{"id":560257,"date":"2022-01-14T02:06:22","date_gmt":"2022-01-14T01:06:22","guid":{"rendered":"http:\/\/antonpotocnik.com\/?p=560257"},"modified":"2022-01-14T02:06:25","modified_gmt":"2022-01-14T01:06:25","slug":"transmon-qubit-calculator","status":"publish","type":"post","link":"https:\/\/antonpotocnik.com\/?p=560257","title":{"rendered":"Transmon Qubit Calculator"},"content":{"rendered":"

With this calculator you can convert between charging energy (E<\/em>c<\/sub>) and total capacitance (C<\/em>\u03a3<\/sub>), between Josephson energy (E<\/em>J<\/sub>), Josephson inductance (L<\/em>J<\/sub>), junction critical current (I<\/em>c<\/sub>) or room temperature junction resistance (R<\/em>), and from charging and Josephson energies calculate relevant qubit transition frequencies. Other calculations are added below including Qubit’s Purcell decay, chi-shift, conversion between spectral linewidth and coherence times, etc.<\/p>\n

All frequencies are in natural units.<\/p>\n

\n\n\n\n\n
\n

Capacitance<\/h4>\n<\/td>\n

\n

Al Josephson Junction\/SQUID<\/h4>\n<\/td>\n<\/tr>\n

\n\n\n\n\n
E<\/em>c<\/sub>\/2\u03c0 =<\/td>\n GHz<\/td>\n<\/tr>\n
C<\/em>\u03a3<\/sub> =<\/td>\n fF<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n\n\n\n\n\n\n\n\n\n
E<\/em>J<\/sub>\/2\u03c0 =<\/td>\n GHz<\/td>\n<\/tr>\n
L<\/em>J<\/sub> =<\/td>\n nH<\/td>\n<\/tr>\n
I<\/em>c<\/sub> =<\/td>\n nA<\/td>\n<\/tr>\n
R<\/em> =<\/td>\n \u03a9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n
\n

Qubit frequencies<\/h4>\n<\/td>\n

 <\/td>\n<\/tr>\n
\n\n\n\n\n\n\n\n\n
f<\/em>ge<\/sub> \u2248<\/td>\n GHz<\/td>\n<\/tr>\n
f<\/em>ef<\/sub> \u2248<\/td>\n GHz<\/td>\n<\/tr>\n
f<\/em>gf<\/sub>\/2 \u2248<\/td>\n GHz<\/td>\n<\/tr>\n
\u03b1<\/em> \u2248<\/td>\n GHz<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n\"\" Schematic of a transmon qubit.<\/td>\n<\/tr>\n
\n

Purcell decay and \u03c7-shift<\/h4>\n<\/td>\n<\/tr>\n

\n\n\n\n\n\n\n
f<\/em>res<\/sub> =<\/td>\n GHz<\/td>\n<\/tr>\n
Q<\/em>res<\/sub> =<\/td>\n k<\/td>\n<\/tr>\n
\u03ba<\/em>res<\/sub> = <\/td>\n MHz<\/td>\n<\/tr>\n
g<\/i>\/2\u03c0 = <\/td>\n MHz<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n\n\n\n\n\n\n\n\n\n\n\n
\u0394<\/em> = f<\/em>ge<\/sub> – f<\/em>res<\/sub> = <\/td>\n GHz<\/td>\n<\/tr>\n
T<\/em>Purcell<\/sub> \u2248 <\/td>\n \u03bcs<\/td>\n<\/tr>\n
\u03ba<\/em>Purcell<\/sub> \u2248 <\/td>\n kHz<\/td>\n<\/tr>\n
\u03c7<\/em>-shift \u2248 <\/td>\n MHz<\/td>\n<\/tr>\n
n<\/em>crit<\/sub> \u2248 <\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n
  <\/td>\n<\/tr>\n
\n

Coherence times \/ spectral linewidth<\/h4>\n<\/td>\n<\/tr>\n

\n\n\n\n\n\n
T<\/i>1<\/sub> =<\/td>\n \u03bcs<\/td>\n<\/tr>\n
T<\/i>2<\/sub> =<\/td>\n \u03bcs<\/td>\n<\/tr>\n
T<\/i>\u03c6<\/sub> =<\/td>\n \u03bcs<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n\n\n\n\n\n\n
f<\/em> =<\/td>\n GHz<\/td>\n<\/tr>\n
df<\/em>fwhm<\/sub> =<\/td>\n kHz<\/td>\n<\/tr>\n
Q<\/em>-factor =<\/td>\n M<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/form>\n\n\n
<\/div>\n\n\n\n

