Coherent states for ladder operators of general order related to exceptional orthogonal polynomials

Coherent states for ladder operators of general order related to exceptional orthogonal polynomials

Scott E. Hoffmann1, Véronique Hussin2, Ian Marquette1 and Yao-Zhong Zhang1 1School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

2Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, H3C 3J7, Canada
Abstract

We construct the coherent states of general order, for the ladder operators and , which act on rational deformations of the harmonic oscillator. The position wavefunctions of the eigenvectors involve type III Hermite exceptional orthogonal polynomials. We plot energy expectations, time-dependent position probability densities for the coherent states and for the even and odd cat states, Wigner functions, and Heisenberg uncertainty relations. We find generally non-classical behaviour, with one exception: there is a regime of large magnitude of the coherent state parameter, where the otherwise indistinct position probability density separates into distinct wavepackets oscillating and colliding in the potential, forming interference fringes when they collide. The Mandel parameter is calculated to find sub-Poissonian statistics, another indicator of non-classical behaviour. We plot the position standard deviation and find squeezing in many of the cases. We calculate the two-photon-number probability density for the output state when the coherent states (where labels the lowest weight in the superposition) are placed on one arm of a beamsplitter. We find that it does not factorize, again indicating non-classical behaviour. Calculation of the linear entropy for this beamsplitter output state shows significant entanglement, another non-classical feature. We also construct linearized versions, of the annihilation operators and their coherent states and calculate the same properties that we investigate for the coherent states. For these we find similar behaviour to the coherent states, at much smaller magnitudes of but comparable average energies.

I Introduction

In a recent publication Hoffmann2018a (), the present authors constructed the coherent states associated with the ladder operators and Marquette2013b (); Marquette2013a () for the particular choice of the order and investigated their properties. In this paper we construct the coherent states for general order, (even). The systems are rational, nonsingular, deformations of the harmonic oscillator, obtained through supersymmetric quantum mechanics (SUSY QM), with position wavefunctions involving type III Hermite exceptional orthogonal polynomials GomezUllate2010 (); Marquette2013b (); Marquette2013a (). The ladder operators and are distinguished in that they are currently the only ones that connect all states of the partner Hamiltonians, including the ground states.

A coherent state vector is a coherent superposition of the energy eigenvectors of a system. Coherent states are of widespread interest Glauber1963a (); Glauber1963b (); Klauder1963a (); Klauder1963b (); Barut1971 (); Perelomov1986 (); Gazeau1999 (); Quesne2001 (); Fernandez1995 (); Fernandez1999 (); GomezUllate2014 (); Ali2014 () because they can, in some cases, be the quantum-mechanical states with the most classical behaviour.

Since the ladder operators and change the energy index by there are distinct, orthogonal coherent states, each with a different lowest weight state. For these ladder operators, we construct the coherent states (defined as eigenvectors of the annihilation operator with complex eigenvalue, ). Then we calculate energy expectations, time-dependent position probability densities for the coherent states and for the even and odd cat states, Wigner functions and Heisenberg uncertainty relations. The Mandel parameter is calculated to find the number statistics of these coherent states. The position standard deviation is plotted as a function of and to investigate the possibility of squeezing. We place one of our coherent states on one arm of a beamsplitter and calculate the two-photon-number probability distribution to look for factorization. The linear entropy is calculated for this output state to look for entanglement.

For comparison, we also construct linearized versions of the ladder operators, and and the coherent states of investigate the same properties and compare with the coherent states.

In the previous paper Hoffmann2018a (), we compared coherent states labelled and (for lowest weights ) at the same value of the complex coherent state parameter, and found significantly different behaviours. However the average energies of these coherent states have significantly different dependencies on . In this paper, we will compare the two cases, and at comparable average energies, and we will find similar rather than different behaviours.

We have used the Barut-Girardello definition Barut1971 () of coherent states as eigenvectors of the annihilation operator with complex eigenvalue, Note that there are other possible definitions, for example using a displacement operator Perelomov1986 ().

Coherent states for various choices of ladder operators in systems that are the rational extensions of the harmonic oscillator were also constructed in Bermudez2015 ().

