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S(ω)dω in the quasistatic limit [27].
The
simulation parameters and details are given in the Sup-
plemental Material.
This order of magnitude increase
in the dephasing time obtained by placing an additional
screening layer under thin Si QW and SiGe layers is quite
promising.
The example above is one of the TLS realizations in
which the screening is very effective (∼ top 1% of the
simulated realizations). The PSD for the spin qubit fre-
quency is typically reduced by one to two orders of magni- | [
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] |
quency is typically reduced by one to two orders of magni-
tude for d = 5 nm, and varies a lot in the given frequency
range depending on details such as TLS positions, orien-
tations, and switching rates. To capture this variability,
we simulate many possible TLS realizations to determine
how much the dephasing time can be enhanced on aver-
age. Table I shows the results for average dephasing time
(Tϕ) and the increase factor (IF), which is the ratio be- | [
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0.8758208751678467,
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0.08243348449468613,
0.38307902216911316,
0.5018988251686096,
-0.13595324754714966,
0.2800371050834656,
0.27484020590782166,
-0.04313540458679199,
-0.16525691747665405,
0.35995444655418396,
-0.5000736117362976,
-0.32479584217071533,
-0.3486635684967041,
-0.8000002503395081,
-0.390892893075943,
-0.22517529129981995,
0.3987717032432556,
-0.12205635011196136,
0.15688194334506989,
-0.16204483807086945,
0.1893254816532135,
-0.1887524127960205,
0.3025723397731781,
-0.322567343711853,
-0.07693344354629517,
0.019402990117669106,
-0.18491452932357788,
-0.12771396338939667,
0.09358920902013779,
-0.6394937038421631,
-0.11807548254728317,
0.024396220222115517,
-0.04287189990282059,
-0.026159394532442093,
0.48360303044319153,
0.2735521197319031,
-0.34885212779045105,
-0.4937603771686554,
-0.566878616809845,
0.23317615687847137,
-0.3671052157878876,
0.13191033899784088,
0.05925947427749634,
0.16063393652439117,
-0.13734658062458038,
-0.4595066010951996,
-0.4708132743835449,
0.4345702528953552,
0.3202594220638275,
-0.6222510933876038,
-0.45249947905540466,
0.25739505887031555,
0.21601013839244843,
-0.18582351505756378,
-0.019990045577287674,
-0.07843196392059326,
1.0065828561782837,
-0.05621149390935898,
-0.19877807796001434,
-0.8143886923789978,
-0.19968152046203613,
0.3634677827358246,
-0.3875848650932312,
-0.636624276638031,
-0.13499876856803894,
0.06889308243989944,
0.053120244294404984,
0.37883859872817993,
-0.30472156405448914,
-0.4093840718269348,
0.31480100750923157,
-0.40472620725631714,
0.12199272960424423,
-0.5368173718452454,
0.3150847256183624,
0.6544827222824097,
0.18554940819740295,
0.2320876121520996,
-0.4998493790626526,
0.32361364364624023,
-0.514005720615387,
-0.17026425898075104,
-0.04169730097055435,
-0.6889165043830872,
-0.0507800318300724,
0.18983030319213867,
0.22531186044216156,
-0.07700163125991821,
0.13465318083763123,
-0.22170691192150116,
-0.2547423243522644,
0.07360716164112091,
-0.18920183181762695,
0.40916523337364197,
-0.6013798117637634,
0.1653178483247757,
0.02458658069372177,
0.38565534353256226,
0.5056312084197998,
0.5877134203910828,
-0.0952707976102829,
-0.36116454005241394,
0.020820342004299164,
0.42467817664146423,
0.9499596357345581,
-0.5190753936767578,
0.3027578592300415,
-0.04013720527291298,
0.3474005162715912,
0.07598093152046204,
-0.2625761330127716,
-0.14314879477024078,
0.6078358292579651,
0.49394893646240234,
0.10488995909690857,
0.33432909846305847,
-0.10690982639789581,
-0.48532378673553467,
0.048272356390953064,
-0.06009194999933243,
0.5969626903533936,
0.5687911510467529,
-0.028651531785726547,
0.7471646666526794,
0.8885579109191895,
-0.2657514810562134,
-0.8420764803886414,
0.5887560248374939,
0.18701669573783875,
-0.09326811879873276,
-0.014770942740142345,
0.3430268466472626,
0.31215155124664307,
0.4628387689590454,
-0.4606543481349945,
0.39945411682128906,
0.09570175409317017,
-0.20734649896621704,
0.1652686446905136,
-0.005570701789110899,
-0.04109815135598183,
0.06561152637004852,
0.810158908367157,
0.6740769743919373,
-0.4817206859588623,
0.2887488007545471,
-0.047232355922460556,
0.37497836351394653,
-0.07516054064035416,
-0.3218655288219452,
0.16647648811340332,
0.18216565251350403,
0.18524865806102753,
0.05543217808008194,
-0.35166940093040466,
-0.3061669170856476,
-0.10860396176576614,
0.22784732282161713,
0.22426630556583405,
-0.4628942310810089,
-0.25190234184265137,
0.12163489311933517,
0.410930335521698,
0.1759602278470993,
-0.13398371636867523,
0.020258190110325813,
-0.5900837779045105,
-0.3987754285335541,
0.3423677384853363,
0.08547281473875046,
0.08214732259511948,
-0.19697825610637665,
-0.3123526871204376,
0.08568603545427322,
-0.19201983511447906,
-0.13534389436244965,
0.37833860516548157,
-0.8844594359397888,
0.14193513989448547,
0.13226579129695892,
-0.16777007281780243,
0.002169502666220069,
-0.17817513644695282,
0.05769124627113342,
-0.1597127765417099,
-0.3793409466743469,
-0.13206015527248383,
0.36337581276893616,
0.09587724506855011,
0.05370762571692467,
-0.4465206265449524,
0.4946058690547943,
0.5738839507102966,
-0.48194074630737305,
0.04200118035078049,
-0.06384710967540741,
-0.06305573135614395,
-0.11690595746040344,
0.08497277647256851,
0.4048660397529602,
0.5750587582588196,
-0.12991692125797272,
0.3235861361026764,
-0.2671983242034912,
0.11044866591691971,
0.1908738762140274,
-0.38218972086906433,
0.4010489284992218,
-0.3097341060638428,
-0.11969505250453949,
-0.09455186873674393,
0.10523771494626999,
0.798444926738739,
0.14731143414974213,
-0.161950945854187,
0.06653380393981934,
-0.24266637861728668,
-0.3289429843425751,
0.30912452936172485,
-0.1779647320508957,
-0.28947052359580994,
-0.4866240620613098,
-0.3002023696899414,
-0.1994602084159851,
0.6742038130760193,
-0.2970082759857178,
0.6879638433456421,
0.08517926931381226,
0.06376532465219498,
0.2632752060890198,
-0.030155103653669357,
0.11207729578018188,
-0.19204527139663696,
-0.43518438935279846,
-0.40461280941963196,
0.7937783002853394,
-0.44382229447364807,
-0.2304830551147461,
-0.04612688347697258,
-0.2057540863752365,
0.05861808732151985,
-0.2030489593744278,
-0.3082195520401001,
-0.39439061284065247,
-0.16101038455963135,
0.27739375829696655,
-0.11842452734708786,
0.45533064007759094,
0.06809590011835098,
-0.7623530030250549,
-0.6784144043922424,
0.07696306705474854,
0.18330880999565125,
-0.291186660528183,
-0.5423436164855957,
0.19328716397285461,
-0.22924590110778809,
-0.10672218352556229,
-0.040149297565221786,
0.02936648018658161,
0.2672065496444702,
-0.011606457643210888
] |
(Tϕ) and the increase factor (IF), which is the ratio be-
tween the dephasing times with and without screening,
Case
No sc.
