Analysis of the effects of nonextensivity for a generalized dissipative system in the SU(1,1) coherent states | Scientific Reports – Nature.com

Basics of the general CK oscillator with nonextensivity

Various physical systems subjected to a friction-like force which is a linear function of velocity can be modeled by the formal CK oscillator. The Hamiltonian of the CK oscillator is given by34,35

$$begin{aligned} {hat{H}} = e^{-gamma t} frac{{hat{p}}^2}{2m} + frac{1}{2} e^{gamma t} m omega ^2 {hat{x}}^2, end{aligned}$$

(1)

where (gamma) is a damping constant. This Hamiltonian can be generalized by replacing the ordinary exponential function with a deformed one that is defined by1,37

$$begin{aligned} exp _q {(y)} = [1+(1-q)y]^{1/(1-q)}, end{aligned}$$

(2)

with an auxiliary condition

$$begin{aligned} 1+(1-q)y ge 0, end{aligned}$$

(3)

where q is a parameter indicating the degree of nonextensivity. This generalized function is known as the q-exponential and has its own merit in describing non-idealized dynamical systems. The characteristic behavior of the q-exponential function is shown in Fig.1. In the field of thermostatistics, a generalization of the Gaussian distribution through the q-exponential is known as the Tsallis distribution that is well fitted to many physical systems of which behavior does not follow the usual BG statistical mechanics38.

q-exponential function for several different values of q.

In terms of Eq.(2), we can express the generalized CK Hamiltonian in the form1

$$begin{aligned} {hat{H}}_q = frac{{hat{p}}^2}{2m exp _q{(gamma t)}} + frac{1}{2} exp _q{(gamma t)} m omega ^2 {hat{x}}^2. end{aligned}$$

(4)

This Hamiltonian is Hermitian and, in the case of (q rightarrow 1), it recovers to the ordinary CK one that is given in Eq.(1). From the use of the Hamiltons equations in one dimension, we can derive the classical equation of motion that corresponds to Eq.(4) as

$$begin{aligned} ddot{x} + frac{gamma }{1+(1-q)gamma t}{dot{x}} + omega ^2 x = 0. end{aligned}$$

(5)

In an extreme case where (q rightarrow 0), Eq.(2) reduces to a linear function (1+y). Along with this, Eq.(5) reduces to

$$begin{aligned} ddot{x} + frac{gamma }{1+gamma t}{dot{x}} + omega ^2 x = 0. end{aligned}$$

(6)

If we think from the pure mathematical point of view, it is also possible to consider even the case that q is smaller than zero based on the condition given in Eq.(3). However, in most actual nonextensive systems along this line, the value of q may not deviate too much from unity which is its standard value. So we will restrain to treating such extreme cases throughout this research.

In general, for time-dependent Hamiltonian systems, the energy operator is not always the same as the given Hamiltonian. The role of the Hamiltonian in this case is restricted: It plays only the role of a generator for the related classical equation of motion. From fundamental Hamiltonian dynamics, we can see that the energy operator of the generalized damped harmonic oscillator is given by26,39

$$begin{aligned} {hat{E}}_{q} = {hat{H}}_q/exp _q{(gamma t)}. end{aligned}$$

(7)

Let us denote two linearly independent homogeneous real solutions of Eq.(5) as (s_1(t)) and (s_2(t)). Then, from a minor mathematical evaluation, we have40,41

$$begin{aligned} s_1(t)= & {} {s}_{0,1}sqrt{frac{pi omega }{2gamma (1-q)}} [exp _q{(gamma t)}]^{-q/2} J_nu left( frac{omega }{(1-q)gamma } + omega t right) , end{aligned}$$

(8)

$$begin{aligned} s_2(t)= & {} {s}_{0,2}sqrt{frac{pi omega }{2gamma (1-q)}} [exp _q{(gamma t)}]^{-q/2} N_nu left( frac{omega }{(1-q)gamma } + omega t right) , end{aligned}$$

(9)

where (J_nu) and (N_nu) are the Bessel functions of the first and second kind, ({s}_{0,1}) and ({s}_{0,2}) are constants which have dimension of position, and (nu = {q}/{[2(1-q)]}). From Fig.2, we see that the phases in the time evolutions of (s_1(t)) and (s_2(t)) are different depending on the value of q. Now we can represent the general solution of Eq.(5) in the form

$$begin{aligned} x(t) = c_1 s_1(t) + c_2 s_2(t), end{aligned}$$

(10)

where (c_1) and (c_2) are arbitrary real constants.

Time evolution of (s_1(t)) (A) and (s_2(t)) (B) for several different values of q. We used (omega =1), (gamma =0.1), and (s_{0,1}=s_{0,2}=0.1).

