A mixed squid array

Abstract

The present invention relates to a quantum magnetic field detecting device comprising a two-dimensional array of superconducting loops, acting as a high- efficiency transformer-free Magnetically Small Antenna (MSA), comprising Superconducting Quantum Interference Device (SQUID) loops with Josephson junctions and bare loops without Josephson junctions.

Description

A Mixed SQUID array Technical Field

[0001] The present invention relates to a quantum magnetic field detecting device comprising a two-dimensional array of superconducting loops, acting as a high- efficiency transformer-free Magnetically Small Antenna (MSA), comprising Superconducting Quantum Interference Device (SQUID) loops with Josephson junctions and bare loops without Josephson junctions. The quantum magnetic field detecting device is called a ‘Mixed-SQUID array’. Background of Invention

[0002] A central property of superconductors is that they are highly sensitive to an applied external magnetic field. It was realised that this feature can be utilised to fabricate magnetic field detecting devices – devices which experience a distinctive voltage drop when a magnetic field is incident. A prototypical example of a superconducting device used to construct these magnetic field detectors is a solid, closed loop of superconducting material, i.e. a superconducting loop. When an external magnetic field is incident on one of these loops, the magnetic flux through the loop area is denoted by the magnetic flux ^^ as follows in equation 1. 1) Area

[0003] By a physical effect, known as the Meissner effect, the magnetic flux induces a direct current which flows around the loop. To distinguish this current from say an externally applied current, the current induced by the magnetic flux will be known as the ‘screening current’. It is important to note that this screening current is a super-current in that it lacks any resistance; hence, no measurable voltage drop due to the screening current can be detected.

[0004] To measure a voltage, a ‘Superconducting QUantum Interference Device’, more commonly known by the acronym ‘SQUID’, can been used. The SQUID has the addition of two Josephson junctions – heterostructures consisting of the non- superconducting material sandwiched between the two superconducting regions – attwo points in the superconducting loop. That is, the Josephson junctions being small barriers separating two superconducting regions.

[0005] In Fig.1, an existing DC SQUID is shown, where an external magnetic flux is being applied normal to the plane leading to a total flux ^^.

[0006] The induced screening current must pass through the non-superconducting regions of the Josephson junctions of the DC SQUID. As it does so, it produces a measurable voltage drop which is schematically depicted as V in this Fig.1. This is the manner through which the DC SQUID acts as a magnetic flux to voltage transducer, as it converts the screening currents induced by the external applied magnetic fields into an electrical voltage which can be measured. Thus, it acts as a quantum magnetic field detecting device (magnetometer).

[0007] Fig.2 shows the prototypical periodical DC-SQUID Magnetic Flux Φ to (Normalized) Voltage response. This explain why the DC SQUID is the prototypical device for measuring a magnetic field as a magnetometer.

[0008] Although the periodic voltage response depicted in Fig.2 is the standard response for a relative DC SQUID magnetometer, the ideal response for an absolute magnetometer is instead aperiodic. This ideal absolute response is depicted in Fig.3. The strong ‘dip’ in the voltage response shown in Fig.3 at zero external magnetic field, followed by a uniform voltage at other flux values, is a feature known as an ‘anti- peak’. The appearance of this anti-peak is the ideal response all devices seek to produce, under the effects of an external magnetic field, to be used for quantum sensing applications.

[0009] SQUIDs are generally known as the building block of superconducting based magnetometers. This technology has matured to the level of many commercial applications, such as their use in magnetoencephalography (MEG) machines. However, SQUIDs are highly noise sensitive making them impractical for some technologies. Because of this, as fabrication techniques improve, interest has turned to two-dimensional arrays of SQUID loops as they promise greater device sensitivity while preserving all the other properties associated to superconductivity.

[0010] Two-dimensional arrays of SQUIDs are known to produce a voltage response that features an anti-peak at zero magnetic field. Fig.4a shows a prototypical two-dimensional SQUID array, also known as Superconducting Quantum Interference (SQIF).

[0011] A SQUID array, such as a DC SQUID array, increases the effectiveness of SQUIDs to act as a flux to voltage transducer, as shown in Fig.4b. In this specification, an array consisting of solely DC SQUID loops is referred to as a ‘SQUID array’. SQUID arrays take into advantage the principle of ergodic systems that the signal-to-noise ratio (SNR) will be proportional to √^^ with ^^ the number of loops within the array. Hence, if ^^ can reach a sufficient large number (100000 to 1000 000), a SQUID array can theoretically sense at the quantum limit of noise.

[0012] There are difficulties, however, in designing an effective SQUID array. The most significant difficulty is that, for optimum performance, the SQUID arrays require a very large number ^^ of single SQUID loops. However, another difficulty is that for a device to observe this anti-peak behaviour, the areas of individual SQUID loops must be different – this shown in the device of Fig.4c.

[0013] That is, a SQIF, as shown in Fig.4c, may be formed with SQUIDs arranged in an array with non-equal-sized loops. It may be crucial for device operability that this condition is satisfied. It is, however, technically difficult to fabricate such a device.

