigger
like in Wigner’s original argument). The Wigner’s friend scenario with the
timeless formalism has been extensively discussed in a recent work of Baumann,
Brukner and collaborators Baumann  _et al._ (2021), where several forms of
probabilities are considered. The notation presented here leads, in contrast,
to a unique form of the probability. Its derivation is presented in detail in
the following paragraphs applying the results presented above.

As in Ref. Baumann  _et al._ (2021), without losing in generality and to keep
simple expressions, we consider no-free dynamics for three systems: the
studied system $S$, Wigner’s friend $F$, and Wigner $W$. The corresponding
Hamiltonian, relative to a measurement $M$ at the time $t_{M}$ between the
system and the friend, and a measurement $N$ at the time $t_{N}$ between
Wigner and the ensemble friend-system, is

$H=\hat{p}_{T}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{F}\otimes\mathbb{1}_{W}+\mathbb{1}_{T}\otimes\hat{H}_{S}\otimes\mathbb{1}_{F}\otimes\mathbb{1}_{W}\\\
+\hat{V}_{SF}\delta(\hat{T}-t_{M})\otimes\mathbb{1}_{W}+\hat{V}_{SFW}\delta(\hat{T}-t_{M})\\\
+\mathbb{1}_{T}\otimes\mathbb{1}_{S}\otimes\hat{H}_{F}\otimes\mathbb{1}_{W}+\mathbb{1}_{T}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{F}\otimes\hat{H}_{W}.$
(14)

We consider a system initial condition described by

$\ket{\psi}_{S}=a\ket{\uparrow}_{S}+b\ket{\downarrow}_{S}$ (15)

with $|a|^{2}+|b|^{2}=1$. For Wigner, after the friend’s measurement, $S$ and
$F$ are described by

$\ket{\psi}_{S}\otimes\ket{\varphi}_{F}=a\ket{\uparrow}_{S}\otimes\ket{\uparrow}_{F}+b\ket{\downarrow}_{S}\otimes\ket{\downarrow}_{F}.$
(16)

We consider also that Wigner’s measurement on $F$ and $S$ consists in the
detection of the ancillary state $\ket{yes}_{W}$, which corresponds to the
operator on the system-friend space
$\hat{\Pi}_{N}^{yes}=\ket{yes}_{SF}\,{}_{SF}\bra{yes}\in\mathcal{H}_{S}\otimes\mathcal{H}_{F}$
with

$\ket{yes}_{SF}=\alpha\ket{\uparrow}_{S}\otimes\ket{\uparrow}_{F}+\beta\ket{\downarrow}_{S}\otimes\ket{\downarrow}_{F},$
(17)

with $|\alpha|^{2}+|\beta|^{2}=1$, and its complementary operator
$\hat{\Pi}_{N}^{no}=\mathbb{1}_{S}\otimes\mathbb{1}_{F}-\hat{\Pi}_{N}^{yes}$.

The global wavefunction for these two measurements is similar to that of Eq.
(LABEL:eq:final_psi_2):

$\ket{\Psi}\rangle=\int_{-\infty}^{t_{M}}dt\ket{t}_{T}\otimes\ket{\psi}_{S}\otimes\ket{r}_{F}\otimes\ket{r}_{W}\\\
+\int_{t_{M}}^{t_{N}}dt\ket{t}_{T}\otimes\sum_{f}\hat{K}_{M}^{f}\ket{\psi}_{S}\otimes\ket{f}_{F}\otimes\ket{r}_{W}\\\
+\int_{t_{N}}^{\infty}dt\ket{t}_{T}\otimes\sum_{f,w}\hat{K}_{N}^{w}\hat{K}_{M}^{f}\ket{\psi}_{S}\otimes\ket{f}_{F}\otimes\ket{w}_{W}.$
(18)

The probability of having the measurement results $w$ and $f$ at the time $t$
is associated to the operators
$\hat{\Pi}^{f,w,t}_{MN}=\ket{t}_{T}{{}_{T}}\\!\bra{t}\otimes\mathbb{1}_{S}\otimes\ket{f}_{F}{{}_{F}}\\!\bra{f}\otimes\ket{w}_{W}{{}_{W}}\\!\bra{w}$.
Its values are

$\wp(f\land w\land
t)=tr(\hat{\rho}\,\hat{\Pi}^{f,w,t}_{MN})=||(_{T}\bra{t}\otimes_{F}\\!\bra{f}\otimes_{W}\\!\bra{w})\ket{\Psi}\rangle||^{2}=\\\
\begin{cases}||_{F}\\!\braket{f}{r}_{F}\
{}_{W}\\!\braket{w}{r}_{W}\ket{\psi}_{S}||^{2}&\text{ if }t<t_{M},\\\
||_{W}\\!\braket{w}{r}_{W}\hat{K}^{f}_{M}\ket{\psi}_{S}||^{2}&\text{ if
}t_{M}\leq t<t_{N},\\\
||_{F}\\!\bra{f}\hat{K}^{w}_{N}\sum_{f^{\prime}}\hat{K}^{f^{\prime}}_{M}\ket{\psi}_{S}\otimes\ket{f^{\prime}}_{F}||^{2}&\text{
if }t\geq t_{N},\end{cases}$ (19)

where $\ket{r}_{F}$ and $\ket{r}_{W}$ are the “ready” state of the friend and
of Wigner. The associated Kraus operators are

$\begin{split}&\hat{K}_{M}^{\uparrow}=\ket{\uparrow}_{S}{}_{S}\\!\bra{\uparrow},\\\
&\hat{K}_{M}^{\downarrow}=\ket{\downarrow}_{S}{}_{S}\\!\bra{\downarrow},\\\
&\hat{K}_{N}^{yes}=\ket{yes}_{S}\ket{yes}_{F}{}_{S}\\!\bra{yes}_{F}\\!\bra{yes}\\\
&=(\alpha\ket{\uparrow}_{S}\ket{\uparrow}_{F}+\beta\ket{\downarrow}_{S}\ket{\downarrow}_{F})(\alpha^{*}_{S}\\!\bra{\uparrow}_{F}\\!\bra{\uparrow}+\beta^{*}\
_{S}\\!\bra{\downarrow}_{F}\\!\bra{\downarrow}),\\\
&\hat{K}_{N}^{no}=\ket{no}_{S}{}_{S}\\!\bra{no}\\\
&=(-\beta^{*}\ket{\uparrow}_{S}\ket{\uparrow}_{F}+\alpha^{*}\ket{\downarrow}_{S}ket{\downarrow}_{F})(-\beta_{S}\\!\bra{\uparrow}_{F}\\!\bra{\uparrow}+\alpha_{S}\\!\bra{\downarrow}_{F}\\!\bra{\downarrow}),\end{split}$
(20)

where we omitted the tensor product symbol to keep the formulas compact.
Differently from the general case of subsequent measurements treated in Sec.
III, where all $\hat{K}$ operators were acting only on $\mathcal{H}_{S}$ (like
$\hat{K}_{M}^{f}$ as well), $\hat{K}_{N}^{w}$ is operating on
$\mathcal{H}_{S}\otimes\mathcal{H}_{F}$. This implies the more complex
expression of $\wp(f\land w\land t)$ where the tensor product between
$\ket{\psi}_{S}$ and $\ket{f^{\prime}}_{F}$ is still present because,
differently from $\ket{w}_{W}$ basis, the orthogonality on the friend basis
${}_{F}\\!\braket{f^{\prime}}{f}_{F}=\delta_{ff^{\prime}}$ cannot be directly
used. More explicitly, for $t\geq t_{N}$ we have

