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spin-singlet s-wave superconductor and examined the
unique characteristics of bogolon Cooper pairs near the
interface. A central difference between the two cases is
the frequency dependences of the Cooper pairs. The bo-
golons form the odd-frequency pair in bulk and induce
the even-frequency pair at the interface. On the other
hand, in the s-wave superconductor, the even-frequency
pair is realized in bulk and the odd-frequency pair is in-
duced near the interface.
This difference leads to the | [
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] |
duced near the interface.
This difference leads to the
different θ-dependences of Green functions. As recapitu-
lated in Table I, the bogolon Cooper pair shows distinc-
tive features in physical quantities such as the LDOS and
the current-phase relation.
We have studied properties of bogolons (such as bo-
golons’ LDOS, Cooper pairs, and quasiparticle current)
in this paper as a first step to understanding the junc-
tions with BFS. To gain more quantitative insights rel- | [
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] |
tions with BFS. To gain more quantitative insights rel-
evant to existing superconductors, the relation between
physical quantities in terms of bogolons and experimen-
tal observables needs to be clarified, which remains to be
clarified. A combination of first principles calculations
[37] with junctions presents an intriguing avenue for fu-
ture research [115, 116].
ACKNOWLEDGMENTS
TM is grateful to R. Iwazaki for fruitful discussions.
This work was supported by JSPS with Grants-in-Aid for | [
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] |
This work was supported by JSPS with Grants-in-Aid for
Scientific research No. 23KJ0298 (TM), No. 23K17668
(YT and SH), No. 24K00583 (YT), No. 21K03459
(SH), and No. 23H01130 (SH). ST was supported by
the W¨urzburg-Dresden Cluster of Excellence ct.qmat,
EXC2147, project-id 390858490, the DFG (SFB 1170),
and the Bavarian Ministry of Economic Affairs, Regional
Development and Energy within the High-Tech Agenda
Project “Bausteine f¨ur das Quanten Computing auf Basis
topologischer Materialen.”
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] |
Project “Bausteine f¨ur das Quanten Computing auf Basis
topologischer Materialen.”
15
Appendix A: Correspondence between bogolons and
original electron degrees of freedom
To see the origin of the pair potential of bogolons, we
start with the Hamiltonian of original electrons. We here
consider the impurity effect, which is significant in the
low energy region in the presence of the BFS [21]. For
concreteness, we employ the j = 3/2 model, which has
been discussed in previous studies on the superconduc- | [
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] |
been discussed in previous studies on the superconduc-
tors with BFS [3, 4, 16, 17, 29]. The impurity potential
part of the Hamiltonian is defined in terms of the original
electrons:
Himp =
�
i
�
dr
�
η
Uη(r − Ri)⃗c†(r) ˆOη⃗c(r)
(A1)
with ⃗c(r) = (c3/2(r), c3/2(r), c1/2(r), c−1/2(r), c−3/2(r))T.
In the above expression, we consider the isotropic (η = 1)
and anisotropic (η = xy, yz, zx, z2, x2 − y2) scattering
centers located at Ri.
The operators of electrons and those of bogolons are | [
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] |
centers located at Ri.
The operators of electrons and those of bogolons are
connected by the Bogoliubov transformation, which is
given by
ckm = u∗
kmαk + v−k,mα†
−k,
(A2)
where ckm is a Fourier component of cm(r) in Eq. (A1).
Using this transformation, we rewrite Eq. (A1) as
Himp = 1
V
�
k,q
ρqU1(k, q)α†
k+qαk
+ 1
V
�
k,q
ρqU2(k, q)α†
k+qα†
−k + H.c.
+ Const.
(A3)
with
U1(k, q) =
�
η
�
m,m′
Uη(q)[uk+q,mOη
mm′u∗
k,m′
− v∗
k,mOη
mm′vk+q,m′],
(A4)
U2(k, q) =
�
η
�
m,m′
Uη(q)uk+q,mOη
mm′v−k,m′.
(A5) | [
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0.46200963854789734,
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0.18482962250709534,
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] |
mm′u∗
k,m′
− v∗
k,mOη
mm′vk+q,m′],
(A4)
U2(k, q) =
�
η
�
m,m′
Uη(q)uk+q,mOη
mm′v−k,m′.
(A5)
The full list of 4 × 4 matrices is defined by using the ˆ
J
matrix in Ref. [17]. The particle number of bogolons is
not conserved because of the presence of U2(k, q). We
here provide the expression of the pair potential with the
Born approximation from Ref [21]:
Γ2k = 4πD0nimp⟨U1(k, q)U2(k, q)⟩q,
(A6)
where ⟨· · ·⟩q =
�
dq · · · /
�
dq1, nimp = V −1 �
i 1, and
D0 is a DOS at Fermi energy. Therefore, we need to take | [
0.1087101548910141,
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0.8324908018112183,
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0.43519845604896545,
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0.7003680467605591,
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0.85616534948349,
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0.4516199827194214,
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] |
�
dq · · · /
�
dq1, nimp = V −1 �
i 1, and
D0 is a DOS at Fermi energy. Therefore, we need to take
into account U2(k, q) for the presence of pair potential
Γ2k. Since the phase of the superconductor is given by
arg uv, the phase of Γ2 for bogolons is identical to the
phase of the superconductor as discussed in the main
text.
We note that a similar discussion is possible for the
self-consistent Born approximation [114] by considering
that the Green’s function is determined self consistently | [
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0.09114392101764679,
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0.17631514370441437,
0.03755722939968109,
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0.5859683156013489,
0.20717066526412964
] |
that the Green’s function is determined self consistently
including the anomalous Green’s function [21], and the
above conclusion does not change.
Appendix B: McMillan formalism
In this Appendix, we briefly follow the formalism by
McMillan [30, 94, 97–102]. Firstly, we impose the bound-
ary condition for the Green’s function. For x < x′, the
boundary condition at x → ±∞ is given by
ˆG(iωn, x, x′ → ∞) = ˆG(iωn, x → −∞, x′) = 0,
(B1)
while for x > x′, the condition is given by | [
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0.2885635197162628,
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0.08270835131406784,
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0.3001215159893036,
0.0009568912209942937,
-0.43226557970046997,
-0.4508155882358551,
0.2952992022037506,
-0.2096705287694931,
0.5073884129524231,
0.047368839383125305,
0.495064377784729,
0.1515725553035736,
-0.4135691821575165,
-0.8665004372596741,
0.6710987687110901,
0.1870821714401245,
-0.05804848670959473,
0.09835325181484222,
0.07170538604259491,
-0.17563572525978088,
-0.21873772144317627,
-0.1558404564857483,
-0.18843959271907806,
-0.10335765779018402,
0.040538106113672256
] |
ˆG(iωn, x, x′ → ∞) = ˆG(iωn, x → −∞, x′) = 0,
(B1)
while for x > x′, the condition is given by
ˆG(iωn, x → ∞, x′) = ˆG(iωn, x, x′ → −∞) = 0.
(B2)
We chose the wave functions Eq. (10) and Fig. 2 to satisfy
Eqs. (B1) and (B2) [30, 94, 97–102].
