*Wei-Min Zhang *

**Physics Department, National Cheng Kung University, Taiwan **

**wzhang@mail.ncku.edu.tw **

NTU Joint Colloquium, June 13_{, 2017 }

## Physics behind

**Physics of Open Systems **

**Collaborators: **

** Matisse W. Y. Tu ** ** (HKU) **

** Pei-Yun Yang** ** (NCKU) **
** Ping-Yuan Lo (NCTU) **

** Jinshuang Jin ** ** (Hongzhou Normal Univ) **
** Jin-Hong An ** ** (Lauzhou Univ) **

** Hua-Tang Tan (Huachung Normal Univ) **
** Heng-Na Xiong ** ** (Zhejiang Univ of Tech) **

** Md M. Ali** ** (NCTS) **

** Chun U Lei ** ** (Yale) **
** …….. **

** **

**Outline **

² **Introduction to general open systems **

² **A general quantum theory of open systems **

² **General analytical solution of non-Markovianity **

² **Measurement of non-Markovianity (a new way) **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **

**Smiling Face of Universe
**

**Our Physical World
**

** All the Physics: **

**cosmology, gravity,**
** particle physics,**
** nuclear physics,**
** condensed matter,**
** atomic physics,**

** molecules, optics…**

### L = (i ·@ m) = 0 ^{
}

### All the Physics we know

### + |(@ ^{µ} +ig ^{0} A _{µ} ) | ^{2}

### 2

### 4 ( | | ^{2} v ^{2} ) ^{2}

### + 1

### 16⇡ (R 2⇤) op

### gd ^{4} x S = Z n

### g _{↵} ·A ^{↵} ^{1}

### 4 F _{µ⌫} F ^{µ⌫}

### ⇢ = 1

### Z exp { (H µN ) }

*s*

^{(time) }*m*

^{(space) }

^{10}*26*
*10*^{-6}^{
} *10*^{-10}^{
} *10*^{-18}^{
}

*10*^{17}^{
}

*10*^{-18}^{
}

*10*^{-44}^{
}

### S = Tr[⇢ ln ⇢]

**? **

**Nonequilibrium Statistics **
**of open systems
**

### wow

**How to describe **
**information and its **
**processing in states ?
**

### That’s the Physics we don’t know !

Tr[⇢] = 1, hHi = const.

hN i = const.

### if

Thermo-Statistics:

**(Information)
**

## 物理學

過去

(Force)⼒力

現在 將來

(Energy) 能量

### 資訊

### (Information)

### F = ma E = mc ^{2}

### E = ~! S = tr[⇢ ln ⇢]

**Closed and open system dynamics, the essence **

### | (t)i

### d

### dt ⇢(t) = 1

### i ~ [H , ⇢(t)]

### i ~ d

### dt | (t)i = H| (t)i

### ⇢(t) = | (t)ih (t)|

### ⇢(t) 6= | (t)ih (t)|

### Mixed state: loss of coherence

Schrödinger equation, 1925

von Neumann equation, 1927 (also Landau, 1927)

Ø For closed systems, quantum state

### :

In terms of density matrix

### :

Ø For open systems, in general

### :

*S=0 (no information)
*

*S≠0 (rich information)
*

### S = tr[⇢ ln ⇢]

**A complete description of ** **physical systems **

**is **

### ⇢

### which is the density matrix that fully characterizes

### all possible quantum states of physical systems

### Ø Underlying physics of open quantum systems:

ü Dissipation and fluctuations ü Non-Markovian memories

**Main Physics of open quantum systems **

### Non-equilibrium Physics

Matter exchange

Information exchange

Energy exchange

particle transport involves the phenomena of mater and energy

exchanges (dissipation) and
information exchange (fluctuations)
*ΔS *

*ΔE *

*ΔW *

à Open quantum systems:

**cover All problems in physics ! **

### quantum foundation: e.g. origin of probability …

### quantum information and computation: decoherence …

### nonequilibrium physics: new physics …

### quantum thermodynamics: mesoscopic scale

### Foundation of quantum phase transition, topological orders of matter, and even quantum gravity: quantum entanglement …

### Origin of the life, origina of spece and time: also entanglement ?…

### Unification of various physics at different scales …

### ……

**Physical significances of open systems **

### Typical open systems:

l The system interacting with many other systems surrounding

**Modeling open quantum systems **

l One or a few particles interacting with other particles in many-body systems

l A subspace of the Hilbert space correlating with others states in the space

Controls Measurements Environments

Open system

### H

^{-space }

^{Λ}

Λ_{1}
Λ_{2}
Λ_{3}^{
}

l Physics is governed by master equation of the reduced density matrix

**Theory of open quantum systems **

l Physics is characterized by various correlation functions in many-body systems

l Physics at different scales is related by renormalization group

Controls Measurements Environments

Open system

### H

^{-space }

^{Λ}

Λ_{1}
Λ_{2}
Λ_{3}^{
}

**Building the Theory of Open Quantum Systems:
**

Ø The theory of open quantum systems must go beyond quantum mechanics ?

Ø The theory of open quantum systems can be established within quantum mechanics ?

Answer: upon how you define open systems

l A1: open systems must contact with reservoirs.

----a definition from Statistical Mechanics (19 century) l A2: systems of interest couple to the outside world which we

are not interested.

