Systems of coupled \mathcal{PT}-symmetric oscillators

Systems of coupled -symmetric oscillators

Carl M. Bender cmb@wustl.edu    Mariagiovanna Gianfreda Maria.Gianfreda@le.infn.it    S. P. Klevansky spk@physik.uni-heidelberg.de Department of Physics, Washington University, St. Louis, MO 63130, USA
Department of Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK
Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505, Japan
Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany111Permanent address. and Department of Physics, University of the Witwatersrand, Johannesburg, South Africa
July 12, 2019
Abstract

The Hamiltonian for a -symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken -symmetry disappears entirely. However, if is localized in the sense that it decreases for more distant oscillators, then the unbroken--symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a -symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudo-bound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken symmetry.

pacs:
11.30.Er, 03.65.-w, 02.30.Mv, 11.10.Lm

I Introduction

A previous paper R1 () considered a system consisting of a pair of coupled oscillators, one with loss and the other with gain. Such a system is symmetric if the loss and gain parameters are equal. The energy of this -symmetric system is exactly conserved because this system is described by a Hamiltonian. In the current paper we examine the systems that arise when the number of pairs of coupled oscillators is extended from to , where can be arbitrarily large.

Let us review the case . A single pair of coupled oscillators, the first with loss and the second with gain, is described by the equations of motion

(1)

To treat this system at a classical level, we seek solutions to (1) of the form . The classical frequency then satisfies the quartic polynomial equation

(2)

This classical system becomes symmetric if the loss and gain are balanced; that is, if we set . In this case the frequencies are given by

(3)

Note that there are four real frequencies when is in the range

(4)

This defines the unbroken -symmetric region. In the broken--symmetric region there are two pairs of complex-conjugate frequencies and in the broken--symmetric region there are two real frequencies and one complex-conjugate pair of frequencies.

When , the system (1) is not Hamiltonian. However, when the system is symmetric (), (1) can be derived from the two-coupled-oscillator Hamiltonian

(5)

This Hamiltonian is symmetric because under parity reflection the loss and gain oscillators are interchanged R2 (),

(6)

and under time reversal the signs of the momenta are reversed,

(7)

The Hamiltonian is symmetric but it is not invariant under or separately R3 (). Because the balanced-loss-gain system is Hamiltonian, the energy (that is, the value of ) is conserved. However, the total energy (5) is not the usual sum of kinetic and potential energies (such as ).

If we set the coupling parameter to zero, describes the system studied by Bateman R4 (). Bateman showed that an equation of motion having a friction term linear in velocity could be derived from a variational principle. To do this he introduced a time-reversed companion of the original damped harmonic oscillator. This auxiliary oscillator acts as an energy reservoir and can be viewed as a thermal bath. The classical Hamiltonian for the Bateman system was constructed by Morse and Feschbach R5 () and the corresponding quantum theory was analyzed by many authors, including Bopp R6 (), Feshbach and Tikochinsky R7 (), Tikochinsky R8 (), Dekker R9 (), Celeghini, Rasetti, and Vitiello R10 (), Banerjee and Mukherjee R11 (), and Chruściński and Jurkowski R12 (). Only the noninteracting () case was considered in these references.

The noteworthy feature of -symmetric systems is that they exhibit transitions; the classical system described by exhibits two transitions. The first occurs at . If , the energy flowing into the resonator cannot transfer fast enough to the resonator, where energy is flowing out, so the system cannot be in equilibrium. However, when , the energy flowing into the resonator transfers to the resonator and the entire system is in equilibrium. The frequencies of a classical system in equilibrium are real and the system exhibits Rabi oscillations (power oscillations between the two resonators) in which the two oscillators are out of phase. Complex frequencies indicate exponential growth and decay and are a signal that the system is not in equilibrium. A second transition occurs at ; when , the classical system is no longer in equilibrium. This transition is difficult to see in classical experiments because in the strong-coupling regime the loss and gain components would have to be so close that they would interfere with one another. For example, in the pendulum experiment in Ref. R13 () the pendula would be so close that they could no longer swing freely, and in the optical-resonator experiment in Ref. R14 () the solid-state resonators would be damaged. This strong-coupling region is discussed for the case of coupled systems without loss and gain in Ref. R15 (), where it is called the ultrastrong-coupling regime.