Although the coefficients α1, · · · , α4, β1, · · · , β4 are
uniquely
determined
by
the
boundary
conditions
Eqs. (16) and (17), these are rewritten in a simpler form
by using Eqs. (B20) and (B21). Consequently, we obtain
the following forms [30, 94, 98–102]: | [
-0.30451858043670654,
-0.22528517246246338,
0.6128208041191101,
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0.5886232256889343,
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0.1240905374288559,
0.2920226454734802,
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0.037000611424446106,
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0.6834441423416138,
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0.15375639498233795,
0.22966887056827545,
0.2773095369338989,
0.5533131957054138,
0.21819999814033508,
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0.049918320029973984,
0.23735088109970093,
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0.30494409799575806,
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0.23910437524318695,
0.34953296184539795,
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0.13976819813251495,
0.011285720393061638,
0.7219537496566772,
0.03786547854542732,
0.06749727576971054,
0.2341422289609909,
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0.38388684391975403,
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0.19116349518299103
] |
by using Eqs. (B20) and (B21). Consequently, we obtain
the following forms [30, 94, 98–102]:
α1 = m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
c(−)
in
c(+)
in c(−)
in
− d(+)
in d(−)
in
,
(B3)
α2 = −m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
d(−)
in
c(+)
in c(−)
in
− d(+)
in d(−)
in
,
(B4)
α3 = −m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
d(+)
in
c(+)
in c(−)
in
− d(+)
in d(−)
in
,
(B5)
α4 = m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
c(+)
in
c(+)
in c(−)
in
− d(+)
in d−
in
,
(B6)
and
β1 = m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
˜c(−)
in
˜c(+)
in ˜c(−)
in
− ˜d(+)
in ˜d(−)
in
,
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] |
in
,
(B6)
and
β1 = m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
˜c(−)
in
˜c(+)
in ˜c(−)
in
− ˜d(+)
in ˜d(−)
in
,
(B7)
β2 = −m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
˜d(−)
in
˜c(+)
in ˜c(−)
in
− ˜d(+)
in ˜d(−)
in
,
(B8)
β3 = −m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
˜d(+)
in
˜c(+)
in ˜c(−)
in
− ˜d(+)
in ˜d(−)
in
,
(B9)
β4 = m(iωn + iΓ1L)
ikFℏ2ΩL(iωn)
˜c(+)
in
˜c(+)
in ˜c(−)
in
− ˜d(+)
in ˜d(−)
in
,
(B10)
where we have assumed k±
L ≃ k±
R ≃ kF. Solving Eq. (13)
and Eq. (14), we obtain the coefficients a(±)
out, a(±)
in , b(±)
out,
16
b(±)
in , c(±) | [
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0.4379218518733978
] |
and Eq. (14), we obtain the coefficients a(±)
out, a(±)
in , b(±)
out,
16
b(±)
in , c(±)
out, c(±)
in , d(±)
out and d(±)
in , which are given by
a(±)
out =
Ξ±±
LR Ξ±∓
LR
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B11)
b(±)
out =
−Z(Z ± i)Ξ∓±
LL Ξ∓±
RR
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B12)
c(±)
out =
(±iZ − 1)Ξ±∓
LR Ξ∓±
LL
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B13)
d(±)
out =
±iZΞ±±
LR Ξ∓±
LL
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B14)
and
a(±)
in
=
Ξ±±
RL Ξ±∓
RL
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B15)
b(±)
in
= | [
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] |
LR Ξ±∓
LR
,
(B14)
and
a(±)
in
=
Ξ±±
RL Ξ±∓
RL
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B15)
b(±)
in
=
−Z(Z ± i)Ξ∓±
RRΞ∓±
LL
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B16)
c(±)
in
=
(±iZ − 1)Ξ±∓
RL Ξ∓±
RR
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
,
(B17)
d(±)
in
=
±iZΞ±±
RL Ξ∓±
RR
Z2Ξ∓±
LL Ξ∓±
RR − Ξ∓±
LR Ξ±∓
LR
(B18)
with
Ξss′
rr′ = us
rvs′
r′ − us′
r′vs
r
(B19)
and s, s′ = ±, r, r′ = L, R.
We note that the coeffi-
cients for wave functions with tilde ˜a(±)
out, ˜a(±)
in , ˜b(±)
out, ˜b(±)
in ,
˜c(±)
out, ˜c(±)
in , ˜d(±) | [
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0.48465800285339355,
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0.7194696068763733,
0.8614563345909119,
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0.03528495132923126,
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0.07147977501153946,
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0.09345896542072296,
0.4419029653072357,
1.0054413080215454,
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0.15904639661312103,
0.11585213989019394,
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0.15540055930614471,
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] |
out, ˜a(±)
in , ˜b(±)
out, ˜b(±)
in ,
˜c(±)
out, ˜c(±)
in , ˜d(±)
out, and ˜d(±)
in
can be obtained by substitut-
ing θ → −θ in a(±)
out, a(±)
in , b(±)
out, b(±)
in , c(±)
out, c(±)
in , d(±)
out, and
d(±)
in , respectively. For Γ1L = Γ1R and |Γ2L| = |Γ2R|, the
coefficients ˜a(±)
out and ˜b(±)
out reduce to Eq. (25).
We here comment on the eigenequation of Eq. (12).
Using Eq. (19)–(21), we obtain the following relations:
˜u±
r u±
r = 1
2
�
1 ±
Ωr(iωn)
iωn + iΓ1rsgn ωn
�
,
(B20)
˜v±
r v±
r = 1
2
�
1 ∓ | [
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0.878198504447937,
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0.40738821029663086,
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0.5068300366401672,
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0.23030784726142883,
0.11381154507398605,
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0.3596678674221039,
0.4463667571544647,
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] |
˜u±
r u±
r = 1
2
�
1 ±
Ωr(iωn)
iωn + iΓ1rsgn ωn
�
,
(B20)
˜v±
r v±
r = 1
2
�
1 ∓
Ωr(iωn)
iωn + iΓ1rsgn ωn
�
.
(B21)
We choose the phase of the eigenvector such that u±
r is
given by
u±
r =
�
�
�
�1
2
�
1 ±
Ωr(iωn)
iωn + iΓ1r
�
(B22)
for the practical calculation. Then, v±
r , ˜u±
r , and ˜v±
r are
uniquely determined under this choice.
Appendix C: Specific expression for s-wave superconductor junction | [
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] |
Appendix C: Specific expression for s-wave superconductor junction
In this Appendix, we summarize the results for electrons in s-wave SC as a reference for comparison with the results
for the bogolon junction. The basis is given by ⃗c(x) = (c↑(x), c†
↓(x))T where cσ(x) is an annihilation operator for the
spin σ =↑, ↓ electron. The self-energy is defined by
ˆΣ(x) =
�
0
∆(x)
∆(x)∗
0
�
(C1)
with
∆(x) =
�
∆L = |∆L|eiθL
(x < 0),
∆R = |∆R|eiθR
(x > 0).
(C2) | [
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] |
ˆΣ(x) =
�
0
∆(x)
∆(x)∗
0
�
(C1)
with
∆(x) =
�
∆L = |∆L|eiθL
(x < 0),
∆R = |∆R|eiθR
(x > 0).