[0014] Fig.4d shows a schematic of a 2D SQUID array as in Fig.4c with the figure schematically demonstrating how the flow of electrons in the Superconducting Materials (and the output Voltage, V, of the overall 2D array) is affected by the external EM signals as they interact via the magnetic flux in each of the loop. In superconducting materials, reasonable change of V can be observed, up to very small changes of the amplitudes of the EM signals, leading to high signal sensitivities for these systems.

[0015] Further, varying the area of individual SQUID loops invariably degrades the absolute performance of the SQUID array. The measure of this performance is a quantity known asand it is highly sensitive to variations in the lengths of thesuperconducting wire segments forming the SQUID loop. ^^^^ is given by ^^^^ = ^^^^^^ / ^^0,where L is the inductance of the SQUID Loop, where ^^^^is the critical current of the component Josephson junctions, and ^^0is the quanta of magnetic flux.

[0016] One problem with standard 2D SQUID arrays, shown in Fig.4b, is although they are sensitive to changes in the magnetic flux, they do not work as absolute magnetometers for the reasons stated above. One existing solution to this problem was developed by making the area of each SQUID loop incommensurate to introduce interference between the loops. As a result, the oscillating voltages of each SQUID loop, in the device of Fig.4b, de-constructively interfere to produce a so called anti- peak response; a sharp dip in the voltage at zero flux which allows the device to function as an absolute magnetometer.

[0017] The incommensurate loop areas required to produce an anti-peak, however, degrade the device performance by altering the ^^^^parameter of each individual SQUID, which is a central problem for these devices. ^^^^is given by ^^^^= ^^^^^^ / ^^0, where ^^^^is the critical current of the component Josephson junctions, and ^^0is the quanta of magnetic flux.

[0018] Indeed, another issue emerges in that, by varying the lengths of the wire segments within the loop such that there is a spread in the areas, the inductance ^^ of each individual SQUID loop is drastically changed. The central parameter this modifies is ^^^^. It has been determined experimentally that precise control of ^^^^is critical to proper device performance. Although there have been proposals to mitigate this, it still posses a significant technological challenge for the fabrication of large 2D SQUID arrays with area spread.

[0019] A reference herein to a patent document or other matter which is given as prior art is not to be taken as an admission that the document or matter was known or that the information it contains was part of the common general knowledge at the priority date of any of the disclosure or claims herein. Such discussion of prior art in this specification is included to explain the context of the present invention in terms of the inventors’ knowledge and experience.Summary of Invention

[0020] According to one aspect of the present invention, there is provided a quantum magnetic field detecting device, comprising: a two-dimensional (2D) array of superconducting loops arranged in a plane, the array comprising: a first subset of the superconducting loops comprises Superconducting Quantum Interference Device (SQUID) loops comprising two Josephson junctions each arranged in opposed first and second arms of each of the SQUID loops; and a second subset of the superconducting loops comprises bare loops without Josephson junctions, wherein the two-dimensional array is configured to function as an active transformer-free Magnetically Small Antenna (MSA).

[0021] Preferably, the two-dimensional array is a high-efficiency Superconducting Magnetically Small Antenna (MSA).

[0022] Superconducting Magnetically small antennas (MSAs) behave in a similar fashion to electrically small antennas (ESAs) but, for these MSAs, the dimensions are completely decoupled from the wavelength of interest.

[0023] The two-dimensional array functions as an active MSA by using SQUID loops and bare loops to amplify and convert magnetic flux directly into voltage.

[0024] The SQUID unit cell used in the 2D array is a DC SQUID. It will be appreciated by those persons skilled in the art, however, that other configurations of superconducting quantum interference devices could be used, for example, a Bi- SQUID unit cell, which will be described below.

[0025] The second subset of the superconducting loops comprises bare loops without Josephson junctions. These loops comprise a solid, closed loop of superconducting material.

[0026] The quantum magnetic field detecting device is thus a novel class of physical devices which, for the purpose of this text, is called a ‘Mixed-SQUID array’. The device is “mixed” as it comprises an array of both SQUID superconducting loops, and bare superconducting loops.

[0027] In another aspect of the present invention, an absolute magnetometer comprising the quantum magnetic field detecting device is provided. The absolute magnetometer manifests an optimal anti-peak response at zero magnetic field. That is, the voltage response of the absolute magnetometer induced by a magnetic flux external to the absolute magnetometer dips when the magnetic flux is zero and is relatively uniform at other magnetic flux values. An absolute magnetometer with a Mixed-SQUID array is configured to produce a voltage response that features an anti- peak at zero magnetic field / flux; thus, a quantum magnetic field detecting device can function as an effective absolute magnetometer.

[0028] In an embodiment, the area of each superconducting loops is the same.

[0029] The quantum magnetic field detecting device thus overcomes the technical difficulties of altering the area of each individual SQUID loop in a 2D SQUID array as described above; for example, in the design of absolute magnetometers. The performance of the quantum magnetic field detecting device will not be degraded, and the device may perform at the theoretically optimal levels.

[0030] The fundamental sensing principal behind the mixed-SQUID array is identical to the standard SQUID array: screening currents are induced by a magnetic flux; these currents can produce a measurable voltage when passing through the Josephson junctions. One difference between the quantum magnetic field detecting device and the existing SQUID array device is that not every SQUID loop is required to have a different area to observe an anti-peak.