Parameter Description<\/h4>\n\n\n\n
E<\/em>c<\/sub><\/td> <\/td>Charging energy<\/td><\/tr>
C<\/em>\u03a3<\/sub><\/td> <\/td>Total capacitance<\/td><\/tr>
E<\/em>J<\/sub><\/td> <\/td>Junction\/SQUID Josephson energy<\/td><\/tr>
L<\/em>J<\/sub><\/td> <\/td>Junction\/SQUID Josephson inductance<\/td><\/tr>
I<\/em>c<\/sub><\/td> <\/td>Junction\/SQUID critical current<\/td><\/tr>
R<\/em><\/td> <\/td>Junction\/SQUID room temperature resistance for 30\/40 nm thin Al films<\/td><\/tr>
f<\/em>ge<\/sub><\/td> <\/td>Maximal qubit ground (g) to first excited state (e) transition frequency<\/td><\/tr>
f<\/em>ef<\/sub><\/td> <\/td>Maximal qubit first (e) to second excited state (f) transition frequency<\/td><\/tr>
f<\/em>gf<\/sub><\/td> <\/td>Maximal qubit ground (g) to second excited state (f) transition frequency<\/td><\/tr>
\u03b1<\/em><\/td> <\/td>Qubit anharmonicity \u2248 Ec<\/sub><\/td><\/tr>
f<\/em>res<\/sub><\/td> <\/td>Readout resonator frequency<\/td><\/tr>
Q<\/em>res<\/sub><\/td> <\/td>Readout resonator Q-factor<\/td><\/tr>
\u03ba<\/em>res<\/sub><\/td> <\/td>Readout resonator: spectral linewidth<\/td><\/tr>
g<\/i><\/td> <\/td>Resonator-qubit coupling constant<\/td><\/tr>
\u0394<\/em><\/td> <\/td>Qubit-resonator transition frequency difference<\/td><\/tr>
T<\/em>Purcell<\/sub><\/td> <\/td>Qubit’s Purcell decay time<\/td><\/tr>
\u03ba<\/em>Purcell<\/sub><\/td> <\/td>Qubit’s Purcell spectral linewidth<\/td><\/tr>
\u03c7<\/em>-shift<\/td> <\/td>Half of the f<\/em>res<\/sub> difference between qubit being in the g<\/em> or e<\/em> state.<\/td><\/tr>
T<\/em>1<\/sub><\/td> <\/td>Energy relaxation time<\/td><\/tr>
T<\/em>2<\/sub><\/td> <\/td>Decoherence time<\/td><\/tr>
T<\/em>\u03c6<\/sub><\/td> <\/td>Dephasing time<\/td><\/tr>
f<\/em><\/td> <\/td>Corresponding frequency<\/td><\/tr>
df<\/em>fwhm<\/sub><\/td> <\/td>Full width at half maximum – spectral linewidth<\/td><\/tr>
n<\/em>crit<\/sub><\/td> <\/td>Critical photon number<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
<\/div>\n\n\n\n

Formulas<\/h4>\n\n\n\n
\"E_c<\/td>\"\Phi_0<\/td><\/tr>
\"E_J<\/td>\"\Delta_0<\/td><\/tr>
\"E_J<\/td>\"e_0<\/td><\/tr>
\"I_c<\/td>\"h<\/td><\/tr>
\"L_J<\/td>\n\"df_\mathrm{fwhm}<\/td><\/tr>
\"hf_{ge}<\/td>\"Q\n<\/td><\/tr>
\"hf_{ef}<\/td>\"1\/T_{2}<\/td><\/tr>
\"\kappa_\mathrm{Purcell}<\/td>\"T_\mathrm{Purcell}<\/td><\/tr>
\"\chi\mathrm{-shift}<\/td>\"n_\mathrm{crit}<\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"

With this calculator you can convert between charging energy (Ec) and total capacitance (C\u03a3), between Josephson energy (EJ), Josephson inductance (LJ), junction critical current (Ic) or room temperature junction resistance (R), and from charging and Josephson energies calculate relevant qubit transition frequencies. Other calculations are added below including Qubit’s Purcell decay, chi-shift, conversion between spectral…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[18],"tags":[19],"class_list":["post-560257","post","type-post","status-publish","format-standard","hentry","category-software","tag-tools"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p3Hjcy-2lKp","_links":{"self":[{"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/posts\/560257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=560257"}],"version-history":[{"count":79,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/posts\/560257\/revisions"}],"predecessor-version":[{"id":627773,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=\/wp\/v2\/posts\/560257\/revisions\/627773"}],"wp:attachment":[{"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=560257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=560257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/antonpotocnik.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=560257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}