(C2)
Following the McMillan’s method discussed in Sec. II and Appendix B, we obtain the Green’s function as follows
17
[30, 94, 97–102, 106]:
ˆG(iωn, x, x′) =
m
2ikFℏ2ΩL(iωn)
��
eik+
L |x−x′| + ˜b(+)
oute−ik+
L (x+x′)� �
iωn + ΩL(iωn)
∆L
∆∗
L
iωn − ΩL(iωn)
�
+ ˜a(+)
oute−ik+
L x+ik−
L x′
�
|∆L|
eiθL[iωn + ΩL(iωn)]
e−iθL[iωn − ΩL(iωn)]
|∆L|
�
+
�
e−ik−
L |x−x′| + ˜b(−)
outeik−
L (x+x′)� �
iωn − ΩL(iωn)
∆L
∆∗
L | [
-0.6185455918312073,
-0.2617141306400299,
0.30836221575737,
-0.5785990357398987,
0.014943507499992847,
0.24568310379981995,
0.07133230566978455,
-0.9199951887130737,
-0.17099690437316895,
-0.520983874797821,
0.017218686640262604,
0.35183292627334595,
-0.5782244801521301,
0.19397467374801636,
0.5641860365867615,
1.0080525875091553,
0.852153480052948,
0.3392390310764313,
-0.8035228252410889,
-0.5635613799095154,
0.33101925253868103,
0.2146853506565094,
0.4364382326602936,
0.4197937548160553,
0.21735960245132446,
0.1997532695531845,
0.521876871585846,
-0.10947858542203903,
0.47551029920578003,
0.1576043963432312,
0.08742925524711609,
0.1941252052783966,
0.3643888831138611,
0.20082662999629974,
0.6907972097396851,
-0.2135925441980362,
0.3154405355453491,
0.11980623751878738,
0.3119050860404968,
0.08826727420091629,
-0.2413541078567505,
0.008618483319878578,
0.7385978102684021,
0.8116279244422913,
-0.3565142750740051,
-0.13814060389995575,
-2.087101459503174,
0.27024370431900024,
0.29042693972587585,
0.10915056616067886,
0.12725837528705597,
0.17414985597133636,
0.4177584946155548,
0.3034415543079376,
-0.557611882686615,
0.4354296922683716,
-0.18975718319416046,
0.20295651257038116,
0.4696411192417145,
-0.5848445296287537,
0.6030053496360779,
-0.010667200200259686,
-0.07473757117986679,
0.8267039060592651,
-0.3965916633605957,
1.4263055324554443,
-0.4524940550327301,
0.1839452087879181,
-0.7023411393165588,
0.19073285162448883,
0.612743079662323,
0.027854960411787033,
-0.38480818271636963,
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] |
e−iθL[iωn − ΩL(iωn)]
|∆L|
�
+
�
e−ik−
L |x−x′| + ˜b(−)
outeik−
L (x+x′)� �
iωn − ΩL(iωn)
∆L
∆∗
L
iωn + ΩL(iωn)
�
+ ˜a(−)
outeik−
L x−ik+
L x′
�
|∆L|
eiθL[iωn − ΩL(iωn)]
e−iθL[iωn + ΩL(iωn)]
|∆L|
� �
(C3)
with
˜a(±)
out(iωn) = −|∆L|[iωn sin2(θ/2) ± iΩL(iωn) sin(θ/2) cos(θ/2)]
Z2ΩL(iωn)2 + (iωn)2 − |∆L|2 cos2(θ/2)
,
(C4)
˜b(±)
out(iωn) = −
Z(Z ± isgn ωn)ΩL(iωn)2
Z2ΩL(iωn)2 + (iωn)2 − |∆L|2 cos2(θ/2).
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] |
,
(C4)
˜b(±)
out(iωn) = −
Z(Z ± isgn ωn)ΩL(iωn)2
Z2ΩL(iωn)2 + (iωn)2 − |∆L|2 cos2(θ/2).
(C5)
We turn to the discussion of the quasiclassical representation, which corresponds to Sec. V for the bogolon junction.
The quasiclassical Green’s function is given by [113]
ˆg±±(iωn, x) =
i
ΩL(iωn)
�
i∆Lˆτ y + iωnˆτ z + ˜a(∓)
oute−ikF(ΩL(iωn)/µ)x�
∓ ΩL(iωn)ˆτ x + iωniˆτ y + |∆L|ˆτ z��
.
(C6)
where we set θL = 0 as done for the bogolon junction in Sec. V. The spatial dependence is determined by the factor
exp(2
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] |
exp(2
�
ω2n + |∆L|2x/ℏvF).
(C7)
In the retarded Green’s function, the factor is given by
exp(−2i
�
ω2 − |∆L|2sgn ωx/ℏvF).
(C8)
Using Eq. (C6), we obtain the pair amplitudes fs(iωn, x) and fp(iωn, x) defined by Eqs. (D1) and (D2), which are
given by
fs(iωn, x) =
2i∆L
ΩL(iωn) +
�
i
�
˜a(+)
out − ˜a(−)
out
�
+ i
�
˜a(+)
out + ˜a(−)
out
�
iωn
ΩL(iωn)
�
e−ikF(ΩL(iωn)/µ)x,
(C9)
fp(iωn, x) =
�
i
�
˜a(+)
out + ˜a(−)
out
�
− i
�
˜a(+)
out − ˜a(−)
out
�
iωn
ΩL(iωn)
�
e−ikF(ΩL(iωn)/µ)x.
(C10) | [
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] |
�
i
�
˜a(+)
out + ˜a(−)
out
�
− i
�
˜a(+)
out − ˜a(−)
out
�
iωn
ΩL(iωn)
�
e−ikF(ΩL(iωn)/µ)x.
(C10)
We also calculate the LDOS and j(iωn, x) in quasiclassical representation. The LDOS is expressed as
D(ω, x) = − 1
π Re
�2(ω + i0+)
Ωret(ω)
+ ˜a(+)
out,ret + ˜a(−)
out,ret
Ωret(ω)
|∆L|e−ikF(Ωret(ω)/µ)x
�
,
(C11)
where a(±)
out,ret is a retarded version of a(±)
out. j(iωn, x) defined by Eq. (D4) is given by
j(iωn, x) =
|∆L|
ΩL(iωn)
�
˜a(+)
out − ˜a(−)
out
�
e−ikF(ΩL(iωn)/µ)x.
(C12) | [
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] |
j(iωn, x) =
|∆L|
ΩL(iωn)
�
˜a(+)
out − ˜a(−)
out
�
e−ikF(ΩL(iωn)/µ)x.
(C12)
Appendix D: Quasiclassical Green’s function for bogolon junction
In this Appendix, we list the specific form of the physical quantities in the quasiclassical representations. The
physical quantities discussed in Sec. IV are expressed as the combination of g++ and g−− [113]. The s-wave and
p-wave components of the pair amplitudes are given by
fs(iωn, x) = [ˆg++(iωn, x) + ˆg−−(iωn, x)]12,
(D1) | [
-0.01493783574551344,
-0.10757482796907425,
0.6415113806724548,
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0.30932360887527466,
0.3989577889442444,
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-0.955180823802948,
0.13618426024913788,
0.3632860481739044,
-0.23141348361968994,
0.05821124091744423,
0.38375523686408997,
0.2786977291107178,
0.700253963470459,
-0.17281346023082733,
-0.12724977731704712,
-0.5202164053916931,
-0.08122213929891586,
0.381064772605896,
-0.0012177653843536973,
-0.19944898784160614,
-0.2101726233959198,
-0.14897292852401733,
0.11192314326763153,
0.12991130352020264,
0.7183969616889954,
-0.3640790581703186,
-0.5650073885917664,
0.1792999804019928,
0.37208253145217896,
-0.06651438772678375,
0.09607113897800446,
-0.20332786440849304,
0.8870976567268372,
-0.5012196898460388,
-0.6055387854576111,
0.19827555119991302,
-0.3613402545452118,
0.15126657485961914,
0.06105703487992287,
-0.28854119777679443,
0.2826530933380127,
0.13282892107963562,
-0.5468029379844666,
-0.519282877445221,
-0.1211632713675499,
0.2183414101600647,
-0.04878292232751846,
-0.4635263681411743,
1.0716698169708252,
0.9463918209075928,
0.13998720049858093,
-0.1000908836722374,
-0.08197399228811264,
0.30847206711769104,
0.6241829991340637,
-0.5932533740997314,
0.47202593088150024,
-0.6866378784179688,
0.3212464451789856,
-0.5974756479263306,
-0.25748196244239807,
0.6464198231697083,
0.2729669213294983,
0.14558415114879608,
0.14759954810142517,
-0.2460840940475464,
-0.1259533017873764,
0.2703247666358948,
-0.2521667182445526,
-0.10096491873264313,
-0.8087202906608582,
-0.7629347443580627,
-0.6980127096176147,
0.574462354183197,
0.44457724690437317,
0.037522051483392715,
-0.12674428522586823,
0.4804006814956665,
0.20489723980426788,
-0.014367373660206795,
0.10122119635343552,
0.33134955167770386,
-0.03587837517261505,
-0.06799585372209549,
0.5548604130744934,
-0.996515691280365,
0.48350951075553894,
-0.26756635308265686,
0.3369235694408417,
0.06000051274895668,
-0.33049967885017395,
0.012366586364805698,
0.0505487360060215,
-0.46166932582855225,
0.5193430781364441,
0.2574928104877472,
0.10554633289575577,
-0.19912417232990265,
-0.3381224572658539,
-0.5390013456344604,
0.6228560209274292,
0.36709433794021606,
-0.13120736181735992,
0.09438298642635345,
-0.14465287327766418,
0.26226606965065,
0.12769830226898193,
0.09230737388134003,
-0.45016592741012573,
-0.04083114489912987,
0.051155198365449905
] |
fs(iωn, x) = [ˆg++(iωn, x) + ˆg−−(iωn, x)]12,
(D1)
fp(iωn, x) = [ˆg++(iωn, x) − ˆg−−(iωn, x)]12.
