# Optimal control of quantum state preparation and entanglement creation in
two-qubit quantum system with bounded amplitude

Xikun Li  School of Physics and Optoelectronic Engineering, Anhui University,
Hefei, Anhui 230601, China  Max-Planck-Institut für Physik komplexer Systeme,
D-01187 Dresden, Germany  Department of Physics and Astronomy, Aarhus
University, DK-8000 Aarhus C, Denmark

###### Abstract

We consider the optimal control problem in a two-qubit system with bounded
amplitude. Two cases are studied: quantum state preparation and entanglement
creation. Cost functions, fidelity and concurrence, are optimized over bang-
off controls for different values of the total duration, respectively. For
quantum state preparation problem, three critical time points are determined
with high precision, and optimal controls are obtained for different
durations. A better estimation of the quantum speed limit is obtained, so is
the time-optimal control. For entanglement creation problem, two critical time
points are determined, one of them is the minimal time to achieve maximal
entanglement (unit concurrence) starting from the product state. In addition,
the optimal control to reach the unit concurrence is found.

##  I Introduction

Quantum optimal control (QOC) is crucial to quantum information processing
tasks, such as quantum computation and quantum communication. In these tasks
complex quantum systems are engineered and manipulated, e.g. to achieve target
quantum gates and target quantum states Glaser  _et al._ (2015); D’Alessandro
(2021); Krotov (1993); Brif  _et al._ (2010). In certain cases, the adiabatic
operations, which are generally executed very slowly, are desired in
experiments, because we wish to avoid heating the sample and to guarantee the
target gate/state is prepared with perfect fidelity Gericke  _et al._ (2007).
However, in experiments the decoherence and noise from the environment often
make such slow operations impossible. Therefore, speedup the time evolution by
applying fast and robust controls is sensible Guéry-Odelin  _et al._ (2019);
Chen  _et al._ (2010).

Quantum optimal control theory, which is proposed to solve the problems
mentioned above, has been widely applied in various physical systems such as
NMR Khaneja  _et al._ (2005), Bose-Einstein condensate van Frank  _et al._
(2016), cold atoms in optical lattices Li  _et al._ (2018); Srivatsa  _et al._
(2021). One of the important topics in QOC theory is to search the time-
optimal control with which the transitions are finished in the minimal time.
In the context of QOC the minimal time is generally called the quantum speed
limit (QSL) Caneva  _et al._ (2009). And the temporal shape of the
corresponding control field is called time-optimal control. Analytic solutions
are available for several cases where the quantum systems considered are in
low-dimensional Lloyd and Montangero (2014); D’Alessandro and Dahleh (2001);
Boscain  _et al._ (2002); Khaneja  _et al._ (2001); Boscain and Chitour
(2005); Boscain and Mason (2006); Boscain  _et al._ (2014); Hegerfeldt (2013,
2014); Boozer (2012); Jafarizadeh  _et al._ (2020). For multiple-level quantum
systems where analytical results are absent, one has to perform numerical
optimization.

Roughly speaking, we rely on two classes of optimization: local optimization
algorithms, like Krotov Sklarz and Tannor (2002), GRAPE Khaneja  _et al._
(2005), CRAB Doria  _et al._ (2011), GROUP Sørensen  _et al._ (2018) and GOAT
Machnes  _et al._ (2018), as well as global ones such as differential
evolution (DE) and covariance matrix adaptation evolution strategy (CMA-ES) Li
_et al._ (2018); Zahedinejad  _et al._ (2014). Machine learning techniques,
especially reinforcement learning is another promising method Bukov  _et al._
(2018a).

In real experiments the range of tuning parameters of apparatus are finite,
thus constraints in general exist on the control field, e.g., the amplitude is
bounded. In such cases the appearance of local suboptimal traps in the quantum
control landscape makes the QOC problem nontrivial Pechen and Tannor (2011);
Larocca  _et al._ (2018).

The time-optimal problem of two-qubit system with unbounded amplitude is
studied in Ref. Jafarizadeh  _et al._ (2022). In this paper we consider the
optimal problem in a two-qubit system with bounded amplitude. We consider two
problems: quantum state preparation and entanglement creation. For the first
one, one wants to achieve the target quantum state with QSL, and to find the
temporal shape of time-optimal control. For latter, we are interested in the
problem that for given total duration, how large the maximal entanglement can
be obtained.

In Ref. Li (2022) a systematic scenario is proposed to estimate QSL and the
time-optimal control by optimizing over the bang-off controls, and the two-
level quantum system is considered as an example. Employing the scenario
proposed, we optimize over the bang-off controls to estimate the optimal
controls. We show in this paper that physics of two-qubit systems is much
richer than that of two-level systems.

##  II Model

Figure 1: Maximal fidelity $F$ as a function of the total duration $T$
obtained with different number of switches from $N_{s}=0$ to $N_{s}=5$. Three
critical time points are $T_{c}=0.37037$, $T_{\mathrm{sb}}=\pi/2$,
$T_{\mathrm{QSL}}\approx 2.775$. For $T\in[0,T_{c}]$, the optimal control
field is $P_{T/2}N_{T/2}$. For $T\in(T_{c},T_{\mathrm{sb}})$, the optimal
control field is $P_{t_{1}}0_{T-2t_{1}}N_{t_{1}}$. For $T=T_{\mathrm{sb}}$,
the optimal control is $0_{\pi/2}$. For
$T\in(T_{\mathrm{sb}},T_{\mathrm{QSL}}]$, the best $F$ increases as $N_{s}$
increases. When $N_{s}\geq 6$, however, the increment of $F$ is too little,
thus are not shown in this figure.

We consider the symmetrically coupled two-qubit Hamiltonian studied in Ref.
Bukov  _et al._ (2018b), which is described in the following:

$H(t)=-2gS_{1}^{z}S_{2}^{z}-h_{z}(S_{1}^{z}+S_{2}^{z})-h_{x}(t)(S_{1}^{x}+S_{2}^{x})$
(1)

where $g=h_{z}=1$ are the interaction strength and static magnetic field along
the $z$ direction, and $h_{x}(t)$ is the time-dependent control field along
the $x$ direction. $S_{1}^{z}=\sigma_{z}/2$ is spin-1/2 Pauli operators for
the first spin, and so on. The bounded control field $h_{x}(t)$ is a real
function under constraint $|h_{x}(t)|\leq M$. The dynamics of the system is
governed by the Hamiltonian
$\mathrm{d}|\psi(t)\rangle/\mathrm{d}t=-\mathrm{i}H(t)|\psi(t)\rangle$, where
we set $\hbar=1$, starting from the initial state $|\psi_{i}\rangle$.

For the quantum state preparation problem, we set the cost function to be the
fidelity $F$ defined as follows:

$\displaystyle F(h_{x}(t),T)$
$\displaystyle=|\langle\psi_{t}|\mathcal{T}\exp(-\mathrm{i}\int_{0}^{T}H(t)\mathrm{d}t|\psi_{i}\rangle|^{2}$
(2) $\displaystyle=|\langle\psi_{t}|\psi_{f}\rangle|^{2}.$

where $\mathcal{T}$ is the time-ordering operator. $T$ is the total duration
of time evolution, and $|\psi_{f}\rangle$ is the final state. The initial
state $|\psi_{i}\rangle$ is prepared in the ground state of Hamiltonian (1)
with $h_{x}=-2$, and the target state $|\psi_{t}\rangle$ is set to be the
ground state of Hamiltonian with $h_{x}=2$.

For the entanglement creation problem, we use concurrence to measure the
entanglement of two-qubit pure state. A general two-qubit pure state can be
expressed as $|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$,
where $a$, $b$, $c$, $d$ are complex numbers with normalization condition
$|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}=1$. The concurrence of two-qubit pure state
is defined in the following:

$C(|\psi\rangle)=2|ad-bc|.$ (3)

Specially, we denote $C((h_{x}(t),T)\equiv C(|\psi_{f}\rangle)$ the
concurrence of final state following the Schrödinger evolution with control
field $h_{x}(t)$ Hill and Wootters (1997).

For quantum state preparation problem (entanglement creation problem), we want
to find the control field $h_{x}(t)$ which maximizes the fidelity
$F(h_{x}(t),T)$ ($C((h_{x}(t),T)$) for given $T$. We refer to such $h_{x}(t)$
as the optimal control for $T$. Particularly, for quantum state preparation
problem, we wish to estimate the quantum speed limit $T_{\mathrm{QSL}}$ with
which the target state is obtained with unit fidelity $F=1$. For entanglement
creation problem, we want to calculate the minimal time $\tau_{\mathrm{min}}$
such that the unit concurrence $C=1$ is reached. Notice that different from
the quantum state preparation problem, the number of two-qubit pure states
with unit concurrence is infinite, while there is, in general, only one target
state for quantum state preparation problem.

Figure 2: The optimal control is $P_{T/2}N_{T/2}$ for $T\in[0,T_{c}]$. (a)
$\Delta F=F_{2}-F_{1}$ versus the total duration $T$. $F_{N_{s}}$ is the best
fidelity obtained with number of switch $N_{s}$. $\Delta F=0$ when $T\leq
T_{c}$. However, $\Delta F>0$ when $T>T_{c}$. $T_{c}=0.37037$ is indicated by
an arrow. (b) The fidelity as a function of $t_{1}$ for all 6 types of
$N_{s}=1$ control with duration vector $\mathbf{t}=[t_{1},T-t_{1}]$ where
$T=0.2$. The best fidelity is obtained with control $P_{0.1}N_{0.1}$ (c) The
quantum control landscape of fidelity as a function of $[t_{1},t_{3}]$ for
$N_{s}=2$ control $P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with $t_{1}+t_{2}+t_{3}=0.3$.
The maximal fidelity is indicated by a blue pentagram whose location is
$[t_{1},t_{3}]=[0.15,0.15]$, which means $t_{2}=0$ and the $N_{s}=2$ control
is reduced to $N_{s}=1$ control $P_{0.15}N_{0.15}$.

