# On Patterns and Re-Use in Bioinformatics Databases

Michael J Bell, and Phillip Lord,
School of Computing Science
Newcastle University

###### Abstract

##  1 Motivation:

As the quantity of data being depositing into biological databases continues
to increase, it becomes ever more vital to develop methods that enable us to
understand this data and ensure that the knowledge is correct. It is widely-
held that data percolates between different databases, which causes particular
concerns for data correctness; if this percolation occurs, incorrect data in
one database may eventually affect many others while, conversely, corrections
in one database may fail to percolate to others.

In this paper, we test this widely-held belief by directly looking for
sentence reuse both within and between databases. Further, we investigate
patterns of how sentences are reused over time. Finally, we consider the
limitations of this form of analysis and the implications that this may have
for bioinformatics database design.

##  2 Results:

We show that reuse of annotation is common within many different databases,
and that also there is a detectable level of reuse between databases. In
addition, we show that there are patterns of reuse that have previously been
shown to be associated with percolation errors.

##  3 Introduction

It is estimated that over 1,500 active databases are currently in existence
(Fernández-Suárez and Galperin, 2013). While these are generally thought of as
containing biological data, they often also contain collected and collated
information about the data they carry, which is described as annotation. There
are many different types of annotation (Wooley and Lin, 2005): some is highly
structured and organised containing, for instance, links through to other
databases, ontology terms, or taxonomic relationships; others include
unstructured or semi-structured free text. The free text, or textual
annotation, is often considered to be the highest value annotation, although,
by its very nature it is also the hardest to represent and analyse
computationally (Lord et al., 2001). In this paper, we will consider this form
of annotation.

Biological databases and annotation mirror the evolution of biological
systems. As highly similar genes occur in many different organisms,
transferred both horizontally and vertically, so the annotation about these
genes is reused between different databases (Richardson and Watson, 2013).
This form of reuse substantially reduces the work required by database
annotators, but also creates a problem; for most databases it is difficult to
determine the source or support for a particular statement (Bolleman et al.,
2010). Although most databases contain out-going references, either to the
primary literature or to other databases, these are generally given at the
level of the database record. For the richest databases, there may be many
statements for each record. In short, databases often lack a formal
representation of their provenance (Bolleman et al., 2010).

While this form of reuse is part of the folk-history of
bioinformatics111http://madhadron.com/posts/2012-03-26-a-farewell-to-
bioinformatics.html, and is apparent from even a short perusal of a few
bioinformatics databases, it has rarely been explicitly studied. In a previous
study (Bell et al., 2013), we have shown that the level of reuse in UniProtKB
is extremely high – the most reused sentence in TrEMBL occurs more than seven
million times, while the most common sentence in Swiss-Prot occurs more than
91,000 times. More over, we have shown that this reuse operates as an informal
indicator of provenance; two identical sentences are likely to share a common
history. This, in turn, allowed us to identify _propagation patterns_ that can
be used to detect inconsistencies and errors in this annotation.

Our previous analysis looked at only a single database; but we also believe
that reuse occurs between different databases, forming a biological knowledge
ecosystem. In this paper, we extend the analysis further, looking at several
different databases, and show that within these there are also high levels of
reuse. The same analysis also allows us to track reuse between databases and
show that, here also, there is significant reuse. Further, we look for tell-
tale signature patterns previously shown to indicate erroneous annotation and
show that these patterns are also present within several databases and can be
seen between several databases. This analysis suggests that as well as reuse
being common-place, that it is possible to detect knowledge flow between
databases, giving an informal mechanism for detection of provenance.

##  4 Methods

###  4.1 Choosing a set of Databases

There are many databases in the bioinformatics ecosystem that we could use to
study. Unfortunately, these vary significantly technologically, both in terms
of their format, their identifiers and their scheme for updates and
maintenance history. Our previous analysis focused exclusively on UniProtKB,
using it an exemplar gold standard. This analysis also benefited from the
organisation of UniProtKB, which consists of two databases: Swiss-Prot, which
is manually curated and reviewed; and TrEMBL, which is computationally
generated and unreviewed. Here, we wish to identify a set of suitable
databases that allow us to extend our analysis further.

We, therefore, have used the following criteria for selection of a database,
firstly on technical grounds: the database must make available historical
versions; contain more than just minimal amounts of textual annotation; and,
be in a form which is relatively easy to obtain and parse. Within this, we
have picked a set of databases of mixed maturity to obtain a reasonable
sample. We chose the following five databases:

  * •

neXtProt (Lane et al., 2012) — focused solely on human proteins, neXtProt
incorporates data from various sources and is built as a participative
platform; the core corpus is based on human proteins from Swiss-Prot. Unlike
many databases, neXtProt provides a classification system that categorises
data based on its quality into gold, silver or bronze.

  * •

PROSITE (Sigrist et al., 2013) — consists of sequence patterns, or motifs,
that are conserved in protein sequences and can be used to help infer
information about a sequence, such as which protein family it belongs to and
its possible function. Each PROSITE entry contains a pointer to a relevant
documentation entry, which provides biological information that can be
inferred by the pattern.

  * •

PRINTS (Attwood et al., 2003) — a collection of sequence motifs, similar to
PROSITE. However, entries in PRINTS are known as fingerprints, as they are
composed of multiple motifs, unlike entries in PROSITE which contain only
single motifs. All PRINTS entries are manually curated and provide cross-
references to the equivalent PROSITE entries, if they exist.

  * •

TIGRFAMs (Haft et al., 2003) — provides a collection of protein families which
are designed to assist with the prediction of protein function. Each TIGRFAMs
entry contains a textual annotation section with additional supporting
information, such as GO annotations and references to relevant Pfam and
InterPro entries.

  * •

InterPro (Hunter et al., 2012) — an integrative database collating information
regarding protein families, domains and functional sites from eleven member
databases, including PROSITE, PRINTS and TIGRFAMs. Each InterPro entry
contains a description, or abstract, which is often supplemented with
references to relevant literature.

The chosen databases are summarised, along with the URL used to access each
database, in Table 1.

Database Name | URL
---|---
UniProtKB (Swiss-Prot & TrEMBL) | http://www.uniprot.org/
InterPro | http://www.ebi.ac.uk/interpro/
neXtProt | http://www.nextprot.org
PROSITE | http://prosite.expasy.org/
PRINTS | http://130.88.97.239/PRINTS/
TIGRFAMs | http://www.jcvi.org/cgi-bin/tigrfams/
Table 1: The databases chosen for our analyses, including the web address
(URL) of each database.

###  4.2 Data extraction and visualisation

For our analysis, we need to extract sentences from the textual annotation of
each database. As each of these has a different format for each of these
necessitates, a custom framework was written for each, which was extended from
the tool described previously (Bell et al., 2013). Fortunately, the
requirements for our analysis are fairly simple: we need only extract the
textual annotation and basic metadata for a record (the identifier or
accession number), so this process is relatively straightforward and robust to
differences (or changes over time) in the database format. Sentences are
intentionally extracted _verbatim_ and stored in lower-case, with only
database-specific formatting removed. For example, the following data from
UniProtKB:

    CC -!- FUNCTION: May be a transcription factor with important functions
    CC     in eye and nasal development.

would be transformed and stored as:

    may be a transcription factor with important functions in eye and nasal
    development.

This form of analysis is intentionally very simple; we performed no stemming
or even stop-word analysis, with white space normalisation the only change
made to sentences. While this form of analysis may seem very blunt, we choose
it for two reasons: it is computationally very attractive, both when parsing
and searching for matches; and, most importantly, we were concerned more with
correctness than recall. When a match between two databases is found, it is
very likely to be a real one.

Following extraction, sentences were stored in a relational database, linked
to a record identifier, database and version. Dates of records are calculated
using the release version in which a record occurs, and therefore reflect an
upper bound, the size of which is reflective of the release frequencies of the
databases, as described previously (Bell et al., 2013).

The visualisation of sentence propagation uses an interactive visualisation
using the Highcharts library 222http://www.highcharts.com/products/highcharts
driven directly from the database generated in the previous step. These
visualisations provide various interactive features such as zooming, narrowing
and so forth. For full details, please see (Bell, 2015).

