Effects of light-wave nonstaticity on accompanying geometric-phase evolutions

Jeong Ryeol Choi

doi:10.1364/OE.440512

1. Introduction

The Berry’s seminal discovery [1] to the appearance of an additional phase evolution in eigenstates of the Hamiltonian during a slow variation of a quantum system in the parameter space had triggered extensive research for such extra phases in both theoretical and experimental spheres. The Berry phase is a geometrical characteristic of quantum waves, which corresponds to a holonomy transformation in state space. It is impossible to gauge out the geometric character in the phase because the Berry phase or, in general, the geometric phase is gauge invariant. For this reason, the geometric phase is non-negligible and, especially, inevitable for analyzing the transmission of light waves in media with time-varying parameters. The geometric phase reflects the geometry of a quantum wave evolution. It has been proved that the original concept of the Berry’s geometric phase can be extended to more general cases which are nonadiabatic, non-cyclic and/or non-unitary evolutions of light waves [2–4].

The geometric phase is a notable research subject that has been widely investigated with the purpose of manipulating quantum phases of light waves and controlling their behaviors. The scientific fields that the geometric phase can be applicable are plentiful: they include a holonomic quantum computation with geometric gates [5], a stellar interferometry [6], the testing of CPT (charge conjugation, parity, and time reversal) invariance in particle physics [7], entanglement of atoms [8], analysis of the Aharonov-Bohm effect [9], etc. Among them, a geometric quantum computation enables us to carry out quantum logic operations by means of multi-qubit gates, which are main techniques in the process of quantum computers [5].

For the case of the Fock state of a quantum system, it is assumed that the geometric phase appears when the state is nonstatic, whereas it always vanishes when the state becomes stationary [10]. As is well known, the ordinary waves in the Fock state are static in time, resulting in no emergence of the geometric phase. However, if we prepare a quantum wave with time-varying eigenfunctions, there will appear geometric phases even for a simple situation where the parameters of the medium do not vary over time. For instance, the eigenfunctions in coherent and squeezed states are expressed in terms of time regardless of whether the parameters of the medium depend on time or not [11]. This leads to the appearance of the geometric phase in such states [11,12].

Meanwhile, it was reported from our recent work [13] that nonstatic quantum light waves can also appear in the Fock state in a static environment associated with a transparent medium. Such waves exhibit a peculiar behavior in time as a manifestation of their nonstaticity, which is that they become narrow and broad in turn periodically in quadrature space. Subsequently, we also analyzed the mechanism underlying such a phenomenon from a fundamental point of view [14].

Even if the environment is static in the above-mentioned case, the periodic variation of the waves is accompanied by the evolution of the geometric phases. This is due to the fact that the eigenfunctions also vary in time according to the nonstaticity of the wave. We will investigate the characteristics of the geometric phases in the Fock state arisen in such a situation in this work. It will be focused on analyzing how the geometric phases evolve in relation with the wave nonstaticity. We will compare the behavior of the geometric phases with that of the dynamical phases. While we regard the Fock state in this work, the nonstatic waves and their phases can also be evaluated in more general circumstances, e.g., in the case of Gaussian states [13].

The Hannay angle [15] of the system, which is the classical analogue of the geometric phases, will also be investigated. Utilizing the relation of this angle with the quantum phases, its physical meanings will be addressed. Based on the Hannay angle, we can obtain an insight on the classical geometric structure of light, including its connection with the quantum geometric-phase structure [16,17].

2. Description of nonstatic waves

To develop the theory of geometric phases for a nonstatic wave, we first show how to describe nonstatic quantum light waves in a static environment. The Hamiltonian for a light is given by

(1)$$\hat{H}= {\hat{p}^{2}}/{(2\epsilon)} + \epsilon\omega^{2} \hat{q}^{2} /2,$$

where $\hat {q}$ is the quadrature operator, $\hat {p} = -i\hbar \partial /\partial q$, and $\epsilon$ is the electric permittivity of the medium. The geometric phases are dependent on the preparation of the wave functions [18]. If we consider elementary static wave functions in the Fock state, of which eigenfunctions are given by $\langle q|\phi _n \rangle = ({\alpha }/{\pi })^{1/4}({\sqrt {2^{n} n!}})^{-1} H_{n} \left (\sqrt {\alpha } {q} \right )\exp \left [-{\alpha } {q}^{2}/{2} \right ]$ where $\alpha = \epsilon \omega /{\hbar }$ and $H_n$ are Hermite polynomials, the geometric phases do not take place [11,19]. However, for the case of the wave functions whose eigenfunctions are time-dependent, the geometric phases are nonzero because they are given in terms of the time derivative of the eigenfunctions. Notice that, for a time-independent Hamiltonian including the case regarded here, there are Schrödinger solutions associated with wave nonstaticity as well as the ones that correspond to static waves.