In Section II we review the ladder operators and and their matrix elements. In Section III we form the coherent states , labelled by the lowest weight, in each superposition. In Section IV we calculate energy expectations for and for all cases of and In Section V we plot the time-dependent position probability densities for several cases. In Section VI we plot time-dependent position probability densities for the even and odd Schrödinger cat states formed from at In Section VII we plot the Wigner functions for all cases of and . In Section VIII we calculate Heisenberg uncertainty relations and find an example of squeezing. In Section IX we calculate the Mandel parameter for and find sub-Poissonian number statistics. In Section X we place the coherent state on one arm of a beamsplitter, show that the output two-quantum number distribution does not factorize and calculate the linear entropy of the output state. Conclusions follow in Section XI.

Ii The ladder operators, c(m), and their matrix elements for general m

The systems we consider are rational deformations of the harmonic oscillator, obtained through supersymmetry Marquette2013b (); Marquette2013a (), with Hamiltonians and , with the dependence on of and understood. The partner potentials are, for any even

 V(m)(x)=x2−2[H′′mHm−(H′mHm)2+1]. (II.1)

The modified Hermite polynomials are

 Hm(x)≡(−i)mHm(ix), (II.2)

which are positive definite for even The spectra of the partner Hamiltonians and the harmonic oscillator Hamiltonians are the same except for an additional ground state of the partner Hamiltonians

 E(+,m)ν =2(ν+m+1)for ν=0,1,2,… and E(−,m)ν=2(ν+m+1)for ν=−m−1,0,1,2,…, (II.3)

with

 H+|ν(+)⟩=E(+,m)ν|ν(+)⟩for ν=0,1,2,… and H−|ν(−)⟩=E(−,m)ν|ν(−)⟩for ν=−m−1,0,1,2,…. (II.4)

The ladder operators, are constructed and their matrix elements derived in Marquette2013b (). Each operator involves a path from to through the supercharge then a return to in steps through intermediate Hamiltonians that may be singular. This is

 c(m)=Am…A2A1A†. (II.5)

We make a small change to the matrix elements compared to Marquette2013b () to simplify subsequent calculations. We rephase the ground state, multiplying it by This does not change the polynomial Heisenberg algebra, which only involves diagonal operators. Then we multiply all the position wavefunctions by which has no physical consequence. Then we find the matrix elements

 ⟨ν−m−1(−)|c(m)|ν(−)⟩=−[2m+1(ν−1)(ν−2)…(ν−m)(ν+m+1)]12≡a(m)ν (II.6)

for all The operator lowers the index by

The polynomial Heisenberg algebra satisfied by these ladder operators is Marquette2013b ()

 [H−,c†] =(2m+2)c†,[H−,c]=−(2m+2)c, [c,c†] =Qm(H−+2m+2)−Qm(H−), (II.7)

with

 Qm(H−+2m+2)=(H−+2m+2)m∏i=1(H−−2i)andQm(H−)=H−m∏i=1(H−−2m−2−2i), (II.8)

polynomials of order in

The position wavefunctions are

 ψ(m)ν(x)=N(m)νe−x2/2Hm(x)y(m)ν+m+1(x), (II.9)

with the Hermite exceptional orthogonal polynomials

 y(m)0(x) =1, y(m)ν+m+1(x) =Hm(x)Hν+1(x)+mHm−1(x)Hν(x), (II.10)

and the normalization factors

 N(m)−m−1 =[2mm!√π]12, N(m)ν =1√√π2ν+1(ν+m+1)ν!. (II.11)

Iii The coherent states of the operators c(m) and ~c(m)

We have seen that the ladder operator connects states with indices differing by This separates the space into distinct, orthogonal ladders, with lowest weights So there must be coherent states of defined with the Barut-Girardello definition Barut1971 () as eigenstates of the annihilation operator with complex eigenvalue with each one being a superposition of the state vectors of one of the ladders of the form

 |z,c,m,μ⟩=∞∑k=0|μ+(m+1)k,m(−)⟩A(m,μ)k(z). (III.1)

