Quantum Hamiltonian model based on continuous-mode operators

3. Theoretical models for a 4WM process in a TWJPA

3.2 Quantum Hamiltonian model based on continuous-mode operators

A standard method to treat quantum superconducting circuits is represented by the lumped element approach [34]. In this latter, the Hamiltonian of the quantum circuit is straightforwardly derived from its classical counterpart by promoting Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

fields to operators and properly imposing commutating relations. In this view, one can proceed by deriving the Lagrangian of a TWJPA composed by the repetition of N unitary cells, under first nonlinear order approximation, as

L ¼X^N�1

n¼0

∂Φn

∂t

� �2

þCJ

∂ΔΦn

∂t

� �2

�EJ0 1�cos 2π Φ0ΔΦn

� �

" #

≈ (19)

≈X^N�1

n¼0

∂Φn

∂t

� �₂

þCJ

∂ΔΦn

∂t

� �₂

� 1

2LJ₀ΔΦ²_n� 1 24LJ₀

2π Φ0

� �₂ ΔΦ⁴_n

" #

(20) where EJ₀ ¼IcΦ0=2π¼IcLJ₀. Under the assumption that a=λ≪1 it is possible, as performed in the previous subsection, to replace the discrete index n with a continuous position x along the line (i.e.,Φnð Þ !t Φðx, tÞ) and approximate, at the first order,ΔΦn!a ∂Φðx, tÞ=∂x. Furthermore, extending the system via two lossless semi-infinite transmission lines (characterized by a constant distributed capacitance c0and a constant distributed inductance l0), the

Lagrangian can be expressed through a space integral extending from x¼ �∞to x¼ þ∞as

L Φ,∂Φ

∂t

� �

¼1 2

ð∞

�∞ c xð Þ ∂Φ

∂t

� �₂

þ 1

ω²_Jð Þl xx ð Þ

∂²Φ

∂x∂t

� �

� 1 l xð Þ

∂Φ

∂x

� �₂

þγð Þx ∂Φ

∂x

� �₄

" #

dx (21) where c xð Þand l xð Þare the distributed capacitance and inductance of the system, defined as

c xð Þ ¼

c0 x<0 C_J=a 0<x<z c0 x>z 8>

>: and l xð Þ ¼

l0 x<0 L_J₀=a 0<x<z l0 x>z 8>

>: (22)

ω_Jð Þ ¼x

∞ x<0 1= ffiffiffiffiffiffiffiffiffiffiffi

LJ₀CJ

p 0<x<z

∞ x>z

>: and γð Þ ¼x

0 x<0

a³EJ₀=12

� �

2π=Φ0

ð Þ⁴ 0<x<z

0 x>z

(23) where z is the length of the TWJPA,ω_Jð Þx is the junction’s plasma frequency, andγð Þx is the term deriving from the nonlinearity of the junctions.

From Eq. (21), one can easily derive the Euler-Lagrange equation whose form, for 0<x<z, is equal to Eq. (9), giving the same dispersion relation (Eq. (10)) under analogous assumptions. Instead, outside the nonlinear region, the wave vector turns out to be kωð Þ ¼x ffiffiffiffiffiffiffiffi

c0l0

p ω².

The Hamiltonian of the system can be derived from the Lagrangian by taking into accountπðx, tÞ ¼δL=δ½∂Φ=∂t�, the canonical momentum of the fluxΦðx, tÞ:

H½Φ,π� ¼ ð_∞

�∞ π∂Φ

∂t

� �

dx� L

¼1 2

ð_∞

�∞ c xð Þ ∂Φ

∂Φt

� �2

þ 1 l xð Þ

∂Φ

∂x

� �2

þ 1

ω²_pð Þl xx ð Þ

∂²Φ

∂x∂t

� �2

" #

dx�γ 2

ð_z

∂Φ

∂x

� �4

" # dx

¼H₀þH₁

(24) Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric…

DOI: http://dx.doi.org/10.5772/intechopen.89305

particular solution in Eq. (9) and assuming that, along the line, the amplitudes are slowly varying (i.e.,∣∂²A_n=∂x²∣ ≪k_n∣∂A_n=∂x∣ ≪k²_n∣A_n∣) and that Aj j_s²and Aj j_i²are negligible (i.e.,∣As,i∣ ≪ ∣Ap∣, strong pump approximation), we obtain a system of three coupled differential equations for the amplitudes Anð Þx that describe the energy exchange between the three waves along the line:

∂A_p

∂x ¼iϑ_p��A_p��²A_pþ2iXpA_p^∗A_sA_ie^iΔkx (12)

∂As ið Þ

∂x ¼iϑs ið Þ��Ap��²As ið ÞþiXs ið ÞA²_pAi sð Þe^iΔkx (13) whereΔk¼2kp�ks�kiis the chromatic dispersion. The termϑ_pis responsible for the self-phase modulation of the pump tone, whileϑ_{s i}_{ð Þ}is responsible for the cross-phase modulation between the pump tone and the signal or idler, respectively.

These terms can be expressed as ϑ_p¼ a⁴k⁵_p

16C0I²_cL³_J₀ω²_p and ϑ_{s i}_{ð Þ}¼ a⁴k²_pk³_{s i}_{ð Þ}

8C0I²_cL³_J₀ω²_{s i}_{ð Þ} (14) while the coupling constants X_n, depending on the circuit parameters, are defined as

X_p¼a⁴k²_pk_sk_i�k_p�Δk�

16C0I²_cL³_J₀ω²_p and X_{s i}_{ð Þ}¼a⁴k²_pk_sk_i�k_{s i}_{ð Þ}þΔk�

16C0I²_cL³_J₀ω²_{s i}_{ð Þ} (15) Expressing the complex amplitudes Anð Þx in a the co-rotating frame

Anð Þ ¼x An0e ⁱ^ϑⁿj^A^p0j²^x (16) it can be demonstrated that, working under the undepleted pump approximation∣Apð Þ∣x ¼A_p₀≫ ∣As ið Þð Þ∣, the amplitude of the signal can be expressed asx

Asð Þ ¼x As0 cosh g� ₁x�

�i Ψ1

2g₁ sinh g� ₁x�

� �

þiX_{s i}_{ð Þ}��A_p₀��²

g₁ A_{s i}^∗_{ð Þ}₀sinh g� ₁x�

" #

e ⁱ^Ψ²¹^x (17) whereΨ1¼Δkþ�2ϑ_p�ϑ_s�ϑ_i�

A_p₀

�� ²¼Δkþϑ��A_p₀��²is the total phase mismatch and g₁is the exponential complex gain factor, defined as

g₁¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XsX_i^∗��Ap₀��⁴� Ψ1

� �₂ s

(18) The total gain of an amplifier, composed by the repetition of N elementary cells, can then be expressed as G_sðaNÞ ¼jA_sðaNÞ=A_s₀j².

3.2 Quantum Hamiltonian model based on continuous-mode operators

L ¼X^N�1

n¼0

∂Φn

∂t

� �2

þCJ

∂ΔΦn

∂t

� �2

�EJ0 1�cos 2π Φ0ΔΦn

� �

" #

≈ (19)

≈X^N�1

n¼0

∂Φn

∂t

� �₂

þCJ

∂ΔΦn

∂t

� �₂

� 1

2LJ₀ΔΦ²_n� 1 24LJ₀

2π Φ0

� �₂ ΔΦ⁴_n

" #

Lagrangian can be expressed through a space integral extending from x¼ �∞to x¼ þ∞as

L Φ,∂Φ

∂t

� �

¼1 2

ð∞

�∞ c xð Þ ∂Φ

∂t

� �₂

þ 1

ω²_Jð Þl xx ð Þ

∂²Φ

∂x∂t

� �

� 1 l xð Þ

∂Φ

∂x

� �₂

þγð Þx ∂Φ

∂x

� �₄

" #

dx (21) where c xð Þand l xð Þare the distributed capacitance and inductance of the system, defined as

c xð Þ ¼

c0 x<0 C_J=a 0<x<z c0 x>z 8>

>: and l xð Þ ¼

l0 x<0 L_J₀=a 0<x<z l0 x>z 8>

>: (22)

ω_Jð Þ ¼x

∞ x<0 1= ffiffiffiffiffiffiffiffiffiffiffi

LJ₀CJ

p 0<x<z

∞ x>z

>: and γð Þ ¼x

0 x<0

a³EJ₀=12

� �

2π=Φ0

ð Þ⁴ 0<x<z

0 x>z

(23) where z is the length of the TWJPA,ω_Jð Þx is the junction’s plasma frequency, andγð Þx is the term deriving from the nonlinearity of the junctions.

c0l0

p ω².