Instead of $\langle q|\phi _n \rangle$, nonstatic waves in this context can be described with generalized eigenfunctions of the form [13]

(2)$$\langle q |\Phi_n \rangle = \left({\frac{\beta(t)}{\pi}}\right)^{1/4} \frac{1}{\sqrt{2^{n} n!}} H_n \left( \sqrt{\beta(t)} q \right) \exp \left[ - \frac{\beta(t)}{2} \left(1-i\frac{\dot{f}(t)}{2\omega}\right)q^{2} \right],$$

where $\beta (t) = {\epsilon \omega }/{[\hbar f(t)]}$ and $f(t)$ is a time function which is given by

(3)$$f(t) = A \sin^{2} \tilde{\varphi}(t)+ B \cos^{2} \tilde{\varphi}(t) + C \sin [2\tilde{\varphi}(t)],$$

under an auxiliary condition, $AB-C^{2} = 1 \label {13}$ with $AB \geq 1$, while $\tilde {\varphi }(t)=\omega (t-t_0) +\varphi$ whereas $\varphi$ is a real constant. It is now possible to establish time-dependent wave functions associated with the nonstatic wave in terms of $\langle q |\Phi _n \rangle$ as follows [13]:

(4)$$\langle q |\Psi_n(t) \rangle = \langle q |\Phi_n(t) \rangle \exp \bigg[{-i\omega (n+1/2) \int_{t_0}^{t} f^{{-}1} (t') dt'} + i\gamma_n (t_0) \bigg],$$

where $\gamma _n (t_0)$ are phases at $t_0$. We note that the wave functions given above satisfy the Schrödinger equation associated with the Hamiltonian represented in Eq. (1), and $f(t)$ used here follows the nonlinear differential equation of the form $\ddot {f} - {(\dot {f})^{2}}/({2f}) + 2\omega ^{2} \left (f- {1}/{f}\right ) =0. \label {4}$

We will restrict the classical angle within the range $-\pi /2 \leq \varphi < \pi /2$, because the research in this range is enough owing to the fact that Eq. (3) is a periodic function with the angle period $\pi$. For $A=B=1$ and $C=0$, Eq. (3) reduces to $f(t) =1$ which corresponds to the case that gives static wave functions; all other choices for the set of $A$ and $B$ give nonstatic wave functions. We have compared static and nonstatic wave packets in Fig. 1. The nonstatic wave packet shown in Fig. 1(b) varies periodically over time. The degree of such variation caused by nonstaticity is determined by the quantitative measure of nonstaticity that is represented as [13]

(5)$$D_\textrm{F} = \frac{\sqrt{(A+B)^{2}-4}}{2\sqrt{2}}.$$

Usually, the nonstatic properties of the light wave become significant as this measure increases. The nonstatic-wave packets described up until now will be used in order to investigate the geometric phases in the subsequent section.

Fig. 1. Comparison between probability densities for static (a) and nonstatic (b) wave packets. We have chosen the time function $f(t)$ for a as 1 and for b as Eq. (3) with $A=2.5$ and $B=0.5$. We take only positive values for $C$ throughout all figures in this work for convenience. Then, the value of $C$ is automatically determined from $A$ and $B$ through the auxiliary condition given below Eq. (3). Other values that we have chosen are $n=5$, $\omega =1$, $\epsilon =1$, $\hbar =1$, $t_0=0$, and $\varphi =0$. All variables are chosen to be dimensionless for convenience; this rule will also be applied to subsequent figures. For the nonstatic case, the probability density undergos cyclic evolution with the period $T=\pi /\omega$, that is, the width of the wave packet in quadrature space becomes broad and narrow in turn over time.

