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This patent relates to quantum metrology and more particularly to quantum metrology based on strongly correlated states.
The ability to interrogate a physical system and precisely measure its observables forms the basis of both fundamental and applied sciences. While certain techniques are based on specially controlled individual particles, in general, large ensembles can be used to enhance measurement sensitivity. However, existing techniques are limited by inter-particle interactions.
According to some embodiments, a method includes: initializing an ensemble of quantum spins in an entangled state; driving the ensemble of quantum spins in the entangled state with periodic electromagnetic pulses at a driving frequency while the ensemble of quantum spins is exposed to an external signal, wherein the driving frequency is tuned based on the frequency of the external signal to be measured; reading out a state of the ensemble of quantum spins, wherein the state of the ensemble of quantum spins is indicative of the strength of the external signal.
According to some embodiments, the driving frequency is approximately twice the signal frequency.
According to some embodiments, the electromagnetic pulses comprise π-pulses.
According to some embodiments, the ensemble of quantum spins comprises a plurality of NV spin centers in diamond.
According to some embodiments, the method further includes polarizing, prior to the initializing, the ensemble of quantum spins by applying an external magnetic field at a first field strength to the ensemble of quantum spins.
According to some embodiments, the external magnetic field has a strength that is stronger than the strength of ferromagnetic interactions between the NV spin centers.
According to some embodiments, the initializing the ensemble of quantum spins in the strongly entangled state comprises reducing the strength of the external magnetic field to a second field strength.
According to some embodiments, the method further includes applying, during the initializing, periodic π-pulses to the ensemble of quantum spins at a frequency that is detuned from the signal frequency.
According to some embodiments, the second field strength is zero.
According to some embodiments, the second field strength is greater than a field strength that would cause a phase transition of the ensemble of quantum spins.
According to some embodiments, the method further includes increasing the external magnetic field strength to a third field strength prior to the reading out the state of the ensemble of quantum spins.
According to some embodiments, the third magnetic field strength is the same as the first magnetic field strength.
According to some embodiments, the increasing the external magnetic field strength to the third field strength is performed at the same rate as the decreasing the magnetic field strength to the second magnetic field strength.
According to some embodiments, the plurality of NV spin centers have a density in the diamond of approximately 10-4 to approximately 0.01 spins/nm2.
According to some embodiments, the strongly entangled state is one or more of a Greenberger-Horne-Zeilinger (“GHZ”) state, a NOON state, a W-state, any other gaussian or non-gaussian squeezed state, or another similarly entangled state.
According to some embodiments, the reading out the state of the ensemble of quantum spins comprises determining the average parity of the ensemble of quantum spins.
According to some embodiments, the reading out the state of the ensemble of quantum spins comprises determining the magnetization of the ensemble of quantum spins.
According to some embodiments, the reading out the state of the ensemble of quantum spins is performed with one or more of an avalanche photodetector (APD) or a single photon counting module (SPCM).
According to some embodiments, the method further includes repeating the initializing, driving, and reading out for at least one second signal frequency to determine an external signal spectrum.
According to some embodiments, the driving frequency is approximately twice the signal frequency minus an offset value of 2Δω.
According to some embodiments, the ensemble of quantum spins comprises a two-dimensional array of quantum spins.
According to some embodiments, the entangled state of the ensemble of quantum spins acts as a two-state system.
24 According to some embodiments, in the entangled state, one or more of Quantum Fisher information or linear susceptibility is enhanced compared to disentangled states.
These and other capabilities of the disclosed subject matter will be more fully understood after a review of the following figures, detailed description, and claims. It is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting.
Various objectives, features, and advantages of the disclosed subject matter can be more fully appreciated with reference to the following detailed description of the disclosed subject matter when considered in connection with the following drawings, in which like reference numerals identify like elements.