[0031] More specifically, in the above-described existing SQUID array, ‘interference’ is introduced between SQUID loops to generate an ideal ‘anti-peak’ response. One existing method to produce this interference is to introduce a spread in the areas of individual SQUID loops in a 2D array of SQUID loops. The Applicant notes that, producing area spread in an array of superconducting loops without strongly degrading device performance is a significant technological challenge.

[0032] In an embodiment, the device generates a synthetic area spread of the superconducting loops, the synthetic area spread being an effective magnetic sensing area greater than a geometric area of the two-dimensional array, achieved by inclusion of the bare superconducting loops in the two-dimensional array.

[0033] Accordingly, the device generates the synthetic area spread of the superconducting loops even in the absence of real, physical spread.

[0034] The device can be tuned in a desired way by the placement of the bare superconducting loops in the device. In an embodiment, the placement of the bare superconducting loops in the two-dimensional array is selected to optimise the synthetic area spread for a target application.

[0035] . That is, the quantum magnetic field detecting device comprises bare superconducting loops within the 2D SQUID array to generate the synthetic area spread. In this manner, a Mixed-SQUID array behaves identically to a device with area spread, when no such physical spread actually exists. The difficulties normally associated with physically introducing area spread in 2D SQUID designs are thus not encountered with a Mixed-SQUID array that produces a purely synthetic area spread. The placement of the bare loops within the Mixed-SQUID array will determine the shape of the synthetic area spread; hence, by determining an optimal placement for these bare loops it is possible to generate the most suitable synthetic area spread valid for different applications.

[0036] That is, in the embodiment, the device generates synthetic area spread of the superconducting loops, where the area of each superconducting loops is the same.

[0037] In an embodiment, a second arm of a first SQUID loop forms a first arm of a second SQUID loop. In the embodiment, the SQUID loops are located in a row of the 2D array. In another embodiment, the SQUID loops are located in a column of the 2D array.

[0038] As the SQUID loops comprises two Josephson junctions at opposed points in each of the SQUID superconducting loops, adjacent loops share a common Josephson junction. Accordingly, locating SQUID loops in a row or a column has manufacturing efficiency.

[0039] In an embodiment, one or more of the SQUID loops comprises a Bi-SQUID loop comprising a third arm electrically connecting the first arm and the second arm, and the third arm comprising a Josephson junction. The Bi-SQUID loop furthercomprises fourth and fifth arms electrically connecting the first arm and the second arms, respectively. This creates first and second sections of the Bi-SQUID. The first section between the first, second, third and fourth arms must be shielded from any external and internal magnetic fluxes. The second section of the area of the Bi- SQUID, between the first, second, third and fifth arms, is sensing magnetic fluxes as it is done by the areas of any conventional DC SQUID.

[0040] In an embodiment, the total area of the Bi-SQUID loop, defined by the first, second, fourth and fifth arms, is twice the area of one of the bare loops, and thus the area of each superconducting loops is the same.

[0041] In an embodiment, a second arm of a first Bi-SQUID loop forms a first arm of a second Bi-SQUID loop. The Bi-SQUID loops may thus be located in a row of the 2D array.

[0042] In an embodiment, a fifth arm of a first Bi-SQUID loop forms a fourth arm of a second Bi-SQUID loop. The Bi-SQUID loops may thus be located in a column of the 2D array.

[0043] As above, locating Bi-SQUID loops in a row or a column has manufacturing efficiency.

[0044] In an embodiment, the 2D array is a N x M array, where values of N and M are based on application of the device. For example, the 2D array is a 1000 x 1000 array. It will be appreciated that it is technically difficult to achieve physical area spread for 1000 x 1000 SQUID loops thus synthetic area spread is advantageous. Brief Description of Drawings

[0045] A preferred embodiment of the present invention will now be described with reference to the accompanying drawings wherein:

[0046] Figure 1 is a schematic of a conventional DC Superconducting Quantum Interference Device (DC-SQUID) loop;

[0047] Figure 2 is a plot illustrating a typical periodical DC-SQUID Magnetic Flux to Voltage response;

[0048] Figure 3 is a plot illustrating an ideal anti-peak response from a quantum magnetic field detecting device;

[0049] Figure 4a is a schematic of a 2D SQUID array with no spread in the areas of the SQUID loops;

[0050] Figure 4b is a schematic of a quantum magnetic field detecting device comprising a 2D SQUID array;

[0051] Figure 4c is a schematic of a quantum magnetic field detecting device comprising a 2D SQUID array with a physical spread in the areas of the SQUID loops;

[0052] Figure 4d is a schematic of a 2D SQUID array in operation as an EM sensor;

[0053] Figure 5 is a schematic of a quantum magnetic field detecting device in the form of a Mixed-SQUID array, according to an embodiment of the present invention;

[0054] Figure 6 is a circuit schematic of a Mixed-SQUID array according to an embodiment of the present invention;

[0055] Figures 7a shows a plot illustrating a response from a quantum magnetic field detecting device;

[0056] Figure 7b shows a plot illustrating an anti-peak response from a device according to embodiments of the present invention;

[0057] Figures 8a and 8b show plots illustrating anti-peak responses from quantum magnetic field detecting devices according to embodiments of the present invention for larger N X N;