We introduce another time function s(t) that will be used later as

$$begin{aligned} s(t) = sqrt{s_1^2(t)+s_2^2(t)}. end{aligned}$$

(11)

This satisfies the differential equation42

$$begin{aligned} ddot{s}(t) + frac{gamma }{1+(1-q)gamma t}{dot{s}}(t) + omega ^2 s(t) - frac{Omega ^2}{[mexp _q{(gamma t)}]^2} frac{1}{s^3(t)} = 0, end{aligned}$$

(12)

where (Omega) is a time-constant which is of the form

$$begin{aligned} Omega = m exp _q{(gamma t)} [s_1 {dot{s}}_2 - {dot{s}}_1 s_2 ]. end{aligned}$$

(13)

By differentiating Eq.(13) with respect to time directly, we can readily confirm that (Omega) does not vary in time.

In accordance with the invariant operator theory, the invariant operator must satisfy the Liouville-von Neumann equation which is

$$begin{aligned} frac{d {hat{I}}}{d t} = frac{partial {hat{I}}}{partial t} + frac{1}{ihbar } [{hat{I}},{hat{H}}_q] = 0. end{aligned}$$

(14)

A straightforward evaluation after substituting Eq.(4) into the above equation leads to24,40

$$begin{aligned} {hat{I}} = hbar Omega left( {hat{b}}^dagger {hat{b}} + frac{1}{2}right) , end{aligned}$$

(15)

where ({hat{b}}) is a destruction operator defined as

$$begin{aligned} {hat{b}} = sqrt{frac{1}{2hbar Omega }} left[ left( frac{Omega }{s(t)} -i m exp _q{(gamma t)} {dot{s}}(t) right) {hat{x}} + i s(t) {hat{p}} right] , end{aligned}$$

(16)

whereas its hermitian adjoint ({hat{b}}^dagger) is a creation operator. If we take the limit (gamma rightarrow 0), Eq.(16) reduces to that of the simple harmonic oscillator. One can easily check that the boson commutation relation for ladder operators holds in this case: ([{hat{b}},{hat{b}}^dagger ]=1). This consequence enables us to derive the eigenstates of ({hat{I}}) in a conventional way.

The zero-point eigenstate (| 0 rangle) is obtained from ({hat{b}}| 0 rangle =0). The excited eigenstates (| n rangle) are also evaluated by acting ({hat{b}}^dagger) into (| 0 rangle) n times. The Fock state wave functions (| psi _n rangle) that satisfy the Schrdinger equation are different from the eigenstates of ({hat{I}}) by only minor phase factors which can be obtained from basic relations24. However, we are interested in the SU(1,1) coherent states rather than the Fock states in the present work.

The SU(1,1) generators are defined in terms of ladder operators, such that

$$begin{aligned} hat{{mathcal {K}}}_0= & {} frac{1}{2} left( {hat{b}}^dagger {hat{b}} + frac{1}{2}right) , end{aligned}$$

(17)

$$begin{aligned} hat{{mathcal {K}}}_+= & {} frac{1}{2} ({hat{b}}^dagger )^2, end{aligned}$$

(18)

$$begin{aligned} hat{{mathcal {K}}}_-= & {} frac{1}{2} {hat{b}}^2. end{aligned}$$

(19)

From the inverse representation of Eq.(16) together with its hermitian adjoint ({hat{b}}^dagger), we can express ({hat{x}}) and ({hat{p}}) in terms of ({hat{b}}) and ({hat{b}}^dagger). By combining the resultant expressions with Eqs.(17)(19), we can also represent the canonical variables in terms of SU(1,1) generators as

$$begin{aligned} {hat{x}}^2= & {} frac{hbar s^2}{Omega } (2hat{{mathcal {K}}}_0 + hat{{mathcal {K}}}_+ + hat{{mathcal {K}}}_-), end{aligned}$$

(20)

$$begin{aligned} {hat{p}}^2= & {} frac{hbar }{s^2} Bigg [ 2 left( Omega + frac{[mexp _q(gamma t)]^2}{ Omega } s^2{dot{s}}^2 right) hat{{mathcal {K}}}_0 -left( sqrt{Omega } - frac{imexp _q(gamma t)}{ sqrt{Omega }} s{dot{s}} right) ^2 hat{{mathcal {K}}}_+ nonumber \&-left( sqrt{Omega } + frac{imexp _q(gamma t)}{ sqrt{Omega }}s{dot{s}} right) ^2 hat{{mathcal {K}}}_- Bigg ]. end{aligned}$$

(21)

The substitution of the above equations into Eq.(4) leads to

$$begin{aligned} {hat{H}}_q = delta _0(t) hat{{mathcal {K}}}_0 + delta (t) hat{{mathcal {K}}}_+ + delta ^*(t) hat{{mathcal {K}}}_- , end{aligned}$$

(22)

where

$$begin{aligned} delta _0(t)= & {} frac{hbar }{s^2} left( frac{Omega }{mexp _q{(gamma t)}} + frac{1}{Omega } mexp _q{(gamma t)} s^2 {dot{s}}^2 right) + frac{hbar }{Omega } mexp _q{(gamma t)}omega ^2 s^2 , end{aligned}$$

(23)

$$begin{aligned} delta (t)= & {} - frac{hbar }{2 mexp _q{(gamma t)} s^2} left( sqrt{Omega } - i frac{mexp _q{(gamma t)}s{dot{s}}}{sqrt{Omega }} right) ^2 + frac{hbar }{2Omega } mexp _q{(gamma t)} omega ^2 s^2 . end{aligned}$$

(24)

In accordance with Gerrys work (see Ref. 43), Eq.(22) belongs to a class of general Hamiltonian that preserves an arbitrary initial coherent state. In the next section, we will analyze the properties of nonextensivity associated with the SU(1,1) coherent states using the Hamiltonian in Eq.(22).