(D2)
18
The LDOS of bogolons and j(iωn, x) in the quasiclassical representations are expressed as
D(ω, x) = − 1
π Im [ˆg++(ω + i0+, x) + ˆg−−(ω + i0+, x)]11,
(D3)
j(iωn, x) =
i
nd
[ˆg++(iωn, x) − ˆg−−(iωn, x)]11,
(D4)
respectively.
Below, we list the expressions of physical quantities derived from the quasiclassical Green’s function Eq. (60). The | [
-0.14815972745418549,
-0.3036278486251831,
0.5345730781555176,
-0.22576437890529633,
0.03617994114756584,
0.446339875459671,
0.0037784173619002104,
-0.707137942314148,
-0.24006664752960205,
-0.2986176013946533,
-0.36441776156425476,
0.6192159652709961,
-0.5557334423065186,
-0.17309068143367767,
0.4150168001651764,
0.9123878479003906,
0.573165237903595,
0.02913038060069084,
-0.6913676857948303,
-0.6337697505950928,
0.33950504660606384,
-0.2173529863357544,
0.5857159495353699,
-0.0526561476290226,
-0.14594431221485138,
-0.022947831079363823,
0.5239703059196472,
0.15244004130363464,
0.19465622305870056,
0.6379901766777039,
-0.18312066793441772,
0.5315021872520447,
0.0015101449098438025,
0.15567776560783386,
0.14848755300045013,
0.16783364117145538,
0.4284515082836151,
0.09053611010313034,
0.12019796669483185,
-0.16838333010673523,
0.048928599804639816,
-0.45190778374671936,
0.1662626415491104,
0.24251370131969452,
-0.23105604946613312,
-0.3274042010307312,
-1.0993657112121582,
-0.2745981812477112,
-0.3441077172756195,
-0.2256677895784378,
-0.39438390731811523,
0.2523547410964966,
-0.25443992018699646,
0.30643847584724426,
-0.23321333527565002,
-0.10446792840957642,
-0.7333425283432007,
-0.23260970413684845,
0.39347052574157715,
-0.3300749957561493,
0.3418034017086029,
-0.09682821482419968,
0.26728495955467224,
0.7674229145050049,
-0.2725059986114502,
1.021159052848816,
-0.3306126892566681,
0.35212600231170654,
-0.6743457913398743,
-0.2544766962528229,
0.2797936201095581,
-0.06595452129840851,
-0.5320135354995728,
0.038329899311065674,
0.11670389771461487,
-0.10134968906641006,
-0.6360519528388977,
-0.33389395475387573,
0.15836335718631744,
0.3879956007003784,
-0.0550059974193573,
0.15067294239997864,
0.46816954016685486,
-0.582663893699646,
0.22083710134029388,
0.4594438374042511,
-0.13205984234809875,
0.7253703474998474,
-0.5384726524353027,
0.4865255355834961,
0.48084521293640137,
-0.1507951319217682,
0.14558152854442596,
0.13498306274414062,
0.1969575583934784,
0.5565380454063416,
-0.32142964005470276,
-0.4073849618434906,
0.40641283988952637,
0.30913272500038147,
0.03948022425174713,
-0.2139068990945816,
-0.4358806014060974,
-0.6302079558372498,
-0.4517725110054016,
-0.741416335105896,
-0.6019740700721741,
-0.7327950596809387,
-0.11928199976682663,
0.0616014339029789,
-1.3168387413024902,
0.19345101714134216,
-0.07292043417692184,
-0.5169748663902283,
-0.04941297695040703,
-0.0741916298866272,
-0.02880999818444252,
-0.2966206669807434,
0.4567802846431732,
0.5108973383903503,
0.17583869397640228,
0.5842493176460266,
0.2343389093875885,
0.6363154649734497,
0.6141546368598938,
-0.27397656440734863,
-0.013685519807040691,
0.7429776787757874,
-0.3018976151943207,
0.07252585142850876,
0.17256613075733185,
0.5175209045410156,
0.4561476707458496,
0.10276677459478378,
-0.2755734324455261,
-0.47860029339790344,
-0.36529457569122314,
-0.3214423656463623,
-0.671299397945404,
0.007908406667411327,
-0.7320302128791809,
-0.08283508569002151,
-0.3978533446788788,
0.0036885912995785475,
0.5744317173957825,
-0.4437764585018158,
-0.16630102694034576,
-0.15844102203845978,
-0.037965163588523865,
0.1229066327214241,
-0.14351478219032288,
-0.005381099879741669,
-0.3231245279312134,
0.48326486349105835,
-0.45784279704093933,
-0.22185228765010834,
0.5220450162887573,
-0.012645290233194828,
-0.37521710991859436,
0.0484614223241806,
0.36685073375701904,
-0.14556455612182617,
-0.11019565910100937,
-0.24442258477210999,
0.5479004979133606,
-0.5761489868164062,
0.6107421517372131,
-0.1252608299255371,
-0.16824787855148315,
0.01876183971762657,
-0.21843306720256805,
-0.4441789388656616,
0.0179133377969265,
0.21032002568244934,
-0.3841312825679779,
-0.17540428042411804,
-0.13002967834472656,
0.12423244118690491,
0.4756699502468109,
-0.059586361050605774,
-0.7554512023925781,
-0.6585865616798401,
0.23051366209983826,
-0.27619534730911255,
0.2045792043209076,
0.3819842040538788,
0.29062554240226746,
0.09091002494096756,
0.5851951241493225,
0.48092034459114075,
-0.302809476852417,
0.1459227353334427,
0.27361419796943665,
-0.8807345032691956,
1.088441014289856,
-0.08882629126310349,
-0.9514947533607483,
0.08979883790016174,
-0.21424026787281036,
0.13053332269191742,
-0.12918496131896973,
-0.0460374541580677,
0.12157727032899857,
0.16491225361824036,
-0.3564620614051819,
0.345796138048172,
-0.12225458025932312,
0.5457720160484314,
-0.6925035715103149,
0.30720633268356323,
0.029282690957188606,
-0.4125174582004547,
0.03426462039351463,
-0.3680517077445984,
0.3251115083694458,
-0.3003356158733368,
-0.1037955954670906,
-0.42169663310050964,
0.24860726296901703,
0.11371511965990067,
0.6328591704368591,
-0.16003067791461945,
-0.013211910612881184,
0.3888930678367615,
-0.42349082231521606,
0.5052307844161987,
0.8869956731796265,
0.2366192638874054,
0.6384736895561218,
-0.6419323086738586,
0.08713498711585999,
-0.10745114833116531,
0.17152747511863708,
0.13504034280776978,
0.3623270094394684,
0.4472789466381073,
0.06822343170642853,
0.06475416570901871,
0.7705134153366089,
-0.5738212466239929,
0.39425352215766907,
0.6764011383056641,
-0.12253345549106598,
0.17482799291610718,
0.1751859188079834,
-0.2302255481481552,
-0.18866565823554993,
0.11346273869276047,
-0.312468558549881,
-0.2918664813041687,
0.03494710102677345,
0.06893420219421387,
0.08665736019611359,
-0.14274165034294128,
-0.20695607364177704,
0.04377543181180954,
0.007472397293895483,
-0.003624280681833625,
0.10305678099393845,
0.2544952630996704,
-0.646648645401001,
-0.27997222542762756,
0.07803612947463989,
-0.3213745057582855,
-0.14668405055999756,
-0.05999019369482994,
-0.8735247254371643,
0.11589118838310242,
0.643485426902771,
0.6232986450195312,
0.19390372931957245,
-0.12227648496627808,
-0.7847565412521362,
-0.31654632091522217,
-0.37501657009124756,
-0.7367258071899414,
0.2051732987165451,
0.24723868072032928,
-1.014504075050354,