Solving the defining condition,

 c(m)|z,c,m,μ⟩=z|z,c,m,μ⟩, (III.2)

and normalizing gives

 A(m,μ)k(z)=1√F(m,μ)(z)zkD(m,μ)k, (III.3)

with

 D(m,μ)k=k∏i=1a(m)μ+(m+1)i (III.4)

and

 F(m,μ)(z)=∞∑k=0|z|2kD(m,μ)2k. (III.5)

Using the form (II.6) for the gives

 D(m,μ)k=(−)k(2m+2)k(m+1)/2[(μ−1m+1+1)k(μ−2m+1+1)k…(μ−mm+1+1)k(μ+m+1m+1+1)k]12 (III.6)

and

 F(m,μ)(z)=|1Fm+1(1;μ−1m+1+1,μ−2m+1+1,…,μ−mm+1+1,μ+m+1m+1+1;|z|2(2m+2)m+1). (III.7)

Here

 (a)k=Γ(a+k)Γ(a) (III.8)

and is a generalized hypergeometric function Gradsteyn1980 ().

We define a linearized version, of the operator (in the intrinsic class of Fernandez1994 ()) to have matrix elements

 ⟨ν−m−1(−)|~c(m)|ν(−)⟩=√2ν, (III.9)

like those of the harmonic oscillator annihilation operator. In operator form, this is (for acting on the subspace with lowest weight )

 ~c(m,μ)=−1[Qm(H−)]12√H−−2m−2−2μm+1c. (III.10)

The operators and satisfy the Heisenberg algebra

 [H−,~c†] =(2m+2)~c†,[H−,~c]=−(2m+2)~c, ~c(m,μ)~c†(m,μ) =H−−2m−2−2μm+1+2,~c†(m,μ)~c(m,μ)=H−−2m−2−2μm+1, [~c(m,μ),~c†(m,μ)] =2. (III.11)

Because of the similarity with the harmonic oscillator, it is a simple matter to find the form of the coherent states, again with orthogonal ladders labelled by the lowest weight

 |z,~c,m,μ⟩=∞∑k=0|μ+(m+1)k,m(−)⟩~Ak(z), (III.12)

with

 ~Ak(z)=e−|z|2/4(z/√2)k√k!. (III.13)

Iv Energy expectations

The energy eigenvalues are given by

 H−|μ+(m+1)k,m⟩=(2μ+(2m+2)(k+1))|μ+(m+1)k,m⟩. (IV.1)

This leads to the energy expectations

 ⟨z,c,m,μ|H−|z,c,m,μ⟩=2μ+2m+2+2m+2(μ+m)(μ+m−1)…(μ+1)(μ+2m+2)××|z|22m+1|1Fm+1(2;μ+mm+1+1,μ+m−1m+1+1,…,μ+1m+1+1,μ+2m+2m+1+1;|z|2/(2m+2)m+1)|1Fm+1(1;μ+mm+1,μ+m−1m+1,…,μ+1m+1,μ+2m+2m+1;|z|2/(2m+2)m+1). (IV.2)

For the linearized operators, we have the much simpler result

 ⟨z,~c,m,μ|H−|z,~c,m,μ⟩=2μ+2m+2+(m+1)|z|2. (IV.3)

In Figure 1, we show these functions of for and in Figure 2 for This allows comparison between different values of We see that the average energy always rises much faster for the coherent states than for the coherent states.

V Time evolution of position probability densities

We study the time evolution of the position probability densities for the coherent states we have constructed. For the states, we find that for low values of the pattern of many peaks and valleys does not take a simple form. However, in all cases, we find that increasing to very large values gives a pattern of a small number of peaks oscillating in the potential and colliding with each other, producing interference fringes when they collide. We call this regime semi-classical, since it is the closest to the appearance of (more than one) classical particle that we have found.

In our previous paper Hoffmann2018a (), we only studied the coherent states to and did not see the semi-classical regime. Taking for (to give the same average energy as with ) shows the semi-classical regime with three wavepackets.

