Patent ID: 12228624

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Exemplary embodiments of the invention will now be described with reference to the drawings.

The SNR in magnetic resonance has the following key dependencies:

SNR∝ω⁢MVm⁢Nrms⁢Q
where ω is the resonance frequency, M is the magnetization, Vmis the mode volume, Q is the resonator quality factor and Nrmsis the Johnson noise. Since the intrinsic signal of the spins is extremely weak (e.g., due to their thermal magnetization distribution of a 1 ppm difference in the aligned versus anti-aligned orientation for nuclei at ambient temperatures and typical magnetic fields of several Teslas) in practice a variety of solutions have been devised to increase the SNR.

For example, by increasing the magnetic field the resonance frequency, ω and the magnetization, M will increase proportionally and over the past several decades one direction of research has been to develop superconducting magnetic field generators that produce greater and greater fields.

In nuclear magnetic resonance (NMR) applications, the magnetic field exceeds 20 Tesla and in magnetic resonance imaging (MRI) applications up to 7-Tesla systems are clinically available, while 11-Tesla research systems are used for specialized studies.

An exemplary aspect of the present invention is directed to increasing magnetization by redistributing the intrinsic thermal distribution on-demand, without the introduction of an external agent to achieve higher polarization using cQED techniques.

The magnetization thermal equilibrium distribution is given by

N+-N-N=tan⁢h⁡(γ⁢ℏ⁢B02⁢KB⁢TS)
where N is the total number of spins while N+and N−are the number of aligned and anti-aligned spins, respectively and γ is the spin gyromagnetic ratio (28.2 GHz/T for electrons and 42.5 MHz/T for protons), h is Planck's constant divided by 2π, B0is the magnetic field, KBis the Boltzmann constant and TSis the spin temperature, as shown inFIG.1.

If the nuclei or spin temperature can be reduced to 1 Kelvin from room temperature, then the magnetization can be enhanced by a factor of 100 or equivalently the SNR can be increased by this same factor. This then motivates cQED cavity cooling techniques to reduce the spin temperature. The scheme requires resonantly coupling the spins to a resonator or cavity that is at sub-Kelvin cryogenic temperatures to enable removing the excess entropy at the higher spin temperature by accelerating the spontaneous emission process so it exceeds the natural thermal relaxation process in order for the spins to thermalize to the lower resonator or cavity temperature. The spontaneous emission acceleration is achieved through an enhanced coupling of the spins to the resonator or cavity by quantum squeezing.

FIG.2shows a spin ensemble10, having frequency ωs, in a static magnetic field (Bo)11enclosed by a thermally insulated squeezing cavity12, having frequency ωa, and a broadband squeezing cavity13, having frequency ωc, in a cryogen14. The spin ensemble10is in a natural thermal distribution state.

InFIG.3, the spin ensemble10is hyperpolarized after squeezed cooling where the spontaneous emission has overcome the natural thermally relaxed state.

Based on the above concept, an exemplary aspect of the present invention is shown inFIG.4, in which, in a hyperpolarizing system1, three distinct cavities are thermally separated.

More specifically, the exemplary hyperpolarizing system1includes the spin ensemble10immersed in a static magnetic field and being subject to hyperpolarization in a cryogen14, a cold cavity15, having frequency ωb, that is at a temperature below a spin temperature of the spin ensemble10, the spin ensemble10being resonantly coupled to the cold cavity15, a squeezing cavity12that is detuned from the cold cavity15, the squeezing cavity12being coupled to the cold cavity15and squeezing the cold cavity15such that coupling of the cold cavity15to the spin ensemble10increases to a sufficient strength (ranging from 3 dB to as much as 100 dB) to enable a spontaneous emission process to exceed a spin thermal relaxation process of the spin ensemble10, and a broadband squeezing cavity13coupled to the squeezing cavity12.

The broadband squeezing cavity13may have a same squeezing parameter as the squeezing cavity12and being phase-matched with the squeezing cavity12such that differences in phase (ϕ) between the squeezing cavity12and the broadband squeezing cavity13are integer odd multiples of pi (π) for suppressing or canceling squeezed noise or heat generated by the squeezing cavity12as Δϕ=nπ; n=±1, ±3, ±5 . . . .

Prior to the resonantly coupling the spin ensemble10, the cold cavity15, squeezing cavity12, and the broadband squeezing cavity13are thermally insulated and through cryocooling are maintained, for example, at temperatures between 10 milli-Kelvin and 10 Kelvin.

The hyperpolarization of the spin ensemble10is operated at a temperature above the cold cavity and up to a room temperature (e.g., ambient).

The hyperpolarization of the spin ensemble10is achieved within a time less than a time of the spin thermal relaxation process of the spin ensemble.

In the squeezing of the cold cavity15, then the cold cavity15couples with the spin ensemble10with sufficient strength to enable the spontaneous emission process to exceed the spin thermal relaxation process by increasing a polarization of the spin ensemble10more than a thermal equilibrium polarization of the spin ensemble10.

The coupling of the cold cavity15to the squeezing cavity12includes quantum squeezing to increase the resonant coupling of the spin ensemble10to the cold cavity15and reach sufficient strength (3 dB-100 dB) to enable spin cooling.

The quantum squeezing of the cold cavity15is implemented by modulating a squeezing cavity resonance frequency.

FIG.5shows another exemplary aspect of the present invention according toFIG.4that is directed to an exemplary method2to hyperpolarize the spin ensemble10.

As shown inFIG.5, the method2includes, in step21, resonantly coupling the spin ensemble10to the cold cavity15that is at a temperature below a spin temperature of the spin ensemble10.

The exemplary method2further includes step22for coupling the cold cavity15to the squeezing cavity12that is detuned from the cold cavity15, the squeezing cavity12squeezing the cold cavity15such that coupling of the cold cavity15to the spin ensemble10increases to a sufficient strength (3 dB-100 dB) to enable a spontaneous emission process to exceed a spin thermal relaxation process of the spin ensemble10.

Finally, in step23, method2includes coupling the squeezing cavity12to a broadband squeezing cavity13having a same squeezing parameter as the squeezing cavity12and being phase-matched with the squeezing cavity12for suppressing or canceling squeezed noise or heat generated by the squeezing cavity12.

It is noted that the above steps are exemplary aspects of the described method2, and the method can be performed with additional or reduced steps.

FIG.6shows cavity frequency relationships according to the exemplary embodiment demonstrated in the exemplary system1shown inFIG.4.

The power versus frequency plot ofFIG.6shows the spin ensemble resonance frequency ωsin relation to the cold cavity resonance frequency ωb, the detuned squeezing cavity ωa, and the broadband squeezing cavity ωc.

Another exemplary aspect of the present invention is shown inFIG.7, in which, in a hyperpolarizing system3, two distinct cavities are thermally separated in a cryogen14and enclose the spin ensemble10.

The exemplary system3includes a spin ensemble10immersed in a static magnetic field, a squeezing cavity12held at a temperature below a spin temperature of the spin ensemble10, the spin ensemble10being resonantly coupled to the squeezing cavity12to enable a spontaneous emission process to exceed a spin thermal relaxation process of the spin ensemble10, and a broadband squeezing cavity13coupled to the squeezing cavity12, the broadband squeezing cavity13having a same squeezing parameter as the squeezing cavity12and being phase-matched with the squeezing cavity12for suppressing or canceling squeezed noise or heat generated by the squeezing cavity12, thereby increasing the polarization of the spin ensemble10more than a thermal equilibrium polarization of the spin ensemble10.

The differences in phase Δϕ between the squeezing cavity12and the broadband squeezing13cavity, for example, are integer odd multiples of pi (π) as Δϕ=nπ; n=±1, ±3, ±5 . . . .

Prior to the resonantly coupling of the spin ensemble10, the squeezing cavity12and the broadband squeezing cavity13are thermally insulated and through cryocooling are maintained, for example, at temperatures between 10 milli-Kelvin and 10 Kelvin.

The hyperpolarization of the spin ensemble10is operated at a temperature above the squeezing cavity12and up to room temperature (e.g., ambient).

The hyperpolarization of the spin ensemble10is achieved within a time less than a time of the spin thermal relaxation process of the spin ensemble.

In the resonantly coupling of the spin ensemble10to the squeezing cavity12, the squeezing cavity12squeezes the spin ensemble10with sufficient strength to enable the spontaneous emission process to exceed the spin thermal relaxation process by increasing a polarization of the spin ensemble more than a thermal equilibrium polarization of the spin ensemble10.

FIG.8shows another exemplary aspect of the present invention according toFIG.7that is directed to another exemplary method4to hyperpolarize the spin ensemble10.

As shown inFIG.8, the exemplary method4includes, in step41, resonantly coupling the spin ensemble10to the squeezing cavity12, which is at a temperature below a spin temperature of the spin ensemble10, to enable a spontaneous emission process to exceed a spin thermal relaxation process of the spin ensemble10.

Method4further includes, in step42, coupling the squeezing cavity12to the broadband squeezing cavity13having a same squeezing parameter as the squeezing cavity12and being phase-matched with the squeezing cavity12for suppressing or canceling squeezed noise or heat generated by the squeezing cavity12.

It is noted that the above steps are exemplary aspects of the described method4, and the method can be performed with additional or reduced steps.

FIG.9shows cavity frequency relationships according to another exemplary embodiment demonstrated in the exemplary system3shown inFIG.7.

The power versus frequency plot ofFIG.9shows the spin ensemble resonance frequency ωsin relation to the squeezing cavity ωaand the broadband squeezing cavity ωc.

Based on the exemplary systems1and3shown inFIGS.4and7, and their cavity frequency relationship demonstrated inFIGS.6and9, respectively, hyperpolarizing systems can be achieved for room temperature polarization in the spin ensemble10faster than the typical relaxation times of the spins through non-radiative processes or thermal processes.

In these systems, for example, quantum squeezing is used to enhance the spin ensemble coupling to the cold cavity15and enable entropy removal or spin cooling.

The cold cavities adapted to be thermally insulated and through cryocooling maintained at temperatures well below ambient or room temperature. The squeezing cavity12, the broadband squeezing cavity13, and the cold cavity15are distinct in the hyperpolarizing system1, and the squeezing cavity12and the broadband squeezing cavity13are distinct in the hyperpolarizing system3.

The broadband squeezing cavity13eliminates the strong squeezed noise generated inside the squeezing cavity12that couples into the cold cavity15. In the hyperpolarizing system1, as shown inFIG.6, the squeezing cavity12is detuned from both the spin ensemble10and cold cavity15, while also being coupled to both of these cavities. The cold cavity15is resonant with the spin ensemble10.

The quantum squeezing of the squeezing cavity12is implemented by modulating a squeezing cavity resonance frequency.

The squeezing of the squeezing cavity12increases coupling of the squeezing cavity12to a cryogen14or to a cold load, thereby increasing a cooling efficiency.

Exemplary system configuration1shown inFIG.4enables strong coupling of the spin ensemble10to the cold cavity15by quantum squeezing. The hyperpolarizing systems1and3further include coupling in a broadband squeezed noise to eliminate the squeezing cavity12induced strong noise.

The technical aspects of the processes in the exemplary systems1and3discussed above are described in more details in the following section.

Resonator (or Cavity)-Spin Coupling

The quantum mechanical description for the resonator-spin coupling as shown in the above-described systems1and3, shown inFIGS.4and7, respectively, is given by the Jaynes-Cummings model. In this model, the resonator is characterized by a lumped-element LC oscillator whose quantized Hamiltonian is analogous to a quantum harmonic oscillator:

H=φˆ22⁢L+qˆ22⁢C
in terms of the flux and charge operators {circumflex over (φ)} and {circumflex over (q)}, respectively. As usual ϕ=LI and q=CV.