In Ref. R1 () it is shown that the classical and the quantum systems described by exhibit transitions at the same two values of the coupling parameter . When and when the quantum energies are complex, but in the unbroken--symmetric region the quantum energies are real.

This paper is organized as follows. In Sec. II we formulate the equations of motion for a linear chain of identical pairs of -symmetric loss-gain oscillators and we construct the Hamiltonians for such systems. We show that there are two ways to represent such Hamiltonians, one that we call a sum representation and another that we call a product representation. In the product representation it is easy to see that the Hamiltonian is not unique and that this nonuniqueness takes the form of a gauge invariance. Next, in Sec. III we construct the Hamiltonians for a general -symmetric system of coupled oscillators in which the coupling parameter and the loss-gain parameter are allowed to vary from oscillator to oscillator. In addition, we consider a system of coupled -symmetric oscillators, where symmetry requires that the central oscillator have neither loss nor gain. We also perform the limit of . In this limit the equations of motion of the oscillators become coupled linear wave equations with balanced loss and gain.

In Sec. IV we ask whether a -symmetric chain of coupled oscillators can have an unbroken--symmetric region. We show that as increases, if and are the same for all oscillators, the region of unbroken symmetry shrinks and disappears entirely as . However, if the loss-gain parameter decreases to 0 for distant oscillators, then such systems always have an unbroken--symmetric region for intermediate values of the coupling parameter surrounded by broken--symmetric regions for small and large values of . Specifically, for the cases in which decreases like or , where is the number of the oscillator measured from the center of the system, we show that an unbroken--symmetric region persists in the limit as . If one views loss-gain as the consequence of an impurity, then a configuration of oscillators for which decreases with increasing distance from the center can be seen as having a localized impurity. Thus, in Sec. V we investigate a special case for the continuum model in which there is a point-like -symmetric impurity localized at the origin. We find that this impurity gives rise to a pseudobound-state solution. In Sec. VI we consider the simplest case of a two-dimensional array of coupled oscillators, namely three oscillators, one with loss, one with gain, and the third with neither loss nor gain. This system is interesting because it can exhibit five distinct regions as a function of the coupling constant, two having unbroken symmetry and three having broken symmetry. Finally, in Sec. VII we make some brief concluding remarks.

Ii -symmetric system of coupled classical oscillators

In this section we describe the properties of a -symmetric one-dimensional chain of coupled oscillators with alternating loss and gain. We begin by making the simplifying assumptions that the natural frequency , the coupling to adjacent oscillators , and the loss-gain parameter are the same for all oscillators. The classical coordinates are () and the equations of motion are

(8)

These equations of motion are symmetric, where the definitions of and are generalized from (6) and (7) to

(9)

The equations of motion (8) imply that there is a conserved quantity. To construct this constant of the motion we multiply the first equation by , the second equation by , the third equation by , the fourth equation by , and so on. If we add the resulting equations, drops out entirely and we obtain a time-independent quantity, which we can identify as the energy of the system:

(10)

The existence of a conserved quantity suggests that (8) is a Hamiltonian system, and indeed one can find a Hamiltonian from which these equations of motion can be derived. There are two ways to express the (nonunique) Hamiltonian that gives rise to (8); we can use what we call a sum or a product representation. We describe these two structures below.

ii.1 Sum representation of the Hamiltonian

In the sum representation consists of four terms. First, there is a pure momentum term of the form . Second, there is a momentum times a coordinate term proportional to : . Third, there is a potential-energy-like term proportional to : . (It is surprising that this term is proportional to because in the equations of motion appears to play the role of a coupling constant; does not appear to be a measure of the potential energy, which one associates with a frequency of oscillation.) Fourth, there is an oscillator coupling term proportional to :

(11)

Note the interesting structure of this term: The jumps in the products in (11) skip 0, 2, 4, 6, … and change sign. A compact expression for is

(12)

To obtain the equations of motion (8) for this Hamiltonian from Hamilton’s equations, we take one derivative of with respect to to find :

(13)

We then take a time derivative,

(14)

and use the one-derivative equation (13) to recover the equations of motion (8).

ii.2 Product representation of the Hamiltonian

In this representation it is easy to understand the nonuniqueness of the Hamiltonian that gives rise to the equations of motion (8). This nonuniqueness is a gauge invariance, where plays the role of an electric charge. Without changing the equations of motion we rewrite the sum representation in (5) so that the momentum terms appear in factored form:

(15)

Similarly, the sum representation for ,

(16)

can be reconfigured in product form as

(17)

without changing the equations of motion. The product representation of has the form

(18)

The general structure for the product representation of is now clear.