[0058] Figures 9a and 9b show extended scans of the responses shown in Figures 8a and 8b, respectively;

[0059] Figure 10 shows the robustness of the VMF response for devices with bare loops according to embodiments of the present invention vs synthetic area, even for N x N close to 64 X 64

[0060] Figure 11 is a schematic of a Mixed-SQUID array comprising Bi-SQUID loops only, according to an embodiment of the present invention;

[0061] Figure 12 is a schematic of a quantum magnetic field detecting device in the form of a Mixed-SQUID array comprising Bi-SQUID loops and bare loops, according to an embodiment of the present invention;

[0062] Figures 13a and 13b show experimental VMF responses for a type A 16×16 quantum magnetic field detecting device according to an embodiment of the present invention;

[0063] Figure 13c shows experimental VMF responses for a quantum magnetic field detecting device with no bare loops;

[0064] Figures 14a shows experimental VMF responses for a type B 16 ×16 quantum magnetic field detecting device according to an embodiment of the present invention; and

[0065] Figure 14b shows experimental VMF responses for a quantum magnetic field detecting device with no bare loops. Detailed Description

[0066] A quantum magnetic field detecting device 10 according to an embodiment of the present invention is shown in Fig.5. The device 10 comprises a two- dimensional (2D) array of superconducting loops 12. The superconducting loops 12 comprise Superconducting Quantum Interference Device (SQUID) loops 14 and bare loops 16 without Josephson junctions. The SQUID loops 14 are DC-SQUID loops, which are superconducting loops 12 comprising two Josephson junctions 18 at opposed first 20 and second 22 arms in the SQUID loop. The 2D array is thus configured to function as a high-efficiency active transformer-free Superconducting Magnetically Small Antenna (MSA).

[0067] The SQUID loops 14 form a first subset of superconducting loops 12 and the bare loops 16 form a second subset. Thus, some, but not all, of the loops 12 of the device 10 are bare loops 16.

[0068] The bare superconducting loops 16 not comprising a SQUID each have a solid, closed loop of superconducting material, such as Niobium (Nb), Aluminium, YBCO, Bi2Sr2CaCu2O8+δ (Bi-2212), Tl2Sr2Ca2Cu3O10+δ (Tl-2223), MgB2 and many others.

[0069] The SQUID loops 14 also have loops of any superconducting material, such as Niobium (Nb), Aluminium, YBCO, Bi2Sr2CaCu2O8+δ (Bi-2212), Tl2Sr2Ca2Cu3O10+δ (Tl-2223), MgB2, just to name a few superconducting materials, with the two Josephson junctions 18, formed by the insulating barriers or the non- superconducting material, at opposed points in each loop 14. The Josephson junctions 18 may be created each time Aluminium Oxide or other insulating or non- superconducting materials is inserted between superconducting parts of the loop 14.

[0070] The device 10 comprising an array of bare superconducting loops 16 and SQUID loops 14 is called a ‘Mixed-SQUID array’. The device 10 is “mixed” as it comprises an array of both SQUID superconducting loops and bare superconducting loops. In an embodiment, an absolute magnetometer comprising the quantum magnetic field detecting device 10 can be constructed with a designated placement of the bare loops in the array 12.

[0071] In Fig.5, a mixed-SQUID array is shown with two rows of bare loops 16 separating a row of SQUID loops 14. As mentioned, this array is configured to function as a high-efficiency active transformer-free Superconducting Magnetically Small Antenna (MSA). The Josephson junctions 18 shown in the embodiment of Fig. 5 illustrate that there are rows containing solely SQUID loops 14, alternating with two rows containing entirely bare loops 12. That is, a second arm of a first SQUID loop thus forms a first arm of a second SQUID loop. It will be appreciated that, in other embodiments, the SQUID loops 14 may not be arranged in rows. For example, the SQUID loops 14 may be arranged in columns.

[0072] The area of each superconducting loop 12 is the same. The placement of bare loops 16 in the array of superconducting loops 12 provides a “synthetic” spread in the areas of the SQUID loops 14. That is, an actual (physical) spread in the areas of SQUID loops is not required to produce a voltage response of the device 10 that features an anti-peak at zero magnetic field.

[0073] There is therefore no spread in the areas of the superconducting loops 12 of the device 10. The resulting equations of motion for the device 10 are the same as a device with no bare loops but possessing an area spread. As a result, the Applicant demonstrates an equivalence between devices with no spread in the areas with some bare loops, and devices with no bare loops but with area spread. Furthermore, the Applicant also demonstrated that a careful addition of bare loops into the 2D SQUID arrays, acting as a high-efficiency active transformer-free MSA, with no physical spread in the areas will lead to optimal anti-peak at zero magnetic field response and absolute magnetometer signature.

[0074] To determine the equivalence between the device 10 comprising a mixed- SQUID array with bare loops and no area spread, and an existing device with area spread and no bare loops, the following equations are used.

[0075] A circuit theory approach utilising the RSJ equations is used with matrix formalism.

[0076] In Fig.6, a schematic of a 2D Mixed-SQUID array according to an embodiment of the present invention is depicted. It consists of ^^ superconducting loops in the vertical axis, and ^^ loops in the horizontal axis. Some of these loops of superconducting wire have two Josephson junctions along their vertical sides, hence forming SQUID loops; the remaining loops which do not contain Josephson junctions are bare superconducting loops.