$\wp(f\land w\land
t)=||_{F}\\!\bra{f}\hat{K}^{w}_{N}\sum_{f^{\prime}}\hat{K}^{f^{\prime}}_{M}\ket{\psi}_{S}\otimes\ket{f^{\prime}}_{F}||^{2}\\\
=||_{F}\\!\bra{f}\hat{K}^{w}_{N}(a\ket{\uparrow}_{S}\otimes\ket{\uparrow}_{F}+b\ket{\downarrow}_{S}\otimes\ket{\downarrow}_{F})||^{2}.$
(21)

Considering the above equation and the explicit form of $\hat{K}^{w}_{N}$
(Eqs. (20)), the different possible values of $\wp(f\land w\land t)$ are
obtained and summarized in Tab. 1.

Table 1:  $\wp(f\land w\land t)$ values for $t>t_{N}$. f w | yes | no
---|---|---
$\uparrow$ | $|\alpha|^{2}|a\alpha^{*}+b\beta^{*}|^{2}$ | $|\beta|^{2}|a\beta-b\alpha|^{2}$
$\downarrow$ | $|\beta|^{2}|a\alpha^{*}+b\beta^{*}|^{2}$ | $|\alpha|^{2}|a\beta-b\alpha|^{2}$

Similarly to Eq. (13), the conditional probabilities $\wp(w|f\land t)$ and
$\wp(f|w\land t)$ can be calculated from

$\wp(w|f\land t)=\frac{\wp(f\land w\land t)}{\wp(f\land
t)}=\frac{tr(\hat{\rho}\,\hat{\Pi}^{f,w,t}_{MN})}{tr(\hat{\rho}\,\hat{\Pi}^{f,t}_{M})}$
(22)

and

$\wp(f|w\land t)=\frac{\wp(f\land w\land t)}{\wp(w\land
t)}=\frac{tr(\hat{\rho}\,\hat{\Pi}^{f,w,t}_{MN})}{tr(\hat{\rho}\,\hat{\Pi}^{w,t}_{N})}$
(23)

where
$\hat{\Pi}^{f,t}_{M}=\ket{t}_{T}{{}_{T}}\\!\bra{t}\otimes\mathbb{1}_{S}\otimes\ket{f}_{F}{{}_{F}}\\!\bra{f}\otimes\mathbb{1}_{W}$
and
$\hat{\Pi}^{w,t}_{N}=\ket{t}_{T}{{}_{T}}\\!\bra{t}\otimes\mathbb{1}_{S}\otimes\mathbb{1}_{F}\otimes\ket{w}_{W}{{}_{W}}\\!\bra{w}$.
To deduce their values, $\wp(w\land t)$ and $\wp(f\land t)$ have to be
evaluated first. For $\wp(w\land t)$ we have

$\wp(w\land
t)=tr(\hat{\rho}\,\hat{\Pi}^{w,t}_{N})=||(_{T}\bra{t}\otimes_{W}\\!\bra{w})\ket{\Psi}||^{2}=\\\
\begin{cases}||_{W}\\!\braket{w}{r}_{W}\ket{\psi}_{S}\otimes\ket{r}_{F}||^{2}&\text{
if }t<t_{M},\\\
||_{W}\\!\braket{w}{r}_{W}\sum_{f}\hat{K}^{f}_{M}\ket{\psi}_{S}\otimes\ket{f}_{F}||^{2}&\text{
if }t_{M}\leq t<t_{N},\\\
||\hat{K}^{w}_{N}\sum_{f}\hat{K}^{f}_{M}\ket{\psi}_{S}\otimes\ket{f}_{F}||^{2}&\text{
if }t\geq t_{N},\end{cases}$ (24)

where we explicitly used the expression of $\ket{\Psi}\rangle$ from Eq. (18).
For the case $t>t_{N}$, we obtain an expression very similar to Eq. (21)

$\wp(w\land
t)=||\hat{K}^{w}_{N}(a\ket{\uparrow}_{S}\otimes\ket{\uparrow}_{F}+b\ket{\downarrow}_{S}\otimes\ket{\downarrow}_{F})||^{2}.$
(25)

where, differently from $\wp(f\land w\land t)$, the projection to a particular
$\ket{f}_{F}$ is absent. The different values of $\wp(w\land t)$ are reported
in Tab. 2.

Table 2:  Values of $\wp(w\land t)$ for $t>t_{N}$. w | $\wp(w\land t)$
---|---
yes | $|a\alpha^{*}+b\beta^{*}|^{2}$
no | $|a\beta-b\alpha|^{2}$
Table 3:  Values of $\wp(f\land t)$ for $t>t_{N}$. f | $\wp(f\land t)$
---|---
$\uparrow$ | $|\alpha|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\beta|^{2}|a\beta-b\alpha|^{2}$
$\downarrow$ | $|\beta|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\alpha|^{2}|a\beta-b\alpha|^{2}$

For $\wp(f\land t)$ we have

$\wp(f\land
t)=tr(\hat{\rho}\,\hat{\Pi}^{w,t}_{M})=||(_{T}\bra{t}\otimes_{F}\\!\bra{f})\ket{\Psi}\rangle||^{2}=\\\
\begin{cases}||_{F}\\!\braket{f}{r}_{F}\ket{\psi}_{S}||^{2}\otimes\ket{r}_{W}&\text{
if }t<t_{M},\\\ ||\hat{K}^{f}_{M}\ket{\psi}_{S}\otimes\ket{r}_{W}||^{2}&\text{
if }t_{M}\leq t<t_{N},\\\
||\sum_{w}{}_{F}\\!\bra{f}\hat{K}^{w}_{N}\sum_{f^{\prime}}\hat{K}^{f^{\prime}}_{M}\ket{\psi}_{S}&\otimes\ket{f^{\prime}}_{F}\otimes\ket{w}_{W}||^{2}\end{cases}\\\
\text{ if }t\geq t_{N}.\hskip 15.6491pt$ (26)

Using the properties of the tensor product and orthogonality of the Wigner’s
basis ${}_{W}\\!\braket{w}{w^{\prime}}_{W}=\delta_{ww^{\prime}}$, we can sort
the sum over $w$ and obtain

$\wp(f\land
t)=\sum_{w}||_{F}\\!\bra{f}\hat{K}^{w}_{N}\sum_{f^{\prime}}\hat{K}^{f^{\prime}}_{M}\ket{\psi}_{S}\otimes\ket{f^{\prime}}_{F}\otimes\ket{w}_{W}||^{2}.$
(27)

The different values of $\wp(f\land t)$ can be obtained from the sum over $w$
of single values of $\wp(f\land w\land t)$ (Tab. 1) and are reported in Tab.
3.

