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**Pedram Roushan**, from Google’s Quantum AI team in California, describes this elusive form of matter – and how it could be simulated on the company’s Sycamore quantum processor

With their enchanting beauty, crystalline solids have captivated us for centuries. Crystals, which range from snowflakes to diamonds, are made up of atoms or molecules that are regularly arranged in space. They have provided foundational insights that led to the development of the quantum theory of solids. Crystals have also helped develop a framework for understanding other spatially ordered phases, such as superconductors, liquid crystals and ferromagnets.

Periodic oscillations are another ubiquitous phenomenon. They appear at all scales, ranging from atomic oscillations to orbiting planets. For many years, we used them to mark the passage of time, and they even made us ponder the possibility of perpetual motion. What is common between these periodic patterns – either in space or time – is that they lead to systems with reduced symmetries. Without periodicity, any position in space, or any instance of time, is indistinguishable from any other. Periodicity breaks the translational symmetry of space or time.

In physics, space and time are often interwoven, so if a collection of many particles can show spatial periodicity, it is perhaps not so strange to wonder if the same symmetry pattern can spontaneously emerge in time. Spontaneous symmetry breaking is when the lowest-energy or ground state of a system does not respect a symmetry that, in principle, is not forbidden. The most common example of such a feat in nature is the very existence of crystals, where their continuous translational symmetry breaks and is replaced by a discrete periodic symmetry in space.

In physics, space and time are often interwoven, so if a collection of many particles can show spatial periodicity, it is perhaps not so strange to wonder if the same symmetry pattern can spontaneously emerge in time

## Periodicity in space versus time

Over the last decade, physicists have been wondering whether systems with ground states where time translational symmetry is broken can exist. It seems there is a major difference between breaking spatial versus temporal translational symmetry. The common examples of spatially ordered systems consist of many interacting particles, while those with stable periodic oscillations have only a few degrees of freedom (figure 1*a*). Indeed, no example of periodic oscillation with many particles readily comes to mind.

It makes us wonder if it is even possible to find a large system of interacting particles that has oscillations for an indefinitely long time. In our search we arrive at many systems that almost, but not quite, fit the bill. An example could be the synchronous collective oscillations observed in a large system of particles – such as phononic oscillations or mass spring systems. These oscillations in isolated many-body systems will not persist; or if they do, it would only be for highly tuned initial configurations, and this would not constitute a new phase of matter.

The stark contrast between large interacting systems with periodicity in space (common) and time (essentially non-existent) may seem unexpected. After all, Einstein’s special relativity unifies space and time into one seamless “space–time” object. However, Lorentz transformations, which relate the space and time co-ordinates of two systems moving relative to each other, do not mean that space and time are completely equivalent, as there is causality.

## Confronting the second law

All laws of physics are invariant with respect to forward or backward flow of time, or the choice of the origin of time for a given equation. The only exception is the second law of thermodynamics, which establishes the concept of entropy. The second law says that any isolated system of many particles spontaneously evolves towards its equilibrium configuration, in which one can no longer detect the passage of time by making local measurements. This homogeneity in time contrasts sharply with our desire to stabilize a temporal order, which implies heterogeneous time instances. The dichotomy poses a fundamental theoretical and experimental challenge that is the root of the elusiveness of time crystals.

Indeed, in an open system, where energy can be added to and dispelled from its surroundings, entropy can be expelled and one could, in principle, stabilize temporal ordering by balancing parameters – a challenging task, though one that can be achieved. To exclude such cases, we define time crystals as “isolated systems of many interacting particles that show oscillations indefinitely”. The existence of many particles and associated degrees of freedom in the system is the key part of this definition. With entropy not on our side, finding stable time crystals is a major pursuit.

Furthermore, these degrees of freedom should be energetically accessible. For example, oscillations observed in Josephson junctions in superconductors do not constitute a time crystal. Although such oscillations are dissipation-free and can go on forever, the entire condensate has a few degrees of freedom. Physically, all Cooper pairs – pairs of electrons bound together at extremely low temperatures – form a coherent condensate. In a sense, Cooper pairs are frozen, just as a coin can flip and all the atoms in it move together, and the energy scale required to make them move independently cannot be accessed with any reasonable definition.

## A decade old quest

In 2012 Nobel-prize-winning physicist Frank Wilczek first proposed a scheme for realizing a perpetual periodic oscillation (*Phys. Rev. Lett*. **109** 160401). He suggested threading a small magnetic field through a superconducting ring, which, in response, would form a current that can circulate indefinitely. However, this spontaneously developed supercurrent would be a perpetual motion and not a perpetual oscillation. A perpetual oscillation would break the translational symmetry of time, and makes time instances distinct from each other. A perpetual current does not break this translational symmetry.

Wilczek, who coined the term “quantum time crystals’’, proposed that introducing a weak attraction between circulating Cooper pairs can make them bunch up. The resulting uneven distribution of circulating particles along the ring could then provide a clear sense of oscillation (figure 1*b*). However, in 2014 physicists Haruki Watanabe and Masaki Oshikawa ruled out this conclusion by considering what being “at equilibrium” implies. Going beyond local observables that are trivially time-independent, they examined temporal correlations between spatially distinct points and found that correlations also cannot show oscillatory behaviour. The pair developed a “no-go theorem” that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions.

Watanabe and Oshikawa asked how to construct a physical ground state from the highly degenerate lowest-energy time-independent eigenstates of a system, and if its associated observables remain non-zero as the system size grows (*Phys. Rev. Lett.* **114** 251603). As a general property of many-body systems, Watanabe and Oshikawa show that being at thermal equilibrium is a strong constraint, and does not permit the emergence of stable, time-dependent responses.