##  5 Results

First, we introduce a number of measures that we have used to analyse reuse of
textual annotation. We focus on the number of sentences within a database. It
would be expected that for the sentences that occur in the database, some will
occur more than once (i.e. the database is redundant) and some only once.
These allows us to distinguish between the three following measures of a
sentence which we reuse throughout the paper.

  * •

Total sentences – A redundant set of all sentences in a database version.

  * •

Unique sentences – A non-redundant set of all sentences in a database version.

  * •

Singleton sentences – A set of sentences that occur only a single time within
an entire database version.

###  5.1 Reuse within Databases

Previously, we have shown that UniProtKB (i.e. Swiss-Prot and TrEMBL) reuse
sentences between multiple records; in the case of TrEMBL this reuse is
extreme with only 8,131 unique sentences from 22,706,421 total sentences.
First, we address the question of how widespread this practice of reuse is
within our chosen databases. Moreover, we ask whether this is a feature of the
overall size and complexity of a database.

To address this question, in Figure 1 we show the total number of sentences in
each database; to recap, this is the number of sentences that occur in all
records, whether they are duplicates or not. This is shown on a log scale as
TrEMBL is much larger in size than all of the others, as shown in Table 2. In
Figure 1, we also show the number of unique and singleton sentences as a
percentage of the total.

Figure 1: Figure showing the percentage of singleton (red) and unique (blue) sentences in each analysed database. The line graph represents the total number of sentences in the database (shown on log scales). Within this graph we can broadly see that the larger the database, the more redundant its annotation. | Total Sentences | Unique Sentences | Singleton Sentences | Total Unique
---|---|---|---|---
Swiss-Prot | $3,304,681$ | $394,233$ | $255,349$ | $531,206$
TrEMBL | $26,706,421$ | $8,131$ | $735$ | $49,665$
InterPro | $139,624$ | $71,755$ | $57,628$ | $100,874$
neXtProt | $158,929$ | $101,822$ | $90,875$ | $110,607$
PROSITE | $22,940$ | $21,902$ | $21,356$ | $29,127$
PRINTS | $27,987$ | $16,953$ | $14,356$ | $17,858$
TIGRFAMs | $13,360$ | $12,155$ | $11,481$ | $13,373$
Table 2: Table showing the total number of sentences, unique (i.e. distinct)
sentences and singleton sentences contained within the latest version of each
analysed database. Additionally, we show the total number of unique sentences
over the lifetime of the entire database.

This analysis shows a number of features. First, TrEMBL is shown to be an
extreme outlier; as a database, it is very large, but has the lowest number of
unique sentences (8,131, followed by TIGRFAMs with 12,155). Of the databases,
the PROSITE database has the highest percentage of unique sentences – over 95%
of sentences are unique. Broadly, this theme is also repeated in the other
databases – the larger the database, the more reuse we see.

From this, we conclude that reuse of sentences is a feature of all of the
databases that we have analysed, and that this reuse is substantial in most
cases.

###  5.2 Patterns of Reuse within databases

Previously, we have shown that there are identifiable patterns of reuse within
UniProtKB. We hypothesised that some of these patterns may be indicative of
low quality or erroneous annotation occurring as a result of a failure to
propagate changes; this was confirmed for one pattern by a close analysis of a
number of examples Bell et al. (2013).

Having confirmed in Section 5.1 that sentence reuse is a feature of all
databases that we have analysed. We now address the question as to whether the
patterns we found in UniProtKB are also present elsewhere.

We analyse the databases here for two patterns, _transient_ and _missing
origin_. The transient pattern is where sentences occur within an entry for
only a single database release (i.e. they are removed from an entry after one
iteration of the database). From this definition, it follows that it is
impossible to classify a sentence as transient when it occurs only in the
current version of a database, so we show these independently as _possibly
transient_ , although we do not consider this to be a separate pattern. A
sentence follows the missing origin pattern if it initially occurs in a
database entry, is later propagated to a secondary entry (or entries) and then
subsequently removed from the origin entry whilst still remaining in the
secondary entries. Table 3 shows the number of sentences identified in each
database which follow each pattern.

Database Name | Missing Origin | Transient | Possibly Transient
---|---|---|---
UniProtKB | $8,355$ | $42,460$ | $25,582$
InterPro | $2,689$ | $4,094$ | $1,293$
neXtProt | $35$ | $5,148$ | $773$
PROSITE | $132$ | $2,644$ | $21$
PRINTS | $81$ | $206$ | $363$
TIGRFAMs | $17$ | $563$ | $63$
Table 3: Table summarising the number of sentences following the transient and
missing origin propagation patterns for each database. Sentences classified as
possibly transient are those which appear a single time in the latest version
of the database.

From these results, we note that all of the databases show incidences of the
patterns that we have previously identified. Of the databases, PRINTS and
TIGRFAMs have the lowest level of all of these patterns. This is consistent
with Figure 1 – as these patterns are a feature of a unique sentence, they are
upper-bounded by the uniqueness, and likely to be affected by the level of
reuse within the databases. To be classified, a sentence only needs to exhibit
a pattern in a single entry. A clear example of this is shown in Figure 2
which shows an example of the missing origin pattern. This sentence
(“pyelonephritogenic e.coli specifically invade the uroepithelium by
expressing between 100 and 300 pili on their cell surface”) initially appears
in InterPro entry IPR004086 in 2001 and later appears in InterPro entry
IPR005430 approximately a year later. However, the sentence is removed from
IPR004086 (the origin) in 2003 while still remaining in the secondary entry
IPR005430 for another release.

We have chosen this example, because it clearly represents an error in the
database, albeit a minor typographical one; namely the presence of a space
between the species and genus in “E. coli”333This modification seems to
reflect a change in the underlying XML representation as taxonomic markup was
removed at the same time. Our analysis explicitly excludes markup early in the
pipeline; we note, however, that were it included, the missing origin pattern
would also have detected the lack of percolation of markup changes.. Obviously
this form of the error is unlikely to cause major challenges for human
consumption of the database annotation, but could cause issues for
computational use.

Figure 2: Example of sentence which follows the missing origin pattern. Here,
the sentence originates in InterPro entry IPR004086 before later appearing in
entry IPR005430. It remains in this entry even when then sentence is removed
from IPR004086. Interestingly, we note that the sentence occurs in PRINTS both
before and after it exists in InterPro.

In this section, therefore, we have demonstrated that the patterns of reuse
that we have previously seen in UniProtKB also occur in other databases and in
some cases, at reasonably high levels. In general, these patterns occur more
in databases with more redundancy.

###  5.3 Reuse between Databases

One common worry about knowledge in biology is that it is circular, as the
knowledge is reused and percolated through the biological database ecosystem.
If this is true, then we should be able to detect this supposed reuse, for
instance by sentence reuse between databases. In fact Figure 2, in addition to
showing a missing original, also shows an example of reuse between databases;
a sentence which appears first in PRINTS and, then, later reappears in
InterPro.

To address this question more systematically, we have looked for identical
sentences that occur between any of the databases in our collection, the
results of which are shown in Table 4.

Database Combination | Total Sentences
---|---
UniProtKB | $526,435$
neXtProt; UniProtKB | $83,868$
InterPro | $82,968$
neXtProt | $26,539$
PROSITE | $23,182$
PRINTS | $10,064$
TIGRFAMs | $9,661$
InterPro; PRINTS | $7,751$
InterPro; PROSITE | $5,790$
InterPro; TIGRFAMs | $3,681$
InterPro; UniProtKB | $435$
InterPro; neXtProt; UniProtKB | $151$
PROSITE; UniProtKB | $71$
InterPro; PROSITE; UniProtKB | $26$
neXtProt; PROSITE; UniProtKB | $20$
InterPro; PRINTS; UniProtKB | $20$
InterPro; neXtProt; PROSITE; UniProtKB | $19$
InterPro; PRINTS; PROSITE | $14$
TIGRFAMs; UniProtKB | $14$
InterPro; TIGRFAMs; UniProtKB | $9$
InterPro; neXtProt; PRINTS; UniProtKB | $4$
neXtProt; TIGRFAMs; UniProtKB | $3$
InterPro; neXtProt; TIGRFAMs; UniProtKB | $2$
InterPro; TIGRFAMs; PROSITE | $2$
InterPro; neXtProt; PRINTS; PROSITE; UniProtKB | $1$
InterPro; PRINTS; PROSITE; UniProtKB | $1$
PRINTS; UniProtKB | $1$
PRINTS; PROSITE | $1$
PRINTS; TIGRFAMs | $1$
Table 4: Table summarising the distribution of all unique sentences shared
between the analysed databases.