We emphasize that Pontryagin’s principle does not necessarily mean that the
optimal control is always of bang-bang type when the Hamiltonian is linear in
the controls. The case of of singular controls exists Boscain and Mason
(2006); Bao  _et al._ (2018). The Hamiltonian (1) is linear in control
$h_{x}(t)$, and is called bilinear Hamiltonian Cong (2014). Therefore, it is
sensible to employ the family of bang-off controls, (or bang-singular
controls), to optimize the cost functions.

Employed the same scenario in Ref Li (2022), we optimize $F$ and $C$ over
bang-off control. The bang-off control refers to a finite concatenation of
bang controls $P$ and $N$, and off control $0$, which is also called singular
control Boscain and Mason (2006). $P$ ($N$) is short for Positive (Negative)
where $h_{x}(t)=M$ ($h_{x}(t)=-M$) and $0$ is $h_{x}(t)=0$. The control field
is represented by the type–a sequence of $P$, $N$ and 0–and vector of
durations $\mathbf{t}=[t_{1},t_{2},...]$. For example, one control field
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ is defined in the following

$h_{x}(t)=\Bigg{\\{}\begin{matrix}M&0\;\;\leq t<t_{1}\\\ 0&\;\;t_{1}\leq
t<t_{1}+t_{2}\\\ -M&\;\;t_{1}+t_{2}\leq t\leq t_{1}+t_{2}+t_{3},\\\
\end{matrix}$ (4)

where the order of letter sequence is from left to right. For the example
above, the switch number is two $N_{s}=2$, and the bang-off control is of type
$P0N$ which is switched from bang ($P$) to off ($0$), then to bang ($N$). The
number of possible types $N_{\mathrm{type}}$ is at most $3\times 2^{N_{s}}$
for a given number of switches $N_{s}$. For certain initial/target quantum
states, $N_{\mathrm{type}}$ can be further reduced. Here we take $M=4$ such
that $|h_{x}(t)\leq 4|$.

For given $T$, we optimize $F$ (and $C$) starting from $N_{s}=0$. For each
type with given $N_{s}$ we optimize the vector of durations $\mathbf{t}$ using
quasi-Newton method. We denote $F_{i}$ the maximal fidelity using control
fields with $N_{s}=i$, and the difference of maximal fidelity $\Delta
F_{i}\equiv F_{i+1}-F_{i}$. Similar notations are defined for $C$.

Once $\Delta F$ ($\Delta C$) is zero or vanishing small as $N_{s}$ increases,
we stop the optimization and estimate the optimal control with the
corresponding optimized control field.

Figure 3: The optimal control is of type $P_{t_{1}}0_{T-2t_{1}}N_{t_{1}}$ for
$T\in(T_{c},T_{\mathrm{sb}}]$. (a) The quantum control landscape of fidelity
as a function of $[t_{1},t_{3}]$ for $N_{s}=2$ control
$P_{t_{1}}0_{t_{2}}N_{t_{3}}$ with $t_{1}+t_{2}+t_{3}=0.8$. The maximal
fidelity is indicated by a red pentagram whose location is
$[t_{1},t_{3}]=[0.1648,0.1648]$. (b) $t_{1}$ as a function of the total
duration $T$ for the optimal control $P_{t_{1}}0_{T-2t_{1}}N_{t_{1}}$.
$t_{1}=0$ when $T=\pi/2$, thus the optimal control is $0_{\pi/2}$.

Figure 4: (a) The differences of best fidelity $\Delta F$ obtained with switch
number from $N_{s}=6$ to $N_{s}=9$ as a function of total duration $T$, where
$\Delta F_{6}=F_{7}-F_{6}$ and so on. (b) The infidelity $1-F$ as a function
of total duration $T$ with optimal control protocol of $N_{s}=4$ (black solid
line), $N_{s}=6$ (magenta dotted line), and $N_{s}=9$ (blue dashed line). The
estimation of quantum speed limit is marked by a vertical asymptotic line
(blue dashed line), where two vertical lines (red solid line and black dash-
dotted line) are the estimation of quantum speed limit obtained by GRAPE
$T\approx 2.775$ (red solid line) and by symmetric ansatz (black dash-dotted
line) $T\approx 2.907$ obtained in Ref. Bukov  _et al._ (2018b), respectively.
Figure 5: Concurrence C of the optimal control as a function of total duration
$T$. Two critical time points are $\tau_{c}=0.380181$, and
$\tau_{\mathrm{min}}\approx 1.778635$.  Figure 6: $\Delta C_{0}\equiv
C_{1}-C_{0}$ versus the total duration $T$. The critical time point is
$\tau_{c}=0.380181$. $\Delta C_{0}=0$ when $T\leq\tau_{c}$, and $\Delta
C_{0}>0$ when $T>\tau_{c}$. For $T\in[0,\tau_{c}]$, the optimal control field
is $P_{T}$ and $N_{T}$.

Figure 7: (a) Inconcurrence $1-C$ as a function of total duration $T$ for
$N_{s}=2$ and $N_{s}=3$ with the initial state $|00\rangle$. The unit
concurrence is obtained by the optimal control field
$P_{t_{1}}0_{t_{2}}N_{t_{3}}0_{t_{4}}$ which is shown in (b). The estimation
of minimal time to reach unit concurrence is $\tau_{\mathrm{min}}\approx
1.778635$.  Figure 8: The optimal trajectory obtained using the optimal
control shown in Fig.7(b). (a) The trajectory of reduced density matrix
obtained using the optimal control on the Bloch sphere by tracing one qubit.
The initial state on the north pole is marked by the blue square, and the
final state which is on the centre is marked by a red pentagram. The optimal
trajectory is shown in black solid line. (b) The corresponding Cartesian
coordinate shown in (a). (c) The coefficients in Eq. (5) with respect to three
Bell states which span the triplet subspace of two-qubit Hilbert space.

##  III Quantum State Preparation

In Ref. Bukov  _et al._ (2018b) the two-qubit Hamiltonian (1) is investigated
from the view point of quantum control phase transition. Three critical time
points $T_{c}$, $T_{\mathrm{sb}}$ and $T_{\mathrm{QSL}}$ are estimated by
studying the behavior of several physical quantities, such as correlation.
Numerically the bang-bang protocol is optimized using stochastic descent (SD)
method to approximate the optimal control fields and to estimate QSL. However,
the values of these critical time points are not accurate by optimizing bang-
bang control using SD. The optimal controls in different phase regions are
approximated by averaging the optimized bang-bang control. However, the
optimal controls obtained do not behave like bang-bang control at all.

To overcome the problems mentioned above, we employ the bang-off controls
instead. We obtain the values of three critical time points by observing the
behavior of $\Delta F_{i}$. In addition, we find that the optimal controls in
overconstrained phase ($T\in(0,T_{c}]$) and correlated and glassy phase
($T\in(T_{c},T_{\mathrm{sb}}]$) are indeed bang-off controls. The values of
critical time points are more accurate than those obtained using bang-bang
control Bukov  _et al._ (2018b). They are $T_{c}=0.37037$,
$T_{\mathrm{sb}}=\pi/2$, and $T_{\mathrm{QSL}}\approx 2.775$; cf.Fig. 1. In
addition, we find that the optimal types found with different $N_{s}$ is of
type $P...N$. This might result from the fact that the initial sate is the
ground state of Hamiltonian with negative value $h_{x}=-2$, while the target
state is that of positive value $h_{x}=+2$. The details are in the following.

###  III.1 Time-optimal control for $T\in(0,T_{c}]$

For $T\in(0,T_{c}]$, the optimal control protocol is $P_{T/2}N_{T/2}$ in the
overconstrained region. The best fidelity obtained with larger $N_{s}$ is
equal to that with $P_{T/2}N_{T/2}$ for $T\in(0,T_{c}]$. In addition, the
optimal control fields with larger $N_{s}$ reduce to $P_{T/2}N_{T/2}$.

In Fig. 2(a) we show $\Delta F=F_{2}-F_{1}$, which is the difference between
the best fidelity obtained with $N_{s}=2$ and that with $N_{s}=1$. It is
observed that $\Delta F=0$ for $T\leq 0.37037$, whereas $\Delta F>0$ for
$T>0.37037$. The same result holds for $N_{s}\geq 3$. In such way we locate
the value of $T_{c}=0.37037$ which is more accurate than the one obtained in
Bukov  _et al._ (2018b) where $T_{c}$ is approximately equal to 0.38.

In Fig. 2(b) we show that the maximal fidelity is obtained with control
$P_{T/2}N_{T/2}$ for $T=0.2$, within all six types of bang-off control with
$N_{s}=1$. The same conclusion is true for all $T\in(0,T_{c}]$. In Fig. 2(c)
we further demonstrate that the optimal control is $P_{0.15}0_{0}N_{0.15}$,
which is one of twelve types of $N_{s}=3$ control, for $T=0.3$. Notice that
$P_{0.15}0_{0}N_{0.15}$ is in fact $P_{0.15}N_{0.15}$, thus belongs to the
type $P_{T/2}N_{T/2}$. Same result holds for $N_{s}\geq 4$.

