11affiliationtext: Department of Physics, Alagappa Chettiar Government College
of Engineering and Technology, Karaikudi, Tamilnadu-630 003,
India22affiliationtext: Department of Physics, Nehru Memorial College,
Puthanampatti, Tiruchirapalli, Tamilnadu - 621 007, India (Affiliated to
Bharathidasan University, Tiruchirapalli, Tamilnadu - 620 024,
India)33affiliationtext: Department of Electrical and Electronic Engineering,
Faculty of Engineering and Technology (FET), University of Buea, P.O. Box 63,
Buea, Cameroon44affiliationtext: Department of Physics, KCG College of
Technology, Chennai - 600 097, India

# Emergence of chaotic hysteresis in a second-order non-autonomous chaotic
circuit 111Preprint submitted to Chaos Solitons & Fractals

G. Sivaganesh  K. Srinivasan T. Fonzin Fozin R. Gladwin Pradeep

###### Abstract

The observation of hysteresis in the dynamics of a third-order autonomous
chaotic system namely, the Chua’s circuit has been reported recently [1]. In
the present work, we make a detailed study on the emergence of dynamical
hysteresis in a simple second-order non-autonomous chaotic system namely, the
Murali-Lakshmanan-Chua (MLC) circuit. The experimental realization of chaotic
hysteresis is further validated by numerical simulation and analytical
solutions. The presence of chaotic hysteresis in a second-order non-autonomous
electronic circuit is reported for the first time. Multistable regions are
observed in the dynamics of MLC with constant bias.

Keywords: Bifurcation; Chaos; MLC circuit; Hysteresis; Multistability

##  1 Introduction

The rise of computers and computing power in the latter part of the 20th
century facilitated the numerical study of many dynamical systems, reigniting
interest in understanding chaotic dynamical systems. Chaotic behavior, which
is prevalent in nature, is characterized by extreme sensitivity to initial
conditions, where even a small variation in initial conditions can result in a
vastly different future state [2]. Understanding chaotic dynamics is crucial
in comprehending a wide range of natural systems. Furthermore, artificial
dynamical systems were created as part of the effort to grasp chaotic
dynamics, which led to the discovery of phenomena such as stochastic
resonance, chaotic synchronization, and chaos control [3, 4, 5, 6]. Chaotic
electronic circuits are important for studying chaos because they are easy to
construct and control [7, 8]. Chua’s circuit is one of the most extensively
studied chaotic circuits and has led to the experimental realization of
stochastic resonance, chaotic synchronization, and control of chaos [9, 10,
11, 12]. It is a piece-wise linear third-order autonomous system that has also
been studied analytically [13]. Recent research has reported hysteresis
behavior in Chua’s circuit with a DC offset voltage [1]. The variation of the
DC offset from negative to positive values yields different dynamical paths,
resulting in hysteresis in the system dynamics. The Murali-Lakshmanan-Chua
(MLC) circuit, a second-order non-autonomous circuit constructed with Chua’s
diode as the nonlinearity [14], has also been instrumental in the experimental
realization of various chaotic dynamics [15, 16]. The MLC circuit and its
variants have wide applications in secure communication, image encryption,
electronic noise generation, chaotic logic gates, and reservoir computing [17,
18, 19, 20, 21]. Furthermore, analytical solutions for a class of second-order
chaotic systems have been reported recently [22].

In this study, we discuss the appearance of chaotic hysteresis in the behavior
of the MLC circuit resulting from the introduction of a DC offset voltage in
the circuit. We examine the variations of the offset voltage that lead to the
hysteresis phenomenon in the system dynamics and validate our electronic
circuit experimental results through numerical and analytical results. The
paper is structured as follows: in section 2, we present the hysteresis
phenomenon observed in the MLC circuit using experimental phase portraits; in
section 3, we present the numerical simulation results that explain the
emergence of hysteresis through bifurcation diagrams and the multistability in
the system is analyzed using offset boosting method; and in section 4, we
report the analytical solutions developed for the normalized circuit equations
to study the hysteresis phenomenon. Finally, we provide our conclusions in
section 6.

##  2 Chaotic hysteresis: Experimental results

The circuit realization of the simple non-autonomous (MLC) circuit is shown in
Fig. 1. It contains a capacitor, an inductor, a linear resister, an external
periodic forcing and only one nonlinear element, namely, the Chua’s diode
$(N_{R})$. In order to measure the inductor current $i_{L}$ in our
experiments, we insert a small current sensing resister $R_{s}$ as shown Fig.
1.

Figure 1: Circuit realization of periodically driven MLC circuit with a DC
offset voltage $E$. Here, $N_{R}$ is the Chua’s diode. The parameter values of
the other elements are fixed as $L=18.0~{}mH$, $C=10.0~{}nF$,
$R=1340~{}\Omega$ and $R_{s}=20~{}\Omega$. Figure 2: a) Schematic realization
of Chua’s diode $N_{R}$ using Op-Amps. (b) Voltage-Current characteristics of
Chua’s diode indicating its piecewise-linear nature.

By applying Kirchhoff’s laws to this circuit, the governing equations for the
voltage $v$ across the capacitor $C$ and the current $i_{L}$ through the
inductor $L$ are represented by the following set of two first-order non-
autonomous differential equations as

$\displaystyle C{dv\over d\tau}$ $\displaystyle=$ $\displaystyle i_{L}-g(v),$
(1a) $\displaystyle L{di_{L}\over d\tau}$ $\displaystyle=$
$\displaystyle-(R+R_{s})i_{L}-v+Fsin(\Omega\tau)+E,$ (1b)

where $F(t)$ is the sinusoidal periodic force with amplitude $F$ and angular
frequency $\Omega$. The term $E$ represents a constant voltage added in
addition to the sinusoidal voltage source ($F(t)$) is called as the $DC$
offset voltage which can be negative or positive. The value of $DC$ offset
voltage shifts the signals to the same amount of its value. The term $g(v)$
represents the $v-i$ characteristics of the _Chua’s_ diode as shown in Fig. 2
and is given by

$\displaystyle g(v)=G_{b}v+0.5(G_{a}-G_{b})[|v+B_{p}|-|v-B_{p}|],$ (2a) or in
the piecewise-linear form it is given as $\displaystyle
g(v)=\begin{cases}{G_{b}}v+({G_{a}}-{G_{b}})&\text{if $v\geq 1$}\\\
{G_{a}}v&\text{if $|v|\leq 1$}\\\ {G_{b}}v-({G_{a}}-{G_{b}})&\text{if
$v\leq-1$}\end{cases}$ (2b)

Figure 3: Chaotic hysteresis observed in MLC circuit with variation of offset
voltage $E$. The system is set for a one-band chaotic attractor state at
$F=0.11~{}V$ with $E=0$. Phase-portraits in the left-half of $v-i_{L}$ plane:
(a) $E=-0.02~{}V$, period-1T LC (b) $E=-15~{}mV$, period-2T LC (c)
$E=-8.0~{}mV$, period-4T LC (d) $E=-2.5~{}mV$, one-band chaos and (e)
$E=0.1~{}mV$, double-band chaotic attractor. Phase-portraits in the right-half
of $v-i_{L}$ plane: (f) $E=2.5~{}mV$, one-band chaos (g) $E=8.0~{}mV$,
period-4 T LC (h) $E=15~{}mV$, period-2T LC (i) $E=0.02~{}V$, period-1T LC.
LC-Limit Cycle.