These results show that there is substantial reuse of sentences in two key
cases. Firstly, there is a very high-level of reuse between UniProtKB and
neXtProt. This is expected as neXtProt explicitly depends on UniProtKB– in
this case, perhaps, it is more surprising that a significant proportion of
neXtProt is unique to it (around 25% of the total sentences in neXtProt). A
second case is shown between the InterPro database and PRINTS, PROSITE and
TIGRFAMs. This is to be expected as InterPro is a federated database,
explicitly depending on the other three databases. We do also see reuse
between other databases, although this occurs at a fairly low-level, compared
to the total number of sentences. There is one sentence which occurs in all
five of the databases which is “visual pigments are the light-absorbing
molecules that mediate vision.”

From this we conclude that knowledge does percolate between different
databases and that it is possible to detect this by using whole sentence
analysis. However, in the majority of cases where identical sentences are
found in large numbers between databases, occur as a result of a formal
relationship between the two – for instance, between UniProtKB and neXtProt.

###  5.4 Patterns between databases

As we have shown previously, and in this paper, it is possible to detect
patterns of reuse within databases, and that in some cases these patterns
appear to be related to errors of percolation. Further, we know that, in some
cases, sentence percolation can also be seen between databases. This raises
the question as to whether we could detect patterns that occur between
databases.

While we do have algorithms for pattern detection within a database, the same
process turns out to be considerably harder between databases, mostly because
of the lack of co-ordinated release dates. If a database record contains a
sentence which is removed between two releases, for example, should it be
considered present only till the first release, or till just before the
second? When comparing two databases, these problems are significant, as the
second database may have undergone several releases subsequently.

As a result of these issues, we have not yet been able to address the question
of pattern occurrence systematically between all databases. However, we have
been able to find specific examples by inspection. We show one of these in
Figure 3. In this case, a sentence appears first in PRINTS (in around 1999),
and then later in 2000 appears, presumably by percolation, in InterPro first
in one record (IRP001055) and then later in 2008 in another (IPR018298).
Around the same time, it disappears from the original entry.

Interestingly, it is not possible to detect the occurrence of this pattern
just by considering a single database. In PRINTS, the sentence occurs at one
point, then stops later. In InterPro, it continues to occur in all records
that it has percolated to. It is only by considering the removal from PRINTS,
and the continued occurrence in InterPro that we see an instance of the
missing origin pattern. This does suggest that cross-database comparisons may
reveal more knowledge than the consideration of a single database.

Figure 3: An example of a sentence in InterPro which does not follow any
propagation pattern. However, if you also consider PRINTS, and the sentence
was copied from PRINTS into InterPro, then the sentence technically follows
the missing origin pattern. This would have significant impact on the
potential correctness of sentences in all databases.

Of course, inspection of this form is not a scalable mechanism for detecting
instances of these patterns; however, without co-ordinated release dates,
automation is hard to achieve. Despite this, our initial analysis indicates
that there are examples of reuse patterns that are detectable between
different databases and, indeed, patterns that are only detectable by
considering multiple databases.

##  6 Discussion

It is often said that knowledge in bioinformatics is frequently reused, and
moves through the database infrastructure. In this paper, we have attempted to
investigate this in as direct a manner as possible, by looking for exact reuse
of sentences between different databases. As a result of this we have found
that, indeed, reuse of knowledge within bioinformatics databases is extremely
common. In the most extreme case for a manually curated database,
UniProtKB/Swiss-Prot, some 91% of the sentences occur more than once. For
TrEMBL, this is even more skewed where unique sentences number less than 1% of
the total sentences. To our knowledge, this demonstrates the first system
attempt to detect, investigate and record the impact of this knowledge flow.

This reuse is, perhaps, a reflection of evolution of these databases. By
itself, it is not necessarily a problem, however, it is a cause for concern.
It does mean that the annotation is heavily denormalised – that is, what is
effectively the same data is stored multiple times within a single database.
This presents significant difficulties during updates; if a duplicated piece
of knowledge needs to be updated with respect to a single record, then perhaps
it also needs to be updated with respect to another.

We have previously shown that it is possible to detect errors or low-quality
annotation, resulting from this denormalisation, by looking for specific
patterns of provenance in the database (Bell et al., 2013). In this paper, we
have shown that two of these patterns, the missing origin and transient
patterns, are also present within other databases; in the case of the missing
origin pattern, this is clearly in the systematic representation of species
names. Further, we have shown that reuse also occurs between databases
although, in general, this happens at a fairly low-level. Even here, though,
it is possible to detect patterns between databases where they are not
detectable from a single database.

The work described here shows the value and importance of historical records,
and that this value is also relevant to the present. We have previously made
extensive use of historical records when looking at trends in database word
usage (Bell et al., 2012), as have others to determine when a database might
be complete (Baumgartner et al., 2007), or to assay the accuracy of predictive
tools (Gross et al., 2009). These analyses have dealt with both the structured
(GO) and unstructured (comments) components of annotation. This demonstrates
that an accurate record of the past is useful to increase our understanding of
the current state of the annotation; truly, understanding the past is useful
to correcting the errors of the present.

However, there are important limitations. In our previous work, we were more
able to investigate some of the instances of annotation patterns in detail,
and demonstrate that they were actually errors. In this work, we were greatly
aided by the existence of UniSave (Leinonen et al., 2006) which allowed us to
rapidly and efficiently investigate the historical record. UniProtKB is
unusual in providing this form of tool however.

We can compare this to Wikipedia which includes a more complete feature set
with respect to versioning than any of the bioinformatics databases that we
have analysed (with UniProtKB coming a notable second best). It does
demonstrate that it is possible to store a fine-grained full version history
for even a very large knowledge base. That it is searchable using the current
schema is an added bonus and would greatly help this form of analysis; in
fact, Wikipedia has been used as the basis for analysis of historical
resources (Viégas et al., 2004). Interestingly, in the last few years, PFAM
has moved toward using Wikipedia as the main mechanism for maintaining their
textual annotation (Punta et al., 2012); while we do not believe this was the
original intention, from the point-of-view of this analysis, this move should
increase the quality of the historical data available.

The second critical limitation of our work is that we are not looking directly
at provenance but inferring from the occurrence of identical sentences. In our
work, we have erred on the side of caution by using direct string matching;
this is a very useful tool for two reasons: firstly, it is computationally
very simple, and extremely scalable and secondly it gives a high-level of
confidence that a match does actually demonstrate knowledge flow. It is,
however, also a very blunt tool, and we are likely to be missing many examples
of information flow. Small changes to sentences, including grammatical or
textual corrections, will break the provenance trail; indeed, we have a direct
example of this happening. Moreover, when tracking provenance between
databases, we suspect that database authors have a positive incentive to alter
text to avoid issues of copyright or plagiarism, inadvertently making the
provenance even harder to track.

A third issue with tracking provenance is the difficulty of dating individual
sentences. Databases are normally developed continuously, but only released
periodically, and it is the releases that we have tracked. These problems are
exacerbated between databases, as the release date is the only information we
have to infer the direction of the travel of knowledge. Taken together, these
limitations mean that our understanding of provenance is heuristic and may be
wrong. In short, our ability to exploit this knowledge is curtailed by the
limited provenance information that is stored.

