The symmetry breaking idea for quantum devices: The pump and the phase battery

1 Introduction

It is ubiquitous that symmetry breaking in physics holds great potential in the definition of quantum devices and their performances. In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system’s fate, by determining which branch of a potential bifurcation is taken. Symmetry is the essential basis of nature, which gives rise to conservation laws. In comparison, the breaking of the symmetry is also indispensable for many phase transitions, like ferromagnetism or superconductivity.

Among various symmetry breaking phenomena, spontaneous symmetry breaking (fig. 1) lies at the heart of many fascinating and fundamental properties of nature. Spontaneous symmetry breaking describes a phenomenon where physical states violate the underlying invariance of the system, which is being concerned in diverse fields, such as Higgs physics, Bose-Einstein condensates, and superconducting states. Meanwhile, spontaneous symmetry breaking in an optical system also holds great potential for the study of fundamental physics and high-performance quantum devices, which, however, is seldom realized.

Here we invoke the symmetry breaking idea to show its consequences on daily life in energetic and technology applications. We are going to discuss two quantum devices where the symmetry breaking concept is relevant: the quantum pump and the quantum phase battery. We will show that the latter is a quantum device that works on the basis of time-reversal and inversion symmetry breaking, while the former works in the absence of mirror symmetry.

2 The Thouless quantum pump

The quantum pump is a device able to generate a particle current by the slow and periodic modulation of two system parameters, in the absence of any external bias. For this reason it is considered as one of the most intriguing effects in quantum mechanics. In a conductor a direct current (dc) is usually associated to a dissipative flow of the electrons in response to an applied bias voltage (fig. 2). In systems of mesoscopic scale a dc current can be generated even at zero bias (e.g. in semiconductor nanostructures of nm size and tens of atoms) in the presence of slow periodic perturbations. In the adiabatic limit, when the applied perturbations are slow in comparison to the escape rate to external contacts, the electronic state of the quantum system is the same after a period, but a net charge has been transferred thanks to a squeezing of the central region wave function. Since the quantum state of the system remains coherent this effect is known as quantum charge pumping.

The quantum pump entails the transport of charge, in the absence of a net external electric or magnetic field, through an adiabatic cyclic evolution of the underlying Hamiltonian, as originally proposed by Thouless. In contrast to classical transport, the transported charge is quantized and purely determined by the topology of the pump cycle, making it robust against perturbations, such as interaction effects or disorder. In fact, Thouless pumping is nothing but the quantum version of the famous Archimede’s screw (fig. 3) where a directional motion of water is generated by slow and periodic movement of the handle.

Consider spinless electrons in a one dimension and subject to a periodic potential $U ( x ) $ with periodicity $a$: $U ( x+a ) = U ( x ) $. If the number of electrons per period $a$ is an integer $N$, or equivalently $na$ is an integer, the lowest $N$ bands of the energy spectrum are full, while the higher bands are empty. To transport charge, let us now slide the potential with some small velocity $v$, $U ( x, t ) = U ( x–vt ) $. Since the potential $U ( x ) $ is translationally invariant, at each point $x$ the potential $ U ( x, t )$ varies periodically. If the electrons can tunnel through the sliding potential, then a current $I = nev$ is induced and over a period $T=a/v$ a quantized charge $Q= IT = Ne$ is transferred through the cross section of the channel. Despite at each position $x$ the potential varies periodically in time, after each cycle something changes in the system: every potential minimum gets shifted by the period a. There would be no charge transfer in the presence of a standing wave alone, a particle transport would be possible only in the presence of traveling waves. However, the traveling wave $U ( x–vt ) $ can be presented as a superposition of two standing waves in the following form: $U ( x–vt ) = U_1 ( t ) \mathrm{sin} ( 2\pi x/a ) + U_2 ( t ) \mathrm{cos} ( 2\pi x/a ) $, with $U_{1,2} ( t ) = U_0 \mathrm{cos} (2 \pi t/T + \phi_{1,2} )$ and $\phi_{1,2} = 0, \pi/2$ ‫in the simplest case of $U ( x ) = U_0 \mathrm{sin} ( 2\pi x/a ) $. Thus, one can conclude that, although each standing wave alone is unable to pump the charge, two of them together can do it. This also implies that at least two parameters are needed to have a particle transport. Let us also note that the sign of the current depends on the phase and the mirror symmetry of the system is broken by a time-dependent phase of the potential.