According to some embodiments, the disclosed techniques facilitate the use of many-body strongly entangled quantum spins for measuring oscillating signals with reduced decoherence caused by noise. For example, the quantum spins can be prepared in a strongly entangled state. While such entangled states of n particles are approximately n times more sensitive than single spin systems, they are also approximately n-times more susceptible to noise. Accordingly, in some embodiments, to reduce the effects of common (or typical) noise while still maintaining some sensitivity gains owing to quantum entanglement, the quantum spins can be driven periodically at a driving frequency, ω0, (e.g., Floquet frequency) that is tuned to the frequency of the signal to be measured, ωs. In some embodiments the driving frequency ω0 is approximately twice the signal frequency ωs. According to other embodiments, the driving frequency ω0 is approximately slightly detuned from this number by a value of Δω. Accordingly, decoherence caused by noise can be reduced, facilitating increased sensitivity to oscillating signals. Furthermore, since the disclosed technique increases sensitivity to a particular frequency oscillation at ωs, the driving frequency can be swept during multiple measurements. The disclosed technique utilizes quantum entanglement in order to perform each measurement faster, thereby increasing the bandwidth of the measurement. This technique allows for simple readout by determining, for example, the parity or magnetization of the resulting quantum spins is also disclosed.
For an ensemble of N uncorrelated two-level systems (e.g., quantum spins), the standard quantum limit (SQL) for measuring a small energy shift, scales as δω∝1/√{square root over (NT2T)}, where T2 is the relevant coherence time and T is the total measurement duration. While this scaling suggests that increasing the number of particles always improves the signal to noise ratio, the effect of inter-particle interactions needs further consideration. Above a certain density, these inter-particle interactions can limit T2 and thus the maximum achievable sensitivity. Without wishing to be bound by theory, this limit may arise from the fact that interactions typically drive phenomena such as thermalization, which can be used to describe a number of different effects such as decoherence and depolarization, where the system loses both its local coherences and any accumulated spectroscopic signal. Accordingly, while increasing the number of quantum spins can increase the sensitivity of a measurement using those quantum spins as sensors, there may be a limit based on the interactions between quantum spins that produces noise (which can also be referred to as interaction-induced decoherence). In certain instances, this noise can negate the effect of adding more quantum spins to the ensemble, such that sensitivity cannot be further increased simply by adding more quantum spins.
Under certain conditions, a many-body quantum system can evade rapid thermalization, and thus the resulting increased noise. Systems including intrinsically out-of-equilibrium systems (e.g., systems that cannot be described by a conventional thermal state equilibrium) can exhibit robust dynamical features that are forbidden in equilibrium. One such example is the discrete time crystal (DTC), which is protected by an interplay between strong interactions and rapid periodic pulses. The spatio-temporal ordering of the DTC phase is robust to arbitrary static perturbations and has been experimentally observed in both a trapped-ion quantum simulator and a dipolar spin ensemble.
According to some embodiments, the disclosed technique can be used to enhance quantum metrology (e.g., measuring AC magnetic fields, oscillating electric fields, or periodic temperature fluctuations). For example, disclosed herein is a class of protocols that allows for circumventions of the limitations on the effective coherence time that may be imposed by many-body interactions, such as the effective internal noise discussed above. Furthermore, the disclosed protocols can leverage these many-body interactions to develop quantum correlations while reducing the negative effects associated with many-body interactions and imperfections in a system, thereby leading to improved performances in both measurement sensitivity and bandwidth. In some embodiments, sensitivity can be enhanced partially from utilization of a higher density of quantum spins in sensors without a reduction in sensitivity caused by thermalization or other density-dependent noise. Furthermore, in some embodiments both sensitivity and bandwidth can be enhanced by preparing and utilizing quantum correlated states.
According to some embodiments, the technique can be realized by implementing a Floquet system where large quasi-energy gaps protect strongly-entangled states from static perturbations, while still ensuring their sensitivity to an oscillating signal. For example,
The multi-body quantum spin system described with reference to
According to two exemplary embodiments, two techniques are disclosed that allow for the precise measurement of AC magnetic fields in Floquet spin ensembles: one with reference to
Exemplary Sensing Protocol:
According to these exemplary embodiments, a sensing protocol can be implemented using an ensemble of N spin ½ particles in a d-dimensional array. The N spin ½ particles participate in ferromagnetic Ising interactions. As shown in
where Ŝiμ(μϵ{x, y, z}) are spin-½ operators, and Jij>0 is the interaction strength between spins at site-i and j with characteristic strength J˜ΣjJij According to some embodiments, the spins can also be driven by fast, global, periodic π-pulses, P=exp[−iπ ΣiŜix], that rotate each spin around the {circumflex over (x)} axis. For example, applications of strong microwave or optical pulses can be used to realize such π-pulses. For B=0 (e.g., for measuring small fields that can be considered a small perturbation on a zero-field scenario), the dynamics can be driven by a Floquet unitary ÛF={circumflex over (P)} e−2πiĤoτ, where τ is the time-duration between π-pulses, setting the Floquet frequency, ωo=2π/τ. Using the sensing protocol described below, the strength of the small magnetic field signal, B can be measured.
