Twofold Transition in -Symmetric Coupled Oscillators
The inspiration for this theoretical paper comes from recent experiments on a -symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two transitions depending on the size of the coupling parameter . For small the symmetry is broken and the system is not in equilibrium, but when becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the symmetry is unbroken. For very large the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of .
pacs:11.30.Er, 03.65.-w, 02.30.Mv, 11.10.Lm
The predicted properties of -symmetric Hamiltonians A1 (); A2 () have been observed at the classical level in a wide variety of laboratory experiments involving superconductivity R1 (); R2 (), optics R3 (); R4 (); R5 (); R6 (), microwave cavities R7 (), atomic diffusion R8 (), nuclear magnetic resonance R9 (), and coupled electronic and mechanical oscillators R10 (); R11 (). Although -symmetric systems were originally explored at a highly mathematical level, it is now understood that one can interpret -symmetric systems simply as nonisolated physical systems having a balanced loss and gain.
In this paper we examine a mathematical model based on recent unpublished experiments. These experiments were performed on a system consisting of two coupled -symmetric whispering galleries (optical resonators) R12 (). Such a system is symmetric if one resonator is optically driven and the other resonator has a balanced loss. We examine here the properties of the mathematical model at a theoretical level and we study both the classical and the quantum versions of the system.
A system of two identical coupled resonators, one with loss and the other with gain, can be modeled as coupled oscillators whose amplitudes are and . Both oscillators have a natural frequency . The first oscillator is subject to a friction force (), while the second oscillator is subject to an antifriction force (). The parameters and are a measure of the loss and gain. The oscillators are coupled linearly and the coupling strength is represented by the parameter . The equations of motion of these oscillators are
To treat this system at a classical level, we seek solutions to (1) of the form . The frequency satisfies the quartic polynomial equation
An important special case arises when the loss and gain are balanced; that is, when . In this case the frequencies are the roots of the quartic polynomial , where
If the coupling parameter of the and oscillators is set to zero, this Hamiltonian reduces to the Hamiltonian considered by Bateman R14 (). In his paper Bateman sought a variational principle to derive an equation of motion having a friction term linear in velocity. To do so, he introduced an additional degree of freedom; namely, a time-reversed version of the original damped harmonic oscillator. This auxiliary oscillator acts as an energy reservoir and can be considered as an effective description of a thermal bath. The classical Hamiltonian for the Bateman system was constructed by Morse and Feschbach R15 () and the corresponding quantum theory was analyzed by many authors, including Bopp R16 (), Feshbach and Tikochinsky R17 (), Tikochinsky R18 (), Dekker R19 (), Celeghini, Rasetti, and Vitiello R20 (), Banerjee and Mukherjee R21 (), and Chruściński and Jurkowski R24 (). We emphasize that in all these references only the noninteracting () case was considered. It is easy to see that the Hamiltonian (4) is symmetric, where the action of parity is to interchange the loss and gain oscillators and its effect is given by R22 ()
whilst the action of time reversal is to change the signs of the momenta
Note that is not symmetric under or separately, but it is symmetric under combined and . For a one-dimensional system, reduces to the usual parity operator and is the usual time-reversal operator. Because the balanced-loss-and-gain system is Hamiltonian, the energy (the value of ) is conserved; that is, its numerical value is constant in time. However, the expression for the energy in (4) is not recognizable as a simple sum of kinetic and potential energy (such as ).
The noteworthy feature of -symmetric systems with balanced loss and gain is that they exhibit phase transitions. When the coupling of the two oscillators is small, the energy flowing into the resonator cannot transfer fast enough to the resonator, where the energy is flowing out. Thus, the system cannot be in equilibrium. However, when the coupling constant exceeds a critical value, all of the energy flowing into the resonator can transfer to the resonator and the entire system can attain equilibrium. The system is in equilibrium only if the frequencies are real because complex frequencies indicate that there is exponential growth and decay.
To understand why there are phase transitions we plot the quartic polynomial in (3) to see whether this polynomial cuts the horizontal axis in four places (in which case there are four real frequencies), two places (in which case there are two real frequencies and two complex frequencies), or not at all (here, there are four complex frequencies). As one can see in Fig. 1, for small values of there are no real frequencies, but as increases there is a transition at to a situation where there are four real frequencies. Interestingly, one can see that when the coupling is sufficiently large, there is a second transition at . This transition is difficult to see in classical experiments because in the strong-coupling regime the loss and gain components must be so close that they overlap and therefore interfere with one another. For example, in the pendulum experiment in Ref. R11 () the pendula would have to be so close that they touch and could no longer swing freely. This strong-coupling region is discussed for the case of coupled systems without loss and gain in Ref. R23 (), where it is referred to as the ultrastrong-coupling regime.
The signal that the system is in equilibrium is that the resonators exhibit Rabi oscillations (power oscillations between the two resonators) as shown in Fig. 2. Note that the Rabi oscillations are out of phase.
