Nonlocal control of the quantum state of light

TL;DR

This paper explores controlling the quantum state of light remotely through entanglement and quantum non-demolition techniques, enabling manipulation of quantum states without destroying their coherence.

Contribution

It introduces a method for nonlocal control of light's quantum state using entanglement and quantum non-demolition impact, advancing remote quantum state manipulation.

Findings

Demonstrates remote quantum state control of light

Utilizes entanglement for nonlocal manipulation

Employs quantum non-demolition impact for state preservation

Abstract

The work is dedicated to the challenge: control and manipulation of the quantum state of a remote physical object. The goal is approached via the entanglement channel between the object and a local physical system - actuator - subjected to a quantum non-demolishing impact. Physical systems under exploration are light beams.

Tables

Table 1. 4.1: / .

$n$	$p\left[0\mathrm{clicks}\right]$	$p\left[1\mathrm{click}\right]$	$p\left[2\mathrm{clicks}\right]$
0	${\left(1-\eta \right)}^0$	0	0
1	${\left(1-\eta \right)}^1$	$\eta$	0
2	${\left(1-\eta \right)}^2$	$2\eta -\frac{3{\eta }^2}{2}$	$\frac{{\eta }^2}{2}$
3	${\left(1-\eta \right)}^3$	$3\eta -\frac{9{\eta }^2}{2}+\frac{7{\eta }^3}{4}$	$\frac{3{\eta }^2}{2}-\frac{3{\eta }^3}{4}$

Equations

$$\hat{a}=\mu {\hat{a}}_1+\lambda {\hat{a}}_2,$$

$${\hat{a}}_1{|\psi ⟩}_{\hat{a}}=({\mu }^{\ast }\hat{a}+\lambda {\hat{a}}_{\perp }){|\psi ⟩}_{\hat{a}}={\mu }^{\ast }\hat{a}{|\psi ⟩}_{\hat{a}},$$

$${\hat{a}}_1{|N⟩}_a\propto {|N-1⟩}_a.$$

$$\frac{\partial \hat{\rho }}{\partial z}\propto {\hat{a}}_1\hat{\rho }{\hat{a}}_1^{†}-\frac{\hat{\rho }{\hat{a}}_1{\hat{a}}_1^{†}+{\hat{a}}_1{\hat{a}}_1^{†}\hat{\rho }}{2},$$

$${\hat{a}}_1^{†}{|N⟩}_a/\propto {|N+1⟩}_a.$$

$${\hat{\rho }}_{\mathrm{Bob}}=$$

$$=$$

$${\hat{\rho }}_{\mathrm{Bob}}=$$

$$=$$

$$p_1=\frac{R_1}{\eta \nu }\approx \frac{1}{150},$$

$$\frac{R_2}{\nu }=p_1^2\ast \frac{{\eta }^2}{2}=2.1\times 10^{-7},$$

$$\hat{H}=i\lambda {\hat{a}}_1({\hat{a}}_1^{\prime }{)}^{†}-i\lambda {\hat{a}}_1^{†}{\hat{a}}_1^{\prime },$$

$${\hat{B}}_2=\exp \left[-i\hat{H}\tau /\hslash \right]\approx 1-i\hat{H}\tau /\hslash .$$

$${\hat{B}}_2\left[|\psi ⟩\otimes |0⟩\right]\approx |\psi ⟩\otimes |0⟩+\zeta \left({\hat{a}}_1|\psi ⟩\right)\otimes |1⟩.$$

$$(1-t^2){⟨\hat{n}⟩}_{\psi }\ll 1,$$

$${\hat{\Pi }}_{\text{off}}(\eta )$$

$${\hat{\Pi }}_{\text{on}}(\eta )$$

$${\mathrm{Tr}}_{a_1^{\prime }}\left[{\hat{\Pi }}_{\text{on}}{\hat{B}}_2(\hat{\rho }\otimes |0⟩⟨0|){\hat{B}}_2^{†}\right].$$

$$\mathcal{F}\left(\hat{\rho },{\hat{\rho }}_{\mathrm{ideal}}\right)={\left(\mathrm{Tr}\left[\sqrt{\sqrt{\hat{\rho }}{\hat{\rho }}_{\mathrm{ideal}}\sqrt{\hat{\rho }}}\right]\right)}^2.$$

$$|\Psi ⟩=\frac{|10⟩+|01⟩}{\sqrt{2}}.$$

$${\hat{\rho }}_{\mathrm{on}}=\mathrm{T}{\mathrm{r}}_{\mathrm{A}}\left[{\hat{\mathrm{\Pi }}}_{\mathrm{on}}|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|\right]=\frac{\kappa }{2}|0⟩⟨0|,$$

$$\Pi_{\rm on}=\left[{\begin{array}[]{cc}0&0\\ 0&\kappa\\ \end{array}}\right].$$

$${\hat{\rho }}_{\mathrm{off}}=\mathrm{T}{\mathrm{r}}_{\mathrm{A}}\left[{\hat{\mathrm{\Pi }}}_{\mathrm{off}}|\mathrm{\Psi }⟩⟨\mathrm{\Psi }|\right]=\frac{1}{2}|1⟩⟨1|+\frac{1-\kappa }{2}|0⟩⟨0|,$$

$${\hat{\Pi }}_{\mathrm{off}}=\hat{1}-{\hat{\Pi }}_{\mathrm{on}}.$$

$$⟨{\hat{n}}_{\mathrm{Bob}}⟩=\mathrm{Tr}\left[{\hat{n}}_{\mathrm{Bob}}{\hat{\rho }}_{\mathrm{on}}\right]+\mathrm{Tr}\left[{\hat{n}}_{\mathrm{Bob}}{\hat{\rho }}_{\mathrm{off}}\right]=\frac{1}{2},$$

$${⟨{\hat{n}}_{\mathrm{Bob}}⟩}_{\mathrm{Alice}\mathrm{ON}}=$$

$$\mathrm{Tr}\left[{\hat{n}}_{\mathrm{Bob}}{\left(\hat{1}\right)}_{\mathrm{Alice}}|\Psi ⟩⟨\Psi |\right]={⟨{\hat{n}}_{\mathrm{Bob}}⟩}_{\mathrm{Alice}\mathrm{OFF}}.$$

$${\hat{U}}^{\ast }={\hat{U}}^{-1}.$$

$${\Psi }_{\zeta }^{\mathrm{TMS}}(X_a,X_b)=\frac{1}{\sqrt{\pi }}\exp \left[-{\left(\frac{X_a-X_b}{2e^{-\zeta }}\right)}^2\right]\ast \exp \left[-{\left(\frac{X_a+X_b}{2e^{+\zeta }}\right)}^2\right],$$

$$\mathcal{F}=\mathrm{Tr}{\left(\left[\sqrt{\sqrt{{\hat{\rho }}_{\mathrm{th}}}{\hat{\rho }}_{\exp }\sqrt{{\hat{\rho }}_{\mathrm{th}}}}\right]\right)}^2=99.9\%.$$

$$\overline{⟨{\Psi }_{\zeta }^{\mathrm{TMS}}\left|{\hat{Q}}_{\theta }^2\right|{\Psi }_{\zeta }^{\mathrm{TMS}}⟩}=(1-R)\frac{e^{-2\zeta }+e^{2\zeta }}{4}+\frac{R}{2}\approx \mathrm{0.6.}$$

$$⟨{\hat{b}}^2⟩⩾g^2⟨{\hat{a}}^2⟩+\frac{|g^2-1|}{2}.$$

Full text

1. 1. 0.1

2. 0.2

3. 0.3

4. 0.4

5. 0.5

6. 0.6

7. 1 :

8. 1.1

9. 1.2

10. 1.2.1

11. 1.2.2

12. 1.2.3

13. 1.3

14. 1.3.1

15. 1.3.2

16. 1.4

17. 1.5

18. 2 – –

19. 2.1 - 1

20. 2.1.1

21. 2.1.2

22. 2.1.3

23. 2.2

24. 2.2.1

25. 2.2.2 –

26. 2.3

27. 2.3.1

28. 2.3.2

29. 2.4

30. 2.4.1

31. 2.4.2

32. 2.4.3 ´

33. 2.4.4

34. 2.5

35. 2.6

36. 3

37. 3.1

38. 3.2

39. 3.2.1

40. 3.2.2

41. 3.3

42. 3.3.1

43. 3.3.2

44. 3.4

45. 4

46. 4.1

47. 4.1.1

48. 4.1.2

49. 4.1.3

50. 4.1.4

51. 4.2

52. 4.2.1

53. 4.2.2

54. 4.2.3

55. 4.3 :

56. 4.3.1 –

57. 4.3.2 –

58. 4.3.3

59. 4.3.4 – –

60. 4.3.5

61. 4.4

62. 4.4.1

63. 4.4.2 -

64. 4.4.3 2

65. 4.4.4 1

66. 4.5 .