Download Full Size | PDF

3. Geometric phase

The geometric phases are examples of holonomy which gives additional phase evolutions of the quantum wave over time. The study for the development of the geometric phases of a wave enables us to clarify topological features in quantum mechanics. The geometric phases for the nonstatic waves can be evaluated from [20,21]

(6)$$\gamma_{G,n}(t) = \int_{t_0}^{t} \langle\Phi_n(t') |i\frac{\partial}{\partial t'}| \Phi_n(t') \rangle dt' +\gamma_{G,n}(t_0). \\$$

These are parts of the phases of quantum wave functions at time $t$, which have geometric origin. On one hand, there is a concept of the geometric phase whose definition is a little different: it is the geometrical part of the phase acquired during only one cycle evolution of the eigenstate through a closed path in the circuit [12]. In what follow, we will use the former concept of the geometric phase associated with Eq. (6) throughout this paper. We assume that the initial phases are zero for convenience from now on: $\gamma _{G,n}(t_0)=0$. It is possible to evaluate Eq. (6) by using Eq. (2) with the consideration of $f(t)$ given in Eq. (3). Hence, as a result, we have (see Methods section which is the last section)

(7)$$\gamma_{G,n}(t) = \frac{1}{2} \left( n+\frac{1}{2} \right) \{(A+B)\omega (t-t_0) -2 [\tan^{{-}1} Z(t) - \tan^{{-}1} Z(t_0)+G(t) ]\},$$

for $t\geq t_0$, where $Z(\tau )= C+A\tan [\omega (\tau -t_0) +\varphi ]$, and $G(t)$ is a time function that is expressed in terms of the unit step function (Heaviside step function) $u[t]$ as $G(t)=\pi \sum _{m=0}^{\infty }u[t-t_0-(2m+1)\pi /(2\omega )+\varphi /\omega ]$. Here, $G(t)$ is necessary in order to compensate the periodic discontinuities (with a period of $\pi$) of tangent functions in Eq. (7) in a way that $\gamma _{G,n}(t)$ become continuous functions. We note that the obtained formula of the geometric phases, Eq. (7), holds within the considered range for $\varphi$, $-\pi /2 \leq \varphi < \pi /2$ (see the previous section for this range). These phases may provide a deeper insight for the understanding of the nature of the nonstatic waves.

The dynamical phases can also be derived from their definition using the same wave functions. Their resulting formula is given by (see Methods section)

(8)$$\gamma_{D,n}(t) ={-}\frac{1}{2}\left( n+\frac{1}{2} \right)(A+B)\omega (t-t_0).$$

Since the measure of nonstaticity, Eq. (5), is nearly proportional to $A+B$ provided that $A+B \gg 4$, $\gamma _{D,n}(t)$ at a certain time is linearly proportional to the measure of nonstaticity for highly nonstatic waves.

Whereas the dynamical phases decrease linearly over time from their initial values, the time behavior of the geometric phases is not so simple. Let us divide the geometric phase in Eq. (7) into two parts for the convenience of analyses. We call the term that involves $(A+B)$ as the first part and the remaining terms the second part. We can readily see that the first part, which increases in a monotonic manner over time, exactly cancels out the dynamical phase. Hence, the second part of the geometric phase is the same as the total phase of the wave. However, it may be not so easy to completely estimate the evolution of the geometric phases since the second part is somewhat intricate.

To understand the time behavior of wave phases rigorously, we plotted the evolution of the geometric phase in Fig. 2 together with the dynamical phase for several different values of parameters. Let us first examine the effects of $A$ and $B$ on the geometric phase. For a trivial case where $A=B=1$ and $C=0$, which corresponds to Fig. 2(a), $f(t)$ becomes unity and the eigenfunctions, Eq. (2), reduce to time-independent ones as we have seen from the previous section. As a consequence, the geometric phases disappear, $\gamma _{G,n}(t) = 0, \label {10}$ whereas the dynamical phases result in the well-known ones, $\gamma _{D,n}(t) = -( n+{1}/{2} )\omega (t-t_0). \label {11}$ The absence of the geometric phases in this case is due to the exact opposite behaviors of their first and second parts: whereas the first part increases linearly over time, the second part decreases linearly. Besides, the second part of the geometric phase is the same as the dynamical phase only when $A=B=1$.