----a definition by Feynman (1960’s)

** Theory for open quantum systems: **

**a long-standing problem **

Openness

Quantum Mechanics

J. von Neumann (1927)

L. D. Landau (1927) Non-Markovian Memory (Non-equilibrium Physics)

However, it has been attempted for many years without a very satisfactory answer to find the exact master equation for an arbitrary open quantum system that is tractable, since Pauli proposed the first master equation in 1928 !

l In general (Master Equation):

**Historical development of Master Equation **

u Pauli master equation (W. Pauli, *Festschrift zum 60. Geburtstage A. *

*Sommerfelds (Hirzel, Leipzig, p.30, 1928) *

Only valid in the weak system-reservoir coupling

⇢_{tot}(t) = P⇢^{tot}(t) + (1 P)⇢^{tot}(t)
d

dt⇢_{tot}(t) = i[H_{tot}, ⇢_{tot}(t)] ⌘ L(t)⇢^{tot}(t)

+ Z t

t0

d⌧PL(t)G(t, ⌧)QL(⌧)P⇢^{tot}(⌧ )
d

dtP⇢^{tot}(t) = PL(t)P⇢^{tot}(t) + PL(t)G(t, t^{0})Q⇢^{tot}(t_{0})

*τ à t
*

u **Nakajima-Zwanzig master equation (S. Nakajima, Prog. Theo. Phys. 20, **
**948, 1958; R. Zwanzig, J. Chem. Phys. 33, 1338, 1960): **

u **Born-Markov master equation (e.g. F. Haake, Z. Phys. 223, 353; 364, 1969): **

**Theory of Open Quantum Systems **

### Ø Dynamics of open quantum systems is one of biggest problems that have not been solved in physics

u people deeply involved such research include

• W. Pauli (1920’s)

• J. Schwinger (1960’s)

• R. Feynman (1960’s)

• A. J. Leggett (1980’s)

• ……

### Ø It still attracts and challenges theorists to build and

### develop the theory of open quantum systems.

**In the literature, one often uses the Lindblad-GKS form ** ** ** ** of an approximated master equation: **

### L

_{a}

i,a^{†}_{j}

### [⇢(t)] ⌘ a

^{i}

### ⇢(t)a

^{†}

_{j}

### 1

### 2 a

^{†}

_{j}

### a

_{i}

### ⇢(t) 1

### 2 ⇢(t)a

^{†}

_{j}

### a

_{i}

where

which only describes Markovian (memoryless) processes, the basis for quantum optics.

### d⇢(t)

### dt = 1

### i [ ˜ H

_{S}

### (t), ⇢(t)] + X

ij

ij

### L

_{a}

j,a^{†}_{i}

### [⇢(t)]

G. Lindblad, Comm. Math. Phys. 48, 119 (1976);

V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17,821 (1976).

**Goes beyond the Markovian dynamics: **

Ø A typical timescale of the system.

Ø A typical timescale of the environment

Ø The inverse of the coupling constant between the system and environment

u The Lindblad-GKS master equation valid only for τ_{s}*>> τ** _{c }*in the weak
system-environment coupling regime.

u Dissipation, fluctuations as well as non-Markovian memory effects strongly rely on the relations among these different timescales.

However, such relationships have yet been established quantitatively ! l Physically, three timescales dominate the dynamics of open quantum

systems

Ø The general behavior of dissipation, fluctuations and non-Markovain nonequilibrium physics of open systems is mainly determined by the energy structure or the spectral density profile of environments.

Ø Universal non-Markovian memory can be extracted from the exact analytical solutions, solved by connecting the exact master equation with the nonequilibirum Green functions.

Ø Although the exact master equations have been derived only for some class of system-environment couplings. Establishing the connection between the exact master equation and the non -equilibrium Green functions provides a general approach to explore the non-Markovian nonequilibrium physics even though the exact master equation is unknown.

**Here I would like to show:
**

**Outline **

² **Introduction to general open quantum systems **

² **A general theory of open quantum systems **

² **General analytical solution of non-Markovianity **

² **Measurement of non-Markovian decoherence **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **

**Two useful theories for nonequilibrium dynamics **

Schwinger-Keldysh’s Green function technique J. Schwinger, J. Math. Phys. 2, 407 (1961)

* L. P. Kadanoff, & G. Baym, Quantum Statistical Mechanics (1962) *
* L. V. Keldysh, Sov. Phys. JETP, 20, 1018 (1965) *

Feynman-Vernon influence functional approach

* R. P. Feynman and F.L. Vernon, Ann. Phys. 24, 118 (1963) *
* A. O. Caldeira and A. J. Leggett, Physics. A 121, 587 (1983) *

However, both theories only solve partially some problems of the nonequilibrium phy sicsof open systems,

Ø The former provides correlations of local operators, but lacks the full state information of the whole system.

Ø The later provides a nice way to treat the environmental degrees of freedom, but still far from the complete understanding of all the states of the system.

**Closed-path approach for nonequilibrium dynamics: **

The correlation function or the evolution of density matrix

### ⇢(t) = U (t, t

_{0}

### )⇢(t

_{0}

### )U

^{†}

### (t, t

_{0}

### )

### t _{0}

**J. Schwinger, J. Math. Phys. 2, 407 (1961) ****R. P. Feynman, F. L. Vernon, Ann. Phys. (N.Y.) 24, 118 (1963) **

n which is very different from the scattering theory in field theory, scattering theory is in general INVALID for nonequilibrium physics !!!

