Exploring quantum phases by driven dissipation

Dephasing and decay are the intrinsic dissipative processes prevalent in any open quantum system and the dominant mechanisms for the loss of coherence and entanglement. This inadvertent effect not only can be overcome but can even be capitalized on in a dissipative quantum simulation by means of tailored couplings between the quantum system and the environment. In this context it has been demonstrated that universal quantum computation can be performed using purely dissipative elements, and furthermore, the efficient preparation of highly entangled states is possible. In this article, we are interested in nonequilibrium phase transitions appearing in purely dissipative systems and the exploration of quantum phases in terms of a dissipative quantum simulation. To elucidate these concepts, we scrutinize exemplarily two paradigmatic models: the transverse-field Ising model and the considerably more complex ${\mathbb{Z}}_2$ lattice gauge theory. We show that the nonequilibrium phase diagrams parallel the quantum phase diagrams of the Hamiltonian “blueprint” theories.

Figure 1

We consider a $D\text{-dimensional}$ rectangular lattice with spins attached to the sites. The system is homogeneously coupled to two tailored Markovian baths with relative coupling strength $\kappa$. The $\left\{P_s\right\}$ ($\left\{F_s\right\}$) jump operators drive the system towards the paramagnetic (ferromagnetic) ground states of the transverse-field Ising model. There is $no$ unitary dynamics involved.

Figure 2

Results for the dissipative transverse-field Ising model. (a) Mean-field phase diagram. We show the magnetization $m_z$ (solid lines) for all steady states as a function of $\kappa$ (thick red and blue lines, stable ferromagnetic; thin green line, unstable paramagnetic; thick green line, stable paramagnetic]. The corresponding purities $|\widehat{\mathbf{m}}|$ are illustrated by dashed lines of the same color and thickness. (b) Dynamical mean-field Lindblad flow $\mathbf{F}\left(\mathbf{m}\right)$ in the $m_x-m_z$ plane of the Bloch ball. Stable (unstable) steady states are labeled by bullets (circles); their paths for $0\le \kappa \le \infty$ are highlighted. We illustrate the flow for three (1, 2, and 3) different ratios $\kappa$ above and below the critical ratio ${\kappa }_c=3$. (c) Relaxation of the Bloch vector $\mathbf{m}\left(t\right)=\delta \mathbf{m}\left(t\right)+\widehat{\mathbf{m}}$ close to the steady state below, at, and above the critical ratio. The relaxation in the $m_x$ and $m_z$ direction becomes algebraic at the phase transition. (d) Quantum jump trajectory of a $3\times 3$ instance with periodic boundary conditions in the ferromagnetic regime with $\sqrt{\kappa }=1/3$. We show the average magnetization $1/9\langle {\sum }_s{\sigma }_s^z\rangle$ [red (dark) line] and the correlation $\langle {\sigma }_1^z{\sigma }_2^z\rangle$ [gray (light) line] starting from a completely polarized state ${\left|\uparrow \right.〉}^{\otimes 9}$. Ferromagnetic (paramagnetic) jumps $F_s$ ($P_s$) are encoded by blue (black) impulses in the lower part.

Figure 3

Conceptual foundation of the dissipative ${\mathbb{Z}}_2\text{-Gauge-Higgs}$ model. (a) Qualitative illustration of the well-known phase diagram of the Hamiltonian ${\mathbb{Z}}_2\text{-Gauge-Higgs}$ theory in the $\omega -\lambda$ plane. There are three characteristic phases: the (I) confined charge, (II) free charge, and (III) Higgs phases. In order to drive the system dissipatively in a distinct phase, combinations of the baths adjacent to the labels I, II, and III are employed. (b) Effects of the six types of jump operators (characterizing the baths) on elementary excitations in two spatial dimensions. Asymmetric arrows denote asymmetric quantum jump probabilities. The symbols in the lower three boxes read as follows: yellow (dark) site $\iff {\sigma }^x=-1$ (electric charge); red (thick) edge $\iff {\tau }^x=-1$ (gauge string). In the upper three boxes: blue (dark) site + (thick) edge $\iff I_e=-1$ (Higgs excitation); blue (filled) face $\iff B_p=-1$ (magnetic flux). The formal definitions are listed in Table 1.