There are practical steps that current database provider could take which
could increase our knowledge. Most software engineering projects make use of
version control, which can store practically unlimited provenance of source
code. Wikipedia (and, therefore, also PFAM) use the same technology for their
textual annotation. This may provide a simple solution for many biological
databases; it would, at least, address the requirement for fine-grained date
information. Alternatively, a more formal model of provenance (such as PROV
(Missier et al., 2013)) might be used, which could potentially provide a more
fine-grained dataset describing the relationships between sentences
explicitly. This is also likely to be necessary for larger databases such as
TrEMBL, which may be less suited to version control systems because of their
size, automatic generation and relatively low levels of textual annotation per
entry.

Despite these limitations, we have shown that knowledge flows between
databases even when there is not a formal link between them. While this raises
the spectre that some of the knowledge in these databases may be circular, we
have also shown that it is possible to detect patterns which may lead to
mechanisms of error detection, which should increase the quality of knowledge
in biology.

##  7 Acknowledgements

MJB was funded by an EPSRC DTA award to Newcastle University.

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# Proof of Principle for Ramsey-type
Gravity Resonance Spectroscopy with qBounce

René I.P. Sedmik1111Correspondence<EMAIL_ADDRESS>1Technische
Universität Wien, Atominstitut, 1020 Vienna, Austria  Joachim Bosina1,2
1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria  Lukas
Achatz1 1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria
Peter Geltenbort2 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Manuel Heiß1 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Andrey N. Ivanov1 1Technische Universität Wien, Atominstitut, 1020
Vienna, Austria  Tobias<EMAIL_ADDRESS>1Technische Universität Wien,
Atominstitut, 1020 Vienna, Austria  Jakob Micko1,2 1Technische Universität
Wien, Atominstitut, 1020 Vienna, Austria  Mario Pitschmann1 1Technische
Universität Wien, Atominstitut, 1020 Vienna, Austria  Tobias Rechberger1
1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria  Patrick
Schmidt1 1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria
Martin Thalhammer1,2 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Hartmut<EMAIL_ADDRESS>1Technische Universität Wien,
Atominstitut, 1020 Vienna, Austria

# Proof of Principle for Ramsey-type
Gravity Resonance Spectroscopy with qBounce

René I.P. Sedmik1111Correspondence<EMAIL_ADDRESS>1Technische
Universität Wien, Atominstitut, 1020 Vienna, Austria  Joachim Bosina1,2
1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria  Lukas
Achatz1 1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria
Peter Geltenbort2 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Manuel Heiß1 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Andrey N. Ivanov1 1Technische Universität Wien, Atominstitut, 1020
Vienna, Austria  Tobias<EMAIL_ADDRESS>1Technische Universität Wien,
Atominstitut, 1020 Vienna, Austria  Jakob Micko1,2 1Technische Universität
Wien, Atominstitut, 1020 Vienna, Austria  Mario Pitschmann1 1Technische
Universität Wien, Atominstitut, 1020 Vienna, Austria  Tobias Rechberger1
1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria  Patrick
Schmidt1 1Technische Universität Wien, Atominstitut, 1020 Vienna, Austria
Martin Thalhammer1,2 1Technische Universität Wien, Atominstitut, 1020 Vienna,
Austria  Hartmut<EMAIL_ADDRESS>1Technische Universität Wien,
Atominstitut, 1020 Vienna, Austria

###### Abstract

Ultracold neutrons (UCNs) are formidable probes in precision tests of gravity.
With their negligible electric charge, dielectric moment, and polarizability
they naturally evade some of the problems plaguing gravity experiments with
atomic or macroscopic test bodies. Taking advantage of this fact, the qBounce
collaboration has developed a technique – gravity resonance spectroscopy (GRS)
– to study bound quantum states of UCN in the gravity field of the Earth. This
technique is used as a high-precision tool to search for hypothetical Non-
Newtonian gravity on the micrometer scale. In the present article, we describe
the recently commissioned Ramsey-type GRS setup, give an unambiguous proof of
principle, and discuss possible measurements that will be performed.

###### pacs:

ultracold neutrons; Ramsey spectroscopy; quantum bouncer

##  1 Introduction

Gravity resonance spectroscopy (GRS) [1, 2, 3, 4, 5] has been developed as a
tool to search for non-Newtonian gravity, combining the virtues of ultracold
neutrons (UCNs) – namely insensitivity to electrostatic and van der Waals
interactions – with spectroscopy, being the method permitting the highest
possible sensitivity known. Massive particles bound between the Fermi
potential of a horizontal mirror from below and the gravitational potential of
the Earth from above form eigenstates of non-equidistant energies. This system
– known as the ‘quantum bouncer’ [6], has been investigated theoretically [7,
8, 9] and experimentally [10, 11]. Early efforts and the classical limit are
reviewed in Ref. [12]. For UCN, the spatial extent of the wavefunctions is of
the order of tens of microns, which makes them very well accessible in the
experiment. As the eigenenergies, are non-equidistant and in the range of a
few peV, it is possible to uniquely address and trigger transitions between
them by acoustic-frequency mechanical oscillations. Furthermore, these
energies solely depend on three parameters: the neutron mass $m$, the reduced
Planck constant $\hbar$, and the Earth’s acceleration $g$. Therefore, by
measuring transition frequencies between UCN bound states, GRS is highly
sensitive to hypothetical deviations from Newtonian gravity at the micron
distance scale. We exploited this fact to set tight limits on chameleon [4,
13, 14] and symmetron [5, 15] dark energy as well as axion [4] interactions.
In recent years, the search for non-Newtonian forces has received much
attention. The development of new techniques such as Cs atom interferometry
[16, 17], spin-relaxation of 3He [18], or the isoelectric Casimir technique
[19], has lead to impressive reductions of the open parameter spaces of most
dark sector models. In order to further reduce these limits, and eventually
exclude candidate models for dark energy and dark matter, an improvement of
the setups in terms of sensitivity is required. In qBounce, such an
improvement can be realized by extending the recently utilized Rabi-type setup
[5] to allow for Ramsey spectroscopy [20] with quantum states in the
gravitational potential of the Earth [2]. Ramsey spectroscopy is implemented
by splitting the central excitation region of a Rabi setup into two parts that
are traversed in sequence by the test particles. These two parts are then
separated by an additional passive region, where the particle states may
evolve either freely or under the influence of external conditions. Thereby, a
long interaction time can be realized without the technical burden of a long
excitation region. Such a change, while potentially reducing the neutron count
rate and increasing the complexity of the setup, allows us to increase the
interaction time and reduce the sensitivity to systematic effects, such as the
spectral distribution of neutron velocities. Motivated by this perspective, we
have developed a new GRS setup implementing a Ramsey-type scheme.
In the present article, we describe this new setup and the underlying
measurement scheme in Section 2, discuss the relevant theory in Section 3, and
give a first set of data demonstrating the working principle in Section 4.
Finally, we present a brief outlook on possible applications in Section 5.

Figure 1: Working principle of Ramsey GRS. a) Illustration of the UCN state
evolution in our setup at hand of the transition
$|2\rangle\leftrightarrow|4\rangle$. b–d) Theoretical dependence of the
relative transmission $1-P$ on frequency, oscillation velocity amplitude
$A\omega$, and the relative phase $\alpha$ between the oscillations in regions
2 and 4.

##  2 Ramsey GRS – scheme and setup

The working principle of GRS relies on the non-equidistant energy levels
$E_{n}$ of UCNs with mass $m$ in a gravitational well. We now illustrate this
scheme qualitatively at hand of the transition between the second and the
fourth eigenstates, $|2\rangle\leftrightarrow|4\rangle$, in Fig. 1. UCNs enter
the setup from the left through a velocity selecting aperture system (not
shown). This system consists of two neutron-absorbing blades with vertical
opening $s$, mounted at a distance $\ell$ and height $h$ in front of the first
region. Region 1 consists of a polished neutron mirror on the bottom with a
rough mirror on top, separated by a slit of width $h_{s}\ll h,s$. The aperture
system selects all neutrons with a horizontal velocity component $v_{x}$
fulfilling the condition $\sqrt{g\ell^{2}/(2h)}\lesssim
v_{x}\lesssim\sqrt{g\ell^{2}/[2(h+s)]}$ (flight parabola between the openings
of the aperture and region 1).