The construction of Chua’s diode using operational amplifiers (Op-Amps) and
its corresponding Voltage-Current characteristics is shown in Fig. 2. The
parameters of the circuit elements are fixed at
$L=18.0~{}mH,C=10.0~{}nF,R=1340~{}\Omega,R_{s}=20~{}\Omega,G_{a}=-0.76~{}mS,G_{b}=-0.41~{}mS$
and $B_{p}=1.0~{}V$ and the frequency $(\nu=\Omega/2\pi)$ of the external
forcing source is $8890~{}Hz$. The parameters $F(t)$ and $E$ are taken as
control parameters. The dynamical state of the circuit can be kept either in
the chaotic or periodic state and the offset voltage $E$ is varied to observe
the changes in the dynamics of the system. Figure 3 shows the experimental
dynamics of the $MLC$ circuit by varying the offset voltage $E$ with the
circuit initially operated at the one-band chaotic attractor state for
$F=0.11~{}V$ when $E=0$. Hence, in the absence of the DC offset, the system
operates in the one-band chaotic attractor state. The DC offset voltage is
increased from negative to positive values and the dynamics observed in the
$v-i_{L}$ plane is reported. A period-1 limit cycle is observed for
$E=-0.02~{}V$ leading to period-2 and period-4 limit cycles for $E=-15~{}mV$
and $E=-8.0~{}mV$ as shown in Fig. 3(a), 3(b) and 3(c), respectively. Further
increase in $E$ leads to an one-band chaotic attractor in the left-half plane
for $E=-2.5~{}mV$ followed by a double-band chaotic attractor at $E=0.1~{}mV$
as shown in Fig. 3(d) and 3(e), respectively. With increase in $E$, the
dynamics of the system jumps to the right-half plane and results in the
evolution of an one-band chaotic attractor shown in Fig. 3(g) at $E=2.5~{}mV$.
Further increase in $E$ indicates a reverse period-doubling sequence
represented by the period-4, period-2 and period-1 limit cycles for the values
$E=8.0~{}mV$, $E=15~{}mV$ and $E=0.02~{}V$ as shown in Fig. 3(g), 3(h) and
3(i), respectively. However, when we start from the period-1 limit cycle at
the right-half plane, the decrease in $E$ results in the system dynamics
indicated by Figs. 3(i), 3(h) and 3(g), 3(f) and gives rise to the double-band
chaotic attractor of Fig. 3(e). Further decrease in $E$ results in the jumping
of the dynamics to the left-half plane through the evolution of the one-band
chaotic attractor shown in Fig. 3(d) followed by the reverse period-doubling
sequence of the periodic attractors in the left-half plane. The region of $E$
encompassing the evolution of one-band chaotic attractor in the left-half
plane followed by the double-band chaotic attractor leading to the one-band
chaotic attractor in the right-half plane when $E$ is increased from negative
values and the similar dynamics observed when $E$ is decreased such that the
one-band chaotic attractor transits from the right-half plane to the left-half
plane through the evolution of the double-band chaotic attractor represents
the region of chaotic hysteresis observed in the circuit system.

##  3 Chaotic hysteresis: Numerical results

In this section, we report the emergence of chaotic hysteresis in the MLC
circuit through numerical simulation of the normalized circuit equations.

###  3.1 Hysteresis evidence

The normalized equations obtained from Eq. (1) is written as

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle y-g(x),$ (3a)
$\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle-\sigma y-\beta
x+fsin(\omega t)+c,~{}~{}~{}~{}~{}\left(\cdot=\frac{d}{dt}\right)$ (3b)

where $t=(\tau G/C)$, $\sigma=(\beta+\nu\beta)$, $\beta=(C/LG^{2})$,
$\nu=GR_{s}$, $a=G_{a}/G$, $b=G_{b}/G$, $f=(F\beta/B_{p})$, $\omega=(\Omega
C/G)$, $c=(E\beta/B_{p})$, $G=1/R$. The nonlinear function $g(x)$ is written
in its normalized form as

$\displaystyle g(x)=bx+0.5(b-a)[|x+1|-|x-1|]$ (4a) In piecewise-linear form
$g(x)$ is $\displaystyle g(x)=\begin{cases}bx+(a-b)&\text{if $x\geq 1$}\\\
ax&\text{if $|x|\leq 1$}\\\ bx-(a-b)&\text{if $x\leq-1$}\end{cases}$ (4b)

Figure 4: Numerical results: One-parameter bifurcation diagram in the $c-x$
plane obtained in the range $-0.025<c<0.002$ for the initial conditions
($x_{0},y_{0}$) =($-0.1,-0.1$) is indicated in violet while the bifurcation
diagram in the range $0.025>c>-0.002$ indicated in red is obtained for the
initial condition $(x_{0},y_{0})=(0.1,-0.1)$. The region of the bifurcation
diagram in violet corresponds to the attractor in the LHP while the region in
red corresponds to the attractors in the RHP of the phase space. The vertical
dashed lines in violet/red indicate the dynamics of the system at that point
of $c$ which is also represented by the corresponding colored phase-portraits
in the $(x-y)$ plane. The system jumps from the LHP to RHP at $c=0.002$ in the
one-band chaotic state while the dynamics jumps from the RHP to LHP at
$c=-0.002$ in the one-band chaotic attractor state. The region of $c$ in the
range $-0.002\leq c\leq 0.002$ indicates the region of chaotic hysteresis.
Figure 5: Numerical result: Maximum Lyapunov exponent when varying parameter
$c\in[-2.5,2.5]\times 10^{-2}$ in the MLC oscillator.

The phenomenon of chaotic hysteresis is studied using the one-parameter
bifurcation diagram obtained through numerical simulation of the circuit
equations. The normalized system parameters are fixed as
$a=-1.02,b=-0.55,\beta=1.0,\nu=0.015,\omega=0.75$. Figure 4 shows the one-
parameter bifurcation diagram obtained for two different initial conditions
and for the existence of the attractors in the left-half plane and right-half
plane. The system is initially set at the one-band chaotic attractor state for
$f=0.1104$ when the offset voltage parameter $c=0$. The period-doubling
sequence leading to chaos obtained for the initial condition
$(x_{0},y_{0})=(-0.1,-0.1)$ by sweeping the parameter $c$ in the range
$-0.025<c<0.002$ is represented in violet while that obtained for the initial
condition $(x_{0},y_{0})=(0.1,-0.1)$ by sweeping the parameter $c$ in the
range $0.025>c>-0.002$ is represented in red. The periodic and the chaotic
attractors plotted at their respective positions are indicated by the vertical
lines of their respective colors. The values of $c$ in the range
$-0.025<c<0.002$ represented in violet indicate the attractors in the left-
half plane (LHP) and the attractors in the right-half plane (RHP) are
indicated in red in the range $0.025>c>-0.002$. In particular, the system
exhibits chaotic attractors in the LHP at the values $c=-0.002$ and $c=0$ and
a double-band chaotic attractor evolves at $c=0.0001$. At $c=0.002$, the
system jumps to the RHP as a one-band chaotic attractor. Similarly, varying
$c$ in the range $0.025c-0.002$ represented in red indicates the existence of
attractors in the RHP with one-band and double-band chaotic attractors
observed at the values $c=0.002$ and $c=-0.0001$, respectively. At $c=-0.002$,
the system jumps to the LHP as a one-band chaotic attractor. Hence, the
parameter range $-0.002\geq c\geq 0.002$ indicates the emergence and existence
of chaotic hysteresis when it is initially operated in the one-band chaotic
attractor state. It has to be noted that the system exists in the one-band
chaotic attractor state for $c=0$ and its location in the phase plane
corresponds to the dynamical path of operation i.e. either from the LHP or
RHP. The maximum Lyapunov exponent of the MLC circuit obtained as a function
of $c$ indicating the regions of periodic and chaotic behavior is as shown in
Fig. 5.