The corresponding conditional probabilities $\wp(w|f\land t)$ and
$\wp(f|w\land t)$ can now be calculated and are reported in Tabs. 4 and 5,
respectively.

Table 4:  Values of $\wp(w|f\land t)$ for $t>t_{N}$. f w | yes | no
---|---|---
$\uparrow$ | $\frac{|\alpha|^{2}|a\alpha^{*}+b\beta^{*}|^{2}}{|\alpha|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\beta|^{2}|a\beta-b\alpha|^{2}}$ | $\frac{|\beta|^{2}|a\beta-b\alpha|^{2}}{|\alpha|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\beta|^{2}|a\beta-b\alpha|^{2}}$
$\downarrow$ | $\frac{|\beta|^{2}|a\alpha^{*}+b\beta^{*}|^{2}}{|\beta|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\alpha|^{2}|a\beta-b\alpha|^{2}}$ | $\frac{|\alpha|^{2}|a\beta-b\alpha|^{2}}{|\beta|^{2}|a\alpha^{*}+b\beta^{*}|^{2}+|\alpha|^{2}|a\beta-b\alpha|^{2}}$
Table 5:  Values of $\wp(f|w\land t)$ for $t>t_{N}$. f w | yes | no
---|---|---
$\uparrow$ | $|\alpha|^{2}$ | $|\beta|^{2}$
$\downarrow$ | $|\beta|^{2}$ | $|\alpha|^{2}$

###  IV.2 Discussion

As can be seen from above equations and the corresponding values in the
different tables, the probability function is defined without ambiguities for
all cases and is automatically normalized. In particular $\sum_{w}\wp(w|f\land
t)=1$ and $\sum_{f}\wp(f|w\land t)=1$. For defined values of $f$ and $w$, the
value of $\wp(f\land w\land t)$ is simply defined by the the initial state
(Eq. (15)) and the states measured by Wigner (Eq. (17)). For $\wp(w\land t)$
and $\wp(f\land t)$ the situation is more complex and interesting. From the
Wigner point of view, the friend and the system are described by a
superimposition of states. The superimposition condition is represented by the
sum of terms $\hat{K}^{f}_{M}\ket{\psi}_{S}\otimes\ket{f}_{F}$ in Eq. (24).
Similarly, the friend sees the state of Wigner as a superposition represented
by the sum of the terms $\hat{K}^{f}_{M}\ket{\psi}_{S}\otimes\ket{f}_{F}$ in
Eq. (26). Because of the presence of the tensor product involving
$\ket{w}_{W}$ basis, for $\wp(f\land|t)$ the square modulus of the sum of
states in Eqs. (26), can be written as a sum of square moduli (Eq. (27)). This
is not true for $\wp(w\land|t)$ because of the more complex expression due to
the operators $\hat{K}^{w}_{N}$ acting on the friend Hilbert space
$\mathcal{H}_{F}$ in addition to the system space $\mathcal{H}_{S}$. The final
expressions of $\wp(w|f\land t)$ and $\sum_{f}\wp(f|w\land t)$ are equivalent
to the ones presented in Baumann et al. Baumann  _et al._ (2021) for a
specific choice of the probability definition (definition 2b). Unlike the work
presented in Ref. Baumann  _et al._ (2021) where different timeless
probability definitions are discussed for the Wigner’s friend case, a unique
definition of probability emerges here, where all systems are considered to be
quantum in nature (without any classical realm) and interacting with each
other by unitary operators.

The presence of different superposition combinations for the probabilities
$\wp(w\land t)$ or $\wp(f\land t)$ (Eqs. (26),(24)) is a manifestation of the
different descriptions of the same system $S$ from Wigner and the friend. The
probabilities depend in fact only on the mutual relations $S-W$, $S-F$ and
$F-W$. In the last years, no-go theorems evoking situations similar to the
Wigner’s friend scenario have been formulated Frauchiger and Renner (2018);
Časlav Brukner (2018); Bong  _et al._ (2020) and experimentally tested
Proietti  _et al._ (2019); Bong  _et al._ (2020). They demonstrated that one
of the following assumptions has to be violated: i) the universal validity of
quantum mechanics, ii) the locality, iii) the freedom of choice on the
measurement settings and iv) the observer-independent experimental outcomes.
In the present work, assumption i) has been accepted from the beginning
considering all systems described by quantum mechanics (including detectors
and human beings). Locality also has been implicitly assumed by considering
only unitary processes (no projections or collapses that could induce “spooky
actions at a distance”) and by the form of the Hamiltonian considered here.
The freedom of choice is also implicitly assumed in the basic properties of
the probability function required from the Gleason-Busch theorem. From these
considerations, it is no surprise that the above probability expressions
indicate a clear violation of the measurement output universality. If we
assume that the experimental outputs are relative to each pair of observer and
observed system, we can have a different description of a same system from
different observers, with no real paradox in the Wigner’s friend scenario.

##  V Conclusions

We presented here a derivation of the conditional probability $\wp$ for
subsequent measurements in the context of the quantum time approach. This is
obtained with a minimal number of assumptions, considering only systems
subject to the rules of Quantum Mechanics, without classical systems, and
interacting with each other by unitary operators with the implementation of
POVM. Applying the Gleason-Busch theorem to the global kinetic Hilbert space,
$\wp$ is built from first principles. With the approach presented here with a
one clock system, it emerges that a two-time probability function cannot be
defined. The causal order of the subsequent measurements is encoded in the
Hamiltonian, whose structure should be considered as prior knowledge in the
probability function.

When the Wigner’s friend scenario is considered, the relativity of
measurements naturally emerges, without contradictions or ambiguities. The
roles of Wigner and his friend are completely interchangeable, and Wigner is
seen by the friend as being in a superposition. This is particularly evident
in the marginal probabilities $\wp(w\land t)$ and $\wp(f\land t)$ and, as
consequence, in the conditional probabilities $\wp(w|f\land t)$ and
$\wp(f|w\land t)$.

Future developments will be focused on the extension of the formalism
presented here for other Wigner’s friend-type scenarios, like those considered
for the no-go theorems in Refs Frauchiger and Renner (2018); Časlav Brukner
(2018); Bong  _et al._ (2020).

###### Acknowledgements.

I would like to thank N. Paul, M. Romanelli and E. Lamour for their support. I
also wish to express my gratitude to the organizers of the QISS group who, by
making available on-line their workshops and seminars presentations, gave me
the initial stimulus for this present work. I would like to thank also
“referee n.2” for the very valuable suggestions that considerably improve the
original manuscript.