Therefore, we have numerically demonstrate that the optimal control is
$P_{T/2}N_{T/2}$ for $T\in(0,T_{c}]$ with $T_{c}=0.37037$.

###  III.2 Time-optimal control for $T\in(T_{c},T_{\mathrm{sb}}]$

For $T\in(T_{c},T_{\mathrm{sb}}]$, the optimal control protocol is of
$N_{s}=2$ type $P_{t_{1}}0_{t_{2}}N_{t_{1}}$ with $2t_{1}+t_{2}=T$. In
addition, the optimal control fields with $N_{s}\geq 3$ reduce to
$P_{t_{1}}0_{t_{2}}N_{t_{1}}$.

For $N_{s}=2$, we have numerically checked that the optimal control is of type
$P0N$ within all 12 types when $T\in(T_{c},T_{\mathrm{sb}}]$. Moreover, the
best fidelity is obtained with a special control field
$P_{t_{1}}0_{T-2t_{1}}N_{t_{1}}$, i.e., the first duration being equal to the
last one $t_{3}=t_{1}$. In Fig. 3(a) it is shown that the quantum control
landscape of control field $P_{t_{1}}0_{T-t_{1}-t_{3}}N_{t_{3}}$ with $T=0.8$.
The maximal fidelity is obtained with $t_{1}=t_{3}=0.1648$. Similar results
hold for $T\in(T_{c},T_{\mathrm{sb}}]$. Therefore, the optimal duration vector
is $[t_{1},T-2t_{1},t_{1}]$. The value of $t_{1}$, shown in Fig. 3(b), is
determined numerically. Notice that $t_{1}=0$ when $T=T_{\mathrm{sb}}$, thus
the optimal control field reduces to $0_{\pi/2}$.

In Ref. Bukov  _et al._ (2018b) the temporal shape of optimal control for
$T\leq T_{\mathrm{sb}}$ is obtained by averaging the optimized bang-bang
controls, which turns out to be approximately bang-off type. Considering the
above results and the ones obtained here, we conjecture that the optimal
controls for $T\in(0,T_{\mathrm{sb}}]$ are bang-off controls.

###  III.3 Time-optimal control for $T\in(T_{\mathrm{sb}},T_{\mathrm{QSL}}]$

In the symmetry-broken phase region $T\in(T_{\mathrm{sb}},T_{\mathrm{QSL}}]$,
the double degeneracy of optimal control field is displayed by two optimal
control fields of same type, but with duration vectors where one is the
flipped vector of another, i.e.
$h_{1}^{\mathrm{opt}}(t)=-h_{2}^{\mathrm{opt}}(T-t)$.

For $T\in(T_{\mathrm{sb}},T_{\mathrm{QSL}})$, in general, the best fidelity
obtained increases as $N_{s}$ increases. See Fig. 1. When $N_{s}\geq 4$, the
increment is smaller than $10^{-4}$. In Fig. 4(a) we show $\Delta F$ as a
function of $T$ for switch numbers from $N_{s}=6$ to $N_{s}=9$. We observe
that while the difference $\Delta F_{6}$ and $\Delta F_{7}$ is of order
$10^{-5}$, $\Delta F_{8}\lesssim 10^{-10}$ is vanishingly small. Therefore, we
estimate the optimal control in the symmetry-broken phase region by using
bang-off control with $N_{s}=9$.

In Ref. Bukov  _et al._ (2018b) the quantum speed limit is estimated by
optimizing the three-pulse symmetric anstaz, which is in fact bang-off control
of one special type
$P_{t_{1}}0_{t_{2}}P_{t_{3}}0_{t_{4}}N_{t_{3}}0_{t_{2}}N_{t_{1}}$ with
$N_{s}=6$. The estimation of quantum speed limit obtained using this anstaz is
$T\approx 2.907$, while another estimation using GRAPE is
$T_{\mathrm{GRAPE}}\approx 2.775$. In fact, the estimation of
$T_{\mathrm{QSL}}$ can be better if the bang-off control is not restricted to
be symmetric. From Fig. 4(b) we observe that the estimation of
$T_{\mathrm{QSL}}$ using $N_{s}=4$ bang-off controls is better than the
anstaz.

In Fig. 4(b) we show the infidelity $1-F$ as a function of $T$ obtained using
the bang-off control. The optimal control protocol with $N_{s}=9$ is of type
$P0N0NP0P0N$. The unit fidelity is reached $F=1-\mathcal{O}(10^{-15})$ with
two optimal duration vectors:
$\mathbf{t}^{*}=[0.232,0.244,0.561,0.317,0.017,0.093,0.858,0.044,0.241,\newline
0.173]$, and another one which is the flipped vector of $\mathbf{t}^{*}$. The
estimation of QSL $T_{\mathrm{QSL}}\approx 2.775$ by using bang-off control
with $N_{s}=9$ is indicated by the vertical asymptote which is equal to the
one obtained with GRAPE, and less than the one obtained using bang-bang
controls or variational ansatz which in fact belongs to one type of $N_{s}=6$
bang-off controls Bukov  _et al._ (2018b). While the control fields obtained
using GRAPE are continuous, the temporal shape of bang-off control is much
simpler than the former.

##  IV Entanglement creation

In this section we investigate the optimal control problem of entanglement
creation. For given $T$, we maximize $C$ starting from the initial state
$|00\rangle$ which is a product state with $C=0$. Similar to the quantum state
preparation problem, two critical time points are found: $\tau_{c}=0.380181$,
and $\tau_{\mathrm{min}}\approx 1.778635$. See Fig. 5 for illustration.

###  IV.1 Time-optimal control for $T\in[0,\tau_{c}]$

For $T\in[0,\tau_{c}]$ the optimal control is $P_{T}$ and $N_{T}$. This is
numerically confirmed by two steps. First, we calculate $\Delta C_{i}\equiv
C_{i+1}-C_{i}$, where $C_{i}$ is the maximal concurrence obtained with
$N_{s}=i$, for various values of switch number. All these $\Delta C_{i}$ are
zero when $T\leq 0.380181$. However, $C_{1}>C_{0}$ when $T>0.380181$. Second,
when $T\in[0,\tau_{c}]$, we find that $C_{0}$ is obtained with the control
field $P_{T}$ and $N_{T}$, and $C_{i}$’s ($i\geq 1$) are obtained with $P_{T}$
and $N_{T}$, too. Therefore, the first critical time point is
$\tau_{c}=0.380181$, and the optimal control is $P_{T}$ for
$T\in[0,\tau_{c}]$.

In Fig. 6 we show $\Delta C_{0}$ as a function of $T$ for example. Following
the terminology in Ref. Bukov  _et al._ (2018b), we call the region with
$T\in[0,\tau_{c}]$ the overconstrained phase, because the optimization is
easy, even though the number of global optima is double, rather than single.

###  IV.2 Time-optimal control for $T\in(\tau_{c},\tau_{\mathrm{min}}]$

We have numerically calculate the concurrence $C$ with different values of
$N_{s}$ for $T>\tau_{c}$. We find that $C$ stops increasing when $N_{s}$ is
larger than three. In addition, the optimized time vectors found by different
$N_{s}$ ($N_{s}\geq 3$) are very close to each other. Therefore, we conjecture
that the optimal control belongs to the control field with $N_{s}=3$ for
$T>\tau_{c}$.

In Fig. 7(a) we show the inconcurrence $1-C$ as a function of $T$ for number
of switches $N_{s}=2$ and $N_{s}=3$. The data for $N_{s}\geq 4$ is not shown,
because the values obtained with $N_{s}\geq 4$ are equal to those with
$N_{s}=3$. The unit concurrence is obtained with the optimal control when
$T=1.778635$, while for $T<1.778635$ the unit concurrence cannot be reached.
See Fig. 7(a) for illustration. Therefore, the minimal time to reach the unit
concurrence is estimated to be $\tau_{\mathrm{min}}\approx 1.778635$. The
corresponding optimal control estimated with $N_{s}=3$ control field is
$P_{t_{1}}0_{t_{2}}N_{t_{3}}0_{t_{4}}$ with the optimal time vector being
$\mathbf{t}=[0.40858,0.52057,8.1384\times 10^{-3},0.84135]$; cf. Fig. 7(b).

In Fig. 8(a) we show the trajectory of reduced density matrix on the Bloch
sphere by tracing one qubit. The initial state is indicated by the blue
square, and the final state in the centre of Bloch sphere is indicated by a
red pentagram, which means the final state of two-qubit state is one of
maximally entangled two-qubit state, i.e. $C=1$. In Fig. 8(b) we further show
the Cartesian coordinate $[x(t),y(t),z(t)]$ of the optimal trajectory on the
Bloch sphere.

We have studied the time evolution of optimal trajectory in a single-qubit
picture. Now we investigate it in the full two-qubit picture. Notice that the
initial state $|00\rangle$ is inside the Hilbert subspace of triplet manifold,
and that the Hamiltonian (1) is invariant by exchanging the two qubits Bukov
_et al._ (2018b). Therefore, the time evolution of two-qubit system is inside
the Hilbert subspace of triplet manifold. The time evolving state is the
superposition of three Bell states, i.e.,
$|\Phi^{+}\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$,$|\Phi^{-}\rangle=(|00\rangle-|11\rangle)/\sqrt{2}$,
and $|\Psi^{+}\rangle=(|01\rangle+|10\rangle)/\sqrt{2}$. We monitor the three
coefficients with respect three Bell states

$|\psi(t)\rangle=C_{1}(t)|\Phi^{+}\rangle+C_{2}(t)|\Phi^{-}\rangle+C_{3}(t)|\Psi^{+}\rangle.$
(5)

In Fig. 8(c) we show the squared coefficients $|C_{i}(t)|^{2}$ of the quantum
state following the optimal trajectory. From Fig. 8(c) we show that the final
state with unit concurrence is not Bell state, because all coefficients are
not zero.

