Abstract:
A topological qubit wire hosts Majorana zero-energy modes and includes a superconductor, which may be an s-wave superconductor, and a quasi-1D nanowire, which may be a semi-conductor. The Majorana zero-energy modes are localized at ends of the quasi-1D nanowire, which may be sized and shaped to provide occupancy of a few transverse modes in a first direction and occupancy of a few transverse modes in a second direction. In some instances, the occupancy in the first direction may be greater than or equal to 3, and the occupancy in the second direction may be 1.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims benefit, under 35 U.S.C. §119(e), of U.S. Provisional Patent Application No. 61/644,010, filed May 8, 2012, which is incorporated herein, by reference, in its entirety. 
     
    
     BACKGROUND 
       [0002]    Majorana fermions, particles which are their own antiparticles, were originally envisioned by E. Majorana in 1937 in the context of particle physics (i.e., the physics of neutrinos). However, the current search for Majorana modes (Majoranas) is mostly taking place in condensed matter systems where Majorana quasi-particles appear in electronic systems as a result of fractionalization, and as emergent modes occupying non-local zero energy states. The non-locality of these modes provides the ability to exchange and manipulate fractionalized quasiparticles and leads to non-Abelian braiding statistics. Hence, in addition to being of paramount importance for fundamental physics, this property of the Majoranas places them at the heart of topological quantum computing schemes. 
         [0003]    Majorana zero-energy modes/quasiparticles can appear quite naturally in 2D chiral p-wave superconductors where these quasiparticles, localized at the vortex cores, correspond to an equal superposition of a particle and a hole. A very simple model for Majorana zero-energy modes/quasiparticles is a one-dimensional (1D) Majorana quantum wire with localized Majorana zero-energy modes/quasiparticles at the ends. Both of the above cases involve spinless p-wave superconductors where the existence of Majorana zero-energy modes can be explicitly demonstrated by solving the corresponding mean field Bardeen-Cooper-Schrieffer (BCS) Hamiltonian. Recently, a way to engineer spinless p-wave superconductors has been suggested using a combination of strong spin-orbit coupling and superconducting proximity effect, thus opening the possibility of realizing Majorana quasiparticles in solid-state systems. 
       SUMMARY 
       [0004]    This disclosure describes a new topological qubit-wire based on semiconductor/superconductor heterostructure. Majorana zero-mode modes, localized at opposite ends of the topological qubit-wire constitute a non-local quantum two level system (quantum bit or qubit), and can be used to encode information. The topological qubit wire includes a superconductor, which may be an s-wave superconductor, and a quasi-1D nanowire, which may be a semi-conductor. 
         [0005]    The quasi-1D nanowire may be sized and shaped to provide occupancy of one or more transverse modes in a first direction and occupancy of one or more transverse modes in a second direction. If the number of modes (or conduction channels) is one in both transverse directions, a nanowire is regarded as strictly one-dimensional. However if the number of channels is larger than 1 but smaller than 10, a nanowire may be regarded as quasi-one-dimensional (quasi-1D). Physical properties of 1D and quasi-1D nanowires may differ substantially. For the purposes of this disclosure, the quasi 1D nanowire discussed herein provides between 2 through 10 transverse modes (or channels). In some instances, the occupancy in the first direction may be greater than or equal to 3, and the occupancy in the second direction may be 1. 
         [0006]    The topological qubit-wire may include an insulative layer between the superconductor and the quasi-1D nanowire. A quantum tunneling transmission coefficient for electrons tunneling between the superconductor and the quasi-1D nanowire is controlled by a thickness of the insulative layer. The quantum tunneling transmission coefficient may be non-uniform. 
         [0007]    In some embodiments, the quantum tunneling transmission coefficient may vary in one direction while remaining constant, or approximately constant, in a second direction. 
         [0008]    This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    The detailed description is described with reference to the accompanying figures. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. The same reference numbers in different figures indicate similar or identical items. 
           [0010]      FIG. 1  is a schematic diagram of an example topological qubit array. 
           [0011]      FIG. 2  is an isometric schematic view of an example topological qubit-wire that may be employed in the topological qubit array. 
           [0012]      FIG. 3  is an isometric schematic view of another example topological qubit-wire that may be employed in the topological qubit array. 
           [0013]      FIG. 4  is schematical cross sectional view of the example topological qubit-wire of  FIG. 2 . 
           [0014]      FIG. 5  is a schematical cross sectional view of the example topological qubit-wire of  FIG. 2 , in accordance with one non-limiting embodiment. 