Figure 2: Simplified view of the Ramsey GRS setup on the granite block.

In the first region, we exploit the fact that the spatial probability densitiy
of neutrons in higher states extends further upwards than the one of lower
states. Therefore, high states interact with high probability with the upper
rough surface, whereupon they are scattered out of the system or absorbed [21,
22]. Hence, only UCNs in the lowest gravitational states can pass through
region 1. The remaining UCNs in region 2 that are in state $|2\rangle$ are
excited by mechanical oscillation into a coherent superposition of states
$|2\rangle$ and $|4\rangle$. For this excitation we make use of the fact that
the resonance frequencies $\omega_{mn}=|E_{n}-E_{m}|/\hbar$ are unique, for
which we can address transitions between specific states $|n\rangle$ and
$|m\rangle$. In region 3 the states are allowed to evolve freely on the
horizontal mirror, before a second excitation in region 4 drives the
$|2\rangle$ to $|4\rangle$ transition once more (at the same frequency
$\nu=\omega/2\pi$ as in region 2). On resonance ($\omega=\omega_{24}$), the
state transition into the target state $|4\rangle$ is thereby completed. The
state selector in region 5 again selects only the lowest gravitational states.
As all neutrons converted by excitation from $|2\rangle$ to $|4\rangle$ are
filtered out in this process, while neutrons in state $|2\rangle$ can pass
region 5 with high probability, the rate measurement after region 5 directly
reflects the transmission probability $P$, and hence also the transition
probability $1-P$. The latter mainly depends on the amplitude $A$, relative
phase $\alpha$, and frequency $\nu$ of the two excitations, and the precise
form of the gravitational potential. These dependencies are depicted in the
lower part of Fig. 1 for the $|2\rangle\leftrightarrow|4\rangle$ transition:
In (b), the frequency $\nu$ is swept while keeping $A$ and $\alpha$ constant,
which leads to the well-known frequency pattern of Ramsey spectroscopy. The
middle plot (c) shows the dependence of the transmission on the amplitude
$\omega A$ for resonant conditions ($\nu=\nu_{24}$ and $\alpha=0$). Plot (d)
on the right side shows a unique feature of Ramsey spectroscopy – the
sinusoidal variation of the transmission with the relative phase $\alpha$
between the oscillations in region 2 and 4. Here, we have also assumed a
resonance condition ($\nu=\nu_{24}$ and $A$ corresponding to a $\pi/2$-flip).
In all plots of Fig. 1, the black curves are calculated for the mean UCN
velocity of $8.8\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$, while the red curves
are convolutions of the theory with the (measured) velocity spectrum (see Fig.
3).
The split of the excitation in two regions with an intermediate free evolution
is the main difference between Rabi and Ramsey spectroscopy. For both methods,
the sensitivity increases with the interaction time. In practice, however, the
implementation of two smaller excitation regions is technically simpler than
one large such region. In addition, Ramsey spectroscopy has the advantage of
being less sensitive to a broad horizontal velocity (time of flight) spectrum.

Similar to previous versions, the qBounce Ramsey GRS setup is located at the
UCN beam position of the ultracold and very cold neutron facility PF2 at the
Institute Laue-Langevin (ILL). In accordance with the scheme described above,
the setup shown in Fig. 2 is comprised of a row of five ‘regions’ (labeled
1–5). Each region consists of a $30\text{\,}\mathrm{mm}$ thick optically flat
BK7 glass mirror coated with a conductive layer of Al and a protective SiO2
layer on top, carried by a 3-axis piezo-electric stage (height, pitch, and
roll). These stages have two tasks. The first one is to control the parallel
alignment of the mirror’s top surfaces to maintain a unified mirror surface
guiding the neutrons at the same height without steps from the entry on the
left over the five regions into the detector behind region 5. The second task
is to create vertical oscillations of the mirrors in regions 2 and 4 to
trigger state transitions (see Section 3). For this purpose, an arbitrary
function generator (AFG) supplies the stages in region 2 and 4 with
independent analog sinusoidal setpoint signals. While both of these signals
oscillate at the same frequency $\nu$, the AFG allows to add a phase shift
$\alpha$ between them. In regions 1 and 5, additional rough glass plates
acting as state selectors are placed at a height
$h_{s}=$22\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ above the smooth mirror
surfaces444The separation is set via precision spacers that are clamped
between the mirror and the upper glass plate.. The five regions and the
detector are rigidly fixed to a massive fine-polished granite block. A tilt
sensor with resolution $1\text{\,}\mathrm{\SIUnitSymbolMicro rad}$ is used to
drive a feedback circuit controlling three piezo-electric transducers below
the granite, thereby leveling the block with respect to the local gravity axis
to $\Delta g/g<10^{-10}$. In Fig. 2, a gantry arches the five regions. Its
purpose is to allow an array of capacitive sensors and pentaprisms to sweep
over the mirror surfaces to detect deviations in the mirror alignment. PISeca
D-510 capacitive sensors are used to detect and correct minor steps between
the mirror surfaces to below $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ via a
feedback circuit controlling the piezo stages (thereby reducing errors in
hight, pitch, and roll of the mirrors). The pentaprisms attached to the gantry
are part of two independent laser interferometers (not shown, Sios GmbH,
SP15000TR and SP2000TR, the latter one being mechanically decoupled from the
granite) monitoring the oscillating movement of regions 2 and 4. The entire
setup is placed inside a large rectangular vacuum chamber (pressure
$\sim${10}^{-5}\text{\,}\mathrm{mbar}$$). Neutrons are supplied from the
Steyerl UCN turbine via a series of Ni-coated glass guides separated by thin
Al foils and two independent shutter systems. The rate of neutrons provided by
the turbine is monitored by a dedicated 3He detector attached to the glass
guides.
Isolation against various disturbances plays an important role in qBounce.
Seismic and acoustic vibrations are reduced by the mass
($840\text{\,}\mathrm{kg}$) of the granite block and the vacuum environment. A
two-layer Mu-metal shielding inside the chamber achieves an attenuation of
external magnetic fields by a factor 100 on the granite surface. Thermally,
the environment at the PF2 installation is controlled within
$\sim$1\text{\,}\mathrm{K}$$ around room temperature.

##  3 Theoretical aspects

The theory of GRS has been discussed extensively in the literature [3, 5], for
which we only give a brief explanation here. UCNs of mass $m$ trapped between
the Fermi energy $E_{F}$ of a vibrating horizontal mirror at
$z_{m}(t)\\!=\\!A\sin(\omega t\\!+\\!\phi)$ and the gravitational potential
$V_{g}(z)\\!=\\!mgz$ of the Earth can be described by the Schrödinger equation
(with usual conventions for $\hbar$ and $g$)

$\displaystyle{\rm i}\hbar\frac{\partial}{\partial t}\psi(z,t)=H\psi(z,t)\,,$
(1) $\displaystyle\text{with }H=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z^{2}}+mgz+E_{F}\theta\left[A\sin(\omega t+\phi)-z\right]\,,$