Fig. 2. Time evolution of the geometric phase and the dynamical phase for several different values of the parameters. The values of ($A$, $B$, $n$, $\omega$) are ($1$, $1$, $0$, $0.5$) for a, ($0.5$, $2.5$, $5$, $0.5$) for b, and ($0.1$, $10.0$, $10$, $1$) for c. We have chosen other parameters as $t_0=0$ and $\varphi =0$. The pink-white graphics in the upper part of the panels are the time evolution of the corresponding probability density. The measure of nonstaticity is 0.00 for a, 0.79 for b, and 3.50 for c.

Download Full Size | PDF

If at least one of $A$ and $B$ is not unity, it becomes the case of a nonstatic wave as shown in Figs. 2(b,c). In this case, the width of the corresponding probability density varies periodically with the period of $T=\pi /\omega$; thus the frequency in this temporal variation is large when $\omega$ is high. We can confirm from Figs. 2(b,c) that the second part of the geometric phase changes depending on the width of the wave packet. The second part nearly monotonically decreases over time when the width is large, but it somewhat abruptly decreases whenever the width becomes small. Consequently, the geometric phase, which is the addition of the first and second parts, varies in a regular way as shown in the figure. We can demonstrate these behaviors more concretely from Fig. 3, which shows that the time derivative of the geometric phase drops abruptly whenever the width of the wave packet is narrow. In contrast to this, the dynamical phases vary in a monotonic manner at all times. The system will pick up a memory of its time evolution in the form of the geometric phase in this way, leading to an observable shift of the wave phase.

Fig. 3. Behavior for the time derivative of the geometric phase $d\gamma _{G,n}(t)/dt$, the dynamical phase $d\gamma _{D,n}(t)/dt$, and the total phase $d\gamma _{n}(t)/dt$. All chosen parameters for a and b are the same as those for b and c in Fig. 2, respectively.

Download Full Size | PDF

The detailed evolution of the second part of the geometric phase is given in Fig. 4. Whereas the first part of the geometric phase becomes large as the value of $A+B$ (or the measure of nonstaticity) increases, the envelope of the second part is not so significantly affected by the values of $A$ and $B$. The gradient of the envelope of the second part is in fact irrelevant to the measure of nonstaticity. Such a gradient is instead determined by $\omega$, i.e., the envelope of the second part decreases more rapidly as $\omega$ grows. The rise of $\omega$ also makes the rate of the increase of the first part large.

Fig. 4. The evolution of the second part of the geometric phase for several values of $A$ (a) and $B$ (b), where $B=1$ for a, $A=1$ for b, $n=5$, $\varphi =0$, and $t_0=0$. The conventions for colors designated for solid lines in the legends are also applied to the dashed lines within the figure panels. The values of the measure of nonstaticity (Eq. (5)) in turn from red to violet curve are 0.00, 0.79, 2.00, 3.82, 7.39, 14.48, and 35.70 for both a and b.

Download Full Size | PDF

We can confirm from Fig. 4(a) that the second part drops abruptly when $\omega t$ is $\pi$, $2\pi$, $3\pi$, etc. provided that $A$ is very large while $B$ is unity; however, except for these moments, the second part almost does not vary over time. The bottom envelope of the second part is always given by $-( n+{1}/{2} )\omega (t-t_0)$ in this case regardless of the value of $A$. Similar behaviors in the phase evolution can also be seen from the curves in Fig. 4(b), which correspond to the case where $B$ is very large whilst $A$ is unity; in this case, the second part drops when $\omega t$ is $\pi /2$, $3\pi /2$, $5\pi /2$, etc.

The dependence of the geometric-phase evolution on $\varphi$ within the considered region $-\pi /2 \leq \varphi < \pi /2$ is shown in Fig. 5. All geometric phases in the figure start from zero, but the phase interval between the adjacent geometric phases is $\Delta \varphi = 0.15 \pi$. Whereas the geometric phase varies periodically as $\varphi$ increases with the period of $\pi$, it is also possible to know the pattern of the geometric phase outside the considered region for $\varphi$. More precisely speaking, the geometric phase for $\varphi =\pi$ is the same as that for $\varphi =0$; the geometric phase for $\varphi =1.15\pi$ is the same for $\varphi =0.15\pi$, etc.