### G(xt; x

^{0}

### t

^{0}

### ) = i hT

^{C}

### [ (xt)

^{†}

### (x

^{0}

### t

^{0}

### )] i

### t ^{0}

### if t > t

^{0}

**see a Review by Yang & WMZ, Front. Phys. 12, 127204 (2017) **

**Nonequilibrium Green-function approach: **

Kadanoff-Baym equation (1962):

where

retarded GF

lesser GF, when
G^{c}(⌧, ⌧^{0}) =

Z Z

G^{r}(⌧, ⌧^{00})⌃^{c}(⌧^{00}, ⌧^{000})G^{a}(⌧^{000}, ⌧^{0})d⌧^{00}d⌧^{000}
L. V. Keldysh (1965)

### G

^{<}

_{0}

### = 0

**J. Schwinger, J. Math. Phys. 2, 407 (1961) **

**Influence functional approach **

a nonperturbation way for fully covering the back-reaction of baths on the system one concerns:

### ρ ^{(t}

^{(t}

_{0}^{) } ρ ^{(t) }

^{) }

^{(t) }### propagating function influence functional

Ø for quantum Brownian motion à

Feynman & Vernon,

** Ann. Phys. 24, 118 (1963) **

Caldeira & Leggett,

** Physica A 121. 587 (1983) **

### t

_{0}

### i

**But, the Feynman-Vernon’s influence functional **

** does not result directly in master equation **

for example, consider a spin-boson coupling:

The influence functional is given

Ø ** None has derive the exact master equation from it !!! **

where

v The leading order system-environment couplings are dominated by particle-particle (energy and information) exchanges:

**Our consideration: **

** General system-environment couplings: **

** ** ** (generalized Fano-Anderson models)
**

### H

_{SB}

### = X

↵ki

### [V

_{↵ki}

### a

^{†}

_{i}

### b

_{↵k}

### + V

_{↵ki}

^{⇤}

### b

^{†}

_{↵k}

### a

_{i}

### ]

Ø *the system contains arbitrary N energy levels *

Ø environment can contain many different reservoirs Ø each reservoir is specified by the spectral density:

### J

_{↵ij}

### (!) = 2⇡ X

k

### V

_{↵ik}

### V

_{↵jk}

^{⇤}

### (! ✏

_{k}

### )

**P. W. Anderson. Phys. Rev. 124, 41 (1961) ****U. Fano, Phys. Rev. 124, 1866 (1961) **

**A. T. Leggett, et al., Rev. Mod. Phys. 59, 1 (1987) **

**Path integral approach to fermion open systems: **

**The System **

**The Environment **

l With the initial state

l taking the fermionic coherent state representation l All matters are made by fermions

and

**Tu & WMZ, Phys. Rev. B 78, 235311 (2008) **

H = H_{S} + H_{B} + H_{SB}

*The Influence Functional *

path integral over the system

(the action of the system)

Initial Fermi-Dirac distribution function

where

WMZ et al., Rev. Mod. Phys. 62, 867 (1990) Tu & WMZ, Phys. Rev. B 78, 235311 (2008)

(obtained after completely integrated over all the environmental degrees of freedom)

**A few crucial steps toward obtaining the master equation **

where

3. A nontrivial transformation:

2. The role of the end points:

1. The classical action in coherent state rep.

Tu & WMZ, PRB 78, 235311 (2008)

**Our Exact Master Equation:
**

Ø Dissipation and fluctuation coefficients in the master equation:

Dissipation

**Dissipation **
**(relaxation) **

**Fluctuations **
**(noises) **

Fluctuation

### d⇢(t)

### dt = 1 i

### h H e

_{S}

### (t), ⇢(t) i

### + X

ij

### n

ij

### (t) h

### 2a

_{j}

### ⇢(t)a

^{†}

_{i}

### a

^{†}

_{i}

### a

_{j}

### ⇢(t) ⇢(t)a

^{†}

_{i}

### a

_{j}

### i

### e"(t) = i 2

### ⇥ ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### ) H.c. ⇤ (t) = 1

### 2

### ⇥ ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### ) + H.c. ⇤ e(t) = ˙v(t, t) ⇥

### ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### )v(t, t) + H.c. ⇤

**renormalization **

### + e

^{ij}

### (t) h

### a

^{†}

_{i}

### ⇢(t)a

_{j}

### ± a

^{j}

### ⇢(t)a

^{†}

_{i}

### ⌥ a

^{†}

_{i}

### a

_{j}

### ⇢(t) ⇢(t)a

_{j}

### a

^{†}

_{i}

### io

upper sign: bosons lower sign: fermions

Tu and WMZ, Phys. Rev. B 78, 235311 (2008)

**Jin, Tu, WMZ & Yan, New J. Phys. 12, 083013 (2010) ****Lei and WMZ, Ann. Phys. (N.Y.) 327, 1408 (2012) **

**WMZ, Lo, Xiong & Nori, Phys. Rev. Lett.109, 170402 (2012) ****Yang, Lin & WMZ, Phys. Rev. B 92, 165403 (2015) **

**Yang & WMZ, Frontiers of Physics 12, 127204 (2017) **

**Nonequilibrium Green functions u(t,t**

_{0}**) and v(t,t) **

**) and v(t,t)**

u Kadanoff-Baym-like equations:

**Tu & WMZ, Phys. Rev. B 78, 235311 (2008) **
**WMZ, et al, Phys. Rev. Lett.109,170402 (2012) **

**Non-Markovian memory kernels **

u

*u(t,t*

_{0}*) *

and* v(t,t) *

are two Green basic functions in Schwinger-Keldysh’s
non-equilibrium theory:
### d

### d⌧ u(⌧, t

_{0}

### ) + i"

_{s}

### u(⌧, t

_{0}

### ) + Z

⌧t0

### d⌧

^{0}

### g(⌧, ⌧

^{0}

### )u(⌧

^{0}

### , t

_{0}

### ) = 0

dd⌧ v(⌧, t) + i"_{s}v(⌧, t) +
Z ⌧

t0

d⌧^{0}g(⌧, ⌧^{0})v(⌧^{0}, t) =
Z t

t0

d⌧^{0}eg(⌧, ⌧^{0})u^{†}(⌧^{0}, t_{0})

### t

_{0}

### ⌧ t

subject to the conditions:

### u(t

_{0}

### , t

_{0}

### ) = 0, v(t

_{0}

### , t) = 0,

### v

_{ij}

### (⌧, t) ⇠ ha

^{†}

_{j}

### (t)a

_{i}

### (⌧ ) i

ü Retarded GF ü Correlated GF

d

d⌧ u(⌧, t0) + i"u(⌧, t0) + X

↵

Z ⌧
t_{0}

d⌧^{0}g_{↵}(⌧, ⌧^{0})u(⌧^{0}, t0) = 0,

### v(⌧, t) = X

↵

### Z

⌧ t_{0}

### d⌧

_{1}

### Z

tt_{0}

### d⌧

_{2}

### u(⌧, ⌧

_{1}

### ) eg

↵### (⌧

_{1}

### , ⌧

_{2}

### )u

^{†}

### (t, ⌧

_{2}

### )

### g

_{↵ij}

### (⌧, ⌧

^{0}

### ) = X

k

### V

_{i↵k}

### (⌧ )V

_{j↵k}

^{⇤}

### (⌧

^{0}

### )e

^{i}

^{R}

^{⌧ 0}

^{⌧}

^{✏}

^{↵k}

^{(⌧}

^{1}

^{)d⌧}

^{1}

eg_{↵ij}(⌧_{1}, ⌧_{2}) = X

k

V_{i↵k}(⌧_{1})V_{j↵k}^{⇤} (⌧_{2})e ^{i}

R_{⌧1}

⌧2 ✏↵k(⌧^{0})d⌧^{0}

hb^{†}_{↵k}(t_{0})b_{↵k}(t_{0})i

dissipation kernel

Nonequilibrium Fluctuation-Dissipation Theorem via v(t,t)
Equation of motion for Nonequilibrium Green Function u(t, t_{0}*) *

fluctuation kernel

Nonequilibrium Fluctuation-Dissipation Kernels

Spectral Density (level-broadening function)

Tu & WMZ, PRB78, 235311 (2008) Jin, Tu, WMZ & Yan, NJP 12, 183013 (2010) Lei & WMZ, Ann. Phys. 327, 1408 (2012)

**Connection of Master Equation with **

** Non-equilibrium Correlators for open systems: **

Relations of dissipation and fluctuation with the retarded and correlation Green’s functions

Keldysh’s Green functions

L. V. Keldysh, JETP 20, 1018 (1965)

### ha

^{†}j

### (t)a

_{i}

### (⌧ ) i ⌘ ⇢

^{(1)}ij

### (⌧, t) = ⇥

### u(⌧, t

_{0}

### )⇢

^{(1)}

### (t

_{0}

### )u

^{†}

### (t, t

_{0}

### ) + v(⌧, t) ⇤

ij

### h[a

^{i}

### (⌧ ), a

^{†}

_{j}

### (t

_{0}

### )] i

±### = u

_{ij}

### (⌧, t

_{0}

### ) ! iG ^{R} (⌧, t _{0} )

### ! iG ^{<} (⌧, t)

Jin, Tu, WMZ & Yan, NJP 12, 183013 (2010) Lei & WMZ, Ann. Phys. 327, 1408 (2012) Yang & WMZ, Front. Phys. 12, 127204 (2017)

** Reproduce NEGF transport theory: **

where

Jin, Tu, WMZ & Yan, NJP 12, 183013 (2010) Lei & WMZ, Ann. Phys. 327, 1408 (2012) Yang & WMZ, Front. Phys. 12, 127204 (2017)

Wingreen, Jauho & Meir, PRB48, 8487 (1993)

Ø We reproduce and further generalize the transient current:

### I

_{↵}

### (t) = 2e

### ~ Re Z

tt0

### d⌧ Tr n

### g

_{↵}

### (t, ⌧ )v(⌧, t) g e

_{↵}

### (t, ⌧ )u

^{†}

### (t, ⌧ )

### + g

_{↵}

### (t, ⌧ )u(⌧, t

_{0}

### )⇢

^{(1)}

### (t

_{0}

### )u

^{†}

### (t, t

_{0}

### ) o

### = 2e

### ~ Re Z

tt0

### d⌧ Tr n

### ⌃

^{R}

_{↵}

### (t,⌧)G

^{<}

### (⌧,t)

### +⌃

^{<}

_{↵}

### (t,⌧)G

^{A}

### (⌧,t) o

G^{<}(⌧, t) =i⇥

u(⌧, t_{0})⇢^{(1)}(t_{0})u^{†}(t, t_{0})+v(⌧, t)⇤

=G^{R}(⌧, t_{0})G^{<}(t_{0}, t_{0})G^{A}(t_{0}, t)
+

Z ⌧
t_{0}

d⌧_{1}
Z t

t_{0}

d⌧_{2}G^{R}(⌧, ⌧_{1})⌃^{<}(⌧_{1}, ⌧_{2})G^{A}(⌧_{2}, t).

**Weak coupling limit à Born-Markov Master Equation: **

u Kadanoff-Baym equation can be rewritten as:

### ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### ) = i✏

_{s}

### Z

t t0### d⌧ g(t, ⌧ )u(⌧, t

_{0}

### )u

^{1}

### (t, t

_{0}

### )

u Take perturbation up to the 2^{nd}-order in coupling strength:

˙u(t, t_{0})u ^{1}(t, t_{0}) ' i✏^{s}

Z t
t_{0}

d⌧

Z d!

2⇡ J(!)e ^{i(! ✏}^{s}^{)(t ⌧ )}

### ˙v(t, t) ' 2 Z

tt_{0}

### d⌧

### Z d!

### 2⇡ J(!)f (!) cos[(! ✏

_{s}

### )(t ⌧ )]

u Dissipation and fluctuation in Born-Markovian master equation:

"^{0}_{s}(t) = Im[ ˙u(t, t_{0})u ^{1}(t, t_{0})] ' !^{c}

Z t t0

d⌧

Z d!

2⇡J(!) sin[(! ✏_{s})(t ⌧ )]

(t) = Re[ ˙u(t, t_{0})u ^{1}(t, t_{0})] '
Z t

t_{0}

d⌧

Z d!

