# Procedural generation of meta-reinforcement learning tasks

Thomas Miconi
ML Collective
<EMAIL_ADDRESS>

###### Abstract

Open-endedness stands to benefit from the ability to generate an infinite
variety of diverse, challenging environments. One particularly interesting
type of challenge is meta-learning (“learning-to-learn”), a hallmark of
intelligent behavior. However, the number of meta-learning environments in the
literature is limited. Here we describe a parametrized space for simple meta-
reinforcement learning (meta-RL) tasks with arbitrary stimuli. The
parametrization allows us to randomly generate an arbitrary number of novel
simple meta-learning tasks. The parametrization is expressive enough to
include many well-known meta-RL tasks, such as bandit problems, the Harlow
task, T-mazes, the Daw two-step task and others. Simple extensions allow it to
capture tasks based on two-dimensional topological spaces, such as full mazes
or find-the-spot domains. We describe a number of randomly generated meta-RL
domains of varying complexity and discuss potential issues arising from random
generation.

##  1 Introduction

A potential key step for open-endedness is the ability to generate a wide
variety of diverse, challenging environments. There is of course much
ambiguity in what counts as “‘challenging”. One possible, reasonably objective
criterion would be that each new episode of this environment requires the
learning and exploitation of new, unpredictable information. For example, if
the environment is “mazes”, each new maze requires learning anew the reward
location and/or maze structure; if the environment is “bandit tasks’, each new
instance requires learning and exploiting a new probability distribution over
returns, etc. Successfully mastering such an environment would then imply not
just learning a fixed set of rules or values, but _learning how to learn_ (and
exploit) the appropriate information, quickly and efficiently, for each new
instance of the domain. This ability, once introduced as “the acquisition of
learning sets” Harlow, (1949), is now commonly called “meta-learning”.

Meta-learning (“learning to learn”) is an important aspect of cognition
(Harlow,, 1949; Hattori et al.,, 2023) and has elicited much interest in
machine learning (Thrun and Pratt,, 1998; Schmidhuber,, 1993; Hochreiter et
al.,, 2001; Finn et al.,, 2017; Wang et al.,, 2016; Duan et al.,, 2016; Bengio
et al.,, 1991; Floreano and Urzelai,, 2000; Ruppin,, 2002; Soltoggio et al.,,
2008; Miconi,, 2016; Miconi et al.,, 2018). Recently, meta-reinforcement
learning (meta-RL) has been a particularly active area of research (Wang et
al.,, 2016, 2018; Duan et al.,, 2016; Miconi et al.,, 2018; Laskin et al.,,
2022). Successful Meta-RL agents must develop an in-built RL algorithm
specifically adapted to the environment, allowing for fast and efficient
acquisition of each new individual task from the domain. Recent research has
often focused on increasing the _complexity_ of meta-RL environments, making
use of combinatorial rule spaces, physical grounding, long horizons, etc. Wang
et al., (2021); Team et al., (2023); or alternatively, making the meta-domain
adversarial by biasing it towards individual tasks difficult for the agent
Jackson et al., (2023).

On the other hand the total _number_ of meta-learning domains in the
literature has remained relatively limited. We note that certain potential
applications (such as robustly evaluating meta-learning algorithms, or perhaps
even training them in a meta-meta-learning process) require a large number of
different meta-learning tasks. More fundamentally, the limited number of known
meta-learning tasks reduces the potential of meta-learning environments for
open-endedness.

One way to greatly expand the number of available meta-tasks is random
procedural generation. Randomly generating meta-tasks requires a _parametrized
space_ of meta-tasks, which can be explored by varying the values of the
parameters. Here we describe one possible parametrization for meta-RL tasks,
which is both simple and expressive enough to capture commonly used meta-RL
tasks from the literature.

We emphasize that we are not proposing a new meta-RL domain, from which one
can generate complex individual RL tasks (as in Wang et al., (2021); Team et
al., (2023)); rather, we are proposing a method for generating an arbitrary
number of simple, but full-fledged _meta_ -RL tasks. The fact that the
generated domains are truly meta-RL tasks is demonstrated by construction, by
showing that the system can generate candidate domains that are commonly used
in the literature and universally regarded as true meta-tasks (such as the
Harlow meta-task, bandit problems, simple mazes, “dark room”, etc.)

All code for this paper is available at github.com/ThomasMiconi/Meta-Task-
Generator

##  2 A parametrized space of meta-learning tasks

Figure 1: Conventions for the graphical description of meta-learning tasks.
Each meta-task is specified as a Partially Observable Markov Decision Process
(POMDP). Each rounded square represents a state of the POMDP. State ID is not
directly accessible to the agent; only the observation/stimulus for this state
(indicated on the left side of the square), which may be null, is provided.
Possible actions are indicated on the right side of the square. Arrows
indicate future possible state for each action; bifurcated arrows indicate
random choice between two possible next states, with equal probabilities
unless stated otherwise. Reward is delivered to the agent whenever one of the
reward conditions is met; if multiple conditions are met, the applicable
reward condition furthest down the list is used. Elements in red are
variables, meaning that they are randomly resampled for each new task
generated from the meta-specification.

###  2.1 Meta-RL tasks as POMDPs with variable quantities

We describe individual RL tasks as partially-observable Markov decision
processes (POMDP). In POMDPs, unlike standard Markov decision processes, the
agent is not explicitly informed of which state it is in. Instead, each state
may provide a certain stimulus (“observation”) to the agent, which in general
will not uniquely identify the state, but can be used (in addition to other
context, past or present) to infer the state.

Accordingly, in our parametrization, an individual RL task is described as a
set of states $S$, each of which may provide a certain stimulus (observation)
$o_{s}$ to the agent. In each state $s\in S$, a certain set of actions
$a_{i}\in A$ are available. Each action causes a transition to a next state
$s^{\prime}=s_{t+1}\in S$ with a certain probability determined by the
transition function $T(s_{t},a_{t})$. In addition, each action may provide a
reward, as determined by the (possibly probabilistic) reward function
$r(s_{t},a_{t},s_{t+1})$.

In our formulation, the reward function is described as a set of rules of the
following form: “if starting from state $s$, taking action $a$, and ending up
in new state $s^{\prime}$, receive reward $r$ with probability $p$”.
Importantly, $s$, $a$ and $s^{\prime}$ may be left unspecified (“don’t care”),
allowing the rule to be triggered for any value of the unspecified variables
(though at least one of them must be specified for each rule).

Furthermore, the rules for a given meta-task are provided in an ordered
sequence, such that later rules in the sequence may override earlier one. For
example, if a rule early in the sequence specifies that taking any action in
state 2 results in reward 1 with probability 1, while a later rule specifies
that taking action 0 in state 2 results in reward 2 with probability 0.5, the
former rule will be applied if the agent (in state 2) takes any action other
than 0, but if it takes action 0 then the latter rule will apply.

A POMDP with fixed rules, as described above, describes a single reinforcement
learning task. To turn this into a meta-learning task specification, we must
allow various quantities in the POMDP to be specified by _variables_. These
variables are then assigned randomly resampled values (from a pre-specified
domain) for each new individual task. Variable quantities may include the
probabilities for each reward rule, the state or action in which it applies,
and the stimulus (observation) provided at any given state (other quantities,
such as transition probabilities, might be specified as variables, but this
will not be explored here). Only some quantities will be variable across
individual tasks, while others will be identical across all tasks and thus be
part of the meta-task specification. Importantly, a given variable may be used
multiple times in a specification, and each occurrence of the variable is
assigned the same value for each new task, allowing for coherent variation
across task instances.

###  2.2 Basic Examples

Figure 2: A simple two-arm bandit task. Probabilities of reward for each arm
($p_{0}$ and $p_{0}$) are randomly resampled for each new task instance, as
indicated by red coloring.

#### Conventions

In all examples in this paper, the reward value for any triggered rule is
always 1, with probability 1 unless specified otherwise. There are two
possible actions (“0” and “1”) for each state. Reward rules do not make use of
the post-action state $s^{\prime}$. Also, state 0 is always the starting state
for each new episode, and episodes are ended after a fixed number of time
steps (removing the need to specify a termination state). This convention
simplifies descriptions while still allowing many meta-tasks to be described.

For graphical representations (Figure 1), each rounded square represents a
state of the POMDP. Each bifurcating arrow indicates a random choice between
two possible outcomes (next states), with equal probability unless stated
otherwise. Furthermore, if only one arrow emerges from a given state (as
opposed to one separate arrow for each action), it means the next-state
transition probabilities for both actions are identical; however, the reward
for both actions may still be different. See Figure 1 for a summary. Variables
are indicated in red, while black lettering indicates values that are fixed
across all individual tasks.

