Open Quantum Systems. Sabrina Maniscalco. Turku Centre for Quantum Physics, University of Turku Centre for Quantum Engineering, Aalto University

Transcription

1 Open Quantum Systems Sabrina Maniscalco Turku Centre for Quantum Physics, University of Turku Centre for Quantum Engineering, Aalto University Turku Quantum Technologies

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3 GOAL at least 3 useful concepts several references

4 Take home message Sit by the window if you fly north

5 The basic stuff and why you should be listening Friday 29th,

6 OQS coolest current trends the Markovian - Non Markovian bloody battle Saturday 30th,

7 Myths and legends about non-markovian dynamics Monday 2nd,

8 How many of you have listened to a talk/lecture on OQS before?

9 How many of you have studied / done research on open quantum systems?

10 How many of you liked the breakfast?

11 WHAT Open Quantum Systems theory Very general theoretical framework!

12 References H.-P. Breuer and F. Petruccione The Theory of Open Quantum Systems, Oxford University Press Recent reviews: H.-P. Breuer, et al., Rev. Mod. 88, (2016) A. Rivas, et al., Rep. Prog. Phys. 77, (2014) I. de Vega and D. Alonso, Rev. Mod. Phys. 89, (2017) my lecture notes

13 ENVIRONMENT S

14 IDENTIFY S

15 ENVIRONMENT S

16 ENVIRONMENT S Examples of environment?

17 the first and main assumption No initial correlations T (0) = S E time generally correlations are created T (t) S (t) =Tr E ( T (t)) PS: Also classical noise

18 the goal of OQS theory Dynamics S (0) S (t) master equation dynamical map

19 WHAT FOR? When and where can we use open quantum systems theory?

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21 Foundations

22 Foundations Schrödinger kitten of light S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, Nature 455, 510 (2008) 2012 Nobel prize in Physics

23 Quantum Technologies Quantum Computers Quantum Simulators Quantum Devices for Communication Cryptography Energy conversion Metrology

24 superconducting quits quantum dots photonics systems NV-centres in diamonds trapped ions nanomechanical oscillators ultracold gases relativistic systems (Unruh effect) cavity QED optical systems

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26 THE PLAN states transformations (quantum channels) dynamics (dynamical map) master equation

27 BREAK!

28 quantum states Quantum system living in a finite dimensional Hilbert space probability distribution = X k w k k ih k X w k > 0 w k =1 positive trace 1 k This decomposition is highly NON-unique

29 linear map transformations :M n (C)! M n (C) convex subset of positive matrices M + n (C) ={A 2 M n (C) A 0} M n (C) PROPERTIES (A )=[ (A)] Hermicity preserving (M + n (C)) M + n (C) positive tr (A) =tra trace preserving why? (I n )=I n unital

30 transformations PROPERTIES (A )=[ (A)] Hermicity preserving (M n + (C)) M n + (C) positive tr (A) =tra trace preserving (I n )=I n unital S (t 1 )= S (0) is that enough? NO

31 consider this. system 1 system 2 H 1 H 2 1 : M n (C)! M n (C), 2 : M m (C)! M m (C) composite map 1 2 : M n m (C)! M n m (C) example (2x2 system) [1l 2 ](A) = 2X i,j=1 e ij (A ij )= (A11 ) (A 12 ) (A 21 ) (A 22 ) T n ( ) = T Transposition

32 consider this. example (2x2 system) [1l 2 ](A) = 2X i,j=1 e ij (A ij )= (A11 ) (A 12 ) (A 21 ) (A 22 ) T n ( ) = T Transposition positive trace preserving quantum entanglement!

33 complete positivity :M n (C)! M n (C) is called k-positive when 1l k :M k (C) M n (C)! M k (C) M n (C) is positive :M n (C)! M n (C) is called completely positive (CP) when 1l k :M k (C) M n (C)! M k (C) M n (C) is positive for all k: 1,2,3,..

34 quantum channels are completely positive and trace preserving (CPTP) linear maps

35 Choi s fantastic theorem is CP if and only if [1l n ](P + n ) 0 P + n = + n ih + n + n i = 1 p n n X k=1 e k e k two remarkable things here!

36 Kraus form is CP if and only if (X) = X K XK Kraus operators for X 2 M n (C)

37 THE PLAN states transformations (quantum channels) dynamics (dynamical map) master equation

38 dynamical map S (t) = t S (0) t-parametrised family of quantum channels (CPTP linear maps) t 0 0 =1l n how is it connected to the total system evolution?

39 What if T (0) 6= S (0) E (0) ANSWER 1: the dynamical map need not be CP (and indeed it may not even be positive)! ANSWER 2: the dynamical map still needs to be CP

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41 master equation d S (t) dt = L t S (t) EXAMPLE 1: Spontaneous emission

42 master equation - dynamical map the connection S (t) = t S (0) d S (t) dt = L t S (t) t = L t t, 0 =1l n solution t =Texp Z t 0 the generator L d

43 the most important OQS theorem Gorini-Kossakowski-Sudarshan-Lindblad (GKSL theorem) also known as the Lindblad theorem. until the moment you say it in front of George Sudarshan

44 Take home message In case of doubt, attack!

45 the most important OQS theorem Gorini-Kossakowski-Sudarshan-Lindblad (GKSL theorem) d S (t) dt = L t S (t) d S (t) dt = L S (t) Z t t =Texp 0 L d t = e Lt

46 the most important OQS theorem Gorini-Kossakowski-Sudarshan-Lindblad (GKSL theorem) characterisation of the master equation of any physical dynamics characterisation of the generator of CPTP dynamical maps

47 the most important OQS theorem Gorini-Kossakowski-Sudarshan-Lindblad (GKSL theorem) L( ) = i[h, ]+ X k k decay rates k 0 V k V k 1 2 {V k V k, } jump (or Lindblad) operators EXAMPLE: Spontaneous emission

48 MARKOVIAN OPEN QUANTUM SYSTEMS

49 Lindblad form Master Equations EXAMPLE: Lossy quantum harmonic oscillator

50 Lindblad form Master Equations EXAMPLE: pure dephasing of a qubit easy to solve

51 Lindblad form Master Equations EXAMPLE: depolarising channel for a qubit 3X L( ) = k( k k ) k=1

52 What if d S (t) dt = L t S (t) time dependent

53 OQS Holy Graal

54 TOMORROW S LECTURE

55 Take home message T (0) = S E

56 Take home message GKSL structure of all Markovian MEs

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