PT-symmetric scattering in flow duct acoustics
Yves Aurégan Vincent Pagneux


 

yves.auregan@univ-lemans.fr
Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613
Av. O Messiaen, F-72085 LE MANS Cedex 9, France
 vincent.pagneux@univ-lemans.fr
Laboratoire d'Acoustique de l'Université du Maine, UMR CNRS 6613
Av. O Messiaen, F-72085 LE MANS Cedex 9, France



We show  theoretically and experimentally that the propagation of an acoustic wave in an airflow duct going through a pair of diaphragms, 
 with equivalent amount of mean-flow-induced effective gain and  loss, 
displays all the features of a parity-time (𝒫𝒯) symmetric system. 
Using a scattering matrix formalism, 
we observe experimentally the  properties which reflect the 𝒫𝒯-symmetry of the scattering acoustical system:  
the existence of a spontaneous symmetry breaking with symmetry-broken pairs of scattering eigenstates showing amplification and reduction, 
and the existence of points with unidirectional invisibility.




11.30.Er, 43.20.Mv, 43.20.Rz, 68.35.Iv

 𝒫𝒯-symmetric scattering in flow duct acoustics
    Vincent Pagneux
    
===============================================







Hydrodynamic instability theory shows that flow can provide energy to small perturbations
<cit.>. 
If, in addition, these perturbations are compressible, then both acoustic wave propagation and energy exchange
with the flow are possible, leading e.g. to the classical whistling phenomena <cit.>.
Thus, in the particular case of  flow duct acoustics, the wave can obviously be  convected  but it also experiences gain or loss due to 
interactions with the flow inhomogeneities <cit.>. 
Consequently, propagation of acoustic waves in ducts with flow is a natural Non-Hermitian system where
loss and gain are available.

Non-Hermitian systems, where energy conservation is broken, lead to dynamics governed by evolution equations with 
non-normal operators, where surprising phenomena can appear due to huge non-normality especially close to 
exceptional points <cit.>.
The particular case of 𝒫𝒯-symmetry, where gain and  loss are delicately balanced,
has attracted a lot of attention in the last two decades
<cit.>. 
It opens the possibility  to obtain purely real spectra from Non-Hermitian Hamiltonians, 
as well as a spontaneous symmetry breaking where real eigenvalues coalesce at an exceptional point to become complex conjugate pair.

From a scattering point of view, another type of spontaneous symmetry breaking for 𝒫𝒯-symmetric systems has been theoretically proposed <cit.>.
It corresponds to the transition of norm-preserving scattering eigenstates, with unimodular eigenvalues, to symmetry broken pairs 
of amplified and lossy scattering eigenstates, with associated pairs of scattering eigenvalues with 
inverse moduli <cit.>.
 It is to be noticed that this type of symmetry breaking is still waiting to be observed experimentally <cit.>.
 
Initiated in the domain of quantum mechanics, many works on 𝒫𝒯-symmetry 
have displayed several intriguing effects such as 
power oscillation <cit.>, 
unidirectional transparency <cit.>, 
single-mode laser <cit.>, spectral singularity and Coherent Perfect Absorber (CPA)-Laser
 <cit.>
or enhanced sensitivity <cit.>.
A majority of the studies has been conducted in optics with
some attempts in acoustics where the difficulty to obtain gain has been recognized.
Actually, whilst losses can be easily  introduced <cit.>, the gain for acoustic waves has until now been obtained owing
to active electric amplification <cit.>. 

In this letter, we report the experimental realization of a purely mechanical scattering 𝒫𝒯-symmetric system 
for the propagation of acoustic waves in a waveguide. 
The loss and the gain are produced by 
two localized scattering units made of  diaphragms, one associated with loss and the other
associated with gain, see Fig. <ref>(a).
In our experiments, the Mach number of the flow is small enough 
(𝑀-.20𝑒𝑚 𝑎≃ 0.01) such that the effect of convection on the sound wave can
be neglected, preserving the reciprocity property, and the only effect of the flow is located at the two diaphragms, 
characterized by  normalized complex impedances C_1 and C_2.
The balance of gain and loss is realized by finely tuning the flow rate and the geometry of each diaphragm, ensuring
a 𝒫𝒯-symmetric system that corresponds to  C_1=C_2^* (note that the real part of the two impedances have to be equal to
get the parity symmetry).
Measurements of the scattering matrix components allow us to demonstrate unidirectional invisibility and to verify the 𝒫𝒯-symmetry properties.
Besides, by changing the distance between the scatterers,  the spontaneous symmetry breaking of the scattering matrix is observed with the transition from
 exact-𝒫𝒯-symmetric  phase to  𝒫𝒯-broken phase.
 In the broken phase, with the experimental gain available, the scattering eigenstates can be simultaneously fourfold amplified or reduced, 
 and we show that this effect might be enhanced by considering a finite periodic collection of the set of two diaphragms, leading to CPA-Laser points.