#### Basic two-arm bandit

A very simple example of a meta-task specification is the simple two-arm
bandit problem, as shown in Figure 2. There is only one state, with no
stimulus (observation is empty), and both actions loop back to the same state.
The reward probabilities depend on which action is taken, and are randomly
resampled (from a uniform distribution over $(0,1)$) for each new task.

Figure 3: Harlow task with sequential, random-order object presentation. State
1 (resp. 2) is the presentation of the rewarded (resp. unrewarded) stimulus.
Action 0 means “select”, while action 1 means “ignore”. See text for details.

#### Harlow task with sequential object presentation

A slightly more elaborate example is the Harlow meta-task Harlow, (1949) with
sequential object presentation, as described by Wang et al., (2016) (end of
Section 3.2.2) and Goudar et al., (2023). In each episode of this task, two
objects (the “rewarded” object and the “non-rewarded” object) are repeatedly
presented in random order. The agent receives a reward for picking the
rewarded object, and for ignoring the non-rewarded object, when either is
presented. Importantly, two completely different objects are used for each new
episode (that is, for each new instance of the meta-task), and each time the
agent must find out which of them is the rewarded one from rewards alone.

This task is described in Figure 3. Actions 0 and 1 correspond to “select” and
“ignore”, respectively. State 1 (resp. 2) represents the presentation of the
rewarded (resp. unrewarded) object. The appearance of the object is
represented by the stimuli (observations) S1 and S2 associated with each
state; as indicated by the red coloring, these are variables, and thus are
randomly regenerated for each new individual task, representing the two novel
objects of each episode.

###  2.3 Special states

Figure 4: The Daw two-step task, in the version described by Wang et al.,
(2016).

Some meta-tasks may require that one specific state be in some sense
“special”, in a way that affects various aspects of the environment
consistently. Which state is to be the “special” one varies from one
individual task to the next. For example, in a typical maze meta-task, the
reward location changes across instances of the task (i.e. for each individual
maze), affecting the reward function. We model this by allowing a meta-task to
declare one or more “special state” variables, that can be used in the reward
rules instead of definite state numbers. Then, when generating each new
individual task, we randomly assign a specific state number to each special-
state variable, and replace each occurrence of this special-state variable in
the rules with its assigned actual state number. Importantly, each occurrence
of a given state variable is replaced by the same state number, and different
state variables are replaced with different states, which allows for
coordinated changes across the system.

As an example, consider Figure 4, which describes the Daw two-step task (in
the version implemented by Wang et al., (2016)). In the initial state, the
agent must choose one of two actions, each of which can lead to either of two
second-step states, with opposite probabilities (action 0 is much more likely
to lead to state 1, while action 1 is much more likely to lead to state 2 -
but both have nonzero probability to lead to both state 1 and 2); these
probabilities are fixed for the meta-task across all task instances. Both
second-step states probabilistically provide reward, but one provides much
more reward than the other. Which of the two states is more rewarded varies
randomly from one task to the next and must be inferred by integrating
actions, stimuli and rewards.

Importantly, a competent learner should take task structure into account: in
some cases, receiving a reward after a certain choice should actually _reduce_
the probability of taking this particular choice in the future, if the choice
was followed by a rare transition to the less frequent next state for this
choice (since it suggests that this less frequent state is likely to provide
reward, and is more reliably accessed by taking the other choice).

This meta-task is represented very simply with special-state variables. The
reward function specifies that either of the two second-step states (1 and 2)
provides reward with some low probability, and then the special state variable
$S$ (which is randomly resampled from the set ${1,2}$ for each new task
instance) indicates which second-step state provides reward with high
probability. Since latter rules override earlier ones, the latter will apply
whenever the agent is in the corresponding state.

###  2.4 Statefulness: flag variables

In POMDPs, statefulness is encoded by the current state (though it may not be
directly perceptible to the agent). However, in addition, it may be convenient
to allow for additional statefulness, in which some trace of previous agent
actions may be stored in readily accessible variables separately from the
actual POMDP state. For example, in a key-door task (e.g. (Laskin et al.,,
2022)), the agent receives reward on reaching the “door” location, but only if
it has picked the “key” beforehand. Such a task can be implemented in a strict
POMDP by by duplicating each state according to whether or not the key has
been picked. However, it is much simpler to explicitly store whether the agent
has picked the key in an accessible variable, on which transition and reward
functions may be conditioned.

We implement a simple type of statefulness in the form of “flag” variables.
The meta-task may specify one or more “flag” variables, which are set to 0 at
initialization, and can be set to 1 under certain conditions. These conditions
are specified as rules conditioned on $s_{t}$ and $a_{t}$, much like rewards.
Furthermore, rewards themselves can now be conditioned on the current value of
flag variables: reward rules include a “flag” field which specifies a
necessary value of the flag for the rule to apply (possibly including “don’t
care”).

As an example, consider the simple T-maze meta-task described in Figure 5. In
the first state, one of two possible stimuli is shown, each of which encodes a
direction (“left” or “right”) to be followed at the end of the maze. The agent
must then go through a number of intermediate states (actions 0 and 1 encoding
“stay put” and “forward”, respectively - optimal behavior is to proceed
forward without delaying). The agent then enters the “T-junction” state, at
which point actions 0 and 1 encode “left” and “right” respectively, and must
choose the one instructed by the initial stimulus. Both actions transition
back to the initial state for a new trial. The content of the “left” and
“right” stimuli are randomly resampled for each new instance of the task.

Figure 5: A simple T-maze task. Either stimulus instructs a different choice
in the final state, and sets the flag variable accordingly. Reward delivery
after the final state depends on action taken and current flag value.

Without flag variables, describing this task requires duplicating each state
depending on which stimulus was shown in state 0. By contrast, with flag
variables, the right-coding stimulus sets the flag to 1. The flag is then used
at the T-junction state, together with which action was taken, to determine
reward for this trial. This considerably simplifies the description of the
meta-task. In turn, this simplification implies that a much larger set of
meta-tasks can be covered, at the cost of minimal additional information.

Optionally, the meta-task may also specify that all flags should be reset to 0
every time the agent re-enters the initial state. This introduces a multi-
trial structure to the episode, with each “trial” starting every time the
system re-enters the initial state, with flags reset at 0.

###  2.5 Implementing novel meta-tasks

As a further illustration, we show that the framework can describe novel meta-
RL tasks not previously reported in the meta-learning literature, though
reminiscent of animal training experiments. We describe a stay/switch bandit
task (in which the agent must decide to stay with the current arm or switch to
a different arm), and a familiarity detection task. See Section A for details.

##  3 Random generation of novel meta-RL tasks

(a) First

(b) Second

Figure 6: First and second randomly generated meta-task.

We now describe several meta-RL tasks randomly generated from the space
described above. The meta-tasks in this section were selected for potential
interest and are shown just as they were generated, without any modification.
All code is available at github.com/ThomasMiconi/Meta-Task-Generator.

#### First meta-task

In Figure 6(a) we show a simple meta-task with three states, including initial
state 0. In state 0, action 0 always lead to state 1, while action 1 randomly
leads to any of the three states. From state 2, the only possible transition
is to state 1. The relevant choice in the task is in state 1, where actions 0
and 1 both lead to either state 0 or 2, but with opposite probabilities.
Furthermore, from the list of reward rules, we see that state 1 is always
rewarded, and in addition the special state (which can be either 0 or 2 for
any given task instance) is always rewarded (note that the last rule is
redundant given the second rule). State 0 and 1 are given different stimuli
(which are fixed across all instances), while state 2 provides no stimulus;
thus all states are identifiable in zero-shot given enough meta-training.

The optimal strategy depends on whether the special state for the current task
is 0 or 2. In state 0, the best strategy is always to take action 0. If the
special (rewarded) state is 0, the optimal strategy in state 1 is to take
action 1, which maximizes occupancy of rewarded states 0 and 1. If the special
state is 2, the optimal strategy in state 1 is to take action 0, which
maximizes occupancy of rewarded states 0 and 2. Since the special state is not
known to the agent a priori, which of these conditions hold must be determined
by exploration for each new task. Note that in state 2, the action taken is
irrelevant. Thus, the task essentially amounts to a bandit task on the two
actions of state 1.