## References

  * Von Neumann (1955) J. Von Neumann,  _Mathematical Foundations of Quantum Mechanics_ (Princeton University Press, 1955).
  * Zurek (2003) W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
  * Ghirardi  _et al._ (1979) G. Ghirardi, C. Omero, A. Rimini, and T. Weber, Nuovo Cim. A 52, 421 (1979).
  * Penrose (1996) R. Penrose, Gen. Relativ. Gravit. 28, 581 (1996).
  * Auletta  _et al._ (2009) G. Auletta, M. Fortunato, and G. Parisi,  _Quantum Mechanics_ (Cambridge University Press, 2009).
  * Busch  _et al._ (2016) P. Busch, P. Lahti, J. Pellonpää, and K. Ylinen,  _Quantum Measurement_ (Springer International Publishing, 2016).
  * Rovelli (1996) C. Rovelli, Int. J. Theor. Phys. 35, 1637 (1996).
  * Rovelli (2018) C. Rovelli, Phil. Trans. R. Soc. A 376, 2017.0312 (2018).
  * Everett (1957) H. Everett, Rev. Mod. Phys. 29, 454 (1957).
  * Dewitt and Graham (2015) B. Dewitt and N. Graham,  _The Many Worlds Interpretation of Quantum Mechanics_ (Princeton University Press, 2015).
  * Griffiths (2003) R. B. Griffiths,  _Consistent Quantum Theory_ (Cambridge University Press, 2003).
  * Omnès (1992) R. Omnès, Rev. Mod. Phys. 64, 339 (1992).
  * Isham (1994) C. J. Isham, J. Math. Phys. 35, 2157 (1994).
  * Caves  _et al._ (2002) C. M. Caves, C. A. Fuchs, and R. Schack, Phys. Rev. A 65, 022305 (2002).
  * Wigner (1995) E. P. Wigner, “Remarks on the mind-body question,” in  _Philosophical Reflections and Syntheses_, edited by J. Mehra (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995) pp. 247–260.
  * Oreshkov  _et al._ (2012) O. Oreshkov, F. Costa, and Časlav Brukner, Nat. Commun. 3, 1092 (2012).
  * Flatt  _et al._ (2017) K. Flatt, S. M. Barnett, and S. Croke, Phys. Rev. A 96, 062125 (2017).
  * Castro-Ruiz  _et al._ (2018) E. Castro-Ruiz, F. Giacomini, and Časlav Brukner, Phys. Rev. X 8, 011047 (2018).
  * Page and Wootters (1983) D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983).
  * Hellmann  _et al._ (2007) F. Hellmann, M. Mondragon, A. Perez, and C. Rovelli, Phys. Rev. D 75, 084033 (2007).
  * Gambini  _et al._ (2009) R. Gambini, R. A. Porto, J. Pullin, and S. Torterolo, Phys. Rev. D 79, 041501 (2009).
  * Giovannetti  _et al._ (2015) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. D 92, 045033 (2015).
  * Castro-Ruiz  _et al._ (2020) E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Časlav Brukner, Nat. Commun. 11, 2672 (2020).
  * Höhn  _et al._ (2021) P. A. Höhn, A. R. Smith, and M. P. Lock, Phys. Rev. D 104, 066001 (2021).
  * Baumann  _et al._ (2021) V. Baumann, F. D. Santo, A. R. Smith, F. Giacomini, E. Castro-Ruiz, and C. Brukner, Quantum 5, 524 (2021).
  * Rovelli (2007) C. Rovelli,  _Quantum Gravity_ (Cambridge University Press, 2007).
  * Ruiz  _et al._ (2017) E. C. Ruiz, F. Giacomini, and Časlav Brukner, Proc. Natl. Acad. Sci. U.S.A. 114, E2303 (2017).
  * Smith and Ahmadi (2019) A. R. Smith and M. Ahmadi, Quantum 3, 160 (2019).
  * Loveridge and Miyadera (2019) L. Loveridge and T. Miyadera, Found. Phys. 49, 549 (2019).
  * Rossi and Soares-Pinto (2020) V. P. Rossi and D. O. Soares-Pinto, arXiv:2009.10218[quant-ph] (2020).
  * Gleason (1957) A. M. Gleason, J. Math. Mech. 6, 885 (1957).
  * Busch (2003) P. Busch, Phys. Rev. Lett. 91, 120403 (2003).
  * Trassinelli (2020) M. Trassinelli, Phys. Rev. A 102, 052224 (2020), arXiv:2010.00216 [quant-ph] .
  * Paunković and Vojinović (2020) N. Paunković and M. Vojinović, Quantum 4, 275 (2020).
  * Kuchar (2011) K. V. Kuchar, Int. J. Mod. Phys. D 20, 3 (2011).
  * Dolby (2004) C. E. Dolby, arXiv:0406034 [gr-qc] (2004).
  * Frauchiger and Renner (2018) D. Frauchiger and R. Renner, Nat. Commun. 9, 3711 (2018).
  * Časlav Brukner (2018) Časlav Brukner, Entropy 20, 350 (2018).
  * Bong  _et al._ (2020) K.-W. Bong, A. Utreras-Alarcón, F. Ghafari, Y.-C. Liang, N. Tischler, E. G. Cavalcanti, G. J. Pryde, and H. M. Wiseman, Nat. Phys. 16, 1199–1205 (2020).
  * Proietti  _et al._ (2019) M. Proietti, A. Pickston, F. Graffitti, P. Barrow, D. Kundys, C. Branciard, M. Ringbauer, and A. Fedrizzi, Sci. Adv. 5, eaaw9832 (2019).


# Dexterous holographic trapping of dark-seeking particles with Zernike
holograms

Jatin Abacousnac  David G. Grier  Department of Physics and Center for Soft
Matter Research, New York University, New York, NY 10003, USA

###### Abstract

The intensity distribution of a holographically-projected optical trap can be
tailored to the physical properties of the particles it is intended to trap.
Dynamic optimization is especially desirable for manipulating dark-seeking
particles that are repelled by conventional optical tweezers, and even more so
when dark-seeking particles coexist in the same system as light-seeking
particles. We address the need for dexterous manipulation of dark-seeking
particles by introducing a class of “dark” traps created from the
superposition of two out-of-phase Gaussian modes with different waist
diameters. Interference in the difference-of-Gaussians (DoG) trap creates a
dark central core that is completely surrounded by light that can trap dark-
seeking particles rigidly in three dimensions. DoG traps can be combined with
conventional optical tweezers and other types of traps for use in
heterogeneous samples. The ideal hologram for a DoG trap being purely real-
valued, we introduce a method based on the Zernike phase-contrast principle to
project real-valued holograms with the phase-only diffractive optical elements
used in standard holographic optical trapping systems. Zernike phase
holography should be useful for improving the diffraction efficiency of
holograms projected with phase-only diffractive optical elements.