The SU(1,1) coherent states for the quantum harmonic oscillator belong to a dynamical group whose description is based on SU(1,1) Lie algebraic formulation. The analytical representation of the SU(1,1) coherent states provides a natural description of quantum and classical correspondence which has an important meaning in theoretical physics. On the experimental side, optical interferometers like radio interferometers that use four-wave mixers as a protocol for improving measurement accuracy are characterized through the SU(1,1) Lie algebra44,45.

According to the development of Perelomov46, the SU(1,1) coherent states are defined by

$$begin{aligned} | {tilde{xi }};k rangle = hat{{mathcal {D}}}(beta )|{{{tilde{0}}}} rangle _k , end{aligned}$$

(25)

where (hat{{mathcal {D}}}(beta )) is the displacement operator, (|{{{tilde{0}}}} rangle _k) is the vacuum state in the damped harmonic oscillator, and k is the Bargmann index of which allowed values are 1/4 and 3/4. The basis for the unitary space is a set of even boson number for (k=1/4), whereas it is a set of odd boson number for (k=3/4). Here, the displacement operator is given by

$$begin{aligned} {hat{D}}(beta )= & {} exp left[ frac{1}{2} (beta ^2 hat{{mathcal {K}}}_+ - beta ^{*2} hat{{mathcal {K}}}_-) right] nonumber \= & {} e^{{tilde{xi }} hat{{mathcal {K}}}_+} exp {-2ln [cosh (|beta |^2/2)] hat{{mathcal {K}}}_0} e^{-{tilde{xi }}^* hat{{mathcal {K}}}_-}, end{aligned}$$

(26)

where (beta) is the eigenvalue of ({hat{b}}) and ({tilde{xi }}) is an SU(1,1) coherent state parameter of the form

$$begin{aligned} {tilde{xi }} = frac{beta ^2}{|beta |^2} tanh (|beta |^2/2). end{aligned}$$

(27)

The above equation means that (|{tilde{xi }}| <1). For (k=3/4) among the two allowed values, the resolution of the identity in Hilbert space is given by47

$$begin{aligned} int dmu ({tilde{xi }};k) | {tilde{xi }} ; k rangle langle {tilde{xi }} ; k| = mathbf{1}, end{aligned}$$

(28)

where (dmu ({tilde{xi }};k)=[(2k-1)/pi ] d^2 {tilde{xi }} /(1-|{tilde{xi }}|^2)^2). More generally speaking, this resolution is valid for (k>1/2). For a general case where k is an arbitrary value, the exact resolution is unknown. Brif et al., on one hand, proposed a resolution of the identity with a weak concept in this context, which can be applicable to both cases of (k>1/2) and (k<1/2)47. In what follows, various characteristics of the damped harmonic oscillator with and without deformation in quantum physics, such as quantum correlation, phase coherence, and squeezing effect, can be explained by means of the SU(1,1) Lie algebra and the coherent states associated with this algebra48,49.

The expectation values of SU(1,1) generators in the states (| {tilde{xi }};k rangle) are50

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_0|{tilde{xi }};krangle= & {} k frac{1+|{tilde{xi }}|^2}{1-|{tilde{xi }}|^2 } , end{aligned}$$

(29)

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_+|{tilde{xi }};krangle= & {} frac{2k{tilde{xi }}^*}{1-|{tilde{xi }}|^2 } , end{aligned}$$

(30)

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_-|{tilde{xi }};krangle= & {} frac{2k{tilde{xi }}}{1-|{tilde{xi }}|^2 }. end{aligned}$$

(31)

Using the above equations, the expectation values of the Hamiltonian given in Eq.(22) are easily identified as50,51

$$begin{aligned} {{mathcal {H}}}_{q,k}= & {} langle {tilde{xi }} ;k | {hat{H}}_q |{tilde{xi }};k rangle nonumber \= & {} frac{k}{1-|{tilde{xi }}|^2} { delta _0(t)(1+|{tilde{xi }}|^2) +2 [delta (t){tilde{xi }}^*+delta ^*(t){tilde{xi }}] } . end{aligned}$$

Go here to see the original:

Analysis of the effects of nonextensivity for a generalized dissipative system in the SU(1,1) coherent states | Scientific Reports - Nature.com