To avoid dealing with quadratic operators, the customary prescription calls for factorizing the Hamiltonian leading to the annihilation and creation operators defined by

a^=12⁢ℏ⁢Z0⁢(φˆ+i⁢Z0⁢qˆ)a^++=12⁢ℏ⁢Z0⁢(φˆ-i⁢Z0⁢qˆ)
respectively

Z0=LC
is the impedance and

ω0=1LC
is the resonance frequency of the circuit. This enables us to rewrite the harmonic oscillator Hamiltonian in the typical form as:

H=ℏ⁢ω0(a^+⁢a^+12)
where the zero-point energy,

12⁢ℏ⁢ω0,
is a consequence of the noncommutativity of the operators [{circumflex over (φ)}, {circumflex over (q)}]=iℏ; a central feature common to all quantum systems and the source of the quantum fluctuations key to quantum coupling mechanisms and the invention to be described below.

In the context of electromagnetics, the eigenstates of the above Hamiltonian are known as the Fock (or number) states {|n>} and obey:

a^⁢❘"\[LeftBracketingBar]"n〉=n⁢❘"\[LeftBracketingBar]"n-1〉a^+⁢❘"\[LeftBracketingBar]"n〉=n+1⁢❘"\[LeftBracketingBar]"n+1〉n^≡a^+⁢a^→n^⁢❘"\[LeftBracketingBar]"n〉=a^+⁢a^⁢❘"\[LeftBracketingBar]"n〉=n⁢❘"\[LeftBracketingBar]"n〉→H⁢❘"\[LeftBracketingBar]"n〉=ℏ⁢ω0(n+12)⁢❘"\[LeftBracketingBar]"n〉

The lowest energy state (the zero-point energy eigenstate) is known as the vacuum state:
â|0>=0
from which the number state can be derived recursively as

❘"\[LeftBracketingBar]"n〉=(a^+)nn⁢❘"\[LeftBracketingBar]"0〉

With these definitions, the voltage between the capacitors and the current in the inductors can be quantum mechanically expressed as

Vˆ=qˆC=i⁢ω0⁢ℏ⁢Z02⁢(a^+-a^)Iˆ=φˆL=ω0⁢ℏ2⁢Z0⁢(a^+-a^)

Then, the rms vacuum fluctuations of the current and voltage in the resonator ground state can be calculated:

δ⁢I2=〈0⁢❘"\[LeftBracketingBar]"Iˆ2❘"\[RightBracketingBar]"⁢0〉=ℏ⁢ω022⁢Z0δ⁢V2=〈0⁢❘"\[LeftBracketingBar]"Vˆ2❘"\[RightBracketingBar]"⁢0〉=ℏ⁢Z0⁢ω022

Due to the factor of ˜1000 difference between the electron and proton gyromagnetic ratio, electrons couple much more strongly than protons, thereby yielding stronger magnetization and enable efficient cooling. Experiments performed using electron spin resonance exploit this strong coupling to implement cavity cooling. The spin-resonator coupling denoted by g is mediated by the quantum fluctuations of the resonator or cavity. Spins, as shown for example inFIG.3, have a dipole magnetic moment which in a magnetic field experience a torque and their energy quantum mechanically is expressed through the interaction Hamiltonian H=−{circumflex over (μ)}·{circumflex over (B)} where {circumflex over (μ)}=γŜ is the quantized moment arising from the spin angular momentum Ŝ=mℏ{circumflex over (σ)} with eigenvalues m=±½. The “+” eigenvalue is the aligned state while the “−” eigenvalue is the anti-aligned orientation.

The {circumflex over (σ)}s are the Pauli spin operators given by

σˆx=(0110),σˆy=(0-ii0)⁢and⁢σˆz=(100-1).

Spin state raising and lowering operators are defined through

σˆ±=12⁢(σˆx±i⁢σˆy):

σˆ+=(0100)⁢and⁢σ^-=(0010)

Since the resonator magnetic field directly depend on the currents flowing through it, Î∝δI(â++â), as shown for example inFIG.3, the spins experience the magnetic field quantum fluctuations of the resonator arising from the zero-point energy as expressed through the current fluctuation SI. For electrons at GHz frequencies, this is on the order of tens of nA and, for protons at MHz frequencies, this is on the order of sub pA. The corresponding coupling strength is derived from the coupling interaction energy
Hint=−{circumflex over (μ)}·{circumflex over (B)}=−γŜ·δ{circumflex over (B)}
and in the spin basis {|+>, |−>} the coupling strength, g, is given by

g=γ⁢❘"\[LeftBracketingBar]"〈+❘"\[LeftBracketingBar]"S^·δ⁢B^❘"\[RightBracketingBar]"-〉❘"\[RightBracketingBar]"=γ2⁢❘"\[LeftBracketingBar]"δ⁢B❘"\[RightBracketingBar]".

For a nominal NMR resonant frequency of 100 MHz, we calculate g as follows:

g=γ⁢❘"\[LeftBracketingBar]"〈+❘"\[LeftBracketingBar]"δ⁢B^·S^❘"\[RightBracketingBar]"-〉❘"\[RightBracketingBar]"=γp2⁢❘"\[LeftBracketingBar]"δ⁢B❘"\[RightBracketingBar]"=γp2⁢μ0⁢δ⁢i2⁢π⁢f⁡(r)=γp2⁢μ0⁢ω02⁢π⁢f⁡(r)⁢ℏ2⁢Z0⁢ℏ2⁢Z0=1⁢0-3⁢4⁢J·s1⁢02⁢J·s/C2=10-1⁢8⁢C=1⁢0-1⁢8⁢A·s

The constants are

g=(4⁢2.5⁢8×1⁢06⁢HzT)⁢(4⁢π×1⁢0-7⁢NA2)⁢(2⁢π×1⁢08⁢Hz)2⁢(2⁢π)⁢(1⁢0-2⁢m)⁢1⁢0-1⁢8⁢A·sg=(2.1)⁢(4⁢π×1⁢08)⁢(1⁢02)⁢(1⁢0-1⁢8)⁢Hz=8⁢π·10-8
given

T=NA·m
and we have assumed ƒ(r)≈10−2m. Therefore, for protons, this is on the order of micro-Hertz. Substituting the electron gyromagnetic ratio and nominal ESR resonant frequency of 10 GHz gives a coupling constant on the order of a milli-Hertz. The regime where radiative processes start to exceed thermal processes is approached when g can reach values of about 1 Hertz and resonator Q-values are in excess of 105. For protons, this implies enhancing g by factors ranging from 3 dB-100 dB depending on operating temperatures and other considerations.

In ESR, by carefully designing experiments where the electrons are as close as possible to the resonator (e.g., within microns so the mode volume is minimized), and using very high Q resonators, a strong enough coupling is established to the resonator and cavity cooling, for example cold cavity15in system1, can be achieved through the Purcell effect: the regime where spontaneous and radiative emissions dominate over thermal processes such as phonon and dipole-dipole interactions that give rise to the Boltzmann distribution. For protons, the coupling is several orders of magnitude weaker and cavity cooling is negligible. However, the quantum fluctuations of the resonator arising from the zero-point energy obey Heisenberg's uncertainty principle:

δ⁢I⁢δ⁢V=ℏ2

If the current fluctuation can be amplified and the voltage fluctuation proportionally suppressed so their product still obeys Heisenberg's uncertainty principle, then the resonator-spin coupling can be enhanced and reach values whereby cavity cooling, for example cold cavity15in system1, can be enabled. This ‘SQUEEZING’ or anti-squeezing of the current fluctuation can be implemented through quantum optical techniques developed over the last decade-and-a-half in the new field of cQED.

Quantum Squeezing of an LC Resonator

If the resonator Hamiltonian changes from one set of LC values to another an infinitesimal time later, then the change appears as if the resonator had been squeezed. This motivates how to bring about the necessary squeezing, for example squeezing cavity12in system1, to amplify the ground state current fluctuation. Starting with the original Hamiltonian and a given set of LC parameters a change to a new set implies:

H1=φˆ22⁢L1+qˆ22⁢C1→Δ⁢HH2=φˆ22⁢L2+qˆ22⁢C2Z01=L1C1→ΔZ02=L2C2

The associated fluctuations change as follows:

δφ1=ℏZ012δ⁢q1=ℏ2⁢Z01}→Δ{Z01Z02⁢ℏ⁢Z022≡er⁢ℏ⁢Z022=er⁢δ⁢φ2Z02Z01⁢ℏ2⁢Z02≡e-r⁢ℏ2⁢Z02=e-r⁢δ⁢q2;er=Z01Z02

Consequently eris the squeezing factor with r as the squeezing parameter. (Note that δφ1δq1=erδΦ2e−rδq2=δφ2δq2=ℏ/2.) The flux and charge operators change similarly:

φˆ1=ℏ⁢Z012⁢(a^1++a^1)qˆ1=-i⁢ℏ2⁢Z01⁢(a^1+-a^1)}→Δφˆ2=ℏ⁢Z022⁢(a^2++a^2)qˆ2=-i⁢ℏ2⁢Z02⁢(a^2+-a^2)

In an infinitesimal time δt<<1, the changes are small enough that the operators approximately equal each other. Therefore

φ^1≈φ^2q^1≈q^2}→{(a^1++a^1)=Z02Z01⁢(a^2++a^2)=e-r(a^2++a^2)(a^1+-a^1)=Z01Z02⁢(a^2+-a^2)=er(a^2+-a^2)

From this set of equations, the individual operators can be found:

a^1=12[(er+e-r)⁢a^2+(er-e-r)⁢a^2+]=cosh⁢r⁢a^2+sinh⁢r⁢a^2+a^1+=12[(er-e-r)⁢a^2+(er+e-r)⁢a^2+]=sinh⁢r⁢a^2+cosh⁢r⁢a^2+a^2=12[(e-r+er)⁢a^1+(e-r-er)⁢a^1+]=cosh⁢r⁢a^1-sinh⁢r⁢a^1+a^2+=12[(e-r-er)⁢a^1+(e-r+er)⁢a^1+]=-sinh⁢r⁢a^1+cosh⁢r⁢a^1+

These set of transformations are known as Bogoliubov-Valatin transforms. From the first set of these equations, we can derive the squeezing operator. Operationally, this can be obtained by considering the action of â1on the ground state |0>(1)which can be related to the action of the â2& â2+operators as follows:
â1|0>(1)=0→(coshr â2+sinhr â2+)|0>(1)=0

To obtain the action of â2& â2+on the vacuum state |0>(1)it can be related to the basis sets {|0>(2)} through a power series perturbation expansion:

❘"\[LeftBracketingBar]"0〉(1)→C0❘"\[LeftBracketingBar]"0〉(2)+C2⁢a^2+⁢a^2+❘"\[LeftBracketingBar]"0〉(2)+C4⁢a^2+⁢a^2+⁢a^2+⁢a^2+❘"\[LeftBracketingBar]"0〉(2)+…≡α⁡(r)⁢exp[-12⁢f⁡(r)⁢a^2+⁢a^2+]⁢❘"\[LeftBracketingBar]"0〉(2)
where only the even terms are included since the ground state has even symmetry. The power series expansion has a succinct exponential representation in anticipation of the solution. The constants α(r) and ƒ(r) are determined through normalization conditions and the actions of the operators on the exponential expression above:

a^1⁢❘"\[LeftBracketingBar]"0〉(1)=(cosh⁢r⁢a^2+sinh⁢r⁢a^2+)⁢❘"\[LeftBracketingBar]"0〉(1)=α⁡(r)⁢(cosh⁢r⁢a^2+sinh⁢r⁢a^2+)⁢exp[-12⁢f⁡(r)⁢a^2+⁢a^2+]⁢❘"\[LeftBracketingBar]"0〉(2)=0
to finally yield

❘"\[LeftBracketingBar]"0〉(1)=1cosh⁢r⁢exp[-12⁢tanh⁢r⁢a^2+⁢a^2+]⁢❘"\[LeftBracketingBar]"0⁢〉(2).
The operator

S⁡(r)=1cosh⁢r⁢exp[-12⁢tanh⁢r⁢a^2+⁢a^2+]
is the squeeze operator which can be recast in the more familiar form

S⁡(r)=exp[-r2⁢(a^2+⁢a^2+-a^2⁢a^2)]≡exp[-r2⁢(a^2+2-a^22)]
by using the Schwinger disentangling theorem.
Squeezing of a Resonator

In hyperpolarizing systems such as system1shown inFIG.4, in cQED, bilinear or quadratic operators such as needed for the squeeze operator are interactions generated by parametric drives and nonlinear couplings. These types of drives or interactions modify the resonator to an extent where the critical resonance condition required for magnetic resonance is drastically altered. An alternative way to generate the squeeze operator is to modulate the resonance frequency. Therefore, a non-adiabatic or sudden change to the resonator impedance is considered that alternates between a lower impedance and a higher one in such a way that when the squeezing, for example by squeezing cavity13in system1, is achieved, the resonance condition is met.