z = −15 nm z = −10 nm z = −5 nm
(A) Tϕ 1.1 (0.7)
2.2 (2.2)
3.1 (3.2)
5.5 (6.2)
IF
1
1.8 (0.6)
2.4 (1.0)
4.3 (2.1)
(B) Tϕ 0.9 (0.7)
1.9 (2.2)
2.5 (3.1)
4.7 (6.0)
IF
1
1.8 (0.7)
2.4 (1.2)
4.4 (2.7)
(C) Tϕ 1.0 (0.7)
3.0 (5.8)
4.5 (8.6)
8.9 (16.3)
IF
1
2.2 (1.5)
3.2 (2.6)
6.1 (5.6)
TABLE I. Mean (standard deviation) of dephasing time (Tϕ) | [
0.12540289759635925,
0.2892482578754425,
0.6462138295173645,
-0.25499412417411804,
-0.08980125933885574,
0.13231238722801208,
0.34958142042160034,
-1.0168286561965942,
0.08185216039419174,
-0.28962647914886475,
-0.03312193229794502,
0.5028786659240723,
-0.06014448404312134,
-0.06286206841468811,
0.16233229637145996,
0.6832901239395142,
0.8188055753707886,
0.4176853597164154,
-0.39379817247390747,
-0.7673230767250061,
0.1767944097518921,
-0.1567791849374771,
0.4970853626728058,
-0.07043008506298065,
-0.18884341418743134,
-0.0502336211502552,
0.0879521444439888,
-0.26136401295661926,
-0.05403958633542061,
0.01264469139277935,
-0.26511192321777344,
0.8193801641464233,
0.41501179337501526,
0.10621953010559082,
0.49229538440704346,
0.34205740690231323,
0.5127575993537903,
0.08451163023710251,
0.09256255626678467,
-0.020377526059746742,
-0.6885120272636414,
-0.25539135932922363,
-0.024260180070996284,
0.31811168789863586,
-0.08693944662809372,
-0.7569589018821716,
-1.7849231958389282,
-0.15132606029510498,
-0.01859431155025959,
-0.2770417034626007,
0.4102416932582855,
-0.03086063265800476,
0.48329487442970276,
0.0174401942640543,
-0.07673748582601547,
0.2716315984725952,
-0.48200422525405884,
0.02428894117474556,
0.5412407517433167,
-0.19061054289340973,
0.4199008643627167,
-0.49358469247817993,
0.1936860978603363,
0.25186213850975037,
-0.19765900075435638,
0.5198220610618591,
-0.1360471099615097,
0.2850033640861511,
0.0004404970968607813,
-0.06438262015581131,
0.5107556581497192,
-0.07174083590507507,
-0.5143163204193115,
-0.035101473331451416,
0.09032583236694336,
0.6034720540046692,
-0.42868131399154663,
-0.2877284586429596,
0.3663535416126251,
0.12868277728557587,
-0.2691788971424103,
-0.3429499566555023,
0.1703614741563797,
-0.5028645992279053,
0.2912648916244507,
-0.09768076986074448,
0.1488642543554306,
0.6273940205574036,
-0.3172747492790222,
0.4189196228981018,
0.18473970890045166,
-0.35099703073501587,
0.36268600821495056,
0.44799134135246277,
-0.21809649467468262,
0.022917572408914566,
0.1359742432832718,
-0.036326173692941666,
0.4930768609046936,
0.1069411113858223,
0.06595193594694138,
-0.14946851134300232,
0.00734039768576622,
-0.03645006939768791,
-0.626396894454956,
-0.4925438165664673,
-0.3193981349468231,
-0.6174677610397339,
-0.08069143444299698,
-0.330646276473999,
-1.0306787490844727,
0.3916790783405304,
-0.17731499671936035,
-0.051067888736724854,
-0.45033320784568787,
-0.08595220744609833,
-0.35302817821502686,
-0.054274916648864746,
0.26476407051086426,
0.7611817717552185,
-0.02037147432565689,
0.9231692552566528,
0.2370484322309494,
0.1518479436635971,
0.4499736726284027,
0.2118186056613922,
0.4378089904785156,
0.5529441237449646,
-0.07709164917469025,
0.5311936140060425,
-0.08089213818311691,
0.4034702777862549,
0.23792998492717743,
0.07788355648517609,
-0.24888676404953003,
-0.38706254959106445,
-0.21501776576042175,
0.2304830700159073,
0.14082495868206024,
-0.45094558596611023,
-0.3318447172641754,
-0.15810786187648773,
-0.3723173141479492,
-0.2944796085357666,
0.09093872457742691,
-0.20261627435684204,
-0.3377735912799835,
0.3485845625400543,
-0.34950146079063416,
0.4606707990169525,
0.2784881889820099,
0.09395337104797363,
-0.517125129699707,
0.10800625383853912,
-0.3578418791294098,
-0.39503398537635803,
-0.2712918221950531,
-0.19599059224128723,
-0.4654845595359802,
0.28074073791503906,
0.24820531904697418,
0.40770748257637024,
-0.4155985713005066,
-0.008373484946787357,
0.9123250246047974,
-0.35640522837638855,
0.24852249026298523,
-0.02958788350224495,
-0.11635203659534454,
0.10110362619161606,
-0.2570478320121765,
0.09759918600320816,
0.2234998494386673,
-0.16480891406536102,
-0.320276141166687,
-0.0033677397295832634,
-0.22744375467300415,
0.14279523491859436,
0.08780942112207413,
0.11077695339918137,
-0.7551369667053223,
-0.030739622190594673,
0.1827743798494339,
-0.04393450915813446,
0.29460763931274414,
0.4280875623226166,
0.1946905106306076,
0.10703073441982269,
0.4465521275997162,
0.2761925458908081,
-0.4416544735431671,
-0.19566109776496887,
0.4724368453025818,
-0.7169139981269836,
0.9786777496337891,
-0.2493353933095932,
-0.060317013412714005,
0.4368433654308319,
0.027064867317676544,
-0.2920014560222626,
0.05304202064871788,
0.22416214644908905,
0.4189516603946686,
0.20984208583831787,
-0.46282297372817993,
0.4627220034599304,
-0.14155259728431702,
0.14062264561653137,
-0.7377506494522095,
0.2631853520870209,
0.2866246998310089,
-0.1718534529209137,
-0.20997865498065948,
-0.3527544140815735,
0.2235121876001358,
-0.17642270028591156,
-0.05517812445759773,
-0.010095268487930298,
0.1487148404121399,
0.02974165417253971,
0.15247394144535065,
-0.13677214086055756,
-0.13577157258987427,
0.5814400911331177,
-0.44508427381515503,
0.801755964756012,
0.7959696054458618,
-0.11710939556360245,
0.6622273325920105,
-0.4709663689136505,
0.1995551884174347,
-0.09670114517211914,
0.31792399287223816,