##  4 Chaotic hysteresis: Analytical solutions

In this section, we report the explicit analytical solutions developed for the
state equations of the MLC circuit with offset voltage and study the
hysteresis of the system dynamics using the analytical solutions. The MLC
circuit system represented by Eq. (3) is solved for the variables $x(t),y(t)$
in each of the piecewise-linear region and the solutions are matched across
the break-point regions for generating the attractors in the phase plane. The
analytical solutions for each region is summarized as follows.

###  4.1 $D_{0}$ region

In this region, $g(x)=ax$ and the state equations are written as

$\displaystyle{dx\over dt}$ $\displaystyle=$ $\displaystyle y-ax,$ (5a)
$\displaystyle{dy\over dt}$ $\displaystyle=$ $\displaystyle-\beta y-\nu\beta
y-\beta x+f\sin(\omega t)+c,$ (5b)

The Jacobian matrix determining the stability of the fixed point
$p_{0}=(-\frac{c}{a+\beta},\frac{ac}{2\beta+\nu\beta})$ in this region is

$J_{0}=\begin{pmatrix}-a&&&1\\\ -\beta&&&-\beta-\nu\beta\\\ \end{pmatrix},$
(6)

and the fixed point $p_{0}$ is a saddle or hyperbolic fixed point. The fixed
point is $(0,0)$ when $c=0$ and its location in the phase-plane changes with
the variation of $c$. Equation (5) written as a second-order differential
equation is given as

${\ddot{y}}+{A_{1}\dot{y}}+B_{1}y=\Delta_{1}+af\sin(\omega
t)+f\omega\cos(\omega t),$ (7)

where $A_{1}=\beta+\nu\beta+a$, $B_{1}=\beta+a(\beta+\nu\beta)$ and
$\Delta_{1}=ac$. For the given values of the system parameters, the roots of
Eq. (7) $m_{1,2}=\frac{-A_{1}}{2}\pm\frac{\sqrt{{A_{1}}^{2}-4B_{1}}}{2}$ are
real and distinct and the state variables are obtained as

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle
C_{1}e^{m_{1}t}+C_{2}e^{m_{2}t}+E_{1}+E_{2}\sin(\omega t)+E_{3}\cos(\omega
t),$ (8a) $\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}(-(\beta+\nu\beta)y-\dot{y}+c+f\sin(\omega t)),$
(8b)

where, the constants $E_{1},E_{2},E_{3},C_{1},C_{2}$ are

$\displaystyle E_{1}=$ $\displaystyle\frac{\Delta_{1}}{B_{1}}$ (9a)
$\displaystyle E_{2}=$
$\displaystyle\frac{f{\omega}^{2}(A_{1}-a)+afB_{1}}{A_{1}^{2}{\omega}^{2}+(B_{1}-{\omega}^{2})^{2}}$
(9b) $\displaystyle E_{3}=$
$\displaystyle-\frac{f\omega(A_{1}a+\omega^{2}-B_{1})}{A_{1}^{2}{\omega}^{2}+(B_{1}-{\omega}^{2})^{2}}$
(9c) $\displaystyle C_{1}=$
$\displaystyle\frac{e^{-m_{1}t_{0}}}{m_{1}-m_{2}}\\{(-(\beta+\nu\beta+m_{2})y_{0}-\beta
x_{0}+c+m_{2}E_{1})$ (9d) $\displaystyle-(\omega E_{2}-m_{2}E_{3})\cos\omega
t_{0}+(\omega E_{3}+m_{2}E_{2}+f)\sin\omega t_{0}\\}$ $\displaystyle C_{2}=$
$\displaystyle\frac{e^{-m_{2}t_{0}}}{m_{2}-m_{1}}\\{(-(\beta+\nu\beta+m_{1})y_{0}-\beta
x_{0}+c+m_{1}E_{1})$ (9e) $\displaystyle-(\omega E_{2}-m_{1}E_{3})\cos\omega
t_{0}+(\omega E_{3}+m_{1}E_{2}+f)\sin\omega t_{0}\\}$

###  4.2 $D_{+1}$ region

In $D_{+1}$ region, $g(x)=bx+(a-b)$ and the state equations are written as

$\displaystyle{dx\over dt}$ $\displaystyle=$ $\displaystyle y-bx-(a-b),$ (10a)
$\displaystyle{dy\over dt}$ $\displaystyle=$ $\displaystyle-\beta y-\nu\beta
y-\beta x+f\sin(\omega t)+c,$ (10b)

The stability of the fixed point
$p_{1}=(\frac{(\beta+\nu\beta)(b-a)+c}{\beta(1+b(\beta+\nu\beta))},\frac{\beta(a-b)+bc}{b(\beta+\nu\beta)+\beta})$
is determined using the Jacobian matrix

$J_{0}=\begin{pmatrix}-b&&&1\\\ -\beta&&&-(\beta+\nu\beta)\\\ \end{pmatrix},$
(11)

and $p_{1}$ is found to be a stable focus fixed point. Equation (10) can be
written as a second-order differential equation in terms of the variable $y$
as

${\ddot{y}}+{A_{2}\dot{y}}+B_{2}y=\Delta_{2}+bf\sin(\omega
t)+f\omega\cos(\omega t),$ (12)

where $A_{2}=\beta+\nu\beta+b$, $B_{2}=\beta+b(\beta+\nu\beta)$ and
$\Delta_{2}=bc+\beta(a-b)$. For the given values of the system parameters, the
roots of Eq. (12) $m_{3,4}=u\pm iv$, where,
$u=\frac{-A_{2}}{2},v=\frac{\sqrt{4B_{2}-{A_{2}}^{2}}}{2}$ are a pair of
complex conjugates and the state variables are obtained as

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle e^{ut}(C_{3}\cos
vt+C_{4}\sin vt)+E_{4}+E_{5}\sin(\omega t)+E_{6}\cos(\omega t),$ (13a)
$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}(-(\beta+\nu\beta)y-\dot{y}+c+f\sin(\omega t)),$
(13b)

The constants $E_{4},E_{5},E_{6}$ are the same as $E_{1},E_{2},E_{3}$ of Eq.
(9) except the constants $\Delta_{1},a$ are replaced with $\Delta_{2},b$,
respectively. The constants of the complementary function $C_{3},C_{4}$ are

