Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
b
a
c
phase-shift ϕ a b=aeiϕ
ϕ →
a b
parametric down-conversion a ηb c
b → i
a
NL
c
parametric up-conversion a b ηc
a c i| →
NL
b
amplification A a b;G,N
G,N i →| i
a b
Table 1: “Toolbox” of lossless elements for quantum interference devices.
20
S 1 b M
✓✏
L
a
✒✑
c P ϕ
S
2 D
1
d ✟
✠
M e
D
✡ ✠2
Figure 3: Mach-Zehnder interferometer. A single quantum (photon, neutron, electron etc) is
emitted in L and meets a lossless beam splitter (half-silvered mirror) S , after which its wave
1
function is in a coherent superposition of b and c. In beam path b a phase shifter shifts the phase
of state b by ϕ. The two beams are then recombined at a second lossless beam splitter (half-
silvered mirror) S . The quant is detected at either D or D , corresponding to the states d and
2 1 2
e, respectively.
In present-day quantum optical nonlinear devices (NL), parametric up- or down-conversion, i.e.,
the production of a single quant (particle) from two field quanta (particles) and the production of
two field quanta (particles) from a single one occurs at the very low amplitude rate of η 10 6.
−
≈
In what follows, a lossless Mach-Zehnder interferometer drawn in Fig. 3 is discussed. The
computation proceeds by successive substitution (transition) of states; i.e.,
S : a (b+ic)/√2 , (83)
1
→
P : b beiϕ , (84)
→
S : b (e+id)/√2 , (85)
2
→
S : c (d+ie)/√2 . (86)
2
→
The resulting transition is
eiϕ+1 eiϕ 1
a ψ =i d+ − e . (87)
→ 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
Assume that ϕ=0, i.e., there is no phase shift at all. Then, equation (87) reduces to a id, and
→
the emitted quant is detected only by D . Assume that ϕ = π. Then, equation (87) reduces to
1
a e,andthe emittedquantisdetectedonlybyD . Ifonevariesthephaseshiftϕ,oneobtains
2
→−
the following detection probabilities:
ϕ ϕ
P (ϕ)= (d,ψ)2 =cos2( ) , P (ϕ)= (e,ψ)2 =sin2( ) . (88)
D1 D2
| | 2 | | 2
For some “mindboggling” features of Mach-Zehnder interferometry, see [12].
D Universal 2-port quantum gate
The elementary quantum interference device Tbs depicted in Fig. (4.a) is just a beam splitter
21
followedbyaphaseshifterinoneoftheoutputports. Accordingtothe“toolbox”rulesofappendix
C, the process can be quantum mechanically described by17
P : 0 0eiα+β , (89)
1
→
T(ω) iR(ω)
17 Alternatively, the action of a lossless beam splitter may be described by the matrix =
iR(ω) T(ω)
(cid:18) (cid:19)
cosω isinω eiϕ 0
. A phase shifter in twodimensional Hilbert space is represented by either or
isinω cosω 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
1 0
. Theactionoftheentiredeviceconsistingofsuchelements iscalculatedbymultiplyingthematrices
0 eiϕ
(cid:18) (cid:19)
inreverseorderinwhichthequantapasstheseelements[98,91].
21