-0.5326645970344543,
0.056775569915771484,
0.5688934922218323,
1.074755311012268,
0.08264075964689255,
-0.04265953227877617,
-0.611947774887085,
0.16230469942092896,
-0.44286414980888367,
-0.5487983822822571,
-0.4425056278705597,
0.1354997456073761,
0.4000207483768463,
-0.14785677194595337,
-0.730252742767334,
0.13773991167545319,
0.4476432800292969,
-0.22573402523994446,
0.3387451171875,
0.20865744352340698,
-0.10382932424545288,
-0.08874665200710297,
-0.35456985235214233,
-0.00015301343228202313,
-0.18440023064613342,
0.1144162267446518,
-0.7276252508163452,
-0.43072572350502014,
0.8062512874603271,
-5.812923431396484,
0.08267389237880707,
-0.7088647484779358,
0.618797779083252,
-0.19526030123233795,
-0.044537294656038284,
0.5400760769844055,
-0.03754835203289986,
0.21135978400707245,
-0.37381744384765625,
-0.15878596901893616,
-0.25534486770629883,
-0.2552313208580017,
0.028276121243834496,
-0.14930929243564606,
0.626571536064148,
0.4381532669067383,
-0.08103682100772858,
0.2668730616569519,
0.39407771825790405,
0.4423227906227112,
-0.27527040243148804,
-0.09208735823631287,
0.08115961402654648,
0.6356703639030457,
0.571774959564209,
-0.6320777535438538,
0.29845014214515686,
-0.3305599093437195,
-0.2500191032886505,
-0.33866915106773376,
-0.32002341747283936,
0.060729555785655975,
-0.8079315423965454,
0.46423327922821045,
0.10953808575868607,
0.8201243877410889,
-0.14532631635665894,
-0.20202450454235077,
-0.6917082071304321,
0.47695815563201904,
0.8633787035942078,
-0.213313028216362,
-0.4657670855522156,
-0.11986090242862701,
0.024220949038863182,
-0.04288625717163086,
0.4246022701263428,
-0.1365060806274414,
0.2991631329059601,
-0.21110372245311737,
-0.02357579953968525,
-0.026114709675312042,
-0.019967123866081238,
0.08483200520277023,
-0.15622974932193756,
0.0473836325109005,
0.14243856072425842,
-0.14518389105796814,
0.8211862444877625,
0.1551039069890976,
-0.3151026666164398,
0.013905627653002739,
-0.14696235954761505,
-0.07077613472938538,
0.5010996460914612,
-0.3785385191440582,
-0.02353152446448803,
0.8304882049560547,
-0.06466110795736313,
-0.5555083751678467,
0.07133110612630844,
0.2353016883134842,
-0.7094542384147644,
0.33532044291496277,
-0.2937738597393036,
0.09339986741542816,
-0.10872229933738708,
-0.18998493254184723,
0.12329135835170746,
-0.2855846881866455,
-0.5988104343414307,
-0.09872882068157196,
0.05483914539217949,
0.13003003597259521,
-0.027963383123278618,
-0.3430384397506714,
0.0961824283003807,
-0.22014860808849335,
0.5802832841873169,
0.2554387152194977,
0.15287595987319946,
0.5798460841178894,
0.30121883749961853,
0.3139798939228058,
0.29920753836631775,
0.5343186259269714,
-0.25483566522598267,
0.1736523061990738,
-0.03589608520269394,
0.22347982227802277,
-0.23234038054943085,
0.10414014011621475,
-0.7110236287117004,
0.11146257072687149,
0.19476819038391113,
-0.5110329985618591,
0.5888746976852417,
0.6008093953132629,
-0.2645817697048187,
0.07939057797193527,
-0.1767105907201767,
-0.5904076099395752,
-0.2768460512161255,
0.8075993061065674,
0.07609958946704865,
-0.6053029298782349,
0.453206330537796,
0.3438287675380707,
-0.4991617500782013,
-0.496735543012619,
0.3448164761066437,
-0.4315149486064911,
-0.6511450409889221,
-0.7953739166259766,
-0.13521523773670197,
0.14473426342010498,
0.39672166109085083,
-0.3701404631137848,
0.29137393832206726,
-0.16997122764587402,
0.4250660538673401,
-0.308323472738266,
0.0009547722875140607,
0.22313067317008972,
-0.4056353271007538,
0.12933768332004547,
0.23309223353862762,
0.23992715775966644,
-0.7011673450469971,
0.4490630328655243,
0.4245949387550354,
0.030959168449044228,
-0.25694602727890015,
0.06375826150178909,
0.3609185516834259,
0.3093401789665222,
0.02271728776395321,
-0.2050633728504181,
0.34539663791656494,
0.5978801250457764,
-0.29982027411460876,
0.6858901381492615,
-0.1703604757785797,
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0.2684714198112488,
-0.008174643851816654,
0.039053335785865784,
0.4669940769672394,
-0.5346821546554565,
0.43711552023887634,
0.33020806312561035,
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0.25992774963378906,
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-0.05036177113652229,
0.06579235941171646,
-0.29550716280937195,
-0.3010335862636566,
-0.05002452805638313,
-0.047295644879341125,
-0.0811343565583229,
-0.278205931186676,
-0.13157008588314056,
-0.2896222770214081,
0.20622149109840393,
0.1131182536482811,
0.2556920647621155,
-0.5212951898574829,
0.21790404617786407,
0.9362519383430481,
0.9099175333976746,
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-0.14457480609416962,
0.20256996154785156,
-0.6258722543716431,
-0.4072461724281311,
0.34314846992492676,
-0.16855782270431519,
0.327635258436203,
-0.052306342869997025,
0.4797515571117401,
-0.17007380723953247,
0.5524927973747253,
0.72294682264328,
-0.31328085064888,
-0.482636034488678,
-0.16540083289146423,
0.7289551496505737,
-0.44136345386505127,
0.38994890451431274,
-0.133387491106987,
-0.12538716197013855,
0.06502946466207504,
-0.29880350828170776,
0.059022266417741776,
-0.25696685910224915,
-0.4051496088504791,
-0.43708470463752747,
-0.5409491062164307,
-0.026244444772601128,
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] |
s-wave and p-wave components of the pair amplitudes Eqs. (D1) and (D2) are reduced to
fs(iωn, x) = −2iΓ2sgn ωn
ΩL(iωn)
+
�
−
�
˜a(+)
out − ˜a(−)
out
�
− (˜a(+)
out + ˜a(−)
out)iωn + iΓ1Lsgn ωn
ΩL(iωn)
�
e−ikF(ΩL(iωn)/µ)x,
(D5)
and
fp(iωn, x) =
��
˜a(+)
out + ˜a(−)
out
�
+
�
˜a(+)
out − ˜a(−)
out
�iωn + iΓ1Lsgn ωn
ΩL(iωn)
�
e−ikF(ΩL(iωn)/µ)x,
(D6)
respectively. We note that ˜a(+)
out + ˜a(−)
out is an even-function with respect to frequency, while ˜a(+)
out − ˜a(−)
out is an odd- | [
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] |
out is an even-function with respect to frequency, while ˜a(+)
out − ˜a(−)
out is an odd-
function. Furthermore, comparing Eq. (D5) with Eq. (D6), the positions of two factors ˜a(+)
out + ˜a(−)
out and ˜a(+)
out − ˜a(−)
out
are reversed. Therefore, we can check that Eq. (D5) corresponds to the odd-frequency pair amplitude, while Eq. (D6)
corresponds to the even-frequency pair amplitude even in the quasiclassical representation.