[0077] This generalised 2D SQUID array can be considered as a standard SQUID array interspersed with bare superconducting loops. In the specific embodiment shown in Fig.6, the first and third rows consist of SQUID loops, whilst the second and ultimate rows contain solely bare loops.

[0078] The numbering systems of the bias currents (^^^^^^), the horizontal wires (^^^^), the vertical wires (^^^^), and the final row of horizontal wires (^^^^^^) are labelled.

[0079] The current is denoted through the ^^thvertical wire segments as ^^^^, where, reading from left to right and top to bottom, each superconducting wire is numberedsequentially; only the first ^^ + 1 vertical wire segments are labelled in Fig. 6. For thehorizontal wire segments, they are denoted by ^^^^and also labelled sequentially, however, the final row of horizontal currents is distinguished by labelling as ^^^^^^.

[0080] The vectors containing each of these elements are denoted by theunderlined quantities ^^, ^^, ^^^^, ^^^^. Concerning dimensions: ^^ contains ^^(^^ + 1)elements, ^^ contains ^^(^^ − 1) elements, ^^^^ contains ^^ elements, and we pad ^^^^ with(^^ + 1)(^^ − 1) zeros (^^^^ ≡ [^^^^1, … , ^^^^ ^^+1, 0, … ,0]^^) such that the resulting vector has the same length as ^^.

[0081] Within a circuit theory, one can construct Kirchhoff’s current conservation laws at each vertex of the circuit. For example, the conservation condition for the vertex located at the intersection between the ^^thcolumn and the ^^throw, assuming it is not lying on the edge of the graph, is given by

[0082] Coefficient matrices ^^^^and ^^^^consisting entirely of −1, 0, and 1 are introduced such that the Kirchhoff conservation condition for every node can be represented as the matrix equation 3) ^^^^^^ = ^^^^^^ + ^^^^ .where ^^^^ has dimensions [^^(^^ + 1) × ^^(^^ + 1)] and ^^^^ has dimensions[^^(^^ + 1) × ^^(^^ − 1)].

[0083] As the final row of horizontal currents is distinguished, further Kirchhoffmatrices ^^^^, with dimension [^^ × ^^(^^ + 1)], and ^^^^, with dimension [^^ × ^^], definedsuch that the current conservation laws for the final row of vertices, are contained within the system of equationswhere ^^^^ ≡^^… , ^^ ^^]^^bundles the currents leaving the device into a vector.

[0084] The current through each wire segment will in-turn produce an inductive flux contribution through each superconducting loop, which is denoted by the vectorcontaining ^^ × ^^ elements. To determine the flux contributions resulting fromthese vertical (^^), horizontal (^^), and final row (^^^^), introduced are inductance matrices ^^^^, ^^^^, ^^^^respectively. Although the precise form of these matrices will be determined by the physical model for the inductances considered i.e. kinetic inductances, geometric inductances, they give the resulting inductive contribution to the flux through each loop by

[0085] Due to the phase quantisation condition for each superconducting loop, this inductive contribution to the flux is balanced by both the external flux through that loop, ^^ext, and any Josephson junction phase differences ^^ present within that loop. However, in a bare superconducting loop there are no Josephson junction phases to balance the external flux; this is the critical distinction between a bare loop and a SQUID loop. Without the loss of generality, for the ^^thloop in the array surroundedgenerically by the ^^ − 1 and ^^ vertical wire segments, the corresponding phasequantisation conditions through the SQUID loop and the bare loops are given by

[0086] At the expense of introducing more notation, it is useful to introduce sets ^^ and ^^ defined as ^^ ≡ {^^ | the ^^th loop is a SQUID} ,^^ ≡ {^^ | the ^^th vertical wire forms part of a SQUID} ,to allow us to distinguish notationally between wires and loops forming SQUID loops. The number of elements of these sets are precisely the number of SQUID loops, and the number of Josephson junctions, which we label as ^^SQUIDand ^^JJrespectively. We can now compactly write the phase quantisation conditions for SQUIDS and bare loops defined by Eq.7 in vector form aswhere introduced is a generalised difference vector ^^ whose elements are defined by

[0087] With the aim of rewriting the inductive contribution within Eq.8 as a function of the currents through the device, the Kirchhoff matrices defined previously are combined with the inductive flux contribution defined in Eq.5 to express the phase quantisation condition as a function of solely the horizontal currents ^^where

[0088] The dynamics of each Josephson junction is a function of not only both the phase difference across that junction, ^^^^, and the current ^^^^, but also the physical parameters such as the resistance ^^, and the critical current ^^^^of each junction. Assuming identical overdamped Josephson junctions, the relationship between the current through each junction and the phase difference ^^ is given by the RSJ model.

[0089] Note that a noise and further capacitive term are omitted, which, although are standard terms to include for the purposes of modelling, are irrelevant in proving the equivalence between arrays containing bare loops and no spread, and arrays containing area spread but no bare loops.

