1/19

11:55 - 12:20

共鳴相互作用する
1次元量子系の普遍性
関野裕太
理研・仁科センター
基研研究会「熱場の量子論とその応用」
Monday, September 2nd, 2019
@Yukawa Institute for Theoretical Physics, Kyoto University

Collaborators

Prof. Yusuke Nishida
Tokyo Tech

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Prof. Shina Tan
Peking Univ./Georgia Tech

Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)
Y. Sekino, & Y. Nishida, in progressOutline of this talk

2/19

1. Introduction:
Universality near resonance & properties in 1D
2. Universal relations for 1D bosons & fermions near2-body resonances
Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)

3. Summary

Outline of this talk

2/19

1. Introduction:
Universality near resonance & properties in 1D
2. Universal relations for 1D bosons & fermions near2-body resonances
Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)

3. Summary

Ultracold atoms
Coldest systems in the universe !!
T = 1 µK~ 10 nK
Very pure & dilute

3/19

High controllability of ultracold atoms

4/19

Ø Statistics & internal d.o.f by atomic species and isotopes

7Li, 23Na, 39K, 41K,…)
Bosons(J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 115307

Fermions (6Li, 40K, 173Yb,V…)
Gokhroo et al

Figure 1. Hyperfine energy level diagram of D2 transition in 39 K and 40 K.

Gokhroo, et.al., (2011)

High controllability of ultracold atoms

4/19

Ø Statistics & internal d.o.f by atomic species and isotopes
Ø Control of parameters

Feshbach resonance → Interaction
40K

Ø Spatial geometry
Array of 1D gases Cubic lattice

Cubic lattice
Bloch Nat. Phys. (2005)

weakstrong
Regol and Jin, PRL (2003)

Highly controllable systems
Ideal ground to study universal physics near resonance5/19

Universalities near resonances
Resonant regimeV(r)r0(λT ~ T-1/2, lmean ~ n-1/d)

r0 << λT, lmean, |a|rr0 → 0Independent of details of V(r)E.g. S-wave resonance in 3Dü Ultracold atoms

Universal !!

ü Neutrons|a|/r0 ≈ 18

ü 4He atoms|a|/r0 ≈ 20

Regal & Jin, PRL (2003)→ Unitary Fermi gas, Efimov effect, …

6/19

1D systems near resonances

Fermions neara p-wave resonance

Bosons near
3D gasesan s-wave resonanceConfined to 1D

Granger & Blume,
PRL (2004).

Olshanii, PRL (1998)

1D bosons nearan even-wave resonance

1D fermions nearan odd-wave resonance

Bose-Fermi
correspondence
VF (x) =<latexit sha1_base64="t6LfOCWBsnTFtSJYRa1ixBVDxqI=">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</latexit>

Lieb & Liniger, Phys. Rev. (1963).

2aF
m0(x)D̃x

Girardeau & Olshanii, PRA (2004).

Regularized differential :7/19

Bose-Fermi correspondence (1)Free fermions

Hard-core bosons

VF (x) = 0<latexit sha1_base64="dEb1yX3nLC0uq5bq/D/C1QqXxFA=">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</latexit>! +1<latexit sha1_base64="kQ6494C0F1T2xfCu2jiyNYKAPJI=">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</latexit>

Bose-Fermi mapping<latexit sha1_base64="dVdiLP4o1+ytVn7QjC3jGZMrXGA=">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</latexit>

B (x1 , · · · , xN ) = AF (x1 , · · · , xN )EB = EF = E<latexit sha1_base64="RJ+AxR49XJtXUddDeNH/w9TtDhQ=">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</latexit>A=Y

sgn(xij )i<jGirardeau, J. Math. Phys. (1960),

Correspondences ofThermodynamics

Density correlations in cord. space8/19

Bose-Fermi correspondence (2)BosonsFermions

aB = aF = a<latexit sha1_base64="BCkHr1KVOw0DjmE5FxEEpktuIfA=">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</latexit>

VF (x) =<latexit sha1_base64="t6LfOCWBsnTFtSJYRa1ixBVDxqI=">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</latexit>

2aFm0(x)D̃x

Cheon & Shigehara, PRL (1999);
Girardeau & Olshanii, PRA (2004).