Fig. 5. Time evolution of the geometric phase for various values of $\varphi$. We have chosen parameters as $A=2.5$, $B=0.5$, $n=0$, $\omega =0.5$, and $t_0=0$.

Download Full Size | PDF

The overall phases of the waves are given by $\gamma _n(t) = \gamma _{G,n}(t) + \gamma _{D,n}(t) . \label {7}$ Hence, by adding our two results for geometric and dynamical phases, we have the total phases such that $\gamma _n(t) = -\omega ( n+ {1}/{2} )\int _{t_0}^{t} f^{-1}(t') dt' . \label {6-1}$ These formulae are exactly the same as the phases that appeared in the wave functions, Eq. (4), under the condition that the initial phases are zero, $\gamma _n(t_0)=0$.

Hannay confirmed that a phase similar to the geometric phase also appears in the classical domain [15]. This is a geometrical angle for a classical wave which allows to estimate its holonomy effect from the corresponding geometrical interpretation. This is an important concept in theoretical physics and, hence, it may be instructive to see the Hannay angle of the system. From the theory of action-angle variables for an integrable classical system, one can confirm that the Hannay angle $\Theta _H (t)$ is related to the geometric phase by $\Theta _H (t)= - {\partial \gamma _{G,n}(t)}/{\partial n} \label {27}$ [22,23]. Based on this, we can easily show that the Hannay angle in this case is of the form $\Theta _H(t) = - 2\gamma _{G,0}(t). \label {28}$ Hence the evolution of the classical additional angle can be represented in terms of the zero-point geometric phase. This simple picture provides a distinctive geometric meaning for unification of the gauge structures of the waves for quantum and classical mechanics, leading in effect to Bohr’s correspondence principle [24]. If a method to generate the generalized wave packet given by Eq. (4) is developed in the future, this consequence may be demonstrated from the measurement of the Hannay angle using a technique related to the averaging theorem in phase space [25–27]. The reason why the Hannay angle takes place in the classical system is that the structure of the classical phase space is quite the same as the quantum Hilbert space [28,29].

Whenever the wave packet collapses as shown in Fig. 1(b), the uncertainty of quadrature $q$ instantaneously reduces below its standard quantum level. From this, we can regard the nonstatic state as a kind of squeezed state. Several methods for producing squeezed states are known until now [30–34]. Accordingly, the nonstatic quantum state may possibly be produced by applying some techniques of squeezed-state generation. Once we are able to produce such a nonstatic wave, the associated geometric phase that is theoretically demonstrated here can be measured by using advanced experimental technologies [35–37].

4. Conclusion

Ordinary Fock-state quantum light waves in a transparent medium in which electromagnetic parameters do not vary are static in time, resulting in no appearance of the geometric phase. However, if one or both of the parameters $A$ and $B$ in that circumstance deviates from unity, the waves become nonstatic and, as a consequence, the geometric phases emerge. As well as the appearance of such nonstatic waves, the characteristics of the resultant geometric phases may be noteworthy [13]. We have analyzed the influence of the wave nonstaticity on the evolution of the geometric phases in such a case.

The geometric phases of the nonstatic light waves exhibit periodic oscillatory behavior, where the center of such an oscillation increases linearly with time. On the other hand, the dynamical phases always show a linear decrease. Because the scale of the geometric phases is smaller than that of the dynamical phases, the total phases are accumulated with a negative sign as time goes by. The higher the measure of nonstaticity, the more rapid the accumulation of both the geometric and dynamical phases with their own sign. Although the geometric phases increase on the whole, they periodically drop provided that the measure of nonstaticity is sufficiently high. Such a phase variation is quite significant when the measure of nonstaticity is extremely large.