As shown in
After polarizing the spins, the transverse field Ω can be subsequently decreased in ramp segment 126 to zero over time duration Tp. During this process, π-pulses can be applied at a driving frequency ω0 (e.g., Floquet frequency) that is sufficiently detuned from 2ωs such that the effect of Ĥsignal on the spin dynamics is negligible. In other words, as the transverse field Ω is ramped down, the π-pulses are applied at a sufficiently different rate such that the signal does not significantly affect the spin states of the many-body spin system. Accordingly, a strongly entangled state, such as the GHZ state discussed in more detail below, can be prepared for the measurement step 130.
As discussed above, the sensitivity of such an entangled state to observable quantities such as AC magnetic fields can be reduced by interactions between the quantum spins and noise. Accordingly, in the measurement step 130, the Floquet frequency is adjusted to be resonant with the signal, ω0≈2ωs while the system evolves for a time duration Ts under zero transverse field 132. During this time evolution, the information about the strength of the signal at frequency ωs becomes recorded as a relative phase ϕ between two many-body quantum states |↑↑ . . . ↑) and |↓↓ . . . ↓, as indicated in
Finally, in the read-out step 140 the initialization step 120 is reversed over a time Tr in ramp segment 146. Afterward, and each spin's polarization is then measured along the {circumflex over (x)} axis in the segment 142. For example, the parity across the entire many-body quantum spin system can be determined by applying laser light (e.g., at 532 nm) to the quantum spins and collecting the responsive fluorescence signal at 637 nm. The parity can be indicative of a number of a number of quantum spins 128 that have flipped (shown in
According to some embodiments, three steps 120, 130, and 140 can be completed within the relevant coherence time of the system (
In order to understand how sensitive parity changes are as a function of signal strength, the dynamics in each of the three exemplary steps 120, 130, and 140 are analyzed in detail. During the initialization step 120, Ising interactions among spins can be used to prepare a quantum state with strong spin-spin correlations between constituent quantum spins. To understand the dynamics during state preparation, the so-called toggling frame can be moved, which rotates with every π-pulse, {circumflex over (P)}, by applying the unitary transformation Ĥ{circumflex over (P)}−1Ĥ{circumflex over (P)}. In this frame, Ĥ0 remains invariant while Ĥsignal changes sign during every Floquet period, modifying the time-dependence of the original signal to Beff(t)=B sin(ωst)θ(ω0t/4π), where θ(x) is a square function with unit period. The dynamics of such a system can be well approximated by an effective, quasi-static Hamiltonian:
where
During the measurement step 130, tuning the driving frequency to ω0=2ωs can give rise to a non-zero time-averaged signal strength
According to some embodiments, the sensitivity can be estimated for an exemplary ideal case without noise. When the protocol described above with reference to
δB−1˜δϕ−1NTs˜N√{square root over (
saturating the Heisenberg limit. Note that the relevant coherence time here can be determined by noise at the probing frequency ωs (and is not limited by interactions between the spins).