2⇡ J(!) cos[(! ✏_{s})(t ⌧ )]

e(t) ' ˙v(t, t) ' 2 Z t

t_{0}

d⌧

Z d!

2⇡ J(!)f (!) cos[(! ✏_{s})(t ⌧ )]

**H. N. Xiong, WMZ, et al. Phys. Rev .82, 012105 (2010) **

**Lindblad-GKS form of the Exact Master Equation: **

u Our exact master equation can be rewritten as:

Ø Lindblad form of the master equation has an excellent symmetric superoperator form but it mixes the dissipation and fluctuation such that the fluctuation-dissipation theorem is not manifested.

**Lei & WMZ, Ann. Phys. 327, 1408 (2012) **
**WMZ et al., PRL 109, 170402 (2012) **

d⇢(t)

dt = 1

i [ ˜H_{S}(t), ⇢(t)] + X

ij

˜_{ij}(t)L_{a}†

i,a_{j}[⇢(t)] + X

ij

[2 _{ij}(t) ± eij(t)]L_{a}

j,a^{†}_{i} [⇢(t)]

### L

_{a}

i,a^{†}_{j}

### [⇢(t)] ⌘ a

^{i}

### ⇢(t)a

^{†}

_{j}

### 1

### 2 a

^{†}

_{j}

### a

_{i}

### ⇢(t) 1

### 2 ⇢(t)a

^{†}

_{j}

### a

_{i}where

### Obtain for the first time the exact master equation

### with Lindblad-GKS form !!!

**Goes beyond the fundamental theories for open systems: **

Schwinger-Keldysh’s Green function technique J. Schwinger, J. Math. Phys. 2, 407 (1961)

* L. P. Kadanoff, & G. Baym, Quantum Statistical Mechanics (1962) *
** L. V. Keldysh, Sov. Phys. JETP, 20, 1018 (1965) **

Feynman-Vernon influence functional approach

R. P. Feynman and F.L. Vernon, Ann. Phys. 24, 118 (1963)
** A. O. Caldeira and A. J. Leggett, Physics. A 121, 587 (1983) **

### However, if there are initial correlations between the system and the environments,

n The Feynman-Vernon’s influence functional approach is not valid, and the Schwinger-Keldysh’s Green function technique is also intractable.

Ø We can solve the problem by the exact quantum Langevin equation
and also other methods.
**Tan & WMZ, Phys. Rev. A83, 032102 (2011) **

**Yang, Lin & WMZ, Phys. Rev. B92, 165403 (2015) **

Yang, Lin & WMZ, Phys. Rev. B 92, 165403 (2015)

**For partition-free initial states involving initial correlations :
**

Ø the dissipation dynamics given by is the same, only the fluctuation is modified with

### u(t, t

_{0}

### )

where

### v(⌧, t)

### v(⌧, t) = X

↵

### Z

⌧ t_{0}

### d⌧

_{1}

### Z

tt_{0}

### d⌧

_{2}

### u(⌧, ⌧

_{1}

### )[g

^{ee}

_{↵}

### (⌧

_{1}

### , ⌧

_{2}

### ) + g

^{se}

_{↵}

### (⌧

_{1}

### , ⌧

_{2}

### )]u

^{†}

### (t, ⌧

_{2}

### )

g^{ee}_{↵ij}(⌧_{1}, ⌧_{2}) = X

↵^{0}

X

kk^{0}

V_{i↵k}(⌧ )e ^{i}^{R}^{t0}^{⌧1} ^{✏}^{↵k}^{(⌧}^{0}^{)d⌧}^{0}

⇥ Vj↵^{⇤} ^{0}k^{0}(⌧_{2})e^{i}^{R}^{t0}^{⌧2} ^{✏}^{↵0 k0}^{(⌧}^{0}^{)d⌧}^{0}hc^{†}_{↵}^{0}_{k}^{0}(t_{0})c_{↵k}(t_{0})i
g^{se}_{↵ij}(⌧_{1}, ⌧_{2}) = 2i X

k

⇥V_{i↵k}^{⇤} (⌧_{2})e^{i}

R_{⌧2}

t0 ✏↵k(⌧^{0})d⌧^{0}

(⌧_{1} t_{0})hb^{†}_{↵k}(t_{0})a_{j}(t_{0})i
V_{i↵k}(⌧_{1})e ^{i}^{R}^{t0}^{⌧1} ^{✏}^{↵k}^{(⌧}^{0}^{)d⌧}^{0} (⌧_{2} t_{0})ha^{†}_{j}(t_{0})b_{↵k}(t_{0})i⇤
**Generalized fluctuation-dissipation theorem in time-domain **

### ⇢

_{tot}

### (t

_{0}

### ) = 1

### Z e

^{H}

^{tot}

**The Theory for Open Quantum Systems we have here **

u The dissipation and fluctuation coefficients in the exact master equation are microscopically and nonperturbatively determined from the Kadanoff-Baym-like equations for nonequilibrium Green functions, where the environment-induced memory kernels take into account all the back-actions from the environment.

u The dissipation and fluctuation coefficients are constrained by the nonequilibrium fluctuation-dissipation theorem so that the positivity of the reduced density matrix is guaranteed during the non-Markovian time evolution. In fact, the exact master equation has the Lindblad-GKS form but it can fully address the non-Markovain physics.

u In the weak-coupling limit, the solution of the Kadanoff-Baym equations with a perturbation expansion up to second order in the coupling constant between the system and the environment reproduces the Born-Markovian master equation from the exact master equation, and then the Lindblad one in the Markov limit.