$\displaystyle C_{3}=$
$\displaystyle\frac{e^{-ut_{0}}}{v}\\{((\beta+\nu\beta+u){y_{0}}+\beta{x_{0}}-c)\sin
vt_{0}+(vy_{0}-vE_{4})\cos vt_{0}-uE_{4}\sin vt_{0}$ $\displaystyle+((\omega
E_{5}-uE_{6})\sin vt_{0}-vE_{6}\cos vt_{0})\cos\omega t_{0}$
$\displaystyle-((\omega E_{6}+uE_{5}+f)\sin vt_{0}+vE_{5}\cos
vt_{0})\sin\omega t_{0}\\}$ $\displaystyle C_{4}=$
$\displaystyle\frac{e^{-ut_{0}}}{v}\\{(c-(\beta+\nu\beta+u){y_{0}}-\beta{x_{0}})\cos
vt_{0}+(vy_{0}-vE_{4})\sin vt_{0}+uE_{4}\cos vt_{0}$ $\displaystyle-((\omega
E_{5}-uE_{6})\cos vt_{0}+vE_{6}\sin vt_{0})\cos\omega t_{0}$
$\displaystyle+((\omega E_{6}+uE_{5}+f)\cos vt_{0}-vE_{5}\sin
vt_{0})\sin\omega t_{0}\\}$

###  4.3 $D_{-1}$ region

In this region, $g(x)=bx-(a-b)$ and the state equations are written as

$\displaystyle{dx\over dt}$ $\displaystyle=$ $\displaystyle y-bx+(a-b),$ (14a)
$\displaystyle{dy\over dt}$ $\displaystyle=$ $\displaystyle-\beta y-\nu\beta
y-\beta x+fsin(\omega t)+c,$ (14b)

The stability of the fixed point
$p_{2}=(\frac{(\beta+\nu\beta)(a-b)+c}{\beta(1+b(\beta+\nu\beta))},\frac{\beta(b-a)+bc}{b(\beta+\nu\beta)+\beta})$
is obtained using Eq. (11) and $p_{2}$ is found to be a stable focus fixed
point. The state equation written in terms of the variable $y$ is

${\ddot{y}}+{A_{3}\dot{y}}+B_{3}y=\Delta_{3}+bfsin(\omega t)+f\omega
cos(\omega t),$ (15)

where $A_{3}=\beta+\nu\beta+b$, $B_{3}=\beta+b(\beta+\nu\beta)$ and
$\Delta_{3}=bc-\beta(a-b)$. The roots of eq. (15) are a pair of complex
conjugates given as $m_{3,4}=u\pm iv$, where,
$u=\frac{-A_{3}}{2},v=\frac{\sqrt{4B_{3}-{A_{3}}^{2}}}{2}$ and the state
variables are obtained as

$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle e^{ut}(C_{5}\cos
vt+C_{6}\sin vt)+E_{7}+E_{8}\sin(\omega t)+E_{9}\cos(\omega t),$ (16a)
$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}(-(\beta+\nu\beta)y-\dot{y}+c+f\sin(\omega t)),$
(16b)

The constants $E_{7},E_{8},E_{9},C_{5},C_{6}$ are the same as
$E_{4},E_{5},E_{6},C_{3},C_{4}$ of $D_{+1}$ region except that $\Delta_{2},a$
is replaced with $\Delta_{3}$. The analytical solutions are used to generate
the trajectories in the $(x-y)$phase plane of the system by varying a control
parameter of the system for the fixed values of other parameters.

The emergence of chaotic hysteresis in the system observed using the explicit
analytical solutions briefed above is depicted in Fig. 6. The MLC circuit is
initially set for the one-band chaotic attractor state at $c=0$. The one-band
chaotic attractor is so chosen such that it exists at the edge of the one-band
chaotic state. The parameter values are kept at the values
$a=-1.02,b=-0.55,\beta=1.0,\nu=0.015,f=0.1105,\omega=0.75$. Because of the
dependence of the fixed points $p_{0},p_{1},p_{2}$ on the parameter $c$, a
smaller shift in the position of the fixed points in the phase plane is
observed while $c$ is varied. However, the stability of the fixed points
remains unaffected as they are independent of $c$. Increasing $c$ value from
$c=-0.02$ with the system operating initially ($c=0$) at the one-band chaotic
attractor state around the fixed point $p_{2}$ in clockwise direction in the
LHP for the initial condition $x_{0},y_{0}=-0.1,-0.1$ yields the dynamical
path $ABCDEFJIHG$. The violet colored region indicates the dynamics of the
existing in the LHP while the red colored corresponds to the dynamics in the
RHP. The phase-portraits existing in the LHP corresponding to the points of
the dynamical path is presented near each point. The different dynamical
behaviors of the system at each of the points is briefed in Table 1. From Fig.
6 it is observed that the one-band chaotic attractor existing in the LHP (E)
existing around the fixed point $p_{2}$ at $c=0$ jumps to the RHP as an one-
band chaotic attractor (J) evolving around the fixed point $p_{1}$ in the
clockwise direction at $c=0.0025$ through the evolution of the double-band
chaotic attractor (F) at $c=0.0001$ and the system exhibits reverse period-
doubling sequence with the attractors existing in the RHP represented by
attractors and path in red. Similarly, decreasing $c$ from $c=0.02$ with the
system operating initially ($c=0$) at the one-band chaotic state in the RHP
around the fixed point $p_{1}$ yields the dynamical path $GHIJKLDCBA$. The
transition of the one-band chaotic attractor state existing around $p_{1}$ in
the RHP at $c=0$ to an attractor existing around $p_{2}$ in the LHP at the
value $c=-0.0025$ (D) is observed through the evolution of a double-band
chaotic attractor state at c=-0.0001 (L). The intersection of the paths
$ABCDEFJIHG$ and $GHIJKLDCBA$ yields the region of chaotic hysteresis
represented by $DEFJKL$ which is in confirmation with the dynamical region
represented in the numerical simulation results shown in Fig. 4.

Dynamical Point | $c$ | Dynamical state | LHP/RHP
---|---|---|---
A | -0.02 | period-1T LC | LHP
B | -0.015 | period-1T LC | LHP
C | -0.0078 | period-4T LC | LHP
D | -0.0025 | One-band chaos | LHP
E | 0.0 | One-band chaos | LHP
F | 0.0001 | Double-band chaos | Both
G | 0.02 | period-1T LC | RHP
H | 0.015 | period-1T LC | RHP
I | 0.0078 | period-4T LC | RHP
J | 0.0025 | One-band chaos | RHP
K | 0.0 | One-band chaos | RHP
L | -0.0001 | Double-band chaos | Both
Table 1: Dynamics of the MLC system at the dynamical points specified in Fig.
6. LC-Limit Cycle. Figure 6: Analytical results: Chaotic hysteresis observed
in $MLC$ circuit through explicit analytical solutions. The system is
initially operated in the one-band chaotic attractor state ($c=0$) existing in
the left-half or right-half plane. The increase of the parameter $c$ from in
the range $-0.02\leq c\leq 0.02$ with the attractor fixed in the left-half
plane ($c=0$) yields the dynamical path $ABCDEFJIHG$. The chaotic attractors
in the $x-y$ plane is plotted adjacent to each of the dynamical path points.
Conversely, the dynamical path $GHIJKLDCBA$ is observed when the attractor is
initially set at the right-half plane ($c=0$) and for $c$ varying in the range
$0.02\geq c\geq-0.02$. The intersection of these dynamical paths reveals the
existence of the chaotic hysteresis represented by the path $DEFJKL$. The
jumping of the attractor from the left-halt to right-half plane and vice-versa
occurs through the evolution of the double-band chaotic attractor state.
Figure 7: Analytical results indicating the phenomena of chaotic hysteresis in
MLC circuit for the offset voltage factor $c$ as the control parameter. (a)
Period-T attractor for $c=-0.04$, (b) Period-2T attractor for $c=-0.03$ and
(c) one-and chaotic attractor for $c=-0.01$ obtained in the LHP around the
fixed point $p_{1}$; (d) double-band chaotic attractor for $c=0$; (e) one-and
chaotic attractor for $c=0.01$, (f) Period-2T attractor for $c=0.03$ and (g)
Period-T attractor for $c=0.04$ obtained in the RHP around the fixed point
$p_{2}$. Figure 8: Analytical results indicating chaotic hysteresis in the MLC
circuit with the variation of DC offset voltage factor $c$. The labels $a-g$
in the figure corresponds to the phase-portraits in Fig. 7(a)-7(g).