According to some embodiments, the initialization process in 120 can be closely related to adiabatic preparation of strongly correlated (entangled) quantum states. In such a method, the Hamiltonian of a quantum system can be slowly tuned, such that the system belongs to a certain quantum phase of matter (e.g. paramagnetic phase) at the beginning and to another quantum phase (e.g. ferromagnetic phase) at the end. If the system starts out in the ground state of the first phase, it can be evolved so as to end in the ground state of the second phase. The fidelity of this method of preparing a quantum state (the ground state of the second phase) can depend on how fast the Hamiltonian parameters are tuned compared to the energy gap (see below). In particular, when the system size, N, is large, the fidelity of the state preparation is limited by the small energy gap of the effective Hamiltonian D at the phase transition point (e.g., the boundary between ferro- and para-magnetic phases). This energy gap can be defined by the difference in energies between the ground state(s) and the first excited states of the Hamiltonian D, and can be distinct from quasi-energy gap in
According to some embodiments, the protocol described above may be limited by the adiabatic preparation fidelity of the GHZ state. The phase transition can be between paramagnetic phase when Ω is large and ferromagnetic phase when Ω is small. When Ω is ramped sufficiently slow, a strongly correlated quantum state can be prepared. The fidelity of this state preparation method can depend on so-called the energy gap, which is in turn related to the system size N. The energy gap, or the transition energy from one state to another as shown in
In this exemplary scenario, the initialization step 120 prepares an even parity state of the form |ξ+=(|ξ)+P|ξ)/√2, where |ξ is ferromagnetically ordered with correlation length ξ. The state accumulates a collective phase ϕ=4BTSΣi|ŜZi during the measurement stage 130, leading to a parity expectation value {circumflex over (P)}=(ξ|cos(2{circumflex over (ϕ)})|ξ. For a weak signal, {circumflex over (P)} varies quadratically as: 1−{circumflex over (P)}≈2|{circumflex over (ϕ)}{circumflex over (ϕ)}|ξ˜ξdN(BTs)2, which results in a sensitivity scaling
δB−1˜√{square root over (T/(Tp+Ts+Tr))}√{square root over (ξdN)}Ts. (4)
This scaling can be understood as follows: the state |ξ+ can be viewed as multiple copies of a GHZ state with size ˜ξd. While each GHZ state allows Heisenberg-limited sensitivity ˜ξd√T
Given a limited coherence time in which the measurement can take place, the relative duration of each step (e.g., segment 126, measurement step 130, and segment 146) can be optimized. According to some embodiments, the optimum can be achieved when β=Tp/
δB−1˜√{square root over (NTT
This scaling can improve upon the SQL by a factor ˜(J
Accordingly, as discussed above and further below, periodic π-pulses as used in this exemplary protocol can assist in suppressing low frequency noise and preventing changes to the parity of the spin ensemble. This protection originates from the quasi-energy gap between pairs of Floquet eigenstates (eigenstates of a Floquet unitary evolution over one cycle) with opposite parity as shown in
Robustness:
The following characterizes the exemplary robustness of some embodiments of the disclosed protocol in the presence of imperfections and/or noise. As discussed in more detail below, embodiments of the disclosed protocol can still present significant improvements in sensitivity in the face of such imperfections and noise.
First, considering the quasi-static local perturbations, ∈ΣiδĤi, which can be decomposed into parity-preserving and parity-changing terms: δĤi=δĤi++δĤi− with δĤi±≡(δĤi±{circumflex over (P)}δĤi{circumflex over (P)})/2. According to some embodiments, the parity-preserving term, δĤi+, does not affect the nature of the phase transition nor the sensitivity scaling of the exemplary protocol. The parity-changing term, δĤi−, can, in some embodiments, hinder both the state preparation and the measurement fidelity of the magnetic field signal. However, this effect can be parametrically suppressed by the presence of the periodic π-pulses discussed throughout the present disclosure, which effectively “echoes” out this contribution to leading order. More specifically, higher order corrections to the effective Hamiltonian appear only as ˜∈J/ω0 and can be safely neglected assuming ∈J/ω0<<(ξd TP)−1 (initialization) and ∈J/ω0<<B (measurement) Under these conditions, the effects of higher order corrections can be small enough compared to the effects of smallest signal to be measured.
Second, according to some embodiments, the presence of inhomogeneities in δΩi, δJi, and δθ (slight variations in the transverse field, ferromagnetic interactions between spins, and spin rotations with respect to average values) can lead to (Anderson) localization, which limits the maximum correlation length of the system. In general, the localization length at the critical point (e.g., boundary between paramagnetic and ferromagnetic phases) scales as ξloc˜(W/J)−μ, where W is the disorder bandwidth, or degree of deviations (in this context, it can be said that W is given as the standard deviations of δΩi, δJi or δθ ω0) of the coupling parameters and μ is the corresponding critical exponent. When the localization length ξloc is shorter than the original correlation length ξ˜√JTP, the state preparation time should be reduced to Tp*=(W/J)−2μ/J (so that more time can be allocated for the measurement step). This leads to a modified sensitivity scaling as summarized by the disordered system 230 in
Finally, the effect of noise from the environment can also be considered, which can limit the coherence time,
where α is a power describing how spectral density decays as a function of frequency ω, and A01+α is a suitable normalization constant, the periodic π-pulses described throughout the present disclosure can decouple the system from low frequency noise ω<ωs, implying that the decoherence rate is determined by the noise density at the probe frequency, S(ωs). If the noise on each spin is independent, then the relevant coherence time of the entangled spin state is shortened to
Exemplary Sensing Protocol without Parity Measurements:
According to some embodiments, parity measurements can become challenging in an ensemble experiment where resolution of individual spin projections cannot be performed. To this end, an alternative exemplary approach to implementing the disclosed protocol can be based upon measuring an extensively scaling observable, which, in this context, can refer to an observable with a value that scales linearly with the system size. For example, a parity (the value of which can be plus or minus one) of spins does not scale with system sizes, according to some embodiments, but the number spin up states can scale with system sizes. Embodiments of this exemplary modified protocol are shown in
As shown in
In the measurement step 330, the transverse field remains at the level in segment 332. In addition, rather than setting the Floquet frequency equal to 2ωs, as discussed in relation to
Finally, in the segment 346 of the read-out phase 340, Ω is slowly brought back to its original value in the segment 346. After the segment 346, the number of spin-flip excitations, Ne, encodes the signal strength B. This can be read from the magnetization ΣiŜix. The measurement of magnetization can be achieved by, for example, differential fluorescence measurements, or any other exiting methods to read out spin states of a quantum spin ensemble. For example, reading out the state of the ensemble of quantum spins can be performed with one or more of an avalanche photodetector (APD) or a single photon counting module (SPCM).