Starting with the LC resonator Hamiltonian

H0=φ^22⁢L0+q^22⁢C0=1L0⁢(φ^22+L0C0⁢q^22)=12⁢L0⁢(φ^2+Z02⁢q^2)
at time (t=0) and operators

q^=i⁢ℏ2⁢Z0⁢(a^+-a^)&⁢φ^=ℏ⁢Z02⁢(a^++a^)
as previously defined. When the impedance Z0is changed suddenly to anew value Z(t), the Hamiltonian is given by, after letting (ℏ=1),

H⁡(t)=12⁢L⁡(t)[φ^2+Z2(t)⁢q^2]=12⁢L⁡(t)[(Z02)⁢(a^++a^)2-Z2(t)⁢(12⁢Z0)⁢(a^+-a^)2]=14⁢L⁡(t)⁢{Z0(a^+2+a^+⁢a^+a^⁢a^++a^2)-[Z2(t)Z0]⁢(a^+2-a^+⁢a^-a^⁢a^++a^2)}=14⁢L⁡(t)⁢{(Z0+[Z2(t)Z0])⁢(a^+⁢a^+a^⁢a^+)+(Z0-[Z2(t)Z0])⁢(a^+2+a^2)}=14⁢L⁡(t)⁢Z⁡(t)⁢{([Z0Z⁡(t)]+[Z⁡(t)Z0])⁢(a^+⁢a^+a^⁢a^+)+([Z0Z⁡(t)]-[Z⁡(t)Z0])⁢(a^+2+a^2)}
H(t) is now in a form to be diagonalized by a Bogoliubov-Valatin transform when the following substitutions are made

1L⁡(t)⁢Z⁡(t)=1L⁡(t)⁢L⁡(t)C⁡(t)=1L⁡(t)⁢C⁡(t)=ω⁡(t)&⁢r⁡(t)=-12⁢ln[Z⁡(t)Z0]
so that

H⁡(t)=ω⁡(t)2⁢{[e2⁢r⁡(t)+e-2⁢r⁡(t)2]⁢(a^+⁢a^+a^⁢a^+)+[e2⁢r⁡(t)-e-2⁢r⁡(t)2]⁢(a^+2+a^2)}=ω⁡(t)2⁢{cosh[2⁢r⁡(t)]⁢(a^+⁢a^+a^⁢a^+)+sinh[2⁢r⁡(t)]⁢(a^+2+a^2)}.

The new operators {{circumflex over (b)}, {circumflex over (b)}+} in the diagonal frame are then expressed as

B=T-1⁢A→[b^⁢(t)b^+⁢(t)]=[cosh[r⁡(t)]-sinh[r⁡(t)]-sinh[r⁡(t)]cosh[r⁡(t)]][a^a^+]=[cosh[r⁡(t)]⁢a^-sinh[r⁡(t)]⁢a^+cosh[r⁡(t)]⁢a^+-sinh[r⁡(t)]⁢a^]=[S⁡(t)⁢a^⁢S+(t)S⁡(t)⁢a^+⁢S+(t)]
and S(t) is the squeeze operator:

S⁡(t)=exp[-r⁡(t)2⁢(a^2-a^+2)].

Finally,

H⁡(t)=ω⁡(t)2[b^+(t)⁢b^(t)+b^(t)⁢b^+(t)].

This Hamiltonian, in the original frame, can be written as

S+(t)⁢H⁡(t)⁢S⁡(t)-i⁢S+(t)⁢∂∂tS⁡(t)=ω⁡(t)2⁢a^+⁢a^+i⁢r.(t)⁢(a^2-a^+2).

The time-dependence of the squeezing parameter is

r˙(t)=-12⁢∂∂t{l⁢n[Z⁡(t)Z0]}=-12⁢Z⁢(t).Z⁡(t)

When the squeezing modulation is turned off, the time-dependence goes to zero and the squeezing parameter reaches its final asymptotic value.

Enhanced Coupling of Spin-Ensembles to a Resonator

In hyperpolarizing systems, such as system1shown inFIG.4, the full Hamiltonian of spins magnetically coupled to a resonator and coherently interacting with it are well described by the Tavis-Cummings model:

H0=ω0⁢a^+⁢a^+ωs2⁢∑j=1Nσˆzj+g2⁢∑j=1N(a^+⁢σˆj¯+a^⁢σˆ+j)+HE+HS⁢E=ω0⁢a^+⁢a^+ωs⁢Jˆz+g⁡(a^+Jˆ-+a^⁢Jˆ+)+HE+HS⁢E
where N is the number of spins and Ĵzand Ĵ±represent the collective spin interactions.

The last two terms represent the interaction of the resonator with an external, larger environment such as a cryostat. A standard bosonic environment is assumed and represented by the set of modes {{circumflex over (d)}k+, {circumflex over (d)}k}. These interactions are given by

HE=∑k=1Mgk⁢dˆk+⁢dˆk&⁢HS⁢E=∑k=1Mgk(dˆk+⁢a^+dˆk⁢a^+).

When the resonator is subjected to sudden mode or impedance changes, it is squeezed and the instantaneous Hamiltonian effectively transforms to
H(t)=S+(t)H0S(t)−iS+(t)Ŝ(t)=ω(t)â+â+ωsĴz+i{dot over (r)}(t)(â2−â+2)+gS+(t)(â++â)S(t)Ĵx+HE+S+(t)HSES(t).

Note that the resonator is now at the frequency ω(t) and the term i{dot over (r)}(t)(â2−â+2) is the time-dependent squeezing. When the squeezing modulation is turned off the time-dependence goes to zero and the squeezing parameter has reached its final asymptotic value. With the relations

[S+(t)⁢a^⁢S⁡(t)S+(t)⁢a^+⁢S⁡(t)]=[cos⁢h[r⁡(t)]⁢a^+sin⁢h[r⁡(t)]⁢a^+cos⁢h[r⁡(t)]⁢a^++sin⁢h[r⁡(t)⁢a^]
the field operators (â++â) in the Hamiltonian transform to
S+(t)(â++â)S(t)=cosh[r(t)]â++sinh[r(t)]â+cosh[r(t)]â+sinh[r(t)]â+=er(t)(â+â).

Plugging this into the Hamiltonian results in
H(t)=ω(t)â+â+ωsĴz+ger(t)(â++â)Ĵx+HE+HSE(t)
revealing the exponentially enhanced spin-resonator coupling. The resonator-environment interaction is also enhanced as

HS⁢E(t)=S+(t)⁢HS⁢E⁢S⁡(t)=∑k=1Mgk[dˆk+⁢S+(t)⁢a^⁢S⁡(t)+dˆk⁢S+(t)⁢a^+⁢S⁡(t)]=∑k=1Mgk⁢{dˆk+[cos⁢h[r⁡(t)]⁢a^+sin⁢h[r⁡(t)]⁢a^+]+dˆk[cos⁢h[r⁡(t)]⁢a^++sin⁢h[r⁡(t)]⁢a^]}=∑k=1Mgk⁢{dˆk+[(er⁡(t)+e-r⁡(t)2)⁢a^+(er⁡(t)-e-r⁡(t)2)⁢a^+]}+dˆk[(er⁡(t)+e-r⁡(t)2)⁢a^++(er⁡(t)-e-r⁡(t)2)⁢a^]}=er⁡(t)2⁢(a^++a^)⁢∑k=1Mgk(dˆk++dˆk)-e-r⁡(t)2⁢(a^++a^)⁢∑k=1Mgk(dˆk++dˆk).

Finally, after dropping the last term because it is exponentially suppressed

HS⁢E(t)=er⁡(t)2⁢(a^++a^)⁢∑k=1Mgk(dˆk++dˆk).

Therefore, squeezing the resonator through a sudden change in its mode or impedance does enhance the spin-resonator coupling, for example by squeezing cavity12in system1shown inFIG.4as well as the resonator-environment coupling.

Dissipative Squeezing

As discussed above, by minimizing the mode volume and increasing the Q-factor of resonators, researchers have been successful at achieving strong spin-resonator coupling whereby the radiative relaxation dominates thermal relaxation mechanisms.

In the present invention, an exemplary object is to enhance the resonator coupling through squeezing, as shown for example in systems1and3, and to reach regimes where the radiative processes dominate the thermal relaxation interactions.

The dissipative evolution of the hyperpolarizing systems, such as above-mentioned systems1and3, without the broadband squeezing cavities13, is governed by the Master equation and it is assumed to be Markovian so that the environment, be it the cryogen14or the molecular environment of the spin ensemble10, is memory-less, and it does not influence the system. In the standard prescription of dissipative evolution, the full density matrix (the state of the system) is given by:
ρ=ρσ⊗ρc⊗ρR
where, ρσis the density matrix for the spins (e.g., such as spin ensemble10), ρcthe cavity or resonator (e.g., such as squeezing cavity12) and ρRthe reservoir or environment (e.g., such as cryogen14). ρ is initially assumed to be factorizable and the three subsystems are represented as being in a simple tensor product state. The Master equation is derived by moving into the triply rotating interaction frame of the system-environment and then tracing away the environment (reservoir) degrees-of-freedom:
ρs=ρσ⊗ρc=TrR{ρ}=TrR{ρσ⊗ρc⊗ρR}.

This is obtained as follows. First, the dynamics of the system is given by the Liouville-von Neuman equation:

i⁢ℏ⁢∂∂tρ=[H⁡(t),ρ].

The interaction frame transformation is implemented by the unitary-similarity transform:
U′=ei/ℏH′t
where H′ is a triply rotating frame

H′=ω⁡(t)⁢a^+⁢a^+ωs⁢Jˆz+∑k=1Mgk⁢dˆk+⁢dˆk.
In the new frame we get

i⁢ℏ⁢∂∂tρ˜=[H~(t)-H′,ρ˜].

The transformed Hamiltonian is

H~(t)-H′=ger⁡(t)⁢ei⁢ω⁡(t)⁢a^+⁢at^(a^++a^)⁢e-i⁢ω⁡(t)⁢a^+⁢at^⁢ei⁢ωs⁢Jˆz⁢t⁢Jˆx⁢e-i⁢ωs⁢Jˆz⁢t+er⁡(t)2⁢ei⁢ω⁡(t)⁢a^+⁢a^⁢t(a^++a^)⁢e-i⁢ω⁡(t)⁢a^+⁢a^⁢∑k=1Meit⁢∑j=1Lgj⁢dˆj+⁢dˆj(dˆk++dˆk)⁢e-it⁢∑j=1L⁢gj⁢dˆj+⁢dˆj.