67. 4.5.1

68. 4.5.2

69. 4.6

70. 4.6.1

71. 4.6.2

72. 4.6.3

73. 4.6.4

– – , - . , . . .

0.1

– , , [1], [2] [3] – [4] “” . [5], [6], [7].

1920- [8, 9]. – (1945 .), (1947 .) (1954 .) – . , – , – , . , . : “* . , , ; … , *” [10]. , , ; , .

20 . , , , . , , . , - .

“” “” . , . , , : , . , , - “”, “ ” (. quantum - ); , , . [11, 12]; “” , – – .

, : , , “ ”, . , , () . . , “, , . ” [13].

, . ? ? . - , , – – [14, 15, 16, 17, 18, 19, 20, 21].

0.2

, : [22, 23], [24], [25], “ ” [26]. . : . – – , . “, , ” [27] – .

– – , . , – [28], ´ .

( , ), . – , , [29, 25, 30]. , : . , , [32, 33] …

– – , – , [11, 34]. – – , ; [35].

, , , , . , , . , , ; , .

. , , [36]. [37]. ; , . , , , [36, 37, 38, 39].

. , , ; , . , , – . “ ” – [40, 9]. , , , [41, 42, 43].

, ; - - . , [44, 45, 46], [47, 48, 49, 50, 51, 52, 53, 54], : $N$- [55, 56, 57], [58, 59], .

•  *  *

; . – – , . ; , [60, 61]. . , [62, 63, 64, 65] [66, 67, 56, 57, 68, 69, 70, 71].

. ; [72]. - – , , , – , [73, 74, 75]. - .

0.3

, :

1. 1.

**.

** . , . . , -, . , . 2. 2.

**—-.

** , . [60, 61], [76]; , , . 3. 3.

**.

** , ; [81]. . , , , .

4, , - . . (1.5, 2.6, 3.4).

0.4

1. 1.

**.

** : / . , - , . . , , [77] , . 2. 2.

**—-.

** [78, 79]. , - – , . - [80]. 3. 3.

**.

** . : – [73], – [75, 81], , – [74]. .

0.5

. - , - . 2. 2.

• • . -, , $\approx 0.6$. , - 0.65dB, 5 20 . , , .

; . , –. 5%, 95%.

0.6

, .. , .. , .. , .. , .. , .. . , / , .

1 :

• . , – . * *, ; , , – – .

. , , . , (. 4.3); , – , . [82, 83, 33]; , , , . : .

, - - [84]. , , [40, 9, 85]. : [86], [87] (. ). , – .

• [88], - [77], . . , , . -, , . , ; .

1.1

$\left|\psi\right\rangle$, , $\hat{a}$. , ( ) . , ( ), $\hat{a}_{1}$ $\hat{a}_{2}$,

[TABLE]

$|\mu|^{2}$ $|\lambda|^{2}$ [. 1.1]. $\left|\psi\right\rangle$ – , – [89].

.

[TABLE]

$\hat{a}_{\perp}=\lambda^{*}\hat{a}_{1}-\mu^{*}\hat{a}_{2}$ , $\hat{a}$. , . , , , $\left|\psi\right\rangle$, . $\hat{a}$ $N$, (1.2)

[TABLE]

• , $\hat{a}$, . 1.2. $1$. , . $\hat{a}_{1}$, . , – . , (1.2), ; , $\hat{a}$ . , , . , .

, . , , .

[TABLE]

$\hat{\rho}$ $z$ , , . , , , :

[TABLE]

. (1.2): $\hat{a}$ $\hat{a}^{\dagger}$, $\lambda\hat{a}_{\perp}^{\dagger}\left|\psi\right\rangle_{\hat{a}}$ .

, , . 1.2 . , , ( , , [91, 92, 93]). , , , $\left|\psi\right\rangle$ . , . , , – , .

1.2

1.2.1

$\hat{a}$ - - , . $45^{\circ}$ . , $\hat{a}_{1}$ $\hat{a}_{2}$, (1.1) $\mu=\lambda=1/\sqrt{2}$. , , () - (. 1.1). $\hat{a}_{1}$ $\hat{a}_{2}$ , ( ) ( ) ; , , , , . 1.1.

, - . , , $45^{\circ}$. $\approx 60^{\circ}$, ($>99.8\%$), 6% - . 3, $\hat{a}_{1}$.

• $\lambda/4$ $\lambda/2$. , , . , - . , $\hat{a}_{1}$ $\hat{a}_{2}$, $\hat{a}_{1}$, $\hat{a}$. , $\hat{a}$ .

, - $\hat{\rho}_{\mathrm{Bob}}$ ; - . – -, - $\hat{a}$ – - . , $\hat{\rho}_{\mathrm{out}}$, . , - -, $\hat{\rho}_{\mathrm{out}}$

[TABLE]

[TABLE]

, (1.3) , – $\left|0\right\rangle$ $\left|1\right\rangle$. , - (1.3).

1.2.2

. 1.4. $\hat{a}$ () . , () (. 4.3), 1.

$\hat{a}$ 4.4.3 4.6.1. Excellitas, . , 5-6 14-15 (. 1.4), DFG, $\hat{a}$ .

, DFG , 1 2, . : , 50%. , , . – – (. 4.6.2).

1 2 $R_{1}=5\times 10^{4}$ . ,

[TABLE]

$\eta=0.12$, $\nu=76$ – . (. 4.6.2)

[TABLE]

$R_{2}\approx 16$ .

1.2.3

, 3. [77, 94]. : $\hat{B}_{2}$ (. 1.1) [95]

[TABLE]

$\hat{a}^{\prime}_{1}$ , $\lambda$ $t$ $t=\cos(\zeta)$ $\zeta=\lambda\tau/\hbar$, $\tau$ . $|\zeta|\ll 1$,

[TABLE]

$a^{\prime}_{1}$ , $a$ $\left|\psi\right\rangle$, (1.11)

[TABLE]

$a^{\prime}_{1}$, $a_{1}$ $\hat{a}_{1}\left|\psi\right\rangle$.

, $\hat{a}_{1}$ , 3 1 / 2. , , . , - , .

, 3 ; (1.11) 4.6.2.

[TABLE]

$\left\langle\hat{n}\right\rangle_{\psi}$ .

(1.13) , / 3 POVM (4.119), [96]

[TABLE]

$\eta$ , $p_{\rm bg}$ $p_{\rm sub}$ 3. 3, $a_{1}$ $\hat{\rho}=\left|\psi\right\rangle\!\left\langle\psi\right|$

[TABLE]

1.3

Agilent DSO9254A. 1–3. , $\hat{a}$ , 1 , 2 1 . , . 500 , $2\times 10^{-10}$ . 3 . 1.5. , , ; , . 4.2.2.