The time evolution of the potential can be described in terms of a closed “trajectory” $C$ of the system in the parameters space $U_1$, $U_2$, usually a circle (see fig. 4). For a nonzero phase difference $\phi = \phi_1 - \phi_2$ between $ U_1$ and $U_2$, the trajectory encircles a nonzero area in the parameter. This is an important condition for the pumping to occur. This condition has been clearly demonstrated by Switkes et al. in their experiment on a quantum dot. The meaning of this condition is clear if one considers a quantum particle subject to a weak magnetic field. After each winding along a closed trajectory, it acquires a phase equal to the magnetic flux through the trajectory in units of $h/e$ (the Aharonov-Bohm effect). This phase can be measured directly and causes, for example, the anomalous magnetoresistance. It represents a macroscopic quantum phenomenon reminiscent of the quantum Hall effect.

The Aharonov-Bohm phase is nothing but an example of a more general phenomenon, the so-called Berry phase. Consider the Hamiltonian of an isolated quantum system and suppose it depends on the parameters $V_i ( t )$, which evolve adiabatically with time. After a period $T$ let them return to their original values encircling a closed path in the parameter space $C$. Simultaneously, the wave function of the system may acquire an additional phase (Berry phase), which depends on $C$. Similar to the Aharonov-Bohm phase, the Berry phase can be thought of as a flux of an effective “magnetic field”, known as Berry curvature in the mathematical literature, through the contour $C$ (as detailed in Box 1).

Of course, both the contour and the “magnetic field” do not exist in real space rather in the parameter space. The requirement that the wave function is single valued requires that the Berry curvature integrated over a closed manifold is quantized in the unit of $2\pi$. Also, for a system with Bloch bands, the pumped charge can be interpreted as a “magnetic flux”. It can be expressed as an integral of the effective “magnetic field” over the area inside $C$, which gives the Chern number $\nu$ defined over a (1+1)-dimensional periodic Brillouin zone formed by quasi-momentum $k$ and time $t$. This quantized value is given by a topological invariant, the sum of the Chern numbers of the occupied energy bands. Interestingly, the quantization of pumped charge shares the same topological origin as the Integer Quantum Hall Effect (IQHE), i.e., it can be considered as the dynamical version of IQHE. However, the quantization of the pumped charge is valid as long as the potential is varied adiabatically. Studies away from the adiabatic limit show that, despite its topological nature, this phenomenon is not generically robust to non-adiabatic effects. Indeed the mean value of the pumped charge shows a deviation from the topologically quantized limit which is quadratic in the driving frequency for a sudden switch-on of the driving.

Only very recently two groups, see Lohse et al. and Nakajima et al., reported the realization of a Thouless pump with ultracold bosonic and fermionic atoms in an optical superlattice. A superlattice is formed by superimposing a long and a short lattice with periodicities $d_L$ and $d_S = \alpha d_L$, $\alpha < 1$, whose relative position is determined by the phase $\phi$, as shown in fig. 5. A cyclic pumping scheme is realized when the phase $\phi$ is adiabatically varied in time by changing it by $2\pi$, i.e. moving the long lattice by $d_L$.

Completing a pumping cycle the lattice potential returns to its initial configuration and the band gap never closes during the whole pumping procedure so, ideally, the atoms stay in the lowest band during the adiabatic pumping process. However, the phase sweep breaks time-reversal symmetry of the Hamiltonian and the energy bands can acquire a non-zero Chern number $\nu$. Such a case can be realized with both fermions, by placing the Fermi energy in a band gap, and bosons, by preparing a Mott insulator in the n-th band.

The shift of the center of mass (CoM) of the atomic cloud position during a pump cycle in a topologically non-trivial band is given by $\nu d_L$. Since $\nu$ can take only integer values, the CoM position is quantized in units of $d_L$. As the displacement is proportional to a topological invariant, it neither depends on the pumping speed, as long as adiabaticity holds, nor on the specific lattice parameters, as long as band crossings do not occur. This ensures that the quantization of transport is highly robust.