and $\theta$ denoting the Heaviside theta function. For $A\to 0$, the
solutions of Eqn. (1) are known to be the Airy functions $\langle
z|n\rangle=\psi_{n}(z)=\kappa_{n}{\rm Ai}\left[z/z_{0}-E_{n}/E_{0}\right]$
with normalization constant $\kappa_{n}$ and the characteristic scales
$z_{0}^{3}=\hbar^{2}/2m^{2}g$ and $E_{0}^{3}=\hbar^{2}mg^{2}/2$ [7, 10]. In
general, a neutron assumes a mixed state that can be written as a coherent
superposition $\psi(z,t)=\sum_{n}C_{n}(t){\rm e}^{-{\rm
i}E_{n}t/\hbar}\psi_{n}(z)$, with coefficients $C_{n}(t)$ that fulfill
$\sum_{n}|C_{n}(\tau_{1})|^{2}=1$. The corresponding energy levels
$E_{n}=\\{1.407,\,2.459,\,3.321,\,4.083,..\\}\,$peV are non-equidistant, which
opens the possibility to trigger transitions between specific states
$|m\rangle$ and $|n\rangle$ by applying an excitation at circular frequency
$\omega_{mn}=2\pi\nu_{mn}=|E_{m}-E_{n}|/\hbar$. In this article, we only
consider the transition $|2\rangle\leftrightarrow|4\rangle$. Depending on the
amplitude $A$ of the oscillation, and the interaction time
$\Delta\tau_{i}=\tau_{i}-\tau_{i-1}$ of the neutron with the excitation in
region $i$, transitions will occur with a certain probability (denoted
generically $1-P$ in Fig. 1 having its maximum at
$A\omega\Delta\tau_{i}\left|\langle m|\partial_{z}|n\rangle\right|=\pi$
(defining a ‘$\pi$ flip’). By detecting only states up to $n^{\prime}$ with
$m\leq n^{\prime}<n$, we only detect those neutrons that do not undergo a
transition to higher states. Hence, we can relate the transition probability
$1-P$ to the measured transmission rate $r$. As for the UCNs in our experiment
the Fermi energy is large compared to the vertical kinetic energy of the
neutron, we may assume $E_{F}\to\infty$, leading to the boundary condition
$\psi(z=z_{m},t)=0$. The parameter $\phi$ describes an unknown phase (or time)
offset between the arrival of the neutron and the zero position of the
continuous vibration. For Ramsey spectroscopy, $\phi$ is irrelevant but the
difference $\alpha=\phi_{4}-\phi_{2}$ between the oscillations in region 4 and
2, respectively, plays an important role, as we will explain further below.

After traversing the first state selector (region 1) neutrons entering region
2 at $t=\tau_{1}$ from the left (see Fig. 1), arrive in exactly one energy
eigenstate $E_{n}$ [21, 22]. Statistically, these eigenstates have relative
probabilities of occurrence $a_{n}$ with $\sum_{n}a_{n}=1$, for which we
describe the total state as an incoherent sum
$|\psi(z,t)|^{2}=\sum_{n}a_{n}|\psi_{n}(z,t)|^{2}$. From separate measurements
of the $a_{n}$ and the respective normalized coefficients $b_{n}$ for the
second state selector in region 5 (the method is documented in [23]), we know
that only the first 3 states have significant probabilities of occurrence. For
this reason, we focus in the following on the transmission probability of the
state $|2\rangle$, which (apart from an offset given by the unaffected states
$|1\rangle$ and $|3\rangle$) is the quantity measured in our setup. This
effectively reduces the incoherent sum to one single term,
$|\psi(z,t)|^{2}=|\psi_{2}(z,t)|^{2}$.

The evolution of $\psi(z,t)$ for $t>\tau_{1}$ is determined by the evolution
of the coefficients $C_{n}(t)$, which is given by

$\displaystyle\partial_{t}C_{p}(t)=A\omega\cos(\omega
t+\phi)\sum\limits_{n}{\rm e}^{{\rm i}\omega_{pn}t}T_{pn}C_{n}(t)\,,$ (2)

with the transition matrix elements $T_{np}=\langle
n|\partial_{z}|p\rangle=-T_{pn}$. Analytical solutions of Eqn. (2) for two and
three contributing states are documented in Refs. [24, 25]. As discussed
below, for the $|2\rangle\leftrightarrow|4\rangle$ transition considered in
this article, we limit ourselves to the contribution of exactly these two
states. Then, the solution of Eqn. (2) yields a $2\times 2$ coefficient matrix
$M(\Delta\tau,\phi)$, such that
$\mathbf{C}(\tau_{2})=M(\Delta\tau_{2},\phi)\mathbf{C}(\tau_{1})$, where we
use the notation $\mathbf{C}=\\{C_{2},C_{4}\\}$. We can thus derive the state
at $t=\tau_{2}$ after passing region 2. The same procedure is repeated for
region 3, where $M_{3}\to\mathbb{1}$. In region 4,
$M_{4}(\Delta\tau_{4},\phi)$ is again a solution of Eqn. (2). Concatenation
yields

$\displaystyle\mathbf{C}(\tau_{4})=M_{4}(\Delta\tau_{4},\phi+\alpha)M_{3}M_{2}(\Delta\tau_{2},\phi)\mathbf{C}(\tau_{1})\,,$
(3)

for the state evolution from $\tau_{1}$ to $\tau_{4}$. The total transmission
probability $P(t)=\int_{0}^{\infty}\\!{\rm
d}z\,|\psi(z,t)|^{2}\approx\int_{0}^{h_{s}}\\!{\rm d}z\,|\psi(z,t)|^{2}$ of
state $|2\rangle$ through the setup is then obtained from 555Note that the
mixed terms in $|\psi(z,t)|^{2}$ vanish due to averaging
$|\psi(z,t)|^{2}=\lim_{T\to\infty}(1/T)\int_{-T/2}^{T/2}{\rm
d}t_{0}|\psi(z,t-t_{0})|^{2}$ over the unknown arrival time $t_{0}$ of the
neutron.

$\displaystyle P(\tau_{5})\approx\int\limits_{0}^{h_{s}}\\!{\rm
d}z\,|\psi(z,\tau_{5})|^{2}$
$\displaystyle=b_{2}|C_{2}(\tau_{4})|^{2}a_{2}\,.$ (4)

We would like to stress that the phase $\alpha$ between the excitations in
regions 2 and 4 is a free parameter of the experiment that influences $C_{2}$.
The approximation of a two-state system applied here and in Refs. [3, 4, 5] is
convenient to obtain analytic expressions for the $M_{i}$, but neglects
contributions by multiple transitions. Such contributions can be important if
several transition frequencies are in close vicinity to each other [4]. As a
measure for this ‘vicinity’ we use the half width of the Rabi fringe (one
oscillating region), which can be approximated by the inverse of the
interaction time $\Delta\nu_{\rm fringe}\approx
1/(2\Delta\tau_{2})\approx$29.4\text{\,}\mathrm{Hz}$$. In the present case, we
investigate the $|2\rangle\leftrightarrow|4\rangle$ transition
($\nu_{24}=$392.57\text{\,}\mathrm{Hz}$$) with closest neighbours being the
transitions $|3\rangle\leftrightarrow|5\rangle$
($\nu_{35}=$352.54\text{\,}\mathrm{Hz}$$) and
$|1\rangle\leftrightarrow|3\rangle$ ($\nu_{13}=$462.93\text{\,}\mathrm{Hz}$$)
666There can be transitions between higher states $n>5$ that have closer
resonance frequencies, such as $|8\rangle\leftrightarrow|11\rangle$
($\nu_{8\,11}=$390.27\text{\,}\mathrm{Hz}$$) or
$|7\rangle\leftrightarrow|10\rangle$
($\nu_{7\,10}=$405.64\text{\,}\mathrm{Hz}$$). Such transitions, however, have
vanishing small occupation numbers, for which their contribution can be
neglected.. Both of these neighbours are spaced by more than $\Delta\nu_{\rm
fringe}$ from $\nu_{24}$. Another way to estimate the ‘vicinity’ is the
spectator shift [25] of $\nu_{24}$ for our setup, which is negligible with
$0.9\text{\,}\mathrm{Hz}$. Using these arguments, we validate the use of a
two-state approximation to solve Eqn. (2) for $C_{2}$ and $C_{4}$. In the
computation of $P$, we take into account the spectrum $\eta_{v}$ of horizontal
velocities of the UCNs (see Section 4) by averaging Eqn. (4) over $\eta_{v}$.

