Interferometric Geometric Phases of $\mathcal{PT}$-symmetric Quantum Mechanics

The introduction of non-Hermitian quantum mechanics (NHQM) [1, 2, 3] has uncovered many fascinating phenomena, including the Anderson localization [4], gapless quantum phase transitions [5], unconventional behavior of quantum emitters [6, 7], tachyonic dynamics [8, 9], and distinctive topological properties [10, 11, 12]. A major branch of NHQM includes systems with non-Hermitian Hamiltonians obeying parity-time reversal ($\mathcal{PT}$) symmetry, which can possess real-valued eigenvalues, making them a relevant extension of conventional quantum mechanics. Therefore, $\mathcal{PT}$-symmetric quantum mechanics (PTQM) has attracted considerable research attention in many aspects [13, 14, 15, 16, 17, 18, 19, 20] and has been experimentally realized across different fields, including acoustics, optics, electronics and quantum systems [21, 22]. It has also catalyzed extensive investigations into physical and topological characteristics of non-Hermitian systems [23, 24, 25, 26, 27, 28, 29, 30, 31].

Geometric phases in quantum systems, including the Berry phase [32] and the Aharonov-Anandan phase [33], have advanced our understanding of the geometric structures behind interesting physical systems and shown significant influence across various fields. For instance, the Berry phase is fundamental in the study of topological matter since it connects geometric objects from the underlying mathematical structure to measurable physical quantities [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. Recently, the notion of geometric phase has been generalized to non-Hermitian quantum systems [46], which is further applied to the construction of the quantum geometric tensor for non-Hermitian systems [47]. On the other hand, the geometric phase has also been generalized to mixed quantum states via different approaches [48, 49, 50, 51, 52, 53, 54, 55, 56]. In this work, we will generalize the one proposed by Sjöqvist et al [57] based on an extension of the optical process in the Mach-Zehnder interferometer, which is referred to as the interferometric geometric phase (IGP). Numerous studies [58, 59, 60, 61, 62, 63] have been dedicated to this field, and the IGP has been observed by various techniques, including the nuclear magnetic resonance [64, 65], polarized neutrons [66], and the Mach-Zehnder interferometer [67]. A different approach was introduced by Uhlmann [68, 69, 70] soon after the discovery of the Berry phase and the phase is usually called the Uhlmann phase. This approach incorporates a full mathematical structure based on fiber bundles and has gained attention due to its relevance to condensed matter and quantum information [71, 72, 73, 74, 75, 76, 77, 78].

We aim at generalizing the concept of IGP to thermal states in PTQM. In the beginning, we establish a formalism for pure-state geometric phase in $\mathcal{PT}$-symmetric systems by using the conventional derivation and then introducing the parallel-transport conditions of quantum states. In contrast to conventional QM, a PTQM system is shown to allow two distinct geometric phases, called $\theta^{1}$ and $\theta^{2}$, which are differentiated by the states undergoing parallel-transport. On the one hand, $\theta^{2}$ exactly coincides with the known result in Ref. [46] and is the real part of $\theta^{1}$. On the other hand, $\theta^{1}$ is a complex-valued phase with its imaginary part adjusting the amplitude of the wavefunction due to the lack of Hermiticity. Since $\theta^{1}$ will be shown to be associated with the non-Hermitian Hamiltonian, the generalization to thermal states in PTQM will be based on it.

Following the construction of the IGP of thermal states in conventional QM and the derivation of the geometric phase $\theta^{1}$, we develop a framework of the IGP of thermal states in PTQM. In general, the IGP is the argument of the thermal-weighted sum of the geometric phase factor for each individual energy level. The imaginary part of the generalized IGP will be shown to alter the relative thermal weights, which introduces effective temperatures to the thermal states. Consequently, there may be quantized jumps of the IGP at certain temperatures and system parameters. This phenomenon signifies a geometric phase transition at finite temperature in PTQM. To illustrate our findings and visualize the results, we study a $\mathcal{PT}$-symmetric two-level system and present its generalized IGP. The geometric phase transitions of the model at finite temperatures are located and analyzed.