In some embodiment, the resonant magnetic field signal creates, on average, a single collective excitation within the correlation volume (in number of quantum spins), ξd. In other words, each correlation volume comprising multiple quantum spins over the correlation distance scale is excited into a collective excitation state. The probability of creating such an excitation, p˜(χξd/2BTS)2, in a given correlation volume depends on the proximity of the transverse field to the critical point during the measurement stage 330, which leads to the enhancement factor (e.g., how the susceptibility has been enhanced due to quantum correlation), χ=(ΔΩ/Ω)−η, where η is the scaling dimension of the operator Ŝiz. Since there are N/ξd correlated spin segments in the system, the average number of excitations Ne˜pN/ξd. This number fluctuations fluctuates such that δNe˜√{square root over ((p(1−p)N/ξd))}. These fluctuations result in a signal-to-noise ratio of ∂BNe/δNe˜√{square root over (NTs(JTp)η/(zv+1))}. As before, when this procedure is repeated over a total duration T with an optimal Tp (with respect to
δB−1˜√{square root over (NT
For nearest neighbor interactions in 1D (Ising universality class), the scaling dimension is η=⅜ and δB−1˜√{square root over (NT
Exemplary Implementations:
the measurement protocol, including preparation into a strongly correlated state, driving at a Floquet frequency during exposure to an external oscillating signal source, and observing the resulting system as described throughout the present specification can be used in a number of applications. For example, it can be implemented as an AC magnetic field sensor using a 2D array of spin centers within a crystal lattice, such as shallowly implanted nitrogen-vacancy (NV) color centers in diamond. According to some embodiments, the sensitivity per unit area in this exemplary approach can be limited by the dipolar interactions between the S=1 NV centers, which are difficult to decouple using conventional NMR techniques. However, the protocol of the present disclosure provides a way to circumvent this interaction-induced limitation and enable significant improvements to the sensitivity by preparing the NV centers in a strongly correlated state, such as the GHZ state, and driving them with π-pulses as described above to stabilize them for measurement of a particular frequency signal, such as an AC magnetic signal. Since NV centers are also sensitive to signals such as AC electric fields and rapidly oscillating temperatures, such a method could also be implemented to measure such signals.
A second exemplary platform for implementing the disclosed protocol can include nuclear spin ensembles in layered materials, such as, but not limited to hexagonal boron-nitride (for example with boron isotope quantum spins) or 13C enriched graphene (for example with 13C quantum spins), or other materials which may include quantum spin degrees of freedom. Similar techniques in which the quantum spins are prepared in a strongly correlated state, driven with periodic microwave or radio-frequency (RF) pulses to reduce the destabilizing effects of noise during a measurement phase, and subsequently read out. A person of ordinary skill in the art would understand based on the present disclosure that such quantum spins may be sensitive to different oscillating signals, and thus that the steps in the disclosed protocol can be adapted to analogous control techniques for different quantum spins in order to measure such oscillating signals. According to some embodiments, applications of such systems can include the detection of time-varying signals (such as vibrations caused by scattering) resulting from weakly interacting massive particles such as axions.