0.30957528948783875,
-0.053364142775535583,
0.36650171875953674,
0.38230928778648376,
0.3363932967185974,
0.7618688344955444,
-0.4523305296897888,
-0.031764838844537735,
0.009994817897677422,
0.206667959690094,
0.04449132829904556,
-0.04772820696234703,
0.4622926414012909,
-0.18205635249614716,
0.1874966323375702,
-0.38250359892845154,
-0.08189515769481659,
0.4309605062007904,
-0.07391351461410522,
0.22015248239040375,
-0.601557731628418,
-0.015873661264777184,
-0.1939317286014557,
0.016996392980217934,
-0.10018432885408401,
-0.10270913690328598,
0.013008318841457367,
-0.12385176122188568,
-0.2609177529811859,
0.5436596870422363,
-0.3779672086238861,
0.2080317586660385,
-0.18134038150310516,
-1.2241615056991577,
-0.5004357695579529,
0.6227976679801941,
-0.15634183585643768,
0.2514616847038269,
0.0564647912979126,
-0.7768084406852722,
-0.08586635440587997,
-0.6500306129455566,
-0.683239221572876,
0.6744542121887207,
0.6537654399871826,
-1.037095069885254,
-0.34751656651496887,
0.009877967648208141,
0.4688456654548645,
0.6616108417510986,
0.24317359924316406,
-0.3360995054244995,
-0.542695164680481,
0.09198588132858276,
-0.4335099756717682,
-0.4002383053302765,
-0.387281596660614,
-0.3035936951637268,
1.0155082941055298,
0.028060246258974075,
-0.48737412691116333,
0.3675869107246399,
0.22186240553855896,
-0.08405868709087372,
0.2206984907388687,
0.8340818881988525,
-0.23431900143623352,
0.123959019780159,
-0.5885853171348572,
-0.528541088104248,
0.05861237272620201,
-0.4007089138031006,
-0.6820595860481262,
-0.801300048828125,
0.08529350161552429,
-5.916778087615967,
-0.1565147042274475,
-0.3607236444950104,
0.23065900802612305,
-0.701373815536499,
-0.23508653044700623,
0.9549366235733032,
-0.24193139374256134,
-0.13677260279655457,
0.09035152941942215,
0.20153634250164032,
-0.16625988483428955,
0.30309823155403137,
-0.10648756474256516,
0.029776979237794876,
0.4741155505180359,
0.6881506443023682,
-0.12763571739196777,
0.511944591999054,
0.2870938777923584,
0.22047069668769836,
0.42982354760169983,
0.09320961683988571,
0.30159878730773926,
0.5292702913284302,
0.2743649184703827,
-0.31951114535331726,
0.6127864122390747,
-0.6171032786369324,
-0.21052834391593933,
-0.01895802654325962,
-0.1548229157924652,
-0.135296031832695,
-0.06257122755050659,
0.26083993911743164,
0.34985673427581787,
0.3432372808456421,
-0.3559459149837494,
0.18983231484889984,
-0.036760587245225906,
0.3679986596107483,
0.5482006072998047,
-0.12697502970695496,
-0.26851218938827515,
0.09639963507652283,
-0.04900376498699188,
-0.2532067596912384,
-0.14359112083911896,
-0.16518902778625488,
0.34667104482650757,
-0.16980458796024323,
0.0011580128921195865,
-0.24656206369400024,
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] |
IF
1
2.2 (1.5)
3.2 (2.6)
6.1 (5.6)
TABLE I. Mean (standard deviation) of dephasing time (Tϕ)
in units of µs and its dimensionless increase factor (IF). No
sc. and z = −d nm denote no screening layer and screening
layer at z = −d nm respectively.
for three different cases: Case (A): TLSs located at the
interface (z = 50 nm) with random orientations; Case
(B): TLSs in the oxide layer (30 < z < 50 nm) with
random orientations; and Case (C): TLSs at the inter- | [
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] |
random orientations; and Case (C): TLSs at the inter-
face with vertical orientations (along ±z directions). In
all cases, the TLSs are assumed to have random posi-
tions and switching rates, consistent with the random
formation of defects in the devices. The physical models
relevant to Case (B) and (C) correspond to the random
dipole and trap model respectively, which are discussed
in Ref. [31].
The mean values of Tϕ are not so important because it
is not valid to compare them between different screening | [
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] |
is not valid to compare them between different screening
scenarios; they can be used to validate whether the simu-
lation parameters are reasonable or not. In all cases, they
are about 1 µs without screening, which agrees with val-
ues for natural Si devices [8]. Case (B) has larger mean
values than Case (A) since TLSs can be closer to the QD.
The standard deviations in parentheses look very large,
which is not surprising because the dephasing times have | [
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] |
which is not surprising because the dephasing times have
positively skewed distributions (see Supplemental Mate-
rial). If we change the representative values from mean
to median, they decrease in all cases.
A more important quantity is the IF since it is ob-
tained for each realization and can be directly compared
with other IFs. Cases (A) and (B) have similar IF values,
while those for Case (C) have the largest IFs. This in-
dicates that the screening is most effective for vertically | [
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] |
dicates that the screening is most effective for vertically
oriented TLSs. The IFs for d = 10 nm are about 2-3,
corresponding to a slightly less than one order of mag-
nitude reduction in the PSD. The IFs for d = 5 nm are
promising, with values ranging from 4 to 6, which means
there is a more than one order of magnitude reduction in
the PSD.