1.3.1

40 , 1,2 3, . 1.6. , , , $\eta=53\%$. (. 4.6.4). (. 4.2.3), $\eta$ , $3\%$ : - $\hat{a}$.

3 1 2 ; $\sim 10$ . 12600 . $53\%$ – . . $\hat{a}$ , (1.3). (1.6), .

, $\hat{a}$ . 1 2. , .

1.3.2

$10^{4}$ (4.6.4). , , , ; 3 1 2. . (1.2).

, (. 1.6, N=2) 3 $p_{\rm bg}=6\times 10^{-4}$ , 3, 1 2, $p_{\rm sub}=1.2\times 10^{-2}$. , $p_{\rm bg}/p_{\rm sub}=5\%$ 3 ; , $\sim 5\%$ , . 1.6, N=2, .

. 1.7

[TABLE]

(1.15) (1.2) ( . 1.6). , . , . , $t$ 3, $t$.

$p_{\rm bg}/p_{\rm sub}=0.05$, , ($\mathcal{F}=0.965$) $t=0.07$. $t$ : $t\rightarrow 0$ $\mathcal{F}\rightarrow 0.85$.

1.4

, , . , , . , ?

,

[TABLE]

$\kappa$. ,

[TABLE]

$\hat{\Pi}_{\rm on}$ (4.119), , ,

[TABLE]

(1.18) , $\kappa/2$.

, .. $1-\kappa/2$, . ,

[TABLE]

$\hat{\Pi}_{\rm off}$ -

[TABLE]

(1.18) (1.20) , :

[TABLE]

(1.17).

, , $r$ $\eta$, $\kappa=r\eta\ll 1$. (1.18), , , ; , (1.20). , (1.22).

(1.22) $\Psi$:

[TABLE]

, , ; , , , - .

$\hat{U}$ :

[TABLE]

, , [94]. , – – [98, 99].

1.5

, , ; : , , , .., . . -, . (1.2), .

; , , . [50, 51, 61, 58]. , [100, 101]. , , , .

$\hat{a}_{1}$ $\hat{a}_{2}$, , $\left|\psi\right\rangle_{a}$ . , $\left|\alpha\right\rangle_{a}$, $\hat{a}_{1}$ $\hat{a}_{2}$ $\mu\alpha$ $\nu\alpha$, . , $\hat{a}_{1}$ $\hat{a}_{1}$, $\hat{a}$, . , , – , [102]. , $\hat{a}_{1}\tilde{\rho}_{N}\hat{a}_{1}^{\dagger}$ - $\tilde{\rho}_{N-1}$, $\tilde{\rho}_{N}$ $\left|N\right\rangle_{a}\!\left\langle N\right|$. , : / . , - : .

:

Ilya A. Fedorov, Alexander E. Ulanov, Yury V. Kurochkin, and A.I. Lvovsky, Quantum vampire: collapse-free action at a distance by the photon annihilation operator. Optica 2 (2), 112-115 (2015) [103].

:

Quantum 2014 – Workshop ad memoriam of Carlo Novero, 25–31 2014, , 2. 2.

23rd annual International Laser Physics Workshop, 14–18 2014, , 3. 3.

Quantum Optics VII, 27–31 2014, —-, 4. 4.

Third International Conference on Quantum Technologies, 13–17 2015, ,

2 – –

. , , , (. . 1). - , (. 4.3.1)

[TABLE]

$X_{a}$, $X_{b}$ , $\zeta$ – . (2.1) . / . / . . [40, 9], - $a$ $b$, (2.1).

, [104, 105], (2.1) ( – ) , . – [106], [107], [108], [109], , [110, 111].

-. , , – . , . , [78, 79, 80]. , [112, 113] – , (2.1) , .

, - , . .

2.1 - 1

, - , , .

1 . , - [114], - [115]. , , -, .

• -  .  ,   . [4.3.4](#Ch4.S3.SS4),           ,           - ([2.1](#Ch2.E1)).
  

  [116] .

2.1.1

. 2.1. - (. 4.3) - , 2.95 (1 2). 1, , (. 4.4.4).

, $90^{\circ}$. , , , . , . 2.2. - .

, - - , - - , . 3, .

, – “” (. seed – ), , . 4.4.4. 1 2, .. . , ; , $\approx 10\%$ .

, - , 2 . , $\sim 1$ , .

, , . . 4.14. , . .

• , , , . $45^{\circ}$, . . . , . 0.7 , 0.85 , $57\%$ (4.15), 1.5 . $w_{0}=14$ $15\%$. . 2.2.

• , , 2 3 , 0.62 . , (. . 4.4.2). $\mathrm{CaCO_{3}}$ () $1.8$ , . , , .

• , - . , , , , . $22.5^{\circ}$. (. . 2.3 2.5).

. , , . , 12–14 16–18 , ( . 4.8, $i=j=0$), , (. 4.3.3). , - . , 12–14 16–18 , - .

1 2 (. 4.2). , . 2 3 , 12–14 16–18 .

2.1.2

[117, 118] (. 4.3.1 4.3.2):

.

(. [4.6.4](#Ch4.S6.SS4)),         ([4.32](#Ch4.E32)).   ,        .

2. 2.

(. 4.7 4.6).

  (. [4.6.4](#Ch4.S6.SS4)),         ;         ,     .

3. 3.

-.

, .

. 2.3 $10^{6}$ , . (4.53) , - 3. $\approx 0.35$, $1.5\mathrm{dB}$ (4.56) ($1.6\mathrm{dB}$, . 4.5).

(4.54), . 2.3 $\theta$, (. 4.2), . , - , ; , / . , . 2.3 . . 2.4.

, (4.84). $\left|1\right\rangle\left\langle 1\right|$ $\left|2\right\rangle\left\langle 2\right|$ $R$ (4.84) $0.5$, (. . 4.3.5). , , . 2.4. $\zeta=-0.44$, , c

[TABLE]

. 2.5 $1.33\times 10^{6}$ , - . , - 3. , . . , (4.41).

, . 2.5 , - . , $\theta_{1}+\theta_{2}=0$ ($\pi$) () (4.40).

. $0.36$, 1.4dB (4.80) . , , , . (. 4.3.5) $\zeta=-0.44$ $1-R=0.52$, . 2.1.2:

[TABLE]

• , . (4.85) , . 2.5, $\theta_{1}+\theta_{2}$ .

• . 2.6. , () (4.86, ), - (. 2.1.2): $\zeta=-0.44$ $\tau_{1}^{2}=\tau_{2}=0.52$. . (2.2) $99.5\%$.

2.1.3

•         1.       ,     - ,    (. [4.5](#Ch4.S5)).    2.36dB  ,    1.5dB,        2.
  

, . , . , , – $\approx 80$ ; , , . , , , .

, - . 4.3.1, - , (). , 1, - .

• : Ilya A. Fedorov, Alexander E. Ulanov, Yury V. Kurochkin, and A.I. Lvovsky, Synthesis of the Einstein–Podolsky–Rosen entanglement in a sequence of two single-mode squeezers. Optics Letters 42 (1), 132-134 (2017) [119].

2.2

2.2.1

[120, 121, 122]. , , . , $g$ , $|g^{2}-1|$ [123, 124]. $\hat{a}$ $\hat{b}$,

[TABLE]

$\hat{a}$ $\hat{b}$

[TABLE]

, .

(2.4) , .. , . , , , (2.4) - .

: , , . , $P<1$. , , :

[TABLE]

$\left|\emptyset\right\rangle$ . , , (2.6) () .

[TABLE]

$\left|\rm ON\right\rangle$ $\left|\rm OFF\right\rangle$ – .