[0090] As the RSJ equations require only the currents through the Josephson junctions, rather than through every wire segment, introduced is a mask matrix ^^ withdimensions [^^JJ × ^^(^^ + 1)] which can project vectors defined over every wiresegment onto vectors defined only over wire segments containing a Josephson junction. The elements of this projection matrix are defined as13) ^^^^^^ = ^^^^,^^^^ ,such that the vector of currents through each Josephson junction is given by ^^^^. Substituting in Eq.10 and the Kirchhoff matrices, and through the use of the projection matrix ^^, we can now express the RSJ equations defined in Eq.12 aswhere we have definedto simplify the resulting equation. Eq.14 is the equation of motion for a 2D array containing bare loops, whereupon all dynamics can be evaluated by solving this equation. Synthetic area spread

[0091] In this section, the equivalence between arrays of superconducting loops containing bare loops and no spread in areas of the loops, and arrays containing area spread of loops and no bare loops is described. As above, for arrays containing bare loops, the area spread is denoted as a synthetic area spread.

[0092] A 2D array containing both SQUID and bare loops as per Figure 5 is provided, where the area of each loop is contained within the vector ^^; for the case of equal areas ^^ will simply be a vector of ones. Also considered is a separate 2D SQUID array which does not contain any bare loops, where all of its quantities are denoted by primed coordinates; i.e., ^^^^′ denotes the vertical inductive elements. The inductances of each wire segment are set, and the strength of each bias current, in this new device such that

[0093] It is important to note that whilst ^^^^−1 has dimension [^^JJ × ^^^^], thedimensions of ^^′^^′−1 are [^^JJ × ^^SQUID] as we are only selecting the columns thatcorrespond to a SQUID loop. Although both matrices map input vectors into ℝ^^JJ, because Kirchhoff’s laws are overdetermined then the rank of both of these matrices are ^^SQUIDrather than ^^JJ; physically, the ^^JJwire segments we are solving for are shared between ^^SQUIDloops which constrains the set of currents which are possible. We now introduce a vector ^^′ defined bywhere + denotes the Moore-Penrose inverse. As the rank of ^^′^^′−1and ^^^^−1are the same, and ^^′^^′−1is simply a subset of the columns of ^^^^−1, then theMoore-Penrose inverse has the property that ^^′^^′−1(^^′^^′−1)+^^^^−1 = ^^^^−1.

[0094] As a result, we can rewrite the RSJ equations governing the bare loop device given in Eq.14 as

[0095] Assuming that the area spread of the non-bare loop device is identically given by ^^′, then we can immediately write the equation of motion for that separate device as

[0096] As a result, the equations governing the bare loop device given by Eq.18 and the equations governing the non-bare loop device given by Eq.19 are identical. Importantly these are two different physical devices with the same dynamics: an array with bare loops and area spread ^^, and an array with no bare loops and a modified area spread ^^′, albeit with slightly varied inductances and bias current strength. In theextreme case where ^^^^ ≡ 1 for any ^^, such that the bare loop device has no areaspread, then Eq.17 states that there exists an equivalent device with a non-trivial area spread given by the (synthetic) non-unity vector ^^′; it is for this reason we denote it as the synthetic area.

[0097] To explore this equivalence numerically, we now look to solving the dynamics from the equation of motion. Generically, given the physical parameters of each Josephson junction, Eq.14 can be solved numerically to determine the phase difference across each junction. We can then substitute this solution back into Eq.14 to determine the voltage across each junction by the Josephson relation

[0098] Finally, by averaging the voltage over parallel junctions 21)we can compute the experimentally measurable time-averaged voltage by 22) ^^‾^ ^ ≡^l^i→m∞

[0099] To explore the two equivalent devices further, we consider a small device consisting of six SQUID loops. In Fig.7(a), shown is a plot the time-averaged voltage for the array when every loop area is the same size and with no bare loop present in the array. The voltage response is of a 2D SQUID array with no spread in the areas of the SQUID loops and no bare loops and is characteristically periodic as there is no interference induced by the different SQUID loops.

[0100] In Fig.7 (b), the voltage response of two different devices is plotted: one on the left containing two random rows of bare loops between rows of SQUID loops, where all loops have the same area; and, on the right is the equivalent device with a spread in the areas of the loops. The voltage response of the device on the left is shown with a black diamond in the plot and the voltage response of the device on the right is shown with a black circle in the plot. The voltage-flux response for the deviceon the right of Fig. 7(b) is for a 3 × 2 array with different loop areas. The voltage-fluxresponse for the device on the left of Fig. 7(b) is for a 5 × 2 array containing two rowsof solely bare loops, and each loop has an equal area. The voltage-flux response for the synthetic area was determined by solving for ^^′ in Eq.17.

[0101] The array in Fig.7(a) does not have bare loops per the Mixed-SQUID array, and hence operates as a relative magnetometer. The Mixed-SQUID array device of the embodiment shown in Fig.7(b), on the other hand, acts as a high- efficiency active transformer free Magnetically Small Antenna (MSA) and, as a result of the addition of bare loops, the device provides an absolute magnetometer response even without physical spread in the area.

[0102] Figure 8(a) shows the voltage response for another example of a Mixed- SQUID array with a larger N X N, up to 10 X 10, and one line of bare loops. Figure 8(b) shows the voltage response of another example of a Mixed-SQUID array with a larger N X N, up to 10 X 10, and three lines of bare loops.