The rest of the paper is organized as follows. Sec. II briefly reviews the basics of PTQM and its statistical physics. We also review the geometric phases of pure and mixed quantum states in Hermitian systems via parallel-transport. In Sec. III, we generalize the formalism of geometric phase to PTQM, first by deriving two different expressions due to their associated evolution equations or parallel-transport conditions. We then generalize the results to thermal states in PTQM and derive the generalized IGP. Sec. IV presents the IGP of a $\mathcal{PT}$-symmetric two-level system and its geometric phase transitions at finite temperatures. Sec. V concludes our work. Some details and derivations are summarized in the Appendix.

To understand the IGP of PTQM more clearly, we study a $\mathcal{PT}$-symmetric two-level system introduced in Refs. [84, 46] and calculate its IGP. The Hamiltonian is given by

$\displaystyle H=\epsilon\mathbf{1}_{2\times 2}+\left(a\mathbf{n}^{r}+\mathrm{i}b\mathbf{n}^{\theta}\right)\cdot\bm{\sigma},$

(38)

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})^{T}$ is the collection of Pauli matrices and $\mathbf{n}^{r}\equiv(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{T}$, $\mathbf{n}^{\theta}\equiv(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)^{T}$ are the unit vectors respectively along the radial and tangent directions of a meridian on a unit sphere. The eigenvalues are $E_{\pm}=\epsilon\pm\sqrt{a^{2}-b^{2}}$. We limit our discussion to the regime of $a^{2}>b^{2}$, where the $\mathcal{PT}$-symmetry is not broken and $E_{\pm}$ is real. Without loss of generality, we let $a>0$. The two eigen-vectors are

		$\displaystyle\left|\Psi_{+}\right\rangle=n_{+}\left(\begin{array}[]{c}\left(\cos\frac{\theta}{2}-\mathrm{i}\alpha\sin\frac{\theta}{2}\right)\mathrm{e}^{-\mathrm{i}\phi}\\ \mathrm{i}\alpha\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\end{array}\right),$		(41)
		$\displaystyle\left|\Psi_{-}\right\rangle=n_{-}\left(\begin{array}[]{c}-\left(\mathrm{i}\alpha\cos\frac{\theta}{2}+\sin\frac{\theta}{2}\right)\mathrm{e}^{-\mathrm{i}\phi}\\ \cos\frac{\theta}{2}-\mathrm{i}\alpha\sin\frac{\theta}{2}\end{array}\right),$		(44)

where $\alpha=\frac{b}{a+\sqrt{a^{2}-b^{2}}}$ and $n_{\pm}=\mathrm{e}^{-\mathrm{i}\frac{\theta}{2}}\sqrt{\frac{a^{2}+a\sqrt{a^{2}-b^{2}}}{2\left(a^{2}-b^{2}\right)}}$ are normalization coefficients. The metric operator $W$ of this case is

$\displaystyle W=1-\frac{b}{a}\mathbf{n}^{\phi}\cdot\bm{\sigma},$

(45)

where $\mathbf{n}^{\phi}=(-\sin\phi,\cos\phi,0)^{T}$ is the unit tangent vector of a latitude. In what follows, we will fix $a$ and $b$, thus the parameters $(\theta,\phi)$ form the parameter manifold $S^{2}$, a unit spherical surface. Using Eq. (18), $\theta^{2}_{\pm}$ associated with a loop $C$ on $S^{2}$ is given by [46]

	$\displaystyle\theta^{2}_{\pm}(C)=$	$\displaystyle\mp\frac{1}{2}\frac{a}{\sqrt{a^{2}-b^{2}}}\Omega(C)+\left(1\pm\frac{a}{\sqrt{a^{2}-b^{2}}}\right)\pi$
	$\displaystyle\pm$	$\displaystyle\frac{\pi}{4}\left(1-\frac{a}{\sqrt{a^{2}-b^{2}}}\right),$		(46)

if the north pole is enclosed by $C$, or

$\displaystyle\theta^{2}_{\pm}(C)=\mp\frac{1}{2}\frac{a}{\sqrt{a^{2}-b^{2}}}\Omega(C)\pm\frac{\pi}{4}\left(1-\frac{a}{\sqrt{a^{2}-b^{2}}}\right)$

(47)

if the north pole is not enclosed by $C$. Here $\Omega(C)={\oint}_{C}\mathrm{d}\phi(1-\cos\theta)$ is the solid angle of the surface enclosed by $C$ with respect to the origin.

