Note: Phase Space
Transcript of Note: Phase Space
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• ☐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
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Number (Fock) states
|0i |n = 1i
non-classical states
negative probability withLudmilaPraxmeyer
single-photon
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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
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Cat States in Phase Space
SCHRöDINGER’S CAT STATE
byLudmilaPraxmeyer
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Squeezed StatesCoherent to Squeezed states
Courtesy: Roman Schnabel (2017).
byPopoYang
7withPopoYang,OleSteuernagel(U.Hertfordshire,UK)
Quantum Optics in Phase Space
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Real Spectrum in PT Hamiltonian
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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.
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PT in Phase space: Ground states
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PT in Phase space: at the Exceptional Point
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Phase space: Wigner flow
O.Steuernagel,D.Kakofengitis,andG.Ritter,Phys.Rev.Lett.110,030401(2013).
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Wigner flow of PT : Ground states
L.Praxmey,PopoYang,andRKL,Phys.Rev.A93,042122(2016).
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Wigner flow of PT : 1st/2nd Excited states
L.Praxmey,PopoYang,andRKL,Phys.Rev.A93,042122(2016).