[125, 126]. “ ” [127], (2.7) $\alpha\ll 1$,

[TABLE]

, . , , - (2.1) , . , (2.8) (4.86) $\tau_{1}=1$

[TABLE]

$g=1/\tau_{2}$, (2.9) (4.44). $1/g$ , (4.86).

2.2.2 –

• , “ ” [76], , “”, (2.8). , , , .

. 2.7. , – $\left|\alpha\right\rangle\approx\left|0\right\rangle+\alpha\left|1\right\rangle$ – $\tau\ll 1$ , . , (2.8). , , :

. $\alpha\sqrt{1-\tau^{2}}\approx\alpha$. . 2. 2.

; $\tau$. $\left|\tau\alpha\right\rangle\approx\left|0\right\rangle$.

, [128]: $\tau\left|0\right\rangle+\alpha\left|1\right\rangle\propto\left|0\right\rangle+\alpha/\tau\left|1\right\rangle$. , - (2.8) :

[TABLE]

-.

2.3

2.3.1

. 2.8. , . 2.1.

• 1, 2. 1: 1 1, 2 2.

, 2 . . 4.3.5, . , - .

• , , , 2. , . 3, , 2 . , (. 2.2.2), 2 (. 4.6.1).

2.3.2

. 2.9. , 1–6. , 5, . , $\approx 80$ , 1, $\sim 2/3$. 4 1 2.

, . , , , .

1: 1 (. 4.4.3).

1 1 2, 2 1. , 1, .

2: 2 (. 4.6.1).

2 2 3, 3 3. 2, – DFG. 9, 14–16 4 5, 1.

2. , 10, 11 12, . 2 2 12; 9, 12 4.

3: -, 1 (. 2.1.2).

C DFG, -, 1. 1 3,6 2. 1 1,2 7.

1, -, 2. 1 2 8, 10 5. 1 13.

4: .

• •

-.

 ,      ([4.41](#Ch4.E41)) 0.43,   0.65dB  ([4.56](#Ch4.E56)).    0.6,        $1-R=0.5$    $\zeta=0.33$ ([4.83](#Ch4.E83)).
 ,     1   2      .

• •

, 2 10.

    1        (. [2.4.1](#Ch2.S4.SS1)).

• •

. , 2 , .

• •

. 11 11 1, $\tau^{2}$ (2.10).

2.4

. 2.2, ; , . , - : 1 , 2 – .

1 2, 50 100 , $p_{1}\approx 10^{-3}$. , 2 - $\zeta$, $T$, $g$, $\eta$. , 2

[TABLE]

$\zeta=0.16$, $T=0.05$, $g=10$, $\eta=0.1$, $p_{2}=7.5\times 10^{-4}$. $90\%$ - (2.11), .. $p_{2}=\eta/g^{2}$.

[TABLE]

$\nu=76$ . $p_{1}p_{2}=0.75\times 10^{-6}$, $R_{\mathrm{dist}}=50$ .

2.4.1

, -, . 2.1.2. , , , . , , .

(2.12) – -. ( , ), .

, , , , , $\tau^{2}\ll 1$ -. 1, .

, - ; 95%, $g=12$, 2 $\approx 1/3000$ 2 -. ; (4.85) , $\zeta=0.16$, $\tau_{1}^{2}=\sqrt{1/2}$, $\tau_{2}=1/75$, 1.016 1.010 .

• () – . 2, :

.

, 10. , $80\%$ . , 2 $1/180$ 2; 0.49 0.52, (. 2.4.2).

, -, , $\pi/2$. . . 2. 2.

.

2      ,       10  .   ,   -     ,     .

, - 3 , .

. . 2.10. (. 2.4), , - 3. . . 2.10, 0.06 0.69 . , $130$ , .

2.4.2

, : (1) (2) , . , :

• () Agilent U1084A. ,

[TABLE]

$\hat{Q}_{1,2}$ , $\left\langle...\right\rangle$ $5\times 10^{5}$ $16$ ; - $\nu^{-1}\approx 13.07$ .

(2.13) (4.85), ; (2.13) , - .

(2.13) $\sim 10^{-2}$ . -, 1 (. 2.4.1). 2. 2.

Agilent DSO9254A. . : 1 2 , $-2$ , (. 2.10).

, , 8192. , , $\sim 10^{5}$ . , . , , .

. 2.11. 2 , - 3. 1, , ; . 2.10.

1 №4 . 1 2 №3 .

. , , . , ; 3 /.

2.4.3 ´

. , , . , -. . 2.12 .

2.4.4

´ . 2.13. (2.13) . 3 ; - (. 2.10), , . , , . , , , , , . . 2.13.

. , , 8192 10 , $\sim 10^{3}$ -. - , . , , . :

• •

´ . ( , (2.12)), 10 . , / .

• •

( ). , , .

• •

. , . -. ( ).

. , , - ( . 2.13),

[TABLE]

$t_{0}$ ( ), $a$, $x$, $y$, $z$ – . -, , , , ; , , . . : - , (2.14).

(2.14) . 2.13 . , $t_{i}$, (4.85),

[TABLE]

, . 4.2.2. , (2.9), $\sim 15\%$ . , $\{Q_{1},Q_{2},\theta_{1}+\theta_{2}\}_{i}$, (. 2.1.2).

2.5

. 2.14. (4.41) $\theta_{1}+\theta_{2}$ , - 20 . , $\{Q_{1},Q_{2},\theta_{1}+\theta_{2}\}_{i}$ 10 , ; $\sim 10^{4}$ . 2.14.

• , , . , .

. 2.14, $g=6$ 10, . , 80%. 95%, .

(4.86), “”, , ; , , - , $g=8...12$. , , (4.86) , - , .

. 2.15 : , – . , (. . 4.11 . 2.4.1). , .

2.6

, , ; , . , . 4.11, , , 95%. (4.87) 0.037 0.24, .. 6.5 .

[129] , / . - [130], . , . , – [131].

, (2.9) . . . , - (2.9) [132, 133]. , c- - .

:

Alexander E. Ulanov, Ilya A. Fedorov, Anastasia A. Pushkina, Yury V. Kurochkin, Timothy C. Ralph and A. I. Lvovsky. Undoing the effect of loss on quantum entanglement. Nature Photonics 9, 764–768 (2015). [134].

:

14th International Conference on Squeezed States and Uncertainty Relations, , , 29 – 3 2015 2. 2.

22nd Central European Workshop on Quantum Optics, , , 6–10 2015 3. 3.

Third International Conference on Quantum Technologies, , , 13–17 2015 4. 4.

XII International Workshop on Quantum Optics, , , 11–15 2015 5. 5.

58 , , , 23–28 2015

3

, . “ ”: . “ ” (QPT) [135, 136].

QPT , . , [73]. , , , – , -, . [74] , , . , [137], , .

, QPT [72, 139, 138] (csQPT). [140, 141], . csQPT , -. , [72, 139, 138] , . , MaxLik csQPT. , QPT , (MaxLik) [142]. , . MaxLik csQPT [81] [94].

, MaxLik csQPT “ – ”, . , ; (. ), . [143, 144].

. , . , , – , , [145, 146]. , , , QPT.

, : - – , , , .. – [147]. , [41]; , . , - , , [148, 146]. [149].

: ; , . –: , , [150]. , .

, [151] [75]. , . : , $N$. $N=4$, .

[152, 153], , - . , , , .

3.1

MaxLik csQPT [81] . , $M$– $\mathcal{E}$ $4M$–, $\hat{\rho}^{\rm in}$ $\hat{\rho}^{\rm out}$:

[TABLE]

$|\underline{i}\rangle=|i_{1},\dots,i_{M}\rangle$ . , .

, .. . . , $N+1$ , $i_{k}\in 0\dots N$. , $N\times\hbar\omega$, $\hbar\omega$ .