System description and 1D model.—

The description of the set-up is shown in Fig. <ref>. 
We consider an acoustic waveguide where only plane waves can propagate (k A < 1.841 <cit.>, where A is the tube radius, k=ω/c_0 is the wavenumber, ω is the frequency and c_0 is the sound velocity). The propagation for the acoustic pressure p is then governed by the 1D Helmholtz equation. 
Two diaphragms are inserted into the tube and are separated with a distance D (Fig. <ref>(a)). 
As their thicknesses t are small (k t ≪ 1), the acoustic velocity is conserved while the pressure jumps between the two sides of the discontinuities. Thus the propagation is governed by

    p”  +   k^2    p = 0,


 with the point scatterer jump conditions at the diaphragms: 

    p'  _x=± D/2   =    0 ,
    
       p   _x=- D/2= C_1/k  p'                     and          p   _x= D/2 = C_2/k  p'.

where prime is the derivative  with respect to x. 
The real part of the dimensionless parameters C_1,2 is associated to reactive effects while its imaginary part is linked to the dissipative or gain effects. 
We have thus a very simple 1D reciprocal wave model with two point scatterers at x=± D/2 (Fig. <ref>(b)).
The effect of the flow on acoustic propagation is only and entirely contained in the complex impedances
C_1 and C_2 that reflect the mean-flow-induced effective gain and loss.

The system is 𝒫𝒯-symmetric if and only if the two impedances are complex conjugated: C_2=C_1^* <cit.>. 
 With the exp(-iω t ) convention, there is absorption if (C_i) >0 and gain if (C_i) <0.

The overall behavior of the acoustical system can be described by the transfer matrix 𝖬


    ( [ p_2^+; p_2^- ]) = 
    [ [ M_11 M_12; M_21 M_22 ]]
    ( [ p_1^+; p_1^- ])


where p^+_1,2 and p^-_1,2 are defined in Fig.  <ref>(b).

After some algebra, the component of the overall transmission matrix are found to be:


    M_11   =    -isin( k D) C_1 C_2/2+e^i k D( 1+iC_1/2+iC_2/2)
    
    M_12   =   isin( k D) C_1 C_2/2 -e^i k DiC_1/2 -e^-i k DiC_2/2
    
    M_21   =    -isin( k D) C_1 C_2/2+e^-i k DiC_1/2+e^i k DiC_2/2
    
    M_22   =   isin( k D) C_1 C_2/2+e^-i k D( 1-iC_1/2-iC_2/2)


where in the case of a 𝒫𝒯 symmetric system <cit.>: M_11=M^*_22 and
[M_12] = [M_21] = 0. The transmission and reflection coefficients for waves coming from left and right are defined by


    t_L= det(𝖬) /M_22             r_R=M_12/M_22
    
    r_L= - M_21/M_22             t_R= 1/M_22


Due to reciprocity we have  det(𝖬)=1 and then t=t_L=t_R.

As discussed in detail in <cit.>,  by permutation of the outgoing waves, two different scattering matrices 
with different sets of eigenvalues can be defined, leading to distinct symmetry breaking. 
These two scattering matrices are 

    𝖲_𝗋 = [ r_L   t;   t r_R ]   and   𝖲_𝗍 = [   t r_L; r_R   t ]



    where[ p_1^-; p_2^+ ] 
     =𝖲_𝗋[ p_1^+; p_2^- ] ,       and   𝖲_𝗍=𝖲_𝗋σ_𝗑,


σ_𝗑 is one of the Pauli matrices. 