**WMZ, et al, Phys. Rev. Lett.109,170402 (2012) **

**Outline **

² **General open quantum systems **

² **A general theory of open quantum systems **

² **General analytical solution of non-Markovianity **

² **Direct measurement of non-Markovianity **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **

**Non-Markovain dissipation and fluctuations **

u Retarded GF and dissipation:

u Correlated GF and fluctuation GF:

### e"(t) = i 2

### ⇥ ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### ) H.c. ⇤ (t) = 1

### 2

### ⇥ ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### ) + H.c. ⇤

### e(t) = ˙v(t, t) ⇥

### ˙u(t, t

_{0}

### )u

^{1}

### (t, t

_{0}

### )v(t, t) + H.c. ⇤ u(t, t

_{0}

### ) = T exp n Z

^{t}

t_{0}

### d⌧ ⇥

### i e"(⌧) + (⌧) ⇤o

### ha

^{†}j

### (t)a

_{i}

### (t) i = u(t, t

^{0}

### ) ha

^{†}j

### (t

_{0}

### )a

_{i}

### (t

_{0}

### ) iu

^{†}

### (t, t

_{0}

### ) + v(t, t)

The full solution of noise spectrum The full solution of energy spectrum

**General solution of dissipation dynamics **

u Equation of motion for retarded GF:

u Discontinuity of self-energies:

u General solution

isolated poles branch cut discontinuity

### u(z)

### z

⌃(z) = X

↵

Z d!

2⇡

J↵(!)

z !

discontinuity principal value

### d

### d⌧ u(⌧, t

_{0}

### ) + i"

_{s}

### u(⌧, t

_{0}

### ) + Z

⌧t_{0}

### d⌧

^{0}

### g(⌧, ⌧

^{0}

### )u(⌧

^{0}

### , t

_{0}

### ) = 0

z=!±i0^{+}

! (!) ⌥ iX

↵

J↵(!) 2

### u(t t

_{0}

### ) = X

i

### Z

^{i}

### e

^{i!}

^{i}

^{(t t}

^{0}

^{)}

### + X

k

### Z

B_{k}

### d!

### 2⇡

### h u(! + i0

^{+}

### ) u(! i0

^{+}

### ) i

### e

^{i!(t t}

^{0}

^{)}

dissipation

**WMZ, et al, Phys. Rev. Lett.109,170402 (2012) **

**Universality of non-Markovian dissipation **

u different open systems with different environmental spectral densities have a universal structure of the non-Markovian solution:

where the environmental-modified spectrum:

### D(!) = X

j

### Z

^{j}

### (! !

_{j}

^{0}

### ) + J(!)

### [! "

_{s}

### (!)]

^{2}

### + J

^{2}

### (!)/4

localized modes continuous spectrum part

"_{s}

### u(t t

_{0}

### ) =

### Z

_{1}

1

### d!

### 2⇡ D(!) exp{ i!(t t

_{0}

### ) }

**WMZ, Lo, Xiong & Nori, Phys. Rev. Lett.109,170402 (2012) **

**can be directly measured in experiments !! **

u general structure of two- point correlation functions.

Peskin & Schroesder,

* An introduction to quantum field *
*theory, (Westview Press, 1995) *

**Universality of non-Markovian fluctuations **

u generalized non-equilibrium fluctuation-dissipation theorem:

the steady-limit:

when :

localized modes conventional FDT part

### v(t, t ! 1) = Z

### V(!)d!

V(!) = n(!, T )J(!)h⇣ Z

! !_{b}

⌘2

+ 1

[! "_{s} (!)]^{2} + J^{2}(!)/4
i

### Z ! 0, !

^{00}

### (!) = n(!, T ) D

^{c}

### (!)

* A. O. Caldeira and A. J. Leggett, Physics. A 121, 587 (1983) *
which reproduces the early result of FDT by, e.g. Nyquist (28) , Callen

& Wilton (51), and Kubo (65), and taking high T à Einstein’s FDT.

**Lo, Xiong, WMZ, Sci. Rep. 5, 9423 (2015) ****Xiong, Lo, WMZ, Feng & Nori, Sci. Rep. 5, 13353 (2015) **

### v(⌧, t) = X

↵

### Z

⌧ t_{0}

### d⌧

_{1}

### Z

tt_{0}

### d⌧

_{2}

### u(⌧, ⌧

_{1}

### ) eg

↵### (⌧

_{1}

### , ⌧

_{2}

### )u

^{†}

### (t, ⌧

_{2}

### )

**How to measure dissipation and fluctuation **

u Physical consequences:

1. Experimental measuring *D(ω) and V(ω) à one can *
determine the spectral density *J(ω) *.

2. Knowing *J(ω)*, one can predict the dissipation and fluctuation
through *D(ω) *and *V(ω). *

3. One can also justify the dissipation-fluctuation theorem through the above relationship.

**WMZ, et al., Phys. Rev. Lett. 109, 170402 (2012) **

### D(!) = X

j

### Z

^{j}

### (! !

_{j}

^{0}

### ) + J(!)

### [! "

_{s}

### (!)]

^{2}

### + J

^{2}

### (!)/4

V(!) = n(!, T )J(!)h⇣ Z

! !_{b}

⌘2

+ 1

[! "_{s} (!)]^{2} + J^{2}(!)/4
i
l Environment-modified spectrum of the system

l Environment-induced noise spectrum of the system

**Noise spectrum and spectral density **

u Noise spectrum from current-current correlations:

Noise spectrum of a system is quantum mechanically determined by the Fourier transform of two-time correlation functions

**Ali, Lo & WMZ, NJP. 16, 103010 (2014) **

### S(!) = lim

t!1

### Z

_{1}

1

### e

^{i!⌧}

### ha

^{†}

### (t + ⌧ )a(t) i d⌧.

### = Z

^{2}

### (! !

_{b}

### ) ha

^{†}

### (t

_{0}

### )a(t

_{0}

### ) i + Z

^{2}

### J(!)¯ n(!, T ) (! !

_{b}

### )

^{2}

### + J(!)¯ n(!, T )

### [! !

_{0}

### (!)]

^{2}

### +

^{2}

### (!)

### = S

_{1}

### (!) + S

_{2}

### (!) + S

_{3}

### (!)

S_{↵↵}^{0}(!) = lim

t!1

Z _{1}

1

e^{i!⌧}h I^{↵}(t + ⌧ ) I_{↵}^{0}(t)i d⌧.