The chaotic hysteresis phenomenon observed in the $MLC$ circuit with the
system initially operating in the double-band chaotic attractor state is
depicted in Fig. 7. The circuit is operated at the double-band chaotic state
for $c=0$ and the other parameters are set as
$a=-1.02,b=-0.55,\beta=1.0,\nu=0.015,f=0.14,\omega=0.75$. The phenomenon of
chaotic hysteresis is observed as the parameter $c$ is increased in the range
$-0.04\leq c\leq 0.04$. The system exhibiting a period-T limit cycle attractor
at $c=-0.04$ about the fixed point $p_{2}$ evolves into a period-2T limit
cycle at $c=-0.03$ before settling down into an one-band chaotic attractor at
$c=-0.01$ as shown in Figs. 7(a), 7(b) and 7(c), respectively. Increasing the
value of $c$ results in a double-band chaotic attractor at $c=0$ as shown in
Fig. 7(d). Further increase in $c$ results in a one-band chaotic attractor
evolving around the fixed point $p_{2}$ in the RHP at $c=0.01$ leading to a
period-2T attractor at $c=0.03$ as shown in Figs. 7(e) and 7(f), respectively.
The system finally settles down to a period-T limit cycle attractor at
$c=0.04$ as observed n Fig. 7(g). The chaotic hysteresis behavior is also
observed when the parameter is decreased from $0.04$ to $-0.04$. In either
case, the hopping of the attractors around the fixed points $p_{1}$ to $p_{2}$
and vice versa about the double-band chaotic attractor state for equal and
opposite values of $c$ indicating the phenomenon of chaotic hysteresis is
observed. The evolution of chaotic hysteresis around the double-band chaotic
attractor state is depicted in Fig. 8. The dynamical points $a,b,c,d,e,f,g$
given in Fig. 8 indicates the dynamical state of the system as given in the
phase portraits of Fig. 7(a)-7(g), respectively. Chaotic hysteresis occurs in
the system through hopping of the one-band chaotic attractors across the
planes and by encompassing the double-band attractor state.

##  5 Multistability analysis

Figures 4 and 5 illustrate the bifurcation and maximum Lyapunov exponent as
functions of the control parameter $c$. It is observed that small periodic
windows appear around $c=-0.007$ and $c=0.007$. To provide a closer view of
these regions, we have included magnified versions of the bifurcation diagrams
and the maximum Lyapunov exponent plot in Fig. 9. In this figure, we have
plotted the data for $c$ ranging from -0.025 to 0.025 and backwards, using
different colors to represent the two sets of data superimposed on top of each
other. The results show that tiny regions of multistability exist in the
bifurcation space (Fig. 9(a),9(c)) and maximum Lyapunov exponent plot (Fig.
9(b),9(d)), indicating the coexistence of periodic and chaotic behavior. To
investigate the existence of these multistability regions in equation (3), we
have used an offset boost in the $y$ variable.

(a) (b)

(c) (d)

Figure 9: Magnified region of Fig.4 and 5 showing the coexistence of different
dynamics in system (3). (a) Bifurcation diagrams (red and black colors) and
(b) graphs of maximum Lyapunov exponents for $c\in[-9.8,-6]\times 10^{-3}$.
(c) Bifurcation diagrams (in blue and green colors) and (d) graphs of maximum
Lyapunov exponents for $c\in[6,9.8]\times 10^{-3}$. Various regions with the
coexistence of two qualitative and quantitative dynamics are clearly visible.
(color online)

### Offset boosted MLC circuit

Offset boosting is an easy and reliable method to identify any coexisting
dynamics in the phase space [22, 23, 24]. Offset boosted systems are obtained
by introducing a constant vector into the state variables of an investigated
system. This causes a shift in the opposite direction of the sign of the
introduced offset boosting variable in the state space with no alteration on
the system dynamics. System (3) is offset boosted over the variable $y$ with
$m$ as the boosting controller as depicted in Eq(17).

$\left\\{\begin{array}[]{lcl}\vspace{0.1cm}\displaystyle\dot{x}&=&\displaystyle
y+m-g(x)\\\
\vspace{0.1cm}\displaystyle\dot{y}&=&\displaystyle-\sigma(y+m)-\beta
x+f\sin(\omega t)+c\end{array}\right.$ (17)

(a) (b)

Figure 10: (a) Bifurcation diagram of system (17) versus the offset boosting
parameter $m\in[-0.5,0.5]$ for $c=-6.77\times 10^{-3}$ showing the
periodic/regular oscillations labeled with $P$ and chaotic dynamics denoted by
$C$. (b) Phase portraits showing the coexisting attractors in the plane
$(x,y)$ for various sets of $(m,c)$ (i.e,the offset boosting parameter and
bifurcation term respectively). The color code of the coexisting attractors
are receptively those of Fig.9 for $c=\mp 6.77\times 10^{-3}$ (color online).

When $c$ is fixed at $-6.77\times 10^{-3}$ and the boosting controller $m$ is
varied within $[-0.5,0.5]$, the bifurcation diagram in Fig.10(a) illustrates
two distinct dynamics, one regular/periodic and the other chaotic. Two
discrete offset boosting controller values, $m=-0.4$ and $m=0.5$, yield
captured phase portraits in red and black colors, respectively, as shown in
Fig.10(b). The coexisting phase portraits in green (chaotic) and blue
(periodic) colors are obtained for the respective values of
$(m,c)=(0.5,6.77\times 10^{3})$ and $(m,c)=(-0.4,6.77\times 10^{3})$, as shown
in Fig.10(b).

##  6 Conclusion

Chaotic hysteresis is observed in Murali-Lakshmanan-Chua (MLC) circuit when a
DC offset voltage is introduced and varied. The MLC circuit initially is set
to operate in the one-band chaotic state, in the absence of an offset voltage
parameter. In this one-band chaotic state, the chaotic motion moves from left-
half plane of the phase space to the right-half plane of the phase space
through the evolution of a double-band chaotic attractor for a certain value
of the offset voltage as the offset voltage parameter is increased. Further,
when the offset voltage parameter is decreased, chaotic motion moves from
right-half plane to the left-half plane again through the evolution of a
double-band chaotic attractor but for a different value of an offset voltage,
producing a hysteresis loop. The existence of hysteresis over a certain region
of the offset voltage parameter encompassing the initial state is observed and
reported. The circuit can be set to operate at different initial dynamical
states to study the emergence of chaotic hysteresis through various routes.
The phenomenon of chaotic hysteresis observed by electronic circuit
experiments is validated through numerical and analytical results. Multistable
regions are also observed in the MLC circuit with DC offset voltage and
studied using offset boosting.