, $|{\underline{\alpha}}\rangle=|\alpha_{1},\dots,\alpha_{M}\rangle$, . , .. . $\left|N\right\rangle$ $N+1$, $M$– $M$ $\sqrt{N+1}$. , $\sqrt{1/2}$. , , , $(2N+1)^{M}$.

[137], ( ) $\mathcal{E}$ , $\mathcal{H}$ $\mathcal{K}$:

[TABLE]

[135, 136, 118]. $\mathcal{E}$ $\hat{E}$ :

.

  (.. ,      ),     .     $\hat{E}$.

2. 2.

.

 $\hat{\rho}_{\rm in}$   $\mathrm{Tr}\left[\hat{\rho}_{\rm out}\right]=\mathrm{Tr}\left[\hat{\rho}_{\rm in}\right]$.   $\mathrm{Tr}_{\mathcal{K}}[\hat{E}]=\hat{I}_{\mathcal{H}}$,  $\hat{I}$ –  .   , $\mathrm{Tr}\left[\hat{\rho}_{\rm out}\right]/\mathrm{Tr}\left[\hat{\rho}_{\rm in}\right]<1$    .   ,         $\mathcal{K}_{\rm total}=\mathcal{K}\otimes\mathcal{K}_{\rm fail}$,  $\mathcal{K}_{\rm fail}$     $\left|\emptyset\right\rangle$,   - .

, . $\{\underline{X}_{i},\underline{\theta}_{i}\}$, ${\underline{\theta}}_{i}=(\theta_{i1},\dots,\theta_{iM})$ $M$ $i$– .

MaxLik $\hat{E}$, $\{\underline{X}_{i},\underline{\theta}_{i}\}$. ,

[TABLE]

$\Lambda$ , ,

[TABLE]

$i$– $\left|\alpha\right\rangle$. $\hat{\Pi}_{\underline{\theta}_{i}}(\underline{X}_{i})$ , $i$– . , $\mathcal{K}_{\rm fail}$ ; $\left|\emptyset\right\rangle$ .

( ) $\hat{E}$, (3.3), [81, 142]

[TABLE]

[TABLE]

(3.5)–(3.7) , $\hat{E}^{(0)}{=}\hat{\mathcal{I}}_{\mathcal{H}\otimes\mathcal{K}}/\mathrm{dim}\left[\mathcal{K}\right]$. $\hat{E}$ (3.7) $\hat{\Lambda}$, $\hat{E}$ , (3.5).

$\hat{R}$ $\hat{\Lambda}$ , $\hat{E}$ . , . (3.3) , . $N+1$ $M$; , , .

. , QPT ( ) , .

3.2

1 2, , , , 1 . 3.1. , , - - . 1 2 1; - .

• () (). $45^{\circ}$ . , - , – . - , ; . , $\hat{a}_{1}$ $\hat{a}_{2}$:

[TABLE]

, $\lambda/2+\lambda/4$.

() (. 4.2). , (. 3.1). , ; $25^{\circ}$ , .

$\{Q_{1},...Q_{M}\}$, , (. 4.2.1). (4.24), , $\{\theta_{1},...\theta_{M}\}$ $2M-1$ . . , . , , : .

, , ; . 0.1, 1 2. , , , .

3.2.1

. , . :

• •

, , , .

• •

, , / , . 1 2 3 4.

• •

, , .

BK Precision 4010A-ND , $\sim 300$ .

( , $\alpha_{1}$) ( , $\alpha_{2}$) ($\theta$) - ($\phi$) . . (3.3), ; , 16

[TABLE]

“” , .. , .

, (3.9), . 3.3 . . 3.3, ; , . , , . , $\{\theta,\phi\}=\{30^{\circ},15^{\circ}\}$, $\{45^{\circ},45^{\circ}\}$. $\{\alpha_{1},\alpha_{2}\}\propto\{1,i\}$.

, - . , . . $\sim 10^{-8}$ (. . 4.1) 40dB 1 2 1, 2, . : , . , 2 $\left|\alpha\right|^{2}=0.9$ .

. . , .

3.2.2

, $\theta_{1}$ $\theta_{2}$ (. 4.2.1). , -

[TABLE]

. (3.10), $\Delta$ .

, , 1 2 (. 3.1) $\Delta$. , , $\Delta$. , $\Delta$ .

$\Delta$ . , , , , $\Delta$. , 3 : $\Delta=0.67$, $2.64$ $5.29$, 2, .

, . 4.2, , :

• •

:

[TABLE]

(3.11) - , .

•  ,    ,        .      ,       ,         .               ([3.11](#Ch3.E11)),         .
  

  , - ; , .. .
• •

:

[TABLE]

, , . , . - $n$:

[TABLE]

. 3.4. (3.12). . 3.4 . $n=4.7\times 10^{3}$ , :

[TABLE]

$\sim 3$ -, $\approx 6\pi$. , ; : . 3.4, $0.8$ $1.67$ . , .

10 . 3.4 , .. . ; / 4/5, ´ / ´ . 3.4. . 3.4; , , , (. 3.5).

. , , 10 , $10\times 4700\sim 10^{5}$ ; , $\sim 5$ - (. 3.3.1). , $10^{6}$ , - $1-3$. , (. 3.3) 1 – .

, , $k\in\{1,2\}$, ,

[TABLE]

$\phi_{k}$ , $\phi_{p}(t)$ , -, . (3.11)

[TABLE]

$\theta_{k}$ . , (3.16) $\theta_{k}-\phi_{p}(t)$ . , (3.6).

3.3

3.3.1

, , ; , . , . , , .

, $k$, , ,

[TABLE]

$O_{k}$, $A_{k}$, $\sigma_{k}$, $\omega$ – , $t$ – . (3.17) (3.16) $\phi_{p}(t)=\omega t$. . 3.4 . 3.5.

, (. 3.3.2), . , , $\phi_{1}=0$. (3.16) (3.17),

[TABLE]

$\Delta$ . , 1 . 3.1, . , , 1 2, .

. $k\in\{3,4\}$,

[TABLE]

[TABLE]

j “” , $C_{k}e^{i\theta_{k}}$ $B_{k}e^{i(\phi_{k}+\omega t)}$. $B_{k}$ $C_{k}$,

[TABLE]

1 2 (. 3.2), , (3.16).

, , . 3.6. , / . , 2 1 - 4 6.

. 3.7. , , , .

, (3.8) , . MaxLik , . , , (3.8).

3.3.2

, . - : . : $\delta\theta$ $\delta\theta/\omega$, $\omega$ . - , . , $\theta$:

[TABLE]

(3.1), , ,

[TABLE]

$\mathcal{E}^{\underline{n}\,\underline{m}}_{\underline{j}\,\underline{k}}$.

(3.23). , ($M=2$) , $N=4$ $(N+1)^{4M}=5^{8}=390625$ 352460. , 3.3, (3.22) $\approx 10$ . ; .

, [81], - . , $N$ , (, – [150]), , . , . , $N=4$.

, , 3.1. $8$ [154], . Intel Core i7; 4 8 , 2 . $30-80$ .

, $N^{\prime}=2$. , , [81]. $N^{\prime}=2$ $6561$, – $1107$.

, $N^{\prime}=2$, . 3.8. , . . 3.9. . , – $\left|1,1\right\rangle\rightarrow\left|1,1\right\rangle$, ( ) $\approx 0.01$ ().

, . 3.8, , . 3.9 . ; , . , : $\left|i,j\right\rangle\left\langle k,l\right|=\left|i\right\rangle_{1}\left|j\right\rangle_{2}\left\langle k\right|_{1}\left\langle l\right|_{2}$. , .

–, $\left|1,1\right\rangle\left<1,1\right|$. $(N^{\prime}+1)^{2M}=81$ , , ; , . 3.8 . 3.9.