Using the properties:
eiωsĴztĴxe−iωsĴzt=eiωsĴzt(Ĵ++Ĵ−)e−ωsĴzt=(Ĵ+e−ωst+Ĵ−eiωst)
eiω(t)â+ât(â++â)e−iω(t)â+ât=[â+e−ω(t)t+âeiω(t)t]
and for the reservoir term

eit⁢∑j=1L⁢gj⁢dˆj+⁢dˆ(dˆk++dˆk)⁢e-it⁢∑j=1L⁢gj⁢dˆj+⁢dˆj=[dˆk+⁢e-it⁢∑j=1L⁢gj⁢δ⁡(j-k)+dˆk⁢eit⁢∑j=1L⁢gj⁢δ⁡(j-k)]=dˆk+⁢e-i⁢gk⁢t+dˆk⁢ei⁢gk⁢t
the interaction Hamiltonian can be written as (only keeping the energy conserving or flip-flop terms)

H~(t)-H′=ger⁡(t)[a^+⁢e-i⁢ω⁡(t)⁢t+a^⁢ei⁢ω⁡(t)⁢t]⁢(Jˆ+⁢e-i⁢ωs⁢t+Jˆ-⁢ei⁢ωs⁢t)+er⁡(t)2[a^+⁢e-i⁢ω⁡(t)⁢t+a^⁢ei⁢ω⁡(t)⁢t]⁢∑k=1Mgk(dˆk+⁢e-i⁢gk⁢t+dˆk⁢ei⁢gk⁢t)=g⁢er⁡(t)[a^+J^-⁢ei⁢Δ⁢t+a^⁢Jˆ+⁢e-i⁢Δ⁢t]+er⁡(t)2⁢∑k=1Mgk(dˆk+⁢a^⁢e-i⁢vk⁢t+dˆk⁢a^+⁢ei⁢vk⁢t).

Note that fast terms (rotating-wave approximation (RWA)) have been dropped.

The difference frequencies that remain are
Δ=ωs−ω(t) & υk=gk−ω(t).

As is usual, the dynamical equation in the interaction frame can be integrated to second order, the reservoir traced away, the environment correlation functions calculated and leave for the reservoir portion a Lindblad form for the dynamics:

∂∂tρ˜s=-i⁢g⁢er⁡(t)[a^+J^-⁢ei⁢Δ⁢t+a^⁢Jˆ+⁢e-i⁢Δ⁢t,ρ˜s]+κ⁢e2⁢r⁡(t)[2⁢a^⁢ρ˜s⁢a^+-a^+⁢a^⁢ρ˜s-ρ˜s⁢a^+⁢a^].

Having removed the reservoir degrees-of-freedom, we now focus on removing the cavity or resonator from the dynamics (sometimes called “adiabatic elimination of the cavity”), so we get the final Master equation for the reduced spin operators. In the ‘bad cavity’ limit (e.g., the regime where κ>>g), the cavity losses are greater than the coherent dynamics of the spins, and the spins irreversibly lose energy to the reservoir.

Further, given that the Lindbladian (2âρsâ−â+â{tilde over (ρ)}s−{tilde over (ρ)}sâ+â) makes calculations cumbersome, the complexity can be reduced by vectorizing the density matrix or working with superoperators

∂∂tρ˜ˆ^s=Λ⁢(t)⁢ρ˜ˆ^s=(ℒ+Dc)⁢ρ˜ˆ^sℒ=-i[Iˆ⊗H-HT⊗Iˆ]&⁢𝒟c=κ′[(a^+)T⊗a^-12⁢I^⊗a^+a^-12⁢(a^+⁢a^)T⊗Iˆ]
with
H=ger(t)[â+Ĵ−eiΔt+âĴ+e−iΔt]κ′=κe2r(t).

Removing the cavity implies transforming to the dissipator frame of the cavity

ρ~^^s′=e-𝒟c⁢t⁢ρ~^^s→ℒ~(t)=e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢t
so

∂∂tρ~^^s′=ℒ~(t)⁢ρ~^^s′
since

∂∂tρ~^^s′=∂∂t(e-𝒟c⁢t⁢ρ˜ˆˆs)=-𝒟c⁢e-𝒟c⁢t⁢ρ~^^s+e-𝒟c⁢t⁢∂∂tρ~^^s=-𝒟c⁢ρ~^^s′+e-𝒟c⁢t(ℒ+𝒟c)⁢e𝒟c⁢t⁢e-𝒟c⁢t⁢ρ~^^s.

A standard method for eliminating the cavity dynamics is to use projection techniques. Accordingly

∂∂tρ~^^s′=ℒ~(t)⁢ρ~^^s′→∂∂t℘⁢ρ~^^s′=℘⁢ℒ~(t)⁢ρ~^^s′
where[ ]→ρcTre[ ] and the equation above can be cast in the following integro-differential form:

dd⁢t⁢℘⁢ρ~^^s′=∫0∞du⁢℘⁢ℒ~(t)⁢ℒ~(t-u)⁢℘⁢ρ~^^s′(t).

To return to the original interaction frame, we first consider the reduced density matrix under the projection operation:
′s(t)=ρcTrc[e−cts(t)]=Trc[e−c+t[Î]s(t)]⊗ρc.

Here, we invoked quantum channels and their adjoint properties. Sincec+[Î]=0
e−c+t[Î]=Î→Trc[e−c+t[Î]s(t)]⊗ρc=s(t)⊗ρc.

This allows us to write

dd⁢t⁢ρ~^^s(t)⁢ρc=∫0∞du⁢℘⁢ℒ~(t)⁢ℒ~(t-u)⁢ρ~^^s(t)⁢ρc.

Considering the integrand:

℘⁢ℒ~(t)⁢ℒ~(t-u)⁢ρ~^^s(t)⁢ρc=ρc⁢T⁢rc[ℒ~(t)⁢ℒ~(t-u)⁢ρ~^^s(t)⁢ρc]=T⁢rc[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢t⁢e-𝒟c(t-u)⁢ℒ⁡(t-u)⁢e𝒟c(t-u)⁢ρ~^^s(t)⁢ρc]⁢ρc=T⁢rc[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢e𝒟c(t-u)⁢ρ~^^s(t)⁢ρc]⁢ρc.

The dissipator acting on the density matrix will become

T⁢rc[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢e𝒟c(t-u)(ρ~^^s(t)⁢ρc)]⁢ρc=T⁢rc[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢e𝒟c+(t-u)[Iˆ]⁢ρ~^^s(t)⁢ρc]⁢ρc=T⁢rc[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢ρ~^^s(t)⁢ρc]⁢ρc
and further continuing this procedure

T⁢rc⁢[e-𝒟c⁢t⁢ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢ρ~^^s(t)⁢ρc]⁢ρc=Trc[ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢ρ~^^s(t)⁢ρc⁢e-𝒟c⁢t]⁢ρc=T⁢rc[ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢e-𝒟c+⁢t(ρ~^^s(t)⁢ρc)]⁢ρc=Trc⁢[ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢e-𝒟c+⁢t[Iˆ]⁢ρ~^^s(t)⁢ρc]⁢ρc=T⁢rc[ℒ⁡(t)⁢e𝒟c⁢u⁢ℒ⁡(t-u)⁢ρ~^^s(t)⁢ρc]⁢ρc
where the cyclic property of the trace operation was repeatedly used. Finally, the action of the dissipator on the Liouvillian becomes, following similar procedures
Trc[(t)ecu(t−u)s(t)ρc]ρc=Trc[e−c+u((t))(t−u)s(t)ρc]ρc=Trc([e−κ′u/2(t)(t−u)s(t)ρc]ρc
since

𝒟c+[a^]=-κ′2⁢a^&⁢𝒟c+[a^+]=-κ′2⁢a^+

The integro-differential equation will then take the form

dd⁢t⁢ρ~^^s⁢(t)⁢ρc=∫0∞du⁢℘⁢ℒ~⁢(t)⁢ℒ~⁢(t-u)⁢ρ~^^s⁢(t)⁢ρc=∫0∞d⁢u⁢T⁢rc[e-κ′⁢u/2⁢ℒ⁢(t)⁢ℒ⁢(t-u)⁢ρ~^^s⁢(t)⁢ρc]⁢ρc.

After expanding the superoperators, it becomes

dd⁢t⁢ρ˜σ(t)=-∫0∞du⁢e-κ′⁢u2⁢T⁢rc⁢{[H⁡(t),[H⁡(t-u),ρ˜σ(t)⁢ρ˜c]]}.

To solve this equation, we start by expanding the integrand:

dd⁢t⁢ρ˜σ⁢(t)=-∫0∞du⁢e-κ′⁢u2⁢T⁢rc⁢{[H⁢(t),[H⁢(t-u),ρ˜σ⁢(t)⁢ρ˜c]]}=-∫0∞du⁢e-κ′⁢u2⁢T⁢rc⁢{[H⁡(t),[H⁡(t-u)⁢ρ˜σ(t)⁢ρ˜c-ρ˜σ⁢(t)⁢ρ˜c⁢H⁢(t-u)]]}=-∫0∞du⁢e-κ′⁢u2⁢T⁢rc⁢{H⁢(t)⁢H⁢(t-u)⁢ρ˜σ⁢(t)⁢ρ˜c-H⁢(t)⁢ρ˜σ⁢(t)⁢ρ˜c⁢H⁢(t-u)-H⁢(t-u)⁢ρ˜σ⁢(t)⁢ρ˜c⁢H⁢(t)+ρ˜σ⁢(t)⁢ρ˜c⁢H⁢(t-u)⁢H⁢(t)}.

In the Hamiltonian
H(t)=ger(t)[â+Ĵ−eiΔt+â+e−iΔt]
we define γ=ger(t)and recalling Δ=ωs−ω(t)=0 (i.e. resonance) let
A(t)=âe−iΔt&A+(t)=â+eiΔt
A(t−u)=âe−iΔ(t−u)=A(t)eiΔu&A+(t−u)=â+eiΔ(t−u)=A+(t)e−iΔu
so, the following terms in the integrand can be obtained:

γ⁢∫0∞du⁢e-κ′⁢u2⁢T⁢rc⁢{-[∑α⁢β,β⁢αAα(t)⁢Jβ][∑α⁢β,β⁢αAα(t-u)⁢Jβ]⁢ρ˜σ(t)⁢ρ˜c+[∑α⁢β,β⁢αAα(t)⁢Jβ]⁢ρ˜σ(t)⁢ρ˜c[∑αβ,βαAα(t-u)⁢Jβ]-[∑α⁢β,β⁢αAα(t-u)⁢Jβ]⁢ρ˜σ(t)⁢ρ˜c[∑α⁢β,β⁢αAα(t)⁢Jβ]+ρ˜σ(t)⁢ρ˜c[∑α⁢β,β⁢αAα(t-u)⁢Jβ][∑α⁢β,β⁢αAα(t)⁢Jβ]}
where
Aα(t)=A+(t) &Aβ(t)=A(t)
Jβ=Ĵ−&Jα=Ĵ+

∑α⁢β,β⁢αAα(t)⁢Jβ≡Aα(t)⁢Jβ+Aβ(t)⁢Jˆα=A+(t)⁢Jˆ-+A⁡(t)⁢Jˆ+

These terms are further expanded as follows:

ddt⁢ρ~σ(t)=-γ[∑ββ,βα,αβ,αααα,αβ,βα,ββJβ⁢Jβ⁢ρ~σ(t)⁢ℳαα]+γ[∑ββ,βα,αβ,αααα,αβ,βα,ββJβ⁢ρ~σ(t)⁢Jβ⁢𝒩αα]-γ[∑ββ,βα,αβ,αααα,αβ,βα,ββJβ⁢ρ~σ(t)⁢Jβ⁢𝒩αα*]+γ[∑ββ,βα,αβ,αααα,αβ,βα,ββJβ⁢Jβ⁢ρ~σ(t)⁢ℳαα*]
where terms are defined as