## Acknowledgement

The authors are grateful to Dr.K. Murali, Professor, Department of Physics,
Anna University, Chennai, India for his valuable suggestions and discussions.

## References

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# Dimensional crossover in a quantum gas of light

Kirankumar Karkihalli Umesh111ORCID: 0000-0002-3644-1233 Institut für
Angewandte Physik, Universität Bonn, Wegelerstrasse 8, 53115 Bonn, Germany
Julian Schulz222ORCID: 0000-0003-4630-4117 Physics Department and Research
Center OPTIMAS, RPTU Kaiserslautern Landau, 67663 Kaiserslautern, Germany
Julian Schmitt333ORCID: 0000-0002-0002-3777 Institut für Angewandte Physik,
Universität Bonn, Wegelerstrasse 8, 53115 Bonn, Germany  Martin Weitz444ORCID:
0000-0002-4236-318X Institut für Angewandte Physik, Universität Bonn,
Wegelerstrasse 8, 53115 Bonn, Germany  Georg von Freymann555ORCID:
0000-0003-2389-5532 Physics Department and Research Center OPTIMAS, RPTU
Kaiserslautern Landau, 67663 Kaiserslautern, Germany  Fraunhofer Institute for
Industrial Mathematics ITWM, 67663 Kaiserslautern, Germany  Frank
Vewinger666ORCID: 0000-0001-7818-2981 Institut für Angewandte Physik,
Universität Bonn, Wegelerstrasse 8, 53115 Bonn, Germany

###### Abstract

We experimentally study the properties of a harmonically trapped photon gas
undergoing Bose-Einstein condensation along the dimensional crossover from one
to two dimensions. The photons are trapped inside a dye microcavity, where
polymer nanostructures provide a harmonic trapping potential for the photon
gas. By varying the aspect ratio of the trap we tune from an isotropic two-
dimensional confinement to an anisotropic, highly elongated one-dimensional
trapping potential. Along this transition, we determine caloric properties of
the photon gas, and find a softening of the second-order Bose-Einstein
condensation phase transition observed in two dimensions to a crossover
behaviour in one dimension.

††preprint: APS/123-QED

##  I  Introduction

The dimensionality of a system profoundly influences its physical behaviour,
leading to the emergence of different states of matter in many-body quantum
systems at low dimensions (i.e., less than three dimensions). For bosonic
gases, as an example, Bose-Einstein condensation (BEC) is possible in lower
dimensions only for a sufficiently strong confinement of a power-law trapping
potential [1]. For example, while in the two dimensions (2D) a harmonically
trapped gas can undergo a phase transition to a Bose-Einstein condensate at
finite temperature, this transition in one dimension (1D) is only observed
with tighter confining power-law traps. When confining a 1D system with a
harmonic trap there is no sharply defined phase transition in the
thermodynamic limit to the condensate phase [2], but rather a smooth
crossover. In this regime, large thermal and quantum fluctuations in one
dimensions lead only to the formation of a quasi-condensate without true long-
range order [3, 4]. For finite-size systems, the change from a phase
transition to a crossover is less pronounced. Moreover condensation can occur
in 1D depending on the strength of interactions different regimes for quantum-
degenerate Bose gases are expected [5]. It is therefore of great interest to
study the crossover from three-dimensional (3D) or two-dimensional (2D)
systems to the 1D case. For ultracold atomic gases, both thermodynamic and
coherence properties associated with the emergence of quasi-long range order
along the dimensional crossover from three to one dimension have been studied
experimentally [6, 7].

Optical quantum gases have in recent years emerged as an alternative platform
for quantum gas experiments [8], which are well-suited for the study of the
dimensional crossover from 2D to 1D due to the weak or even negligible
interaction. In exciton-polariton condensates, correlations in 1D, 2D and 3D
have been studied theoretically for an interacting gas [9]. Here, one crosses
from long-range order in 3D via a power law in 2D to exponentially decaying
first-order phase correlations in the 1D case. Experimentally, e.g. the
formation dynamics has been studied for 1D systems [10, 11], and in 1D coupled
lattices, a Kardar–Parisi–Zhang (KPZ) scaling for the phase has been observed
[12].

In weakly- or non-interacting photon gases, thermalization and condensation in
1D has been observed in doped optical fibers, where the photons thermally
populate the longitudinal degree of freedom in a long single mode fibre [13].
To achieve condensation, the initially linear dispersion relation is altered
to a sub-linear dispersion using chirped gratings, together with a spectral
filter to provide a nontrivial ground state at finite energy. In contrast to
this, one can confine photons in a microcavity, effectively freezing out the
longitudinal degree of freedom [14]. The transverse degrees of motion can be
restricted by in-plane trapping potentials induced by transverse variations in
the optical path lengths, either by structuring the cavity mirror surface [15,
16] or the local refractive index [17, 18]. By this, variable potential
geometries can be realized, which has, e.g., allowed to study the
thermodynamics of 2D Bose gases which contain a few photons or are trapped in
box or double well potentials [19, 20, 21]. For this system, a continuous
change in the thermodynamic properties has been predicted for the harmonically
trapped gas when crossing from a 2D to 1D configuration [22, 23], where in
contrast to the 2D case no Bose-Einstein condensation is expected [1].

In this work, we study the transition from two to one dimension for a
harmonically trapped gas of photons by varying the aspect ratio of the
trapping potential. In our system, thermalization occurs via radiative contact
of the photons to a bath of dye molecules, and correspondingly the
thermalization mechanism is decoupled from the dimensionality of the trapping
potential, in contrast to atomic Bose gases [24, 25]. For all investigated
aspect ratios a macroscopic occupation of the ground state is observed as the
photon number is increased, which only in two dimensions is accompanied by a
sharp transition in the chemical potential, while in one dimension we only
observe a smooth crossover. For the intermediate cases, we observe a gradual
softening of the phase transition, which can be associated to an effective
(non-integer) dimension of the system.

##  II Realization

Figure 1: A) Dye-filled microcavity experimental setup. The photon gas is
created by pumping the intracavity dye solution using a pump laser beam
spatially shaped using a spatial light modulator (SLM), and focused with a
10$\times$ objective into the microcavity. The cavity consists of two plane
mirrors, with a polymer structure printed on one of them to provide a
potential for the photons. The leakage radiation from the cavity is sampled
using an imaging objective and subsequently analysed either spatially or
spectrally. B) The polymer structure with refractive index $n_{s}$ surrounded
by dye solution with refractive index $n$, resulting in a potential for the
photon gas. C) Direct laser writing scheme of polymer structure on a mirror
surface and microscope image of the resulting parabolic structures are shown
below. D) Density distribution of photons in the quantum degenerate regime in
a 1D, 2D-1D and 2D harmonic oscillator potentials.