This paper is organized as follows: In Sec. II we examine the classical solutions to (1). Next, in Sec. III we examine the quantized version of the system described by the Hamiltonian (4). We identify the quantum analogs of the phase transitions and show that the quantum and classical transitions occur at exactly the same values of the physical parameters. Finally, in Sec. IV we make some brief concluding remarks.
Ii Classical interpretation
ii.1 Balanced loss and gain
There are four real frequencies when is in the range
This is the unbroken classical -symmetric region.
We plot the real and imaginary parts of the frequency in Fig. 3 for the values and . For these parametric values the phase transition occurs at . When is below this critical value, the real part of has one positive value, which is shown in Fig. 3, and one negative value. Also, below the critical value, the imaginary part of is nonzero, as shown in Fig. 3. As approaches the critical value from below, the imaginary part of vanishes and the real part of bifurcates.
A second transition occurs when . Above this transition point there is now only one super mode, as shown in Fig. 4, instead of two pairs of real frequencies.
ii.2 Unbalanced loss and gain
Let us consider the general case (2) in which (that is, the loss and gain are not exactly balanced. In this case the sharp transition from a region of broken symmetry to a region of an unbroken symmetry disappears and there is only an approximate transition. To see this approximate transition, we take and and plot the classical frequencies in Fig. 5. In contrast with Fig. 3, the frequency is never exactly real. Rather, there is one region of in which the difference between the imaginary parts of the frequencies is big and the difference between the real parts of the frequencies is small but nonzero, and a second region in which the difference between the imaginary parts of the frequencies is small but nonzero and the difference between the real parts of the frequencies is big. Unlike the behavior shown in Fig. 3, at the approximate transition in Fig. 5 the real parts of the frequencies do not separate in an orthogonal direction but rather separate smoothly.
Iii Quantum interpretation
When the loss and gain parameters in (1) are equal, the coupled oscillator system is described by the Hamiltonian in (4). To quantize this classical Hamiltonian we replace the classical variables , , , and with the corresponding quantum operators that satisfy the commutator equations and . In Subsec. III.1 we discuss the eigenfunctions of and in Subsec. III.2 we discuss the eigenvalues.
iii.1 Eigenfunctions of
The eigenfunctions of the Hamiltonian (4) satisfy the time-independent Schrödinger equation
The eigenvalues correspond to the eigenfunctions . The eigenfunctions have the general form
and is a solution to the quartic equation , where
The quantities are polynomials in and . The index is a nonnegative integer while the index is an integer that runs from to . Thus, the polynomials form a Pascal-like triangle in which the first index labels the row and labels the column:
In terms of the quantity
the first seven polynomials are
The polynomials satisfy satisfy two three-term recursion relations, one in the first index with the second index held fixed at (at the left edge of the Pascal triangle),
and another with both indices being equal (at the right edge of the Pascal triangle),
The operators and are lowering operators for the polynomials and :
These equations are the analogs of the relation for the Hermite polynomials .
from which we obtain the differential equation satisfied by the polynomials at the left and right edges of the Pascal triangle:
We can also construct operators that connect the polynomials on a given horizontal level in the Pascal triangle. To do this we define the left shift operator as
The effect of on is
We also define the right shift operator as
The effect of on is
iii.2 Eigenvalues of
The eigenvalues have the general form
where and . Note that there are four possible spectra of eigenvalues corresponding to the four possible solutions for , which are the roots of in (14):
Although there are four possible sets of eigenvalues, we will see that only one of these sets is physically acceptable; that is, there is only one set of eigenvalues that is real and bounded below. This set only occurs in the classical unbroken -symmetric region of in (8).
Corresponding to the two classical phase transitions discussed earlier, there are also two quantum transitions at the same values of the coupling and as the classical transitions. To locate the quantum phase transitions we plot in Fig. 6 the quartic polynomial in (14) as a function of for various values of and observe whether this polynomial cuts the horizontal axis in four places, two places, or not at all.
It is important to understand why there are four possible sets of quantum eigenvalues. This comes about because there are four possible pairs of Stokes wedges in the complex domain in which the eigenfunctions in (12) vanish exponentially. To explain what is going on, we use, as an elementary example, the quantum harmonic oscillator, whose Hamiltonian is . One set of eigenfunctions of this Hamiltonian in complex- space have the form , where is a Hermite polynomial. These eigenfunctions vanish exponentially in a pair of Stokes wedges of opening angle centered about the positive and negative real axes in the complex- plane. The eigenvalues () associated with these eigenfunctions are real and bounded below. There is a second set of eigenfunctions of the form , where is again a Hermite polynomial. These eigenfunctions vanish exponentially in a pair of Stokes wedges of opening angle centered about the positive and negative imaginary axes in the complex- plane. The eigenvalues () associated with these eigenfunctions are real and bounded above. A full description of these two classes of eigenfunctions and eigenvalues is given in Ref. A1 ().