Physical implementation — Although the above results
indicate that ballast charges are effective at screening | [
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] |
indicate that ballast charges are effective at screening
charge noise, this of course comes at the cost of additional
fabrication steps to implement the screening layer. For
GaAs devices, it is possible to epitaxially grow a metallic
layer such as Al [32] to serve this purpose. In-situ growth
of epitaxial metal layers may also be possible for silicon
devices [33]. Epitaxial layers are likely to be highly ef-
4
fective sources of ballast charges. However, creating such | [
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] |
4
fective sources of ballast charges. However, creating such
layers would require significant fabrication development
and patterning of such layers will likely be challenging.
However, there are a few other options to consider.
One promising option involves using double quantum
well structures.
In this structure, the lower electron
(or hole) gas serves as a screening layer for the upper
2DEG. Such structures have been fabricated and oper-
ated in GaAs/AlGaAs [34], Si/SiGe [35], and Ge/SiGe | [
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ated in GaAs/AlGaAs [34], Si/SiGe [35], and Ge/SiGe
platforms [36]. Similarly, adding a Ge spike to a Si QW
could be worth trying to form the structure in a single Si
layer [37]. It may also be possible to create a double-well
structure for SiMOS devices by using silicon-on-insulator
substrates, together with a doped silicon substrate that
could function as a back gate.
Another promising technique, which is especially rele-
vant for donor-qubit devices, is scanning tunneling micro- | [
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] |
vant for donor-qubit devices, is scanning tunneling micro-
scope lithography. By using this technique, one can cre-
ate degenerately doped regions that function like metal-
lic areas. One could then regrow Si over those areas and
then pattern the qubit and gates on top. In principle,
this technique could also be applied to other silicon-based
spin-qubit systems like Si/SiGe and SiMOS qubits.
A final approach could be to use ion implantation and
epitaxial regrowth to create doped metallic regions in- | [
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epitaxial regrowth to create doped metallic regions in-
side the semiconductor for silicon spin qubits. Challenges
with this approach involve minimizing crystal damage
due to the implantation.
Discussion — The principle of ballast water is to in-
crease the inertia of a vessel to fight against external fluc-
tuations (or similarly imagine that the vessel is changed
from a small boat to a large cruise ship).
This is at
the expense of controllability over the vessel.
Ballast | [
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] |
This is at
the expense of controllability over the vessel.
Ballast
charges act in the same way, making spin qubits more
stable against charge fluctuations while requiring more
energy (or gate voltage) to perform gate operations on
the qubits. This is because ballast charges will also work
to counteract these intentional changes to gate voltages.
Just as ballast water can be discharged or loaded de-
pending on what the circumstances require, here too it
might be beneficial to turn on and off the function of | [
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] |
might be beneficial to turn on and off the function of
ballast charges during a quantum circuit. For example,
ballast charges could be turned on (off) when the qubits
are idle (undergoing gate operations). This would call
for switchable discrete patches rather than the always-on
monolithic layer considered above. Such patches would
be particularly useful for modular quantum dot systems
[38, 39], where each module is mainly affected by nearby
local TLSs [40]. | [
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] |
[38, 39], where each module is mainly affected by nearby
local TLSs [40].
Another issue related to the controllability is weak-
ened confining potential at the QD plane. In the case
of a grounded screening layer, for example, a confining
potential, V0, becomes roughly V0d/(l + d) where l (d)
is the distance between the QD plane and the gate layer
(screening layer). With l = 50 nm and d = 5 nm, the
potential becomes V0/11, requiring 11 times larger gate | [
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] |
potential becomes V0/11, requiring 11 times larger gate
voltage for sustaining the original potential value. This
may cause dielectric breakdown of overlapping gates. A
patterned screening layer that exactly mirrors the poten-
tial by applying an offset voltage, −V0l/d, to the layer
can be used, but patterning without introducing more
defects is a big technical challenge.
One practical so-
lution for this is to adopt an unpatterned, uniform layer
with a non-overlapping-gate architecture [6, 41]. Such ar- | [
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] |
with a non-overlapping-gate architecture [6, 41]. Such ar-
chitectures are more robust against dielectric breakdown
because the inter-gate dielectric layers are much thicker
than in overlapping-gate devices.
As mentioned earlier, adding the screening layer should
not incur additional TLSs near the qubits. The effects of
such TLSs would be more critical for the dephasing of the
qubits than the remote TLSs in the oxide layer because
of their short distances from the qubits. Although the | [
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of their short distances from the qubits. Although the
horizontal component of the TLSs will be screened by
the layer, the vertical one will be antiscreened, leading
to larger fluctuations. The screening layer will thus be
effective when the dominant charge noise sources are the
remote TLSs. In addition, most studies have suggested
the charge noise originates from interfaces or materials
above the surface of the semiconductor.
The qubits in the vicinity of a metallic layer can be | [
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] |
above the surface of the semiconductor.
The qubits in the vicinity of a metallic layer can be
affected by evanescent-wave Johnson noise (EWJN) [42].
Electric fields due to EWJN would have a larger impact
on the qubits than magnetic fields, because the magni-
tude of the former is proportional to 1/d3, while the lat-
ter is proportional to w/d2 [43]. Placing the layer close
to the qubits results in an increased contribution from
EWJN; it is estimated to be T EW JN
ϕ
= 4.7 ms for d = 5 | [
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] |
EWJN; it is estimated to be T EW JN
ϕ
= 4.7 ms for d = 5
nm with reasonable thickness w = 30 nm and conductiv-
ity σ = 1.6 × 107 S/m [44]. The relaxation time due to
EWJN is about T EW JN
1
= 27 ms, which is small com-
pared to the measured T1 of conventional Si devices, and
so EWJN could become the main source of relaxation.
For T EW JN
ϕ
, the electric part is one order of magnitude
larger than the magnetic part, while only the magnetic
part contributes to T EW JN
1
. | [
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] |
larger than the magnetic part, while only the magnetic
part contributes to T EW JN
1
.
Heating effects from the metallic layer should also be
considered, because they can shift the spin qubit fre-
quency during gate operations [45, 46]. Increasing the
conductivity of the layer can reduce the heat from dissi-
pation. Since the magnetic and electric parts of EWJN
are proportional to σ and 1/σ, respectively [31], at some
point the magnetic part will exceed the electric part and | [
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] |
point the magnetic part will exceed the electric part and
become the dominant dephasing noise source.
A superconducting thin film may be able to serve as the
layer to minimize the heat from dissipation. Its screening
effect would be comparable to a normal metal [47, 48],
but this needs careful consideration because EWJN may
play a significant role in dephasing. The superconductiv-
ity in a Ge QW induced by a proximity effect could also
be employed [49].