The eigenvalues of 𝖲_𝗋 and  𝖲_𝗍 may have both an exact and broken phases but the symmetry-breaking points are not the same. 
In this paper, we have chosen to consider both 𝖲_𝗋 and  𝖲_𝗍 and the different phase transitions they imply.
When computing the scattering eigenvalues, it is useful to remind the 𝒫𝒯-symmetry conservation relations <cit.> that can be written for instance as 𝖲_𝗍^*=𝖲_𝗍^-1 and leads to

    r^*_L r_R   =    1-|t|^2
    
    r_L t^* + r^*_L t   =   0
    
    r_R t^* + r^*_R t   =   0

The eigenvalues of the scattering matrix 𝖲_𝗍 are given by λ_1,2=t±√(r_R r_L))=t (1 ±√(1-|t|^-2)). Then if |t|<1, the modulus of the eigenvalues is equal to 1. The case |t|=1 corresponds to symmetry-breaking and  |t|>1 correspond to the 𝒫𝒯-broken phase.
The eigenvalues of the other scattering matrix 𝖲_𝗋 are given by s_1,2=(r_R+r_L ±√(Δ))/2 where Δ = (r_R-r_L)^2+4t^2. 
The broken phase condition can be written Δ=0 which leads to r_R-r_L=± 2 i t. In term of the transmission matrix coefficients, it is equivalent to M_12-M_21 = ± 2 i or (C_1) sin( k D) = ± 1. 




Experimental set-up.—
As described in Fig. <ref>, the 𝒫𝒯 symmetric system is mounted in a rigid circular duct between two measurement sections, upstream and downstream. 
Each measurement section consists in a  hard walled steel duct (diameter 30 mm) where two microphones are mounted. 
Two acoustic sources on both sides of the system give two different acoustic states and the four elements of the scattering matrix (transmission and reflection coefficient on both directions) for plane waves can be evaluated. 
A more detail description of the measurement technique can be found in <cit.>. 
The desired gain scatterer is realized by a finely designed diaphragm submitted to a steady flow. In this geometry, a shear layer is formed on its upstream edge and the flow is contracted into a jet with an area smaller than the hole of the diaphragm, see Fig. <ref>(c). 
This shear layer is very sensitive to any perturbations like an oscillation in the velocity due to the acoustic wave. 
The shear layer convects and amplifies these perturbations (see the marked zone in Fig. <ref>(c) and a strong coupling between acoustic and flow occurs when the acoustical period is of the order of the time taken by the perturbations to go from the upstream edge of the diaphragm to the exit of the diaphragm. 
This corresponds to a Strouhal number of the order of S_h=f t/U_d ∼ 0.2 <cit.> 
where f is the frequency of the acoustic perturbation, t is the thickness of the diaphragm and U_d is the mean velocity in the diaphragm U_d=U_0 (A/a)^2 with U_0 the mean velocity in the duct 
and a the radius of the diaphragm (Fig. <ref>(c)). 
Eventually, this gain diaphragm has been chosen with an internal radius a = 10 mm and a thickness t = 5 mm (see Fig. <ref> and the inset in Fig. <ref>). 
The other diaphragm, that has to be lossy, has been chosen with an internal radius a = 12 mm and a thickness t = 4.3 mm. 
Two resistive metallic tissues have been glued to produce the dissipation by viscous and turbulent effects.

In a first step, the scattering coefficients of the two diaphragms have been measured separately, 
allowing us to deduce the values of the impedance  C_1,2.
These parameters,
 that have to  verify C_2=C_1^* to get a 𝒫𝒯-symmetric system, 
are plotted on Fig. <ref>. With the chosen geometry and flow parameters, it can be observed that there is
a frequency f_m where the desired equality (C_2=C_1^*) is achieved.

In a second step, the scattering matrix of the system composed by the two balanced diaphragms is measured. 
All the subsequently reported measurements are made at the frequency f_m = 1920 Hz and at the Mach number 𝑀-.20𝑒𝑚 𝑎= 0.01 for which C_2=C_1^*= 1.83 - 1.36i, allowing the system to be 𝒫𝒯 symmetric. 
In order to be able to observe the symmetry breaking, the distance between the two diaphragms D is varied 
from 312 mm to 417 mm by inserting 22 rigid metallic tubes of different lengths.  
The minimal distance is chosen to minimize the hydrodynamical interactions between the two diaphragms. 
The maximal D is chosen to have points over half a wavelength at the measurement frequency with
a value of kD/2 π approximately in the range 1.7 – 2.4.