In the experiment of Lohse et al. and Nakajima et al. topological pumping is observed through the shift of the CoM of the atomic cloud measured with in situ imaging. They can measure the Chern number of the pumping procedure from the average shift of the CoM per pumping cycle, which is consistent with the ideal value $\nu = 1$. What is surprising is that, while classically it is quite intuitive that a sliding lattice is able to transfer atoms because the potential minima are moving in space, in the quantum pump even though the potential minima are not moving in space as shown in fig. 1, the pumping is achieved by a sequence of quantum tunneling events between the double wells. This is an amazing result coming from topologically equivalent Hamiltonians that share the same Chern number of the occupied band.

Topological pumping also holds when the lattice explicitly depends on the spin and the Hamiltonian can be written in terms of an effective 1D Heisenberg (spin-chain) Hamiltonian with a spin-dependent tilt $\Delta$ and exchange coupling dimerization $\delta J_{ex}$. As demonstrated in the experiment of Schweizer et al. the pump can be realized by a spin-dependent modulation so that the time-reversal symmetry is retained and the Berry curvature of the two spin subbands is reversed. In the tight-binding limit this system is described by the Rice Mele model which comprises staggered on-site energies and alternating tunnel couplings with dimerization parameter $\delta J$. Pumping can be induced by an adiabatic modulation of the potential which corresponds to a closed curve in the parameter space around the degeneracy point.

Spin pump cycle is shown in fig. 6 (green) in the parameter space of the spin-dependent tilt $\Delta$ and exchange coupling dimerization $\Delta J_{ex}$. The path can be parametrized by the angle $\phi$, the pump parameter. As we see, between 0 and $\pi$ the spins exchange their position in the double well. The insets in the quadrants show the local mapping of globally tilted double wells to the corresponding local spin-dependent superlattice tilts with the black rectangles indicating the decoupled double wells. Between $\phi$ equal to 0 and $\pi, | \uparrow\rangle$ and $|\downarrow \rangle$ spins exchange their position, which can be observed by site-resolved band mapping images detecting the spin occupation on the left (L) and right (R) sites, respectively. Two spins exchange their positions via the delocalized triplet state. After a full cycle, the two spin components have each moved by $2d_S$ in opposite directions. The total particle current can be expressed in terms of the $Z_2$ Chern number which is defined as: $C_{SC} = \nu\uparrow - \nu\downarrow$.

A further generalization of the Thouless pump paradigm is represented by the peristaltic pump. It is a quantum device that generates a particle flux as the effect of a sliding finite-size microlattice. The peristaltic pump idea has been inspired by recent experiments in which two superfluids of Li⁶ atoms (reservoirs) are contacted through a one-dimensional quantum wire. The wire is created by intersecting the dark planes of two orthogonal repulsive laser beams. The vertically and horizontally laser beams are Gaussian envelopes with frequencies of kHz forming the two reservoirs that act as source and drain. The mesoscopic lattice between the two reservoirs is realized by projecting thin optical barriers on top of the ballistic conductor.

A one-dimensional tight-binding Hamiltonian model of this quantum machine (fig. 7) has been formulated and analyzed within a lattice Green’s function formalism. The pump observables, as e.g. the pumped particles per cycle, have been studied as a function of the pumping frequency, the number of minima of the lattice potential and temperature. The results show that the number of atoms pumped to the right is not necessarily quantized but depends on the parameters of the potential and the density of quantum states localized within the potential.

The realization of a topological pump will have a far-reaching impact on modern condensed-matter physics. These experiments introduce a new experimental platform to study topological quantum phenomena in many-body systems. Furthermore, introducing interaction effects, which is feasible with ultracold atoms, will open a door for experimental exploration of the interplay of topological and correlation effects. It will also have important applications, such as topological quantum computing.

3 The phase battery

A “conventional” chemical battery converts chemical energy into a persistent voltage, which can be used to supply electronic circuits. Instead, a phase battery can be viewed as a quantum device that allows to “fuel” the wave function of a quantum circuit with a persistent phase bias, making it a key element for boosting the development of quantum technologies based on phase coherence.