The redistribution of charges itself can cause some | [
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] |
be employed [49].
The redistribution of charges itself can cause some
noise, but the relevant time scale is less than picosec-
5
onds. This can be ignored because this noise contributes
to the PSD at frequencies beyond the THz range, and so
the effect on Tϕ and T1 is limited.
An important question is how close to the qubits can
the ballast charges be placed. If they are too close, their
wave functions will overlap with those of the spin qubits,
and unwanted interactions may occur.
Since the bal- | [
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] |
and unwanted interactions may occur.
Since the bal-
last charges would be numerous and interact with one
another, any unwanted entanglement with the qubits
through the charge degree of freedom is expected to de-
cay fast [50].
Indirect interactions between the qubits
mediated by the ballast charges, such as Ruderman-
Kittel-Kasuya-Yosida interactions, may need to be ac-
counted for to achieve precise gate operations in this
regime [51, 52].
Summary and outlook — We have introduced the idea | [
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] |
regime [51, 52].
Summary and outlook — We have introduced the idea
of ballast charges for semiconductor spin qubits. This
amounts to placing an additional screening layer below
the QD plane where spin qubits reside. We have shown
that the PSD for the spin qubit precession frequency can
be reduced by 1-2 orders of magnitude in a Si/SiGe de-
vice. Through simulations over many TLS realizations,
we have confirmed that the reduction gives rise to a 4
to 6-fold increase in dephasing times on average for thin | [
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] |
to 6-fold increase in dephasing times on average for thin
Si QW and SiGe layers. We have discussed the physi-
cal implementation of the layer in various semiconductor
devices. We have also explored other features of ballast
charges and potential issues.
A major benefit of using ballast charges is their pas-
sive character. This approach does not require compli-
cated control protocols or computational resources for
active feedforward. Therefore, ballast charges are com- | [
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] |
active feedforward. Therefore, ballast charges are com-
patible with other techniques such as quantum error cor-
rection codes and error mitigation. It would also be in-
triguing to apply this idea to other qubit platforms as
well. While physical realizations may vary, the idea of
exploiting the innate property of an additional system
to passively counteract external fluctuations could have
wide applicability.
ACKNOWLEDGMENTS
JMN acknowledges funding from the Army Research
Office through Grant No. | [
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] |
ACKNOWLEDGMENTS
JMN acknowledges funding from the Army Research
Office through Grant No.
W911NF-23-1-0115 and the
Air Force Office of Scientific Research through Grant No.
FA9550-23-1-0710. EB acknowledges support from the
Army Research Office through Grant No. W911NF-23-
1-0115.
SUPPLEMENTAL MATERIAL
Analytic calculations using the method of image charges
Suppose that a spin qubit is located at the origin of the coordinate system, and a two-level system (TLS) is located | [
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] |
at ⃗r = (x, y, z) with z > 0 with dipole vector ⃗p = (px, py, pz). The electric potential at the qubit position due to the
TLS is
Vdip =
−⃗p · ⃗r
4πϵrϵ0|⃗r|3 ,
(1)
where ϵr is the dielectric constant, and ϵ0 is the vacuum permittivity. The screening layer is assumed to be at z = −d
(d > 0), and so the position and the dipole vector of its image TLS are ⃗rim = (x, y, −z −2d) and ⃗pim = (−px, −py, pz).
The potential at the qubit position is the sum of the potentials of the real and image TLSs: | [
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] |
The potential at the qubit position is the sum of the potentials of the real and image TLSs:
Vsc = Vdip + Vim =
1
4πϵrϵ0
�−⃗p · ⃗r
|⃗r|3
+ −⃗pim · ⃗rim
|⃗rim|3
�
(2)
=
1
4πϵrϵ0
�−pxx − pyy − pzz
(x2 + y2 + z2)3/2 + pxx + pyy + pz(z + 2d)
(x2 + y2 + (z + 2d)2)3/2
�
.
(3)
When d = 0, they perfectly cancel out so the potential Vsc is exactly zero.
The electric field at the qubit position due to the TLS is
⃗Edip = 3(⃗p · ⃗r)⃗r − |⃗r|2⃗p
4πϵrϵ0|⃗r|5
≡ (Edip,x, Edip,y, Edip,z).
(4)
6
(a)
(b)
(c)
(d)
(e)
(f)
0
1 | [
0.05638886243104935,
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4πϵrϵ0|⃗r|5
≡ (Edip,x, Edip,y, Edip,z).
(4)
6
(a)
(b)
(c)
(d)
(e)
(f)
0
1
2
3
4
0.02
0.04
0.06
0.08
0.1
Tφ (μs)
Probability
0
1
2
3
4
0.02
0.04
0.06
0.08
0.1
Tφ (μs)
0
1
2
3
4
0.04
0.08
0.12
0.16
Tφ (μs)
0
5
10
15
0.01
0.02
0.03
0.04
IF
Probability
0
5
10
15
0.01
0.02
0.03
IF
0
5
10
15
0.01
0.02
0.03
0.04
IF
FIG. 3. Probability histograms of dephasing time (Tϕ) and increase factor (IF). (a), (b), and (c) show histograms of dephasing | [
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] |
time for Case (A), (B), and (C), respectively. (d), (e), and (f) show IFs with d = 5 nm for Case (A), (B), and (C), respectively.
The full ranges of Tϕ and IF are larger than those shown in the histograms.
The electric field at the qubit position is also the sum of the fields from the real and image TLSs:
⃗Esc = ⃗Edip + ⃗Eim =
1
4πϵrϵ0
�3(⃗p · ⃗r)⃗r − |⃗r|2⃗p
|⃗r|5
+ 3(⃗pim · ⃗rim)⃗rim − |⃗rim|2⃗pim
|⃗rim|5
�
≡ (Esc,x, Esc,y, Esc,z).
(5)
When d = 0, Esc,x = Esc,y = 0 and Esc,z = 2Edip,z. | [
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] |
|⃗rim|5
�
≡ (Esc,x, Esc,y, Esc,z).
(5)
When d = 0, Esc,x = Esc,y = 0 and Esc,z = 2Edip,z.
Simulation parameters
The position of a spin qubit is (x, y, z) = (0, 0, 0) nm. TLS positions are randomly sampled from the uniform
distribution over −150 < x, y < 150 nm with z = 50 nm for Case (A) and (C), and with 30 < z < 50 nm for Case (B).
