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Phase transitions with global/ local dissipation Michael Konopik 26.06.2019

Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

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Page 1: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Phase transitions with global/local dissipation

Michael Konopik

26.06.2019

Page 2: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

• Properties of phase transitions/master equations

• Local/global master equations

• Phase transition consideration for the weak coupling Dicke Hamiltonian

Outline

Page 3: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Phase transitions

Page 4: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

• Symmetry (breaking) of the Hamiltonian cause phase transition

• Scaling parameters characterise the PT

• Critical exponent determined by the system’s PT-class

• For closed systems

Phase transitions

4

f(g) / |�� �c|�⌫<latexit sha1_base64="brUo9gpXZTafiutPKopbMlcp2YQ=">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</latexit>

Textbook: continuous phase transition

⌫<latexit sha1_base64="wsTT5dVElwr+o9tIvm23EYZWf10=">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</latexit>

Page 5: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Nonequilibrium phase transitions

5

• Generally experiments are done with open systems

• (Driven) dissipative effects compete with symmetries: new/change of phase transition

• Example: Cold atoms (Dicke model)

Zoller et al, Nature Phys. 4, 878 (2008)Zoller et al, PRL 105, 015702 (2010)Esslinger et al, PNAS 110, 11763 (2013)

Weatheril et al, Phys. Rev. Lett. 111, 113901 (2013)Ott et al, Phys. Rev. Lett. 116, 235302 (2016)

Experiments: Theory:Nagy, Domokos, PRL 115, 043601 (2015)

Page 6: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Dicke phase transitions

6

Scaling parameter:

Dicke model:

Domokos et al. PRA 84 0436 (2011)

ha†ai / (�c � �)��a , hJzi / (�c � �)��n<latexit sha1_base64="LbPvs1Mt71vUUUywPmVAyW7LEN4=">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</latexit>

Hs = !1a†a+ !2

NX

i=1

�z,i +NX

i=1

�(a+ a†)(�+

i + ��i )

<latexit sha1_base64="ZsM6FXlJh2rTiTKDKn4qoLIYEh8=">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</latexit>

�c =p!1!2/2

<latexit sha1_base64="2uXTlWZ+CVTNubNUET9XrmZq3HI=">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</latexit>

R. H. Dicke, Phys. Rev. 93, 99 (1954).

Page 7: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Change of critical exponents

7

Nagy, Domokos, PRL 115, 043601 (2015)

Closed dynamics:

Open dynamics:

�a = �n = 1<latexit sha1_base64="iFBKirTQM59X/BCydgEjoGmukes=">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</latexit>

Esslinger et al, PNAS 110, 29, 11763-11767 (2013)

�a = �n =1

2<latexit sha1_base64="FnSxa45SkNbr9WvRzuuTeq3Iiyo=">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</latexit>

Theory Experiment

Page 8: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Master equations

Page 9: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Master equations

• Weak system bath coupling

• Lindblad master equation

9

⇢̇ = �i![a†a, ⇢] +Da⇢+Da†⇢<latexit sha1_base64="kofNHJrrCN+fR9XWM9h0qLVe2oY=">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</latexit>

⇢̇ = � i!0

2[�z, ⇢] +D�+⇢+D��⇢

<latexit sha1_base64="xG0sM6U0iNehdJ2j+WKUCODRJoc=">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</latexit>

Two-level system Laser mode

⇢̇ = �i[Hs, ⇢] +X

i

DAi [⇢]

DAi [⇢] = �i

✓Ai⇢A

†i �

1

2{A†

iAi, ⇢}◆

<latexit sha1_base64="EMpZvVjteeF7KweunBOcQ1YmrNc=">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</latexit>

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Local/ global approach

Page 11: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

• Involved derivation.

• -dependent dissipators.

Local Master equations

11

Problems:

Hs<latexit sha1_base64="adNdNOQ9uFRcI05VbV2TSHJxCuQ=">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</latexit>

Solution:

• Derive dissipators locally according to expected processes.

