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Phase transitions with global/local dissipation
Michael Konopik
26.06.2019
• Properties of phase transitions/master equations
• Local/global master equations
• Phase transition consideration for the weak coupling Dicke Hamiltonian
Outline
Phase transitions
• 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>
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)
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=">AAAIGniclVXLbttGFGXSVErVl9Muu7mIoUICTVdU3SZeCEjQLIwughSokwAaiRhSI2qg4SOcYRCbmP5GN/2VbrJoEHRXdNO/6Qw5pCmHdhsCgob33Me5j7n0U0a5mEz+uXHzg1sf9vq3Pxp8/Mmnn32+d+eLpzzJs4CcBglLsuc+5oTRmJwKKhh5nmYERz4jz/ztDxp/9pJknCbxz+IsJYsIhzFd0wALJfLu9CbD9SgcozRLUpGMEFOmK+yYfy8YLwsHxbkstYa1WuiEF9BgiFaJQNkmga9nDp2fHOjzwkY8jzyKIiw2AWbFI+kVDz0q5yWKkDLrhpQXQCGOIqysGVmLESgQygDqsEQrHIYkAwfQOsNB4cpiKlHRhtS5JIEkymi4EeNLHFESkRDPcWOAa85tSlhq4SVZbVJiUPmtuM3AqQgZ995EEwOYI05Dlcx5FQNs2PFoULsjmIGcEhoMTzyuKqOrWmxnrlwWjyWYUFswusX24FzqEJUaiskLYErN9FPBTI6M7nbp1GZsaZdGRm5fyJ3xpUYhQV6JYkU5l6Zhsxanx10JqEBXtd14I+mmy5vK0EzCtpqEdxmWSehWdyIKKzqR3fGAsriqg6acLrRGA2wjnYLfSH1drqqoMMItwG6Z+tpxR7osCaQ0M/OeE4cQ7MJ+h4m/Y6Io1LdJT0P9Ut/18eC61GvxtOoKLXvcDNv5Ab0YthprqmI3nsamcUtbqcm6D0tHv4EesHrdaB78RSaKmk0dXpZKcchImx/KKpHZSzBqHNUbrFxSK8IElge/VAcVw73O2384qzbeu9bwP8lU5S/pGCc/eufvZxxrBo/KXK7VN1zjfFZHhWa1xsaytUOv7gM0G+2bqbe3Pzk8vu8eTY7BPZyUDyjJ98ffHd1rJPuWeZ54e3+pHRnkEYlFwDDnc3eSikWBM0EDRuQA5ZykONjikMzVMcYR4Yui/LRJGCrJCtZJpn6xgFLatihwxPlZ5CtNfQP4ZUwLu7B5Ltb3FwWN01yQOKgCrXMGqpb6OwkrmpFAsDN1wEFGFVcINljVSqiv6UAVocn96sPT6aH77eH0p6P9BzNTjtvWV9Zda2S51j3rgXViPbFOraD3a+/33h+9N/3f+q/7b/t/Vqo3bxibL62dp//3v5k7yww=</latexit>
�c =p!1!2/2
<latexit sha1_base64="2uXTlWZ+CVTNubNUET9XrmZq3HI=">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</latexit>