##  V Conclusions

In this paper we investigate the optimal control problem in a symmetrically
coupled two-qubit system with bounded amplitude. By optimizing over the family
of bang-off controls, the problems of quantum state preparation and
entanglement creation are studied. Given the initial states, the cost
functions, fidelity and concurrence, are optimized for various durations for
the problems mentioned above. By studying the difference of best cost function
obtained with different types of control field, optimal controls and critical
time points are determined more accurately than the previous work.

For the quantum state preparation problem, we have shown that for durations
before symmetry-broken phase occurs, the optimal control fields are indeed of
type bang-off, which is taken as an ansatz in the previous work. For durations
in the symmetry broken phase, we estimate the optimal control using bang-off
controls up to $N_{s}=9$. As the number of switch increases, the increment of
best fidelity is decreasing and approaching vanishingly small. Therefore, we
estimate the optimal control field in the symmetry-broken phase with bang-off
controls with $N_{s}=9$. Furthermore, we estimate the quantum speed limit and
time-optimal control field using the same bang-off controls. The QSL obtained
is equal to that obtained with GRAPE, but the temporal shape of time-optimal
control field is simple.

For the entanglement creation problem, we start from the product state and
maximize the concurrence using bang-off controls for different durations. Two
critical time points are obtained. For durations in the overconstrained phase,
the optimal control is simple: the control field takes either the maximal
value or the minimal. As the duration increases, the optimal control is of
bang-off type with $N_{s}=3$. The minimal duration to reach the unit
concurrence is estimated, and time-optimal control is obtained. The optimal
trajectory is also shown using different methods, e.g., the trajectory of the
reduced density matrix on a Bloch sphere, and the coefficients of time-
evolving with respect the basis of triplet states.

In the previous work it has been proved that the time-optimal control with a
bilinear Hamiltonian in a two-level quantum system with bounded amplitude is
indeed bang-off control. Considering the results in the previous work and the
ones in this paper, it is interesting to ask whether the same conclusion holds
for the two-qubit system with bounded amplitude. An analytical proof is
desired.

###### Acknowledgements.

We are grateful to Marin Bukov and Anne Nielsen for helpful discussions.

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# The Rüchardt experiment revisited: using simple theory, accurate measurement
and python based data analysis.

Chris Shearwood and Peter A. Sloan  Department of Physics, University of Bath,
Bath, BA2 7AY, United Kingdom<EMAIL_ADDRESS>
###### Abstract

This project uses the Rüchardt experiment to determine the ratio of specific
heats and hence the number of degrees of freedom $f$ of different gases by
measuring the frequency of damped simple harmonic motion where the gas
provides the Hooke’s law like spring of a cylinder-piston system. This project
links mechanics, electromagnetism, thermodynamics, statistical mechanics and
quantum mechanics making it an excellent synoptic experiment for a mid year
undergraduate student. We present simple derivations of the main relationships
that govern the experiment, a detailed data analysis of the physics of the
apparatus and of the experimental data. We find $f(\textrm{He})=3.48\pm 0.14$,
$f(\textrm{N}_{2})=4.92\pm 0.24$, $f(\textrm{Air})=4.96\pm 0.25$,
$f(\textrm{CO}_{2})=6.46\pm 0.39$ at room temperature and atmospheric
pressure. The results for CO2 requires a statistical analysis of its
vibrational modes. These results show that the expected results can be
measured using fairly simple apparatus, coupled with careful analysis of large
data sets.

††: Phys. Educ.

  *

##  1 Introduction

###  1.1 Teaching aims

The Rüchardt experiment links quantum mechanics, Newtonian mechanics,
statistical physics, thermodynamics, and due to our set-up electromagnetism.
It is an ideal mid-years undergraduate physics experiment allowing students to
use the wide variety of their taught physics, experimental techniques and
computational analysis they have already mastered (or at least passed
assessment on). This version of implementing and analysing the Rüchardt
experiment therefore aims to bridge the gap between the more scripted and
well-defined experiment they typically meet in their early years laboratory
courses and the typically more open-ended, less defined projects, students
tackle in their latter and final years.

Here we present an exemplar lab-report of our implementation of the Rüchardt
experiment with particular emphasis on understanding the experimental
apparatus and how it transforms the physical parameter in question, to a
measurable quantity. We present detailed experimental results and explore the
range over which they are valid before presenting the final results. We
highlight places were simple student written computer simulation would
underpin and strength the student understanding of the underlying physics.
Critical to the learning success of this project as a teaching activity is
that the students are already familiar to the physics concepts required for
the project. It is, instead, the realistic combination of, to the students,
seemingly disparate aspects of their course into a coherent scientific
narrative that is the ultimate learning outcome of this project. We aim to
publish more details, including the team based aspect of our implementation
and the specifically designed authentic assessments, of this more realistic
and holistic approach.

###  1.2 The Rüchardt experiment

The central element of the Rüchardt experiment is using a gas as the spring in
a piston-cylinder apparatus to produce simple-harmonic-motion (SHM) of the
piston when displaced from equilibrium. The frequency of that SHM can then be
related to the adiabatic ratio, and hence the quantum mechanics of the gas
particles.

From its initial reporting in 1929 [1] there has been a steady set of
experimental [2, 3, 4, 5] and theoretical investigations [6] into this simple
experiment. There is a propensity to update the analysis and data-capture are
more sophisticated data-capture and computer techniques become more common
place in undergraduate teaching laboratories. For example on the advent of
digital computing in teaching laboratories [7]. It is available to purchase
through the usual laboratory supply companies for example, LabTrek or LD
Didactic. We have used it as both a year 1 and year 2 physics experiment here
at the University of Bath. In the recent COVID instigated pivot to online
laboratory courses it was one of the experiments to tuned into ‘online’
experiment both at our home institutions and at Uppsala University in Sweden
[8].

##  2 Theory

The apparatus is lightly damped so we actually measure damped SHM (dSHM) and
use the regular mathematical solutions for this system.

###  2.1 Damped SHM

From any introductory course we know that the solution to dSHM is

$z(t)=z_{0}\sin(\omega_{d}t-\alpha)\exp\left(-t/\tau\right)$ (1)

where $z_{0}$ is the starting position away from the equilibrium position, $t$
is the time, $\alpha$ is some initial $t=0$ phase offset. The piston release
is usually taken as $t=0$ so in general $\alpha=0$. Finally $\tau$ is the
lifetime, the time it take for damping to reduce the amplitude to $1/e$ of the
initial amplitude. The damped angular frequency $\omega_{d}$ is related in the
normal way to the undamped angular frequency $\omega_{0}$ by

$\omega_{d}=\sqrt{\omega_{o}^{2}-\left(1/\tau\right)^{2}}$ (2)

with

$\omega_{o}=\sqrt{\frac{k}{M}}$ (3)

where $M$ is the mass of the object doing the motion (the piston) and $k$ is
the spring constant related to the SHM and that we need to relate to the
properties of the gas.

###  2.2 Adiabatic process and degrees of freedom

An adiabatic process has no change in the total energy, heat does not flow
into or out of the system. To minimise the heat flow in or out of the
apparatus the cylinder and piston are make of glass, a poor thermal conductor.
The experiments themselves only last a few seconds at most. As Fourier’s law
states, any heat loss will be proportional to the temperature difference from
the gas to the glass. Below we’ll propose that only in the most extreme
experimental conditions (very small volume) would there be a large enough
temperature difference to drive substantial heat transfer. There is some
suggestion of this non-adiabatic nature in the volume dependence of the
lifetime at small volumes.

If there is no heat transfer into or out of the gas system, then the work done
by the piston on the gas (or vice versa) will be an adiabatic process. We
therefore start with the well know expression for an adiabatic process,

$pV^{\gamma}=\textrm{constant},$ (4)

where $\gamma$ is heat capacity ratio,

$\gamma=\frac{C_{p}}{C_{V}}$ (5)

between the constant pressure heat capacity $C_{p}$ and constant volume heat
capacity $C_{V}$. Leading to the expression for $\gamma$ that depends on the
identity of the gas particles through its number of degrees of freedom $f$,

$\gamma=1+\frac{2}{f}.$ (6)

We will discuss the significance of $f$ and its temperature and molecular
dependence in the General Discussion section 5, but fundamentally it means
that a measurement of macroscopic and thermodynamic property $\gamma$, gives
measurable insight into the structure and properties of the individual gas
atoms or molecules. This is why this experiment is ideally suited to bridge
the gap between scripted experiment and open-ended projects. It links easy to
measure properties (the plunder bobbing up and down) to the microscopic, or
quantum, properties of the individual atoms and molecules.

To estimate the accuracy required to determine significant difference between
the four gases measured, Table 1 presents the number of degrees of
translational and rotational freedom each gas has. Vibrational degrees of
freedom are discussed in section 5. Here we assume air is predominately
nitrogen and oxygen, both diatomic molecules. This simple analysis suggest
$\gamma$ will have to be measured to at least an uncertainty of 10 % and
preferably closer to the 1 % level.

Table 1: Translational and rotational degrees of freedom and associated
$\gamma$ value for the four gases measured (does not include vibrations).