. 3.9 (3.8), $0.8$ . , , , , .

, (3.2):

[TABLE]

$(N=4)$ $N^{\prime}=3$ $N^{\prime}=2$, (3.24) $\mathcal{F}\left(\mathcal{E},\mathcal{E}_{est}\right)=0.88$ $0.95$, . ($N^{\prime}=N=4$), 0.7.

. -, (. 3.3) . , ; $0.502$. , , $0.998$, , ( , ), . , . , , , . $\mathcal{E}^{\prime}_{{\rm est},i}$, $\mathcal{F}\left(\mathcal{E},\mathcal{E}^{\prime}_{{\rm est},i}\right)\sim 0.95$ $i$. $\mathcal{F}\left(\mathcal{E}^{\prime}_{{\rm est},i},\mathcal{E}^{\prime}_{{\rm est},j}\right)$, $\mathcal{F}\left(\overline{\mathcal{E}^{\prime}_{{\rm est},i}},\mathcal{E}\right)$. , $0.95$ , .

3.4

, . – . , –. 5%, 95%. , , , .

:

Ilya A. Fedorov, Aleksey K. Fedorov, Yury V. Kurochkin and A.I. Lvovsky, Tomography of a multimode quantum black box. New Journal of Physics 17, 043063 (2015) [155].

:

Quantum 2014 – Workshop ad memoriam of Carlo Novero, 25–31 2014, , 2. 2.

23rd annual International Laser Physics Workshop, 14–18 2014, , 3. 3.

Quantum Optics VII, 27–31 2014, –, 4. 4.

Third International Conference on Quantum Technologies, 13–17 2015, ,

4

4.1

• Coherent Mira 900, $\lambda\approx 780$ . Coherent Verdi G12, 530 10-12 . Mira , $\tau\approx 2$ $T\approx 13.07$ , $\nu=76$ . $W\approx 1.5$ , $T/\tau\approx 6.5\times 10^{3}$ .

4.1.1

. , . 4.1, . - , - , . $\approx 5$ , 100 1, SHG (. 4.4.4).

[TABLE]

$E_{1,2}$ , $\beta$ (. 4.4.4). (4.1) , – . , (4.1), .

$A$ , . $\Delta$ $\delta$. (4.1) $E_{1}$ $E_{2}$:

[TABLE]

$\sigma\approx 2$ . $E_{1,2}^{2}(t)$

[TABLE]

$\sigma^{-1}$,

[TABLE]

$A(\delta)_{\Delta=0}$ $A(\delta)_{\Delta=\sigma\sqrt{2\ln 2}}$ . $\sigma\sqrt{2\ln 2}$ $\sqrt{2}$ (4.3). $\Delta$

[TABLE]

$c$ .

GW Instek GOS 620FG, . $\approx 15$ , . . 30 , $\approx 1$ . $\Delta=0.35$ , (4.5) $\tau_{\rm FWHM}=1.65$ .

4.1.2

$\lambda$ $\Delta\lambda$ APE PulseCheck. $\lambda=780$ $\Delta\lambda=0.7$ , [156]

[TABLE]

$\omega=2\pi c/\lambda$ . $c=3\times 10^{11}$ /, (4.6) $\tau_{\rm coh}\approx 2.9$ .

[TABLE]

0.44 [157], . , [158]. $P$ $\alpha$

[TABLE]

$\hbar=10^{-34}$ $*$ . $P=6$ , (4.8) $\alpha\approx 1.8\times 10^{4}$.

. , . 4.1, $a=1.5$ , $m=1$ . [156]

[TABLE]

$N$ , $d\approx 1$ , $\alpha\approx 0.2$ – .

4.1.3

• , . . 4.1, (1–2, 75+100 ), 1.

[156]

[TABLE]

$k=2\pi/\lambda$, $a$ – , $\theta$ – . . 4.2, (4.10)

[TABLE]

$F$ 2, 1 . , $R$, (4.10),

[TABLE]

$99.7\%$ (4.11). $e$ ,

[TABLE]

$F=100$ , $a=50$ , $\lambda=780$ , (4.12) (4.13) $R=1.9$ , $w=1.24$ .

c $f$, [156]

[TABLE]

$z$ , , $w_{0}$ – , $z_{R}$ – . $\phi$

[TABLE]

4.1.4

• • , . $\phi$ $\theta$. $\hat{a}_{\mathrm{h}}$ $\hat{a}_{\mathrm{v}}$,

[TABLE]

$W_{\pi/2}$ $W_{\pi}$ , - - .

[TABLE]

$\Delta$ ,

[TABLE]

$x$, . $x$ $x+\pi/2$ (4.17) .

, , $\alpha_{\mathrm{v/h}}$ $\beta_{\mathrm{v/h}}$ (4.16). , , , . 3.

4.2

4.2.1

. – , – , .

, [158]. $\hat{X}$ $\hat{P}$

[TABLE]

, , [118].

(4.19), – – . , $\hat{a}_{\mathrm{s}}e^{i\mu}$ $\hat{a}_{\mathrm{o}}e^{i\nu}$ [159]

[TABLE]

, , .

[TABLE]

$\left|\alpha_{\mathrm{o}}\right\rangle$ $\alpha_{\mathrm{o}}\sim 10^{4}$. , (4.21)

[TABLE]

[159]

[TABLE]

(4.22) (4.19):

[TABLE]

, (4.24) (4.19), $\theta=\nu-\mu$ .

• , (4.20) – (4.22), () [118, 160].

$M$– ,

[TABLE]

$Q_{\mathrm{k}}^{i}$ $i$– $k$, $\theta_{\mathrm{k}}^{i}$ . , (4.25) $\left[0...2\pi\right]^{M}$.

; . , -, . , . (4.24), $\theta$ $\mu$, $\nu$; , - .

(4.25) . , . , , . . , 3.2.2, 4.6.4 2.1.2.

4.2.2

, , $\sim 2$ (. 4.1.1). , $\sim 100$ [160], $\sim 5$ , , 13 .

Agilent DSO9254A 2.5 , - Agilent Acqiris U1084A 4 . , , . 4.3. (. 4.1).

, . , , , ; , ( . 4.3).

. 4.3, , , , . , , , .

, 1/2 [159] . . 4.4 , . 4.3.

[161, 162], (. 1).

4.2.3

, . , ?

(4.24) , (4.19), $\alpha_{o}$. , , ; , . , . [163]

[TABLE]

$\mathrm{\Phi}_{\mathrm{s,o}}(\mathbf{r},t)$ , .

(4.26) . , :

“” .

    ,      .    $A$,       $\Delta$

[TABLE]

$\eta$ (4.26) . :

[TABLE]

. :

• •
• •
• •

– “”.

, ; . . 2. 2.

“” .

   . ,           .          ,         .

, . , , , , , . , , .

, , . [164] , .

, $\omega_{s}$ $\omega_{o}$:

[TABLE]

, ,

[TABLE]

$\Delta$ $t=0$, , $m$ : $\tau=2m\pi/\omega_{o}$.

$I(\Delta)$, (4.28),

[TABLE]

$\eta(\delta\lambda)$ . 4.5 $\tau\approx 2$ (. 4.1.1), $m\sim 10^{3}$.

“” , . , , . / 7-12dB [160]. , $91\%$, $\sim 80\%$.

4.3 :

() - . , . 80 – 90 20 [165, 166, 167, 168, 169, 170, 171, 172].

, . , , ; , , . , , - [174] [175]. , [175, 176]; . .

4.3.1 –

, , , . , . , .

[TABLE]

$\hat{a}$ $\hat{b}$ – [158]. (4.32) , – .