##  I Introduction

Holographic optical trapping uses the forces and torques exerted by computer-
generated holograms to manipulate microscopic objects. Most of the literature
of holographic trapping focuses on micromanipulation of dielectric particles
with refractive indexes higher than the refractive index of the medium,
$n_{p}>n_{m}$. Such high-index particles tend to be drawn toward regions of
high light intensity, such as the focal point of strongly focused optical
tweezers. Low-index particles, reflecting particles and particles that absorb
light all tend to be repelled by bright light, and therefore are difficult to
manipulate with standard optical traps. Successful two-dimensional
manipulation of dark-seeking particles has been achieved in the dark regions
of interference patterns [1] and the dark core of optical vortices [2, 3].
Full three-dimensional control has been demonstrated with cages of light
created with rapidly scanned optical tweezers [4], and with optical bottles
created from superpositions of Bessel beams [5, 6, 7]. The bright cages that
define these traps all have dark gaps through which trapped particles can
escape. Proposals to close the gaps have focused on the properties of vector
beams of light with non-trivial polarization structure [8, 9]. These vector
traps, however, cannot be projected with standard holographic trapping
systems.

Here, we report a class of optical traps for dark-seeking particles that is
based on scalar diffraction theory and so is compatible with standard
holographic trapping techniques. The ideal hologram encoding these traps is
entirely real-valued, which poses a challenge for standard implementations
that rely on phase-only spatial light modulators. We therefore introduce a
method to transform amplitude-only holograms into phase-only holograms for
convenient projection at high diffraction efficiency. The resulting dark
tweezers can be combined with conventional bright tweezers in a standard
holographic trapping system to enable manipulation of heterogeneous colloidal
dispersions. We demonstrate the dark traps’ capabilities through experimental
studies on a model system composed of micrometer-scale silica spheres and
polystyrene spheres co-dispersed in an aqueous solution of dimethylsulfoxide
(DMSO). The silica spheres have a lower refractive index than this medium and
thus are dark-seeking while the polystyrene spheres have a higher index and
are light-seeking.

##  II Dark optical tweezers

Conventional optical tweezers are created by bringing a Gaussian laser beam to
a diffraction-limited focus with a high-numerical-aperture (NA) lens.
Analogous dark optical tweezers can be created by superposing two confocal
Gaussian beams with equal amplitudes, different waist diameters and a relative
phase of $\pi\text{\,}\mathrm{rad}$. The scalar amplitude profile of such a
superposition in the focal plane of the lens is

$u(\mathbf{r})=u_{0}\left[\exp\left(-\frac{r^{2}}{4\alpha^{2}}\right)-\exp\left(-\frac{r^{2}}{4\beta^{2}}\right)\right],$
(1)

where $\beta$ is the radius of the dark core and $\alpha$ is the radius of the
enclosing region of light. As in the case of conventional optical tweezers,
$\beta$ is constrained by the Abbe diffraction limit to $\beta\geq\lambda/2$
for light of wavelength $\lambda$ in the medium. The dark trap furthermore
requires $\alpha>\beta$, with the difference ideally exceeding $\lambda/2$.
Similar modes have been described previously [10] but do not appear to have
been used to create optical traps.

Figure 1: (a) Schematic representation of an integrated instrument for
holographic optical trapping at $1064\text{\,}\mathrm{nm}$ and holographic
video microscopy at $447\text{\,}\mathrm{nm}$. The sample consists of a
heterogeneous dispersion of low-index colloidal spheres, high-index spheres
and particles that strongly absorb the trapping light. (b) Zernike phase
hologram, $\phi_{Z}(\mathbf{r})$, encoding a difference-of-Gaussians dark
optical trap. (c) Measured in-plane intensity, $I(\mathbf{r})$, at the axial
position indicated by the dashed line in (d). The extent of the dark trap in
the $(x,y)$ plane can be tuned by selecting the widths of the component
Gaussian fields. (d) Reconstructed axial intensity profile, $I(\mathbf{r})$,
projected by the hologram in (b). This slice in the $xz$ plane features a dark
central focal volume completely surrounded by light.

Although the superposition described by Eq. (1) could be implemented with
conventional optical elements, it is more conveniently projected with a
holographic optical trapping system [11, 12] such as the example shown
schematically in Fig. 1(a). The standard implementation imprints a hologram on
the wavefronts of a conventional laser using a phase-only spatial light
modulator (SLM) and then relays the modified beam to an objective lens that
focuses it into a sample. The ideal hologram for the difference-of-Gaussians
trap described by Eq. (1) therefore may be computed as the Fourier transform
of the field in the focal plane [13],

$\displaystyle h(\mathbf{r})$ $\displaystyle=\frac{1}{u_{0}}\int
u(\mathbf{x})\,\exp\left(-i\frac{k}{f}\mathbf{r}\cdot\mathbf{x}\right)d^{2}x$
(2a)
$\displaystyle=\alpha^{2}\exp\bigg{(}-\frac{k^{2}}{f^{2}}\alpha^{2}r^{2}\bigg{)}-\beta^{2}\exp\bigg{(}-\frac{k^{2}}{f^{2}}\beta^{2}r^{2}\bigg{)},$
(2b)

where $f$ is the focal length of the lens and $k=2\pi n_{m}/\lambda$ is the
wave number of light in a medium of refractive index $n_{m}$. Unfortunately,
the hologram in Eq. (2b) is purely real-valued and so cannot be projected with
a standard phase-only SLM.

##  III Zernike holograms

A complex-valued hologram may be factored into real-valued amplitude and phase
profiles,

$h(\mathbf{r})=u(\mathbf{r})\,\exp(i\phi(\mathbf{r})),$ (3)

both of which can be imprinted onto the wavefronts of a laser beam using
advanced projection techniques [14, 15]. Most holographic trapping systems,
however, rely on phase-only diffractive optical elements that only modify the
phase profile. A common expedient is to ignore the amplitude profile by
setting $u(\mathbf{r})=1$, and to imprint only the phase profile,
$\phi(\mathbf{r})$, onto the laser beam’s wavefronts [16, 17, 18]. The
resulting phase-only hologram,

$H(\mathbf{r})=\exp\left(i\phi(\mathbf{r})\right),$ (4)

is a superposition of the ideal hologram, $h(\mathbf{r})$, with an error
field,

$\Delta h(\mathbf{r})=\left[1-u(\mathbf{r})\right]\,\exp(i\phi(\mathbf{r})).$
(5)

The effect of $\Delta h(\mathbf{r})$ on the projected optical trapping pattern
depends on the nature and symmetries of the ideal pattern [18].

The hologram that projects one optical trap can be combined with holograms for
other traps by superposing their fields in the hologram plane,

$h(\mathbf{r})=\sum_{n=1}^{N}\alpha_{n}h_{n}(\mathbf{r}),$ (6)

where $h_{n}(\mathbf{r})$ is the ideal hologram for the $n$-th trap and
$\alpha_{n}$ is a complex coefficient setting the relative amplitude and phase
of that trap. Traps can be translated by $\Delta\mathbf{r}$ in three
dimensions by adding a suitable parabolic profile [17],

$\phi_{\Delta\mathbf{r}}(\mathbf{r})=\left(-\frac{k}{f}\mathbf{r}+\frac{kr^{2}}{2f^{2}}\hat{z}\right)\cdot\Delta\mathbf{r},$
(7)

to the phase of the mode-forming hologram [17, 19].