As a concrete example, we take $a=3$ and $b=\sqrt{5}$, so the eigenvalues become $E_{\pm}=\epsilon\pm 2$. To calculate the second term of $\theta^{1}_{\pm}(C)$ in Eq. (26), we choose a proper $S$, which can be constructed via the procedure of Ref. [46] summarized in Appendix C. Explicitly, it is given by

$\displaystyle S_{\text{proper }}=\begin{pmatrix}\frac{1}{2}\sqrt{\frac{15}{2}}\mathrm{e}^{\frac{\mathrm{i}\phi}{4}}&-\frac{1}{2}\mathrm{i}\sqrt{\frac{3}{2}}\mathrm{e}^{-\frac{5\mathrm{i}\phi}{4}}\\ \frac{1}{2}\mathrm{i}\sqrt{\frac{3}{2}}\mathrm{e}^{\frac{5\mathrm{i}\phi}{4}}&\frac{1}{2}\sqrt{\frac{15}{2}}\mathrm{e}^{-\frac{\mathrm{i}\phi}{4}}\end{pmatrix}.$

(48)

Under this proper transformation, the original non-Hermitian Hamiltonian is converted into a Hermitian one:

$\displaystyle H_{0}=S^{-1}_{\text{proper }}HS_{\text{proper }}=\begin{pmatrix}\epsilon+2\cos\theta&2\mathrm{e}^{-\frac{3\mathrm{i}\phi}{2}}\sin\theta\\ 2\mathrm{e}^{\frac{3\mathrm{i}\phi}{2}}\sin\theta&\epsilon-2\cos\theta\end{pmatrix}.$

(49)

The eigenvector associated with $\epsilon$+2 is

$\displaystyle\left|\Psi^{0}_{+}(\theta,\phi)\right\rangle=\begin{pmatrix}\frac{\mathrm{e}^{-\frac{3\mathrm{i}\phi}{2}}(\cot\theta+\csc\theta)}{\sqrt{(\cot\theta+\csc\theta)^{2}+1}}\\ \frac{1}{\sqrt{(\cot\theta+\csc\theta)^{2}+1}}\end{pmatrix}.$

(50)

It can be shown that

	$\displaystyle\langle\Psi^{0}_{+}|S^{-1}\mathrm{d}S|\Psi^{0}_{+}\rangle$	$\displaystyle=-\frac{1}{4}\sqrt{5}\sin\theta\mathrm{d}\phi,$
	$\displaystyle\mathrm{i}\oint_{C}\langle\Psi^{0}_{+}|S^{-1}\dot{S}|\Psi^{0}_{+}\rangle\mathrm{d}t$	$\displaystyle=-\mathrm{i}\frac{\pi}{2}\sqrt{5}\sin\theta,$		(51)

where the loop $C$ is chosen as a circle of latitude $\theta$. Similarly, the imaginary part of $\theta^{1}_{-}$ is

$\displaystyle\mathrm{i}\oint_{C}\langle\Psi^{0}_{-}|S^{-1}\dot{S}|\Psi^{0}_{-}\rangle\mathrm{d}t=\mathrm{i}\frac{\pi}{2}\sqrt{5}\sin\theta.$

(52)

Since the north pole is enclosed by $C$ (a circle of latitude), $\theta^{2}_{\pm}(C)$ is evaluated by Eq. (51). Using Eqs. (28), (51), and (52), the geometric phases associated with the two eigenstates are

	$\displaystyle\theta^{1}_{+}(C)$	$\displaystyle=-\frac{3\pi}{2}(1-\cos\theta)+\frac{3\pi}{8}-\mathrm{i}\frac{\pi}{2}\sqrt{5}\sin\theta,$
	$\displaystyle\theta^{1}_{-}(C)$	$\displaystyle=\frac{3\pi}{2}(1-\cos\theta)-\frac{3\pi}{8}+\mathrm{i}\frac{\pi}{2}\sqrt{5}\sin\theta,$		(53)

respectively. Here an extra factor $2\pi$ is dropped from $\theta^{1}_{+}(C)$. Accordingly, the IGP is

		$\displaystyle\theta_{\text{G}}(C)=\arg\left[\frac{\mathrm{e}^{-2\beta}\mathrm{e}^{\mathrm{i}\theta^{1}_{+}(C)}+\mathrm{e}^{2\beta}\mathrm{e}^{\mathrm{i}\theta^{1}_{-}(C)}}{\mathrm{e}^{2\beta}+\mathrm{e}^{-2\beta}}\right]$
		$\displaystyle=\arg\left[\mathrm{e}^{-2\beta+\frac{\sqrt{5}\pi}{2}\sin\theta}\mathrm{e}^{\mathrm{i}\theta^{2}_{+}(C)}+\mathrm{e}^{2\beta-\frac{\sqrt{5}\pi}{2}\sin\theta}\mathrm{e}^{\mathrm{i}\theta^{2}_{-}(C)}\right],$		(54)

where $\theta^{2}_{\pm}$ is the real part of $\theta^{1}_{\pm}$, as shown by Eq. (28). The imaginary part of $\theta^{1}_{\pm}$ actually changes the thermal weight of each energy-level.