#### Second meta-task

Figure 6(b) shows a different randomly generated meta-task. State 1 is always
rewarded, while state 0 may be rewarded if it is the special state (as
randomly sampled for each new instance of the meta-task). Furthermore, the
reward probabilities are also randomly resampled for each new task. If the
special state is 1 (meaning that only state 1 provide reward), then the
optimal policy is to get to state 1 and stay there (by action 1). If the
special state is 0, and if the reward for state 0 is much higher than that for
1, the optimal policy is to simply stay in state 0 by always invoking action 0
in state 0. Again, to assess which of these two conditions hold, the agent
must explore the environment and adequately integrate received rewards. It
must then decide which state to come back to (0 or 1), successfully navigate
to this state and stay there. Building these exploration and exploitation
strategies is a non-trivial challenge for the outer loop of the meta-learning
process.

#### Third meta-task: spontaneous key-door task

Finally, we show a slightly more elaborate meta-task that involves both flags
and special states. Coincidentally, this randomly generated meta-task
implements a crude “key-door” environment. The agent collects reward any time
it enters state 2 (“door”), but only if the flag is set to 1 (the “key” has
been picked). The flag is set to 1 when the agent enters a special (“key”)
state, which can be any of the non-initial states - including state 2 itself.
Note that, coincidentally, the transition matrix indicates that after entering
state 2 the agent “teleports” to a random state other than 2, reminiscent of
maze tasks where the agent is teleported on reaching the reward location
(Miconi et al.,, 2018). Two stimuli are variable, but each state is clearly
identifiable for a trained agent (notice that variable state 0 is always the
first one seen for each episode).

Figure 7: Third randomly generated meta-task. Note that this meta-task
coincidentally implements a simple type of “key-door” task, with the “key” in
special variable state $S_{1}$ (randomly assigned as state 1, 2 or 3 for each
new task) and the “door” in state 2.

##  4 Extension: tasks with topological structure

In theory, the above framework can include tasks with two-dimensional (2D)
topological environments, such as mazes, Dark Room (“find-the-spot”), key-
door, etc. However, such task possess considerable regularities that would be
unlikely to occur by chance in random generation. Fortunately, this structure
can easily be incorporated in the framework as a potential extension.

To accommodate 2D structure, the states should be conceptually arranged in a
2D grid. Each state must possess enough actions to represent all possible
movements within a neighborhood (e.g. 4 actions for up, down, left and right).
Furthermore, the transition probabilities should reflect the topology,
connecting neighboring states through the appropriate actions. Finally, in
addition to whatever stimuli / observations the task requires, each state $s$
may provide its $x$ and $y$ coordinates in the overall grid as part of its
associated observation $o_{s}$.

In Figure 8, we represent the simple “Dark Room” experiment described in
Laskin et al., (2022). The goal of the task is simply to find the rewarded
spot in a two-dimensional environment, with no other stimulus than the current
$(x,y)$ position (it may be seen as a discrete variant of the Morris maze). We
define the rewarded spot as a special state, to be randomly chosen (from the
entire set of states) for each new task instance. Again, the framework can
easily be extended to allow special states to influence the transition matrix
in addition to rewards, though we do not explore this further in this paper.

Figure 8: The Dark Room task described in Laskin et al., (2022), as a simple
example of tasks with 2D topological structure. One location, randomly
resampled for each new task instance, provides reward. Each state provides its
X and Y coordinates as observation to the agent.

##  5 Machine-readable specifications of meta-tasks

The formal space allows for the description of meta-tasks in machine-readable
format. In Appendix B, we provide an example of a machine-readable (JSON)
specification for the third randomly generated meta-task above (see Figure
S1). This automatically-generated JSON description fully specifies the meta-
task, allowing for automated sampling of individual task from the
specification.

##  6 Potential issues with random generation of meta-RL tasks

The space described above includes meta-tasks that are not all of equal
interest. For example, if no variable element is included, then obviously all
tasks from the specification will be identical. Even if some variable elements
are specified, the resulting set of tasks may turn out to be equivalent, in
the sense that any given policy would obtain the exact same returns on all
tasks from the specification.

A less obvious risk is that the individual tasks would not be equivalent, but
_iso-optimal_ \- that is, some meta-tasks might happen to have the same
_optimal_ policy for all individual task instances. This may occur even if the
actual returns of this single optimal policy are different across task
instances, as long as it is still the optimal policy for each new task
instance. These single-optimum meta-tasks are clearly of reduced interest,
because they remove the need for learning-to-learn : once the optimal strategy
is discovered, any additional task instances can be solved by the exact same
policy without any further in-task learning.

Fortunately, this risk can be addressed by filtering. Because the individual
tasks are simple POMDPs, they can be solved exactly in reasonable time by
known algorithms. Therefore, a filtering process for iso-optimal meta-tasks
would be to sample $N$ individual tasks from the meta-task specification,
solve them (producing an optimal policy $\mathbf{\pi}_{n}$ for each task $n$),
and then test the performance $R_{k}(\mathbf{\pi}_{j\neq k})$ of each optimal
policy $j\neq k$ on each task $k$. If $R_{k}(\mathbf{\pi}_{j\neq
k})=R_{k}(\mathbf{\pi}_{k})$ for all tasks $k,j\neq k$, then the optimal
policy for any task $j\neq k$ is also an optimal policy for each task $k$, and
the meta-task can be flagged as having a single optimal strategy.

##  7 Conclusion

We have introduced a formal parametrization for meta-RL tasks, which can be
used to randomly generate a wide variety of meta-tasks. Although the
specification is conceptually and computationally simple (POMDPs with variable
quantities), it is expressive enough to cover commonly used meta-tasks from
the literature, such as the Harlow (meta-)task, bandit problems, the Daw
2-step meta-task, simple mazes etc. Importantly, this ability confirms that
the problems generated by sampling from this parametrization are truly meta-RL
tasks, rather than just basic RL tasks sampled from a single, complex meta-RL
domain (as in Wang et al., (2021); Team et al., (2023)).

The ability to generate arbitrary numbers of different meta-RL tasks (provided
that care is taken in ensuring this difference, see Section 6) has various
applications. One possible application could be to robustly evaluate and
compare various meta-learning algorithms. Another would be to train systems
capable of autonomous meta-learning, that is, self-contained systems that can
learn to learn when faced with a novel domain never seen before in training or
evolution, much as humans and animal subjects do in behavioral experiments
Harlow, (1949); Morcos and Harvey, (2016).

We note that the specification offers multiple parameters that directly
influence the complexity of the resulting meta-task, such as the number of
states, the number of variable quantities, the dimensionality of the stimuli,
etc. As such, it seems particularly amenable to automatic curricula,
generating tasks of variable and controllable complexity to accompany progress
in the (meta-meta-)learning process.

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## Supplementary Materials

##  Appendix A Description of novel meta-learning tasks

Here we show that the framework can describe novel meta-RL tasks not
previously reported in the meta-learning literature, though reminiscent of
animal training experiments.

#### Stay/switch bandit task

We describe a stay/switch bandit task. Each of three stimuli is associated
with a different probability of reward. On perceiving a given stimulus, the
agent can choose to “stay”, and experience this stimulus again (with its
associated reward probability) in the next time step; or “switch” to a
randomly chosen stimulus among the three. The three stimuli, and their
associated reward probabilities, are randomly resampled for each new task
instance. Solving each instance of this meta-task requires identifying the
stimulus with the highest reward probability, which requires exploration
(“switches”) and integration of rewards (“stays”). As shown in Figure S3, this
meta-task is easily described in our formalism.

#### Familiarity detection

In addition, we describe a simple familiarity detection meta-task. For each
task, the agent sees two “cue” stimuli in turn, then is repeatedly exposed to
a “probe” stimulus, which may be either one of the two previously seen cue
stimuli, or another one. The agent must simply reply “1” if the probe stimulus
is one of the two cue stimuli, or ”0” otherwise. The three stimuli are
randomly generated for each new task instance. Note that this task is a
simplified version of the well-known “Omniglot” task, in which the agent must
associate arbitrary labels to a small set of novel stimuli and retrieve the
correct label for presentation of an additional stimulus belonging to one of
the previously seen classes. The difference is that this simplified task tests
memorization of shown stimuli by familiarity detection rather than explicit
label recall. This task is also easily described in our formalism (see Figure
S2).

##  Appendix B Machine-readable description of a randomly generated meta-task

In Figure S1 we provide the JSON specification for the meta-task described in
Figure 7. The JSON structure describes the transition probability table, the
range of each special-state variable (that is, for each special state
variable, the possible values it may take), the flag conditions (as a list of
rules), the reward conditions, and the stimulus associated with each of the
states.