∑ββ,βα,αβ,αααα,αβ,βα,ββJβ⁢Jβ⁢ρ~σ(t)⁢ℳαα=Jβ⁢Jβ⁢ρ~σ(t)⁢ℳαα+Jβ⁢Jα⁢ρ~σ(t)⁢ℳαβ+Jα⁢Jβ⁢ρ~σ(t)⁢ℳβα+Jα⁢Jα⁢ρ~σ(t)⁢ℳββℳαα=∫0∞du⁢e-κ′⁢u2⁢Trc[Aα(t)⁢Aα(t-u)⁢ρ~c]&⁢ℳαα*=∫0∞du⁢e-κ′⁢u2⁢Trc[ρ~c⁢Aα(t-u)⁢Aα(t)]𝒩αα=∫0∞du⁢e-κ′⁢u2⁢Trc[Aα(t)⁢ρ~c⁢Aα(t-u)]&⁢𝒩αα*=∫0∞du⁢e-κ′⁢u2⁢Trc[Aα(t-u)⁢ρ~c⁢Aα(t)]
and similar summation rules with the other terms. Using the cyclic property of the trace and the cavity correlation functions, all four terms can be collected:

ddt⁢ρ~σ(t)=-γ[2⁢J^-ρ~σ(t)⁢J^--J^-⁢J^-⁢ρ~σ(t)-ρ~σ(t)⁢J^-⁢J^-]⁢𝒞++(t)+γ[2⁢J^+⁢ρ~σ(t)⁢J^--J^-⁢J^+⁢ρ~σ(t)-ρ~σ(t)⁢J^-⁢J^+]⁢𝒞+-(t)+γ[2⁢J^-⁢ρ~σ(t)⁢J^+-J^+⁢J^-⁢ρ~σ(t)-ρ~σ(t)⁢J^+⁢J^-]⁢𝒞-+(t)+γ[2⁢J^+⁢ρ~σ(t)⁢J^+-J^+⁢J^+⁢ρ~σ(t)-ρ~σ(t)⁢J^+⁢J^+]⁢𝒞--(t)
where on resonance (Δ=0) these correlation functions reduce to

𝒞++(t)=∫0∞du⁢e-κ′⁢u2⁢Trc[A+(t)⁢A+(t-u)⁢ρ~c]=ei⁢2⁢Δ⁢t⁢∫0∞du⁢e-(κ′+i⁢Δ)⁢u2⁢Trc[a^+⁢a^+⁢ρ~c]=2κ⁢Trc[a^+⁢a^+⁢ρ~c]𝒞+-(t)=∫0∞du⁢e-κ′⁢u2⁢Trc[A+(t)⁢A⁡(t-u)⁢ρ~c]=∫0∞du⁢e-(κ′-i⁢Δ)⁢u2⁢Trc[a^+⁢a^⁢ρ~c]=2κ⁢Trc[a^+⁢a^⁢ρ~c]𝒞-+(t)=∫0∞du⁢e-κ′⁢u2⁢Trc[A⁡(t)⁢A+(t-u)⁢ρ~c]=∫0∞du⁢e-(κ′+i⁢Δ)⁢u2⁢Trc[a^⁢a^+⁢ρ~c]=2κ⁢Trc[a^⁢a^+⁢ρ~c]𝒞--(t)=∫0∞du⁢e-κ′⁢u2⁢Trc[A⁡(t)⁢ρ~c⁢A⁡(t-u)]=e-i⁢2⁢Δ⁢t⁢∫0∞du⁢e-(κ′-i⁢Δ)⁢u2⁢Trc[a^⁢a^⁢ρ~c]=2κ⁢Trc[a^⁢a^⁢ρ~c]

Therefore,

ddt⁢ρ~σ(t)=2⁢γκ′[2⁢J^-⁢ρ~σ(t)⁢J^--J^-⁢J^-⁢ρ~σ(t)-ρ~σ(t)⁢J^-⁢J^-]⁢Trc[a^+⁢a^+⁢ρ~c]+2⁢γκ′[2⁢J^+⁢ρ~σ(t)⁢J^--J^-⁢J^+⁢ρ~σ(t)-ρ~σ(t)⁢J^-⁢J^+]⁢Trc[a^+⁢a^⁢ρ~c]+2⁢γκ′[2⁢J^-⁢ρ~σ(t)⁢J^+-J^+⁢J^-⁢ρ~σ(t)-ρ~σ(t)⁢J^+⁢J^-]⁢Trc[a^⁢a^+⁢ρ~c]+2⁢γκ′[2⁢J^+⁢ρ~σ(t)⁢J^+-J^+⁢J^+⁢ρ~σ(t)-ρ~σ(t)⁢J^+⁢J^+]⁢Trc[a^⁢a^⁢ρ~c]

Lastly, we calculate the cavity correlations. The cavity, such as squeezing cavity12in the exemplary system3, is in a squeezed thermal equilibrium state. Therefore, the cavity correlations are calculated as follows:
ρc→{tilde over (ρ)}cSq=S(t)ρTHS+(t)→<μ{tilde over (ρ)}cSq>≡Tr[S+(t)μS(t)ρTH] & μ={â+2,â+â,ââ+,ââ}.

With

[a^a^+]→Sqz[S+(t)⁢a^⁢S⁡(t)S+⁢(t)⁢a^+⁢S⁡(t)]=[a^(t)a^+(t)]=[cosh[r⁡(t)]⁢a^+sinh[r⁡(t)]⁢a^+cosh[r⁢(t)]⁢a^++sinh[r⁡(t)]⁢a^]
the correlations become:

Trc[a^+⁢a^+⁢ρ~c]→Sqz⁢THTr[S+(t)⁢a^+⁢a^+⁢S⁡(t)⁢ρTH]S+(t)⁢a^+⁢a^+⁢S⁡(t)=S+(t)⁢a^+⁢S⁡(t)⁢S+(t)⁢a^+⁢S⁡(t)≡a^+(t)⁢a^+(t)=cosh2[r⁡(t)]⁢a^+2+(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(a^+⁢a^+a^⁢a^+)+sinh2[r⁡(t)]⁢a^2
and recognizing terms like Tr[â+2ρTH]=Tr[â2ρTH]=0 & [â,â+]=1 we get

Tr[a^+(t)⁢a^+(t)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢Tr[(a^+⁢a^+a^⁢a^+)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢Tr[(2⁢a^+⁢a^+1)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(2⁢n_+1)

The cavity average occupation isn=<{circumflex over (n)}>=Tr({circumflex over (n)}ρTH)=Tr(â+âρTH) and

n_=1[exp⁡(ℏ⁢vkB⁢T)-1].

Following the same procedure, then the remaining cavity correlations are:

Trc[a^+⁢a^⁢ρ~c]→Sqz⁢THTr[S+(t)⁢a^+⁢a^⁢S⁡(t)⁢ρTH=Tr[a^+(t)⁢a^(t)⁢ρTH]=cosh2[r⁡(t)]⁢Tr⁡(a^+⁢a^⁢ρTH)+sinh2[r⁡(t)]⁢Tr⁡(a^⁢a^+⁢ρTH)=cosh2[r⁡(t)]⁢n_+sinh2[r⁡(t)]⁢Tr⁡((a^+⁢a^+1)⁢ρTH)=cosh2[r⁡(t)]⁢n_+sinh2[r⁡(t)]⁢(n_+1)Trc[a^⁢a^+⁢ρ~c]→Sqz⁢THTr[S+(t)⁢a^⁢a^+⁢S⁡(t)⁢ρTH=Tr[a^(t)⁢a^+(t)⁢ρTH]=cosh2[r⁡(t)]⁢Tr⁡(a^⁢a^+⁢ρTH)+sinh2[r⁡(t)]⁢Tr⁡(a^+⁢a^⁢ρTH)=cosh2[r⁡(t)]⁢Tr⁡((a^+⁢a^+1)⁢ρTH)+sinh2[r⁡(t)]⁢n_=cosh2[r⁡(t)]⁢(n_+1)+sinh2[r⁡(t)]⁢n_Trc[a^⁢a^⁢ρ~c]→Sqz⁢THTr[S+(t)⁢a^⁢a^⁢S⁡(t)⁢ρTH]=Tr[a^(t)⁢a^(t)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢Tr[(a^⁢a^++a^+⁢a^)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢Tr[(2⁢a^+⁢a^+1)⁢ρTH]=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(2⁢n_+1)

To summarize:

Trc[a^+⁢a^+⁢ρ~c]→Sqz⁢TH(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(2⁢n_+1)=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(n_+1)+(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢n_Trc[a^+⁢a^⁢ρ~c]→Sqz⁢THcosh2[r⁡(t)]⁢n_+sinh2[r⁡(t)]⁢(n_+1)=sinh2[r⁡(t)]⁢(n_+1)+cosh2[r⁡(t)]⁢n_Trc[a^⁢a^+⁢ρ~c]→Sqz⁢THcosh2[r⁡(t)]⁢(n_+1)+sinh2[r⁡(t)]⁢n_Trc[a^⁢a^⁢ρ~c]→Sqz⁢TH(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(2⁢n_+1)=(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢(n_+1)+(cosh[r⁡(t)]⁢sinh[r⁡(t)])⁢n_

Substituting these cavity correlation functions into the differential equation then yields an equation that can be written into the following Lindblad form (explicitly with, for example, the spin ensemble10only having eliminated the dependence on the squeezing cavity12):

ddt⁢ρ~σ(t)=∑l=12[2⁢Rl⁢ρ~σ(t)⁢Rl+-Rl+⁢Rl⁢ρ~σ(t)-ρ~σ(t)⁢Rl+⁢Rl]
where

R1=(γκ′⁢(n_+1))⁢R&⁢R2=(γκ′⁢n_)⁢R+
and
R=cosh[r(t)]Ĵ−+sinh[r(t)]Ĵ+.

The steady state solution results in an intelligent state

ρ~σ(∞)=eβ⁢J^z⁢e-i⁢π2⁢J^y⁢❘"\[LeftBracketingBar]"J,M=0〉〈J,M=0❘"\[RightBracketingBar]"⁢ei⁢π2⁢J^y⁢e-β⁢J^z
whereby a nonunitary squeezing transformation is part of the final state demonstrating that the spin ensemble state has been transformed to a squeezed spin ensemble10. Therefore, the cavity squeezed state, for example for the squeezed cavity12in system3, has been dissipatively transferred to the strongly coupled spin ensemble.

However, the final steady-state, a squeezed spin ensemble, such as spin ensemble10in system3, is described by the cavity squeezing characteristics therefore placing the ensemble in a thermal state with mean photon number given by
<N>Sqz=sinh2[r(t)](n+1)+cosh2[r(t)]n.

These squeezing characteristics imply a highly mixed state even beyond the equilibrium Boltzmann distribution.

A time-dependent analysis is useful to ascertain the temporal characteristics. To that end, we expand the master equation and collect terms as follows:

ddt⁢ρ~σ(t)=2⁢γκ′⁢(n_+1)⁢cosh2[r⁡(t)]⁢(J^-ρ~σ⁢J^+-12⁢{J^+⁢J^-,ρ~σ})+2⁢γκ′⁢n_⁢sinh2[r⁡(t)]⁢(J^-ρ~σ⁢J^+-12⁢{J^+⁢J^-,ρ~σ})+2⁢γκ′⁢(n_+1)⁢sinh2[r⁡(t)]⁢(J^+⁢ρ~σ⁢J^--12⁢{J^-⁢J^+,ρ~σ})+2⁢γκ′⁢n_⁢cosh2[r⁡(t)]⁢(J^+⁢ρ~σ⁢J^--12⁢{J^-⁢J^+,ρ~σ})+[2⁢γκ′⁢(n_+1)+2⁢γκ′⁢n_]⁢cosh[r⁡(t)]⁢sinh[r⁡(t)]⁢(J^+⁢ρ~σ⁢J^++J^-⁢ρ~σ⁢J^--12⁢{J^+⁢J^++J^-⁢J^-,ρ~σ}),

Defining the constants
N={cosh2[r(t)]+sinh2[r(t)]}n+sinh2[r(t)]
M=cosh[r(t)]sinh[r(t)](2n+1)
we write

ddt⁢ρ~σ(t)=2⁢γκ′⁢(N+1)⁢(J^-⁢ρ~σ⁢J^+-12⁢{J^+⁢J^-,ρ~σ})+2⁢γκ′⁢N⁡(J^+⁢ρ~σ⁢J^--12⁢{J^-⁢J^+,ρ~σ})+2⁢γκ′⁢M⁡(J^+⁢ρ~σ+J^-⁢ρ~σ⁢J^--12⁢{J^+⁢J^++J^-⁢J^-,ρ~σ})

Now it is possible to obtain the diagonal population dynamics. Inspection of the above equation reveals that only the first two terms (e.g., the ones multiplying N+1 and N) involve populations while the last term (e.g., the one multiplying M) represents the coherences.