Since 10 TLSs are used for each realization, the number density corresponds to 1.11 × 1010 cm−2 for Case (A) and | [
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] |
(C) [53, 54], and 5.56 × 1015 cm−3 for Case (B), respectively. The switching rates of TLSs are log-uniformly sampled
from ν ∈ (10−5, 1) Hz to ensure their power spectral density has a 1/f-like form. The dielectric constant ϵr used in
the simulation is 11. The dipole magnitude is set to p0 = 0.5 |e|·nm ≈ 24 Debye, which is a representative value with
elementary charge e hopping between two locations separated by a distance of 0.5 nm. | [
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0.4403165876865387,
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] |
elementary charge e hopping between two locations separated by a distance of 0.5 nm.
For the spin qubit precession frequency, we need to clarify the artificial spin-orbit coupling introduced by magnetic
field gradients of the micromagnets on top of the Si/SiGe device. The gradients convert electric fields to effective
magnetic fields at the qubit position. We follow the same convention used in Ref. [55]: The change in the spin qubit
frequency (in the units of Hz) is
δfspin = λQδEx,
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frequency (in the units of Hz) is
δfspin = λQδEx,
(6)
where δEx is the change in the x component of the electric field at the qubit position and λQ = γe(∂BQ/∂x)κQ
with electron gyromagnetic ratio γe = −28 GHz/T, magnetic field gradient ∂BQ/∂x = 0.3 mT/nm, and a factor
κQ = e/(mω2
orb). The factor κQ is related to the orbital confinement energy ℏωorb, which is estimated to be 1 meV.
The effective electron mass is set to m = 0.2me with electron rest mass me. Since the change in the electric field is | [
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] |
purely due to TLSs for our purpose, δEx = �
k E(k)
dip,x (δEx = �
k E(k)
sc,x) without (with) screening with TLS index k.
7
Case
No sc.
z = −15 nm z = −10 nm z = −5 nm
(A) Tϕ 0.9 (0.8)
1.5 (1.9)
1.9 (2.7)
3.2 (5.1)
IF
1
1.7 (0.6)
2.1 (1.0)
3.7 (2.3)
(B) Tϕ 0.7 (0.8)
1.1 (1.8)
1.4 (2.5)
2.4 (4.8)
IF
1
1.6 (0.7)
2.1 (1.2)
3.5 (2.7)
(C) Tϕ 0.7 (0.7)
1.2 (2.0)
1.6 (2.9)
2.8 (5.9)
IF
1
1.7 (0.9)
2.3 (1.6)
4.0 (3.8) | [
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] |
3.5 (2.7)
(C) Tϕ 0.7 (0.7)
1.2 (2.0)
1.6 (2.9)
2.8 (5.9)
IF
1
1.7 (0.9)
2.3 (1.6)
4.0 (3.8)
TABLE II. Median (interquartile range) of dephasing time and its increase factor. The dephasing time (Tϕ) is in the units of
microseconds, and the increase factor (IF) is dimensionless. No sc. and z = −d nm denote no screening layer and screening
layer at z = −d nm, respectively.
Simulation details
For each case of (A), (B), and (C), we generated 104 realizations with the parameters above. The power spectral | [
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] |
density (PSD) for the spin qubit frequency is defined as
S(f) ≡
� ∞
−∞
⟨δfspin(t)δfspin(t + τ)⟩e−i2πfτdτ = λ2
Q
�
k
E2
k
νk
(πf)2 + ν2
k
,
(7)
where ⟨·⟩ is the statistical average of the correlation function for δfspin, Ek is either E(k)
dip,x or E(k)
sc,x depending on
the existence of the screening layer, and the frequency dependence has a Lorentzian form for random telegraph noise
[56]. It is assumed that the TLSs do not interact with each other. Given the analytical expression for S(f) above, | [
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] |
the dephasing time (Tϕ) can be computed assuming Gaussian noise [30]. The attenuation function for free induction
decay is
χ(t) = 2π t2
2
� ∞
−∞
df S(f) sinc2(πft),
(8)
where sinc(x) ≡ sin(x)/x. Tϕ can be obtained numerically by imposing χ(Tϕ) = 1.
The histograms of dephasing time and increase factor (IF) for 104 realizations are shown in Fig. 3. Figs. 3(a)–
(c) correspond to Tϕ without a screening layer for Cases (A)–(C), respectively. The IFs in Figs. 3(d)–(f) are the | [
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] |
results with a screening layer 5 nm below Si quantum well (d = 5 nm) for Cases (A)–(C), respectively. Most of the
dephasing times and IFs are less than 4 µs and 15, respectively. Both quantities have skewed distributions. This
may be attributed to the sparseness of the bath of TLSs [57]. In our simulations, the spectral density of TLSs is
NTLS/ ln(νM/νm) = 0.87, where NTLS = 10 is the number of TLSs and νM = 1 Hz (νm = 10−5 Hz) is the maximum | [
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] |
(minimum) switching rate that the TLSs can have. Since the Gaussian approximation is valid when the spectral
density is comparable to or larger than 1, Tϕ could be a little bit off from the exact value. Nevertheless, it should
suffice to assume Gaussian noise because the purpose of our work is not to obtain Tϕ accurately, but to confirm the
increase in Tϕ due to the screening effect. We may be able to increase NTLS while decreasing p0 for more accurate Tϕ | [
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] |
with the Gaussian assumption. This will not be the case when a couple of TLSs dominate the charge noise [15, 17, 40].
The median (interquartile range) values of dephasing times and IFs are shown in Table II. They are smaller than the
corresponding mean (standard deviation) due to the skewed distribution. The dephasing time can still be increased
by a factor of 3-4 for thin Si QW and SiGe layers.
The dephasing times of the example in the main text were computed using the attenuation function given above. | [
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] |
The PSDs of the example were produced by averaging over 100 time series generated by Poisson processes with
sampled switching rates.
Evanescent-wave Johnson noise from metallic screening layer
The dephasing time and relaxation time due to evanescent-wave Johnson noise (EWJN) were calculated based on
the formulas in Ref. [31]. One needs to be cautious about the presence of some typos that should be corrected: | [
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0.344686359167099,
0.1729392409324646,
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0.008026927709579468,
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0.003808670211583376,
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] |
1/2kj → 1/kj in Eq. (7), 1/√4πϵ0 → 1/4πϵ0 in Eq. (10), and the equations that are relevant to Eq. (7) and (10).
We assume that the external magnetic field is applied along the x direction, so the relevant coherence times are T (x)
ϕ
8
and T (x)
1
in Ref. [31]. They change little even if the magnetic field direction is along the y direction or other in-plane
directions.
The simulation parameters given above and in the main text were used to obtain the coherence times due to EWJN. | [
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] |
Other parameters needed for the simulation are the qubit operating frequency ωop = 2π × 16.3 × 109 s−1 and the
electron temperature T = 50 mK [55]. If other magnetic field gradients are not negligible as in Ref. [58], the relaxation
time due to EWJN can also be affected by the electric part, reducing it to a few milliseconds.