As explained in Sections 2 and 4, in the experiment we determine $P$ from a
measurement of the rate $r$ of neutrons transmitted through the setup. While
in recent investigations [4, 5] of hypothetical non-Newtonian potentials
$V(z)$, the transition frequencies between states were determined from
$r(\nu)$, in the present article we focus on the dependence $r(\alpha)$, which
provides a unique feature of Ramsey spectroscopy. This can be understood as
follows. The first ‘$\pi/2$ flip’ in region 2 brings the neutrons into a
coherent superposition of the two states. This superposition then freely
evolves during the passage of region 3 over a time $\Delta\tau_{3}$. We could
now speculate that the information about the phase $\phi_{2}$ of the
oscillators in region 2 is lost while traversing region 3. In this case, the
oscillation in region 4 would represent a statistically independent excitation
(in the Markov sense) inducing an independent $\pi/2$ flip. Hence, the initial
phases $\phi_{2}$ and $\phi_{4}$ entering in the solutions of Eqn. (2) in
regions 2 and 4, respectively, would be independent as well. Then, the total
transmission probability would depend on

$\displaystyle|C_{2}(\tau_{4})|^{2}\\!=\\!\frac{1}{4\pi^{2}}\iint{\rm
d\phi_{2}}\,{\rm
d\phi_{4}}\left|\left[M_{4}(\phi_{4})M_{2}(\phi_{2})\right]_{11}\\!{C_{2}}(\tau_{1})\right|^{2},$
(5)

which just represents two Rabi oscillators in series, independent of the
relative phase $\phi_{4}-\phi_{2}$. If, on the other hand, the phase
information is preserved over $\Delta\tau_{3}$, then we only have one single
unknown phase $\phi$, resulting in a single integration of Eqn. (3) over
$\phi\in[0,\,2\pi]$. Here, an experimentally introduced offset $\alpha$ is
retained in the final transmission rate. For $\alpha\to\pi$, the second
$\pi/2$ flip inverts the first one, resulting in a revival of the initial
state preparation and hence no drop in transmission. Inversely, for $\alpha\to
0$, we have two synchronized $\pi/2$ flips, resulting in a complete state
inversion, and hence the maximum drop in rate. For $-\pi<\alpha<\pi$, the
transmission rate shows roughly a sinusoidal dependence $-\cos\alpha$ [26].
Classically, one would expect that the transmission is independent of the
relative phase between mechanical oscillations. We have confirmed this
expectation by means of Monte Carlo simulations (see appendix A).
Therefore, an experiment in which a sinusoidal dependence of the rate $r$ on
the relative phase between the oscillations in regions 2 and 4 is observed,
can be considered as an unambiguous demonstration of Ramsey spectroscopy with
purely mechanical excitation in the acoustic frequency range.

##  4 Proof of principle

In the experiment, we record the rate $r$ of UCNs transmitted through our
setup as a function of the frequency, amplitude, and relative phase of the
oscillation. In the absence of any oscillation, the transmission rate
$r_{0}=\left.r\right|_{A=0}$ (‘zero rate’) is defined by the reactor power
(neutron flux), the state selectors in regions 1 and 5, loss due to imperfect
vertical alignment of our mirrors, loss off the sides of the mirrors, and the
detector efficiency. As these factors normally stay constant within $\sim 1$%
over the duration of an experimental cycle, we normalize all results with
respect to $r_{0}$. During the ILL’s experimental cycle no. 183 we have
recorded 15 data points for $r_{0}$ and 17 data points with oscillation at
frequency $\nu=$392.625\text{\,}\mathrm{Hz}$\approx\nu_{24}$ and nominal
oscillation amplitude $2\pi
A\nu=$1.47\text{\,}\mathrm{mm}\text{/}\mathrm{s}$$, for several different
settings of $\alpha$. Over the whole cycle, we observed a constant detector
background rate $r_{bg}=0.594\pm 0.053\,$mcps (124 neutrons,
$58\text{\,}\mathrm{h}$) that is subtracted from all recorded data points
prior to normalization, $r_{rel}=(r-r_{bg})/(r_{0}-r_{bg})$.
Due to software- and electricity problems over a period of several days we
were forced to restart the controllers of the piezo-electric stages within our
setup, leading to offsets in the neutron mirror positions before and after the
error. While we were able to realign the edges of the mirrors to within
$\sim$$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ using capacitive and
interferometric measurements, slight residual differences in this alignment
have changed $r_{0}$. We thus discard any data taken during the anomaly, and
normalize our results with respect to two different rates, $r_{0,pre}=17.40\pm
0.35\,$mcps (9 data points, 2911 neutrons, for data taken before the anomaly),
and $r_{0,post}=20.05\pm 0.58\,$mcps (6 data points, 1912 neutrons, for data
taken after the anomaly). The actually applied phase $\alpha$ was determined
from the linearized output of our laser interferometer (Sios SP2000TR). Two of
the interferometer’s beams were constantly monitoring the surfaces of regions
2 and 4, respectively. The resulting displacement data were Fourier-filtered
to a bandwidth of $20\text{\,}\mathrm{Hz}$ around $\nu_{24}$. We then obtained
$\alpha$ and $A$ from the filtered data by multiplication with normalized sine
and cosine functions at frequency $\nu$ and subsequent temporal averaging over
the length of one interferometric measurement ($1\text{\,}\mathrm{s}$). This
procedure represents a software implementation of a four-quadrant lock-in
amplifier.

Figure 3: Measured horizontal velocity distribution of neutrons at the exit of
the first state selector. The solid line is a spline interpolation intended as
a guide to the eye. The boxes represent the selected velocity interval and the
uncertainty in the measurement.

The sum of the interaction times $\Delta\tau_{i}$ of the neutrons with the
mirrors in regions 2–4 is a critical parameter that defines the width of the
transition curves shown in Fig. 1b and, thereby, directly scales the
achievable precision with which the resonance frequencies can be mesasured. In
our experiment, the neutrons are not monochromatic, but posess a certain
velocity distribution $\eta_{v}$. This distribution is measured by detecting
the UCN rate for different settings of the height $h$ and slit opening $s$ of
the aperture mechanism in front of region 1. The result is depicted in Fig. 3.

Figure 4: Measured dependence of the relative transmission $r_{rel}$ on the relative angle between the two excitations, together with a theory fit (orange line). A constant (expected classically and if coherence would be lost) is excluded by $>99.99\%$. The grey band shows the average relative zero rate and its uncertainty. Table 1: Results of a least-squares fit of two-state Ramsey theory to the data in Fig. 4. The fit has 15 degrees of freedom and we compute the fit probability $p$. fit parameters | goodness of fit
---|---
$r_{off}$ | 0. | $718\pm 0.022$ | $\chi^{2}_{red}$ | 0. | 915
$\gamma$ | 0. | $184\pm 0.036$ | $p$ | 0. | 55

Finally, we fit the experimental data on the relative transmission $r_{rel}$
in dependence on $\alpha$ by a simple model, as described below. The
transmission not only depends on the phase, frequency, and amplitude of the
oscillation, but also on the coefficients $a_{n}$ and $b_{n}$ determined by
the state selectors, and the horizontal velocity distribution $\eta_{v}$.
Here, we assume $a_{n}\approx b_{n}$, which is valid in our experiment within
$\sim 10\%$. Then, from $\sum_{n}a_{n}=1$ and
$\sum_{n}|C_{n}(\tau_{1})|^{2}=|C_{2}(\tau_{1})|^{2}=1$ after the first state
selector, it follows that the relative amplitude $\gamma$ of the
$|2\rangle\leftrightarrow|4\rangle$ Ramsey fringes is mainly determined by the
fraction $(a_{2}+a_{4})/\sum_{j}a_{j}$ of states at $t=\tau_{1}$ that
participate in the excitation process. As noted in Section 3 already, we know
that $a_{4}\approx 0$, leaving $a_{2}$ as single parameter determining the
amplitude of the Ramsey fringe. As is illustrated in Fig. 1, other states
(mainly $|1\rangle$ and $|3\rangle$) will pass through the setup unaffected by
the oscillation. These states create a background (offset) in the relative
transmission. As argued in Section 3, we can restrict ourselves to a two-state
approximation [27]. Hence, only $C_{2}(\tau_{4})$ and $C_{4}(\tau_{4})$ are
determined by Eqn. (3), describing the Ramsey fringe dependence on $\alpha$,
while all other states are ignored. Under these assumptions, $P$ would result
in a Ramsey fringe of amplitude $\sim 1$. In order to account for the
contributions of other states (again, mainly $|1\rangle$ and $|3\rangle$) as
well as losses, we introduce an amplitude parameter $\gamma$, and an offset
$r_{off}$ to scale and shift the theoretical curve 777Losses occuring due to
vibration create an offset between the maximum of the fringe and 1. For a
perfect state preparation of only neutrons in state $|2\rangle$, the fringe
would then extend from this maximum towards zero ($=2\gamma$). However, the
sum of transmitted neutrons in states other than $|2\rangle$ reduces the
amplitude, for which we adjust $r_{off}$ and $\gamma$ in the fit.. Eventually,
this results in the fit function

$\displaystyle P_{fit}(\alpha,\gamma,r_{off})=r_{off}+\gamma
P_{th}(\nu,A,\alpha,\mathbf{a}\\!=\\!\\{0,1,0,0,...\\}),$ (6)
$\displaystyle\text{ with
}P_{th}(\nu,A,\alpha,\mathbf{a})=\frac{1}{2\pi}\iint{\rm d}v\,{\rm
d}\phi\,\eta_{v}P(\tau_{5}(v),\phi,\nu,A,\alpha,\mathbf{a}),$

and the transmission probability $P$ defined in Section 3.