We prepare our photon gas in an optical microcavity consisting of two highly
reflective plane mirrors (reflectivity above 99.995% at
$580\text{\,}\mathrm{nm}$) spaced by approximately
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ filled with a dye solution, see Fig.
1A. The confinement in longitudinal direction effectively freezes out this
degree of freedom, yielding a minimum energy of
$\hbar\omega_{c}\approx$2.1\text{\,}\mathrm{eV}$$ for the photons in the
cavity, corresponding to a wavelength of $580\text{\,}\mathrm{nm}$. To confine
the photons in the transversal direction we printed a polymer microstructure
of height $d{(x,y)}$ on one of the flat mirrors. The refractive index of the
polymer $n_{s}$ exceeds the refractive index $n$ of the dye solution, which
locally increases the optical path length, leading to an attractive potential
energy landscape for the photons in the regions where the polymer is deposited
with a potential $V\propto d{(x,y)}(n-n_{s})/n$ [18]. The geometry of the
potential is determined by that of the printed structure, $d{(x,y)}$, where
the surface curvature of the printed structure translates to the curvature of
the trapping potential and the absolute height to its trap depth. For the used
dye solvent ethylene glycole ($n\approx 1.44$) and polymer ($n_{s}\approx
1.55$) [26], we find a trap depth $V\approx 1.27k_{\mathrm{B}}T$ for the used
maximum structure height of approximately $325\text{\,}\mathrm{nm}$. The
polymer nanostructuring allows us to fabricate parabolic structures with
sufficiently strong curvatures in the tightly-confining $y$-direction, along
which the corresponding harmonic oscillator potential contains only one bound
state making the photon gas system effectively 1D (see Fig. 1B).

The polymer structures were fabricated out of the negative-tone photoresist
IP-Dip by a direct laser writing (DLW) system (Nanoscribe Photonic
Professional GT) [27]. The substrate was prepared in the immersion
configuration, which means that during the writing process, the laser was
focused by the objective first through an immersion medium (in our case again
IP-Dip) and the substrate of the cavity mirror and at last through the thin
dielectric Bragg layers into the photo resist (see Fig. 1C). The writing
trajectory followed parallel lines along the longer axis of the potentials
($x$-direction) with a line distance of $100\text{\,}\mathrm{nm}$ (along the
$y$-direction). At each point the laser focus height was adjusted according to
$d(x,y)=h_{0}-c_{x}x^{2}-c_{y}y^{2}$ , where $h_{0}$ denotes the maximum
height of the structure and $c_{x}$ and $c_{y}$ were the curvatures along
$x$\- and $y$-direction, respectively. For the different potentials, we keep
the geometry approximately fixed along the $x$-direction and vary the
curvature along the $y$-direction. In the following, we label the different
potentials by their aspect ratio
$\Lambda=\omega_{y}/\omega_{x}=\sqrt{c_{y}/c_{x}}$, with the trap frequencies
$\omega_{x}$ and $\omega_{y}$. Microscope images of resulting polymer
structures, with $\Lambda=22$, $\Lambda=5$ and $\Lambda=1$ are shown in Fig.
1.C. The ratio $\Lambda$ quantifies the effective dimension of the photon gas
[22], where $\Lambda=1$ corresponds to an isotropic 2D harmonic oscillator
with equal trapping frequency along $x$ and $y$ directions, and quasi-1D is
reached when the first excited mode of the strongly confined dimension is not
trapped in the potential, which in our case is achieved at $\Lambda\approx
22$.

To thermalize the photons they are coupled to a thermal bath at ambient
temperature $T=$300\text{\,}\mathrm{K}$$, realised by a dye solution filled
between the cavity mirrors (see Fig. 1A), similar to previous work [14, 28].
By repeated absorption-emission cycles the photons acquire the temperature of
the dye solution provided thermalization is sufficiently faster than the
photon losses, as is the case in our system [29, 28, 30]. Correspondingly, the
photons populate the energy levels of the transversal degrees of freedom,
i.e., the harmonic oscillator levels in $x$\- and $y$-direction, leading to a
spectrum with equidistant frequency spacing above the lowest energy mode
(called the ’cutoff’ energy or frequency) [14]. To prepare the initial photon
population and to compensate for losses out of the system we exploit the low
reflectivity of our mirrors at $532\text{\,}\mathrm{nm}$ to inject dye
molecular excitation using a frequency-doubled laser at
$532\text{\,}\mathrm{nm}$, which fixes the total photon number $N$. The pump
light is time modulated with $500\text{\,}\mathrm{ns}$ pulse width and
$50\text{\,}\mathrm{Hz}$ repetition rate to prevent bleaching of the dye
molecules [14]. A spatial light modulator is used to shape the pump laser
profile to match the structure size on the mirror, thus preventing
fluorescence from unconfined modes from outside the polymer-based structure.

To analyse the photon gas within the cavity we collect the light emitted
through one of the cavity mirrors, and split the radiation into two paths
after lifting the polarization degeneracy using a polarizer. About 70% of the
light is collected by a spectrometer, and 30% are used for spatial imaging of
the photon gas. The density distribution of the photon gas trapped in the 1D
to 2D potential is shown in Fig 1.D, where one can see the density tightly
squeezed along the $y$ direction for 1D ($\Lambda=22$), elliptic for the 1D-2D
($\Lambda=5$) potential, and radially symmetric for the isotropic 2D potential
($\Lambda=1$). We focus on the spectroscopy of the cavity emission to study
the caloric properties of the gas. For this, the emission is dispersed
energetically along the $y$-direction using an optical grating and imaged onto
a sCMOS camera, which allows us to measure both the population in and the
spatial profile of individual modes simultaneously, with a spectral resolution
of $\approx$$0.08\text{\,}\mathrm{THz}$ for the lowest modes. For higher modes
with quantum numbers $n_{y}\gg 1$ the modes start to spatially overlap,
correspondingly lowering the resolution for highly excited modes. We extract a
spectrum by integrating the obtained raw spectra along the non-dispersed
direction, averaging over 20 realizations for a specific total photon number.

Figure 2: Photon gas spectroscopy. A) The integrated spectrum of a cavity
fluorescence for 2D ($\Lambda=1$), 2D-1D ($\Lambda=5$) and 1D ($\Lambda=22$)
harmonic oscillator potential and the black curve is the expected Bose-
Einstein distribution of photons at $T=$300\text{\,}\mathrm{K}$$. B) The
energy-resolved mode shape of the first few modes, showing the energy states
associated with the tightly confined dimension at around the 5th mode in the
2D-1D potential. The photon number $N$ in each case is chosen in the quantum
degenerate regime, such that all modes are visible, i.e. the population
$N_{0}$ in the ground mode does not significantly exceed that of the excited
modes, i.e. $N_{0}/N\approx 0.3$, $0.18$ and $0.3$ for 2D, 2D-1D and 1D
respectively.

Exemplary measured spectra are shown in Fig. 2A for three different aspect
ratios, together with the corresponding raw spectra in Fig. 2B from which they
were obtained. For the isotropic 2D harmonic oscillator potential with
$\Lambda=1$ (top panel), one observes equidistantly spaced modes with a
frequency spacing of $0.223\text{\,}\mathrm{THz}$. In the raw spectra,
individual modes $(n_{x},n_{y})$ are not resolved, as harmonic oscillator
modes with equal $n_{x}+n_{y}$ have the same energy, and correspondingly they
overlap spatially.