Their temporal evolution can be treated independently. Letting Pm(t)=(m|{tilde over (ρ)}σ(t)|m) for each subspace J of the state |J, m> (using the shorthand |J, m>→|m>)

we get

ddt⁢Pm(t)=2⁢γκ′⁢(N+1)⁢(〈m⁢❘"\[LeftBracketingBar]"J^-⁢ρ~σ(t)⁢J^+❘"\[RightBracketingBar]"⁢m〉-12⁢〈m⁢❘"\[LeftBracketingBar]"J^+⁢J^-,ρ~σ(t)}⁢❘"\[LeftBracketingBar]"m〉)+2⁢γκ′⁢N(〈m⁢❘"\[LeftBracketingBar]"J^+⁢ρ~σ(t)⁢J-′⁢❘"\[LeftBracketingBar]"m〉-12⁢〈m⁢❘"\[LeftBracketingBar]"J^-⁢J^+,ρ~σ(t)}❘"\[RightBracketingBar]"m〉).

The terms are calculated as follows:

〈m⁢❘"\[LeftBracketingBar]"J^∓⁢ρ~σ(t)⁢J^±❘"\[RightBracketingBar]"⁢m〉=[J⁡(J+1)-m⁡(m±1)]⁢〈m±1⁢❘"\[LeftBracketingBar]"ρ~σ(t)❘"\[RightBracketingBar]"⁢m±1〉=[J⁡(J+1)-m⁡(m±1)]⁢Pm±1(t)
and

〈m⁢❘"\[LeftBracketingBar]"J^±⁢J^∓,ρ~σ(t)}❘"\[RightBracketingBar]"⁢m〉=〈m⁢❘"\[LeftBracketingBar]"[J^±⁢J^∓⁢ρ~σ(t)+ρ~σ(t)⁢J^±⁢J^∓]❘"\[RightBracketingBar]"⁢m〉=〈m⁢❘"\[LeftBracketingBar]"J^±⁢J^∓⁢ρ~σ(t)❘"\[RightBracketingBar]"⁢m〉+〈m⁢❘"\[LeftBracketingBar]"ρ~σ(t)⁢J^±⁢J^∓❘"\[RightBracketingBar]"⁢m〉=J⁡(J+1)-m⁡(m∓1)[〈m±1⁢❘"\[LeftBracketingBar]"J^∓⁢ρ~σ(t)❘"\[RightBracketingBar]"⁢m〉+〈m⁢❘"\[LeftBracketingBar]"ρ~σ(t)⁢J^±❘"\[RightBracketingBar]"⁢m∓1〉]=2[J⁡(J+1)-m⁡(m∓1)]⁢〈m⁢❘"\[LeftBracketingBar]"ρ~σ(t)❘"\[RightBracketingBar]"⁢m〉=2[J⁡(J+1)-m⁡(m∓1)]⁢Pm(t)

Therefore

ddt⁢Pm(t)=-2⁢γκ′[N⁢Mm++(N+1)⁢Mm-]⁢Pm(t)+2⁢γκ′[N⁢Mm-1+⁢Pm-1(t)+(N+1)⁢Mm+1-⁢Pm+1(t)]
where
Mm±=[J(J+1)−m(m±1)]

The result is a hierarchy of coupled rate equations. As stated above, the steady-state yields a distribution based on the characteristics of N and can be solved from the resulting two-term recursion relation. The temporal decay or buildup is governed by the constant

2⁢γκ′
where γ=ger(t)as defined above. If the squeezing is not used, then the temporal evolution is characterized by g only and the population statistics is governed byn.
Squeezing Cavity Mediated Cavity Cooling

As shown for example inFIG.7for system3, the dissipative squeezing strongly couples the spin ensemble10to the squeezing cavity12. This strong or exponentially enhanced g-coupling or resonator-spin coupling leads to faster radiative processes compared to thermal processes.

The squeezing, for example performed by squeezing cavity12in system3, while strongly coupling the spin ensemble10, heats up the squeezing cavity12and consequently the spins in the spin ensemble10. To cool them, coupling can be applied to another cavity, for example the broadband squeezing cavity13, maintained at a cold temperature. The Hamiltonian for such a configuration can be written as follows:

H=H0+Hint+HdampH0=ωa⁢a^+a^+ωb⁢b^+⁢b^+ωs⁢J^z+HEHint=ga(a^+⁢J^-+a^⁢J^+)+gb(b^+⁢J^-+b^⁢J^+)+Γ⁡(a^+⁢b^+a^⁢b^+)HE=∑k=1Mgkc⁢c^k+⁢c^k+∑k=1Mgkd⁢d^k+⁢d^k&⁢Hdamp=∑k=1Mgkc(c^k+⁢a^+c^k⁢a^+)+∑k=1Mgkd(d^k+⁢b^+d^k⁢b^+)
where, as before, a standard bosonic environment (e.g., such as the cryogen14) is assumed and represented by the set of modes {ĉk+, ĉk} for cavity â and {{circumflex over (d)}k+, {circumflex over (d)}k} for cavity {circumflex over (b)}.

Let's designate the â cavity as the squeezing cavity12and the {circumflex over (b)} cavity as the cold cavity15that couples the spins in the spin ensemble10to the cold load or cryogen14. However, since the coupling gbis known to be weak, on the order of micro-Hertz and is unaffected by the squeezing, we can neglect its direct effect and focus on the indirect coupling through the electromagnetic coupling between the two cavities12and15encapsulated in their interaction Γ(â+{circumflex over (b)}+â{circumflex over (b)}+). As before, the squeezing on cavity â enhances the couplings to it and we get the following:

H0sqz=ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^+ωs⁢J^z+HEHintsqz=ga⁢er⁡(t)(a^+⁢J^-+a^⁢J^+)+Γer⁡(t)(a^+⁢b^+a^⁢b^+)=γa(a^+⁢J^-+a^⁢J^+)+Γ′(a^+⁢b^+a^⁢b^+)with⁢γa=ga⁢er⁡(t)&⁢Γ′=Γ⁢er⁡(t)Hdampsqz=∑k=1Mγkc(c^k+⁢a^+c^k⁢a^+)+∑k=1Mgkd(d^k+⁢b^+d^k⁢b^+)with⁢γkc=gkc⁢er⁡(t)

Further, let's consider the dispersive regime ωasqz<<ωband

ε=Γ′Δ≪1
where Δ=ωb−ωasqz. In this regime, the two cavities are approximately diagonal, and we can transform the full H by Ud=exp[−ε(â+{circumflex over (b)}−â{circumflex over (b)}+)] to obtain an effective Hamiltonian. The operator Udimplements a Schrieffer-Wolff transform. The calculations will use the Baker-Campbell-Hausdorf (BCH) formula

eA^⁢B^⁢e-A^=B^+[A^,B^]+12![A^,[A^,B^]]+…
for any two non-commutative Hermitian operators Â & {circumflex over (B)} (i.e. [Â, {circumflex over (B)}] ≠0).

First, transforming the cavity coupling term:

Γ′(a^+⁢b^+a^⁢b^+)→Γ′⁢Ud(a^+⁢b^+a^⁢b^+)⁢Ud-1=Γ′⁢{(a^+⁢b^+a^⁢b^+)-ε[(a^+⁢b^-a^⁢b^+),(a^+⁢b^+a^⁢b^+)]+σ⁡(ε2)}=Γ′(a^+⁢b^+a^⁢b^+)-εΓ′([a^+⁢b^,a^⁢b^+]-[a^⁢b^+,a^+⁢b^])⁢=Γ′(a^+b^+a^⁢b^+)-εΓ′(a^+⁢a^[b^,b^+]-[a^,a^+]⁢b^+⁢b^)⁢=Γ′(a^+⁢b^+a^⁢b^+)-εΓ′(a^+⁢a^-b^+⁢b^)

Next, transforming the cavity-free Hamiltonians:

(ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^)→Ud(ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^)⁢Ud-1=ωasqz⁢Ud⁢a^+⁢a^⁢Ud-1+ωb⁢Ud⁢b^+⁢b^⁢Ud-1=ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^-ε[(a^+⁢b^-a^⁢b^+),(ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^)]+ο⁡(ε2)=ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^-=ε⁢{ωasqz([a^+⁢b^,a^+⁢a^]-[a^⁢b^+,a^+⁢a^])+ωb([a^+⁢b^,b^+⁢b^]-[a^⁢b^+,b^+⁢b^])}=ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^-ε⁢{-ωasqz(a^+⁢b^+a⁢b^+)+ωb(a^+⁢b^+a^⁢b^+)=ωbsqz⁢a^+⁢a^+ωb⁢b^+⁢b^-ε[(ωb-ωasqz)⁢(a^+⁢b^+a^⁢b^+)]=ωbsqz⁢a^+⁢a^+ωb⁢b^+⁢b^-Γ′(a^+⁢b^+a^⁢b^+)=ωbsqz⁢a^+⁢a^+ωb⁢b^+⁢b^-εΔ⁡(a^+⁢b^+a^⁢b^+)

The cavity-only terms are now the sum of the transformed free and coupled Hamiltonians:

Heffcav=ωasqz⁢a^+⁢a^+ωb⁢b^+⁢b^-Γ′2Δ⁢(a^+⁢a^-b^+⁢b^)=Ωa⁢a^+⁢a^+Ωb⁢b^+⁢b^
with

Ωa=ωasqz-Γ′2Δ&⁢Ωb=ωb+Γ′2Δ

Therefore, the cavities â and {circumflex over (b)} are now in a diagonal frame accompanied by shifts in their energy states. The squeezing cavity â (e.g., squeezing cavity12in system1ofFIG.4) is coupled to the spins and transforming this coupling creates the indirect coupling to the cavity {circumflex over (b)} (e.g., cold cavity15in system1ofFIG.4), the sought-after result:

γa(a^+⁢J^-+a^⁢J^+)→γa⁢Ud(a^+⁢J^-+a^⁢J^+)⁢Ud-1=γa(a^+⁢J^-+a^⁢J^+)-ε⁢γa[(a^+⁢b^-a^⁢b^+),(a^+⁢J^-+a^⁢J^+)]+ο⁡(ε2)=γa(a^+⁢J^-+a^⁢J^+)-ε⁢γa([a^+⁢b^,a^⁢J^+]-[a^⁢b^+,a^+⁢J^-])⁢=γa(a^+⁢J^-+a^⁢J^+)+ε⁢γa(b^+⁢J^-+b^⁢J^+)

Finally, the effect of Udon the squeeze enhanced channel of Hdampsqzcan be calculated:

∑k=1Mγkc(c^k+⁢a^+c^k⁢a^+)→∑k=1Mγkc⁢Ud(c^k+⁢a^+ck⁢a^+)⁢Ud-1=∑k=1Mγkc(c^k+⁢a^+c^k⁢a^+)-ε⁢∑k=1Mγkc[(a^+⁢b^-a^⁢b^+),(c^k+⁢a^+c^k⁢a^+)]+ℴ⁡(ε2)=∑k=1Mγkc(c^k+⁢a^+c^k⁢a^+)-ε⁢∑k=1Mγkc([a^+⁢b^,c^k+⁢a^]-[a^⁢b^+,c^k⁢a^+])=∑k=1Mγkc(c^k+⁢a^+c^k⁢a^+)+ε⁢∑k=1Mγkc(c^k+⁢b^+c^k⁢b^+)

This shows that the squeezed coupling between the â and {circumflex over (b)} cavities induces a strong squeezed noise source in the cooling cavity {circumflex over (b)} detrimental to the cooling channel {circumflex over (b)}.