  * Gas | Translational $f$ | Rotational $f$ | $\gamma$
---|---|---|---
He | 3 | 0 | 1.667
Air | 3 | 2 | 1.4
N2 | 3 | 2 | 1.4
CO2 | 3 | 2 | 1.4

###  2.3 From adibatic process to Hooke’s law

At the initial conditions we have that,

$\textrm{constant}=p_{0}V_{0}^{\gamma},$ (7)

here $p_{0}=p_{\textrm{atm}}+Mg/A$ is a combination of the atmospheric
pressure $p_{\textrm{atm}}$ bearing down on the top of the piston and pressure
due to the weight of the piston itself $M$, gravitational acceleration $g$ and
the cross-sectional area of the piston $A$. Atmospheric pressure was measured
at the beginning of each experimental session, see experimental section 3. Due
to the weight of the piston being relatively lighter then the ‘weight’ of the
atmosphere, the pressure correction is negligible, $\sim 1$ kPa, compared with
typical atmospheric pressure of $101$ kPa.

To find the restoring force caused by a displacement we rearrange to give the
pressure as a function of volume,

$p(V)=p_{0}\left(\frac{V_{0}}{V}\right)^{\gamma}.$ (8)

A computer simulation could then compute this full expression, through for
example the Verlet algorithm, at each piston position and so numerically find
the time-dependence of the piston. However, here to allow a match to SHM we
consider small displacements to so see how the pressure varies to 1st order -
the linear approximation. A Taylor expansion of the pressure with a small
volume change $\Delta V$ around the initial volume $V_{0}$ gives,

$p(V_{0}+\Delta V)\approx p(V_{0})+\left.\frac{dP}{dV}\right|_{V=V_{0}}\Delta
V\\\ +\frac{1}{2}\left.\frac{d^{2}P}{dV^{2}}\right|_{V=V_{0}}(\Delta
V)^{2}+\textrm{small terms}.$ (9)

We can see that $p(V_{0})=p_{0}$, the next term

$\displaystyle\frac{dP}{dV}$ $\displaystyle=-\gamma
p_{0}\frac{V_{0}^{\gamma}}{V^{\gamma+1}}$ (10)
$\displaystyle\therefore\left.\frac{dP}{dV}\right|_{V=V_{0}}$
$\displaystyle=-\gamma\frac{p_{0}}{V_{0}}.$ (11)

Similarly

$\left.\frac{d^{2}P}{dV^{2}}\right|_{V=V_{0}}=(\gamma^{2}+\gamma)\frac{p_{0}}{V_{0}^{2}}.$
(12)

The total pressure difference across the piston, the equivalent of subtracting
$p_{0}$, gives the pressure imbalance by a small change in volume as

$p(\Delta V)\approx-\gamma p_{0}\left(\frac{\Delta
V}{V_{0}}\right)+\frac{1}{2}(\gamma^{2}+\gamma)p_{0}\left(\frac{\Delta
V}{V_{0}}\right)^{2}.$ (13)

Taking the 1st order, $\Delta V$, term and typical values for helium, at $100$
ml initial volume the maximum increase in pressure due to the oscillations
will be $\approx 4$ kPs and at the smallest initial volume of $10$ ml a
pressure increase of $\approx 40$ kPa. Well within the safety limits of the
glass apparatus.

Finally to mirror a Hooke’s law spring, we convert to force and setting
$\Delta V=A\Delta z$ and $V_{0}=z_{0}A$ to give

$F(\Delta z)\approx\underbrace{-\gamma p_{0}A\left(\frac{\Delta
z}{z_{0}}\right)}_{\textrm{Linear in }\Delta
z}+\underbrace{\frac{1}{2}(\gamma^{2}+\gamma)p_{0}A\left(\frac{\Delta
z}{z_{0}}\right)^{2}}_{\textrm{Quadratic in }\Delta z}.$ (14)

If we assume $\gamma\sim 1$ for all gas particles, we can see that for say a
$1$ in $100$ deviation from a linear response to displacement from the
equilibrium $z_{0}$, e.g., where the linear term is 100 times larger than the
quadratic term, we need that $\Delta z/z_{0}\sim 1/100$. The normal
experimental procedure was to displace by about $5$ mm and release. The
apparatus is made so that $1$ mm of height corresponds to $1$ ml of volume. At
$100$ ml volume the cylinder is $100$ mm high. Therefore any low volume
measurements, well below $100$ ml the response could significantly deviate
from a linear response. Very low initial volume measurements are, however,
also tricky as from an analysis of such data there is an increase in the
damping (presumable coming close to the static friction regime) and so we are
restricted to rather larger initial displacements that therefore may well
drive our system away form a purely linear response system at lower overall
volumes.

In the linear, or 1st order, regime we can identify that $F=-k\Delta z$ so
that $k=\gamma p_{0}A^{2}/V_{0}$ and with $\omega_{0}=\sqrt{k/M}$ we get the
relationship

$\omega_{0}^{2}=\left(\frac{p_{0}A^{2}}{V_{0}M}\right)\gamma.$ (15)

The pressure $p_{0}$ is fixed by the atmospheric pressure and the mass of the
piston which is also fixed, as is the area $A$ of the piston. This means we
can vary the initial volume $V_{0}$ and measure the response $\omega_{0}$ to
find the constant of proportionality, the adiabatic ratio $\gamma$.

Table 2: Measured atmospheric pressure ($\pm 0.1$ kPa), and laboratory
temperature ($\pm 0.5\ ^{\circ}$C) when the experimental results were
recorded.

  * Gas | Atm Pressure / kPa | Lab temp / ∘C
---|---|---
He | 101.1 | 23.3
Air | 101.4 | 23.5
N2 | 101.1 | 22.5
CO2 | 101.7 | 23.5

From the ideal-gas law and the adiabatic relationship of equation 5 the gas
temperature is given by

$T=T_{0}\left(\frac{V_{0}}{V}\right)^{\gamma-1}$ (16)

where $T_{0}$ is the initial temperature of the gas. For small volume changes
around $V_{0}$ this can be Taylor expanded and so approximated as

$\Delta T\approx T_{0}\left(1-\gamma\right)\frac{\Delta V}{V_{0}}.$ (17)

The temperature change due to the oscillation will therefore be larger for gas
with the larger $\gamma$ value (e.g. helium) and larger at the smallest
volumes. For helium this will be $\Delta T\approx 9\ ^{\circ}$C at $V_{0}=100$
ml and $\Delta T\approx 90\ ^{\circ}$C at $10$ ml. Whereas for carbon dioxide
both are lower $\Delta T\approx 4\ ^{\circ}$C at $100$ ml and $\Delta T\approx
41\ ^{\circ}$C at $10$ ml. These results are broadly in line with, say, the
same effect felt when pumping up tyres on a bike as the pump get hot to the
touch. Therefore, any nonadiabaticity (driven by the larger temperature
gradients from gas to external heat bath) will be more likely at the smaller
volumes investigated.

Although we did not do this, it would be interesting and instructive to
directly measure the time-dependence of the gas temperature and pressure. From
this students could try and use the displacement measurement to reconstruct of
the temperature and pressure and compare that with the measured values. That
is, not only using an equation to compute other parameters, but measure all
the parameters to see if that equation holds.

###  2.4 diabtic SHM

The same analysis for a purely diabatic process, that is one at constant
temperature, gives the result $\omega_{0}^{2}=p_{0}A^{2}/V_{0}M$, which means
that if there is any heat flow from or to the apparatus it will naturally lead
to a lowering of the measured value of $\gamma$ towards 1.

##  3 Experimental

###  3.1 Apparatus and procedures

Figure 1:  (A) Photograph of the apparatus with labels. Gap between top of
magnets and bottom of pick-up coils was 50 mm. (B) Detailed photographs of the
base of the piston and the gas tight seal.

The apparatus consisted of a glass cylinder and movable piston with a good
gas-tight seal due to the exact fitting of the piston within the cylinder.
This large glass-glass interface only introduced minimal frictional damping.
To measure the motion of the piston three small rare-earth magnets were glued
to the top of the piston. As the piston/magnets moved this induced an
electromotive force (emf) in a pick-up coil $(50\pm 5)$ mm above the magnets
and was measured and captured by an oscilloscope. Figure 1 shows (A) a
photograph of the whole apparatus and (B) a close up photograph of the end of
the piston and the gas tight seal.

To ensure a smooth operation and to align the magnet and coil a laser light
was passed from above through the coil onto the top of the stack of magnets.
This allowed adjustment of the clamps and orientation of the coil to align it
the axis of the magnets as accurately as possible. Moreover, as the cylinder
was filled with gas, so the starting point of the magnet would naturally rise.
To keep a consistent average emf in the pick-up coil, the coil was at each
gas-volume moved to ensure a 50 mm gap between top of the magnet stack and the
base of the coil. This level of accuracy was only important if a full
understanding of the magnitude of the emf signal was wanted but isn’t critical
to the measurement of the adiabatic ratio $\gamma$.

A precise procedure for emptying, filling, purging and filling the cylinder
with pure gas was followed. We would estimate that perhaps there could be at
most a 1 % (ratio of volume of pipework open to air to volume of the maximum
fill) contamination with air. This would obviously not effect the air
measurements, nor impact on the nitrogen measurements, but could have some
influence on the helium (lowering $\gamma$) and carbon dioxide (raising
$\gamma$) measurement and there is some evidence of that.

To start the dSHM the piston was pushed down by hand by approximately 5 mm
(reducing the volume by 5 ml) and released. The oscilloscope was set up to
capture a single time-trace based on a trigger. Example time-traces are shown
in figure 4. Each gas was measured over 10 volumes from 10 ml to 100 ml, see
for example figure 5 and each volume had 5 repeat measurements. The volume
measurements were taken by reading from the graduated scale of the cylinder
the position of the black-seal ring. The uncertainty in all initial volume
measurements was estimated to be $\pm 1$ ml. This is the main factor limiting
the accuracy of the measurement of the adiabatic ratio $\gamma$.

