page_title: Squeezed states of light
section_title: Physical meaning of measurement quantity and measurement object
hierarchical_section_title: Squeezed states of light / Physical meaning of measurement quantity and measurement object
caption_attribution_description: English: Normalized Variances &#x0394; 2 X f , &#x0394; f / &#x0394; 2 X f , &#x0394; f , G {\displaystyle \Delta ^{2}X_{f,\Delta f}/\Delta ^{2}X_{f,\Delta f,G}} of modulation states of the same carrier light beam versus modulation frequency f {\displaystyle f} . The measurement band width &#x0394; f {\displaystyle \Delta f} is about 10 kHz in this example, such that each trace describes about 200 mutually independent modulation modes.
context_page_description: In quantum physics, light is in a squeezed state if its electric field strength Ԑ for some phases has a quantum uncertainty smaller than that of a coherent state. The term squeezing thus refers to a reduced quantum uncertainty. To obey Heisenberg's uncertainty relation, a squeezed state must also have phases at which the electric field uncertainty is anti-squeezed, i.e. larger than that of a coherent state.
context_section_description: Quantum uncertainty becomes visible when identical measurements of the same quantity (observable) on identical objects (here: modes of light) give different results (eigen values). In case of a single freely propagating monochromatic laser beam, the individual measurements are performed on consecutive time intervals of identical length. One interval must last much longer than the light's period; otherwise the monochromatic property would be significantly disturbed. Such consecutive measurements correspond to a time series of fluctuating eigen values. Consider an example in which the amplitude quadrature  was repeatedly measured. The time series can be used for a quantum statistical characterization of the modes of light.
  Obviously, the amplitude of the light wave might be different before and after our measurement, i.e. the time series does not provide any information about very slow changes of the amplitude, which corresponds to very low frequencies. This is a trivial but also fundamental issue, since any data taking lasts for a finite time. Our time series, however, does provide meaningful information about fast changes of the light's amplitude, i.e. changes at frequencies higher than the inverse of the full measuring time. Changes that are faster than the duration of a single measurement, however, are invisible again. A quantum statistical characterization through consecutive measurements on some sort of a carrier is thus always related to a specific frequency interval, for instance described by  with  Based on this inside, we can describe the physical meaning of the observable  more clearly:
  The quantum statistical characterization using identical consecutive modes carried by a laser beam confers to the laser beam's electric field modulation within a frequency interval. The actual observable needs to be labeled accordingly, for instance as .  is the amplitude (or depth) of the amplitude modulation and  the amplitude (or depth) of the phase modulation in the respective frequency interval. This leads to the doggerel expressions 'amplitude quadrature amplitude' and 'phase quadrature amplitude'.
  Within some limitations, for instance set by the speed of the electronics,  and  can be freely chosen in course of data acquisition and, in particular, data processing. This choice also defines the measurement object, i.e. the mode that is characterized by the statistics of the eigen values of  and  . The measurement object thus is a modulation mode that is carried by the light beam. – In many experiments, one is interested in a continuous spectrum of many modulation modes carried by the same light beam. Fig. 2 shows the squeeze factors of many neighboring modulation modes versus . The upper trace refers to the uncertainties of the same modes being in their vacuum states, which serves as the 0 dB reference.
  The observables in squeezed light experiments correspond exactly to those being used in optical communication. Amplitude modulation (AM) and frequency modulation (FM) are the classical means to imprint information on a carrier field. (Frequency modulation is mathematically closely related to phase modulation). The observables  and   also correspond to the measurement quantities in laser interferometers, such as in Sagnac interferometers measuring rotation changes and in Michelson interferometers observing gravitational waves. Squeezed states of light thus have ample applications in optical communication and optical measurements.
caption: Fig. 2: Normalized variances 
    
      
        
          
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      {\displaystyle \Delta ^{2}X_{f,\Delta f}/\Delta ^{2}X_{f,\Delta f,g}}
    
   of modulation states of the same carrier light beam versus modulation frequency 
    
      
        
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      {\displaystyle f}
    
  . Here, the measurement band width 
    
      
        
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      {\displaystyle \Delta f}
    
   is about 10 kHz. Each trace therefore describes about 200 mutually independent modulation modes.
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