As the complexity of a trapping pattern increases, the amplitude profile,
$u(\mathbf{r})$, develops increasingly rapid spatial variations. The error
term, $\Delta h(\mathbf{r})$, therefore tends to redirect light away from the
intended trapping pattern and outward toward the edges of the instrument’s
field of view. This means that the phase-only hologram, $H(\mathbf{r})$, can
project a near-ideal rendition of the intended trapping pattern within a
limited volume. In such cases, the error term principally reduces the
hologram’s diffraction efficiency into the desired mode by redirecting light
elsewhere.

While the simple phase-only conversion described by Eq. (4) is fast and
effective, more sophisticated algorithms [19] can refine a phase-only hologram
to improve diffraction efficiency and to mitigate artifacts due to overlap
between the intended and error patterns. These refinements typically are too
slow for real-time operation, however, and are reserved for the most exacting
applications.

Purely real-valued holograms pose a particular challenge to phase-only
projection. Sign changes in $h(\mathbf{r})$ can be absorbed into the phase
profile with Euler’s theorem, thereby ensuring that the amplitude,
$u(\mathbf{r})$, is non-negative [20]. Amplitude variations can be
approximated by multiplexing the desired hologram with another grating to
deflect light. These methods either discard most of the light in the field of
view or discard most information about the amplitude profile. They typically
create holograms with undesirably low diffraction efficiency.

We improve both the fidelity and the diffraction efficiency of both real- and
complex-valued holograms by transferring more information about the amplitude
profile into the phase profile using a variant of Zernike’s phase-contrast
approximation,

$u(\mathbf{r})\approx i\left[1-\exp(iu(\mathbf{r}))\right],$ (8)

under the assumption that $u(\mathbf{r})<1$. Using the phase profile to encode
amplitude information complements Zernike’s original purpose, which was to
explain how phase variations contribute to observable contrast in microscope
images. The phase profile implied by Eq. (8),

$\phi_{Z}(\mathbf{r})=\frac{1}{2}\,u(\mathbf{r}),$ (9a) serves as a phase-only
approximation to the real-valued amplitude profile, yielding the phase-only
Zernike approximation to the ideal complex-valued hologram,
$H_{Z}(\mathbf{r})=\exp\left(i\left[\phi_{Z}(\mathbf{r})+\phi(\mathbf{r})\right]\right).$
(9b) The associated error field, $\Delta
h_{Z}(\mathbf{r})\approx\left[1-\left(1-\frac{i}{2}\right)u(\mathbf{r})\right]e^{i\phi(\mathbf{r})},$
(9c) improves upon the standard result because its complex amplitude tends to
cancel ghost traps and other projection artifacts.

Figure 1(b) shows the phase hologram encoding a dark trap that is obtained by
applying Eq. (9) to the purely real-valued hologram for a DoG trap, Eq. (2b).
Although it superficially resembles a standard Fresnel lens, the pattern of
concentric phase rings has very different behavior.

Figure 1(c) and 1(d) show volumetric reconstructions [21] of the dark trap
projected by the hologram in Fig. 1(b). The trap is created by imprinting the
hologram on the wavefronts of a TEM00 laser beam at a vacuum wavelength of
$1064\text{\,}\mathrm{nm}$ (fiber laser, IPG Photonics, YLR-LP-SF) using a
liquid crystal spatial light modulator (Holoeye PLUTO). The modified beam is
relayed to the input pupil of an objective lens (Nikon Plan Apo, $100\times$,
numerical aperture 1.4, oil immersion) that focuses the light into the
intended optical trap with a focal length of
$f=$180\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The trapping beam is diverted
into the objective lens with a dichroic beamsplitter (Semrock) that has a
reflectivity of $99.5\text{\,}\mathrm{\char 37\relax}$ at the trapping
wavelength.

Images of the projected intensity pattern are obtained by mounting a front-
surface mirror in the focal plane of the objective lens [21]. The reflected
light is collected by the objective lens, and a small proportion passes
through the dichroic mirror. This transmitted light is collected with a
$200\text{\,}\mathrm{mm}$ tube lens and is recorded with a video camera (FliR
Flea3, monochrome) with an effective system magnification of
$0.048\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}$.
Transverse intensity slices, such as the example in Fig. 1(c), are obtained by
translating the trap in steps of $\Delta z=$48\text{\,}\mathrm{nm}$$ along the
axial direction using Eq. (7) [21]. A stack of slices then is combined to
obtain the axial section that is presented in Fig. 1(d). These images confirm
that the Zernike hologram for a DoG trap successfully projects a beam of light
that focuses to a dark volume surrounded by light on all sides.

##  IV Simultaneous manipulation of dark- and light-seeking particles

Figure 2: A low-index (silica) particle and a high-index (polystyrene)
particle are trapped simultaneously in a DoG trap and an optical tweezer,
respectively, and are transported around nominally circular paths in the
vertical $yz$ plane. Discrete points show the particles’ actual positions as
measured by holographic particle tracking applied to Supplementary Video 1,
and are colored by time. The gray-scale image is one hologram from that video.

Figure 2 presents holographically measured trajectories of high- and low-index
particles being translated simultaneously in a conventional optical tweezer
and a DoG trap, respectively. The figure also includes one frame from the
holographic video (Supplementary Video 1) that was used to measure the
particles’ trajectories. The particles are dispersed in an aqueous solution of
dimethylsufoxide (DMSO, Acros Organics, CAS Number 67-68-5, Lot Number
A0277547) with a measured refractive index of $n_{m}=$1.53$$. The low-index
particle is composed of silica (Bangs Laboratories, SS05N, Lot Number 4186)
with a radius of $a_{p}=$1.15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and a
nominal refractive index of $n_{\text{SiO}_{2}}=$1.43$$. The high-index
particle is composed of polystyrene (Thermo Fisher Scientific, 5100B Lot
Number 42393) with a radius of $a_{p}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and a nominal refractive index of $n_{\text{PS}}=$1.60$$. A mixed
dispersion of these spheres is contained in the volume created by sealing the
edges of a glass coverslip to the face of a glass microscope slide. This
chamber has an optical path length of
$H=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

In-line holograms of the particles are recorded by illuminating the sample
with the collimated beam from a diode laser (Coherent Cube) operating at a
vacuum wavelength of $\lambda=$447\text{\,}\mathrm{nm}$$. Light scattered by
the particles interferes with the rest of the beam in the focal plane of the
microscope. The intensity of the magnified interference pattern is recorded by
the video camera at
$30\text{\,}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{s}$.
The camera’s $10\text{\,}\mathrm{\SIUnitSymbolMicro s}$ exposure time is short
enough to avoid motion blurring [22, 23]. Each holographic snapshot can be
analyzed with predictions of Lorenz-Mie theory to measure each particle’s
diameter, refractive index and three-dimensional position relative to the
center of the microscope’s focal plane [24]. We use these capabilities to
differentiate the two types of particles on the basis of their refractive
index and to track their three-dimensional motions as they are manipulated in
holographic optical traps.