## Localization for stabilization

It seems that the search for stable temporal ordering inevitably needs to go beyond equilibrium. Arguably, periodically driven systems are the simplest modification to those in equilibrium. These are systems where the periodic application of pulses keeps them away from equilibrium. At first sight, choosing driven systems seems counterintuitive. Common wisdom suggests that these systems continuously absorb heat from the drive and approach the maximum attainable entropy state, which defies any ordering whatsoever. However, recent research indicates that this fate can be avoided when strong disorder inhibits energy exchange between the energy levels and consequently prevents the disorder from spreading – such systems are “many-body localized” (MBL).

Thanks to entropy, most systems achieve thermal equilibrium, via an exchange of energy with their external environment, thereby erasing local memory of the initial conditions. But MBL systems, due to disorder, fail to reach thermal equilibrium, thereby retaining a memory of their initial states for infinite times. The entropy plateaus at smaller values, allowing for temporal ordering.

Therefore, in spite of the periodic drive, the net flow of energy becomes zero and entropy plateaus below the maximum value (figure 1*c*). Saturation below the maximum attainable entropy does not violate the second law, which states that the entropy of an isolated system cannot decrease in time. This law does not demand the entropy of an isolated system to reach the maximum possible value, although it is difficult to avoid. It only states that the rate of change of entropy cannot be negative, it can be positive or zero.

## Relying on MBL stability

When a stable phase is in equilibrium, it does not exchange net energy with its surroundings and no new entropy is generated. Many theoretical and experimental works suggest that the MBL systems also have these properties, while staying far from equilibrium. They are therefore the only viable contender that could host stable time crystalline phases. To date, all known classical systems cannot sustain oscillations indefinitely.

So how stable are MBL systems, and how can we be confident that they are not reaching maximum entropy and equilibrium? The stability of 2D- or 3D-MBL systems has in fact long been a matter of debate, while evidence for their stability is limited. In 2006 researchers in the US (*Ann. Phys*. **321** 1126) showed that 1D-MBL systems retain their localization for all orders of perturbation expansion. However, this cannot fully rule out the possibility of metastability, meaning that the system only appears localized, while actually thermalizing on long time scales. But in 2016 John Imbrie at the University of Virginia, Charlottesville, made significant progress towards a conclusive proof of an MBL phase, by essentially ruling out all nonperturbative effects for certain systems (*Phys. Rev. Lett. ***117** 027201). It is important to note that his proof is not fully general, but assumes “limited level attraction”. While this is a realistic assumption; it is balanced on a very fine edge.

Even if we were to observe evidence of temporal ordering, establishing it as a phase of matter has its own formal requirements

## Establishing a phase of matter

Even if we were to observe evidence of temporal ordering, establishing it as a phase of matter has its own formal requirements. This is because we need to distinguish it from transient behaviours or fine-tuned corner cases. To do so, we examine the rigidity of the temporal response with a checklist that takes into consideration four factors. (i) To establish a nonequilibrium phase, we are required to consider the infinite time limit as well. (ii) In statistical mechanics, phases are only well defined in the infinite system size limit. (iii) Proving the stability of any phase requires that its signatory pattern is stable to perturbation of its equations of motion, and seen over a range of parameters, and (iv) that it emerges for all initial configurations.

Using Google’s Sycamore quantum processor, we recently provided the first convincing experimental observation of a time-crystalline phase on a Noisy Intermediate-Scale Quantum (NISQ) drive by going through the above mentioned criteria (*Nature* **601** 531). (i) To investigate the infinite time response, and show that the observed oscillations are not transient and oscillate persistently, we devised a time-reversal protocol that discriminates external decoherence from intrinsic dynamics (figure 2*a*). This enables us to observe stable ordering for many cycles across a chain of 20 qubits. We found that all our qubits showed synchronous oscillations. (ii) To show that the oscillations we observed survived even if we had much longer chains, we performed what is known as the finite-size scaling, which also allowed us to locate the phase transition control parameter (figure 2*b*). (iii) We showed that the ordering survives over an extended range of parameter variation, and that the system is indeed localized.

Last, (iv) we make use of the concept of “quantum typicality”, to establish that indeed there are oscillations for all initial states. Typicality asserts that ensemble averages can be accurately approximated by an expectation value with respect to a single state, randomly drawn from the Hilbert space. Remarkably, typicality applies to thermalizing, integrable and many-body localized systems. Therefore, using typicality we could circumvent the exponential cost of sampling the entire spectrum and effectively verify the response for all initial states.

## Realizing a dynamical phase

Our effort to realize a time crystalline phase uses quantum processors in a unique way. That’s because computations are usually carried out on these processors with a set of logical operations called quantum gates. The computation has no relation to the underlying governing dynamics of the system – in technical terms, the desired Hamiltonian is not realized on the processor. In such computations, the processor is used in roughly the same way as in a classical computer.

However, this quest of establishing a dynamical phase is fundamentally different. Here, the question is whether a stable phase can emerge in a many-body driven isolated system. For such questions, our processor and our setting is indeed the natural platform to explore such questions. Our implementation is on the direct path toward realization of this dynamical phase. The challenge that makes our result fall short of a full realization is extrapolating the response to infinite time and system size, which are all ultimately rooted in the finite coherence time of the system. The core of our work and achievements is to devise experimental methods to provide a basis for making such extrapolations.

As our results show, the protocols developed here are general and establish a scalable approach to studying non-equilibrium phases of matter on NISQ processors.

Source: Physics World