In order to verify the demonstration of Ramsey GRS, we test two different
hypothesis against our data on $r_{rel}(\alpha)$ by performing least squares
fits. In this procedure, we weight errors by the experimental standard
deviations. First, we fit a constant function $P_{c}={\rm const}$ with 16
degrees of freedom, resulting in a reduced $\chi^{2}=2.98$ ($p$-value $5\times
10^{-5}$). Hence, a classical explanation (see appendix A) and the case that
the phase information would be lost between the two excitation regions, that
both could be represented by a constant, are excluded by $>99.99$%. If
instead, we fit $r_{rel}(\alpha)$ to $P_{fit}(\alpha,\gamma,r_{off})$ as
defined in Eqn. (6) (with fixed $\nu$, $\alpha$, and $A$ from the
interferometric measurements, and $\eta_{v}$ from Fig. 3), which represents a
simplified quantum mechanical treatment of Ramsey spectroscopy, we obtain the
results shown in Fig. 4 and Tab. 1. These results confirm the observation of
mechanically driven Ramsey GRS.
Note that a slightly less artificial parametrization
$P_{alt}=r_{off}+P_{th}(\nu,A,\alpha,\mathbf{a}=\\{0,a_{2},0,0,...\\})$ is
equivalent and yields exactly the same goodness of fit as the model in Eqn.
(6), with $r_{off}=0.72\pm 0.02$, $a_{2}=0.43\pm 0.04$, which is compatible
with independent measurements of the spatial probability distribution. It is
interesting to note that the theoretical prediction for $P_{fit}$ is very
similar to a sine function. A respective fit to
$P_{lin}=A+B\sin(\alpha+\alpha_{off})$ results in $\chi_{red}^{2}=1.14$
($p-$value 0.32). A sine may thus serve as a convenient test function avoiding
the technical effort of computing the full theory.

##  5 Conclusion and Outlook

During the past decade, we have developed and extended the qBounce experiment
from a UCN quantum bouncing ball measurement of gravitational bound states in
the Earth’s gravitational potential to gravity resonance spectroscopy (GRS).
While with the first Rabi-type version of GRS we have been able to set tight
limits on a variety of non-Newtonian dark sector models, the further
restriction or exclusion of these models demands for an increase in
sensitivity. Following this demand we extended the qBounce GRS setup from a
three-region Rabi implementation to a five-region Ramsey-type configuration.
The four-fold interaction time (length) directly relates to a proportional
increase in sensitivity.
In the present paper we give an unambiguous proof of principle for Ramsey GRS.
Yet, it remains to reduce systematic effects before reaching the full
potential of the new setup. The main limitations at the moment are low
counting rates, imperfect state preparation, and alignment of the mirrors.
Residual transmission of states $n\gtrsim 2$ limits the achievable contrast of
the fringes, and thereby reduces sensitivity. In future experiments, we aim to
achieve contrasts around 70%, as in previous measurements [3, 5]. The shape of
the zero order interference fringes (near transition frequencies) in Ramsey
spectroscopy is far less sensitive to a broad velocity distribution than in
Rabi spectroscopy. This insensitivity provides an option to increase the
neutron rate rate by opening up the velocity-selecting aperture mechanism.
Finally, the alignment and characterization of our mirrors needs to be
improved. Other systematic effects, such as tilt of the experiment with
respect to gravity and vibrations, only give insignificant contributions and
can therefore be neglected for the analysis presented in this article.
Besides the aforementioned precision spectroscopy to limit hypothetical dark
sector models, Ramsey GRS will be used to test the zero charge of the neutron
by adding an electrode above the free interaction region [28]. Another
application would be to search for spin-dependent energy shifts due to axion
or other hypothetical interactions. Such measurements require the addition of
a well-controlled magnetic field and spin-selective detection – both of which
we have already demonstrated in a previous version of qBounce [4].
On the one hand, the inherently low counting rate of a GRS experiment is a
drawback of the present technical realization – a problem that might be solved
in the future. On the other hand, neutrons with their negligible electric
charge and electric dipole moment avoid many of the systematic background
effects plaguing setups using other test objects, including atoms used in
interferometry [16], and macroscopic bodies [29, 19, 30]. Therefore, we
consider GRS with UCNs a promising tool to search for physics beyond the
standard model; a tool that yields significant potential for further
improvement.

## Acknowledgments

We acknowledge financial support from the Austrian Fonds zur Förderung der
Wissenschaftlichen Forschung (FWF) under contract numbers W1252-N27 (doctoral
programme DKPI), I529-N20, 531-N20, W1252, P-26781-N20, and P-31702-N27, and
from the German Research Foundation (DFG) as part of the Priority Programme
(SPP) 1491 ‘Precision experiments in particle and astrophysics with cold and
ultra-cold neutrons’. We also acknowledge support from the French L’Agence
nationale de la recherche (ANR) under contract no. ANR-2011-ISO-007-02,
Programme Blanc International–SIMI4-Physique.

##  Appendix A Classical Monte Carlo simulation

In order to obtain a classical estimate for the expected transmission as a
function of the relative phase between the two excitations, we have simulated
the neutron’s movement as a bouncing ball on a classical trajectory. We
approximated the neutrons as point-like massive particles, which reflect
elastically on rigid horizontal flat mirror surfaces. Classical parabolic
trajectories are then computed analytically between two contacts with mirrors.
As starting parameter each neutron receives a horizontal velocity according to
the distribution in Fig. 3, an initial vertical height evenly distributed
between 0 and $22\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and a vertical
velocity evenly distributed within limits that result in a maximal height of
$22\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The latter two conditions
implement the functionality of the state selector in region 1. In addition,
the time between the entrance of a neutron and the zero-position of the
oscillating mirror in region 2 is chosen to be a random number between 0 and
2$\pi/\omega$. Contacts with the surfaces of regions 2 and 4 change a
neutron’s vertical velocity upon reflection, depending on the actual mirror
position and velocity at time of reflection. Neutrons hitting a side wall of a
mirror or exceeding a trajectory height of
$22\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in region 5 are removed from the
simulation. Eventually, the transmission rate is determined from the ratio of
surviving neutrons after region 5 to the number of initial neutrons. We have
simulated seven different settings for the relative phase between the
oscillations in regions 2 and 4, with a minimum of 10000 trajectories per
setting. The resulting transmission as a function of phase can be fitted with
a constant or a sine function equally well ($p$-values both $>0.95$) but as
shown in Fig. 5, the resulting sine has an amplitude of $(3.8\pm 0.8)\times
10^{-3}\,$mcps (factor 20 smaller than the quantum mechanical prediction), and
a phase offset of $\sim 70^{\circ}$. These results rule out a classical
explanation for the observed dependence in Fig. 4.

Figure 5: Results of a classical Monte Carlo simulation of the transmission
through our setup for the same experimental parameters used in measurements.
Experimental data given for comparison are equivalent to those in Fig. 4.

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