           [0015]      FIG. 6  is a schematical cross sectional view of the example topological qubit-wire of  FIG. 2 , in accordance with one non-limiting embodiment. 
       
    
    
     DETAILED DESCRIPTION 
     Overview 
       [0016]    This disclosure describes a topological qubit-wire that provides topological qubits. The topological qubits involve non-local Majorana modes, and the structure may include a quasi-1D semiconductor nanowire and an s-wave superconductor. The quasi-1D semiconductor nanowire has a strong spin-orbit interaction and is in close contact with the s-wave superconductor. In the superconductor-semiconductor heterostructure, the Majorana mode may exist as a zero-energy state. 
         [0017]    For additional details of Majorana quasi-particles and semiconductor nanowires see: Lutchyn, Stanescu, Das Sarma, “Majorana fermions in multiband semiconducting nanowires,” Physical Review Letters 106, 127001 (2011), and Stanescu, Lutchyn, Das Sarma, “Majorana Fermions in Semiconductor Nanowires,” Physical Review B 84, 144522 (2011). 
       Illustrative Topological Qubit Array 
       [0018]      FIG. 1  is a schematic diagram of an example topological qubit array  100 . The topological qubit array  100  includes an array of topological qubit-wires  102 . 
         [0019]    A topological qubit-wire  102  includes one or more topological phase segments  104 , which are also referred to as Majorana segments, and non-topological phase segments  106 . For the purposes of this disclosure a topological phase segment is defined as being in a topological superconducting phase with localized, unpaired zero-energy or low-energy Majorana modes localized at opposite ends of the segment. The non-topological phase segments  106  do not have localized Majorana modes but can be induced to change into Majorana segments  104  and vice-versa. For example, an electric potential may be applied to a portion of a non-topological phase segment  106  to change the chemical potential in the portion of the non-topological phase segment  106 , and the change in the chemical potential may then cause the portion to enter the topological superconducting phase of a Majorana segment. 
         [0020]    Each Majorana segment  104  has length of Majorana wire  108  with unpaired non-abelian anyons  110  localized at opposite ends of the Majorana wire  108 . The non-abelian anyons  110  are Majorana quasi-particles. 
         [0021]    A topological qubit  112  may be comprised of two or more non-abelian anyons  110 . Calculations may be performed by placing the topological qubits  112  in an initial state, evolving the topological qubits  112  such as by braiding world-lines of two or more of the non-abelian anyons  110 , and determining a final state of the topological qubits  112 . 
       Illustrative Examples of Topological Qubit-Wire 
       [0022]      FIG. 2  shows an isometric schematic view of a topological qubit-wire  200  that may be implemented in the topological qubit array  100 . 
         [0023]    The topological qubit-wire  200  includes a superconductor  202  and a multi-band nanowire  204 . The superconductor  202  may be an s-wave superconductor such as aluminum (Al) or nobium (Nb). 
         [0024]    The multi-band nanowire  204  is a quasi-1D semiconductor wire having a length (L x ), a width (L y ) and a thickness (L z ), where the width is larger than the thickness and the length is larger than the width. For example, the width may be in the range of 50-200 nanometers (nm), the thickness may be in the range of 1-10 nm, and the length may be in the order of microns (μm). In one embodiment, the length may be between 5-10 μm, the width may be approximately 130 nm, and the thickness may be approximately 5 nm. 
         [0025]    The multi-band nanowire  204  is a quasi-1D semiconductor wire because the multi-band nanowire  204  is strongly confined in the z-direction, by its thickness (i.e., the thinness of its thickness), so that only the lowest sub-band is occupied, while the weaker confinement in the y-direction, by its width, provides a few occupied sub-bands. 
         [0026]    The multi-band nanowire  204  may be a semiconductor such as indium antimonide (InSb) or Indium arsenide (InAs) and may be epitaxially grown. The multi-band nanowire  204  may have a large spin-orbit interaction strength α and may have a large Lande g-factor (e.g., g InAs :10-25 and g InSb :20-70). Furthermore, the multi-band nanowire  204  may be of a material (e.g., InSb or InAs) that forms interfaces that are highly transparent for electrons, thereby allowing one to induce a large superconducting gap A via the proximity effect. 
         [0027]    The superconductor-semiconductor heterostructure of the topological qubit-wire  200  under proper conditions hosts Majorana modes at zero (or close to zero) energies. The heterostructure consists of the multi-band nanowire  204  having a strong Rashba coupling and the superconductor  202 . The spin-orbit coupling can be characterized a vector (e.g. the spin-orbit vector may point along y-axis). An in-plane magnetic field applied perpendicular to the spin-orbit vector (e.g. {right arrow over (B)}=B 0 {circumflex over (x)},) may drive the system into a topological phase. 