A careful measurement of the actual volume of the cylinder-piston system to
the graduated values found that there was an additional 6 ml of volume to the
marked volume. This volume was attributed to the gas inlet tube system and
slight curvature in the base of the piston. Therefore for a reported volume
of, say 50 ml, the true, or corrected, volume was 56 ml. For ease of reading
this correction is only made to the final fitting function of figure 7.

The piston and magnets had a combined mass of $M=(106.68\pm 0.01)$ g, and the
piston had a diameter of $(34.16\pm 0.01)$ mm. The pick-up coil has a diameter
of $(13\pm 1)$ mm and $n=(1400\pm 20)$ turns. Gasses were of purity 99.8 % or
better. Local gravitational acceleration was calculated using the
International Gravity Formula and using The University of Bath’s latitude of
51.3811∘ and an elevation of the laboratory of 192 m above sea level giving
$g=9.81134$ ms-2.

###  3.2 Damping and gas leakage

A weakness of this approach and apparatus is the seal between cylinder and
piston. It must be sufficiently gas-tight to prevent leakage (and hence change
of volume) of the gas, yet allow the nearly free movement of the piston within
the cylinder. The apparatus here is simply sealed by the precise fit of the
piston within the cylinder.

At volumes larger than 50 ml we find damping lifetimes consistent at $\sim 1$
s (see figure 6) which when compared with the typical dSHM frequency of
$\omega_{d}\sim 130$ rad s-1 results in only a minor correction to compute an
undamped frequency, in this example, of $+1\times 10^{-3}$ rad s-1. Below an
initial volume of 50 ml there is a much larger damping effect.

Figure 2:  Gas leakage rates from the cylinder-piston apparatus. (A) The time-
dependence of volume of the system from an initial fill, of both Helium and of
Nitrogen gas, to 100 ml as it naturally reduced due to the pressure imbalance
due to the mass of the piston-magnet system. (B) Pair-wise simple differential
of (A) to show the gas leak rate.

Figure 2 shows the volume of the apparatus filled with helium and with
nitrogen gas naturally changes over time. This is due to the mass of the
piston itself producing a pressure difference of $\Delta p=Mg/A=1.14$ kPa
which although much lower than the mean atmospheric pressure of $101.3$ kPa is
enough to drive gas form the piston through the seal. From this we determine
the gas leakage rate as shown in figure 2(B), this is just the simple
neighbouring volume and time differences of the data of figure 2(A). The leak
rate shows a similar low-volume and high-volume regime as the damping rate.

This is unexpected. Given that the pressure difference between gas and
atmosphere should be constant in this gas-leak experiment, and that the only
change in the apparatus is the position of the seal we may expect a volume
independent leakage rate. Therefore, we speculate that there is an increase in
the friction between seal and cylinder below the 50 ml mark. This naturally
leads to an increased, and observed, damping (decrease in the lifetime).
Furthermore, there will be an increase in the friction opposing the piston as
it slowly descends in the cylinder leading to a reduction in the pressure
difference between the gas and atmosphere and so a reduction in the leak-rate.

What we can also see is during a typical oscillating experiment of say, 2 s,
there will be at most a 0.1 ml loss of gas from the system therefore we can
safely ignore such gas loss from the analysis of any one experimental time-
trace measurement. In future experiment the volume before and after each dSHM
run should be explicitly measured to quantify any gas leakage directly.

###  3.3 Magnetic field

There are of course many ways to measure the time-dependent movement of the
piston. Here to ensure a non-invasive measurement, but based on introductory
physics we make use of the electromotive force (emf) generated by a moving
magnet. A set of small magnets glued to the top of the piston and a pick-up
coil connected to an oscilloscope to measure the time-dependence of the
induced electromotive force (emf) $\epsilon$. Through Faraday’s law of
induction

$\epsilon=-\frac{\textrm{d}\Phi_{B}}{\textrm{d}t}$ (18)

where $\Phi_{B}$ is the magnetic flux through some area, we can relate the
measured signal to the piston dynamics. The area is the area $A_{c}$ of the
pick-up coils multiplied by the number of turns $n$. If we take the on-axis
description of the magnetic field from the dipole source of the magnets, we
have $B(z)$ the magnetic field strength

$B(z)=\frac{2\mu_{0}}{4\pi z^{3}}m_{B},$ (19)

where $\mu_{0}$ is the vacuum permeability, $z$ is the distance away from the
centre of the magnet and $m_{B}$ is the magnitude of the magnetic-dipole.
Assuming the flux is the same for all the coils, in effect ignoring the height
of the coil stack and the slight off-axis nature of the total area of the
coils, we have

$\Phi_{B}=A_{c}n\left(\frac{2\mu_{0}}{4\pi z^{3}}\right)m_{B}.$ (20)

Here a simple 3D calculation using standard mathematical models the full
dipole magnetic $B$-field could be done to convince a student that these
approximations are valid.

To find a value for $m_{B}$ the magnetic field strength was measured using a
Hall probe at several distances from the magnet as shown in figure 3.

Figure 3:  Magnetic field strength as a function of distance - note log-log
scale. See main text for details of fitting function that was fitted across
data as shown.

On this log-log graph we would expect a linear dependence, which is evident at
distances greater than 10 mm. As the experimental gas data used an initial
$z_{m}=60$ mm separation (approximate midpoint of coil height) with a
$z_{0}=5$ mm initial SHM amplitude, we only consider this region. (At lower
distances the magnetic field measurement is probably sensitive to slight
misalignment of the Hall probe both off-axis and off angle.) To ensure an
equal weighting of the data within the usual least-square-fitting of SciPi’s
‘curvefit’ module, we first took the logarithm of data to fit the expression
$y=\log_{10}(c)-3x$ and find $m_{B}=(0.41\pm 0.13)$ Am2.

The time-dependence of the measured signal emf $\epsilon$ is therefore
introduced by the time-dependence of the position of the magnet as it
oscillates relative to the pick-up coil with dSHM, this is in essence
combining equations 1, 18 and 20. To solve this fully, the $B\propto 1/z^{3}$
dependence would need to be taken into account. (If that dependence is taken
into account a more sawtooth like emf waveform is computed from simple
simulation and is consistent with some of our early measurements where we used
too large and initial amplitude and at too close a position.) Here we make the
further approximation that the displacement caused by the dSHM is small in
comparison with the overall distance between magnet and coil. Taking a Taylor
expansion to first order of equation 20 about the initial $z_{m}=50$ mm
position gives

$\Phi_{B}(z_{m}+\Delta z)=\left(\frac{m_{B}\mu_{0}A_{c}n}{2\pi
z_{m}^{3}}\right)-\Delta z\left(\frac{m_{B}3\mu_{0}A_{c}n}{2\pi
z_{m}^{4}}\right).$ (21)

Now combining equations 1, 18 and 21 for the dSHM, Faraday’s law and the
magnetic flux $z$-dependence gives

$\epsilon(t)=\left(\frac{m_{B}3\mu_{0}A_{c}n}{2\pi
z_{m}^{4}}\right)\frac{\textrm{d}}{\textrm{d}t}z(t)$ (22)

and so

$\epsilon(t)=\beta\left[\omega_{d}\cos\left(\omega_{d}t+\phi\right)-\frac{1}{\tau}\sin\left(\omega_{d}t+\phi\right)\right]\exp\left(-t/\tau\right)$
(23)

where we combine all the constants into one value
$\beta=m_{B}3\mu_{0}A_{c}nz_{0}/2\pi z_{m}^{4}$. The maximum emf will be just
after at $t=0$ where $\omega_{d}t+\phi=\pi/2$. That would give a predicted
maximum emf of $\sim 4$ mV which matches well the actually measured values
(see fig. 4) of $\sim 6$ mV. Note that to measure the gas properties, as the
relationship of equation 15 shows, the absolute magnitude of the emf is not
required only the angular frequency.

As the signal looked like dSHM and not the slightly more nuanced expression of
eq 23, most students struggled with the linkage of the dSHM through the
magnet/coil system using Ampere’s law.

##  4 Results and Specific Discussion

Figure 4 shows an example emf time-trace, at an initial volume 60 ml, for all
four gasses, (A) carbon dioxide, (B) nitrogen, (C) air and (D) helium. Each
has approximately the same initial emf value as they were all started by a 5
mm compression and so have the same initial conditions.

Figure 4:  Typical raw oscilloscope voltage time-traces of the emf induced in
the pick-up coil generated by the dSHM of the piston containing:(A) Carbon-
dioxide , (B) Nitrogen, (C) Air and (D) Helium. The (nearly perfectly)
overlaid red line is a dSHM model as described in the main text fitted from
$t=0$ to $t=2\tau$. The short black vertical line marks the $t=2\tau$
position. The extract values of the undamped angular frequency and decay
lifetime as given for each time-trace. Each experiment was at an initial
volume 60 ml and other parameters as in the Experimental section.
Uncertainties in fitted value of $\omega_{0}\sim\pm 3$ mrad s-1, uncertainties
in $\tau\sim\pm 7$ ms.

All time-traces display the same main features: that there is an oscillation
of approximately the same frequency, and an overall decay of that oscillation.
To extract precise values for these parameters the SciPi package curvefit was
used with equation 23 with fitting parameters $\beta,\omega_{d},\phi$ and
$\tau$. Since each time-trace experiments was started by hand and therefore
there will be a natural range of initial $z_{0}$ initial displacement, and
since the absolute magnitude of the emf is not required to find the adiabatic
gas parameters, all the amplitude terms were simply combined into the one
fitting parameter $\beta$. The (near perfectly) overlaid red curves in figure
4 are best fits of this model to the various data sets. The fitted values of
$\omega_{d}$ and $\tau$ are displayed on the relevant plots.