Recently, Strambini et al. demonstrated that a phase battery can be realized by a hybrid superconducting circuit consisting of an n-doped InAs nanowire, with unpaired-spin surface states, proximized by two Al superconducting electrodes, see fig. 8a. The ferromagnetic polarization of these unpaired-spin states is converted into a persistent phase $\varphi_0$ across the wire, and this eventually makes possible the emergence of the so-called anomalous Josephson effect. The external in-plane magnetic field permits the tuning of $\varphi_0$, thus allowing the quantum phase battery to be magnetically charged/discharged.

To understand the behavior of a phase-coherent superconducting circuit, we should recall the well-known Josephson effect, that is the quantum phenomenon accounting for the flowing of an electric current with no dissipation through two weakly coupled superconductors, forming the so-called Josephson junction (see Box 2). The phase difference $\varphi$ between the superconductors forming this device is intimately intertwined with its electrical response: the current-phase relation (CPR) is an odd function of $\varphi$ when both time-reversal and inversion symmetries are broken. In this case $I_{J} ( \varphi ) = I_{c} \mathrm{sin} ( \varphi )$, with $I_c$ being the Josephson critical current, i.e., the maximum value of supercurrent that the device can support. In this case, the Josephson free energy is minimal when the phase difference across the JJ is zero or $\pi$ (in the latter case we refer to a $\pi$-junction). In such junctions, the Josephson supercurrent vanishes when the phase difference between two superconductors is zero.

The implementation of a phase battery is prevented by these symmetry constraints, which ensure the “rigidity” of the superconducting phase. In other words, as long as one of these symmetries is preserved, an open Josephson junction, i.e., through which $I_J = 0$, cannot give a phase bias and a JJ enclosed in a superconducting loop (i.e. so that $\varphi = 0$) cannot generate current.

Breaking only the time-reversal symmetry maintains phase rigidity but allows two possible phase shifts, 0 or $\pi$ the CPR: for instance, in JJs with ferromagnetic coupling time-reversal symmetry is broken, but there is still no evidence of anomalous Josephson current. This suggests that some other symmetry “involved” in the system precludes the emergence of an anomalous Josephson current at $\varphi = 0$. In fact, both time and inversion symmetries should be broken to generate a finite phase shift $0 < \varphi_0 < \pi$, in which case the CPR finally becomes $I_J = I_c \mathrm{sin} ( \varphi + \varphi_0 )$. The peculiar junction with this CPR has a ground state corresponding to a finite phase shift, $\varphi_0$, so that a non-vanishing phase drop can appear even in the absence of a flowing current, or conversely, a supercurrent, i.e. the so-called anomalous current, can flow even at a zero phase. Recently, anomalous Josephson currents have been investigated in both theoretical and experimental works, thereby paving the way for concrete applications in superconducting electronics and spintronics.

The constituent elements of the Josephson phase battery can be fully controlled via magnetic field. This is obtained by realizing a superconducting quantum interference device (SQUID), see fig. 8b-c, i.e., an efficient and versatile tool for measuring phase-coherent effects, incorporating the Josephson junctions into a phase-sensitive superconducting loop geometry. At the heart of the operating principle of a SQUID is the interference of superconducting wave functions in the two arms of the device, similar to the two-slit interference in optics, due to the out-of-plane external magnetic flux through the superconducting ring. This leads to a modulation of the critical current of the device with a period of one. In the case of a SQUID embedding anomalous junctions, also the in-plane magnetic field can play a role, see fig. 8c, for it can significantly influence the properties of the weak link. Only recently the anomalous phase shift has been experimentally observed in hybrid SQUID configurations made with the topological insulator Bi₂Se₃ and Al/InAs heterostructures and nanowires, thus paving the way to the first proof-of-concept of a Josephson phase battery by Strambini et al. with a hybrid InAs-nanowire-SQUID interferometer.