For the coupled-oscillator problem discussed in this paper the eigenfunctions have the general form (12). The exponential component of these eigenfunctions can be rewritten as
It is important to determine the Stokes wedges in the complex- plane and in the complex- plane in which the eigenfunctions vanish. We consider each of the four values of in (32) in turn.
First, we consider in (32). In Fig. 7 we plot the real and imaginary parts of as functions of and see that is real in the unbroken- region (8). Furthermore, in (15) is real and positive in this region. Thus, the eigenvalues in (31) are real.
In the unbroken region in (13) and in (34) are both negative (see Fig. 8). Thus, the eigenfunctions vanish exponentially in pairs of -Stokes wedges centered about the imaginary axes in the and planes.
However, since is negative, the eigenspectrum (31) is not bounded below, and thus this case must be rejected on physical grounds.
Next, we consider in (32). In Fig. 9 we plot as a function of and see that is real in both the first broken- region and the unbroken- region (8). Furthermore, in (15) is real and positive in the unbroken region of . Thus, the eigenvalues in (31) are real in the unbroken- region.
In the unbroken region in (13) is negative and in (34) is positive (see Fig. 10). Thus, the eigenfunctions vanish exponentially in pairs of -Stokes wedges centered about the imaginary axis in the plane and centered about the real axis in the plane.
However, is negative, so the eigenspectrum (31) is not bounded below, and again this case must be rejected on physical grounds.
Next, we consider in (32). In Fig. 11 we plot as a function of and see that is real and positive in the unbroken- region (8). Again, in (15) is real and positive in this region. Thus, the eigenvalues in (31) are real and positive.
In the unbroken region in (13) and in (34) are both positive (see Fig. 12). Thus, the eigenfunctions vanish exponentially in pairs of -Stokes wedges centered about the real axes in both the and planes.
Because the eigenspectrum (31) is bounded below and the eigenfunctions vanish exponentially in the appropriate Stokes wedges, we regard this as a physically acceptable case.
Finally, we consider in (32). In Fig. 13 we plot as a function of and see that is real in the first broken- region and in the unbroken- region (8). Furthermore, in (15) is real and positive in the unbroken- region. Thus, the eigenvalues in (31) are real.
In the unbroken region in (13) is positive and in (34) is negative (see Fig. 14). Thus, the eigenfunctions vanish exponentially in pairs of -Stokes wedges centered about the real axis in the plane and centered about the imaginary axis in the plane.
Because is positive the eigenspectrum (31) is bounded below and because the eigenfunctions vanish exponentially in the appropriate Stokes wedges in the and planes, we again regard this case as physically acceptable.
It is interesting but perhaps not surprising that the Hamiltonian (4) has two distinct physically allowed positive spectra, which correspond to the choices and . It is not surprising that in the context of quantum mechanics one Hamiltonian can have two independent positive spectra. This phenomenon was discussed previously for the case of the sextic quantum mechanical Hamiltonian in Ref. R26 (). This sextic Hamiltonian also has two positive spectra, which are associated with two distinct pairs of Stokes wedges in which the eigenfunctions vanish exponentially.
Iv Concluding remarks
In this paper we have studied the behavior of a system of two coupled oscillators, one with gain and the other with loss. If the gain and loss parameters are equal, the system is symmetric. Furthermore, it is described by a Hamiltonian and thus the energy is exactly conserved. Both the classical and the quantum systems exhibit two transitions at exactly the same values of the coupling parameter and .
Specifically, if the coupling is smaller than a critical value , the system is not in equilibrium even though the energy is conserved. At the classical level the lack of equilibrium manifests itself as complex frequencies and exponentially growing and decaying modes; at the quantum level the lack of equilibrium is associated with complex energy levels. Above the critical value the system is in an unbroken--symmetric phase; at the classical level the system is in equilibrium and the oscillators exhibit Rabi oscillations and at the quantum level the system exhibits not one, but two independent sets of real spectra and associated eigenfunctions.
There is also a second transition point above which the system is no longer in equilibrium and the quantum energy levels become complex. This super-strong coupling regime is very hard to study at the classical experimental level because it requires that the oscillators be strongly coupled, so strongly coupled that they are likely to interfere with one another. However, at the quantum level it might be possible to perform experiments using quantum optics techniques and that can actually observe the second phase transition. Such quantum experiments might also prove to be extremely interesting because they may provide a platform on which to study quantum entanglement R23 ().
In future work we will study systems of more than two coupled oscillators. The phase structure of such systems is interesting; we have found that as the number of coupled oscillators increases, there are more and more phase transition points.
Acknowledgements.MG is grateful for the hospitality of the Department of Physics at Washington University. CMB thanks the U.S. Department of Energy and MG thanks the INFN (Lecce) for financial support.
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