The transition table is a matrix of size $N_{s}\times N_{a}\times N_{s}$,
where $N_{s}$ is the number of states and $N_{a}$ the number of actions (here
$N_{s}=4$ and $N_{a}=2$). For each current state, it indicates the probability
of ending in each possible state, for each of the actions that can be taken.
Values of the form $100+k$ denote the $k$-th state variable; $1000+k$ denote
the $k$-th probability variable; and $10000+k$ denote the $k$-th stimulus
variable. Each of these is to be replaced with an actual value from the
appropriate domain for each new individual task sampled from the meta-task
specification.

Note that the length of the stimuli list indicates the number of states in the
meta-task, and similarly the length of the special-state variable range.
Similarly, the length of the state-variable ranges list indicates the number
of special-state variables.

    {
    "T": [[[0.0, 0.5, 0.5, 0.0], [0.0, 0.0, 1.0, 0.0]],
    [[0.0, 0.5, 0.5, 0.0], [0.0, 0.0, 0.0, 1.0]],
    [[0.33333, 0.33333, 0.0, 0.33333], [0.33333, 0.33333, 0.0, 0.33333]],
    [[0.0, 1.0, 0.0, 0.0], [0.0, 1.0, 0.0, 0.0]]],
    "statevariableranges": [[1, 3, 2]]
    "flagconditions": [[100, -1, -1, 1]],
    "rewardconditions": [[2, -1, -1, 1.0, 1.0, 1]],
    "stimuli": [10000, -1, 10001, 1],
    }

Figure S1: The JSON description automatically generated for the meta-task
described graphically in Figure 7. T is the state transition matrix. This
description fully specifies the meta-task, allowing for automated sampling.
See text for details. Figure S2: Familiarity detection meta-task. The agent
must repeatedly determine whether incoming “probe” stimuli are among the two
“cue” stimuli presented during the initial phase, or are a different stimulus.
Figure S3: Stay/switch bandit task. Action 0 is “switch”, action 1 is “stay”.
The agent must repeatedly explore the various states to estimate their
respective reward probabilities.


††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Collective Excitation of Spatio-Spectrally Distinct Quantum Dots Enabled by
Chirped Pulses

Florian Kappe  Institut für Experimentalphysik, Universität Innsbruck,
Innsbruck, Austria  Yusuf Karli  Institut für Experimentalphysik, Universität
Innsbruck, Innsbruck, Austria  Thomas K. Bracht  Institut für
Festkörpertheorie, Universität Münster, 48149 Münster, Germany  Condensed
Matter Theory, Department of Physics, TU Dortmund, 44221 Dortmund, Germany
Saimon Covre da Silva  Institute of Semiconductor and Solid State Physics,
Johannes Kepler University Linz, Linz, Austria  Tim Seidelmann  Theoretische
Physik III, Universität Bayreuth, 95440 Bayreuth, Germany  Vollrath Martin Axt
Theoretische Physik III, Universität Bayreuth, 95440 Bayreuth, Germany
Armando Rastelli  Institute of Semiconductor and Solid State Physics, Johannes
Kepler University Linz, Linz, Austria  Gregor Weihs  Institut für
Experimentalphysik, Universität Innsbruck, Innsbruck, Austria  Doris E. Reiter
Condensed Matter Theory, Department of Physics, TU Dortmund, 44221 Dortmund,
Germany  Vikas Remesh<EMAIL_ADDRESS>Institut für
Experimentalphysik, Universität Innsbruck, Innsbruck, Austria

(August 30, 2024)

###### Abstract

For a scalable photonic device producing entangled photons, it is desirable to
have multiple quantum emitters in an ensemble that can be collectively
excited, despite their spectral variability. For quantum dots, Rabi rotation,
the most popular method for resonant excitation, cannot assure a universal,
highly efficient excited state preparation, because of its sensitivity to the
excitation parameters. In contrast, Adiabatic Rapid Passage (ARP), relying on
chirped optical pulses, is immune to quantum dot spectral inhomogeneity. Here,
we advocate the robustness of ARP for simultaneous excitation of the biexciton
states of multiple quantum dots. For positive chirps, we find that there is
also regime of phonon advantage that widens the tolerance range of spectral
detunings. Using the same laser pulse we demonstrate the simultaneous
excitation of energetically and spatially distinct quantum dots. Being able to
generate spatially multiplexed entangled photon pairs is a big step towards
the scalability of photonic devices.

††preprint: APS/123-QED

## Introduction

Entangled photon pairs that are encoded in polarization [1, 2, 3, 4, 5] or
time-bins [6, 7], are key to testing the foundations of quantum mechanics [8]
and implementing secure quantum key distribution protocols [9]. To construct a
scalable quantum photonic architecture for such purposes, it is desirable to
have an ensemble of quantum emitters that can be collectively manipulated [10,
11]. Semiconductor quantum dots are high brightness sources for photon
generation [12, 13, 14, 15], accomplishing high degrees of entanglement [4,
16, 6, 17], low multiphoton rate [18] and near-deterministic operating nature
[19]. Despite these achievements, the fabrication of quantum dots via self-
assembly growth processes inevitably introduces some variability in physical
properties like size, alloy and strain distribution, as well as distribution
of impurities in the environment, resulting in inhomogeneous broadening and
spectral diffusion of emission lines. A possible solution could be electric,
magnetic or strain field tuning [20] to manipulate the emission properties,
but tuning on length scales comparable to the inter-dot distance is still a
colossal task [21]. Hence, an emerging question is how to develop an optical
excitation scheme that excites spatially and energetically distinct quantum
dot sources of entangled photon pairs.

The standard approach of resonant two-photon excitation (TPE) of the biexciton
state [22, 23, 24] is disadvantageous, because of its sensitivity to the
transition dipole moment of individual quantum dots, and fluctuations in laser
parameters like intensity and frequency. Accordingly, in reality, a tailored
laser pulse that fully inverts the population in one quantum dot will likely
not do so in another.

This calls for a robust excitation scheme, i.e., one whose efficiency is
insensitive against small fluctuations in the excitation parameters. One such
scheme is based on chirped excitation via Adiabatic Rapid Passage (ARP), which
already has been demonstrated on single quantum dots [25, 26, 27, 28, 29]. For
the production of single photons, ARP can work simultaneously on several
quantum dots [30, 31, 32], where just the exciton was addressed.

In this paper, we demonstrate the simultaneous preparation of the biexciton
states in a quantum dot ensemble via ARP without modifying the excitation
parameters. Interestingly, we also find a regime of phonon advantage, where a
combination of positively chirped excitation and phonon-assisted state
preparation provides an extended spectral range for high efficiency excitation
of biexciton states in multiple quantum dots at moderate pulse areas. This is
different from a pure phonon assisted excitation [33, 34, 35, 36, 37, 38, 39],
as it requires high pulse areas. Our results pave the way for the realization
of spatially multiplexed entangled photon pairs from the same source.

## Pulse chirping

Figure 1: Experiment overview: (a) Dispersion control: The folded grating
stretcher consisting of a ruled grating (G), a lens (L) and a folding mirror
(M). A motorized slit mounted on the folding mirror enables laser spectral
tuning. The GDD values are varied by translating the grating from the focal
plane of the lens. (b) Cryo-microscopy setup: The excitation laser pulses are
sent through a variable optical attenuator (VOA) to control pulse power and
through the polarizing optics (LVP- Linear Vertical Polarizer, PBS- Polarizing
Beamsplitter, HWP- Half Wave Plate, QWP- Quarter Wave Plate) towards the
optical cryostat that holds the quantum dot ensemble at
$1.5\text{\,}\mathrm{K}$. The emitted photons are collected via the same path,
cross-polarization filtered with Linear Horizontal Polarizer (LHP) and sent
either towards the spectrometer or through a home-built monochromator equipped
with notch filters (NF) to the superconducting nanowire single photon
detection (SNSPD) channels to record the exciton (X) and biexciton (XX) photon
coincidences. (c) Characterization: The two chosen quantum dots are labelled
QD1 and QD2, with their raster scan TPE images, colour-coded red (QD1) and
blue (QD2) respectively, alongside their energy level diagrams. Also displayed
are their resonant TPE emission spectra, showing spectrally shifted X and XX
lines from the scattered resonant laser light.