         [0028]    An insulative layer  206 , such as, but not limited to, an aluminum oxide layer, may interpose the superconductor  202  and the multi-band nanowire  204 . Quantum tunneling of electrons, between the superconductor  202  and the multi-band nanowire  204 , occurs through the insulative layer  206 , and features of the insulative layer  206  may be used to regulate the quantum tunneling of the electrons. For example, the thickness, as measured in the z direction, of the insulative layer  206  may be used to control the amount of quantum tunneling of the electrons. 
         [0029]    In some embodiments, the multi-band topological qubit-wire  200  may include a top array of gates  208  positioned proximal to a surface of the multi-band nanowire  204 . A second insulative layer  210  may interpose the top array of gates  208  and the multi-band nanowire  204 . The array of gates  208  may selectively provide an electric field to the multi-band nanowire  204  to induce changes to the chemical potential of the multi-band nanowire  204 , and thereby change portions of the multi-band nanowire  204  between topological phase (i.e., having Majorana modes) and non-topological phase (i.e., having no Majorana modes). 
         [0030]    In the illustrated embodiment of  FIG. 2 , the insulative layer  210  and the top array of gates are distal from the superconductor  202 , disposed on an upper surface of the multi-band nanowire  204 . However,  FIG. 3  shows an isometric schematic view of a topological qubit-wire  300  that may be implemented in the topological qubit array  100 . The topological qubit-wire  300  is similar to the topological qubit-wire  200 . Features of the topological qubit-wire  300  that have a reference numeral in the 200&#39;s (i.e.,  202 - 208 ) are the same as previously described with respect to the topological qubit-wire  200 . In the interest of brevity, these features are not discussed again. 
         [0031]    The topological qubit-wire  300  also includes a side array of gates  302  and an insulative layer  304 . The side array of gates  302  and the insulative layer  304  are disposed on a side surface of the multi-band nanowire  204 , with the insulative layer  304  interposing the multi-band nanowire  204  and the array of gates  302 . The array of gates  302  may selectively provide an electric field to the multi-band nanowire  204  to induce changes to the chemical potential of the multi-band nanowire  204 , and thereby change portions of the multi-band nanowire  204  between topological phase (i.e., having Majorana modes) and non-topological phase (i.e., having no Majorana modes). 
         [0032]    In other embodiments (not shown), a topological qubit-wire  300  may include additional, or fewer, arrays of gates. For example, such topological qubit-wires may include pairs of side arrays of gates  302  that are disposed on opposite sides of the multi-band nanowire  204 . As another example, such topological qubit-wires may have one or more side arrays of gates  302 , but might not include the top array of gates  208 . 
         [0033]      FIG. 4  is a schematical cross sectional view of the example topological qubit-wire  200 . The insulative layer  206  has a thickness, T, which may be thin, e.g., in the order of a 5-30 angstroms. The thickness, T, and/or the composition of the insulative layer  206  may be utilized to moderate quantum tunneling between the superconductor  202  and the multi-band nanowire  204 . In some embodiments, the insulative layer  206  may have a thickness and/or composition so that the rate of quantum tunneling is not too strong, i.e. broadening of the energy levels in the nanowire due to tunneling into a metal (i.e. not a superconductor) is smaller than the superconducting gap. 
         [0034]    The quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  may be non-uniform. In some embodiments, the quantum tunneling transmission coefficient (T QT ) may vary in both the x-direction and the y-direction, i.e., T QT =T QT (x,y). In some embodiments, variations of the quantum tunneling transmission coefficient (T QT ) in the x-direction may be small in comparison to variations in the y-direction. 
         [0035]    In some embodiments, the quantum tunneling transmission coefficient (T QT ) may vary in the y-direction while remaining constant in the x-direction, i.e., T QT =T QT (x,y)=T QT (y). 
         [0036]    In some embodiments, transverse variations (i.e., variations in the y-direction) in the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  may have a characteristic length comparable to the width, L y , of the multi-band nanowire  204 . In some embodiments, the characteristic length in transverse variations in the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  may range from approximately one-quarter of the width (L y ) to being approximately comparable to the width (L y ). 
         [0037]    It should also be noted that while the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  may vary transversely (i.e., in the y-direction), the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  of the insulative layer  206  may be constant or approximately constant along the longitudinal length (i.e., in the x-direction). In some embodiments, the characteristic length of longitudinal variation of the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative layer  206  may be larger than the width, L y , of the multi-band nanowire  204 . 