[0103] That is, Figures 8(a) and 8(b) shows plots illustrating anti-peak responses for two more embodiments of Mixed-SQUID array devices, both acting as a high- efficiency active transformer free Magnetically Small Antennas (MSA), and both demonstrating how 2D SQUID arrays with bare loops manifest a behaviour equivalent to the one of a device with a spread in the areas of the SQUID loops.

[0104] Figures 9a and 9b show extended scans for each of the two devices in Figure 8(a) and (b), demonstrating that even after 250 periods the central peak is dominating the Voltage Magnetic Flux (VMF) response;

[0105] Figure 10 quantifies how the difference between the bare loop and the synthetic areas VMF responses is still relatively small even for 2D Mixed Squid Arrays as big as 64 X 64. Bi-SQUID loops

[0106] In an embodiment, the array of superconducting loops 12, acting as a high- efficiency active transformer-free MSA, may comprise Bi-SQUID loops 24 as shown in Fig.11. Each Bi-SQUID loop 24 comprising a third arm 26 electrically connecting the first arm 20 and the second arm 22, and the third arm 26 comprising a Josephson junction 28.

[0107] As indicated by the Bi-SQUID loop 24 in Fig.11, a single unit cell for this kind of array 12 is made of DC SQUID loop 14 with Josephson Junctions 18 (as for the conventional DC SQUID) plus a third Josephson junction 28 which is shared witha second loop. The Bi-SQUID operates in a similar fashion to the DC SQUID but because it has the DC SQUID loop section 12 (in black in the Figure) shielded from magnetic flux the later penetrates and is sensed only via the second loop. And can present better performances in terms of Spurious Free Dynamical Range (SFDR) linearity if compared to a conventional DC SQUID.

[0108] A conventional 2D Bi-SQUID array that does not contain bare loops can manifest optimal anti-peak at zero magnetic field response and absolute magnetometer signature only if there is a spread in the distribution of the areas of the second loops of each Bi-SQUID. The 2-D SQUID array shown in Fig.11 has no bare loops and no spread in areas and, as such, no ideal anti-peak response in the Voltage-External Magnetic flux curve is expected for this system

[0109] The Bi-SQUID loop 24 in Fig.11 comprises fourth 30 and fifth 32 arms also electrically connecting the first arm 20 and the second arm 22, respectively. The area of the Bi-SQUID loop, defined by the first 20, second 22, fourth 30 and fifth 32 arms, is twice the area of one of the bare loops or a DC SQUID loop, and the area of each superconducting loops 12 is the same.

[0110] Fig.12 shows an embodiment of the device 10 with the addition of extra bare loops 16 into a 2D array of loops 12 containing Bi-SQUID loops 24, and thus the device 10 acts as a high-efficiency active transformer-free MSA. The bare loops 16 have been placed in the 2D array 12 optimally to generate an ideal anti-peak response Voltage-External Magnetic flux even with no physical spread of the loop areas.

[0111] The Bi-SQUID loops 24 are located in a row of the array 12, such that a second arm 22 of a first Bi-SQUID loop forms a first arm 20 of a second Bi-SQUID loop. In addition, two Bi-SQUID loops 24 are located in a configuration separated by two rows of bare loops 16. For the column of Bi-SQUID loops, a fifth arm 32 of a first Bi-SQUID loop forms a fourth arm 30 of a second Bi-SQUID loop. The fifth arm 32 of the Bi-SQUID loop without a Josephson junction forms the top arm of a bare loop.

[0112] As mentioned, the single unit cell of a Bi-SQUID loop 24 is formed by a conventional SQUID loop 14 with two Josephson Junctions 18 with a second loop with the third Josephson junction 28 added to the cell. The first SQUID loop is markedin black inside the loop 14 to indicate that, to operate close to ideal, no magnetic flux should be penetrating the DC SQUID section of the BI-SQUID cell while all the magnetic flux sensing should be happening within the second loop of each unit cell. The second section of the area of the Bi-SQUID not marked in black senses all the magnetic flux.

[0113] As for the existing 2D SQIF shown in Fig.4c, a 2D Bi-SQUID array geometry will act a relative magnetometer and, to operate as an absolute magnetometer, the array requires second loops in each of the cells to have incommensurate area (as shown in Fig.4c for the 2D SQIF). However, similar to the device 10 of Fig.5 for the 2D-SQIF of Fig.4c, the addition of extra bare loops 16 caused the 2D array of BI-SQUID to operate as an absolute magnetometer with no spread in the areas of the loops 12 required. Fig.12 shows a possible example of a 2D BI-SQUID array operating as an absolute magnetometer, where bare loops 16 are included in the array 12 of superconducting loops.

[0114] Hence, an equivalence between 2D BI-SQUID arrays containing bare loops and no spread, and arrays containing area spread but no bare loops (i.e., Synthetic area spread) can be provided as in the section above, and similar equations as in the previous section can be utilized to analytically outline this equivalence.