### I

_{↵}

### (t) = I

_{↵}

### (t) hI

^{↵}

### (t) i

with

**Yang, Lin & WMZ, PRB 89, 115411 (2014) **

u Noise spectrum from particle-particle correlations:

**Example: bosonic open systems**

where

u Such as nanophotonic or optomechanical resonator, etc.

u General reservoir: J(!) = 2⇡⌘!⇣ !

!_{c}

⌘s 1

exp⇣ !

!_{c}

⌘

u General solution:

and

sub-Ohmic

Ohmic

super-Ohmic (!) = 1

2[⌃(! + i0^{+}) + ⌃(! i0^{+})]

⌃(!) = 8>

>>

><

>>

>>

:

⌘!c

⇥⇡p e

!e ^{!}^{e}erfc(p
e

!) p

⇡⇤

s = 1/2

⌘!c⇥ e

! exp( !)Ei(e !)e 1⇤

s = 1

⌘!_{c}⇥
e

!^{3}e ^{!}^{e}Ei(!)e !e^{2} !e 2⇤

s = 3

### u(t) = Ze

^{i!}

^{0}

^{t}

### + 2

### ⇡

### Z

_{1}

0

### d! J(!)e

^{i!t}

### 4[! "

_{s}

### (!)]

^{2}

### + J

^{2}

### (!)

**A. J. Leggett, et al. RMP 59, 1 (1987) **

**WMZ, LO, Xiong, Tu & Nori, Phys. Rev. Lett. 109, 170402 (2012).
**

**Markovian and non-Markovian processes**

**WMZ, LO, Xiong, Tu & Nori, Phys. Rev. Lett. 109, 170402 (2012).
**

Markovian dynamics

Short-time

non-Markovian dynamics

Long-time

non-Markovian dynamics Dissipation

dynamics

Dissipation coefficient

Fluctuation

coefficient for sub-Ohmic bath

**Dissipation to dissipationless transition **

** for Ohmic-type spectral densities **

boundary line:

### ⌘

_{c}

### (!

_{c}

### ) = !

_{S}

### /[!

_{c}

### (s)]

Dissipationless regime

**Xiong, Lo, WMZ, Feng & Nori, Sci. Rep. 5, 13353 (2015) **

### ⌘

_{c}

### (!

_{c}

### )

### J(!) = ⌘!(!/!

_{c}

### )

^{s 1}

### exp( !/!

_{c}

### )

**Dissipation to dissipationless transition via temperature:**

Ø thermal-like

**General non-Markovian dissipation and fluctuations **

u In the weak coupling region:

1. The localized bound states have little contribution.

2. The non-exponential decays are reduced to exponential decays (mainly observed in Markovian dynamics).

u General solution of the spectral Green function contains two parts:

1. Localized bound states, arisen from the band gaps of the
*spectral density, J(ω)=0, à Dissipationless oscillations. *

2. Non-exponential decays, due to the nonanalyticity of the self-energy induced from the environment.

*WMZ, et al., Phys. Rev. Lett. 109, 170402 (2012) *

u Establishing a rigorous connection of the exact master equation with the Schwinger-Keldysh’s nonequilibrium Green functions à provides a new way to understand nonequilibrium physics of open quantum systems.

u Quantum dissipation and fluctuations (relaxation and dephasing) can be obtained through the computation of nonequilibrium Green functions and obey the generalized nonequilibrium fluctuation-dissipation theorem.

u Universal non-Markovianity contains localized modes plus non-exponential decays. The former comes from the band gaps of the spectral density and the latter arises from the non-analyticity of the environment-induced self-energy corrections.

u It shows that experimentally, by measuring the environment-modified spectrum of the system, one is able to understand non-Markovian effects and decoherence phenomena, and the structure of the environment.

u As long as one can compute nonequilibrium correlation Green functions in the real-time domain, one can obtain the full information about non-Markovian decoherence for more complicated open quantum systems, no need of the knowledge of master equation.

**Conclusions on General Theory of OQS:
**

**Outline **

² **General open quantum systems **

² **A general theory of open quantum systems **

² **General analytical solution of non-Markovianity **

² **Measurement of non-Markovianity (a new point of view) **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **

**Non-Markovianity measure: Memory effects **

l Math of non-Markovianity: divisibility of dynamical map, monotonic decrease of distinguishability of states using trace distance, monotonic decrease of entanglement between the system and an ancilla, negative decay rate, using mutual information

l Two-time correlation functions measuring non-Markovianity: physically, two- time correlation functions correlating a past event with its future provide direct information about the system-environment back-action processes revealing the memory dynamics.

H. P. Breuer, et al. Rev. Mod. Phys. 88, 021002 (2016)

l Physics of Non-Markovianity: characters of memory effect due to strong back action (memory effects), significant in the short and long transient regime, strong SE coupling, low temperature, finite size structured environment, etc.

M. M. Ali, P. Y. Lo, M.W. Y. Tu, WMZ, Phys. Rev. A 92, 062306 (2015)

**Two-time correlation functions: **

### N (t, ⌧) = |C(t, ⌧) C

_{BM}

### (t, ⌧ ) | ,

### C(t, ⌧ ) = hA(t)B(t + ⌧)i

### p hA(t)B(t)ihA(t + ⌧)B(t + ⌧)i

directly measurable in experiments or
calculated with exact master equation
l ** A quantitative measure of Non-Markovianity:
**

### C

_{BM}

### (t, ⌧ )

where is the correlation without memory effect, which can be calculated under Quantum Regression Theorem (QRT),

and QRT is valid only under Born-Markov approximation so that it can be calculated from the Born-Markov master equation under

QRT.