The advantage of the product representation is that if we consider the Hamiltonian to be quantum mechanical, we can identify a gauge invariance. Each momentum factor in the product representation has the form . This term resembles the structure in electrodynamics, which suggests that we can make a unitary (canonical) transformation analogous to a gauge transformation in electrodynamics. By virtue of the Heisenberg algebra , it follows that , where is a constant. Therefore, if we perform the unitary transformation

(19)

on the Hamiltonian, the only terms that will be affected are the product terms because they contain the momentum operators. The only changes that will occur are that the momentum operators and will be shifted by terms that are linear in the coordinates and . There are independent gauge transformations that can be performed on , and therefore we can introduce arbitrary constants into . Furthermore, since the transformation is unitary, it leaves the equations of motion invariant R16 ().

ii.3 Lagrangian

Having found a Hamiltonian for the system (8), it is easy to construct a Lagrangian:

(20)

Iii General case of nonconstant , ,

We can construct a Hamiltonian (in the sum representation) for a -symmetric system of oscillators even if the parameters , , and vary from oscillator to oscillator

(21)

We can also construct a Hamiltonian for oscillators:

(22)

The even Hamiltonian leads to the equations of motion

(23)

and the odd Hamiltonian gives the equations of motion

(24)

iii.1 Continuum limit

In this subsection we show how to take the limit as the number of oscillators approaches infinity. For simplicity, let us consider two rows of identical particles of mass . These masses are coupled by springs, as illustrated in Fig. 1.

Figure 1: Infinite -symmetric array of identical particles coupled by springs. The masses in the top row, whose position coordinates are , experience loss and the masses in the bottom row, which are located at , experience gain.

The top row of particles is subject to damping (friction) forces and the bottom row is subject to undamping forces. Each particle in the top row is coupled by horizontal springs (of force constant per unit length ) to the adjacent particles to the left and right. Thus, the particle at is coupled to its neighbors at and at . The neighboring particles exert a net force on the th mass of strength , where is the equilibrium spacing. The constant is the tension in the horizontal chain of masses. Also, there are fixed springs above the top row of masses that exert a restoring force per unit length of on each of the masses. This force tends to pull the masses back to their equilibrium positions. The parameter has dimensions of mass density (mass per unit length) and the parameter is a frequency having dimensions of . The force on the th mass due to these vertical springs is . Finally, the particle at in the top row is coupled to the particle at the position in the bottom row by a vertical spring of force per unit length . (Here, is a mass density and is a frequency.) The force exerted on the mass at due to the particle at is . The particles in the top row lose energy due to friction (drag), where the dissipation per unit length is given by . Thus, the equation of motion of the th particle is

(25)

Let , where is the horizontal mass per unit length. We then divide (25) by and take the limit as to get the continuum wave equation

(26)

Finally, we divide by and define the quantities , , , and . This leads to the wave equation

(27)

Similarly, from the equation for the particle at we obtain the wave equation

(28)

These equations are the continuous analogs of (8).

In anticipation of the calculation in Sec. V, we rewrite these equations in a more convenient form by defining and . The coupled wave equations satisfied by and are

(29)

where we have now taken the loss-gain parameter to depend on .

Iv Existence of an unbroken--symmetric region

The question addressed in this section is whether a region of unbroken symmetry persists as the number of oscillators increases. We consider first the case in which the loss-gain parameter is the same for all oscillators and show that the unbroken region disappears as increases. Next, we demonstrate numerically that if decreases for the more distant oscillators, a region of unbroken symmetry persists as .

iv.1 Case of constant

To find the frequencies of the system (8), we seek solutions of the form . The frequencies can then be found by imposing the condition that (Cramer’s rule), where is the tridiagonal matrix

(30)

and and are given by and .