R. H. Dicke, Phys. Rev. 93, 99 (1954).
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=">AAAK1XictVZbb+NEFPYut8bcussjLyOqoERuShy67PbB0iJWouIhKohuV8o01tiZOKP4VnsC23oHCYR45b/xxm/gT3DGnnGcxC3qA5aijM93zplz+eZ4vDRkOR8O/37w8K2333n3vb2O+f4HH3708f6jxy/zZJX59NxPwiR75ZGchiym55zxkL5KM0oiL6QX3vIbiV/8RLOcJfGP/DqllxEJYjZnPuEgch/t/dOd94I+TrMk5UkPh2A6IwP17/r9aTHA8UqUWl2tFgyCNWR28SzhOFsk6HNnwCanh3J9aeF8FbkMR4QvfBIWL4RbfO0yMSlRjMGsHQIvCH9LooiAdUjnvIcAROUGsJjiGQkCmqEBwvOM+IUtipHARROCdRkEFjhjwYL3t2LESUQDMiG1AdExN0MiQgq3ZNqkxFDlt4rNQYMqIOXeHcrAEJrgnAWQzE21B7LQhkeFWi2bKWhQQmb31M2hMrKqxdKxxbQYC6S2WiKlWywPb4TcolLDMb1CIaipfgIcip7SXU4H2iycWqWRkltr+aC/1SjM6WtezFieC9UwpxHTuC0B2Oi2titvNF20eYMMFROWFRN2IyyTkK1uRQArWpFNeqCyuNBBVU4bNaiBLCUdIa+WerJcVVFRjzQAq2HqScct6YaJL4TizD0ZhzHahL0WE2/DBEII1GmSbNAv+qz3785di0dVW1jZ5JptN4dszTaN1WWxak991bmpBWpCN2I6kG9IMkzPGxlHfpXxQkejtxelUhyEtBkfziqRGkyoVzvSI6ycUlXKRBz+opewj32Xx/9wWI29XWt0/4CUk+/cm/sZxzKCFzTk5G59Fes6b22vrey7yl+3X3wx+v+IskmO2u0OS9QML78ybt4+T4maI7edIoWirVPUZuRtGZndLadWu129mbXrQSUN4+yqnso+VBbKSsJ0Ac08W7DpcWO8lAIovv4mOnrROMFaJLuzJGkql72xyyy7DxSrwKL+Poq1WlM4br7VlCmzdOqXxqBbJ+cMzeY14s3ONeLN+tDoLPwkzjm8xyunhZxO49u+Zq9Tw0DacTW8SZYlP2MWz/m1uaMnPa4dufsHw6OTZ/bx8ATZR8PyQSD56uTJ8dNacmCo58zd/wso568iGnM/JHk+sYcpvyxIxpkfUmHiVU5T4i9JQCewjElE88uivJUJ1AXJDM2TDH4xR6W0aVGQKM+vIw80JXnybUwK27DJis+fXRYsTlecxn610XwVIjjL8oqHZiyjPg+vYUH8jEGsyF8QqAKHi6AJRahzv33xcnRkf3k0+v744LmjyrFnfGp8ZvQM23hqPDdOjTPj3PA7P3Red37t/GZemML83fyjUn34QNl8Ymw85p//AtAewgA=</latexit>
Theory Experiment
Master equations
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>
Local/ global approach
• 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
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
Phase transition differences
Weak coupling Dicke model
14
Hepp, Lieb, Ann. Phys. 76 360 (1973): �c =p!1!2
<latexit sha1_base64="QdbnnfQq2SX/FzCxh2hHMf3ZOww=">AAAGWXiclVRbb9owFE43YF12a9fHvVirmEABlNAbPCB12h76VHXSepEwRI4xwSIXGptprZU/uYdJ0/7KHuYkDk0Ke5glxMn3nXP8neNjOwuPMm6av7aePK1Ua8+2n+svXr56/WZn9+0VC5cRJpc49MLoxkGMeDQgl5xyj9wsIoJ8xyPXzvxTwl9/IxGjYfCV3y3IyEduQKcUIy4he7cS1uEk5DCaheDDoE2HZ63EHhmQLX2bQh/xGUae+Bzb4qNN42HKQqjX1yiVA0AX+T6SsR6Z8gaQFEgpaYzhBLkuiUAbwGmEsLBi0Y2hKFLSTiVAmZC6M97Uywph6BMXDdEqAOWKi4JQKucRlodkUrO8mbYBaGeCVHrbTIQBMISMurKY+2wPYIBSRsUaGzZTVDul9LqA6VkNI9cZCbNjHR0c9Pots9Pry39TGtaxNHrxmc1kB5Pei/nAisfiPAZK0hyonGLeuo8TKZkbDMgt8KSbJ499ktJe3FC+83E7D/PGRhqkcOMBbzfjR+cJOfnOxYQyFqcFDAqSzjfVKfdJHdcmQ2Uii9l6JlmcGpZ5Nizr4lL9Sd6NjOTERqY8QfKoZV/lIatOWqAwPcBQaBc4K9RJOpX1EzRQgTAKoU6SeEOxXoizWpMd/28oIQRl2tkQ4pRCpIT8wiWDkH80lPxmufZy5dkI0dVg3bdoXEDVQFHZgsxjbNiCxsZDDgW3Exg09TwCJ/ux24iLvON5j2N7Z9/s9HvWodkHVsdMF5DIcf/o8GSF7GtqXdg7P+Q1xUufBBx7iLGhZS74SKCIU+yRWIdLRhYIz5FLhtIMkE/YSKS3LQZ1iUzANIzkL+AgRYsRAvmM3fmO9Ew6zB5zCbiJGy75tDcSNFgsOQlwttF06QEeguSZBRMaEcy9O2kgHFGpFeAZki8Ml4+xLpuwqv3fxlW3Yx10ul8O908Hqh3b2jvtvdbQLO1EO9XOtAvtUsOVn5U/1Wq1Vv1d26pt1/TM9cmWitnTSqu29xe41iSn</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:
Global treatment
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
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
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
Conclusion
19
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