The LDOS of bogolons is given by
D(ω, x) = − 1
π Re
�2(ω + iΓ1L)
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] |
The LDOS of bogolons is given by
D(ω, x) = − 1
π Re
�2(ω + iΓ1L)
Ωret(ω)
+ ˜a(+)
out,ret + ˜a(−)
out,ret
Ωret(ω)
i|Γ2L|e−ikF(Ωret(ω)/µ)x
�
,
(D7)
where a(±)
out,ret is a retarded version of a(±)
out. j(iωn, x) is expressed as
j(iωn, x) = −|Γ2L|sgn ωn
2iΩL(iωn)
�
˜a(+)
out − ˜a(−)
out
�
e−ikF(ΩL(iωn)/µ)x.
(D8)
[1] G. E. Volovik, JETP Lett. 49, 790 (1989).
[2] G. E. Volovik, Phys. Lett. A 142, 282 (1989).
[3] D. F. Agterberg, P. M. R. Brydon, and C. Timm, Phys.
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[105] Note that, for the Hermite case with s-wave SC, u±
r and
v±
r are identical to ˜u±∗
r
and ˜v±∗
r , respectively, but it is
not the case for bogolon model.
[106] Y.
Tanaka,
Physics
of
Superconducting
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] |
not the case for bogolon model.
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[111] The same expression of Green’s function for the simi- | [
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] |
486 (1963).
[111] The same expression of Green’s function for the simi-
infinite superconductor can be obtained considering
only the normal reflection and normal transmission
parts of the wave functions in Fig. 2. Namely, we set
a(±)
out = 0, b(±)
out = 1 in Ψ(±)
out and c(±)
in
= 1, d(±)
in
= 0 in
Ψ(±)
in , and impose the boundary condition for the wave
function Ψ(±)
out (x = 0) = 0.
[112] M. Ashida, S. Aoyama, J. Hara, and K. Nagai, Phys.
Rev. B 40, 8673 (1989). | [
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Pentacene-Doped Naphthalene for Levitated Optomechanics
Marit O. E. Steiner,∗ Julen S. Pedernales,† and Martin B. Plenio‡
Institut f¨ur Theoretische Physik und IQST, Albert-Einstein-Allee 11, Universit¨at Ulm, D-89081 Ulm, Germany
(Dated: May 24, 2024)
We introduce pentacene-doped naphthalene as a material for diamagnetic levitation, offering
compelling applications in matter-wave interferometry and nuclear magnetic resonance. Pentacene- | [
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] |
compelling applications in matter-wave interferometry and nuclear magnetic resonance. Pentacene-
doped naphthalene offers remarkable polarizability of its nuclear spin ensemble, achieving polariza-
tion rates exceeding 80% at cryogenic temperatures with polarization lifetimes extending weeks. We
design a multi-spin Stern-Gerlach-type interferometry protocol which, thanks to the homogeneous
spin distribution and the absence of a preferential nuclear-spin quantization axis, avoids many of | [
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] |
spin distribution and the absence of a preferential nuclear-spin quantization axis, avoids many of
the limitations associated with materials hosting electronic spin defects, such as nanodiamonds con-
taining NV centers. We assess the potential of our interferometer to enhance existing bounds on the
free parameters of objective collapse models. Beyond matter-wave interferometry, we analyze the
prospects for implementing magic angle spinning at frequencies surpassing the current standard in | [
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] |
prospects for implementing magic angle spinning at frequencies surpassing the current standard in
NMR, capitalizing on the exceptional rotational capabilities offered by levitation. Additionally, we
outline a novel protocol for measuring spin ensemble polarization via the position of the nanoparticle
and conduct an analysis of dominant noise sources, benchmarking the required isolation levels for
various applications.
I.
INTRODUCTION
Levitated optomechanics has opened a new window | [
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various applications.
I.
INTRODUCTION
Levitated optomechanics has opened a new window
into the quantum mechanics of mesoscopic systems [1, 2].
The levitation of nano- and microscale solids in a vac-
uum, pioneered by Ashkin more than 50 years ago [3],
started its incursion into the quantum regime barely 10
years ago, with the first proposals for cooling the me-
chanical degrees of freedom of levitated particles [4–6].
Today, with the recent demonstration of motional ground | [
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] |
Today, with the recent demonstration of motional ground
state cooling [7–9], it is generally acknowledged that the
first stage of this incursion has been completed, establish-
ing levitated optomechanics as a quantum platform that
promises a plethora of scientific as well as technologi-
cal applications. Among others, the quantum control of
microscopic objects may open the door to testing the lin-
earity of quantum mechanics in unprecedented regimes, | [
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] |
earity of quantum mechanics in unprecedented regimes,
and offer a platform for ultrasensitive force measurements
with both commercial and fundamental-science applica-
tions.
In
comparison
to
clamped
optomechanical
plat-
forms [10], levitated systems distinguish themselves by
their inherent lack of direct material contact with their
surroundings. This unique characteristic affords excep-
tional isolation conditions, rendering them mechanical
quantum systems with ultra-low dissipation.
Further- | [
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] |
quantum systems with ultra-low dissipation.
Further-
more, from a mechanical point of view, levitated sys-
tems possess the capability to undergo large displace-
ments of their center of mass, including displacements
larger than their size. They also exhibit controllable ro-
tational degrees of freedom [11–13], with the ability to
spin at unprecedented frequencies, only limited by the
tensile strength of the levitated material [14–16].
Ad-
∗ [email protected]
† [email protected] | [
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] |
Ad-
∗ [email protected]
† [email protected]
‡ [email protected]
ditionally, the capacity to dynamically adjust trapping
fields over time or deactivate them entirely offers avenues
for exploring dynamics in complex potential landscapes
as well as free dynamics in the absence of external fields.
All these capabilities render levitated systems as unique
quantum platforms, offering unparalleled opportunities
to explore the realm of quantum mechanics at the macro-
scopic scale. | [
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] |
to explore the realm of quantum mechanics at the macro-
scopic scale.
In terms of levitation methods, these have evolved
from the original optical tweezer technique to encom-
pass diamagnetic levitation as well as electric levitation
in Paul traps, each presenting distinct advantages and
drawbacks. The catalog of commonly levitated materi-
als includes dielectric materials such as silica, diamond
with color centers [17–19], magnets [20–22], or even su-
perfluid helium [23].
A notable subset comprises ma- | [
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] |
perfluid helium [23].
A notable subset comprises ma-
terials containing accessible spin degrees of freedom, as
they offer a pathway for harnessing the motion of the
levitated particle through spin-dependent forces [24–27].
For instance, diamond nanoparticles hosting color cen-
ters hold promise for implementing matter-wave interfer-
ometry, enabling advancements in force sensing and fun-
damental physics applications, based on Stern-Gerlach-
effect [28–30] or photon recoil [31]. | [
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] |
damental physics applications, based on Stern-Gerlach-
effect [28–30] or photon recoil [31].
However, the presence of color centers such as the
nitrogen-vacancy (NV) center in these setups comes with
important challenges. In particular, they are anticipated
to introduce significant channels of decoherence, poten-
tially rendering the most advanced matter-wave interfer-
ometry experiments unattainable [32]. For instance, the
negatively charged NV center induces an electric dipole | [
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] |
negatively charged NV center induces an electric dipole
moment in the neutrally charged diamond nanoparticle,
making it susceptible to interaction with stray electric
fields. Moreover, the preferential axis of quantization of
the NV center leads to undesired torques as it seeks align-
ment with the background magnetic field.
