^{1,a)}, Bjorn K. Berntson

^{2,b)}, David Parker

^{3,c)}and E. Samuel

^{3,d)}

### Abstract

If a quantum-mechanical Hamiltonian is symmetric, there are two possibilities: either all of the eigenvalues are real, in which case the Hamiltonian is said to be in an *unbroken-* *-symmetric phase*, or else some of the eigenvalues are real and some are complex, in which case the Hamiltonian is said to be in a *broken-* *-symmetric phase*. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an elementary experiment in which the phase transition is easily observed.

The authors thank Professor R. Pike and Professor J. Schilling for many useful discussions. The authors also thank Professor Pike for the use of his laboratory facilities. C.M.B. thanks the U.K. Leverhulme Foundation and the U.S. Department of Energy for financial support.

I. INTRODUCTION

II. INTUITIVE EXPLANATION OF THE PHASE TRANSITION

III. DESCRIPTION OF THE EXPERIMENT

A. An overly simple mathematical model of a two-oscillator system

B. An improved energy-conserving oscillator model

C. Description of the experimental setup

IV. FINAL REMARKS

### Key Topics

- Eigenvalues
- 19.0
- Phase transitions
- 18.0
- Oscillators
- 9.0
- Energy transfer
- 6.0
- Antennas
- 4.0

##### G09B

## Figures

A simple schematic -symmetric physical system: A box located at *x* = −*a* contains a sink (an antenna that absorbs) and a box located at *x* = +*a* contains a source (an antenna that radiates at an equal rate). Under space reflection the boxes interchange, and under time reversal the sink becomes a source and the source becomes a sink. Thus, the system is symmetric.

A simple schematic -symmetric physical system: A box located at *x* = −*a* contains a sink (an antenna that absorbs) and a box located at *x* = +*a* contains a source (an antenna that radiates at an equal rate). Under space reflection the boxes interchange, and under time reversal the sink becomes a source and the source becomes a sink. Thus, the system is symmetric.

Oscillatory motion of a pair of coupled pendula with no loss or gain [system (11) with coupling parameter ]. The *x*-displacement is shown in the top panel and the *y*-displacement is shown in the bottom panel. Rabi power oscillations occur in which the maximum and minimum displacements *x*(*t*) and *y*(*t*) are out of phase.

Oscillatory motion of a pair of coupled pendula with no loss or gain [system (11) with coupling parameter ]. The *x*-displacement is shown in the top panel and the *y*-displacement is shown in the bottom panel. Rabi power oscillations occur in which the maximum and minimum displacements *x*(*t*) and *y*(*t*) are out of phase.

Numerical solution to the overly simple coupled-oscillator system (12) for the parameter choice , which is deep in the broken- -symmetric region. As expected, the *y*-oscillator exhibits gain but, after an initial period of decay, the *x*-oscillator also exhibits gain. This behavior is a result of the system not conserving energy, and the excess energy in the *y*-oscillator leaks into the *x*-oscillator.

Numerical solution to the overly simple coupled-oscillator system (12) for the parameter choice , which is deep in the broken- -symmetric region. As expected, the *y*-oscillator exhibits gain but, after an initial period of decay, the *x*-oscillator also exhibits gain. This behavior is a result of the system not conserving energy, and the excess energy in the *y*-oscillator leaks into the *x*-oscillator.

Numerical simulation of the improved energy-conserving oscillator model in the unbroken- region. In these graphs and *g* = 0.01. That is, 1% of the energy in the *x*-oscillator is removed at each peak and this exact amount of energy is then transferred to the *y*-oscillator when it reaches a peak. This transfer reduces the peak *x*-amplitude to 99.5% of its former value, which is too small to be seen on this graph. We can see that the symmetry is not broken—the Rabi oscillations persist and the amplitudes of these oscillations remain constant.

Numerical simulation of the improved energy-conserving oscillator model in the unbroken- region. In these graphs and *g* = 0.01. That is, 1% of the energy in the *x*-oscillator is removed at each peak and this exact amount of energy is then transferred to the *y*-oscillator when it reaches a peak. This transfer reduces the peak *x*-amplitude to 99.5% of its former value, which is too small to be seen on this graph. We can see that the symmetry is not broken—the Rabi oscillations persist and the amplitudes of these oscillations remain constant.

