Editing Charge qubit (section)

== Hamiltonian ==
If the Josephson junction has a junction capacitance <math>C_{\rm J}</math>, and the gate capacitor <math>C_{\rm g}</math>, then the charging (Coulomb) energy of one Cooper pair is:

:<math>E_{\rm C}=(2e)^2/2(C_{\rm g}+C_{\rm J}).</math>

If <math>n</math> denotes the number of excess Cooper pairs in the island (i.e. its net charge is <math>-2ne</math>), then the Hamiltonian is:<ref name=":0" />

:<math>H=\sum_n \big[E_{\rm C}(n-n_{\rm g})^2 |n \rangle \langle n| - \frac{1}{2} E_{\rm J} (|n \rangle \langle n+1|+|n+1 \rangle \langle n|) \big],</math>

where <math>n_{\rm g}=C_{\rm g}V_{\rm g}/(2e)</math> is a control parameter known as effective offset charge (<math>V_{\rm g}</math> is the gate voltage), and <math>E_{\rm J}</math> the Josephson energy of the tunneling junction.

At low temperature and low gate voltage, one can limit the analysis to only the lowest <math>n=0</math> and <math>n=1</math> states, and therefore obtain a two-level quantum system (a.k.a. [[qubit]]).

Note that some recent papers<ref>{{Cite journal|last1=Didier|first1=Nicolas|last2=Sete|first2=Eyob A.|last3=da Silva|first3=Marcus P.|last4=Rigetti|first4=Chad|date=2018-02-23|title=Analytical modeling of parametrically-modulated transmon qubits|journal=Physical Review A|volume=97|issue=2|pages=022330|doi=10.1103/PhysRevA.97.022330|issn=2469-9926|arxiv=1706.06566|bibcode=2018PhRvA..97b2330D|s2cid=118921729 }}</ref><ref>{{Cite journal|last1=Schreier|first1=J. A.|last2=Houck|first2=A. A.|last3=Koch|first3=Jens|last4=Schuster|first4=D. I.|last5=Johnson|first5=B. R.|last6=Chow|first6=J. M.|last7=Gambetta|first7=J. M.|last8=Majer|first8=J.|last9=Frunzio|first9=L.|last10=Devoret|first10=M. H.|last11=Girvin|first11=S. M.|date=2008-05-12|title=Suppressing Charge Noise Decoherence in Superconducting Charge Qubits|journal=Physical Review B|volume=77|issue=18|pages=180502|doi=10.1103/PhysRevB.77.180502|issn=1098-0121|arxiv=0712.3581|bibcode=2008PhRvB..77r0502S|s2cid=119181860 }}</ref> adopt a different notation, and define the charging energy as that of one electron:

:<math>E_{\rm C}=e^2/2(C_{\rm g}+C_{\rm J}),</math>

and then the corresponding Hamiltonian is:

:<math>H=\sum_n \big[4E_{\rm C}(n-n_{\rm g})^2 |n \rangle \langle n| - \frac{1}{2} E_{\rm J} (|n \rangle \langle n+1|+|n+1 \rangle \langle n|) \big].</math>