Additional
torques arise in the presence of magnetic field gradients,
as these exert forces on the NV center which will, typi-
arXiv:2405.13869v1 [quant-ph] 22 May 2024
2 | [
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] |
arXiv:2405.13869v1 [quant-ph] 22 May 2024
2
cally, be offset from the center of mass of the nanopar-
ticle. Besides these drawbacks directly associated with
the presence of the NV center, despite recent advances
[33], the achievable purity of diamond crystals remains
a significant concern. Even the purest diamond samples
harbor a substantial number of electronic impurities both
in the bulk of the nanodiamond, e.g. P1 centers, as well
as in its surface, e.g.
dangling bonds.
These can re- | [
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] |
as in its surface, e.g.
dangling bonds.
These can re-
sult in a significant source of decoherence, for which the
possibility of mitigation is uncertain. These challenges
underscore the need for innovative approaches in the use
of spin-dependent forces including the exploration of al-
ternative materials.
Here, we examine the potential of a material hith-
erto unexplored in the realm of levitated optomechanics,
one that offers solutions to several of the aforementioned | [
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one that offers solutions to several of the aforementioned
challenges: diamagnetically levitated, pentacene-doped
naphthalene. Remarkably, even in macroscopic samples,
the nuclear spins of the hydrogen atoms in naphthalene
crystals can be polarized to rates exceeding 80%, with
ultralong decay times surpassing 900 h at 25 K [34–
38].
This polarization is achieved through the short-
lived photo-excited triplet state of embedded pentacene
molecules. The transient nature of this state allows for | [
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] |
molecules. The transient nature of this state allows for
the elimination of associated noise channels from the ex-
periment following the polarization of the nuclear spin en-
semble. Additionally, naphthalene crystals demonstrate
experimentally the near-absence of magnetically active
impurities, as evidenced by the extended polarization
times. Beyond naphthalene, these advantageous aspects
point towards the possibility of exploring much broader
classes of materials with embedded photo-excitable elec- | [
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] |
classes of materials with embedded photo-excitable elec-
tron spins for applications in high-mass matter-wave in-
terferometry.
As an exemplary application, we design a multi-spin
matter-wave interferometry protocol that leverages the
high polarization rates of the nuclear spin ensemble. Ben-
efiting from a homogeneous spin distribution and the ab-
sence of a preferential axis of quantization, we eliminate
undesired torques, enhancing the visibility of the interfer- | [
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] |
undesired torques, enhancing the visibility of the interfer-
ometer. Beyond matter-wave interferometry, levitating
naphthalene suggests exciting prospects for pushing the
boundaries of NMR applications. The ability to rotate
naphthalene nanoparticles, or any other material, at un-
precedented frequencies opens avenues for attaining un-
matched nuclear spin coherence times through the tech-
nique of magic-angle spinning thus opening novel avenues
for solid-state nuclear magnetic resonance. Furthermore, | [
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] |
for solid-state nuclear magnetic resonance. Furthermore,
we design a novel technique for measuring nanoparticle
magnetization by observing displacements induced by the
interaction with a background magnetic field gradient.
The manuscript proceeds as follows: Section (II) pro-
vides a comprehensive analysis of levitated naphthalene
crystals and relevant parameter regimes, including pro-
cedures for hyperpolarization of the nuclear spin ensem-
ble and for magic angle spinning. Section (III) presents | [
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] |
ble and for magic angle spinning. Section (III) presents
a novel multi-spin matter-wave interferometry protocol
and its applicability in testing the linearity of quantum
mechanics and constraining free parameters of collapse
models. Furthermore, a novel measurement scheme for
detecting the polarization rate of the nuclear spin ensem-
ble is introduced. Section (IV) explores sources of noise
and benchmarks the required isolation for a successful
implementation of the ideas presented in the previous | [
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] |
implementation of the ideas presented in the previous
sections. Finally, Section (V) concludes with a summary
and outlook for future lines of research.
II.
NAPHTHALENE - A MATERIAL FOR
LEVITATION
Naphthalene is the simplest of the polycyclic aromatic
hydrocarbons, consisting of two fused benzene rings as
shown in fig. 1. At room temperature, it forms a white,
crystalline solid held together by Van der Waals forces. It
has a distinct, pungent odor, and, while it is best known | [
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] |
has a distinct, pungent odor, and, while it is best known
as the main ingredient of mothballs, its usage is wide-
ranging [39–41]. Naphthalene can be doped with pen-
tacene, another polycyclic aromatic hydrocarbon, where
each pentacene molecule substitutes two naphthalene
molecules without perturbing the lattice.
Pentacene-
doped naphthalene has gained attention in the nuclear
magnetic resonance community due to its large nuclear
spin polarizability.
When doped with pentacene, the | [
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] |
spin polarizability.
When doped with pentacene, the
spins of the hydrogen atoms in naphthalene can be po-
larized [34–38] to over 80% even in a macroscopic crystal.
Hyperpolarized naphthalene is used as a filter in neutron
scattering [42] and as a means to enhance sensitivity in
nuclear magnetic resonance [43].
In this section, we discuss the particularities of a hyper-
polarized naphthalene nanoparticle for levitation. Specif-
ically, we explore the crucial aspects of nuclear spin hy- | [
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] |
ically, we explore the crucial aspects of nuclear spin hy-
perpolarization protocols, the potential enabled by lev-
itation for conducting magic angle spinning at remark-
able rotation frequencies, and the constraints imposed by
sublimation rates on the permissible bulk temperature of
naphthalene.
A.
Hyperpolarisation
The nuclear spins of the hydrogen atoms in naphtha-
lene can be hyperpolarized by using embedded pentacene
molecules as a source of polarization. Pentacene, in its | [
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] |
molecules as a source of polarization. Pentacene, in its
electronic ground state, is a spin-0 system while in its
optically excited electronic state is a short-lived triplet
state, where selection rules strongly favor the occupation
of one of its three triplet states. This offers a mechanism
for the preparation of a highly polarized spin state of the
electron, whose polarization can then be transferred to
the surrounding nuclear spins via dipole-dipole magnetic
interactions. | [
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] |
the surrounding nuclear spins via dipole-dipole magnetic
interactions.
It is noteworthy to mention that the optical excita-
tion of the electron spin in pentacene requires a good
alignment of the pentacene molecule with the external
magnetic field. Nevertheless, a few degrees of deviation
3
(a)
(b)
(c)
FIG. 1. (a) and (b): The structure of an individual naph-
thalene molecule and naphthalene in its crystalline form. (c)
Structure of pentacene. The carbon atoms are colored green | [
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] |
Structure of pentacene. The carbon atoms are colored green
while the hydrogen atoms are colored gray.
The hydrogen
atoms carry a nuclear spin of 1/2. Naturally, the 13C isotope
of carbon, which carries a spin, constitutes only 1.1 % of all
carbon atoms, whereas the 12C isotope does not possess a
spin. To ensure that only the hydrogen atoms carry a spin,
12C enriched naphthalene can be used.
against perfect alignment is admissible, as this results in
only a few percent loss of the polarization of the pen- | [
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] |
only a few percent loss of the polarization of the pen-
tacene electron spin. However, for larger deviations, the
achievable polarization drops quickly.
For a more de-
tailed analysis, we refer the reader to Ref. [44].
The
alignment might be achieved by suitable trapping con-
ditions.
Alternatively, it has been shown that instead
of orientating the crystal in the magnetic field, the pen-
tacene molecule can be modified with polarizing agents
such as thiophene.
Thiophene-modified pentacene ex- | [
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] |
such as thiophene.
Thiophene-modified pentacene ex-
hibits a sharper and stronger electron spin resonance
spectrum compared to unmodified pentacene. Moreover,
the triplet electron distribution is more isotropic with-
out loss of triplet polarization, making it more robust
to deviations from the ideal orientation in the external
magnetic field [45].
