Fitting over and undercoupled resonators in reflection

Ql: loaded quality factor, Qc: coupling quality factor, Qi: internal quality factor. overcoupled regime: Qc<Qi, for Qc<<Qi we have Qc~Ql and the S11 shows only a phase signal.



In [1]:

    
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from IPython.display import display
%matplotlib inline

Case1: Undercoupled resonator Qc>Qi

First, we generate some fake data which we can fit later:



In [2]:

    
from resonator_tools import circuit
fr = 7e9  #resonance frequency in Hz
Qi = 200e3
Qc = 400e3
freq = np.linspace(fr-0.5e6, fr+0.5e6, 1000)
port1 = circuit.reflection_port()  #define a reflection port
noise = np.random.normal(loc=1.0,scale=0.01,size=(len(freq),))
S11 = noise * port1._S11_directrefl(freq,fr=fr,Ql=Qi*Qc/(Qc+Qi),Qc=Qc,a=1.,alpha=0.,delay=.0)
port1.add_data(freq,S11)

Let's plot the data:



In [3]:

    
plt.rcParams["figure.figsize"] = [10,5]
port1.plotrawdata()

...and fit it: (since we know the electric delay, we tell it the program, this makes it more accurate)



In [4]:

    
port1.autofit(electric_delay=0.)

...and plot the result:



In [5]:

    
port1.plotall()

Next, let us have a look at the fit results. Here, we convert the dictionary of results into a dataframe to display it in a nicer way.



In [6]:

    
display(pd.DataFrame([port1.fitresults]).applymap(lambda x: "{0:.2e}".format(x)))









    






  
    
      
      Qc
      Qc_err
      Qi
      Qi_err
      Ql
      Ql_err
      chi_square
      fr
      fr_err
      theta0
    
  
  
    
      0
       3.99e+05
       1.25e+03
       2.00e+05
       1.10e+03
       1.33e+05
       5.89e+02
       8.95e-05
       7.00e+09
       9.33e+01
       1.22e-03

Finally, we can calculate the single photon limit, i.e., the input power necessary to maintain one photon on average in the resonator:



In [7]:

    
print 'Single photon limit: %.2f dBm' % port1.get_single_photon_limit()









    



Single photon limit: -149.39 dBm

Or, we can compute the photons in the resonator for a given power:



In [8]:

    
print 'At -100dBm, we have %.2e photons in the resonator' % port1.get_photons_in_resonator(-100)









    



At -100dBm, we have 8.70e+04 photons in the resonator

Case2: Strongly over coupled resonator Qc<<Qi

Here, we study the case where internal losses become negigible, and the amplitude signal vanishes.

First, we generate some fake data which we can fit later:



In [9]:

    
from resonator_tools import circuit
fr = 7e9  #resonance frequency in Hz
Qi = 10000e3
Qc = 100e3
freq = np.linspace(fr-0.5e6, fr+0.5e6, 1000)
port2 = circuit.reflection_port()  #define a reflection port
noise = np.random.normal(loc=1.,scale=0.04,size=(len(freq),))
S11 = noise * port1._S11_directrefl(freq,fr=fr,Ql=Qi*Qc/(Qc+Qi),Qc=Qc,a=1.,alpha=0.,delay=.0)
port2.add_data(freq,S11)

Let's plot the data:



In [10]:

    
port2.plotrawdata()

Here, we see that the amplitude signal completely vanishes in the noise

Fit the signal:

IMPORTANT: In the presence of a lot of noise, it is necessary to specify the electric delay by hand



In [11]:

    
port2.autofit(electric_delay=0.)









    



WARNING: Calibration using Lorentz fit failed, trying phase fit...






    



C:\Python27\lib\site-packages\scipy\optimize\minpack.py:237: RuntimeWarning: The iteration is not making good progress, as measured by the 
  improvement from the last ten iterations.
  warnings.warn(msg, RuntimeWarning)

...and plot the result:



In [12]:

    
port2.plotall()

Next, let us have a look at the fit results. Here, we convert the dictionary of results into a dataframe to display it in a nicer way.



In [13]:

    
display(pd.DataFrame([port2.fitresults]).applymap(lambda x: "{0:.2e}".format(x)))









    






  
    
      
      Qc
      Qc_err
      Qi
      Qi_err
      Ql
      Ql_err
      chi_square
      fr
      fr_err
      theta0
    
  
  
    
      0
       1.00e+05
       3.99e+02
       1.04e+07
       4.34e+06
       9.91e+04
       5.57e+02
       1.72e-03
       7.00e+09
       1.42e+02
       1.75e-03

Finally, we can calculate the single photon limit, i.e., the input power necessary to maintain one photon on average in the resonator:



In [14]:

    
print 'Single photon limit: %.2f dBm' % port2.get_single_photon_limit()









    



Single photon limit: -152.84 dBm

Or, we can compute the photons in the resonator for a given power:



In [15]:

    
print 'At -100dBm, we have %.2e photons in the resonator' % port2.get_photons_in_resonator(-100)









    



At -100dBm, we have 1.92e+05 photons in the resonator