The discrete plot symbols in Fig. 2 represent the measured three-dimensional
trajectories of a (low-index) silica sphere and a (high-index) polystyrene
sphere as they are transported in opposite directions around nominally
circular trajectories in the $yz$ plane. The particles follow the programmed
trap trajectories at $5.0\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$ given an estimated laser power of
$100\text{\,}\mathrm{mW}$ for the conventional optical trap and
$50\text{\,}\mathrm{mW}$ for the DoG trap.

Although the two traps were programmed to move in circles of equal radius, the
recorded paths are elliptical. The two particles move over equal horizontal
ranges, but the polystyrene particle moves further than expected in the axial
direction and the silica particle moves less far. Similar axial deviations
have been reported for particles in conventional optical tweezers [25] and can
be ascribed to differences in the particles’ buoyant densities and to changes
in the traps’ Rayleigh ranges as they are displaced along the optical axis.

##  V Assessing performance

Thermally-driven fluctuations in the particles’ positions can be used to
measure the stiffness of the traps, and therefore their trapping efficiency.
Treating the sphere’s thermal fluctuations within the trap as an Ornstein-
Uhlenbeck process, an $N$-step discretely sampled trajectory, $\\{x_{n}\\}$,
along $\hat{x}$ yields an estimate for the associated stiffness [19],

$\kappa_{x}\pm\Delta\kappa_{x}=\frac{k_{B}T}{c_{0}}\pm\frac{k_{B}T}{c_{0}}\sqrt{\frac{2}{N}\left(1+\frac{2c_{1}^{2}}{c_{0}^{2}-c_{1}^{2}}\right)},$
(10a) where $k_{B}T$ is the thermal energy scale at absolute temperature $T$
and $c_{j}=\frac{1}{N}\sum_{n=1}^{N}x_{n}x_{(n+j)\bmod N}$ (10b)

is the autocorrelation of $\\{x_{n}\\}$ at lag $j$. Similar estimates can be
computed for the trap stiffness along the other Cartesian coordinates.
Equation (10) is suitable for interpreting holographically measured single-
particle trajectories because the measurement error,
$\sigma_{x}=$5\text{\,}\mathrm{nm}$$, is smaller than the typical scale of
thermal fluctuations, $\Delta
x=\sqrt{2(c_{0}-c_{1})}\geq$70\text{\,}\mathrm{nm}$$, over the range of laser
powers considered [19]. A $2000$-frame holographic video recorded over one
minute provides enough information to measure $\kappa_{x}$ to within
$1\text{\,}\mathrm{\char 37\relax}$.

Figure 3: The performance of a DoG trap is assessed by measuring its
stiffness, $\kappa_{i}$, along the $i$-th Cartesian axis as a function of
laser power, $P$, for low-index silica spheres of three different radii,
$a_{p}$. (a) Transverse stiffness as a function of laser power. Dashed lines
are least-squares fits for the anticipated linear dependence on laser power.
(b) Axial stiffness. (c) and (d) Transverse and axial trapping efficiency,
$Q_{\perp}(a_{p})=\partial_{P}(\kappa_{x}+\kappa_{y})/2$ and $Q_{z}$. The blue
curve in (c) is a comparison with predictions of Eq. (11). (e) Predicted
transverse potentials (dashed blue curves) compared with the parabolic
potential inferred from measurements of trap stiffness. (f) Comparison of the
measured intensity profile for the DoG trap used in this study (filled curve
with black markers) and the planned difference-of-Gaussians profile encoded in
the projected hologram.

Figure 3(a) shows how the measured stiffness of a DoG trap depends on laser
power and particle diameter for low-index silica spheres. The trap used for
these measurements has $\alpha=$1.92\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
and $\beta=\alpha/2$. The power, $P$, projected into the trap is measured with
a slide-mounted thermal power sensor (Thorlabs S175C). As might reasonably be
expected, the trap stiffness increases linearly with $P$ in all three
coordinates for each of the three particle sizes. Values for the transverse
stiffness plotted in Fig. 3(a) reveal no anisotropy, and thus no dependence on
the trap’s polarization, which is directed along $\hat{x}$. The axial
stiffness, plotted in Fig. 3(b), also increases linearly with $P$ over the
range studied and, not surprisingly, differs significantly from the transverse
stiffness. Fig. 3(c) and Fig. 3(d) show respectively how the transverse and
axial trapping efficiencies, $Q_{\perp}=\partial_{P}(\kappa_{x}+\kappa_{y})/2$
and $Q_{z}=\partial_{P}\kappa_{z}$, depend on particle radius, $a_{p}$.

Interestingly, the measured transverse trapping efficiency, $Q_{\perp}$,
depends nonmonotonically on particle size. We account for this observation
with a simplified model of the particles’ interaction with the DoG trap’s
intensity profile. The particle’s potential energy at distance $r$ from the
center of the dark well depends on its overlap with the light, which we model
as $\displaystyle U(r)$
$\displaystyle=A\,\int_{0}^{2\pi}d\theta\,\int_{0}^{a_{p}}dx\,x\,\zeta(x)\,u^{2}(\mathbf{x}-\mathbf{r})$
(11a) $\displaystyle=4\pi
a_{p}^{3}u_{0}^{2}\,A\left[g\left(r,\frac{1}{\alpha^{2}}\right)+g\left(r,\frac{1}{\beta^{2}}\right)-2g\left(r,\frac{1}{2\alpha^{2}}+\frac{1}{2\beta^{2}}\right)\right],$
(11b) where $\zeta(x)=2\sqrt{a_{p}^{2}-x^{2}}$ is the chord length a distance
$x$ from the center of a sphere of radius $a_{p}$, and where we have defined
$g\left(r,\frac{1}{\alpha^{2}}\right)=e^{\frac{r^{2}}{4\alpha^{2}}}\int_{0}^{1}y\sqrt{1-y^{2}}\,I_{0}\left(\frac{ra_{p}}{2\alpha^{2}}\,y\right)\,e^{-\frac{a_{p}^{2}}{4\alpha^{2}}\,y^{2}}\,dy.$
(11c) The integrand in Eq. (11c) includes the modified Bessel function,
$I_{0}(\cdot)$, and is most readily computed numerically. We treat the overall
scale of the trapping potential, $A$, as an adjustable parameter. Typical
potential energy curves are plotted in Fig. 3(e).