Numerical simulation of the improved energy-conserving oscillator model in the broken- region. Here, the coupling is reduced to and the energy transfer is increased to *g* = 0.3. For this value of *g*, 30% of the energy in the *x*-oscillator is removed each time *x* reaches a peak and this exact amount of energy is then transferred to the *y*-oscillator when it reaches a peak. This transfer reduces the peak amplitude to 0.837 of previous value, and this change can be seen in the plot of *x*(*t*). Observe that the Rabi oscillations cease and that the *x*-oscillations die down to a limiting amplitude and correspondingly the *y*-oscillations increase to a limiting amplitude. This is the characteristic behavior of an oscillator system having a broken symmetry.

Numerical simulation of the improved energy-conserving oscillator model in the broken- region. Here, the coupling is reduced to and the energy transfer is increased to *g* = 0.3. For this value of *g*, 30% of the energy in the *x*-oscillator is removed each time *x* reaches a peak and this exact amount of energy is then transferred to the *y*-oscillator when it reaches a peak. This transfer reduces the peak amplitude to 0.837 of previous value, and this change can be seen in the plot of *x*(*t*). Observe that the Rabi oscillations cease and that the *x*-oscillations die down to a limiting amplitude and correspondingly the *y*-oscillations increase to a limiting amplitude. This is the characteristic behavior of an oscillator system having a broken symmetry.

A photograph of the two-pendulum experiment. Two pendula are suspended from a horizontal rope and the tension in the rope is adjusted to increase or decrease the coupling of the pendula. The horizontal rope runs around a wheel to the left (not shown) and is attached to a tray upon which weights can be added or subtracted to change the tension. Electromagnets near the top of the strings supporting the bobs apply brief impulses to small iron nails attached by white tape to the strings. The electromagnets are triggered by pairs of optical sensors just above the pendulum bobs. The electromagnets are timed so that on each swing a small amount of kinetic energy is subtracted from the left pendulum and a roughly equal amount of kinetic energy is added to the right pendulum.

A photograph of the two-pendulum experiment. Two pendula are suspended from a horizontal rope and the tension in the rope is adjusted to increase or decrease the coupling of the pendula. The horizontal rope runs around a wheel to the left (not shown) and is attached to a tray upon which weights can be added or subtracted to change the tension. Electromagnets near the top of the strings supporting the bobs apply brief impulses to small iron nails attached by white tape to the strings. The electromagnets are triggered by pairs of optical sensors just above the pendulum bobs. The electromagnets are timed so that on each swing a small amount of kinetic energy is subtracted from the left pendulum and a roughly equal amount of kinetic energy is added to the right pendulum.

Experimentally measured motion of the pendula with the magnets turned off. The tension in the string is maintained by a 200-g mass. To produce these graphs, we use a camera that records the instantaneous position of each pendulum (plus marks on the graph) 15 times per second. We then fit a curve through these data points. The motion of the pendula is qualitatively similar to that in the theoretical curves shown in Fig. 2 . In this configuration, the symmetry is unbroken, which is signaled by the presence of Rabi power oscillations that are out of phase. One can observe a slight decay in the amplitudes of the pendula due to friction.

Experimentally measured motion of the pendula with the magnets turned off. The tension in the string is maintained by a 200-g mass. To produce these graphs, we use a camera that records the instantaneous position of each pendulum (plus marks on the graph) 15 times per second. We then fit a curve through these data points. The motion of the pendula is qualitatively similar to that in the theoretical curves shown in Fig. 2 . In this configuration, the symmetry is unbroken, which is signaled by the presence of Rabi power oscillations that are out of phase. One can observe a slight decay in the amplitudes of the pendula due to friction.

Experimental data showing the motion of the pendula when the magnets are turned on weakly and the coupling of the pendula is decreased (the tension in the supporting rope is maintained using a mass of 400 g). Observe that the Rabi power oscillations in Fig. 7 persist, meaning that the system remains in a region of unbroken symmetry.

Experimental data showing the motion of the pendula when the magnets are turned on weakly and the coupling of the pendula is decreased (the tension in the supporting rope is maintained using a mass of 400 g). Observe that the Rabi power oscillations in Fig. 7 persist, meaning that the system remains in a region of unbroken symmetry.

Experimental data showing the motion of the pendula when the magnets are turned on more strongly and the coupling of the pendula is weak (the tension in the supporting string is maintained by a mass of 600 g). Observe that the Rabi oscillations have ceased; this is the signal that the system is in a region of broken symmetry.

Experimental data showing the motion of the pendula when the magnets are turned on more strongly and the coupling of the pendula is weak (the tension in the supporting string is maintained by a mass of 600 g). Observe that the Rabi oscillations have ceased; this is the signal that the system is in a region of broken symmetry.

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