Let () be the polynomial obtained by computing the determinant of the matrix . The first five of these polynomials are

(31)

where . These polynomials satisfy the recursion relation

(32)

where we take .

Given these polynomials, we can calculate the frequencies to see what happens to the unbroken--symmetric region as increases. In Fig. 2 we plot the imaginary part of for , and 4 for fixed and . It is clear that as increases, the size of the unbroken region in the coupling parameter shrinks and at it disappears entirely.

Figure 2: Imaginary parts of the frequencies for and 4 as functions of the coupling constant for and . The frequencies are the zeros of the polynomials in (31). Observe that the extent of the unbroken--symmetric region (where the frequencies are all real) decreases as increases and disappears entirely when .

To study analytically the shrinking of the unbroken region with increasing , we solve the constant-coefficient recursion relation (32). The exact solution is

(33)

Substituting and , we express these polynomials more simply:

(34)

The zeros of are the roots of the equation . Since is negative, we substitute . The equation for then reads , whose solutions are

(35)

Consequently, the equation for becomes , whose roots are

(36)

We consider two cases. For there are four roots, with . These correspond to the six values . The solutions are spurious, and the only admissable solutions are . Substituting into (36), we obtain the four roots of the polynomial in (31).

For the case there are six roots,

with . These correspond to the ten values

The solutions are spurious and there are only four genuine roots and two values to substitute into (36) for getting the eight roots of the polynomial in (31).

In general, in the region of unbroken symmetry the roots in (36) are all real. Thus,

(37)

where and . Condition (37) identifies the region in the parameter space where the symmetry is unbroken. Note that as , and . Thus, as the only allowed is 0 (so that there is no loss and gain), and the range of shrinks to .

Let us examine further how the allowed decreases as a function of increasing . We can see from Fig. 2 that at the lower end of the unbroken region the curves open to the left and at the upper end of this region the curves open to the right. For fixed and fixed , if we increase , the left opening curves will eventually touch the right opening curves and the unbroken region in will disappear. We designate as the critical value of at which the unbroken region in disappears. If we compute as a function of and plot in Fig. 3 these values of versus , we see clearly that the critical value of decreases to 0. Thus, if there are too many oscillators, there cannot be a region of unbroken symmetry in a system with uniform nonzero loss and gain.

Figure 3: Plot of as a function of . This sequence evidently converges to 0 with increasing . Thus, a system of coupled oscillators with a uniform loss-gain parameter has no unbroken -symmetric region if is sufficiently large.

The only way for an unbroken region of symmetry to survive as is for the loss-gain parameter to decrease with increasingly distant oscillators. Our numerical calculations show that if the loss-gain parameter is (where ranges from 1 to ), there will be an unbroken region if is less than about (Fig. 4, left panel), and if the loss-gain parameter is (where ranges from 1 to ), there will be an unbroken region if is less than about (Fig. 4, right panel).

Figure 4: Analog of Fig. 3: Oscillatory convergence of when the loss-gain parameter decreases like (left panel) and like (right panel). Evidently, if the loss-gain parameter decays to zero for more distant oscillators, a region of unbroken symmetry can persist as .

V Localized impurity in the continuum model

In Sec. IV we showed that if the effect of loss and gain is localized about the central oscillators and decays for more distant oscillators, then the unbroken--symmetric region can survive as . This suggests that for the continuum model developed in subsection III.1 it would be interesting to examine what happens when decreases with increasing . The simplest case to study is that for which ; that is, the case of a localized point-like -symmetric loss-gain impurity at the origin. Studies of this type have been performed for tight-binding models by Joglekar et al R17 (); R18 () and Longhi R19 ().

Let us assume that the loss-gain parameter is a localized function of at the origin, , and seek a solution to (29) with frequency :

(38)

If we assume that and that , where and are positive, the coupled wave equations become coupled ordinary differential equations:

(39)

The functions and are continuous at and the delta function gives rise to a discontinuity in the derivatives of and at :