Eq. (14) shows that the IGP is the argument of $\text{Tr}\left[\rho(0)U(t)\right]$, which is the “returning amplitude” between the initial state $\rho(0)$ and the instantaneous state $\rho(t)$ [48, 78]. It can also be thought of as a generalization of the Loschmidt amplitude in mixed quantum states. At its zeros, the IGP exhibits discontinuities and nonanalytical behavior, signaling a change of the geometric nature of the system reflected by the IGP. In this example, the second line of Eq. (IV) shows that $\theta_{\text{G}}(C)$ may become singular if $\beta=\frac{\sqrt{5}\pi\sin\theta}{4}$. To examine the IGP of PTQM, we visualize our findings in Figs. 1, 2, and 3.

In Fig. 1, we present the contour plot of $\theta_{\text{G}}(C)$ as a function of $\beta$ and $\theta$. Indeed, there are three singular points A, B and C lying on the arc $\beta=\frac{\sqrt{5}\pi\sin\theta}{4}$, which correspond to the latitudes $\theta_{\text{A,B,C}}=\arccos\left(\frac{5}{12}\right)\approx 1.14$, $\arccos\left(-\frac{1}{4}\right)\approx 1.82$, and $\arccos\left(-\frac{11}{12}\right)\approx 2.73$, respectively. The IGP changes rapidly near A, B and C, indicating that the value of $\theta_{\text{G}}(C)$ jumps discretely when crossing these singular points. Notably, a jump of the IGP at finite temperature has been ruled out in any two-level model of Hermitian quantum systems [78].

To grasp the physical significance of the arc $\beta=\frac{\sqrt{5}\pi\sin\theta}{4}$, we revisit the corresponding Hermitian quantum system, where the thermal weight of each level is proportional to $\mathrm{e}^{\mp 2\beta}$ at temperature $T=\frac{1}{\beta}$. As $T\rightarrow 0$, the relative weight between the excited and ground states becomes $\lim_{\beta\rightarrow+\infty}\frac{\mathrm{e}^{-2\beta}}{\mathrm{e}^{2\beta}}=0$, leading the IGP to converge to the geometric phase of the ground state. In the infinite temperature limit ($\beta\rightarrow 0$), the relative weight becomes $\lim_{\beta\rightarrow 0}\frac{\mathrm{e}^{-2\beta}}{\mathrm{e}^{2\beta}}=1$. In this case, the Hermitian density matrix corresponds to the maximally mixed state, where each level has equal thermal weight, and the IGP loses its resemblance to the ground-state geometric phase. Turning to $\mathcal{PT}$-symmetric systems, the parallel-transport condition eliminates the dynamic phase from the total phase, leaving a complex IGP. The imaginary part of the IGP (or $\theta^{1}_{\pm}$) modifies the thermal weights of the two levels to exp$\left[\mp\left(2\beta-\frac{\sqrt{5}\pi\sin\theta}{2}\right)\right]$, which will be referred to as the “effective thermal weights”. Notably, in the low-temperature limit, the behavior of the IGP can still mirror that of the corresponding Hermitian system. In Fig. 1, the domain where $\beta>\frac{\sqrt{5}\pi\sin\theta}{4}$ corresponds to the phase at “effective” positive temperatures for the non-Hermitian quantum system. The arc $\beta=\frac{\sqrt{5}\pi\sin\theta}{4}$ signifies the “effective” infinite-temperature threshold. Conversely, the regime where $\beta<\frac{\sqrt{5}\pi\sin\theta}{4}$ corresponds to the phase at “effective” negative temperatures. In this scenario, the original temperature $T$ alongside with the imaginary part of $\theta^{1}_{\pm}$ determines the relative thermal distribution between the excited and ground states.