To allow automated and unbiased fitting, the following procedure coded into a
python script. The maximum emf value was found and this point was set at
$t=0$, all data points before that were removed from the fitting data-set. The
lowest (most negative) data point was found, and by comparing that time with
the time of the maximum emf value an approximate value of $\omega_{d}$ was
determined. To determine an approximate value for $\tau$ a simple single
exponential decay was fitted to the absolute value of the emf signal as a
function of time. Finally using these parameters an initial value of $\beta$
was determined to ensure the $t=0$ value of equation 23 returned the correct
$t=0$ value of the measured signal. Using these initial inputs for the curve-
fitting routine equation 23 was fitted to the data-set.

It was clear (not shown here) that at very low amplitude, below 1 mV, there
was a decrease in the damping life-time. At these very low amplitudes it maybe
that there is a increase of the damping friction as the system begins to feel
the effect of static friction. Therefore, finally the same model was fitted
again, but using the best-fit parameters found from that first fit and only
fitting from $t=0$ to $t=2\tau$. This can be seen in the extent of the red-
line fit and vertical bar for each of the data-sets shown in figure 4.

Figure 5:  Series of typical emf time-traces for nitrogen gas over 10 cylinder
volumes from 100 ml to 10 ml as indicated. To aid clarity each has been
shifted vertically by 20 mV. The overlaid red line is, again, a line of best
fit to a dSHM model that was fitted over a time scale of $2\tau$ for each
individual time-trace. The short black vertical line marks the $t=2\tau$
positions.

Figure 5 shows emf time-traces for nitrogen gas for all 10 measured initial
volumes ranging from 100 ml to 10 ml in 10 ml steps. The (near perfectly)
overlaid red line of the fitting function shows that the model developed
culminating in the expression of eq. 23 accurately describes the experiment
across all gasses and experimental parameters. What is evident (for all
gasses) is a near constant decay lifetime from 100 ml to 60 ml followed by a
dramatic reduction below that volume. There is also an obvious increase in the
rate of oscillation as the initial volume decreases exactly in line with th
expectation of the dSHM and adiabatic gas properties derived in section 2.3.
As the initial volume shrinks, so the adiabatic compression results in a
stiffer response (spring constant) and hence higher oscillating frequency.

Figure 6 presents the three fitting parameters, amplitude of emf, decay
lifetime and undamped angular frequency, for all four gasses across all
volumes. Each data point and associated error bar is the result of averaging
across six time-traces for each set of experimental parameters.

Figure 6:  Gas and volume dependence of the parameters (A) emf amplitude
$\beta$, (B) decay lifetime $\tau$ and (C) undamped angular frequency
$\omega_{0}$. Each point and associated error bar is the average over 6 time-
traces with parameters taken from the best fit of equation 23. Gases are:
carbon dioxide, green solid line; nitrogen black dotted line; air, red dashed
line; and helium, blue dash-dot line.

The emf amplitude $\beta$, shown in panel (A), which is proportional to the
initial displacement $z_{0}$, is remarkably constant given the human
instigation of the oscillations. At very small volumes it becomes quite
difficult to depress the piston by hand and this may lead to an inadvertent
reduction of the amplitude at low ($V<25$ ml) volumes.

What is striking is the volume dependence of the decay lifetime as shown in
Fig. 6(B). It is reasonably constant and high for all gasses for initial
volumes in the range $100$ ml to $50$ ml. Below $50$ ml, however, there is a
stark reduction of the lifetime. This is the equivalent of an increase in the
damping friction. This friction to volume dependence, is similar to that
proposed to explain the earlier gas leakage rate measurements. It is therefore
likely that there is a significant difference in the friction between cylinder
and piston within the apparatus for the piston below or above the 50 ml volume
mark.

The damping lifetime seen in figure 6 shows a clear trend with carbon dioxide
having the longest lifetime (least damping), air and nitrogen almost identical
and helium the shortest lifetime (most damping). For a dSHM system the quality
$Q$-factor can be related to the energy loss per cycle $\Delta E$ and the
damping lifetime by

$Q=2\pi\frac{E}{\Delta E}=\frac{\omega_{0}\tau}{2}$ (24)

giving,

$\tau=\frac{4\pi E}{\omega_{0}\Delta E}$ (25)

where $E$ is the total energy of the SHM system. It is therefore possible to
analyse the time-traces by Fourier analysis [9] to extract both the central
frequency $\omega_{0}$ and the full width half max of the frequency domain
response to extract $\tau$. Care should be taken with such analysis as the emf
measurement is not simply dSHM, but its differential.

The usual assumption is that the damping is due to mechanical frictional that
will be constant across gasses. Another analysis [10] suggests that the energy
dissipation is dominated by heat transfer (nonadiabaticity) of the apparatus.
This is somewhat in line with the thermal conductivity (at $300$ K) of helium
$156$ mW m-1 K-1 is much greater than that of nitrogen $26$ mW m-1 K-1, air
$26.4$ mW m-1 K-1 and carbon dioxide $17$ mW m-1 K-1 [11]. Therefore any
nonadiabatic effects would be more pronounced with Helium. It would be
interesting to try heavier Nobel gasses, still with $f=3$ but with thermal
conductivities closer to the other gasses studied, e.g., Argon $18$ mW m-1
K-1. This may explain why helium’s measured $\gamma$ is lower than expected as
any nonadiabatic processes will skew the measured value of $\gamma$ towards 1.
Although that can also be the case for any contamination by air etc.

The gas and volume dependence of undamped angular frequency is presented in
figure 6(C). The rather long lifetimes, relative to the typical period of
oscillation, for volumes at or above 50 ml means there is little difference,
typically less then $+1$ rad s-1, between damped and the corrected undamped
angular frequency, see eq. 2. The frequency of oscillation was higher for
helium at all volumes, nitrogen and air were closely matched, while carbon
dioxide always had the lowest frequency. To restate for convenience eq. 15

$\omega_{0}^{2}=\left(\frac{p_{0}A^{2}}{V_{0}M}\right)\gamma,$ (26)

where the dependence of the angular frequency on the identity of the gas must
be due to difference in the adiabatic ratio $\gamma$ and so by extension the
number of degrees of freedom of the individual gases. As expected, see table
1, helium should have the highest $\gamma$ and so highest $\omega_{0}$,
nitrogen and air should be near equal and lower than helium.

In simple terms, the work done by the piston moving a certain distance is, for
helium, all converted into kinetic energy of the atoms as there are no other
degrees of freedom. This leads to the largest pressure increase which is the
restoring force of the SHM like spring system of this Rüchardt experiment. For
nitrogen and air, that mechanical work is stored as both kinetic energy and
rotational energy. Assuming an ideal gas like gas-gas (lack of) interactions,
pressure is dependent on the kinetic energy, not rotational energy, therefore
for those gasses the same piston displacement will result in a smaller
pressure response and hence appear as a softer-spring and associated reduced
frequency of oscillation. This kinetic and rotational only analysis suggests
that carbon dioxide should be identical to nitrogen and air - this is
obviously and consistently not the case.

To quantitatively determine accurate values for $\gamma$ for each gas figure
7(A) shows the same angular frequency data as figure 6(C), but now with the
true initial volume that includes the extra 6 ml offset due to the gas tubing
etc. also, note the log-log scale.

Figure 7:  (A) The same data as figure 6(C) but here with a log-log scale to
demonstrate the linear and nonlinear dependence of the frequency of
oscillation and the true volume (an additional 6 ml). (B) The extracted
fitting parameter $\gamma$ found from fitting to figure 6 across ranges of
volumes: smallest range from 100 ml to 90 ml, largest range from 100ml to 20
ml. (C) The number of degrees of freedom $f$ determined from (B). Note for
ease of viewing, the data for Air has been slightly shifted in volume so as to
not completely overlap with the nitrogen data.

It is clear that as the initial volume decreases, there is a deviation from
the linear relationship found at larger volumes. This is plain in the fitted
value of $\gamma$ shows in figure 7(B) where the fitting was performed over a
range of experimental results from 100 ml to 90 ml, to 100 ml to 20 ml (the 10
ml data was dramatically more nonliner than the 20 ml data and so omitted or
all analysis). Figure 7(C) shows the number of degrees of freedom $f$
associated with each $\gamma$ value. For the range of fitting up to and
including the 50 ml value, there is a near invariant value for $\gamma$ (and
hence $f$). This range matches the range of volumes found earlier that do not
suffer large damping effects, or possible large nonadiabatic effects. We
therefore take our best values as those found from fitting from 100 ml to 50
ml data, table 3 shows these values and the values that would have been
attained with and without the $6$ ml correction to the volume.

Table 3: Adibatic ratio $\gamma$ and uncertainty found from best fit over
$100$ to $50$ ml data, for both the corrected volume that incudes 6 ml offset,
and the uncorrected volume.