Figure 4

Action of the Higgs brane fragility $D_e^{\left(2\right)}$ and the flux string tension $B_e$ in three spatial dimensions. (a) Closed dual plane (Higgs brane) $\mathcal{B}$ on the three-dimensional torus ${\mathbb{T}}^3$, which defines the topologically nontrivial brane operator ${\prod }_{e\in \mathcal{B}}{\tau }_e^x$ (filled blue circles). Such excitations cannot be annihilated by the Higgs brane tension $D_s^{\left(1\right)}$ and flux string tension $B_e$ since $\mathcal{B}$ wraps once around the torus and there are no flux strings present. (b)–(d) A cross section of the lattice parallel to $\mathcal{B}$; we show the first few jumps to get rid of the excitations. (b) A Higgs brane fragility jump $D_e^{\left(2\right)}$ acts on the red edge $e$. (c) The Higgs excitation $I_e=-1$ is no longer present. One has to pay for this with four magnetic fluxes $B_p=-1$ on the adjacent faces $p\in e$ (black plaquettes). The latter define a flux loop (red line). (d) Three applications of the flux string tension $B_{e^{\prime }}$ and/or Higgs brane fragility $D_{e^{\prime }}^{\left(2\right)}$ on the edges $e^{\prime }$ (filled red circles) enlarge the hole in the Higgs brane. This process allows the retraction and subsequent annihilation of the formerly closed Higgs brane around the torus.

Figure 5

Mean-field phase diagram for the dissipative ${\mathbb{Z}}_2\text{-Gauge-Higgs}$ model with two separate mean fields. (a) Plot of the maximal $z$ polarization $\langle {\sigma }^z\rangle$ of all stable physical steady states for the matter field color-coded in the $\omega -\lambda$ plane (light $\to \langle {\sigma }^z\rangle =0$, dark $\to \langle {\sigma }^z\rangle =1$). (b) The same for the gauge field, that is, $\langle {\tau }^z\rangle$. (c) Quantitative results for $\langle {\tau }^z\rangle$ (solid lines) and $\langle {\sigma }^z\rangle$ (dashed lines) on the colored and labeled paths in (b) and (a). (d), (e) Cross sections of the Bloch ball ($g_x/m_x-g_z/m_z$ plane) for the gauge field (d) and matter field (e) with the dynamical mean-field flow $\mathbf{F}$ as flux lines and the stable physical fixed points shown by (small and large) circles. The corresponding parameters $(\omega ,\lambda )$ for each vertical pair of cross sections are highlighted by numbers in the 2D plots (a), (b). The flux lines shown for the Bloch vector of one mean field depend on the Bloch vector of the other since the mean-field equations couple all six degrees of freedom. Each depicted gauge-field flux corresponds to a fixed-point Bloch vector for the corresponding matter field, and vice versa (shown by large red and cyan circles). A discussion of the results is given in the text.

Figure 6

Mean-field phase diagram for the dissipative ${\mathbb{Z}}_2\text{-Gauge-Higgs}$ model in unitary gauge. (a) Plot of the maximal $z$ polarization $\langle {\tau }^z\rangle$ of all stable physical steady states color-coded in the $\omega -\lambda$ plane (light $\to \langle {\tau }^z\rangle =0$, dark $\to \langle {\tau }^z\rangle =1$). (b) Quantitative results for $\langle {\tau }^z\rangle$ on the colored and labeled paths in (a). (c) Four characteristic cross sections of the Bloch ball ($g_x-g_z$ plane) with the dynamical mean-field flow $\mathbf{F}$ as flux lines and the stable physical fixed points shown by cyan circles. The corresponding parameters $(\omega ,\lambda )$ for each cross section are highlighted by numbers in the 2D plot (a). A discussion of the results is given in the text.