∗ [email protected]
[1] G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R.
Petta, Semiconductor spin qubits, Reviews of Modern
Physics 95, 025003 (2023). | [
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] |
Odd-frequency pairing of Bogoliubov quasiparticles in superconductor junction
Tatsuya Miki,1 Yukio Tanaka,2, 3 Shun Tamura,4 and Shintaro Hoshino1
1Department of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan
2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
3Research Center for Crystalline Materials Engineering, Nagoya University, Nagoya 464-8603, Japan
4Faculty for Physics and Astronomy (TP4), Universit¨at W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany | [
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] |
(Dated: May 24, 2024)
We study a superconductor Josephson junction with a Bogoliubov Fermi surface, employing
McMillan’s Green’s function technique. The low-energy degrees of freedom are described by spinless
fermions (bogolons), where the characteristic feature appears as an odd-frequency pair potential.
The differential equation of the Green’s function is reduced to the eigenvalue problem of the non-
Hermitian effective Hamiltonian.
The physical quantities such as the density of states and pair | [
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Hermitian effective Hamiltonian.
The physical quantities such as the density of states and pair
amplitude are then extracted from the obtained Green’s function. We find that the zero energy
local density of states at the interface decreases as the relative phase of the Josephson junction
increases. This decrease is accompanied by the generation of an even-frequency pair amplitude near
the interface. We also clarify that the π-junction-like current phase relation is realized in terms of | [
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] |
bogolons. In contrast to conventional s-wave superconductor junctions, where even-frequency pairs
dominate in the bulk and odd-frequency pairs are generated near the interface, our findings illumi-
nate the distinct behaviors of junctions with Bogoliubov Fermi surfaces. We further explore spatial
dependences of these physical quantities systematically using quasiclassical Green’s functions.
I.
INTRODUCTION
The superconductors (SCs) with gapless fermionic ex- | [
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] |
I.
INTRODUCTION
The superconductors (SCs) with gapless fermionic ex-
citations, known as Bogoliubov Fermi surfaces (BFSs)
[1–4], have been the subject of both theoretical [5–37]
and experimental [38–42] studies. BFSs have intriguing
physics due to their potential to exhibit characteristics
distinct from the Fermi surface consisting of electrons in
normal metals [16, 17, 21, 22]. Such an exotic state of
quantum matter is potentially realized in Fe(Se,S), where | [
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] |
quantum matter is potentially realized in Fe(Se,S), where
the remaining density of states (DOS) and gapless quasi-
particle behaviors below the transition temperature are
observed [38, 39, 41, 42]. Residual DOS is observed in
other superconductors [43], which are also candidate sys-
tems with BFSs.
Our previous works have demonstrated that BFSs host
purely odd-frequency Cooper pairs composed of Bogoli-
ubov quasiparticles (bogolons), which is a distinct feature | [
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] |
ubov quasiparticles (bogolons), which is a distinct feature
absent in conventional SCs [21, 37]. Specifically, we fo-
cused on the bulk state of bogolons near the BFS, which
are described by spinless fermions [17, 21, 22, 44]. We
have evaluated the self-energy of bogolons by consider-
ing the impurity and interaction effects, where a standard
perturbative technique is employed as used in conven-
tional Fermi liquid theory. Since the number of bogolons | [
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] |
tional Fermi liquid theory. Since the number of bogolons
is not a conserved quantity in contrast to that of elec-
trons in the normal metal, the anomalous part of the
self-energy, i.e., the pair potential, is finite, which makes
the bogolon system different from normal Fermi liquid.
The time dependence of this pair potential has a purely
odd functional form [45], which is interpreted as the gen-
eration of the odd-frequency Cooper pair composed of
bogolons in bulk. | [
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] |
eration of the odd-frequency Cooper pair composed of
bogolons in bulk.
Up to now, pursuing the odd-frequency pairing has
been an important issue in strongly correlated systems,
and there have been theoretical proposals in (multi-
channel) Kondo lattice models [26, 46–59], itinerant cor-
related electron models [60–68], disordered systems [69–
72], electron-phonon coupled systems [73, 74], SCs with
BFS [21, 27, 28] and other systems [75–78]. If the stan-
dard relation F †
12(iωn) = F ∗ | [
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] |
BFS [21, 27, 28] and other systems [75–78]. If the stan-
dard relation F †
12(iωn) = F ∗
21(−iωn) for the anomalous
Green’s function is taken, the generation of pure odd-
frequency pairs in bulk has a difficulty from the view-
points of stability [44, 79]. Also, the pure odd-frequency
pairing with another relation F †
12(iωn) = −F ∗
21(−iωn)
[80, 81] has a difficulty when it coexists with the odd-
frequency pairing [82, 83] generated by the external sym-
metry breaking from conventional even-frequency one | [
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] |
metry breaking from conventional even-frequency one
[84–94].
On the other hand, the odd-frequency pair-
ing of bogolons discussed in this paper is more naturally
induced by the self-energy effect for the systems with
BFS [21, 37], where the standard relation F †
12(iωn) =
F ∗
21(−iωn) holds in bulk.
Thus, the low-energy bogolon model is a suitable
platform to study the physical properties of the odd-
frequency Cooper pair [21, 44]. In contrast to the previ- | [
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] |
frequency Cooper pair [21, 44]. In contrast to the previ-
ous discussions focused on bulk properties [21, 37, 44], it
is noteworthy that the translational and inversion sym-
metries are broken at surfaces and interfaces. Namely,
odd-frequency pairs can be induced at the surface or in-
terface of conventional even-frequency SCs as a result
of lack of translational symmetry [86–88]. Hence, it is
interesting to study induced even-frequency pairs at the
interface of the odd-frequency SC. | [
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] |
interesting to study induced even-frequency pairs at the
interface of the odd-frequency SC.
In this paper, we study the junction of the SC with the
BFS based on the bogolon model. The schematic figure
is illustrated in Fig. 1 (a). As shown in the next sec-
tion, we begin with the Gor’kov equation with different
self-energies used for left- (x < 0) and right-side (x > 0)
systems. Here, we use the techniques of non-Hermitian
quantum mechanics [95, 96] and McMillan’s formalism | [
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] |
quantum mechanics [95, 96] and McMillan’s formalism
for Green’s function [97], which has been used in the
arXiv:2405.14181v1 [cond-mat.supr-con] 23 May 2024
2
(a) Bogolon junction
(b) s-wave SC junction
FIG. 1. Schematic figure of one-dimensional SC junction. We
consider two systems: (a) Bogolon junction and (b) s-wave
SC junction.
conventional SC junctions with even-frequency pair po-
tential [30, 94, 98–102]. We study a spatial dependence
of the physical quantities such as the local density of | [
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] |
of the physical quantities such as the local density of
states based on the Green functions, and also quasiclas-
sical Green’s function, which extracts a slowly varying
component. Since the relation between physical quan-
tities in terms of bogolon and experimental observables
is not trivial and depends on specific details of each su-
perconductor, we focus on properties of bogolons in this
paper as a first step to understanding the junction with
BFS. We emphasize that the bogolon junction is regarded | [
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] |
BFS. We emphasize that the bogolon junction is regarded
as a Josephson junction of bulk odd-frequency pairing
state, which has never been explored and is a foundation
to understanding superconductor junctions with BFS.