Induced Squeezed Noise Suppression

The induced squeezed noise, for example by squeezing cavity12inFIG.4, is very strong and requires eliminating completely. To do so, a squeezing technique is used that should only eliminate the noise, without affecting the enhanced coupling induced by the squeezing cavity â (e.g., squeezing cavity12in system1ofFIG.4). Therefore, a squeezed reservoir, such as broadband squeezing cavity13in system1, is designed that has the same (or substantially the same) amplitude as the squeezing cavity â, but only cancels the induced squeezed noise.

This is achieved by applying a broadband squeezed reservoir ĉk(e.g., broadband squeezing cavity13in system1ofFIG.4) on cavity â. To maintain the same amplitude as cavity â, yet be broadband, multiple resonant cavities ĉkcan be used so their resonant frequencies are slightly different but overlap and create the effect of a broadband resonator centered on the frequency of cavity â or implement impedance-engineered broadening using a feedline impedance transformer.

Such a squeezed reservoir in effect transforms a as follows:
â→âs=cosh[rφ(t)]â+eiφsinh[rφ(t)]â+

The phase is relevant and crucial in keeping track of the quadrature that is being squeezed. Putting the phase reference back in the original squeezing, the overall transformation becomes

[a^Sa^S+]=[a^(t)a^+(t)]→Sqz⁢BB⁢Res[cos⁢h[rθ⁢(t)]⁢{cos⁢h[rφ⁢(t)]⁢a^+ei⁢φ⁢sin⁢h[rφ⁢(t)]⁢a^+}+ei⁢θ⁢sin⁢h[rθ⁢(t)]⁢{cos⁢h[rφ⁢(t)]⁢a^++e-i⁢φ⁢sin⁢h[rφ⁢(t)]⁢a^}cos⁢h[rθ⁢(t)]⁢{cos⁢h[rφ⁢(t)]⁢a^++e-i⁢φ⁢sin⁢h[rφ⁢(t)]⁢a^}+e-i⁢θ⁢sin⁢h[rθ⁢(t)]⁢{cos⁢h[rφ⁢(t)]⁢a^+ei⁢φ⁢sin⁢h[rφ⁢(t)]⁢a^+}]Collecting⁢terms,we⁢geta^S={cos⁢h[rθ(t)⁢cos⁢h[rθ(t)]+ei⁡(θ-φ)⁢sin⁢h[rθ(t)]⁢sin⁢h[rθ(t)]}⁢a^+{ei⁢φ⁢cos⁢h[rθ(t)]⁢sin⁢h[rφ(t)]+ei⁢θ⁢sin⁢h[rθ(t)]⁢cos⁢h[rφ(t)]}⁢a^+a^S+={cos⁢h[rθ(t)]⁢cos⁢h[rθ(t)]+e-i⁡(θ-φ)⁢sin⁢h[rθ(t)]⁢sin⁢h[rφ(t)]}⁢a^++{e-i⁢φ⁢cos⁢h[rθ(t)]⁢sin⁢h[rφ(t)]+e-i⁢θ⁢sin⁢h[rθ(t)]⁢cos⁢h[rφ(t)]}⁢a^[a^Sa^S+]=[(𝒳+η)⁢a^+(λ+μ)⁢a^+(𝒳+η*)⁢a^++(λ*+μ*)⁢a^]
with the constants as defined below

𝒳=cos⁢h[rθ(t)]⁢cos⁢h[rφ(t)]⁢η=ei⁡(θ-φ)⁢sin⁢h[rθ(t)]⁢sin⁢h[rφ(t)]⁢λ=eiφ⁢cos⁢h[rθ(t)]⁢sin⁢h[rφ(t)]⁢μ=ei⁢θ⁢sin⁢h[rθ(t)]⁢cos⁢h[rθ(t)]

The corresponding reservoir correlations are therefore modified to

Tr[a^+(t)⁢a^+(t)⁢ρTH]→Sqz⁢BB⁢ResTr[a^S+⁢a^S+⁢ρTH]=(𝒳+η*)⁢(λ*+μ*)⁢Tr[(a^+⁢a^+a^⁢a^+)⁢ρTH]=(𝒳+η*)⁢(λ*+μ*)⁢(2⁢n_+1)Tr[a^+(t)⁢a^(t)⁢ρTH]→Sqz⁢BB⁢ResTr[a^S+⁢a^S+⁢ρTH]=(𝒳+η*)⁢(𝒳+η)⁢Tr⁡(a^+⁢a^⁢ρTH)+(λ*+μ*)⁢(λ+μ)⁢Tr⁡(a^⁢a^+⁢ρTH)=(𝒳+η*)⁢(𝒳+η)⁢n_+(λ*+μ*)⁢(λ+μ)⁢(n_+1)Tr[a^(t)⁢a^+(t)⁢ρTH]→Sqz⁢BB⁢ResTr[a^S⁢a^S+⁢ρTH]=(𝒳+η)⁢(𝒳+η*)⁢Tr⁡(a^⁢a^+⁢ρTH)+(λ+μ)⁢(λ*+μ*)⁢Tr⁡(a^+⁢a^⁢ρTH)=(𝒳+η)⁢(𝒳+η*)⁢(n_+1)+(λ+μ)⁢(λ*+μ*)⁢n_Tr[a^(t)⁢a^(t)⁢ρTH]→Sqz⁢BB⁢ResTr[a^S⁢a^S+⁢ρTH]=(𝒳+η)⁢(λ+μ)⁢Tr[(a^⁢a^++a^+⁢a^)⁢ρTH]=(𝒳+η)⁢(λ+μ)⁢(2⁢n_+1)

Focusing on the combined squeezing channels implies that this dissipator can be represented by the transformed squeezing characteristics as

[NM]=[{cos⁢h2[r⁡(t)]+sin⁢h2[r⁡(t)]}⁢n_+sin⁢h2[r⁡(t)]cos⁢h[r⁡(t)]⁢sin⁢h[r⁡(t)]⁢(2⁢n_+1)]→Sqz⁢BB⁢Res[N1,2MS]NS1=(𝒳+η*)⁢(𝒳+η)⁢n_+(λ*+μ*)⁢(λ+μ)⁢(n_+1)=(𝒳2+𝒳⁢η+η*⁢𝒳+η*⁢η)⁢n_+(λ*⁢λ+λ*⁢μ+μ*⁢λ+μ*⁢μ)⁢(n_+1)=cos⁢h[2⁢rθ(t)]⁢cos⁢h[2⁢rθ(t)]⁢n_+12⁢cos⁡(θ-φ)⁢sin⁢h[2⁢rθ(t)]⁢sin⁢h[2⁢rθ(t)]⁢(2⁢n_+1)+cos⁢h2[rθ(t)]⁢sin⁢h2[rφ(t)]+sin⁢h2[rθ(t)]⁢cos⁢h2[rφ(t)]NS2=(𝒳+η)⁢(𝒳+η*)⁢(n_+1)+(λ+μ)⁢(λ*+μ*)⁢n_=(𝒳2+𝒳η*+η⁢𝒳+ηη*)⁢(n_+1)+(λλ*+λμ*+μλ*+μμ*)⁢n_=cos⁢h[2⁢rθ(t)]⁢cos⁢h[2⁢rθ(t)]⁢n_+12⁢cos⁡(θ-φ)⁢sin⁢h[2⁢rθ(t)]⁢sin⁢h[2⁢rθ(t)]⁢(2⁢n_+1)+cos⁢h2[rθ(t)]⁢cos⁢h2[rφ(t)]+sin⁢h2[rθ(t)]⁢sin⁢h2[rφ(t)]MS=(𝒳+η*)⁢(λ*+μ*)⁢(2⁢n_+1)=e-i⁢φ⁢{cos⁢h[rθ(t)]⁢cos⁢h[rφ(t)]+e-i⁡(θ-φ)⁢sin⁢h[rθ(t)]⁢sin⁢h[rθ(t)]}⁢{cos⁢h[rθ(t)]⁢sin⁢h[rφ(t)]+e-i⁡(θ-φ)⁢sin⁢h[rθ(t)]⁢cos⁢h[rφ(t)]}⁢(2⁢n_+1)

To nullify the noise, we phase match by letting:
rθ(t)=rφ(t)=r
θ−φ=Φ
which implies

NS1=cos⁢h2(2⁢r)⁢n_+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+2⁢cos⁢h2⁢r⁢sin⁢h2⁢r=[sin⁢h2(2⁢r)+1]⁢n_+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+12⁢sin⁢h2(2⁢r)=12[sin⁢h2(2⁢r)+1]⁢(2⁢n_)+12⁢sin⁢h2(2⁢r)+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)=12⁢(1+cos⁢Φ)⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+n_NS2=cos⁢h2(2⁢r)⁢n_+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+(cos⁢h2⁢r)2+(sin⁢h2⁢r)2=[sin⁢h2(2⁢r)+1]⁢n_+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+(cos⁢h2⁢r)2+(sin⁢h2⁢r)2=[sin⁢h2(2⁢r)+1]⁢n_+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+12⁢sin⁢h2(2⁢r)+1=12[sin⁢h2(2⁢r)+1]⁢(2⁢n_)+12⁢sin⁢h2(2⁢r)+12⁢cos⁢Φ⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+1=12⁢(1+cos⁢Φ)⁢sin⁢h2(2⁢r)⁢(2⁢n_+1)+(n_+1)In⁢this⁢state,NS1=NS2=MS=0⁢when⁢Φ=n⁢π;n={±1,±3,±5,…}.Therefore,with⁢the⁢noise⁢nullified,the⁢full⁢effective⁢Hamiltonian⁢is⁢thenHsqz=H0sqz+Hintsqz+Hdamp→UdHeffcav+ωs⁢J^z+γa(a^+⁢J^-+a^⁢J^+)+ε⁢γa(b^+⁢J^-+b^⁢J^+)+HdampeffIn⁢this⁢state,NS1=NS2=MS=0⁢when⁢Φ=n⁢π;n={±1,±3,±5,…}.Therefore,with⁢the⁢noise⁢nullified,the⁢full⁢effective⁢Hamiltonian⁢is⁢thenHsqz=H0sqz+Hintsqz+Hdamp→UdHeffcav+ωs⁢J^z+γa(a^+⁢J^-+a^⁢J^+)+ε⁢γa(b^+⁢J^-+b^⁢J^+)+Hdampeff
and keep only the effect of cavity {circumflex over (b)} on the damping of the spins

Hdampeff=∑k=1Mgkd(d^k+⁢b^+d^k⁢b^+)

Moving into the free Hamiltonian co-frames further transforms Heff:

Heffsqz=→Heffcav+HE+ωs⁢J^zHeffsqz′=γa(a^+⁢J^-⁢ei⁢υa⁢t+a^⁢J^+⁢e-i⁢υa⁢t)+ε⁢γa(b^+⁢J^-⁢ei⁢υb⁢t+b^⁢J^+⁢e-i⁢υb⁢t)+∑k=1Mgkd(d^k+⁢b^⁢ei⁢υk⁢t+d^k⁢b^+⁢e-i⁢υk⁢t)
where
υa=ωs−Ωa; υb=ωs−Ωb& υk−ωs−gkd

As before, the damping channel can be represented by the Linbladian
[{circumflex over (b)}]ρ=κa[2{circumflex over (b)}ρ{circumflex over (b)}+−{{circumflex over (b)}+{circumflex over (b)},ρ}] with κa∝gkd

The frequencies are adjusted so υb=0 and υa>>1. This ensures that the spins are dispersively and most importantly, non-resonantly coupled to the squeezing cavity â (e.g., squeezing cavity12in system1ofFIG.4), while they are resonantly coupled to the auxiliary cavity {circumflex over (b)} (e.g., cold cavity15in system1) to generate the necessary strong coupling to it.