Results.—
The measured transmission and reflection coefficients are displayed in Fig. <ref>(a). 
They are compared to the theoretical values obtained by using the measured value of C_1=C^*_2 and the 1D modeling of Eqs. (<ref>). 
The reflections from left r_L (impinging on the loss) and right r_R (impinging on the gain) appear as deeply asymmetric, with two points
with |t|=1 and r_R=0 or r_L=0. 
These two points correspond to the unidirectional transparency phenomenon where 
the wave passes unreflected with no amplitude change through the scatterers form one side, and is strongly reflected from the other side.
In order to verify experimentally the 𝒫𝒯 symmetry of the system,  in Fig. <ref>(b), we plot the 2-norm of 
the matrix 𝖲_𝗍𝖲_𝗍^* - 𝖨 corresponding to the shift from the  𝒫𝒯 symmetry conservation relations in
Eqs (<ref>)-(<ref>). 
For comparison the norm of the matrix 𝖲_𝗍 ^t𝖲_𝗍^* - 𝖨  which represents the deviation to the energy conservation is also displayed. 
It appears that 𝖲_𝗍𝖲_𝗍^* - 𝖨 is nearly equal to zero in the whole range of paramaters  which unambiguously demonstrates
 that the system is  𝒫𝒯 symmetric; meanwhile 𝖲_𝗍 ^t 𝖲_𝗍^* - 𝖨  can take large values confirming 
 that our system strongly violates conservation of energy.
It can be noticed that for kD multiple of π, the system is simultaneously 𝒫𝒯-symmetric and conservative; 
it can be verified (see Eqs. <ref>) that in these cases the scattering is only sensitive to the real part of the impedances C_1 and C_2
ignoring thus the effect of gain and loss.


By  varying the length of the duct between the two diaphragms, we can also inspect the spontaneous symmetry breaking
of the scattering matrix of the system  <cit.>. 
In Fig. <ref>, we show the eigenvalues of S_r and S_t that, since they  are different, 
lead to different symmetric and broken phases <cit.>. 
We represent also the singular value decomposition (SVD) of the scattering matrices. These two SVD are identical for  𝖲_𝗍
and 𝖲_𝗋 (since 𝖲_𝗍 ^t 𝖲_𝗍^*=𝖲_𝗋 ^t 𝖲_𝗋^*) and correspond respectively to the maximum
and minimum outgoing wave for any incoming waves with unit flux; by definition they are upper and lower bound of the modulus of the eigenvalues, and thus
must be different from one to allow the broken phase.
For each choice of scattering matrix, the experimental measurements, very close to the theoretical predictions, 
display clear signatures of the spontaneous symmetry breaking
with different broken phases for 𝖲_𝗍 and 𝖲_𝗋. 
In the symmetric phase the eigenvalues of the scattering matrices
remain on the unit circle in the complex plane, and the symmetry breaking corresponds to pairs of non-unimodular scattering eigenvalues
i.e. where the moduli are the inverse of each other and different from 1.
To the best of our knowledge, it is the first experimental demonstration of the symmetry breaking of the scattering matrix 
for 𝒫𝒯-symmetric systems as proposed in <cit.>.


In the broken phase, a particularly interesting case is the CPA-Laser where one eigenvalue of the 𝖲 matrix goes to infinity (Laser) and the
other goes to zero (Absorber). From the experimental results of Fig. <ref> we can see that this  Laser-Absorber is not achieved because the
maximum eigenvalue corresponds to  a 3.5 amplification.
From Eqs. (<ref>), it can be shown that the CPA-Laser condition can be obtained for larger values of the gain parameter ((C_2)≃2.5) which cannot be achieved  with our current experimental setup. Nevertheless, in Fig. <ref>, we show that quasi-CPA-Laser could be theoretically achieved by taking 
a finite periodic array of N cells of our 𝒫𝒯-symmetric system with a  distance W between each cell (Fig. <ref>(a)).
The use of the 1D model  shows that very near CPA-Laser can be obtained by just tuning the number of cells and the intercell dimensionless frequency kW (N=25 and kW/2π=2.1 in Fig <ref>(b-c)). 
Fig. <ref>(c) indicates that, by using interference Bragg effect in finite periodic case, it is possible to approach very closely the conditions of CPA-Laser.

Conclusion.— 
Owing to vortex-sound interaction providing gain and loss in an acoustical system, 
we have obtained the experimental signatures of the spontaneous 𝒫𝒯-symmetry breaking in scattering systems. 
The scattering matrix eigenvalues can remain on the unit circle in the complex plane despite the Non-Hermiticity and the symmetry breaking results in pairs of scattering eigenvalues with inverse moduli.  The unidirectional transparency has also been observed.
It is noteworthy that this mechanical gain medium does not require to be electronically powered and that this 𝒫𝒯-symmetric system is very simple to manufacture: one tube, two diaphragms and a small flow inside the tube.
Therefore, this kind of acoustic system
can be seen as a building block to study wave propagation with more complex 
𝒫𝒯-symmetry (for instance in periodic systems), 
and, more generally, we believe it provides an important connection 
between hydrodynamic instability theory, acoustic wave propagation and Non-Hermitian physics.


 
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