The disclosed protocol can also be extended along several directions. While the above examples focused on probing the strength of field signals, such as magnetic fields, similar protocols can enable the detection of phase fluctuations in the external control driving itself. For example, the effects of fluctuations in control driving is equivalent to those of time-dependent signals, which can be detected by the disclosed technique. Moreover, the protocol allows for the suppression of symmetry breaking perturbations at leading order via periodic pulses, such as π-pulses.
Exemplary Characterization of the Effects of Disorder
The protocols described above can utilize quantum phase transitions to develop long-range spin correlations. In an exemplary ideal case, the spin correlation length ξ reaches the linear system size (e.g., the full many-body quantum spin system size) at the critical point of the phase transition, and our initialization step prepares Greenberger-Horne-Zeilinger (GHZ) states, allowing Heisenberg limited quantum sensing. In the presence of disorder, however, ξ may be limited as the disorder can, in some embodiments, prevent the propagation of spin correlations by localizing quasi-particle excitations. In turn, the limited spin correlations lead to diminished sensitivity enhancement compared to Heisenberg limit. As an example, below the effects of such disorder for a spin chain with nearest neighbor Ising interactions are quantified.
According to some embodiments, it can be assumed that the transverse field strength at each spin is weakly disordered such that the transverse field Ωi at any given quantum spin can be given by Ωi=Ω+δΩi, where Ω is the average field strength and δΩ is a random variable uniformly distributed from [−W/2, W/2]. Likewise, it can be assumed that the interaction strength for any given spin can be given by Ji=J+δJi, where δJi is uniformly distributed from [˜W/2, W/2]. Under such assumptions, local fluctuations in Ji would also be random. Without loss of generality, the imperfection and disorder in the spin rotation angle δθi can be regarded as a part of δΩi because their effects can be equivalent in the effective Hamiltonian description.
According to some embodiments, in order to quantify the localization length ξloc of quasi-particles excitations at the critical point, the effective Hamiltonian can be numerically diagonalized. For example, the effective Hamiltonian during the initialization step, D=˜Σi JiSizSi+1z−ΣiΩiŜix, can be mapped to a quadratic fermionic Hamiltonian using Jorndan-Wigner transformation: ci†=(Πj=1i-1Six(Siz−iSiy) and ci=(Πj=1i-1Six) (Siz+iSiy). This leads to D=Σij (ci†)Hij(cici†)T, where the single particle Hamiltonian Hij can be written as
In an example, the system can include an even number of particles up to N=3000 in the even parity sector P=1 with a periodic boundary condition (which can correspond to the anti-periodic boundary condition for free fermions, i.e. CN+1=−C1). The single particle Hij can be numerically diagonalized at the critical point Ω=J/2.
As a proxy for an inverse localization length ξloc−1, the average inverse participation ratio (IPR) ξloc−1≈IPR=Σi|ψi|4 can be computed to quantify the (inverse) localization length (which can limit achievable correlation length), where ψi is the quantum amplitude of an eigenstate at site-i, and ⋅ denotes averaging over 50 instances of disorder realizations, which imply various sets of randomly selected values, and over 50 energy eigenstates (closest to zero energy) per disorder realization.
The effects of disorder can become favorable during the read-out step. According to some embodiments, without disorder, domain wall excitations in one dimension can be mobile, can hinder coherent accumulation of phase information over a long time. If the localization lengths during the measurement step (Ω=0) are much shorter than those at the critical point, domain wall excitations become immobile, allowing more stable accumulation of the phase information from the signal. Repeating the numerical calculations at Ω=0, the localization length can be multiple orders of magnitude shorter during the measurement step than at the phase transition point (see
Exemplary Broad-Band Sensing Using Correlated Spin States
As discussed in more detail below, the disclosed protocol presents significant improvements in broad-band sensing. This is because the disclosed protocol maintains increased sensitivity over a broad range of frequency values. Thus, the disclosed protocols are well suited for applications that require both high sensitivity to faint signals and a large bandwidth to detect unknown signals that may be present in a large frequency range.