The Hamiltonian of the system can be derived from the Lagrangian by taking into accountπðx, tÞ ¼δL=δ½∂Φ=∂t�, the canonical momentum of the fluxΦðx, tÞ:

H½Φ,π� ¼ ð_∞

�∞ π∂Φ

∂t

� �

dx� L

¼1 2

ð_∞

�∞ c xð Þ ∂Φ

∂Φt

� �2

þ 1 l xð Þ

∂Φ

∂x

� �2

þ 1

ω²_pð Þl xx ð Þ

∂²Φ

∂x∂t

� �2

" #

dx�γ 2

ð_z

∂Φ

∂x

� �4

" # dx

¼H₀þH₁

(24) Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric…

DOI: http://dx.doi.org/10.5772/intechopen.89305

where the term H0represents the linear contributions to the energy of the system, while H₁is the first-order nonlinear contribution.

This Hamiltonian can be converted to its quantum form promoting the field Φðx, tÞto the quantum operatorΦ^ðx, tÞ.

In direct analogy with Eq. (11), one can express the flux operator in terms of continuous-mode functions [34], such asH^₀is diagonal in the plane-waves unperturbed modes decomposition:

Φ^ðx, tÞ ¼ X

ν¼L, R

ð_∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏl xð Þ 4πkωð Þx s

a^νωe ⁱ^ð^�k^ω^{ð Þx�}^x ^ω^t^ÞþH:c:

" #

dω (25)

where the subscript R denotes a progressive wave, while L denotes a regressive wave (i.e.,a^Rωrepresents the annihilation operator of a right-moving field of frequencyω). In [35], it is demonstrated that by replacing the definition given in Eq. (25) into the linear Hamiltonian,H^₀takes the form

H^₀¼ X

ν¼R, L

ð_∞

0�ℏω^a^†_νωa^νω�

dω (26)

(where the zero-point energy, which does not influence the dynamics of the amplifier, has been omitted).

Using the expansion ofΦ^ðx, tÞintroduced above, under the hypothesis of a strong right-moving classical pump centered inω_p, and that the fieldsa^νωare small except for frequencies closed to the pump frequency (i.e., replacinga^νωwith a^νωþbð Þ, where bω ð Þω is a complex valued function centered inω_p), the nonlinear HamiltonianH^1can be expressed under strong pump approximation, at the first order in bð Þ, as the sum of three different contributions:ω

H^1¼H^CPMþH^SQþHSPM (27) In the expressions of these contributions, the fast rotating terms and the highly phase mismatched left-moving field have been neglected:

H^CPM¼ � ℏ 2π

ð∞

0dωsdωidΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p β^∗� �Ωp

β�Ωp⁰�

ϒ ω� s,ω_i,Ωp,Ωp⁰� a^^†_Rω_sa^Rωi

þH:c

(28) describes the cross-phase modulation,

H^SQ ¼ � ℏ 4π

ð_∞

0dω_sdω_idΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffi kωskωi

p β� �Ωp

β�Ωp⁰�

ϒ ω� s,Ωp,ω_i,Ωp⁰�

a^^†_Rω_sa^^†_Rω_iþH:c:

(29) describes the broadband squeezing, and

HSPM¼ � ℏ 4π

ð∞

0 dωsdωidΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffiffi kωskωi

p β^∗� �Ωp

β�Ωp⁰�

ϒ ω� s,ω_i,Ωp,Ωp⁰�

b^∗ð Þbω_s ð Þω_i þH:c:

(30) Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

describes the self-phase modulation.βð ÞΩ is the dimensionless pump amplitude, proportional to the ratio between the pump current IJ�Ωp�

and the critical current of the junctions Ic:

β� �Ωp

¼I_J� �Ωp

4Ic (31)