To achieve ARP excitation, we require laser pulses of time-varying frequency
(chirping). In the frequency domain this is described by

$E(\omega)=E_{0}\exp\left[-\frac{\left(\omega-\omega_{c}\right)^{2}}{\Delta\omega^{2}}\right]\exp\left[i\frac{\phi_{2}}{2}\left(\omega-\omega_{c}\right)^{2}\right],$

where $E_{0}$ is the amplitude of the Gaussian frequency envelope centered at
$\omega_{c}$ with a frequency bandwidth of $\Delta\omega$ and $\phi_{2}$
denotes the group delay dispersion (GDD) or in general, the linear chirp.
Introducing $\phi_{2}$ has two effects: it stretches the temporal duration of
the laser pulse from the transform limit ($\tau_{0}$) to $\tau_{p}$ according
to the relation $\tau_{p}^{2}=\tau_{0}^{2}+\left(\frac{4\ln
2\left(\phi_{2}\right)}{\tau_{0}}\right)^{2}$, and dictates the frequency
ordering in the pulse. For positive $\phi_{2}$, the frequency increases over
time, meaning the red part of the spectrum arrives before the blue part and
vice versa for negative $\phi_{2}$.

The experimental implementation of the chirping process is sketched in Fig.
1(a). Initially, a Ti:Sapphire laser producing pulses of
$2.6\text{\,}\mathrm{p}\mathrm{s}$ time duration (measured as intensity FWHM,
Tsunami, SpectraPhysics), is tuned to a wavelength of
$793\text{\,}\mathrm{n}\mathrm{m}$. To render $\phi_{2}$ we rely on a folded
grating stretcher, that consists of a diffraction grating (1200 lines/mm,
Newport) to disperse the beam, a lens ($f=$750\text{\,}\mathrm{mm}$$) to focus
the spectral components to its Fourier plane where a folding mirror is mounted
[40, 41, 42, 43]. A motorized slit mounted on the folding mirror enables laser
frequency tuning with respect to the TPE resonance. If the distance between
the grating and the lens is $f$, laser pulses leave the stretcher dispersion-
free, whereas the displacement of the grating towards the lens induces
positive chirp [41, 40]. Pulse durations corresponding to various positions of
the grating are characterized via spectral and nonlinear autocorrelation
measurements (PulseCheck, APE GmbH). The maximum average laser power measured
before the entrance of the stretcher is $400\text{\,}\mathrm{mW}$, which
corresponds to $\approx$ $5\text{\,}\mathrm{nJ}$ energy per pulse and a peak
pulse power of $\approx$ $1.8\text{\,}\mathrm{kW}$.

## ARP Process

The chirped laser pulses exciting the quantum dot induce the ARP process to
achieve high-fidelity inversion of the biexciton state. The ARP mechanism
relies on a sweep of the instantaneous frequency of the chirped laser pulse
across the quantum dot resonance (or electrical Stark tuning under a constant
laser pulse [44]) to achieve a complete population inversion via an avoided
energy level crossing in the dressed state picture [45, 46]. In quantum dots,
the ARP mechanism can be applied to excite both the exciton [27, 25, 29] and
the biexciton [47, 48, 28] states. Because quantum dots are embedded in a
solid-state environment, the interaction with the lattice vibrations (phonons)
can cause a deterioration of the ARP process, which for low temperatures
depends on the sign of the chirp [49]. We performed calculations in a standard
three-level model consisting of ground $\ket{g}$, exciton $\ket{x}$ and
biexciton state $\ket{xx}$ including the exciton-phonon interaction for
longitudinal acoustic phonons [28, 50, 51] (for a detailed description see
SI). With this, we obtain the biexciton generation for a chirped pulse at
$T=$1\text{\,}\mathrm{K}$$ in Fig. 2(a), assuming a biexciton binding energy
of $4\text{\,}\mathrm{meV}$. At
$\phi_{2}=$0\text{\,}\mathrm{p}\mathrm{s}^{2}$$ we observe Rabi rotations. For
$\phi_{2}$ > 0, we obtain a large biexciton population in a large range of
pulse area and chirp, while for $\phi_{2}$ < 0, phonons hinder the efficient
population of the biexciton.

Figure 2: Theoretical calculations: (a) Biexciton (XX) population under ARP at
$T=$1\text{\,}\mathrm{K}$$ as function of $\phi_{2}$ and pulse area, starting
from a transform limited pulse of
$\tau_{0}=$2.72\text{\,}\mathrm{p}\mathrm{s}$$ tuned to the TPE resonance. (b)
Biexciton (XX) population as a function of detuning for chirped excitation
($\phi_{2}=\pm$40\text{\,}{\mathrm{ps}}^{2}$$) for
$\Theta=20\pi,\tau_{0}=$2.72\text{\,}\mathrm{p}\mathrm{s}$$ without phonons
(blue shaded area) and including phonons at $T=$1\text{\,}\mathrm{K}$$ (black
and red-dashed lines) The green shaded area shows the pure phonon-assisted
excitation without chirp at $\tau_{0}=$41\text{\,}\mathrm{p}\mathrm{s}$$ and
$\Theta=78\pi$.

In the experiments, as shown in Fig. 1(b), the laser beam coming from the
stretcher is fiber-coupled and directed to a closed-cycle cryostat (base
temperature $1.5\text{\,}\mathrm{K}$, ICEOxford), where the quantum dot sample
is mounted on a three-axis piezoelectric stage (ANPx101/ANPz102, attocube
systems AG). A programmable electronic variable attenuator (VOA, V800PA,
Thorlabs) helps sweeping the pulse power. The setup employs a cross-
polarization filtering configuration for efficient laser scattering rejection.
The pulse powers are monitored with a 1% reflector [not shown in Fig. 1(b)]
before the polarizing beamsplitter.

The excitation laser beam is focused onto the quantum dots with a cold
objective (NA = 0.81, attocube systems AG). The quantum dot emission is
collected via the same path, and the exciton (X) and biexciton (XX) photons
are spectrally and spatially separated by a home-built monochromator equipped
with four narrow-band notch filters (BNF-805-OD3, FWHM
$0.3\text{\,}\mathrm{nm}$, Optigrate). The separated photons are routed to a
single-photon sensitive spectrometer (Acton SP-2750, Roper Scientific)
equipped with a liquid Nitrogen cooled charge-coupled device camera (Spec10
CCD, Princeton Instruments) or superconducting nanowire single-photon
detectors (SNSPD, Eos, Single Quantum) for lifetime, cross-correlation and
Hanbury Brown-Twiss (HBT) measurements.

Figure 3: Sensitivity of resonant TPE and versatility of ARP: Excitation and
emission-frequency-resolved TPE Rabi measurements (no chirp, i.e.,
$\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$) represented by measured X photon
counts for QD1 [panel (a)] and QD2 [panel (b)]. Corresponding results at ARP
condition ($\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$) for QD1 [panel (d)] and
QD2 [panel (e)]. The respective insets show the linecuts at respective TPE
resonances (indicated by the black dashed lines on the two-dimensional maps).
The integrated photon counts are normalized for QD1 and QD2 individually, but
are the same at Rabi ($\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$) and ARP
($\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$) conditions. (c) Theoretical
calculation of the biexciton preparation efficiency under Rabi
($\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$) and (f) ARP condition
($\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$).

## Robustness for simultaneous excitation

The main goal of this work is to excite multiple quantum dots simultaneously.
On the single quantum dot level, as evident in Fig. 2(a), ARP is robust
against fluctuations in pulse area and chirp. To excite spectrally distinct
quantum dots, we further investigate the efficiency of the biexciton
occupation as a function of detuning as shown in Fig. 2(b) for a chirp of
$|\phi_{2}|=$40\text{\,}{\mathrm{ps}}^{2}$$ and a pulse area of
$\Theta=$20\text{\,}\pi$$. The detuning $\Delta$ is defined as difference of
the central laser frequency to the TPE resonance, i.e.,
$\Delta=\hbar(\omega_{c}-\omega_{\text{TPE}})$.

Without phonons [blue area in Fig. 3(b)], the biexciton occupation does not
depend on the sign of $\Delta$ and we find that for detunings for a detuning
range of
$$-0.45\text{\,}\mathrm{meV}$\lesssim\Delta\lesssim$0.45\text{\,}\mathrm{meV}$$,
the biexciton state still gets fully occupied, relying on the ARP process.

If we now include phonons, the behaviour of the final occupation changes
drastically. For negative chirp [black curve in Fig. 3(b)], phonons destroy
the ARP process [see also Fig. 2(a)], and the laser pulses do not lead to any
biexciton occupation for a detuning range of
$$-0.45\text{\,}\mathrm{meV}$\lesssim\Delta\lesssim$0.45\text{\,}\mathrm{meV}$$,
while for $|\Delta|\approx$0.8\text{\,}\mathrm{meV}$$, the biexciton state can
get $\approx$80% occupied, due to phonon-assisted processes (for details and
an explanation in the dressed states, see SI).