The detection of a weak signal at an unknown frequency can require a highly sensitive spectroscopic method with a large bandwidth. Such technique is often needed in the study of fundamental physics, such as the detection of gravitational waves or weakly interacting massive particles. In a conventional spectroscopic method, the increase in bandwidth entails the decrease in detection sensitivity. In other words, sensitivity (the ability to detect small signals) and bandwidth (the ability to detect signals over a large range of frequencies) are considered a tradeoff. For example, as discussed above, the standard quantum limit (SQL) of N non-interacting particles can lead to the sensitivity scaling:
δB−1˜√{square root over (NTT2)}, (S2)
where δB is the minimum detectable signal strength, T2 is the duration of each measurement cycle, and T is the total integration time. The bandwidth δω of this exemplary method is Fourier limited to the measurement duration δω˜1/T2, leading to the relation between the bandwidth and the signal sensitivity:
δω/(δB)2˜NT(conventional method SQL). (S3)
By utilizing quantum entanglement among many quantum spins, the bandwidth of the detection can be increased while maintaining the same signal sensitivity. In the protocol described throughout the present disclosure, ferromagnetic interactions among quantum spins can be used to develop quantum entanglement among X>>1 spins (where X=ξd in d-dimensional systems). In such a scenario, the sensitivity would scale according to:
(δB′)−1˜√{square root over (NT
where
δω′/(δB′)2˜χNT(quantum correlated method). (S5)
Therefore, the detection bandwidth can be improved by a factor of X while maintaining the same sensitivity. When the disclosed protocol is used, the number of correlated spins X is also determined from the maximum coherence time
For example, in the case of dipolar interactions (Jij˜J0/rij3) in a two-dimensional array of quantum spins, the phase transition can be described by the mean-field theory with the critical exponents v=1 and z=½, leading to X˜ξ2˜(JT2)4/7. In some embodiments, however, the noise bath often exhibits spatial correlations, which can modify the simple estimates provided in this section. Thus, as discussed in the next section, magnetic field fluctuations with microscopic origins such as dipolar spin impurities can exhibit spatial anti-correlations, which may be favorable.
When multiple spins are entangled, the coherence
In this section, exemplary magnetic noise generated from a fluctuating dipole (fluctuator) is shown to exhibit spatial anti-correlations at short distances. Such anti-correlations lead to a relatively longer coherence
Sμv(ω)≡∫eiωtBμ(t)Bv(0)dt. (S9)
Where Szz is a particular component of Sμv, where mu=nu=z.
When multiple spins and fluctuators are located far from one another, the noise fields at distant positions originate from different fluctuators; in such case, it can be assumed that spins are coupled to their own noise bath, leading to the relation
The above-described spatial correlations play a role in determining the coherence time of GHZ states, and particularly for improving the same, since the collective spins state interacts with its environment via the effective magnetic field noise
where {right arrow over (B)}({right arrow over (r)}i,t) is the magnetic field experienced by a single spin positioned at {right arrow over (r)}i. The corresponding spectral density function Seffμv:
which sensitively depends on the spatial correlations, such as Bμ({right arrow over (r)}i,t)Bv({right arrow over (r)}j,0) for ≠. In order to quantify the spatial correlation, a single magnetic dipole {right arrow over (m)} located at a distance z from a two-dimensional array of sensing spins with average spacing α0 can be considered (see
where the integration is performed over the area that covers correlated spins and mz=m cos α and mq=m sin α are the projections of the dipole moment in the vertical and planar {circumflex over (q)} directions, respectively. The corresponding spectral density can scale as:
According to some examples, when <<ξ, the noise density Seffcorr can be strongly suppressed. This suppression arises from the spatial profile of the magnetic field created from a dipole moment, as depicted in
where the nz is the linear density of fluctuators along the perpendicular direction {circumflex over (z)}. Integration along {circumflex over (x)} and ŷ directions can be neglected in this embodiment due to the symmetry.
This result can be compared to a fiducial spectral density Seffcorr in the absence of spatial correlations, for example, assuming Bμ({right arrow over (r)}i, t)Bμ({right arrow over (r)}j,0)=0 for ≠. This condition is equivalent to the assumption that each spin is coupled to an independent noise source. For a single classical dipole at depth z,
Integrated over the entire depth z∈(α0,∞),
where it was assumed that a fluctuator cannot be located closer than the spacing α0. Thus, for a sufficiently large correlation length, ξ>>α0, the integrated noise density of the correlated case is smaller by a factor α0/ξ compared to the uncorrelated case. This result implies that an effective coherence time of:
This sensitivity scaling implies that the reduction of the coherence time due to the enhanced susceptibility to (external) noise is less severe if noises from a fluctuator is correlated compared to when quantum spins experience independent noise.