The rest part of this paper is organized as follows. In
Sec. II, we introduce Green’s function following McMil-
lan’s method.
Before showing the results for the sys-
tem with the BFS, we first summarize the result of the
s-wave SC junction for reference in Sec. III. Section IV | [
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] |
s-wave SC junction for reference in Sec. III. Section IV
provides the result of Green’s function and physical quan-
tities at the interface of the bogolon junction. In Sec. V,
we evaluate the quasiclassical Green’s function to study
the slowly-varying spatial component of both even and
odd-frequency pair amplitudes. We summarize the paper
in Sec. VI. The connection between bogolon and original
electronic degrees of freedom is explained in Appendix A.
The detailed calculation of McMillan Green’s function is | [
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] |
The detailed calculation of McMillan Green’s function is
given in Appendix B. The detailed results for the con-
ventional spin-singlet s-wave SC case are listed in Ap-
pendix C as a reference. The specific forms of physical
quantities in quasiclassical representations are given in
Appendix D.
II.
MCMILLAN GREEN’S FUNCTION
A.
Setup of bogolon junction
We consider a one-dimensional SC junction by com-
bining the systems with BFSs, where low-energy behav- | [
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] |
bining the systems with BFSs, where low-energy behav-
iors are described by spinless bogolons [17, 21, 22, 44]
(bogolon junction).
We also consider the conventional
s-wave SC junction in order to discuss the unique prop-
erties of the bogolon junction by comparing the two sys-
tems. Figure 1 shows schematics of these two systems.
The interface is located at x = 0. Although we focus
on one-dimensional systems, our formulation can be ex-
tended to a three-dimensional junction system straight- | [
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] |
tended to a three-dimensional junction system straight-
forwardly by considering quantum numbers (ky, kz) with
translational symmetry along y and z directions.
The
self-energies take different values between the left side
(x < 0) and the right side (x > 0). In the following of
this paper, we concentrate mainly on the bogolon model.
The results for the conventional s-wave SC are summa-
rized in Sec. III and Appendix C.
We consider the BFS in a time-reversal symmetry bro- | [
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] |
rized in Sec. III and Appendix C.
We consider the BFS in a time-reversal symmetry bro-
ken superconductor [3, 4, 13, 14], where elementary exci-
tations near the BFS are described by spinless fermions
[17, 21, 22, 44]. Since the pair of bogolons is realized by
taking into account the anomalous self-energy [21, 37],
the Green’s function formalism is suitable for the descrip-
tion of bogolon junctions. The Green’s function is defined
in terms of bogolons’ creation/annihilation operators as | [
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] |
in terms of bogolons’ creation/annihilation operators as
ˆG(τ, x, x′) = −⟨T ⃗α(τ, x)⃗α†(x′)⟩,
(1)
where T is a time ordering, and A(τ) is a imaginary-time
Heisenberg representation of an operator A. The vector
operator is given by ⃗α(x) = (α(x), α†(x))T where α(x)
is an annihilation operator for the low-energy bogolon.
The hat symbols (ˆ) indicate 2 × 2 matrices in Nambu
space.
We require the Green’s function to satisfy the
two (left- and right-) Gor’kov equations in the fermionic | [
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two (left- and right-) Gor’kov equations in the fermionic
Matsubara frequency (ωn) domain, which are explicitly
given by
[iωnˆ1 − ˆH0(x) − ˆΣ(iωn, x)] ˆG(iωn, x, x′) = δ(x − x′),
(2)
ˆG(iωn, x, x′)[iωnˆ1 − ˆH0(x′) − ˆΣ(iωn, x′)] = δ(x − x′),
(3)
where ˆΣ(iωn, x) is a self-energy, and ˆ1 is a two dimen-
sional identity matrix.
ˆH0(x) in Eqs. (2) and (3) is a
Hamiltonian of ideal bogolon gas with the barrier poten-
tial:
ˆH0(x) =
�
− ℏ2
2m
d2
dx2 − µ + VBδ(x)
�
ˆτ z,
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] |
tial:
ˆH0(x) =
�
− ℏ2
2m
d2
dx2 − µ + VBδ(x)
�
ˆτ z,
(4)
where m and µ are effective mass and chemical potential
of bogolons, respectively. In the above equation, we in-
troduce the barrier potential VB at the interface x = 0.
Since we consider the same ˆH0(x) between the right side
3
(x > 0) and the left side (x < 0), we assume that both
sides are the same materials.
The model of bogolon for bulk was introduced in our
previous works [21, 37]. The low-energy bogolons form | [
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] |
previous works [21, 37]. The low-energy bogolons form
pure odd-frequency pairs in bulk. Here, we phenomeno-
logically introduce the spatial dependence of the self-
energy ˆΣ(iωn, x) given by
ˆΣ(iωn, x) = −i
�
Γ1(x)
Γ2(x)
Γ2(x)∗ Γ1(x)
�
sgn ωn.
(5)
The presence of the sign function represents the odd-
frequency pair potential in the off-diagonal part.
The
spatial dependence of Γ1(x) and Γ2(x) are given by
Γ1(x) =
�
Γ1L
(x < 0),
Γ1R
(x > 0),
(6)
Γ2(x) =
�
Γ2L = |Γ2L|eiθL
(x < 0),
Γ2R = |Γ2R|eiθR
(x > 0),
(7) | [
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] |
�
Γ1L
(x < 0),
Γ1R
(x > 0),
(6)
Γ2(x) =
�
Γ2L = |Γ2L|eiθL
(x < 0),
Γ2R = |Γ2R|eiθR
(x > 0),
(7)
where Γ1r (r = R, L) is a positive constant and corre-
sponds to the quasiparticle dumping of bogolon.
On
the other hand, Γ2r is a complex number.
We note
that Γ1r and Γ2r must satisfy the relation Γ1r > |Γ2r|
(0 ≤ |Γ2r|/Γ1r < 1) to guarantee the positive DOS in
bulk [21, 22, 44]. The schematic figure of our setup is
illustrated in Fig. 1 (a).
B.
Origins of phase of pair potential | [
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