The transfer of electron spin polarization from the pen-
tacene molecule to the nuclear spin ensemble can be ac- | [
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] |
tacene molecule to the nuclear spin ensemble can be ac-
complished by the application of a suitable microwave
drive (or, a sequence of pulses) that brings the electron
spin in resonance with the nuclear spin lattice [35]. How-
ever, the eigenfrequency of the electron spin depends on
the spatial orientation of the pentacene molecule with
respect to the external magnetic field and thus this will
also require a good alignment between the magnetic field
and the pentacene molecule. In this case, the effect of | [
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] |
and the pentacene molecule. In this case, the effect of
imperfect alignments might be mitigated by using pulse
sequences that exhibit robustness against detuning errors
such as sweep schemes [35] or robust pulsed sequences
such as PulsePol [46].
The triplet state of pentacene has a lifetime in the or-
der of tens of microseconds, which is sufficiently long to
allow for the transfer of polarization to the nuclear spin-
lattice. After the transfer, the electron quickly decays | [
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] |
lattice. After the transfer, the electron quickly decays
back to form a singlet ground state, which does not in-
teract magnetically with its environment. This differenti-
ates pentacene-doped naphthalene apart from persistent
embedded spins, such as color centers in diamond, where
the color center can also be used as a source of polariza-
tion of the nuclear spin-lattice. However, in contrast to
pentacene-doped naphthalene, persistent electronic spins | [
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] |
pentacene-doped naphthalene, persistent electronic spins
used for polarization remain magnetically active, typi-
cally in the form of a ground state triplet. This leads to
spurious undesired interactions with the spin-lattice and
possible environmental fields even after the polarization
transfer has been completed, turning into an unwanted
source of decoherence and dissipation. In contrast, the
short lifetime of the photo-excited triplet states in pen- | [
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] |
short lifetime of the photo-excited triplet states in pen-
tacene allows the opening of a channel for polarization
transfer that is subsequently closed, thus removing an im-
portant source of dissipation, and resulting in unmatched
nuclear polarization lifetimes. In bulk, up to 80% nuclear
polarization was achieved with an ultralong T1 time of up
to 920 h when operating at 25 K and 500 mT [38]. In
a nanoparticle of 70 nm radius, with just one or two em- | [
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] |
a nanoparticle of 70 nm radius, with just one or two em-
bedded pentacenes, even higher polarization rates can be
expected.
As will be discussed in the next chapter, the abil-
ity to polarize the nuclear spins in naphthalene offers
a promising alternative to create macroscopic superpo-
sitions, diverging from the conventional use of single
electron spins like NV centers in diamond.
Despite
the electron spin’s higher gyromagnetic ratio, the collec-
tive strength of naphthalene’s hydrogen spins, numbering | [
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] |
tive strength of naphthalene’s hydrogen spins, numbering
around 6 · 107 in a 70 nm particle, surpasses that of an
individual electron spin by more than 4 orders of magni-
tude.
B.
Magic Angle Spinning
While the longitudinal relaxation time T1 is very long,
the coherence time of each nuclear spin is constrained by
the, generally much shorter, T2 time. For each nuclear
spin, the main source of decoherence is provided by its
interaction with the rest of the nuclear spin lattice, which | [
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] |
interaction with the rest of the nuclear spin lattice, which
constitutes its immediate environment. Thus, to achieve
extended coherence times, it would be desirable to sup-
press the inter-nuclear spin interactions.
A technique,
well known in the field of nuclear magnetic resonance
spectroscopy, that can achieve this is that of magic an-
gle spinning (MAS), which is routinely used in solid-state
NMR to increase the T2 time of spins in a crystal by sup- | [
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] |
NMR to increase the T2 time of spins in a crystal by sup-
pressing their dipole-dipole interaction. The technique
works by rotating the probe sufficiently fast around an
axis that is tilted with respect to the direction of the mag-
4
netic field by a specific angle of arccos
�
1/
√
3
�
≈ 54.74°,
referred to as the magic angle [47]. The achievable T2
time is approximately determined by the rotation fre-
quency νMAS and is given by [48]
T2 ≈ νMAS
πGd2rss
,
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] |
quency νMAS and is given by [48]
T2 ≈ νMAS
πGd2rss
,
(1)
where drss is the effective dipolar coupling. The effec-
tive dipolar coupling experienced by a given proton j
can be written as drss,j = (�
k̸=j d2
j,k)1/2, where dj,k
is the dipolar coupling between the jth and kth spin.
G is a geometric factor given by the structure of the
lattice, with a value that can range between 0.04 and
0.11 [48].
While the exact value for the naphthalene
crystal is not known, for our calculations we will use the | [
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] |
crystal is not known, for our calculations we will use the
maximum value of G = 0.11, leading to the most conser-
vative estimations of the T2 time. To estimate drss, we
first compute the inter-nuclear dipolar coupling strength
dj,k =
µ0γ2
Hℏ
4π|⃗rjk|3 (1 − 3 cos2 θjk) between a nuclear spin j
and all the spins k in its lattice cell as well as in the first-
neighbor cells, and use it to compute drss,j.
We then
average this value over all the spins j within a lattice | [
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0.4645604193210602,
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] |
We then
average this value over all the spins j within a lattice
cell. Here, γH ≈ (2π)42.576 MHz/T is the gyromagnetic
ratio of the hydrogen nucleus, ⃗rjk is the vector joining the
spins j and k, and θjk is the angle formed by the mag-
netic field and the vector ⃗rjk. We take the positions of
the hydrogen atoms in naphthalene from X-ray scatter-
ing data [49]. The computation yields drss ≈ (2π)8.1 kHz,
resulting in the following relation between the achievable | [
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] |
resulting in the following relation between the achievable
T2 time and the rotation frequency of the crystal
T2 ≈ 4.4 νMAS · 10−8 s.
(2)
Thanks to their inherent rotational freedom, nanopar-
ticles levitated in a vacuum offer an unprecedented op-
portunity for MAS with rotational frequencies well be-
yond the state of the art [50] in solid-state NMR. The
most advanced rotation methods in this field typically
operate at frequencies on the order of 100 kHz, with | [
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] |
operate at frequencies on the order of 100 kHz, with
the largest demonstrated rotation frequencies being on
the order of 200 kHz [51].
This is achieved by pneu-
matic techniques, which employ compressed air to spin
an air-bearing rotor containing the sample of interest. In
contrast, levitated nano- and microparticles, exploit the
transfer of angular momentum from circularly polarized
light to the rotational degrees of freedom of the particle.
This transfer can occur through photon absorption, or | [
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] |
This transfer can occur through photon absorption, or
due to the birefringence or geometric anisotropy of the
particles [50, 52]. Exploiting this mechanism, microparti-
cles levitated in a vacuum can attain rotational frequen-
cies up to the MHz range [53, 54] while a nanoparticle
can achieve frequencies as high as GHz [14, 55]. Notably,
naphthalene crystals exhibit strong birefringence with
three optical axes, with refractive indices along these axes | [
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] |
three optical axes, with refractive indices along these axes
measured at 1.442, 1.775, and 1.932 respectively [56], sur-
passing even the birefringence of vaterite [57]. This sug-
gests that under similar conditions, naphthalene could
achieve rotational frequencies as high as those observed
in the rotation of vaterite [53] or potentially higher.
Nevertheless, the ultimate rotation frequencies achiev-
able with any material are constrained by the yield
strength of the material.
The maximal rotation fre- | [
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] |
strength of the material.
The maximal rotation fre-
quency a sphere of naphthalene can survive before break-
ing is given by νmax =
�
σUTS/(r2ρ)/(2π) where σUTS =
49 MPa is its ultimate tensile strength [58, 59]. For ex-
ample, for a 70 nm naphthalene particle with a density
of ρ = 1145 kg/m3 we find νmax = 470 MHz, which sug-
gests that nuclear spins in naphthalene can reach coher-
ence times of T2 ≈ 20 s. As will be discussed in the next
section, this exceeds by several orders of magnitude the | [
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