λT1 T2a b

Page 12: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Shortcomings of the local approach

12

• Local approach breaks down for strong inter-system coupling

• Possible violation of second law of thermodynamics

Adesso et al, Open Systems & Information Dynamics 24, 04, 1740010 (2017)Brunner et al, New J. Phys. 19, 123037 (2017)Levy, Kosloff, EPL 107, 2 (2014)

• Quantitatively small heat current from hot to cold

• Reconciliation: Gabriele De Chiara’s talk

λT1 T2a b

Heat flow

Page 13: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Phase transition differences

Page 14: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Weak coupling Dicke model

14

Hepp, Lieb, Ann. Phys. 76 360 (1973): �c =p!1!2

<latexit sha1_base64="QdbnnfQq2SX/FzCxh2hHMf3ZOww=">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</latexit>

Hs = !1a†a+ !2

NX

i=1

�z,i +NX

i=1

�(a�+i + a

†��i )

<latexit sha1_base64="1CITZScNkEIwg4SP5z8JeCi2fpE=">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</latexit>

RWA Dicke Hamiltonian:

Map of system in limit Holstein-Primakoff transformation:

λT1 T2a b

Hs = !1a†a+ !2b

†b+ �(ab† + a

†b)

<latexit sha1_base64="ILg2MUY7/5eh42cXgd3Egd+9QP0=">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</latexit>

, � �c<latexit sha1_base64="fPPJm9knkbPnPC79IBNIwJBcM9c=">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</latexit>

⇢̇ = �i[Hs, ⇢] +Da[⇢] +Da† [⇢] +Db[⇢] +Db† [⇢]<latexit sha1_base64="Xol9NM1WKiwYW3oIgd1XG3ssGL4=">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</latexit>

N ! 1<latexit sha1_base64="KWDG6UW/ChVVH/IuV3IkbWYeuns=">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</latexit>

Emary, Brandes, PRE 67, 066203 (2003)

Local ME:

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Global treatment

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Global phase transition

16

• Derive ME mathematically

• Global RWA Dicke master equation

Additional nonlocal dissipators

⇢̇ = �i[Hs, ⇢] +Da[⇢] +Da† [⇢] +Db[⇢] +Db† [⇢]<latexit sha1_base64="np8oyP929HBXOWfTXV0kwh8s+HQ=">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</latexit>

Modification:

�i = �i(�)<latexit sha1_base64="mNzFX8nznWDxehiATKYyOUKlTNI=">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</latexit>

+Da+b[⇢] +Da†+b† [⇢]<latexit sha1_base64="6XMf/0qyNciEg297cluNRxuJpro=">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</latexit>

λT1 T2a b

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Measures of phase transitions

17

0 1 2 3 4 5 6 70

2

4

6

8

10

Coupling strength λ

Meanoccupatoinnumber<a†a>

ha†ai / (�c � �)��a , �a = 1<latexit sha1_base64="zxhQOopWn+W/36/h4Xql+5Oz+Hg=">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</latexit>

Red: global Blue: local

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Measures of phase transitions

18

0 1 2 3 4 5 6 70.0

0.5

1.0

1.5

2.0

2.5

3.0

Coupling strength λ

MutualinformationI

Red: global Blue: local

Subsystems become correlated close to phase transition

Discord: Quantum correlations, Mutual information: correlations

Fazio et al, Nature 416, 608–610 (2002) Dillenschneider, Phys. Rev. B 78, 224413 (2008)

0 1 2 3 4 5 6 7

0.00

0.05

0.10

0.15

Coupling strength λ

DiscordD

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Conclusion

19

Page 20: Phase transitions with global/ local dissipation · 2019-06-27 · Ott et al, Phys. Rev. Lett. 116, 235302 (2016) Experiments: Theory: Nagy, Domokos, PRL 115, 043601 (2015) Dicke

Conclusion

20

• Phase transitions may be described in the global approach, but local may fail to do so

• Local and global ME can have different effects on the structure of the phase transitions of the system