1

• ☐Quantum Distribution Theory• ☐Ordering • ☐Normally Ordering, P-representation (Glauber-Sudarshan)

• ☐Anti-Normally Ordering, Q-representation (Husimi)

• ☐Symmetrically Ordering, W-representation (Wigner-Weyl)

• More on Phase Space• ☐Fokker-Planck equation• ☐Quantum Liouville equation• ☐Moyal function • ☐Homodyne detection• ☐Wigner Flow

Note: Phase Space

2

Number (Fock) states

|0i |n = 1i

non-classical states

negative probability withLudmilaPraxmeyer

single-photon

3

Coherent states

withPopoYang

PopoYang,IvanF.Valtierra,AndreiB.Klimov,Shin-TzaWu,RKL,LuisL.Sanchez-Soto,andGerdLeuchs,PhysicaScriptafortheNewFocusissue:QuantumOpticsandBeyond-inhonourofWolfgangSchleich.

wave-nature

4withPopoYang

PopoYang,IvanF.Valtierra,AndreiB.Klimov,Shin-TzaWu,RKL,LuisL.Sanchez-Soto,andGerdLeuchs,PhysicaScriptafortheNewFocusissue:QuantumOpticsandBeyond-inhonourofWolfgangSchleich.

Non-classicality

Cat States in Phase Space

5

Cat States in Phase Space

SCHRöDINGER’S CAT STATE

byLudmilaPraxmeyer

6

Squeezed StatesCoherent to Squeezed states

Courtesy: Roman Schnabel (2017).

byPopoYang

7withPopoYang,OleSteuernagel(U.Hertfordshire,UK)

Quantum Optics in Phase Space

8

Real Spectrum in PT Hamiltonian

9

L.Praxmey,PopoYang,andRKL,Phys.Rev.A93,042122(2016).

• Consider a family of di↵erential equations parameterized by a continuous

parameter ✏ > 0 in the form:

@2 (x)

@x2+ V✏(x) (x) + 2E (x) = 0, (1)

• Here, let us specify the definition of V✏(x) = �(ix)✏, by stating explicitly

the branch of logarithm :

V✏(x) = �(ix)✏ = e✏ log(ix) =

8><

>:

�|x|✏⇥cos(✏⇡2 ) + i sin(✏⇡2 )

⇤, for x > 0;

0, for x = 0;

�|x|✏⇥cos(✏⇡2 )� i sin(✏⇡2 )

⇤, for x < 0 .

(2)

• In a Fock state basis, an analytical formula for the matrix element anm(✏) =

hm|H✏|ni of H✏ =12

@2

@x2 +V✏(x)

2 can be constructed for any natural number

n, m and positive ✏:

anm(✏) =

pn(n�1)

4 �m,n�2 +

p(n+1)(n+2)

4 �m,n+2 � 2n+14 �m,n +

+

h1�(�1)en+fm

4 cos(✏⇡2 ) +1+(�1)en+fm

4 i sin(✏⇡2 )i(�1)b

n2c+bm

2c2en+emn!m!

bn2c! bm2 c!

⇥

⇥ ��1+✏+en+em

2

�FA�1+✏+en+em

2 ;�bn2c,�bm2c;2en+1

2 , 2em+12 ; 1, 1

��m,n (3)

where � is an Euler gamma function; FA is a Lauricella hypergeometric

function; symbol b c denotes a floor function: bkc is the largest integer not

greater than k; character tilde e denotes a binary parity function: ek is 0

for an even k and 1 for an odd k.

• By using truncated Fock state basis, we diagonalize the matrix Mnn(✏)numerically, having truncated the basis to the first 31, 51, or 71 elements.

<latexit sha1_base64="m0ejowKuN9yERl1stZqgTJuIm9Y=">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</latexit>

10

PT in Phase space: Ground states

11

PT in Phase space: at the Exceptional Point

12

Phase space: Wigner flow

O.Steuernagel,D.Kakofengitis,andG.Ritter,Phys.Rev.Lett.110,030401(2013).

13

Wigner flow of PT : Ground states

L.Praxmey,PopoYang,andRKL,Phys.Rev.A93,042122(2016).

14

Wigner flow of PT : 1st/2nd Excited states

L.Praxmey,PopoYang,andRKL,Phys.Rev.A93,042122(2016).