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Quantum Noise of Resonant Cooper Pair Tunneling

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					Mesoscopic Detectors and the
      Quantum Limit
                      (cond-mat/0211001)

       A. A. Clerk, S. M. Girvin, and A. D. Stone
      Departments of Applied Physics and Physics,
                     Yale University
   (and many discussions with M. Devoret & R. Schoelkopf)

Q:What characterizes an “ideal” quantum detector?
       Generic Weakly-Coupled Detector


                               I
              Q                      “gain”



1. Measurement Rate: How quickly can we distinguish the two
   qubit states?
                  P(m,t)




                       0
                           m
2. Dephasing Rate: How quickly does the measurement decohere
   the qubit?
                               SQQ ´ 2 s dt h dQ(t) dQ(0) i
        The Quantum Limit of Detection
                        I
            Q
Quantum limit: the best you can do is measure as fast as you
dephase:




 •Measurement? Need               distinguishable from
 •Dephasing? Need               orthogonal to

 • What symmetries/properties must an arbitrary detector
    possess to reach the quantum limit?
       Why care about the quantum limit?
• Minimum Noise Energy in Amplifiers:
  (Caves; Clarke; Devoret & Schoelkopf)




  • Minimum power associated with Vnoise?

• Detecting coherent qubit oscillations (Averin & Korotkov)
             sz                             SI
                   Q                 I

                                                              w
                            Quantum Limit
       How to get to the Girvin & Stone, cond-mat/0211001
                      A.C.,
                                          Averin, cond-mat/0301524



                        I
             Q
 •Now, we have:



•Quantum limit requires:
   •                             (i.e. no extra degrees of freedom)
   • λ’ vanishes (monitoring output does not further dephase)

   • λ’ is the “reverse gain”:     Q               I
                      What does it mean?
  • To reach the quantum limit, there should be no unused
    information in the detector…
Mesoscopic Scattering Detector:
(Pilgram & Buttiker; AC, Girvin & Stone)


                                    I      mL
                   Q                                        mR
      L                           R
                       What does it mean?
   • To reach the quantum limit, there should be no unused
     information in the detector…
 Mesoscopic Scattering Detector:
 (Pilgram & Buttiker; AC, Girvin & Stone)


                                     I      mL
                    Q                                        mR
       L                           R



Transmission probability
depends on qubit:
         The Proportionality Condition
 •   Need:                      Q      I




     Not usual symmetries!

Phase condition?
•Qubit cannot alter relative
phase between reflection and
transmission
•No “lost” information that
could have been gained in an   L           R
interference experiment….
        Transmission Amplitude Condition
                                  Q     I
                                             L            R

   Ensures that no information is lost when averaging over energy
1) mL
                          mR

                             versus
2) mL
                           mR
       The Ideal Transmission Amplitude



     Necessary energy dependence to be at the quantum limit
Corresponds to a real system-- the adiabatic quantum point contact!
                (Glazman, Lesovik, Khmelnitskii & Shekhter, 1988)


                                                         1   T
                                                       0.8
                                                       0.6
                                                       0.4
                                                       0.2

                                           -4     -2                2   4
                                                                            e - e0
            Information and Fluctuations
     Reaching quantum limit = no wasted information
 • No information lost in phase changes:

 • No information lost when energy averaging:


Look at charge fluctuations:

           Q        I
    L              R




  Gmeas for current experiment             Gmeas for phase experiment
  Measurement Rate for Phase Experiment



Gmeas for current experiment   Gmeas for phase experiment



  t
            r
         Information and Fluctuations (2)
     Reaching quantum limit = no wasted information

Can connect charge fluctuations to        Q        I
information in more complex cases: L             R
1. Multiple Channels

                                          Extra terms due to
                                          channel structure


2. Normal-Superconducting Detector



 Gmeas for current experiment          Gmeas for phase experiment
          Partially Coherent Detectors
 • What is the effect of adding dephasing to the mesoscopic
   scattering detector? Look at a resonant-level model…


mL                             • Symmetric coupling to
                      mR         leads  no information
                                 in relative phase


                                                f
                                                     If = 0
• Assume dephasing due to
  an additional voltage            L                          R
  probe (Buttiker)
         Partially Coherent Detectors
• Reducing the coherence of the detector enhances charge
  fluctuations… total accessible information is increased
• A resulting departure from the quantum limit…
 Charge Noise (SQ)

  2                             1
1.8                           0.8
1.6                           0.6
1.4                           0.4
1.2                           0.2

      0.2 0.4 0.6 0.8    1             0.2 0.4 0.6 0.8
Conclusions
                                      I
                           Q
 • Reaching the quantum limit requires that there be no
 wasted information in the detector; can make this
 condition precise.

 • Looking at information provides a new way to look at
 mesoscopic systems:
    • New symmetry conditions
    • New way to view fluctuations

 • Reducing detector coherence enhances charge
 fluctuations, leads to a departure from the quantum limit

				
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posted:4/9/2010
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