It is now clear that and coherent states show very similar behaviour when compared at the same average energy, but not at the same value of the parameter

For general the position probability density is given by

 ρ(x,t;z,m,μ) =|⟨x|e−iH−t|z,c,m,μ⟩|2 =|∞∑k=0ψ(m)μ+(m+1)k(x)A(m,μ)k(ze−i(2m+2)t)|2. (V.1)

For the linearized states , the corresponding expression is

 ~ρ(x,t;z,m,μ)=|∞∑k=0ψ(m)μ+(m+1)k(x)~Ak(ze−i(2m+2)t)|2. (V.2)

We find that the coefficients fall off very rapidly with for the values of we are using. So in practice, we only take the sum in (V.1) to a small cutoff value of

The state vectors and have the same average energy and similar behaviour for much less than as can be seen in Figures 1 and 2. For the states at low we can also cut off the sums over at low values.

For at and at (giving the same average energy ) we show the probability densities in Figure 3 and note similar behaviour, both systems being in the semi-classical regime. As noted above, in this regime we see isolated wavepackets oscillating and colliding.

For , we display the time-dependence of the position probability density at a set of discrete times. These plots, unlike the 3D plots, do not smear the interference fringes. We note that at times when the wavepackets are well-separated, we count wavepackets, at least for these two choices of and recalling the result for . For we started with low values of (real, positive) and found indistinct position probability densities. It was necessary to take to the extremely large value of before the pattern shown below emerged.

Vi Schrödinger cat states

A cat state is a superposition of two macroscopically distinguishable states. We construct even (+) and odd (-) states by (for real)

 |z,c,m,μ,±⟩=1√2{|+z,c,m,μ⟩±|−z,c,m,μ⟩}. (VI.1)

A measure of the distinguishability of the two component states is (for )

 D(|z|,μ)=⟨+z,c,6,μ,+|−z,c,6,μ⟩=|1F7(1;μ+67,μ+57,…,μ+17,μ+147;−|z|2/147)|1F7(1;μ+67,μ+57,…,μ+17,μ+147;|z|2/147). (VI.2)

This is shown in Figure 5, where we see that is certainly sufficient for distinguishability.

The time-dependent position probability densities for these two cat states are shown in a density plot in Figure 6. We see a great deal of structure at this very large value of The odd state has a central nodal line, consistent with the parity of the odd wavefunctions.

Vii Wigner functions

The Wigner function for a single particle is an everywhere real function of position, and momentum, so can be considered a distribution in phase space. For the coherent states of the harmonic oscillator, the Wigner functions are everywhere positive Gaussians centred on and This fact is used to reinforce the assertion, that we arrived at by looking at the evolution of the position probability density, that the coherent states of the harmonic oscillator are the most closely classical states of a one-dimensional system. It is therefore judged that a state vector does not behave classically, does not have a classical analogue, if its Wigner function is anywhere negative. We will use this criterion to judge the coherent states of our operators.

For a general state vector the Wigner function is defined as

 W(x,p;ψ)=1π∫∞−∞dy⟨ψ|x−y⟩⟨x+y|ψ⟩e−i2py. (VII.1)

For we calculate the Wigner function using

 W(x,p:z,c,m,μ)=∞∑k1=0∞∑k2=0A(m,μ)∗k1(z)w(m,μ)k1k2(x,p)A(m,μ)k2(z), (VII.2)

with

 w(m,μ)k1k2(x,p)=1π∫∞−∞dyψ(m)∗μ+(m+1)k1(x−y)ψ(m)μ+(m+1)k2(x+y)e−i2py. (VII.3)

As we noted in Section V, the coefficients fall off rapidly with for the small values of that we consider. We found that it was sufficient to take the sums over to

For we found the seven Wigner functions shown in Figure 7. The flat, monotone areas in each graph show where the Wigner function takes on negative values. We conclude that these coherent states do not conform to classical expectations (except for ), as do the harmonic oscillator coherent states.