[0115] Figures 13(a) and 13(b) show the Experimental VMF for a type A 16×16 mixed SQUID device, comprising the unit-cell depicted in Fig.6 and with two lines of bare loops. The VDMF response is measured at 4.2K. Although all the SQUID loops and all the bare loops have an identical area, a strong anti-peak response visible indicates the appearance of synthetic loops areas induced by the bare loops present in the circuit. These two devices in Fig.13 (a) and 13 (b) have an identical design and demonstrate an identical response for several different current biases even if these have been measured days apart. The device as in Fig.13 (c) has a similar design, although not identical, as it has no bare loops. As expected, the VMF of the device of Fig.13 (c) behaves as Fig.7(a) as a relative magnetometer rather than an absolute one.

[0116] Figure 14(a) shows Experimental VMF for a type B 16 ×16 mixed SQUID device, measured at 4.5K, whereby shunting resistance in each junction is 9.6 Ohmas opposed to 7.2 Ohm in the device of Fig.13(a). That is, the device still comprises the unit-cell depicted in Fig.6 with two lines of bare loops incorporated between each line of DC SQUIDs. Same as before, a strong anti-peak response is visible indicating the appearance of synthetic loops areas induced by the bare loops present in the circuit. The device in Fig.14(b) has a similar design to Fig.14 (a) but with no bare loops. As expected, the VMF of the device in Fig.14(b), measured at 5.1K, is not anti- peaked.

[0117] Where any or all of the terms "comprise", "comprises", "comprised" or "comprising" are used in this specification (including the claims) they are to be interpreted as specifying the presence of the stated features, integers, steps or components, but not precluding the presence of one or more other features, integers, steps or components.

[0118] Finally, while the invention has been described in conjunction with a limited number of embodiments, it will be appreciated by those skilled in the art that many alternative modifications and variations in light of the foregoing description are possible. Accordingly, the present invention is intended to embrace all such alternative, modifications and variations as may fall within the spirit and scope of the invention as disclosed, such as 2D arrays containing both rows of DC SQUIDs, BI- SQUID and bare loops all having the same area.

Claims

The claims defining the invention are as follows 1. A quantum magnetic field detecting device, comprising: a two-dimensional array of superconducting loops arranged in a plane, the array comprising: a first subset of the superconducting loops comprises Superconducting Quantum Interference Device (SQUID) loops comprising two Josephson junctions each arranged in opposed first and second arms of each of the SQUID loops; and a second subset of the superconducting loops comprises bare loops without Josephson junctions, wherein the two-dimensional array is configured to function as an active transformer-free Magnetically Small Antenna (MSA).

2. A quantum magnetic field detecting device of claim 1, wherein the two- dimensional array is high-efficiency Superconducting MSA.

3. A quantum magnetic field detecting device of claim 2, wherein a second arm of a first SQUID loop forms a first arm of a second SQUID loop.

4. A quantum magnetic field detecting device of claim 3, wherein the SQUID loops are located in a row of the two-dimensional array.

5. A quantum magnetic field detecting device of any one of claims 1 to 4, wherein the SQUID loops are located in a column of the two-dimensional array.

6. A quantum magnetic field detecting device of claim 1 or 5, wherein the area of each of the superconducting loops is the same.

7. A quantum magnetic field detecting device of claim 1 or 2, wherein one or more of the SQUID loops comprises a Bi-SQUID loop comprising a third arm electrically connecting the first arm and the second arm, and the third arm comprising a Josephson junction.

8. A quantum magnetic field detecting device of claim 7, wherein the Bi-SQUID loop comprises fourth and fifth arms electrically connecting the first arm and the second arms, respectively.

9. A quantum magnetic field detecting device of claim 8, wherein the total area of the Bi-SQUID loop, defined by the first, second, fourth and fifth arms, is twice the area of one of the bare loops, and the area of each superconducting loops is the same.

10. A quantum magnetic field detecting device of claim 7 or 8, wherein a second arm of a first Bi-SQUID loop forms a first arm of a second Bi-SQUID loop.

11. A quantum magnetic field detecting device of claim 10, wherein the Bi-SQUID loops are located in a row of the two-dimensional array.

12. A quantum magnetic field detecting device of any one of claims 7 to 11, wherein a fifth arm of a first Bi-SQUID loop forms a fourth arm of a second Bi-SQUID loop.

13. A quantum magnetic field detecting device of claim 12, wherein the Bi-SQUID loops are located in a column of the two-dimensional array.

14. A quantum magnetic field detecting device of and one of claims 1 to 11, wherein the device generates a synthetic area spread of the superconducting loops, the synthetic area spread being an effective magnetic sensing area greater than a geometric area of the two-dimensional array, achieved by inclusion of the bare superconducting loops in the two-dimensional array.

15. A quantum magnetic field detecting device of claim 14, wherein placement of the bare superconducting loops in the two-dimensional array is selected to optimise the synthetic area spread for a target application.

16. A quantum magnetic field detecting device of any one of claims 1 to 15, wherein the two-dimensional array is a N x M array, where values of N and M are based on applications of the device.

17. An absolute magnetometer comprising the quantum magnetic field detecting device of any one of claims 1 to 16.

18. An absolute magnetometer of claim 17, wherein voltage response of the absolute magnetometer induced by a magnetic flux external to the absolute magnetometer dips when the magnetic flux is zero and is relatively uniform at other magnetic flux values.

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