For $\Lambda=5$, corresponding to an anisotropic 2D harmonic oscillator
potential with $\omega_{y}=5\omega_{x}$ (middle panel), one observes
equidistantly spaced modes except for the few lowest modes due to slight
distortions in the polymer structure. The mode degeneracy increases every 5th
mode, as can be seen by the step-like increase in intensity at those modes. In
the raw spectrum, this is reflected by the emergence of multiple parabolas
corresponding to the different quantum numbers $n_{y}$ along the strongly
confined $y$-direction. One correspondingly finds the emergence of the second
dimension ($n_{y}=1$) around the 5th mode ($n_{x}=5$) along the relaxed
$x$-direction.

The potential with $\Lambda=22$ displays 1D harmonic oscillator modes (bottom
panel) in the raw spectrum, with an energy spacing of
$0.37\text{\,}\mathrm{THz}$, and the integrated spectrum correspondingly shows
discrete peaks with a degeneracy of one, characteristic for the absence of the
second dimension in the recorded energy interval and which is confined by the
trap for the strongly asymmetric harmonic oscillator potential, demonstrating
that a photon gas trapped in this potential can be considered as being
effectively 1D.

The photon distribution in all three cases follows the Bose-Einstein
distribution, the black markers in Fig. 2A indicate the expected photon
distribution. The theory estimations were calculated using a Bose-Einstein
distribution $g(E)\left(\text{e}^{(E-\mu)/k_{B}T}-1\right)^{-1}$, with the
degeneracy $g(E)$. We use the energies of a quantum harmonic oscillator
spectrum,
$E=\hbar(\omega_{x}n_{x}+\omega_{y}n_{y}+(\omega_{x}+\omega_{y})/2)$. The trap
frequencies $\omega_{x}$ and $\omega_{y}$ are extracted from the mean mode
spacing from the measured spectra, and we truncate the theory spectra by
excluding the energy levels which exceed the trap depth inferred from the
measured spectrum, and the photon number at each mode is determined by
calibrating the signal of the sCMOS camera. The experimental data well follow
the theory expectations, apart from a slightly lower mode population for the
2D case, which we attribute to a mismatch in photon number between theory and
experiment.

Figure 3: The change from a phase transition in 2D to a crossover in 1D is
visible in the chemical potential. A) Symbols represent the measured change in
the absolute of chemical potential $|\mu|$ in the units of thermal energy
($k_{\mathrm{B}}T$), from 1D to 2D (1D, 2D-1D and 2D) harmonic oscillator
potentials as function of the normalised photon number $N/\tilde{N}$, where
the zero-point energy is set to zero. Solid lines are theory expectations for
corresponding harmonic oscillator potentials. B) The measured internal energy
per particle (photon), in units of $k_{\mathrm{B}}T$, as a function of the
normalised photon number $N/\tilde{N}$, from top to bottom, 2D, 2D-1D and 1D
harmonic oscillator potential, with their corresponding theory expectations in
solid lines.

The influence of the dimension on the phase transition can be explored by
studying the order parameter when tuning the system parameters. As for our
case the temperature is fixed at room temperature, the tuning parameter is the
total photon number, and we use the absolute value of the chemical potential
$|\mu|$ as an order parameter. To find the chemical potential $\mu$, we first
extract the internal energy from spectra as in Fig. 2 for different total
photon numbers. For each spectrum, we set the ground mode energy to be the
zero point energy of the harmonic oscillator,
$E_{0}=\hbar(\omega_{x}+\omega_{y})/2$, count the number of photons in each
energy level, multiply by the corresponding mode energy and sum over the whole
spectrum, yielding the internal energy as shown in Fig. 3B. For all shown
aspect ratios, the internal energy per photon, $U/N$, decreases for increasing
photon numbers indicating an increasing population in the low-energy states,
i.e. the ground mode. For better comparison, each data set is scaled to the
photon number $\tilde{N}$ where the inner energy per particle $U/N$ changes
slope, which is found to be 611.22, 63.20 and 19.15 photons for the 2D, the
2D-1D and the 1D harmonic oscillator potentials respectively. In the isotropic
2D potential ($\Lambda=1$), the curve changes slope sharply around
$N/\tilde{N}\approx 1$, indicating the presence of phase transition, while in
the 1D potential ($\Lambda=22$), the slope changes monotonically, and shows
the absence of phase transition features. As for the anisotropic 2D potential
($\Lambda=5$), the $U/N$ slope indeed has, although less strong, a sharp
change as in 2D isotropic potential Fig. 3.

This is visible more strongly in the chemical potential, by numerically taking
the partial derivative of the internal energy $U$ with respect to the photon
number $N$. The numerical derivative was done by first binning the photon
number data for $U$ (bins of photon number in a geometric series spacing with
a common ratio of 1.2, 1.3 and 1.2 for the 2D, the 2D-1D and the 1D harmonic
oscillator potentials respectively) to suppress numerical noise. The absolute
value of $\mu$ as expected decreases for all three aspect ratios with
increasing photon number. The sharp drop followed by the saturation at the
ground state energy for the isotropic 2D case $\Lambda=1$ indicates the
expected phase transition in two dimensions, while for increasing $\Lambda$ we
observe a gradual softening of the transition to a continuous crossover for
the 1D potential. Thus, the change in the dimension of the potential from 2D
to 1D leads to a crossover between the phases of Bose gas instead of a sharp
phase transition.

##  III Conclusions

To conclude, we have experimentally studied the dimensional crossover from a
2D isotropic harmonically trapped photon gas to a photon gas confined to 1D
around the transition from a thermal to the quantum degenerate case. This
crossover is accompanied by a softening of the phase transition, which crosses
from a true second-order phase transition to a BEC in 2D to a continuous
behaviour in 1D, indicated by the behaviour of the chemical potential and the
internal energy of the photon gas for different photon numbers.

For the future, it will be interesting to study different trapping potentials
for the photons, and investigate the spatial correlations [31]. While in 1D no
true long-range order is possible, in our finite-size system the correlations
can extend over the whole system. In the cavity platform the losses can be
tuned from a nearly lossless system to the case of a driven-dissipative
condensate by modifying the low-frequency cutoff [32], which is expected to
alter the correlations in the system. Also, the structuring method presented
here allows great flexibility in the design of potentials for photons, ranging
from continuous potentials like the ones presented here to tunnel-coupled
lattice structures with large tunneling rates. For example, potentials with a
logarithmic level spacing have been proposed for factorization of large
numbers [33], and in tunnel-coupled potentials, the influence of loss and
drive e.g. leads to stable vortices [34, 35], clustering [36] or the emergence
of a KPZ-like scaling in the correlations [12], and for 1D chains the
emergence of surface states is possible in the presence of a retarded
thermooptic interaction [37].

Acknowledgements

We acknowledge valuable discussions with E. Stein and A. Pelster. This work
has been supported by the Deutsche Forschungsgemeinschaft through
CRC/Transregio 185 OSCAR (project No. 277625399). J. Schmitt acknowledges
financial support by the EU (ERC, TopoGrand, 101040409), M. Weitz and J.
Schmitt by the Cluster of Excellence ML4Q (EXC 2004/1-390534769).

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