The trapping efficiency is obtained from Eq. (11) by fitting the potential
minimum to a parabola using the planned intensity distribution,
$u^{2}(\mathbf{r})$, and particle radius, $a_{p}$, as inputs. Planned and
measured intensity distributions are compared in Fig. 3(f) for the trap used
for Fig. 3. The dependence of the in-plane trapping efficiency on particle
radius is plotted as a blue dashed curve in Fig. 3(c), using only $A$ as an
adjustable parameter. The model predicts that the DoG trap is optimally stiff
when trapping low-index dielectric particles that are slight larger than
trap’s dark core, in agreement with experimental observations.

##  VI Manipulating absorbing particles

In addition to low-index particles and high-index particles, the DoG trap can
manipulate particles that absorb light strongly. Such particles are repelled
by conventional optical tweezers through transfer of momentum from absorbed
light. Absorption also mediates heating, which can propel the particles
through self-thermophoresis [26, 27], and in extreme cases can destroy the
particles or boil the fluid medium [28, 29].

We have demonstrated the ability of DoG traps to stably trap and transport
absorbing particles through experiments on composite particles made of
hematite cubes embedded in dielectric spheres and superparamagetic particles
composed of hematite nanoparticles dispersed within dielectric spheres. The
dielectric spheres used for these demonstrations are created by emulsion
polymerization of 3-methacryloxypropyl trimethoxysilane (TPM) [30], an
organosilicate with a refractive index of $n_{p}=$1.495$$ [31]. Hematite
absorbs infrared light strongly, and both types of hematite-loaded particles
tend to be ejected from conventional optical tweezers. Both types of particles
are readily trapped and transported in three dimensions with DoG traps.
Supplementary Video 2 shows the simultaneous three-dimensional control over a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$-diameter polystyrene particle and a
$1.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ TPM particle enclosing a
$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ hematite cube.

##  VII Discussion

This study introduces a class of “dark” optical traps that are created by
superposing two Gaussian modes of different waist diameters and opposite
phases. The hologram encoding these Difference-of-Gaussians (DoG) traps is
purely real-valued. We therefore introduce a technique based on the Zernike
phase-contrast approximation to project real-valued holograms with phase-only
diffractive optical elements, such as the spatial light modulators commonly
used in holographic optical trapping systems. We have demonstrated DoG traps’
ability to trap dark-seeking particles and to move them in three dimensions.
We further have characterized the efficiency of DoG traps for localizing low-
index particles and have explained the observed nonmonotonic dependence of the
transverse trapping efficiency on particle size. When a DoG trap is optimally
matched to the trapped particle, its trapping efficiency is comparable to that
of conventional optical tweezers for light-seeking particles.

## Funding

This work was supported by the National Science Foundation through Award No.
DMR-2104836. The integrated instrument for holographic trapping and
holographic microscopy used in this study was constructed with support of the
MRI program of the NSF under Award Number DMR-0922680 and is maintained as
shared instrumentation by the Center for Soft Matter Research at NYU. The
authors gratefully acknowledge the support of the nVidia Corporation through
the donation of the Titan Xp and Titan RTX GPUs used for this research.

## Acknowledgments

We are grateful to Shahrzad Zare, who contributed to the initial stages of
this work.

## Data Availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

The open-source software used to project holographic optical traps, record in-
line holographic microscopy data and analyze those data is available online at
https://github.com/davidgrier/.

## References

  * [1] M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863–865 (2001).
  * [2] K. T. Gahagan and G. A. Swartzlander, Jr, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
  * [3] K. T. Gahagan and G. A. Swartzlander, “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” JOSA B 16, 533–537 (1999).
  * [4] K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
  * [5] J. Arlt and M. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
  * [6] D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Three-dimensional arrays of optical bottle beams,” Optics Communications 225, 215–222 (2003).
  * [7] Z. Liu, X. Tang, Y. Zhang, Y. Zhang, L. Ma, M. Zhang, X. Yang, J. Zhang, J. Yang, and L. Yuan, “Simultaneous trapping of low-index and high-index microparticles using a single optical fiber bessel beam,” Opt. Lasers Eng. 131, 106119 (2020).
  * [8] N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007).
  * [9] X.-L. Wang, J. Ding, J.-Q. Qin, J. Chen, Y.-X. Fan, and H.-T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
  * [10] D. Yelin, B. E. Bouma, and G. J. Tearney, “Generating an adjustable three-dimensional dark focus,” Opt. Lett. 29, 661–663 (2004).
  * [11] E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instr. 69, 1974–1977 (1998).
  * [12] D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
  * [13] J. W. Goodman, _Introduction to Fourier Optics_ (Roberts & Company, 2005), 3rd ed.
  * [14] A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16, 4479–4486 (2008).
  * [15] A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Near-perfect hologram reconstruction with a spatial light modulator,” Opt. Express 16, 2597–2603 (2008).
  * [16] J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun.s 185, 77–82 (2000).
  * [17] J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
  * [18] J. E. Curtis, C. H. Schmitz, and J. P. Spatz, “Symmetry dependence of holograms for optical trapping,” Opt. Lett. 30, 2086–2088 (2005).
  * [19] M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
  * [20] Y. Roichman and D. G. Grier, “Projecting extended optical traps with shape-phase holography,” Opt. Lett. 31, 1675–1677 (2006).
  * [21] Y. Roichman, I. Cholis, and D. G. Grier, “Volumetric imaging of holographic optical traps,” Opt. Express 14, 10907–10912 (2006).
  * [22] F. C. Cheong, B. S. R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express 17, 13071–13079 (2009).
  * [23] L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express 19, 4393–4398 (2011).
  * [24] S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. Van Blaaderen, P. Van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
  * [25] M. J. O’Brien and D. G. Grier, “Above and beyond: holographic tracking of axial displacements in holographic optical tweezers,” Opt. Express 27, 25375–25383 (2019).
  * [26] V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Giant optical manipulation,” Phys. Rev. Lett. 105, 118103 (2010).
  * [27] H. Moyses, J. Palacci, S. Sacanna, and D. G. Grier, “Trochoidal trajectories of self-propelled janus particles in a diverging laser beam,” Soft Matter 12, 6357–6364 (2016).
  * [28] H. Rubinsztein-Dunlop, T. Nieminen, M. Friese, and N. Heckenberg, “Optical trapping of absorbing particles,” Adv. Quantum Chem. 30, 469–492 (1998).
  * [29] E. J. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84, 1308–1316 (2003).
  * [30] C. Van Der Wel, R. K. Bhan, R. W. Verweij, H. C. Frijters, Z. Gong, A. D. Hollingsworth, S. Sacanna, and D. J. Kraft, “Preparation of colloidal organosilica spheres through spontaneous emulsification,” Langmuir 33, 8174–8180 (2017).
  * [31] C. Middleton, M. D. Hannel, A. D. Hollingsworth, D. J. Pine, and D. G. Grier, “Optimizing the synthesis of monodisperse colloidal spheres using holographic particle characterization,” Langmuir 35, 6602–6609 (2019).