Viii Heisenberg uncertainty relations and squeezing

For our various coherent states, and we calculate the standard deviations in position and momentum using

 σx =√⟨x2⟩−⟨x⟩2, σp =√⟨p2⟩−⟨p⟩2, (VIII.1)

where the expectation values are in the particular state considered. Then we form the products

 h(z,m,μ)=σx(z,m,μ)σp(z,m,μ). (VIII.2)

These are required by Heisenberg’s uncertainty principle to be, in all cases, greater than or equal to

The expectation are calculated using, with again,

 ⟨ψ|^x|ψ⟩ = ∑k1∑k2A(m,μ)∗k1(z)M(x)k1k2A(m,μ)k2(z), ⟨ψ|^x2|ψ⟩ = ∑k1∑k2A(m,μ)∗k1(z)M(x2)k1k2A(m,μ)k2(z), ⟨ψ|^p|ψ⟩ = ∑k1∑k2A(m,μ)∗k1(z)M(p)k1k2A(m,μ)k2(z), ⟨ψ|^p2|ψ⟩ = ∑k1∑k2A(m,μ)∗k1(z)M(p2)k1k2A(m,μ)k2(z), (VIII.3)

with

 M(x)k1k2 = ∫∞−∞dxψ(m)∗μ+(m+1)k1(x)xψ(m)μ+(m+1)k2(x), M(x2)k1k2 = ∫∞−∞dxψ(m)∗μ+(m+1)k1(x)x2ψ(m)μ+(m+1)k2(x), M(p)k1k2 = ∫∞−∞dxψ(m)∗μ+(m+1)k1(x)(−iddx)ψ(m)μ+(m+1)k2(x), M(p2)k1k2 = ∫∞−∞dxψ(m)∗μ+(m+1)k1(x)(−d2dx2)ψ(m)μ+(m+1)k2(x). (VIII.4)

Again the sums over can be taken to a small finite value depending on the range of values being considered.

In every case we considered, we found a product greater than as expected. In Figure 8 we show the product as a function of and for and for

The Heisenberg uncertainty principle for one-dimensional systems with position and momentum operators, as we have been considering, is This bound is saturated for the coherent states of the harmonic oscillator, where in our units. It is possible for one of the standard deviations to be less that the harmonic oscillator value, provided that the other standard deviation is greater than to preserve the uncertainty relation. In these cases we say that we are dealing with a squeezed state Loudon1987 (); Walls1983 ().

We test our coherent states for squeezing and find that it occurs in many cases. For example, Figure 9 shows the observation of squeezing for the state in at The graph shows a flat area for where falls below

Squeezing has physical applications. Light can be squeezed and used to reduce the noise in measurements Oelker2016 ().

Ix Number statistics

The coherent states of the harmonic oscillator have an excitation number probability distribution (from (III.13))

 Pn(z)=|⟨n|z⟩|2=e−|z|2/2(|z|2/2)nn!, (IX.1)

which is a Poisson distribution with mean number

 ⟨N⟩=|z|22. (IX.2)

It is easily seen that the number variance satisfies

 ΔN2=⟨N2⟩−⟨N⟩2=⟨N⟩. (IX.3)

Then a number distribution with variance less than the average number is called sub-Poissonian, and called super-Poissonian if the variance is greater than the mean. The Mandel parameter measures the classification of an arbitrary number distribution:

 Q=ΔN2−⟨N⟩⟨N⟩=⟨N(N−1)⟩−⟨N⟩2⟨N⟩. (IX.4)

It vanishes for a Poisson distribution, is negative for a sub-Poissonian distribution and is positive for a super-Poissonian distribution.

We investigate the number statistics of the coherent states labelled . We define a number operator, , that acts on the subspace of the order system with lowest weight by

 Nμ|μ+5k,4(−)⟩=k|μ+5k,4(−)⟩ (IX.5)

Then we calculate the Mandel parameter as where the expectations in (IX.4) are in the state vector and

We find

 ⟨Nμ⟩=1(μ+4)(μ+3)(μ+2)(μ+1)(μ+10)|z|232|1F5(2;μ+95,μ+85,μ+75,μ+65,μ+155;|z|2/105)|1F5(1;μ+45,μ+35,μ+25,μ+15,μ+105;|z|2/105) (IX.6)

and

 ⟨Nμ(Nμ−1)⟩=|z|429|1F5(3;μ+145,μ+135,μ+125,μ+115,μ+205;|z|2/105)/|1F5(1;μ+45,μ+35,μ+25,μ+15,μ+105;|z|2/105)(μ+4)(μ+3)(μ+2)(μ+1)(μ+10)(μ+9)(μ+8)(μ+7)(μ+6)(μ+15). (IX.7)

This gives the Mandel parameter as a function of for as shown in Figure 10. We see that the parameter is negative except at the origin, indicating sub-Poissonian statistics. This is considered another sign of non-classical behaviour. For other values curves of similar appearance were produced, but with progressively slower variation as was increased.