Most interestingly, for positive chirp [red-dashed curve in Fig. 3(b)] the
phonons do not hinder the ARP process. Instead, they expand the detuning range
suitable for high fidelity preparation ($\geq$95%) significantly:
$$-0.45\text{\,}\mathrm{meV}$\lesssim\Delta\lesssim$0.75\text{\,}\mathrm{meV}$$.
We refer to this widening of the detuning window as _phonon advantage_. This
implies that, in principle, using chirped pulses, one can excite biexciton
states that are detuned as large as $1.2\text{\,}\mathrm{meV}$ in a quantum
dot ensemble.

To underline the effects of the phonon-assisted preparation further, we
compare the results to the chirp-free scenario
[$\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$, shown as green area in Fig. 2(b)],
but with a much higher pulse area of $78\text{\,}\pi$ and
$\tau_{0}=$24\text{\,}\mathrm{p}\mathrm{s}$$ (while for the ARP calculation we
have used a pulse area of $20\text{\,}\pi$, which corresponds to the same
pulse shape after chirping, see SI). For positive detuning $\Delta>0$, the
phonon-assisted process leads to a high biexciton occupation, which agrees
with the occupation in the chirped case.

In summary, we observe that, for chirped excitation, two different mechanisms
lead to the biexciton occupation for positive detuning: the ARP process (for
$\absolutevalue{\Delta}\lesssim$0.45\text{\,}\mathrm{meV}$$) and the phonon-
assisted effect (for
$$0.45\text{\,}\mathrm{meV}$\lesssim\Delta\lesssim$0.75\text{\,}\mathrm{meV}$$)
(see SI for a detailed discussion).

Given the theoretical background, we now search for a pair of quantum dots,
which are spectrally separated within a window of $1\text{\,}\mathrm{meV}$.
Our sample consists of GaAs-AlGaAs quantum dots with exciton emission centered
around $790\text{\,}\mathrm{nm}$ grown by the Al-droplet etching method [52,
53].

We locate two bright quantum dots that are spatially separated by
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, labelled QD1 (red) and QD2 (blue),
[see Fig. 3(c)], having their characteristic exciton emission lines at
$X_{\text{QD1}}=$792.49\text{\,}\mathrm{nm}$$ and
$X_{\text{QD2}}=$792.37\text{\,}\mathrm{nm}$$, i.e., they are spectrally
separated by $0.12\text{\,}\mathrm{nm}$, or $0.2\text{\,}\mathrm{meV}$. These
quantum dots are not expected to dipole-couple to each other over such a
distance. The corresponding TPE spectra, individually measured, are presented
in Fig 1 (c), alongside their representative energy level schemes.

## Individual excitation

We start with the characterization of the two quantum dots QD1 and QD2 [see
Fig. 1(c)] and investigate the robustness of the ARP in comparison with the
Rabi rotations. To quantitatively illustrate the sensitivity of Rabi rotations
to the excitation conditions and quantum dot inhomogeneous broadening, we
perform the TPE experiment on QD1 and QD2. The excitation- and emission-
frequency resolved results are displayed in Fig. 3(a) and (b). Here, the
central laser frequency is scanned in 64 steps from
$1562.6\text{\,}\mathrm{meV}$ to $1563\text{\,}\mathrm{meV}$, exploiting the
motorized slit in the Fourier plane of the grating stretcher. For every
position of the motorized slit, i.e., for a fixed central laser frequency, the
TPE Rabi experiment is performed, by sweeping the pulse power and recording
the TPE emission spectra for QD1 and QD2, after which the photon counts at X
and XX emission energies are integrated to prepare the spectrally resolved
maps (for brevity, only the photon counts at X emission energy are presented
here). We note that the contribution of phonon-assisted TPE is negligible in
the explored range of parameters. The linecuts (represented by black dashed
lines on the two dimensional maps) denote the respective TPE resonance
conditions of QD1 and QD2. The photon counts are normalized to the individual
maxima for QD1 and QD2.

For the spectral range $1562.6\text{\,}\mathrm{meV}$ to
$1563\text{\,}\mathrm{meV}$ considered in the experiment, we observe a clear
distinction in the X emission landscapes of QD1 and QD2 [Fig. 3 (a) and (b)].
For QD1, the resonance frequency is observed at
$\approx$1562.64\text{\,}\mathrm{meV}$$ and $\pi$ power is found to be
$\approx$ $5\text{\,}\mathrm{\SIUnitSymbolMicro W}$, while for QD2 the
resonance is found at $\approx$1562.80\text{\,}\mathrm{meV}$$ with a $\pi$
power of $\approx$ $2.5\text{\,}\mathrm{\SIUnitSymbolMicro W}$. In other
words, the optimal excitation conditions for QD1 will fail to achieve more
than 60% population of the biexciton state in QD2 (for example at $\approx$
$1562.64\text{\,}\mathrm{meV}$). This clearly demonstrates the deficiency of
Rabi rotation as a universal excitation scheme for multi-quantum dot photon
sources.

In Fig. 3(c) we show the theoretical calculation of the biexciton occupation
under TPE. For low pulse powers, it is clear that, as soon as the laser
frequency shifts by $0.01\text{\,}\mathrm{meV}$, the biexciton occupation
drops rapidly to just 1%, asserting its sensitivity to the detuning. Only for
high pulse areas $\approx$45\text{\,}\pi$$, much beyond the power used in the
experiment, would one benefit from the phonon-assisted preparation scheme,
that provides a high occupation for the same detuning.

We now turn our attention to the applicability of ARP against excitation
spectral shifts and quantum dot spectral shifts. For this we fix
$\phi_{2}\approx$40\text{\,}{\mathrm{ps}}^{2}$$ and perform the same
experiment on QD1 and QD2 [Fig. 3(d) and (e)]. The resulting population
landscape is largely a plateau for QD1 and QD2, despite an excitation
frequency scan of $0.4\text{\,}\mathrm{meV}$, relying on the ARP process. For
QD1, the exciton photon counts largely remain stable after
$15\text{\,}\mathrm{\SIUnitSymbolMicro W}$ onwards, while for QD1, plateau is
achieved from $10\text{\,}\mathrm{\SIUnitSymbolMicro W}$ onwards, which
reflect the differences in their $\pi$ pulse powers.

Therefore, it is clear that despite the energetic separation, through ARP,
both QD1 and QD2 can be excited to the biexciton state with high fidelity
within an excitation frequency scan range of $0.4\text{\,}\mathrm{meV}$.

The robustness of the preparation is again confirmed in the theoretical
calculations in Fig. 3(f) showing an excellent agreement with experiment. It
also clearly demonstrates the region of phonon advantage that is asymmetric
with respect to the sign of detuning as discussed in Fig. 2.

Furthermore, we also investigated the single-photon quality via HBT
measurements and obtained the second-order photon correlation of QD2 at Rabi
condition as $g^{(2)}(0)_{\text{QD1}}=0.02$ and at ARP regime as
$g^{(2)}(0)_{\text{QD2}}=0.05$, respectively, asserting that the single-photon
characteristics are maintained at both regimes (see SI).

## Simultaneous excitation of two quantum dots

Figure 4: Collective excitation at ARP: Integrated photon counts at (a) X and
(b) XX emission energies for QD1 and QD2 recorded under simultaneous
excitation, at $\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$ by positioning the
excitation laser spot between QD1 and QD2. The photon counts counts are
normalized to a common maximum, respectively for X and XX emissions.

Lastly, we investigated the versatility of ARP to collectively excite QD1 and
QD2. For this, after setting $\phi_{2}\approx$40\text{\,}{\mathrm{ps}}^{2}$$,
we position the sample so as to simultaneously excite and collect from both
QD1 and QD2, keeping the objective fixed and the frequency spectrum centered
at $1562.6\text{\,}\mathrm{meV}$. We then perform a laser power sweep,
recording the collective ARP spectra with a single-mode fiber coupler. In Fig.
4 we display the integrated photon counts at X and XX emission energies for
both quantum dots, normalized to the maximum in either case. We observe that
the photon counts of both quantum dots achieve plateaus at high pulse areas
(i.e., at ARP regime). This reflects the response we have witnessed earlier in
Fig. 3.