  * Gas | Corrected volume | Uncorrected volume
---|---|---
| $\gamma$ | $f$ | $\gamma$ | $f$
CO2 | $1.3093(27)$ | $6.465(57)$ | $1.196(10)$ | $10.16(51)$
N2 | $1.4057(16)$ | $4.929(19)$ | $1.284(12)$ | $7.02(29)$
Air | $1.4025(32)$ | $4.968(39)$ | $1.281(12)$ | $7.09(29)$
He | $1.5733(17)$ | $3.488(10)$ | $1.438(14)$ | $4.56(14)$

What is striking is the sensitivity to the volume that the number of degrees
of freedom $f$ has. As this is a systematic uncertainty, or offset, it is not
accounted for by the normal uncertainty analysis of the various linear least
square fitting routines. To determine an error for this volume uncertainty,
the same corrected-volume data was fitted again but with an additional
correction volume of $+1$ ml or of $-1$ ml. The half difference of the final
$\gamma$ and $f$ values from these fitting was used to determine an
uncertainty in the original fitted values of $\gamma$ and $f$, see table 4.

Table 4: Best adibatic ratio $\gamma$ found from fits over $100$ to $50$ ml
data with estimation for a 1 ml systematic uncertainty in the corrected
volume.

  * Gas | $\gamma$ | $f$
---|---|---
CO2 | $1.309(19)$ | $6.46\pm 0.39$
N2 | $1.406(20)$ | $4.92\pm 0.24$
Air | $1.403(20)$ | $4.96\pm 0.25$
He | $1.573(22)$ | $3.48\pm 0.14$

##  5 General Discussion

Heat capacities, from the standard ref [12], at 300 K of the gasses are
presented in table 5.

Table 5: Literature values for the adiabatic ratio for gasses at 300 K[12].

  * Gas | $\gamma$ | $f$
---|---|---
CO2 | $1.293$ | $6.83$
N2 | $1.400$ | $5.00$
Air | $1.402$ | $4.98$
He | $1.667$ | $3.00$
Ar | $1.667$ | $3.00$
C3H8 | $1.127$ | $15.78$

For the four studied gasses the measured experimental values from this work
agree remarkably well with the known literature values. Nitrogen and Air are
well within the experimental uncertainty. The measured value for carbon
dioxide is lower but still within the uncertainty. Only Helium appears outside
its expected range. As the heat capacity of helium has no temperature
dependence (except near $4$ K) we can assign this difference to a combination
of air contamination, and perhaps a large nonadiabatic effect.

Heat capacity is dependent on the number of degrees of freedom a gas particle
has. All gasses have three translational degrees of freedom, each mode
counting $R/2$ to constant volume molar heat capacity $C_{V,m}$. Some have
rotational degrees of freedom up to a maximum of 3, each mode adding $R/2$ to
$C_{V,m}$. Some have vibrational degrees of freedom with the each mode adding
$R$ to $C_{V,m}$. The constant volume molar heat capacity of a gas is
therefore expressed as the sum,

$C_{V,m}=\frac{1}{2}fR=\frac{1}{2}\left(3+n_{R}+2n_{v}\right)R,$ (27)

hence,

$f=3+n_{R}+2n_{v},$ (28)

where $R$ is the Universal gas constant, $n_{R}$ is the number of rotational
degrees of freedom and $n_{v}$ is the number of vibrational degrees of
freedom.

If the moment of inertia of a rotation is dependent on the size of the atomic
nucleus ($\sim 5\times 10^{-4}$ Å), rather than the length of a chemical bond
($\sim 2$ Å) then quantum mechanics dictates that that axis of rotation will
be blocked at room temperature. That is, if the energy level spacing of the
rotational energy states is much larger than the equipartition of energy
$k_{B}T/2$ that mode of rotation will be inaccessible to the molecule and so
cannot effectively store energy - it is blocked - and does not contribute to
the heat capacity. Helium has all $3$-axes of rotation blocked, the linear
molecules nitrogen and carbon dioxide will have one axis blocked and $2$ free
to rotate. As air is predominately nitrogen and oxygen, both linear diatomic
molecules, on average Air will have only two modes of rotation. This analysis
explains the measured values of $f$ for helium, nitrogen and air, but carbon
dioxide, a triatomic linear molecule therefore also with $n_{R}=2$, is higher
than $5$ suggesting it has $n_{v}=0.9$.

Like rotations, molecular vibrational only contribute to the heat capacity
when there is enough thermal energy to excite them. Generally at room
temperature this energy is insufficient to allow the molecule to explore its
full vibrational spectrum. The normal equipartition of energy assumption of
$k_{B}T/2$ will therefore not be valid, instead we must compute the partition
function for each vibration in order to compute a vibration specific $n_{v}$.

Carbon dioxide has four normal modes of vibrations, two energetically
identical bending modes, a symmetric stretch and an asymmetric stretch, see
Table 6. For small vibrational amplitudes we can consider each an example of
the quantum harmonic oscillator (another example of the Taylor expansion) with
a characteristic vibrational energy ladder given by

$E_{n}=\left(n+\frac{1}{2}\right)\hbar\omega,$ (29)

where $n$ is the quantum number of the vibration, $\hbar=$ Plank’s
constant$/2\pi$ and $\omega$ is an angular frequency characteristic of the
vibration. For a stiff vibration this will be high, resulting in more widely
spaced vibrational states in energy, and for a softer vibration the states
will be closer together. Therefore for a fixed temperature, here 300 K, it is
the relative scale of that vibrational ladder to the equipartition of energy
that will dictate if that vibration adds, and by how much, to the number of
degrees of freedom and hence the adiabatic ratio. The standard textbook of
chemical information, the CRC Handbook of Chemistry and Physics, has tables of
these parameters, but as it is written for a chemistry audience, physics
students may struggle in converting from, usually a parameter in cm-1 to the
more typical eV energy unit. From tables [13] and [14] of the CRC Handbook we
can find the information required.

The diatomic molecule nitrogen has only one vibrational mode, namely the
symmetric stretch.

Table 6: Details of the vibrational modes of carbon dioxide and nitrogen
gasses. Schematic digrams of vibrations, the vibrational excitation energy
($\Delta E$), their partitions $q^{V}$ function at 300 K and the corresponding
number of degrees of freedom $n_{v}$ [13, 14].

  * Gas | Vibration | Diagram |  $\Delta E$ / eV | $q^{V}$ | $n_{v}$
---|---|---|---|---|---
CO2 | Bend |  | 0.0827 | 1.043 | 0.454
CO2 | Sym |  | 0.165 | 1.002 | 0.069
CO2 | Asym |  | 0.291 | 1.000 | 0.002
N2 | Sym |  | 0.292 | 1.000 | 0.002

For each mode the partition function $q^{V}$ was computed via

$q^{V}=\frac{1}{1-\exp{\left(-\Delta E/k_{B}T\right)}}$ (30)

at 300 K and the corresponding number of modes,

$n_{v}=\left(\frac{\Delta E}{k_{B}T}\right)^{2}\frac{\exp{\left(-\frac{\Delta
E}{k_{B}T}\right)}}{\left[1-\exp{\left(-\frac{\Delta
E}{k_{B}T}\right)}\right]^{2}}.$ (31)

Figure 8:  Computed temperature dependence of the number of degrees of freedom
of the four vibrational modes of carbon dioxide, the bending modes, the
asymmetric stretch and the symmetric stretch. See main text and table 6 for
details.

see ref [15] for details, and finally

$f=3+2+2\sum_{v}n_{v}.$ (32)

Table 6 shows the computed $n_{v}$ values. As expected at room temperature the
stretching vibration of nitrogen is not excited and so $n_{v}\approx 0$ giving
overall for nitrogen $f=5$ in good agreement with our measured value and the
literature values.

For carbon dioxide the lower energy bending mode is, however, quite excited,
that is, it contributes to the heat capacity, whereas the two stretching modes
contribute only a fractional amount. Fig. 8 show for CO2 the temperature
dependence of the nuber of degrees of freedom for each of its three modes of
vibrations and its overall total number of vibrational degrees of freedom. The
difference in the stiffness of the three modes is evident in the different
computed temperature onsets for each mode. At room-temperature we can see that
although the two identical bending modes are excited they are not full excited
and so do not make a complete (or integer) contribution to the overall number
of vibrational degrees of freedom. Instead for carbon dioxide this analysis
predicts $\sum_{v}n_{v}=0.979$ noting that there are two bending modes with
the same energy. This therefore means a predicted value of $f=6.958$ which is
in near agreement with the literature value of $6.83$ and within the
uncertainty of our measured value of $6.46\pm 0.39$.

A similar analysis could be attempted for the rotational modes, but students
may struggle to understand that quantisation, whereas the quantum harmonic
oscillator is usually covered in most early years physics courses.

###  5.1 Proposed future work

As further work the gasses argon and propane are suggested as they lie at the
extreme ends of the available $\gamma$ values. Argon is a monotonic gas like
helium but its larger mass means it had a much smaller thermal conductivity,
more akin to carbon dioxide, and so it maybe possible to prevent, or measure,
any heat-transfer to of from the apparatus. Propane has a large number of
degrees of freedom and from that it may be able to empirically determine a
$\tau$ vs. $f$ relationship to compare with theory. Finally attaching a small
vacuum pump to the gas inlet system may reduce the chance of any contamination
of the gasses.

##  6 Conclusion

Through careful early years undergraduate level physics analysis of the
experimental apparatus, the collected data and underlying theory, we show that
the Rüchardt experiment is able to measure and differentiate between the
adiabatic ratio, and hence degrees of freedom, of several small gasses. We
show that a complete description of the work requires mechanical dynamics,
electromagnetism, thermodynamics, statistical mechanics and quantum mechanics.
This makes it an ideal project to bridge the gap between scripted experiments
and more advanced open-ended projects students typically tackle in their final
years.

##  7 CRediT author statement

Chris Shearwood: Investigation, Resources, Methodology Peter Sloan
Conceptualization, Software, Formal analysis, Data Curation, Writing,
Visualization

## References

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