It should be clear that since the structure of the linearized coherent states is so like that of the harmonic oscillator coherent states, they have only Poisson statistics. To verify this, we find

 ⟨Nμ⟩=∞∑k=0k|~Ak(z)|2=∞∑k=0ke−|z|2/2(|z|2/2)kk!=|z|22 (IX.8)

and

 ⟨Nμ(Nμ−1)⟩=∞∑k=0k(k−1)|~Ak(z)|2=∞∑k=0k(k−1)e−|z|2/2(|z|2/2)kk!=(|z|22)2. (IX.9)

So

 Q(~c(m),μ,z)≡0for all m,μ. (IX.10)

X A coherent state on a beamsplitter

The optical device called a beamsplitter is, in its simplest form, a partially silvered mirror Walls2008 (). For a 50:50 beamsplitter (the only case we will consider) half the intensity of a beam of light incident on the beamsplitter is transmitted while the other half is reflected. If two perpendicular beams are incident on the beamsplitter, they are mixed coherently and two perpendicular output beams are produced. If just one incident beam is used, the input in the other direction is the vacuum.

We can place one of our coherent states on one arm of a beamsplitter, with the excitations above the ground state taking the role of photons. We confirm below that if the input is a harmonic oscillator coherent state, representing coherent light, the “photon” number distribution from the two output arms is uncorrelated, factorizing into the product of two independent Poisson number distributions. If a squeezed state is the input, there is correlation and entanglement between the two output arms Sanders2012 (); Ourjoumtsev2009 (). Entangled states may have applications in quantum cryptography and quantum teleportation Jennewein2000 (); Bennett1993 (). Thus a beamsplitter calculation can give us more information on whether a coherent state displays classical behaviour (like the harmonic oscillator coherent states) or non-classical behaviour.

If and are linearized ladder operators for the two input arms (for indices and ), respectively, with harmonic oscillator commutation relations, the action of the beamsplitter is represented by the unitary transformation Kim2002 ()

 U=exp(iπ8(~c1~c†2+~c2~c†1)), (X.1)

which is seen to give the transformations

 U~c†1U†=1√2(~c†1+i~c†2),U~c†2U†=1√2(i~c†1+~c†2). (X.2)

The factors of represent phase shifts upon reflection. Then it can be shown that

 U{|n⟩⊗|0⟩}=12n2n∑r=0(nr)12ir|n−r⟩⊗|r⟩, (X.3)

where

 |n⟩=|μ+(m+1)n,m⟩. (X.4)

For a general coherent state with coefficients

 |z⟩=∞∑n=0An(z)|n⟩, (X.5)

we have the result for the output state

 |out⟩=U{|z⟩⊗|0⟩}=∞∑n=0An(z)2n2n∑r=0(nr)12ir|n−r⟩⊗|r⟩. (X.6)

Then the two-photon-number probability distribution for the output arms is

 P(n1,n2;z) =|{⟨n1|⊗⟨n2|}U{|z⟩⊗|0⟩}|2=|An1+n2(z)|22n1+n2(n1+n2n2). (X.7)

We check for a harmonic oscillator coherent state input,

 A(0)k(z)=e−|z|2/4(z√2)k1√k!, (X.8)

that

 P(0)(n1,n2;z)=pn1(|z|24)pn2(|z|24), (X.9)

where

 pn(¯n)=e−¯n¯nnn! (X.10)

is the Poisson distribution for mean number The mean numbers in each arm are half the value for the input state.

For the case we found

 P(4,−5)(n1,n2;z)=1n1!n2!1F(4,−5)(z)(|z|22×105)n1+n2××Γ(−1