Thus, we have successfully produced two spatially separated quantum dot photon
sources that are collectively excited, despite their energetic separation,
without modifying the excitation parameters. This is remarkable: a simple
excitation scheme with displaced beam can achieve a simultaneous and high
efficiency preparation of two distinct biexciton states, in two spatially and
spectrally different quantum dots. Our scheme is much simpler than an
alternative, dual-confocal excitation scheme [54], which, considering the
pulse power requirement at high chirp
($\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$), is challenging, especially when
trying to scale it to more than two quantum dots. On the other hand, in
nanowire quantum dots, where individual quantum dots are separated along the
growth direction [55], one could relatively easily implement our method to
upscale the production of entangled photons.

## Conclusions

To conclude, we have presented that simultaneous, high-fidelity preparation of
biexciton states is possible in multiple, spectrally distinct quantum dots via
chirped laser excitation, relying on adiabatic rapid passage. Initially, we
showed that near-unity preparation efficiency of biexciton states is
maintained in the chosen quantum dots for an excitation energy shift as large
as $0.4\text{\,}\mathrm{meV}$, clearly establishing the versatility of this
scheme against laser frequency fluctuations and quantum dot spectral
detunings. Furthermore, we have validated the robustness of the adiabatic
scheme by simultaneously exciting the two quantum dots to near-unity biexciton
population. We have also presented the theoretical modeling of the scheme
based on dressed states, and determined a regime of phonon advantage, whereby
the spectral detuning range can be greatly expanded in addition to reducing
the pulse power requirements. Our scheme can be generalized to any quantum dot
system and is a significant contribution towards a robust excitation method to
scale-up quantum dot based quantum photonic architecture.

## Acknowledgements

The authors acknowledge Julian Münzberg for insightful discussions. FK, YK, VR
and GW acknowledge financial support through the Austrian Science Fund FWF
projects W1259 (DK-ALM Atoms, Light, and Molecules), FG 5, TAI-556N (DarkEneT)
and I4380 (AEQuDot). TKB and DER acknowledge financial support from the German
Research Foundation DFG through the project 428026575 (AEQuDot). AR and SFCdS
acknowledge C. Schimpf for fruitful discussions, the FWF projects FG 5, P
30459, I 4320, the Linz Institute of Technology (LIT) and the European Union’s
Horizon 2020 research, and innovation program under Grant Agreement Nos.
899814 (Qurope), 871130 (ASCENT+).

##  Appendix A Supporting Information

###  A.1 Theoretical model

The quantum dot is approximated by a three-level system consisting of ground
state $\ket{g}$, exciton state $\ket{x}$ as well as a biexciton state
$\ket{xx}$. The exciton and biexciton are addressed by a linearly polarized,
pulsed laser $\Omega_{X}(t)$, described in dipole and rotating wave
approximation. The Hamiltonian for this system reads

$\displaystyle\begin{split}H=&\hbar\omega_{0}\ket{x}\bra{x}+\hbar(2\omega_{0}-\Delta_{B})\ket{xx}\bra{xx}\\\
&-\frac{\hbar}{2}\left[\Omega_{X}(t)\left(\ket{g}\bra{x}+\ket{x}\bra{xx}\right)+h.c.\right].\end{split}$
(1)

While a single exciton is separated from the ground state by the energy
$\hbar\omega_{0}$, the biexciton exhibits the so-called biexciton binding
energy $\hbar\Delta_{B}$, such that two-photon processes resonant to
$\omega_{0}-\Delta_{B}/2$ are possible. For the dots considered here,
$\hbar\Delta_{B}$ amounts to $4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. To
describe the chirping process, we consider a Gaussian transform-limited pulse

$\Omega_{\text{0}}(t)=\frac{\Theta}{\sqrt{2\pi}\sigma_{0}}e^{-\frac{t^{2}}{2\sigma_{0}^{2}}}e^{-i\omega_{c}t},$
(2)

with the pulse area $\Theta$, the frequency $\omega_{c}$ and the pulse
duration $\sigma_{0}$. The action of the chirper results in a phase of the
pulse

$\phi(t)=\omega_{c}t+\frac{a}{2}t^{2},$ (3)

with the temporal chirp $a=\frac{\phi_{2}}{\phi_{2}^{2}+\sigma_{0}^{2}}$
determined by the GDD $\phi_{2}$. That leads to a pulse of the form

$\Omega_{X}(t)=\frac{\Theta}{\sqrt{2\pi\sigma\sigma_{0}}}e^{-\frac{t^{2}}{2\sigma^{2}}}e^{-i\phi(t)}.$
(4)

Note that the effective pulse area of the chirped pulse is modified due to the
chirping process.

The chirping is accompanied by a change of the pulse duration according to
$\sigma=\sqrt{(\phi_{2}/\sigma_{0})^{2}+\sigma_{0}^{2}}$ [46]. In the
theoretical calculation we consider the pulse duration of the electric field.
In the experimental data, the pulse duration $\tau$ corresponds to the
intensity full width of half maximum (FWHM). Hence, these quantities are
connected via

$\tau_{p}=2\sqrt{\ln 2}\sigma\,.$ (5)

The detuning between the laser pulse and the three-level system is defined
such that resonant excitation corresponds to the two-photon excitation of the
biexciton, i.e.,

$\Delta=\omega_{c}-\omega_{\text{TPE}}=\omega_{c}-(\omega_{0}-\Delta_{B}/2).$
(6)

For the interaction with the phonon environment stemming from the bulk
material which the quantum dot is embedded in, we use the standard pure-
dephasing Hamiltonian describing the coupling to longitudinal acoustic
phonons, such that the additional phonon part reads

$\displaystyle\begin{split}H_{\text{ph}}&=\hbar\sum_{\mathbf{q}}\omega_{\mathbf{q}}b^{\dagger}_{\mathbf{q}}b_{\mathbf{q}}\\\
&+\hbar\left(\ket{x}\bra{x}+2\ket{xx}\bra{xx}\right)\sum_{\mathbf{q}}\left(g_{\mathbf{q}}b_{\mathbf{q}}+g_{\mathbf{q}}^{*}b_{\mathbf{q}}^{\dagger}\right).\end{split}$
(7)

Here, $b_{\mathbf{q}}(b^{\dagger}_{\mathbf{q}})$ is a bosonic operator
destroying (creating) a phonon with wave vector $\mathbf{q}$ and corresponding
energy $\hbar\omega_{\mathbf{q}}$. Coupling between phonons and exciton states
is mediated by the coupling element $g_{\mathbf{q}}$. We consider coupling to
longitudinal acoustic phonons with linear dispersion and deformation potential
coupling. We take the material parameters from Ref. [28] (in the SI).

Since the processes depend on the number of excitons that are present in the
system, the phonon interaction is twice as strong for the biexciton state
involving two exciton states. Also, the energy shift due to the formation of a
polaron is different for exciton and biexciton states. As it depends on the
square of the coupling, it is four times stronger for the biexciton, compared
to the exciton. To compare calculations with and without phonons, in the case
including phonons, the polaron energy shift is subtracted from the system
energies, so that the energy difference between exciton and biexciton again
equals the biexciton binding.

In order to quantify the strength of the carrier-phonon interaction, a useful
measure is the phonon spectral density

$J(\omega)=\sum_{\mathbf{q}}|g_{\mathbf{q}}|^{2}\delta(\omega-\omega_{\mathbf{q}}).$
(8)

This quantity is highly dependent on the size of the dot, while for the
electronic properties only, the specific shape of the dot is not important
[50]. We assume a $5\text{\,}\mathrm{n}\mathrm{m}$ spherical GaAs quantum dot,
which has its maximum of the phonon spectral density at approximately
$1.2\text{\,}\mathrm{meV}$.

The equations of motions are then calculated via the von-Neumann equation and
the Hamiltonian in Eq. (7), which leads to an infinite hierarchy of
differential equations due to the unrestricted dimension of the resulting
Hilbert space. We truncate this hierarchy using a correlation expansion
approach. Neglecting processes involving three or more phonons, corresponding
to a fourth order Born approximation, has been shown to result in an adequate
description of this system [28]. In particular, it achieves good agreement
with a numerically-exact path-integral formalism [56].

###  A.2 Dressed-state picture

Figure 5:  ARP in the dressed-state picture: Dressed state energies during a
pulse with
$\Theta=20\pi,\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$,\sigma_{0}=$1.62\text{\,}\mathrm{p}\mathrm{s}$$,
corresponding to $\tau=$2.72\text{\,}\mathrm{p}\mathrm{s}$$. Solid lines
correspond to detuning $\Delta=0$ The changing color of the line reflects the
composition of the respective dressed states from the bare states. The dashed
blue (red) lines are with a detuning of
$+$0.2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$($-0.2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$)$
(only for $g$ and $xx$).