The realization of anomalous $\varphi_0$ junctions is conveniently achieved through hybrid junctions with strong spin-orbit coupling, with a lateral arrangement that breaks the inversion symmetry giving a natural polar axis orthogonal to the flowing direction of the current. Besides, the electron spin polarization due to Zeeman field of exchange interaction with magnetic impurities is responsible for the breaking of the time-reversal symmetry. This is the reason for using nanowires for the anomalous junctions, see fig. 8d, having oxides or surface defects with unpaired spin acting as ferromagnetic impurities that can be magnetically polarized to supply a persistent exchange interaction. This provides the anomalous shift $\varphi_0$, which depends on the Lifshitz-type invariant in the free energy, $F_{L} \sim f ( \alpha , h ) ( \mathbf{n}_{h} \times \hat{\mathbf{z}} ) \cdot \mathbf{v}_{s} $ with $f ( \alpha , h )$ being an odd function of both the Rashba coupling strength $\alpha$ and the exchange or Zeeman field $h$, $\mathbf{n}_h$ a unit vector in the $h$ direction, and $\mathbf{v}_s$ the superfluid speed. The vector symmetries of $\varphi_0$ are ruled by the triple scalar product in $F_L$, while its magnitude is a function of specific microscopic aspects and macroscopic variables, e.g., the temperature.

The geometry of the Josephson phase battery is designed to maximize the symmetry of the two JJs in order to sum the two anomalous $\varphi_0$ shifts, when a uniform in-plane magnetic field is applied.

When the magnetic field is directed only out of the superconducting ring plane, the magnetic impurities are not polarized and there is no anomalous phase. In this case, the SQUID interference pattern shows a magnetic field modulation simply given by $I_{S} ( \phi ) = 2I_{c} | \mathrm{cos} ( \pi\phi / \phi_{0} ) |$ see fig. 8e. An in-plane magnetic field orthogonally applied to the nanowire axis causes the SQUID interference patterns to shift, see fig. 11a-b, due to the $\varphi_0$ influence. This is a clear hallmark of the magnetically controllable anomalous Josephson effect.

Interestingly, the evolution of the anomalous phase shift extracted from the SQUID interference patterns reveals a remarkable hysteretic behavior, see fig. 11c. This is due to the ferromagnetic coupling between the intrinsic magnetic impurities in the nanowire and cannot be merely ascribed to trapped flux in the superconductor. The observed anomalous phase shift is indeed the result of two competitive mechanisms, i.e., the effective exchange field created by the unpaired surface spins and the Zeeman field generated by the in-plane magnetic field, each of which gives a distinct “intrinsic” or “extrinsic” contribution to the anomalous phase. A nifty strategy for accessing only the “intrinsic” phase shift relies upon a two-step measurement procedure in which the magnetic field is first turned on and then off to allow a measurement that does not depend on the specific value of the magnetic field, but only on its history. Thus, the observed behavior that mimics the initial magnetization curve of a ferromagnet, see fig. 11d-e, definitively confirms the ferromagnetic nature of the ensemble of impurities giving the anomalous intrinsic contribution. The remaining extrinsic anomalous phase shows no hysteresis, see fig. 12, and reveals an odd parity with respect to the in-plane magnetic field. The latter aspect reflects the odd parity of the free energy with respect to the exchange field. Moreover, the main features of this extrinsic anomalous contribution are definitively well captured by modeling the $\varphi_0$-junction setup as a lateral junction treated within a quasi-classical approach, see the inset of fig. 12.

In the end, the concrete implementation of a quantum phase battery is finally demonstrated. We have a quantum element that provides a controllable and localized phase polarization, which can therefore find room in diverse quantum circuits. The magnetic tuning of the superconducting phase opens new possibilities for advanced topological superconducting electronics schemes based on the anomalous Josephson effect.

4 The impact of quantum machines on daily and social life

Starting from the time-reversal and inversion symmetry breaking concept, we have discussed two classes of quantum machines, the quantum pump and the quantum phase battery, that will hold great potential in daily and social life. In fact, the quantum pump may provide novel ways of reducing the dissipation of energy as wasteful heat and define a better current standard in metrology. It can also find applications in quantum computing, as it can be used to generate pairs of spin-entangled electrons. On the other side, we have identified the basic principles of a quantum phase battery. It consists of an $n$-doped InAs nanowire forming the core of the battery (the pile) and Al superconducting leads as poles. The battery is charged by applying an external magnetic field, which then can be switched off. The future of this battery is further being improved and will contribute enormously to quantum technologies that are expected to revolutionize both computing and sensing techniques, as well as medicine, and telecommunications in the near future.

Acknowledgments

We thank the Superconducting Quantum Electronics Lab (SQEL) group headed by Francesco Giazotto at the NEST laboratory in Pisa (Italy), Lucia Sorba, Sebastian Bergeret, and Francesco Romeo.