M. M. Ali, P. Y. Lo, M.W. Y. Tu, WMZ, Phys. Rev. A 92, 062306 (2015)

Different curves for different system-bath coupling strengths

**A bosonic system in a thermal bath
**

### …………

### incomplete in general!

### Te mp era tu re d ep en de nce o f t he non -Ma rko vi an ity

### In iti al st at e de pe nd en ce o f t he non -Ma rko vi an ity

**Conclusion on non-Markovianity: **

### l Short-time non-Markovian memory effect, known for long time ago,

### l Long-time non-Markovian memory effect, just

### discovered recently in our theoretical framework, l Non-Markovian memory effect is decreased with

### increasing the temperature of the reservoir.

### l Non-Markovian memory effect depends sensitively on initial states, in particular, for the long-time non- Markovianity.

M. M. Ali, P. Y. Lo, M.W. Y. Tu, WMZ, Phys. Rev. A 92, 062306 (2015)
**A. O. Caldeira and A. J. Leggett, Physics. A 121, 587 (1983) **

**Outline **

² **General open quantum systems **

² **A general theory of open quantum systems **

² **General analytical solution of non-Markovianity **

² **Measurement of non-Markovianity **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **

### …………

new

non-Markovian

è Solving the exact master equation with arbitrary initial state

**Xiong, Lo, WMZ, Feng and Nori, Sci. Rep. 5, 13353 (2015) **

**Nonequilibrium and Equilibrium Physics:**

(Equilibrium Dynamics)

(Nonequilibrium Dynamics)

**Photon dynamics in cavities: **

### ha(t)i = ha(t

^{0}

### ) ie

^{(i!}

^{C}

^{+)(t t}

^{0}

^{)}

### ! 0

### n(t) = n(t

_{0}

### )e

^{2(t t}

^{0}

^{)}

### + n(!

_{C}

### , T )[1 e

^{2(t t}

^{0}

^{)}

### ]

### ! n(!

^{C}

### , T )

Solving the master equation, one can find

ß fluctuations

ß dissipation

u Bose-Einstein distribution à thermal equilibrium

**Photonic band gap structures: **

*J. D. Joannopoulos, et al, Photonic crystals, Modeling the Flow of Light *

Spectral density

**Lo, Xiong & WMZ, Sci. Rep. 5, 9423 (2015) **

**Photonic dissipative dynamics **

**b**
**a**

«u+tt_{}/«

Z* _{e}*t

^{+}

^{}

^{}

^{/}

G Z*e*

G Z* _{e}*
G

G Z* _{e}* G Z

*e*

'3&'3&'3&

**c** **b** **a** ha (t)i = u (t, t

^{0}

### ) ha (t

^{0}

### ) i

**Lo, Xiong & WMZ, Sci. Rep. 5, 9423 (2015) **

**Photonic fluctuation dynamics: **

(i) (i)

(ii) (ii)

(iii) (iii)

**a** **b**

n (t) = ⌦

a^{†} (t) a (t)↵

= |u (t, t0)|^{2} n (t_{0}) + v (t, t)

### crossover

### crossover

### critical

### transition

**a** **b** **c**

### (i)

### (ii)

### (ii) (ii)

### (iii) (iii)

### (iii)

**Thermalization to Localization transition for initial Fock states : **

**Lo, Xiong & WMZ, Sci. Rep. 5, 9423 (2015) **

**a** **b** **c**

### (i)

### (ii)

### (ii) (ii)

### (iii) (iii)

### (iii)

### (i) (i)

**Lo, Xiong & WMZ, Sci. Rep. 5, 9423 (2015) **

**Thermalization to Localization transition for initial coherent states : **

**Defeats in lattice structures
**

Y. W. Huang and WMZ (2017)

### Sorry, this part is omitted because the result has

### not be published yet……..

**Nonequilibrium photon statistics:
**

### Photon bunching and antibunching statistics are usually characterized by the steady-state ( * ) * second-order correlation function , where an increasing (decreasing) magnitude of with delay-time τ demonstrate antibunching (bunching) statistics of photons.

### g

^{(2)}

### (t, t + ⌧ ) = ha

^{†}

### (t)a

^{†}

### (t + ⌧ )a(t + ⌧ )a(t) i ha

^{†}

### (t)a(t) iha

^{†}

### (t + ⌧ )a(t + ⌧ ) i

### g

^{(2)}

### (t, t + ⌧ ) g

^{(2)}

### (t, t + ⌧ )

### t ! 1

### We studied the exact nonequilibrium transient dynamics of photon statistics for a micro-cavity coupling with a thermal reservoir with Ohmic spectral density.

M. M. Ali and WMZ, Phys. Rev. A 95, 033830 (2017)

In short-time non-Markovian regime, the steady state is a thermal state:

In long-time non-Markovian regime, the steady state cannot be

thermalized

M. M. Ali and WMZ,

Phys. Rev. A 95, 033830 (2017)

l For an initial Fock state

for different transient time

for different temperature

New type of phase transition of photon statistics occurs

at a critical value while passing through weak to strong system-reservoir coupling

**M. M. Ali and WMZ, Phys. Rev. A 95, 033830 (2017) **

**Outline **

² **General open quantum systems **

² **A general theory of open quantum systems **

² **General analytical solution of non-Markovianity **

² **Measurement of non-Markovianity **

² **Non-Markovian nonequilibrium physics **

² **Non-Markovian entanglement decoherence **

² **Non-Markovian decoherence of Majorana qubits **

² **Transient quantum transport and quantum control **

² **Summary **