Finally, from Eq. (S4), the sensitivity scales according to:
(δB′)−1˜√{square root over (NT
˜√{square root over (NTT2)}χ1/4. (S21)
This sensitivity is better than SQL, despite the presence of local magnetic noise sources. Accordingly, even in the presence of local magnetic noise sources, the disclosed protocol can improve upon detection sensitivity. Furthermore, as discussed above, this sensitivity can be obtained across a large bandwidth.
Exemplary Calculation of Sensitivity Enhancement for Magnetic Field Imagers
According to some exemplary embodiments, the amount of sensitivity enhancement for magnetic field imagers that implement the protocols can be calculated. As discussed in more detail below, while traditional sensing techniques are limited by the effects of quantum spin spacing, the disclosed protocol allows for a higher density of quantum spins, and therefore a significantly improved signal for spatially resolved measurements.
In an example, a two-dimensional array of electronic spins such as nitrogen vacancy (NV) color centers can be used to image a spatially resolved AC magnetic field profile. Since quantum states of NV centers are optically read-out, the spatial resolution of this method is given by the diffraction limit of the optical wavelength d˜λ/2≈250 nm, and can be further improved via existing sub-wavelength imaging techniques. Therefore, in order to achieve high precision AC magnetic field sensing without compromising the spatial resolution, the length scale of the probe volume can be configured so as not to exceed d (in this case, the system size). Since optical resolution is din this example, an ensemble of spins can be considered to be within a single optical probe spot (probe volume) and as a single spin ensemble. According to some embodiments, the correlation length ξ may beneficially be as large as d, but need not meet this condition. At such a length scale, however, dipolar interactions among electronic spins, Jdd˜J0/d3≈(2π) 3.3 Hz, is significant compared to the maximum coherence time of a NV center T2≈500 ms at low temperature (e.g., 77 K). According to traditional techniques, more than one NV center per probe volume cannot be used for the purpose of AC field sensing without affecting diffraction limited spatial resolution. Even at room temperature with the coherence time T2≈3 ms, the separation among NV centers should be at least rmin≈100 nm in order to avoid intra-spin interaction-induced decoherence. Accordingly, at most, N0≈6 NV centers can be used per probe volume. Accordingly, using this small increase in the number of NV centers produces a corresponding enhancement in signal-to-noise ratio √{square root over (N0)}≈3 at best in the conventional SQL.
By contrast, embodiments of the disclosed protocols dramatically alleviate the aforementioned limitations. In some embodiments, the minimum distance among NV centers are not bounded by dipolar interactions, allowing much higher particle density. In such examples, the minimum distance is restricted primarily by the length scale of electronic orbitals of NV centers, which is the order of a few nm. In an example with a single layer of NV centers, it is possible to include approximately N1≈2,500 or 6,400 particles to approximately 10,000 particles per probe volume. In some example implementations, such a density allows for an enhancement of a factor of approximately 50 at 77 K, and about 20 at room temperature even without accounting for the effects of quantum correlations. According to some embodiments, NV spin centers can be implemented with a density in diamond of approximately 10−4 to approximately 0.01 spins/nm2
Furthermore, when the coherence time of the sensing spin is limited by correlated magnetic noises, such as that generated by proximal fluctuating dipoles, the disclosed protocol can provide additional sensitivity enhancement as discussed above. Thus, in an example of a two-dimensional array of spins, this sensitivity enhancement amounts to a factor of X1/4 in Eq. (S21). X can be estimated self-consistently similar to Eq. (S8).
This application claims the benefit of priority to U.S. Provisional Application No. 62/616,774, entitled “QUANTUM METROLOGY BASED ON STRONGLY CORRELATED MATTER,” filed on Jan. 12, 2018, the disclosure of which is hereby incorporated by reference in its entirety.
This invention was made with Government support under Grant No. PHY-1654740 awarded by the National Science Foundation (“NSF”), Grant No. N00014-15-1-2846 awarded by the Department of Defense (“DOD”), and Grant No. PHY-1125846 awarded by the NSF. The Government has certain rights in the invention.
Number | Name | Date | Kind |
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20120319684 | Gambetta | Dec 2012 | A1 |
20150316598 | Rogge | Nov 2015 | A1 |
20160061914 | Jelezko | Mar 2016 | A1 |
20180149717 | Jelezko | May 2018 | A1 |
20190049495 | Ofek | Feb 2019 | A1 |
20190235031 | Ibrahim | Aug 2019 | A1 |
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20190219644 A1 | Jul 2019 | US |
Number | Date | Country | |
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62616774 | Jan 2018 | US |