Bose

Bose-Fermi mapping<latexit sha1_base64="dVdiLP4o1+ytVn7QjC3jGZMrXGA=">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</latexit>B (x1 , · · · , xN ) = A

Fermi

F (x1 , · · · , xN )EB = EF = E

A=Ysgn(xij )i<jGirardeau, J. Math. Phys. (1960),<latexit sha1_base64="RJ+AxR49XJtXUddDeNH/w9TtDhQ=">AAACwHichVHLSsNAFD3Gd31V3QhuxKK4kHIrgg8QRFFc+qoKbSlJHOtgmsQkLWjoD/gDLlwpiIh/oRt/wIV/oLjyAW5ceJtGRYt6h5k5c+acO3dmNNuQrkd0W6PU1tU3NDY1R1pa29o7op1da65VcHSR1C3DcjY01RWGNEXSk54hNmxHqHnNEOvazmx5f70oHFda5qq3Z4tMXs2ZckvqqsdUNpr200GSlJPTMj7FKYjhT5AIQWkuO1ManPqmTlSp6Us9X5qay0ZjH0xfNfiwxxDGohU9QxqbsKCjgDwETHiMDahwuaWQAMFmLgOfOYeRDPYFSoiwt8AqwQqV2R0ec7xKhazJ63JON3DrfIrB3WFnHwbohs7pma7pgu7p7ddcfpCjXMsez1rFK+xsx0HPyuu/rjzPHra/XH/W7GEL40Gtkmu3A6Z8C73iL+4fPq9MLg/4g3RCD1z/Md3SFd/ALL7op0ti+QgR/oDEz+euBsmR+ESclkZj0zPhTzShF/0Y4ucewzQWsIgkH3uJOzziSZlVpGIpuxWpUhN6uvEtlP13Dy6lKg==</latexit>

Correspondences of|<latexit sha1_base64="LAj8KHFe3Sbwehsgg6iB1aiCsSM=">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</latexit>

Thermodynamics

2|=|B

2|F

Density correlations in cord. spaceOutline of this talk

9/19

1. Introduction:
Universality near resonance & properties in 1D
2. Universal relations for 1D bosons & fermions near2-body resonances
Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)

3. Summary

10/19Purpose of our study
1D bosons nearan even-wave resonance

1D fermions nearan odd-wave resonanceHow does similarity and difference appear
in universal properties b/w bosons & fermions ??
r0 << λT, n-1, |a|Universal relations
ü Exact constraints for any (a, T, n)ü Characterized by quantities called “contact(s)”Tan, Ann. Phys. (2008).

Universal relations for bosons

11/19

ü Tail of momentum distribution (MD):Olshanii & Dunjko, PRL (2003).

r0 << k-1 << λT, n-1, |a|Probability of“2-body contact”

Contribution from
2 particles at short range~k-1
Few-body d.o.f → short-range scale
k-1
λT, n-1

Universal relations for bosons

12/19

ü Tail of momentum distribution (MD):Olshanii & Dunjko, PRL (2003).

r0 << k-1 << λT, n-1, |a|ü Adiabatic relation:Lieb & Liniger, Phys. Rev. (1963).

ü Energy relation (ER):

Valiente, Europhys. Lett. (2012).

2-body contact C2

Comparison b/w bosons & fermionsü Tail of MD(r0 << k-1 << λT, n-1, |a|):4C2⇢B (k) = 2 4 + O(1/k 5 )
a k

13/19
(boson)<latexit sha1_base64="UL1wjoDPw6SUOUxkdCwitiHcmHw=">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</latexit>(fermion)

Cui, PRA (2016)

ü Adiabatic relation:
ü ER:(boson & fermion)(boson)(fermion)

3-body contact C3

YS, Tan, & Nishida
PRA (2018)

RG analysis in QFT !!

Outline of this talk

14/19

1. Introduction:
Universality near resonance & properties in 1D
2. Universal relations for 1D bosons & fermions near2-body resonances
Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)

3. Summary

Outline of this talk

18/19

1. Introduction:
Universality near resonance & properties in 1D
2. Universal relations for 1D bosons & fermions near2-body resonances
Y. Sekino, S. Tan, & Y. Nishida, Phys. Rev. A 97, 013621 (2018)

3. Summary

19/19

Summary
1D bosons nearan even-wave resonance

1D fermions nearan odd-wave resonance

Bose-Fermi
correspondence

Bose Fermi

ü Universal relations are
1. Exact constraint for any parameters
2. Characterized by C2 & C3, which are identicalb/w bosons & fermions