To understand the processes leading to ARP and phonon assistance, an
illustrative picture is provided by dressed states. For this, the Hamiltonian
in Eq. (1), including the carrier-light interaction but not the phonon
interaction, is transformed into a reference frame rotating with the laser
frequency. By doing this, the coupling terms $\Omega_{X}$ become real
quantities. The transformed Hamiltonian is then diagonalized, leading to the
dressed-state energies and dressed eigenstates of the system.

Figure 5 shows the time evolution of the dressed-state energies for a chirped
excitation with $\phi_{2}=+$40\text{\,}{\mathrm{ps}}^{2}$$. The color of the
lines correspond to how the dressed states are mixed from the bare states.
Note that the exciton state $x$ is also present in this system, but it is
rather far off and therefore is not involved in the ARP process. Before and
after the pulse, the states do not mix. During the pulse, the chirp leads to
an anticrossing of the dressed-state energies and a switching of the
corresponding bare states. This means, the branch that previously consisted of
the biexciton (orange) now consists of the ground state (blue) and vice versa.
If the evolution is adiabatic, i.e., no transitions between the dressed states
happen, this results in a full transfer of the ground state to the excited
state.

Figure 6:  Phonon-assisted preparation in the dressed-state picture: Dressed-
state energies during a pulse without chirp ($\phi_{2}=0$) and
$\Theta=78.16\pi$ and $\sigma_{0}=$24.74\text{\,}\mathrm{p}\mathrm{s}$$. for
(a) positive detuning of
$\Delta=$1.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and (b) negative
detuning of $\Delta=$-1.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. The
dressed state corresponding mainly to the exciton has been omitted here.
During the pulse, phonon emission can occur for positive detuning as indicated
by black arrows.

To understand the influence of phonon-assisted processes, the dressed-state
energies for unchirped excitation are shown in Fig. 6. The corresponding
biexciton occupation as function of detuning and pulse power is shown in the
main text in Fig. 3(c)(top), where we find that for positive detuning and high
powers the biexciton becomes populated. In Fig. 6(a), a positive detuning of
$1.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ is chosen at a pulse area of
$\Theta=78\pi$ and pulse during of
$\sigma_{0}=$24\text{\,}\mathrm{p}\mathrm{s}$$. These values correspond to the
same envelope shape compared to a chirped pulse with
$\phi_{2}=$40\text{\,}\mathrm{p}\mathrm{s}^{2}$,\Theta=20\pi,\sigma_{0}=$1.62\text{\,}\mathrm{p}\mathrm{s}$$.
Before and after the pulse, the upper dressed state corresponds to the ground
state $\ket{g}$ and the lower to the biexciton state $\ket{xx}$. During the
pulse, the system only mixes slightly and the branches do not (anti)cross like
in the case of ARP. Instead, during the pulse phonon assisted processes can
bridge the energy gap between the dressed states. Transitions between the
dressed states are now allowed, converting ground to biexciton state under the
emission of phonons. At low temperature, only phonon emission processes are
possible, while phonon absorption processes cannot happen, because there are
no phonons available to absorb.

Choosing a negative detuning (as shown in Fig. 6(b)), phonon emission
processes are no longer possible. Now, the upper state corresponds to the
biexciton. This explains the asymmetry regrading the detuning as observed in
Fig. 3(c,top).

It is important to note that the splitting of the dressed states is crucial
for processes including phonons: The splitting has to lie within the range of
the phonon spectral density for the effects to happen efficiently. If either
the detuning or the pulse area is too high, the splitting is too large and
phonon influence is suppressed. For resonant excitation, this leads to the so-
called reappearance regime for high pulse area under resonant excitation of a
two-level system.

Figure 7:  Phonon advantage in the dressed-state picture: Same as Fig. 6, but
now for a chirped pulsed with $\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$ (and
$\Theta=20\pi,\sigma_{0}=$1.62\text{\,}\mathrm{p}\mathrm{s}$$).

The most interesting case leading to the phonon advantage is when we use a
chirped pulse with a large detuning. The dressed state energies for these
cases are shown in Fig. 7. Now, the detunings are so large that the dressed-
state energies no longer show an anticrossing, but cross at
$t\approx$60\text{\,}\mathrm{p}\mathrm{s}$$ before (a) or at
$t\approx$-60\text{\,}\mathrm{p}\mathrm{s}$$ after (b) the pulse, depending on
the sign of the detuning. No ARP process can take place for these parameters,
so without phonons no excitation is possible (cf. Fig. 2(b)). Nevertheless,
the splitting of the dressed state energies again allows for phonon assisted
processes. For a positive detuning of
$1.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ shown in Fig. 7(a), as usual the
ground state corresponds to the upper dressed state, so the biexciton can be
prepared under phonon emission. In Fig. 7(b), a negative detuning of
$-1.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ is used. Now the biexciton
corresponds to the upper dressed state such that phonon absorption would be
required. In both cases, the dressed state branch corresponding to the exciton
$x$ intersects with the $g$ or $xx$ branch, but at this point no population is
transferred, because the laser pulse is not active yet.

The dressed state picture can also be used to understand the influence of
phonons on the excitation using a negatively chirped pulse. In the dressed
state picture, changing the sign of the chirp effectively mirrors the time-
axis, so the diagrams have to be read right-to-left instead. For the cases
shown in Fig. 7, this does not change the effects as in (a), still phonon
emission leads to preparation of the biexciton state and in (b) no excitation
is possible. However, the big difference for
$\phi_{2}=-$40\text{\,}{\mathrm{ps}}^{2}$$ is under no or only slight
detuning, the process does not work including phonons. For this, we look back
at Fig. 5 and again read it right-to-left. Now, in the beginning the upper
branch corresponds to the ground state and the lower branch to the biexciton.
During the pulse, phonon emission happens from the upper to the lower branch,
which after the pulse corresponds to the ground state. This explains why ARP
with negative chirp is strongly disturbed by phonons.

###  A.3 Photon characterization

To verify the photon quality, we performed HBT measurements (see Fig. 8) for X
photons for QD2 at the Rabi ($\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$) and the
ARP $\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$ conditions, and observed those
to be $g^{(2)}(0)=0.02$ and $g^{(2)}(0)=0.05$ respectively. Therefore, the
single photon characteristics are preserved in both excitation regimes. The
nominal increase in $g^{(2)}(0)$ for ARP condition is attributed to the laser
scattering at higher pulse powers.

Figure 8: Photon characterization: (a) Measured single-photon characteristics
at Rabi condition, i.e. $\phi_{2}=$0\text{\,}{\mathrm{ps}}^{2}$$ and pulse
power of $2.5\text{\,}\mathrm{\SIUnitSymbolMicro W}$ (b) at ARP condition,
i.e. $\phi_{2}=$40\text{\,}{\mathrm{ps}}^{2}$$ and pulse power of
$7\text{\,}\mathrm{\SIUnitSymbolMicro W}$.

###  A.4 Chirp characterization

Our grating stretcher is constructed in a folded, single grating geometry [41,
40]. In Fig. 9(blue dashed line) we display the calculated GDD values for
various grating positions in our grating stretcher [40]

$\phi_{2}=\derivative[2]{\phi_{c}(\omega)}{\omega}=\frac{-\lambda_{0}^{3}L_{g}}{\pi
c^{2}d^{2}}\left[1-\left(\frac{\lambda}{d}-\sin\theta_{i}\right)^{2}\right]^{-3/2}$

where $\lambda_{0}=$793\text{\,}\mathrm{nm}$$ is the central wavelength,
$L_{g}=-2(f-s)$ is the effective grating distance (for double pass, with $f$
being the focal length of the lens and $s$ being the distance between the lens
and the grating), $d=$1200\text{\,}{\mathrm{mm}}^{-1}$$ is the grating period
and $\theta_{i}=$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$ is the angle of
incidence. We performed the ARP experiments at $s=$20\text{\,}\mathrm{cm}$$,
providing a $\phi_{2}\approx$40\text{\,}\mathrm{p}\mathrm{s}^{2}$$ . The red
stars in Fig. 9 indicate the values of $\phi_{2}$ during the experiments.

Figure 9: Stretcher characterization: Calibrated values of $\phi_{2}$ in the
grating stretcher employed in the experiments. The solid black line is a
linear fit of the measured data points (black dots) while the dashed lines in
blue show the calculated value of $\phi_{2}$. Red stars indicate the
$\phi_{2}$ values at which the experiments were performed.

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