In this manner, conditions are created that strongly couple the spins through the squeezing cavity â to the cold cavity {right arrow over (b)} for the most efficient cooling. It is further assumed that the dominant loss channel is the b cavity that is connected to a cold load.

Under these conditions and the RWA approximation, we have the finally sought-after interaction for performing cavity cooling:
H′eff≈εγ({circumflex over (b)}+Ĵ−+{circumflex over (b)}Ĵ+)+[{circumflex over (b)}]
Solving the ensuing master equation by following the same procedure as the previous section results in

ddt⁢ρeff′(t)=2⁢ε⁢γκb⁢(n_+1)⁢(J^-⁢ρeff′⁢J^+-12⁢{J^+⁢J^-,ρeff′})+2⁢ε⁢γκb⁢n_(J^+⁢ρeff′⁢J^--12⁢{J^-⁢J^+,ρeff′})

The population dynamics once again is obtained from Pm(t)=<m|ρ′eff(t)|m> for each subspace J of the state |J, m>. With the shorthand |J, m>→|m>, we get the dynamic balancing expression

ddt⁢Pm(t)=2⁢ε⁢γκb⁢(n_+1)⁢(〈m⁢❘"\[LeftBracketingBar]"J^-⁢ρeff′(t)⁢J^+❘"\[RightBracketingBar]"⁢m〉-12⁢〈m⁢❘"\[LeftBracketingBar]"{J^+⁢J^-,ρeff′(t)}❘"\[RightBracketingBar]"⁢m〉)+2⁢ε⁢γκb⁢n_(〈m⁢❘"\[LeftBracketingBar]"J^+⁢ρeff′(t)⁢J^-❘"\[RightBracketingBar]"⁢m〉-12⁢〈m⁢❘"\[LeftBracketingBar]"J^-⁢J^+,ρeff′(t)}⁢❘"\[LeftBracketingBar]"m〉)As⁢before,it⁢reduces⁢toddt⁢Pm(t)=-2⁢εγκb[n_⁢Mm++(n_+1)⁢Mm-]⁢Pm(t)+2⁢ε⁢γκb[n_⁢Mm-1+⁢Pm-1(t)+(n_+1)⁢Mm+1-⁢Pm+1(t)]
where
Mm±=[J(J+1)−m(m±1)]

The steady state solution gives
[nMm++(n+1)Mm−]Pm(∞)=[nMm−1+Pm−1(∞)+(n+1)Mm+1−+Pm+1(∞)]
which is a detailed balance equation with equal transition rates Mm±.

Therefore, the equal upward and downward transitions can be written:
nMm+Pm(∞)=(n+1)Mm+1−Pm+1(∞)→nPm(∞)=(n+1)Pm+1(∞)
(n+1)Mm−Pm(∞)=nMm−1+Pm−1(∞)→(n+1)Pm(∞)=nPm−1(∞)

Using the first of these and starting with the state, m=−J since m={−J, −J+1, . . . , +J}, we can recursively and using induction establish

n_⁢P-J(∞)=(n_+1)⁢P-J+1(∞)→P-J+1(∞)=(n_n_+1)⁢P-J(∞)P-J+2(∞)=(n_n_+1)⁢P-J+1(∞)=(n_n_+1)2⁢P-J(∞)⋮Pm(∞)=(n_n_+1)J+m⁢P-J(∞)

To solve for P−J(∞), we use the normalization condition:

1+∑m=-Jm=+JPm(∞)=P-J(∞)⁢∑m=-Jm=+J(n_n_+1)J+m=P-J(∞)⁢∑s=0s=+2⁢J(n_n_+1)s

The sum on the right-hand side is a finite geometric series with

r=n_n_+1
and is summed as

∑s=0s=+2⁢Jrs=1-e2⁢J+11-r→1-(n_n_+1)2⁢J+11-(n_n_+1)=(n_+1)2⁢J+1-n_2⁢J+1(n_+1)2⁢J+11(n_+1)=(n_+1)2⁢J+1-n_2⁢J+1(n_+1)2⁢J
so that

P-J(∞)=(n_+1)2⁢J(n_+1)2⁢J+1-n_2⁢J+1

Finally,

Pm(∞)=(n_n_+1)J+m⁢P-J(∞)=(n_n_+1)J+m⁢(n_+1)2⁢J(n_+1)2⁢J+1-n_2⁢J+1=n_J+m(n_+1)J-m(n_+1)2⁢J+1-n_2⁢J+1
which implies that the spin ensemble, for example the spin ensemble10in system1, is approaching the cavity temperature since the Boltzmann thermal distribution for the spins is given by

ρσ=exp⁡(-β⁢ℋ^σ)𝒵;β=1kB⁢T
and the partition functionis defined as
=Tr[exp(−β)]
with
=ℏωSĴZ
The probability of being in the mth energy level is

Pm=〈m⁢❘"\[LeftBracketingBar]"ρσ❘"\[RightBracketingBar]"⁢m〉=〈m⁢❘"\[LeftBracketingBar]"exp⁡(-β⁢ℋ^σ)❘"\[RightBracketingBar]"⁢m〉𝒵=e-m⁢ℏ⁢ωS⁢β𝒵
and the partition function trace is calculated as follows:

𝒵=Tr[exp⁡(-β⁢ℋ^σ)]=∑m=-Jm=+J〈m⁢❘"\[LeftBracketingBar]"exp⁡(-β⁢ℋ^σ)❘"\[RightBracketingBar]"⁢m〉=∑m=-Jm=+Je-m⁢ℏ⁢ωS⁢β=∑k=02⁢Je-(k-J)⁢ℏ⁢ωS⁢β=e-m⁢ℏ⁢ωS⁢β⁢∑k=02⁢Je-k⁢ℏ⁢ωS⁢β=eJ⁢ℏ⁢ωS⁢β⁢1-e-(2⁢J+1)⁢ℏ⁢ωS⁢β1-e-ℏ⁢ωS⁢β

When the spins are on resonance with the cold cavity {circumflex over (b)}

ωS≈ωb→e-ℏ⁢ωb⁢β=n_n_+1→TS≈Tb

This allows to write:

Pm=e-m⁢ℏ⁢ωS⁢β𝒵=(n_n_+1)m⁢1𝒵𝒵=eJ⁢ℏ⁢ωS⁢β⁢1-e-(2⁢J+1)⁢ℏ⁢ωS⁢β1-e-ℏ⁢ωS⁢β=(n_n_+1)-J⁢1-(n_n_+1)(2⁢J+1)1-(n_n_+1)=(n_n_+1)J⁢(n_+1)2⁢J+1-n_2⁢J+1(n_+1)2⁢J+11(n_+1)=(n_+1n_)J⁢(n_+1)2⁢J+1-n_2⁢J+1(n_+1)2⁢J

Therefore,

Pm=(n_n_+1)m⁢n_J(n_+1)J(n_+1)2⁢J+1-n_2⁢J+1=n_J+m(n_+1)J-m(n_+1)2⁢J+1-n_2⁢J+1

Thus, this verifies that the spin ensemble (e.g.,10in system1ofFIG.4) is in a non-thermal temperature distribution and has been placed in thermal contact with the cold cavity {circumflex over (b)} (e.g., cold cavity15in system1) equilibrating or cooling down the ensemble to the cavity field temperature.

The expectation of Ĵzcan be calculated to view the temporal dynamics from another perspective:

ddt⁢〈J^z(t)⁢ρeff′〉=2⁢ε⁢γκb⁢(n_+1)⁢〈J^z⁢J^-ρeff′⁢J^+-12⁢J^z⁢{J^+⁢J^-,ρeff′}〉+2⁢ε⁢γκb⁢n_⁢〈J^z⁢J^+⁢ρeff′⁢J^--12⁢J^z⁢{J^-⁢J^+,ρeff′}〉

To facilitate this calculation, we evaluate the terms
<ĴzĴ∓ρ′effĴ±>=<Ĵ±ĴzĴ∓ρ′eff>
<Ĵz{Ĵ±Ĵ∓,ρ′eff}>=<Ĵz(Ĵ±Ĵ∓ρ′eff+ρ′effĴ±Ĵ∓)>=<ĴzĴ±Ĵ∓ρ′eff+Ĵ±Ĵ∓Ĵzρ′eff>

Within the expectation value, the operators in the first term are reduced to:

J^±⁢J^z⁢J^∓=(∑lσ±l)⁢(12⁢∑mσzm)⁢(∑nσ∓n)=12⁢∑l,m,nσ±l⁢σzm⁢σ∓n≡12⁢∑kσ±k⁢σzk⁢σ∓k=12⁢∑kσ±k(2⁢σ+k⁢σ-k-𝕀k)⁢σ∓k=12⁢∑k(±E±k)=∓E±
where

E±=12⁢(𝕀±σz)
and the operators in the second term reduce to

J^z⁢J^±⁢J^∓=(12⁢∑lσzl)⁢(∑mσ±m)⁢(∑nσ∓n)=12⁢∑l,m,nσzl⁢σ±m⁢σ∓n≡12⁢∑kσzk⁢σ±k⁢σ∓k=12⁢∑kσzk⁢E±k=12⁢∑k(±E±k)=∓E±
similarly
Ĵ±Ĵ∓Ĵz→Ĵ±Ĵ∓Ĵz=±E±

Letting

τ=2⁢ε⁢γκb,
we can substitute these operators into the expectation value equation to get

ddt⁢〈J^z(t)⁢ρeff′〉=τ⁡(n_+1)⁢(〈-E+⁢ρeff′〉-12⁢〈E+⁢ρeff′+E+⁢ρeff′〉)+τ⁢n_(〈E_ρeff′〉-12⁢〈-E_ρeff′-E_ρeff′〉)=-2⁢τ⁡(n_+1)⁢〈E+⁢ρeff′〉+2⁢τ⁢n_⁢〈E_ρeff′〉=-τ⁡(n_+1)⁢〈(𝕀+J^z)⁢ρeff′〉+τ⁢n_⁢〈(𝕀-J^z)⁢ρeff′〉=-τ⁢〈ρeff′〉-τ⁡(2⁢n_+1)⁢〈J^z⁢ρeff′〉

This expression then clearly shows the time constant τ(2n+1) governing the temporal population dynamics. The enhanced coupling of cavity {circumflex over (b)} (e.g., cold cavity15in system1ofFIG.4) mediated by the squeezing cavity â (e.g., squeezing cavity12in system1) is expressed through the constant τ˜εγ˜εgaer(t). Although there is a reduction of

ε=Γ′Δ
at finite temperaturesn≠0, there is additional enhancement if the overall temperature is sufficiently low for a significant polarization effect well beyond the Boltzmann distribution.
Efficient Squeezed Noise Suppressed Cavity Cooling

As shown in the exemplary system3inFIG.7, another more efficient implementation of the cavity cooling described above is based on eliminating the auxiliary or cold load cavity {circumflex over (b)} (e.g., cold cavity15in system1ofFIG.4) altogether and use just the primary squeezing through cavity â (e.g., squeezing cavity12in system3ofFIG.7) with the broadband squeezed cavity (e.g., broadband squeezing cavity13in system3ofFIG.7) to directly suppress the squeezed thermal bath, thereby coupling to the cold load of the squeezing cavity. The simplified system Hamiltonian is thus

H0=ω0⁢a^+⁢a^+ωs⁢J^z+g⁡(a^+⁢J^-+a^⁢J^+)+HE+HSE⁢HE=∑k=1Mgk⁢d^k+⁢d^k&⁢HSE=∑k=1Mgk(d^k+⁢a^+d^k⁢a^+)

Following the same steps above with the squeezed noise suppression will cool the spins to the squeezing cavity â temperature.

The descriptions of the various exemplary embodiments of the present invention have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Further, Applicant's intent is to encompass the equivalents of all claim elements, and no amendment to any claim of the present application should be construed as a disclaimer of any interest in or right to an equivalent of any element or feature of the amended claim.