         [0038]    In some embodiments, the thickness, T, of the insulative layer  206  may be non-uniform, as illustrated in  FIG. 5 , and in some embodiments, the insulative layer  206  may be non-homogenous, as illustrated in  FIG. 6 . Box  402 , which is shown in dashed line, represents a portion of the topological qubit-wire  200  illustrated in  FIGS. 5 and 6 . 
         [0039]    Referring to  FIG. 5 , at an approximate transverse midpoint  502  of the insulative layer  206 , the thickness of the insulative layer  206  is approximately at a maximum (T max ). The thickness of the insulative layer  206  decreases away from the approximate transverse midpoint  502 , and at ends  504  and  506 , the thickness is approximately at minimum thickness (T min ). The variation in thickness of the insulative layer  206  is one non-limiting example for providing a non-uniform quantum tunneling transmission coefficient in the insulative layer  206 . 
         [0040]    The variation of the thickness of the insulative layer  206  illustrated in  FIG. 5  is one non-limiting example of a variation in thickness. As another example, the thickness of the insulative layer  206  may be greatest at one end, e.g., end  504 , of the insulative layer  206 . It should be noted that variations in the thickness do not need to be monotonic in the y-direction. For example, the thickness of the insulative layer  206  may be approximately the same at both ends  504  and  506 , where it may be approximately the greatest. 
         [0041]    In some embodiments, transverse variations (i.e., variations in the y-direction) in the thickness may have a characteristic length comparable to the width, L y , of the multi-band nanowire  204 . In some embodiments, the characteristic length in transverse variations in the thickness may range from approximately one-quarter of the width (L y ) to being approximately comparable to the width (L y ). 
         [0042]    It should also be noted that while the thickness of the insulative layer  206  may vary transversely (i.e., in the y-direction), the thickness of the insulative layer  206  may be uniform or approximately uniform along the longitudinal length (i.e., in the x-direction). In some embodiments, the characteristic length of a longitudinal variation of thickness may be larger than the width, L y , of the multi-band nanowire  204 . 
         [0043]    Referring to  FIG. 6 , the insulative layer  206  may be non-homogenous and may be comprised of insulative materials  602  and  604  and may have a non-uniform quantum tunneling transmission coefficient. The insulative materials  602  and  604  may provide different quantum tunneling coupling between the superconductor  202  and the multi-band nanowire  204 . For example, the insulative material  602  may be more transparent to electrons than the insulative material  604  (i.e., the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative material  602  may be higher than the quantum tunneling transmission coefficient (T QT ) for electrons tunneling through the insulative material  604 ). 
         [0044]    Thickness, measured in the z-direction, of the insulative material  602  and the insulative material  604  may vary across (in the y-direction) the width of the multi-band nanowire  204  while the thickness of the insulative layer  206  may remain approximately uniform. For example, at an approximate transverse midpoint  606  of the insulative layer  206 , the thickness of the insulative material  602  is approximately at a maximum and the thickness of the insulative material  604  is approximately at a minimum. Away from the approximate transverse midpoint  606 , the thickness of the insulative material  602  decreases while the thickness of the insulative material  604  increases, and at ends  608  and  610 , the thickness of the insulative material  602  is approximately at a minimum and while the thickness of the insulative material  604  is approximately at a maximum. 
         [0045]    The variation of the thicknesses of the insulative materials  602  and  604  illustrated in  FIG. 6  is one non-limiting example for providing a non-uniform quantum tunneling transmission coefficient in the insulative layer  206 . 
         [0046]    In some embodiments, transverse variations (i.e., variations in the y-direction) in the thickness of the insulative material  602  and/or the insulative material  604  may have a characteristic length comparable to the width, L y , of the multi-band nanowire  204 . In some embodiments, the characteristic length in transverse variations in the thickness may range from approximately one-quarter of the width (L y ) to being approximately comparable to the width (L y ). 
         [0047]    It should also be noted that while the thicknesses of the insulative material  602  and/or the insulative material  604  may vary transversely (i.e., in the y-direction), the thickness of the insulative material  602  and/or the insulative material  604  may be approximately uniform along the longitudinal length (i.e., in the x-direction). In some embodiments, the characteristic length of a longitudinal variation of thickness may be larger than the width, L y , of the multi-band nanowire  204 . 
       CONCLUSION 
       [0048]    Although the techniques have been described in language specific to structural features and/or methodological acts, it is to be understood that the appended claims are not necessarily limited to the specific features or acts described. Rather, the specific features and acts are disclosed as exemplary forms of implementing such techniques.