The functionϒ ωð 1,ω₂,ω₃,ω₄Þis the phase matching function, defined as ϒ ωð 1,ω₂,ω₃,ω₄Þ ¼

ð_z

0e^{�i k}ð^ω¹^{ð Þ�k}^x ^ω²^{ð Þþk}^x ^ω³^{ð Þ�k}^x ^ω⁴^{ð Þ}^xÞdx (32) Assuming the nonlinear HamiltonianH^1as a perturbative term of the Hamilto- nianH^0for which the continuous modes are noninteracting, and assuming the initial time of the interaction t0¼ �∞and the final time t1¼ þ∞, it is possible to relate the input field of the system to the output one introducing the asymptotic output field

a^^out_R_ω¼U^a^_RωU^^†, (33) whereU is the asymptotic unitary evolution operator (approximated to the first^ order inH^1)

U^ �U^ð�∞,∞Þ ¼e^�^ℏⁱ^K^{^}¹ where K^1¼K^CPMþK^SQþKSPM (34) Working in the monochromatic degenerate pump limit b�Ωp⁰�

¼b� �Ωp

! b� �Ωp

δ�Ωp�ω_p�

�bp(whereω_pis the pump frequency), the propagators take the form

K^CPM¼ �2ℏ β�� _p��²z ð_∞

0 kωsa^^†_Rω_sa^Rωsdωs (35) K^SQ¼ �iℏ

2��β_p

��

�² ð∞

ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p 1

Δk�e^iΔkz�1�

� �

a^^†_R_ω_sa^^†_R_ω_idωsþH:c: (36) and

K_SPM¼ �ℏz��β_p

��

�²kω_pb_p^∗b_p (37)

whereβ ω� �_p

�β_pandΔk¼2kω_p�kω_s�kω_iare the chromatic dispersions.

Similar to the previous classical treatment, one can introduce the co-rotating framework by replacing the field operators with

a^_Rω_sð Þ ¼z a^~_Rω_sð Þz e²ⁱj j^β^p²^k^ωsz and bpð Þ ¼z b~_pð Þz eⁱj j^β^p²^k^ωpz (38) In this framework, one can derive the following differential equation

∂a^~Rωsð Þz

∂z ¼ i ℏ

dzK^₁,a^~_Rω_s

� �

¼iβ²_p ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p e^iΨ²^{ð Þz}^ω^s a^~^†_Rω_i (39) Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric…

DOI: http://dx.doi.org/10.5772/intechopen.89305

where the term H0represents the linear contributions to the energy of the system, while H₁is the first-order nonlinear contribution.

This Hamiltonian can be converted to its quantum form promoting the field Φðx, tÞto the quantum operatorΦ^ðx, tÞ.

In direct analogy with Eq. (11), one can express the flux operator in terms of continuous-mode functions [34], such asH^₀is diagonal in the plane-waves unperturbed modes decomposition:

Φ^ðx, tÞ ¼ X

ν¼L, R

ð_∞

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏl xð Þ 4πkωð Þx s

a^νωe ⁱ^ð^�k^ω^{ð Þx�}^x ^ω^t^ÞþH:c:

" #

dω (25)

H^₀¼ X

ν¼R, L

ð_∞

0�ℏω^a^†_νωa^νω�

dω (26)

(where the zero-point energy, which does not influence the dynamics of the amplifier, has been omitted).

H^1¼H^CPMþH^SQþHSPM (27) In the expressions of these contributions, the fast rotating terms and the highly phase mismatched left-moving field have been neglected:

H^CPM¼ � ℏ 2π

ð∞

0dωsdωidΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p β^∗� �Ωp

β�Ωp⁰�

ϒ ω� s,ω_i,Ωp,Ωp⁰� a^^†_Rω_sa^Rωi

þH:c

(28) describes the cross-phase modulation,

H^SQ ¼ � ℏ 4π

ð_∞

0dω_sdω_idΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffi kωskωi

p β� �Ωp

β�Ωp⁰�

ϒ ω� s,Ωp,ω_i,Ωp⁰�

a^^†_Rω_sa^^†_Rω_iþH:c:

(29) describes the broadband squeezing, and

HSPM¼ � ℏ 4π

ð∞

0dωsdωidΩpdΩp⁰ ffiffiffiffiffiffiffiffiffiffiffiffiffi kωskωi

p β^∗� �Ωp

β�Ωp⁰�

ϒ ω� s,ω_i,Ωp,Ωp⁰�

b^∗ð Þbω_s ð Þω_i þH:c:

(30) Advances in Condensed-Matter and Materials Physics - Rudimentary Research to Topical …

describes the self-phase modulation.βð ÞΩ is the dimensionless pump amplitude, proportional to the ratio between the pump current IJ�Ωp�

and the critical current of the junctions Ic:

β� �Ωp

¼I_J� �Ωp

4Ic (31)

The functionϒ ωð 1,ω₂,ω₃,ω₄Þis the phase matching function, defined as ϒ ωð 1,ω₂,ω₃,ω₄Þ ¼

ð_z

a^^out_R_ω¼U^a^_RωU^^†, (33) whereU is the asymptotic unitary evolution operator (approximated to the first^ order inH^1)

U^�U^ð�∞,∞Þ ¼e^�^ℏⁱ^K^{^}¹ where K^1¼K^CPMþK^SQþKSPM (34) Working in the monochromatic degenerate pump limit b�Ωp⁰�

¼b� �Ωp

! b� �Ωp

δ�Ωp�ω_p�

�bp(whereω_pis the pump frequency), the propagators take the form

K^CPM¼ �2ℏ β�� _p��²z ð_∞

0kωsa^^†_Rω_sa^Rωsdωs (35) K^SQ ¼ �iℏ

2��β_p

��

�² ð∞

ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p 1

Δk�e^iΔkz�1�

� �

a^^†_R_ω_sa^^†_R_ω_idωsþH:c: (36) and

K_SPM¼ �ℏz��β_p

��

�²kω_pb_p^∗b_p (37)

whereβ ω� �_p

�β_pandΔk¼2kω_p�kω_s�kω_i are the chromatic dispersions.

Similar to the previous classical treatment, one can introduce the co-rotating framework by replacing the field operators with

a^_Rω_sð Þ ¼z a^~_Rω_sð Þz e²ⁱj j^β^p²^k^ωsz and bpð Þ ¼z b~_pð Þz eⁱj j^β^p²^k^ωpz (38) In this framework, one can derive the following differential equation

∂a^~Rωsð Þz

∂z ¼ i ℏ

dzK^₁,a^~_Rω_s

� �

¼iβ²_p ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

p e^iΨ²^{ð Þz}^ω^s a^~^†_Rω_i (39) Superconducting Josephson-Based Metamaterials for Quantum-Limited Parametric…

DOI: http://dx.doi.org/10.5772/intechopen.89305

and

∂~b_pð Þz

∂z ¼ � dKSPM

dz ,b~p

� �

¼0 (40)

whereΨ2ð Þ ¼ω_s Δkþ2��β_p��² kωp�kωs �kωi

� �

is the total phase mismatch. These latter are formally identical to Eqs. (12) and (13), up to a frequency-dependent normalization of the wave-amplitudes, under the undepleted pump approximation.

Reference [11] derives an exact solution for Eq. (39), being

a^out_Rω_sð Þ ¼z cosh g� ₂ð Þzω_s �

�iΨ2ð Þω_s

2g₂ð Þω_s sinh g� ₂ð Þzω_s �

� �

^~ aRω_s

�

þiβ²_p ffiffiffiffiffiffiffiffiffiffiffiffi kω_skω_i

g₂ð Þω_s sinh g� ₂ð Þzω_s � a^~^†_Rω_i�eⁱ^Ψ2^{ð Þ}²^ω^s^z

(41)

where

g₂ð Þ ¼ω_s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β²_p

��

�

��

�²kω_skω_i� Ψ2ð Þω_s 2

� �2

(42) If a state moves inside a TWJPA of length z¼aN, the power gain will be Gðω_s, aNÞ ¼〈a^~^out_R_ω_sðaNÞja^~^out_R_ω^†_sðaNÞ〉=〈a^~_Rω_sja^~^†_R_ω_s〉.

Dalam dokumen Advances in Condensed- Matter and Materials Physics (Halaman 90-94)