, (4.32) ,

[TABLE]

$t=\zeta/\beta$, , (4.33) :

[TABLE]

(4.34) (4.23) :

[TABLE]

, (4.35) , – . $a$ $b$ ,

[TABLE]

, (4.35)

[TABLE]

. (4.37) . 4.6. , , .

, , :

[TABLE]

[TABLE]

(4.35), (4.35) (4.37) (4.39) $e^{2\zeta}$ :

[TABLE]

(4.35) :

[TABLE]

(4.41) (4.37),

[TABLE]

, - (4.37) $\left|nn\right\rangle\equiv\left|n\right\rangle_{a}\otimes\left|n\right\rangle_{b}$ [175]:

[TABLE]

, $\zeta\ll 1$, (4.43)

[TABLE]

(4.43)

[TABLE]

, $a$ , $b$ (4.43):

[TABLE]

(4.46) , . $T$

[TABLE]

4.3.2 –

, , -. , , .

, (4.32)

[TABLE]

, $a$

[TABLE]

, $t=\zeta/\beta$,

[TABLE]

(4.50), (4.34), (4.23):

[TABLE]

, (4.38), (4.51)

[TABLE]

[177] (4.52) . 4.7. , $X$. ,

[TABLE]

, $\theta$ $X$ [118]. , $\cos{2\theta}$, , (4.51) (4.53):

[TABLE]

$\zeta>0$, $X$ (4.39):

[TABLE]

.

[TABLE]

, . (4.7), (4.56) $\mathrm{dB}=\dfrac{20\zeta}{\ln{10}}\approx 8.7\zeta$. , (4.56) - (4.37).

(4.52) [178]

[TABLE]

$\zeta\ll 1$, (4.57) :

[TABLE]

• (4.57) - $\zeta$ (4.45):

[TABLE]

4.3.3

, , ; , , . , , , [179].

, , . [180].

$z$.

[TABLE]

$N(\omega)$ – , $\hat{a}(\omega,k_{x},k_{y},z)$ , . [179]

[TABLE]

$k$ $k_{p}$ , $E_{0}=\int\!\!\,d\omega\,dk_{x}\,dk_{y}\,E_{p}(\omega,k_{x},k_{y})$ $E_{p}$. $L_{\rm NL}$ :

[TABLE]

$d_{\rm eff}$ , $\omega_{p}$ – . , $a$ (4.50), $L$ $\zeta=L/L_{\rm NL}$.

(4.61) , ($z=L/2$) ($z=-L/2$)

[TABLE]

(4.63) c - [181]. $S$ $C$

[TABLE]

$\psi_{n}$ $\phi_{n}$ , $\zeta_{n}$ .

[TABLE]

(4.63)

[TABLE]

, (4.65) - , , , (4.50), . , $E_{p}$ , $\psi_{n}=\phi_{n}^{*}$ [179].

(4.64) , .. $L\ll L_{\rm LN}$. :

[TABLE]

[TABLE]

, $k_{p}(\omega_{p},0,0)=2k(\omega_{p}/2,0,0)$, $\left(k_{x},k_{y}\right)\ll\left(k,k_{p}\right)$, ,

[TABLE]

[TABLE]

c $\tau_{p}$ $d_{p}$ ,

[TABLE]

$\mathrm{sinc}$

[TABLE]

$S$ (4.67):

[TABLE]

[TABLE]

(4.64) [182]. , ${\bf n}=\{l,i,j\}$,

[TABLE]

[TABLE]

[TABLE]

(4.75) . 4.8.

• [174], $\beta_{1}\approx 6.4\times 10^{-9}$ /, $\beta_{1,p}\approx 8.2\times 10^{-9}$ /, $\beta_{2}\approx 2.8\times 10^{-31}$ 2/, $L=1$. $\tau_{p}\approx 2$ , $d_{p}\approx 10$.

[TABLE]

$\Delta_{\omega}\gg\delta_{\omega}$, $\tau_{s}$ $\tau_{p}$ (4.74, 4.76), . , , . ( ) . 4.8 . () $86\%$ [179].

$\delta_{k}$ $\Delta_{k}$, , ; , $d_{s}$ $1/\delta_{k}$ (4.76). , (4.75, $i=j=0$). , , [179, 183].

(4.73), , . 4.9. , (, $\lambda_{1}=\lambda_{2}$) , $\delta_{\omega}$ $\delta_{k}$. , $\Delta_{\omega}$ $\Delta_{k}$ . , $\Delta_{\omega}/\delta_{\omega}\approx 400$, $\Delta_{k}/\delta_{k}\approx 1.6$, .

4.3.4 – –

, - (4.37) . , :

[TABLE]

.(4.40),

[TABLE]

, $a$ $b$ (4.52)! , $a$ , $b$ – . 4.79 [159], (. 2.1, [178]).

4.3.5

. :

• •

. $\sim 5\%$.

• •

. , . 4.2.3, , $86\%$ (. 4.3.3). $\sim 70\%$.

• •

“gray tracking” (. 4.13), , , .

, $35\%$.

–

• $R$,      ([4.54](#Ch4.E54))
  

[TABLE]

/

[TABLE]

(4.82) :

[TABLE]

“” , , .

(4.10) - (. 2.1.2). , , (4.82) .

, - (4.58), $R$

[TABLE]

$\left|1\right\rangle\left\langle 1\right|$ , .

–

• (4.37) $\tau_{1}$ $\tau_{2}$, $\theta_{1}+\theta_{2}$ (4.42) . $\tau_{1}=\tau_{2}$, (4.80), (4.81) ; , (4.82)–(4.83). $\tau_{1}\neq\tau_{2}$,

[TABLE]

$\zeta\ll 1$, - (4.44), ,

[TABLE]

, $\left|11\right\rangle$ $\left|00\right\rangle\left\langle 11\right|$ $\left|11\right\rangle\left\langle 00\right|$, $\left|01\right\rangle\left\langle 01\right|$ $\left|10\right\rangle\left\langle 10\right|$, .

. 4.11 (4.86) $\tau_{1}$ $\tau_{2}$.

[TABLE]

$\hat{\rho}_{\mathrm{pt}}$ , [184]. $\tau_{1}=\tau_{2}$, :

[TABLE]

4.4

, (4.3), :

:

[TABLE]

$\omega_{p}$ , $\omega_{a,b}$ . (4.89) , . 2. 2.

:

[TABLE]

$\vec{\mathbf{k}}_{p,a,b}$ . (4.90) , , . , : , , [175].

, , ,

[TABLE]

(4.89) , (4.90)

[TABLE]

$\lambda=780$ , $n_{\omega,i}$ $\omega$, $i$ . (4.92) [176, 175]. , (4.92), :

  .    ,      -      ([4.3.2](#Ch4.S3.SS2)).

,       .   ,        ([4.3.1](#Ch4.S3.SS1)).

, $P_{\mathrm{pump}}=50-100$ (. 4.4.1) $\sim 1.5\times 10^{5}$ . , , ; . , – “” [185]. [186, 187]. , , 4.4.3 4.4.4.

4.4.1

; , , . , 390 , . (0, . 4.12).

0 $2\times 2\times 17$ 1 $F=100$ ; . , , (. 4.1) ´ . , $\sim 1$ , $200-300$ .

0, 2 3 – F25 F38, , . , , . 4.1.3.

, , . , , , ; 4–5 F25, 5, 25 . 5, , ; 1 2.

2–3. , $200-300$ “” $P_{\mathrm{pump}}=50-100$ .

4.4.2 -

• .               .         ()   ([4.90](#Ch4.E90)).   ,        $\pi$ [[175](#bib.bib175)],      ([4.90](#Ch4.E90))
  

[TABLE]

$D$ . , (4.93)

[TABLE]

$\lambda$ , $n_{p}$ $n_{a/b}$ .

• : .

[TABLE]

$\lambda$ [173, 174].

, $\vec{x}$ . - II, $\vec{y}$, $\vec{y}$ $\vec{z}$. , (4.94)

[TABLE]

$D=7.825$ , (4.96) $\lambda=0.38902$ .

, , . [156]

[TABLE]

, (4.95) (4.97)

[TABLE]

$c$ .

[TABLE]

$L$ . $L=1$ , (4.99) (4.98) $\delta=0.58$ .

4.4.3 2

2 (SHG) – , . : ; – – , ; , 4 2, . 4.12.

SHG:

• •

, 8.

• •

. , .

• •

. ( SHG ), ( , 7.85 ).

, SHG $\sim 1$ . , , , :

. , . (. 4.13), . 2. 2.

, (4.92). , . ; $\lambda=780.1$ , $15^{\circ}$C.

, . , . , . . 4.14; . , , , $40-50$ $85\%$. 3. 3.

, .

6 , SHG 10-20 . , SHG.

, SHG. , , , (DFG) . , , . DFG , 3 1.

:

• •

. 1 , - .

• •

. 2 2 .

• •

, $a$ 3 / 5 $F$ 4 / 9 $F\lambda/a$.

DFG , , DFG. , , (4.89) (4.90), :

DFG.

$\gtrsim 40$ , DFG , , . 5-10 , , - . , , (, , ) . , $\sim 5\%$.

. 1 (. 1) . 4.13. “gray tracking”, , ; , , , - . , , 1 2.

4.4.4 1

(SHG) . 4.4.3, :

• •

, .

• •

1 5-10 2. SHG 70-200 (. . 4.5).

(4.92) SHG [175, 176], .

. 4.14 1, , - (. 2.1).

$W_{\rm shg}$ 10 $F$ . $F=100$ 6 , $W_{\rm shg}=100$ .

SHG. , , . 2, (DFG, . . 4.4.3); 1 , DFG , . , , ; , - 1 . , 1, DFG.

4.5 .

, “” , (. 4.4.3, 4.4.4). (SHG) (DFG) , .

4.5.1

SHG . $E_{\mathrm{seed}}$ $\omega$ , $E_{\rm SHG}$ $2\omega$ [175, 176]

[TABLE]

$d_{\rm eff}$ . (4.100), SHG:

[TABLE]

$L$ .

$W$

[TABLE]

$\nu=76$ , $\tau\approx 2$ , $n$ – (4.98),

[TABLE]

$\theta$ , $r$ – $F$.

(4.100), SHG, $\sqrt{2}$ :

[TABLE]

SHG ,

[TABLE]

(4.102), (4.103) (4.105), (4.101)

[TABLE]

. 2.1 ( 1) $L=1$ , $F=100$ , $W_{\rm seed}=6$ , $r_{\rm seed}=2.5$ , $W_{\rm shg}=100$ , (4.106) $d_{\rm eff}=6.05$ /,

[TABLE]

$d_{33}=10.7$ / - [188]. $2/\pi$ [175].

2, . . 2.3.2, $L=2$ , $W_{\rm shg}=10$ , , . (4.106) $d_{\rm eff}=0.96$ /.

4.5.2

$E_{\rm dfg}$ $E_{\rm pump}$ $E_{\rm seed}$ [175]

[TABLE]

DFG (4.108) :

[TABLE]

$w_{\rm dfg}$ $w_{\rm seed}$ $w_{\rm pump}$ . 4.15 .

2

2, (4.108) (4.33), $a$ $b$ DFG,

[TABLE]

$v_{g}$ (. 4.4.2).

2, DFG , ,

[TABLE]

C (4.102, 4.103), (4.106) (4.109), (4.111)

[TABLE]

, . 4.5.1, $W_{\rm pump}=40$ $r_{\rm pump}=2$ , (4.112) $W_{\rm dfg}=96$ , $W_{\rm dfg}=74$ .

$d_{\rm eff}=0.96$ / (. 4.5.1) , . (4.110), , (4.34):

[TABLE]

(4.106), $\zeta$ :

[TABLE]

, (4.114) $\zeta=0.24$.

1

1, . , , . DFG , (4.108) c

[TABLE]

, $a$, (4.49) (4.110). , / , . , .

, :

[TABLE]

$W_{\rm min}=3.7$ . $\zeta=0.24$ (4.116), $2.1$ dB . $35\%$ (. 4.3.5) , , $1.3$ dB.

2, $d_{\rm eff}=6.05$ / (. 4.5.1), (4.114) , . , . 4.5.1 $W_{\rm pump}=40$ $r_{\rm pump}=2$ , (4.114) $\zeta=0.75$.

4.6

4.6.1

(). , , , 1 2. 4.3.1,

[TABLE]

, , 1. , 1, , 2 , . [44, 186].

; , . , [46].

4.6.2

1 (), , . , , .

$\eta$, . $n$– , -

[TABLE]

POVM [96]

[TABLE]

1 (4.117), 2

[TABLE]

(4.117), $n\geq 2$ (4.120) , (4.120) :

[TABLE]

, $\eta\ll 1$ ( Perkin-Elmer SPCM-AQR-14-FC), (4.120)

[TABLE]

$\left|N\right\rangle$, , , , $N$ $N-1$. $N$ , .

$N=2$. , . $n=2$ , :

.

    $1-(1-\eta)^{2}$.

2. 2.

.

, $2\eta(1-\eta)$.

$1/2$,

[TABLE]

(4.1).

(4.119), POVM

[TABLE]

$p_{n}\left[\rm 2\,clicks\right]$ $n$– . 1 - (4.117),

[TABLE]

(4.120), , :

[TABLE]

, $\eta=0.12$ :

• •

Andover, $\approx 0.25$ (. 4.6.3).

• •

, $\approx 0.8$.

• •

, $\approx 0.60$.

; $N=2$ , (4.123) , (4.119) $\left\langle 2\right|\hat{\Pi}\left|2\right\rangle=2\eta-\eta^{2}$. , $N\rightarrow\infty$, - ; , .

[190, 191].

4.6.3

. 4.3.3, , , . , , , .

, (4.4.3), , . (4.32), . DFG (. 4.4.3, 4.5.2); , .

, . , DFG , $\sim 50$ (. 4.5.2). , , (. 1) , DFG , .

. 4.3.3, , . (. 4.2.3), Andover, $\sim 0.3$ , (. 4.1.2). [186]; , .

. , ; . , , . 4.16. , , 780.1 ; .

4.6.4

(. 4.2) :

-:

[TABLE]

. , $\sim 10$ - . $\sim 10^{4}$ ; , , . 2. 2.

:

[TABLE]

, - . , , 300 . 3. 3.

(. . 4.6.1).

 .            ;          .  ,              .

, , :

• •

:

, $\sim 5$%.

• •

:

, $\lesssim 75$%. 2. 2.

. / , $\approx 12\mathrm{dB}$, – 91% [160].

[175, 118]. $\eta$ ( $\sim 50\%$) [44, 192].

$\hat{a}$, , $1-\eta$. , [175]:

[TABLE]

$\hat{a}_{\mathrm{aux}}$ , .

[TABLE]

,

[TABLE]

, , $\hat{a}$

[TABLE]

(4.132) . , , .. .

, , (4.132), , . $\eta$ .

[TABLE]

(4.132)

[TABLE]

$R=100$ , (4.134) $\sigma\sqrt{\dfrac{2\nu}{R}}=0.014*\sigma$ $\nu=10$ , $\eta$ [192].

:

2. ()

, , , . . , - . 2. ()

. , . 2. 2.

1. ()

—-, . 2. ()

, - 0.65dB, 20 – $\sim 70$– . ; , 0.037 0.24. 3. 3.

1. ()

. , . , . 2. ()

. , –. 95%.

, .

** .**

.

.

   .

.

    .

.

  .

!

2016, .

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