We have shown that Hannay angle, which is the classical analogue of the geometric phase, is represented in terms of the geometric phase associated to the zero-point wave function. This elegant outcome shows a unified picture for the interpretation of the geometric character of light waves in the quantum and the classical regime. This not only bridges the quantum and classical worlds, but can also be extended to more generalized quantum light waves, such as the squeezed-state light and the Gaussian wave packet propagating in time-varying media [38,39]. Lots of research works treated other types of fundamental nonstatic waves are found in the literature [23,38–43]; all the wave functions corresponding to them are actually accompanied by geometric phases. The understanding of the evolution of the geometric phases in this context is highly required, especially in relation with wave modulations. Practical utilization of the geometric phases in diverse scientific areas might be possible under the fundamental knowledge for the behavior of quantal phases.

On one hand, the geometric phase of quantum systems which evolve in a non-unitary way [44] and its relation with the underlying wave nonstaticity may be worthwhile to be explored through subsequent research. The outcome of such research is useful in the analysis of the accompanying non-unitary effects like decoherence and dissipation. In particular, kinematic approach to the geometric phase of non-unitarily evolving systems is important in pursuing robust geometric quantum computation [3,4,45].

5. Methods

Let us see how to evaluate the geometric and the dynamical phases in the nonstatic Fock state. We first derive the geometric phases. From a minor computation of Eq. (6) in the configuration space using Eq. (2), we have

(9)$$\gamma_{G,n}(t) = \frac{1}{2} \left( n+\frac{1}{2} \right) \Gamma_{G}, \\$$

where

(10)$$\Gamma_{G} = \omega[g_1(t)-g_2(t)]+ \frac{g_3(t)}{4\omega}, \\$$

with

(11)$$ g_1(t) = \int_{t_0}^{t} f(t') dt' , $$

(12)$$ g_2(t) = \int_{t_0}^{t} \frac{1}{f(t')} dt' , $$

(13)$$ g_3(t) = \int_{t_0}^{t} \frac{[\dot{f}(t')]^{2}}{f(t')} dt' . $$

Here, the condition $\gamma _{G,n}(t_0)=0$ has been applied for simplification.

Straightforward evaluations of $g_i(t)~(i=1,2,3)$ in Eqs. (11)–(13) using Eq. (3) yield

(14)$$g_i(t) = G_i(t) - G_i(t_0), \\$$

for $t_0 \leq t<t_0+\pi /(2\omega )-\varphi /\omega$, where

(15)$$\begin{aligned} G_1(\tau) &= \frac{1}{4\omega} \{ 2 (A+B)\omega \tau -(A-B)\sin\{2[\omega(\tau-t_0)+\varphi]\}\\ & -2C \cos\{2[\omega(\tau-t_0)+\varphi]\} \}, \end{aligned}$$

(16)$$ G_2(\tau) = \frac{1}{\omega}\tan^{{-}1} \{ C+A\tan[\omega (\tau-t_0) +\varphi] \}, $$

(17)$$\begin{aligned} G_3(\tau) &= \omega \{ 2(A+B)\omega \tau +(A-B)\sin\{ 2[\omega (\tau-t_0)+\varphi] \}\\ & +2C\cos\{ 2[\omega (\tau-t_0)+\varphi] \} - 4\tan^{{-}1}\{ C+A\tan[\omega (\tau-t_0) +\varphi] \} \}. \end{aligned}$$

A readjustment of Eq. (10) with Eqs. (14)–(17) results in

(18)$$\Gamma_{G} = F_G(t)-F_G(t_0),$$

where $F_G(\tau )$ is of the form

(19)$$F_G(\tau) = (A+B)\omega \tau -2 \tan^{{-}1} \{ C+A\tan[\omega (\tau-t_0) +\varphi] \}.$$

Note that Eq. (18) with Eq. (19) holds for the region $t_0 \leq t<t_0+\pi /(2\omega )-\varphi /\omega$, since we have considered $t \geq t_0$ while the first discontinuity in $\tan \theta$ appears at $\theta = \pi /2$.

The above consequence, Eq. (18), can be extend to the whole region ($t \geq t_0$), regarded in the main text, by compensating it with the unit step function $u[t]$, namely

(20)$$\Gamma_{G} = F_G(t)-F_G(t_0)-2\pi \sum_{m=0}^{\infty}u[t-t_0-(2m+1)\pi/(2\omega)+\varphi/\omega].$$

Thus, a rearrangement of Eq. (9) with Eqs. (20) and (19) gives the exact formula of geometric phases represented in Eq. (7).

For the case of the evaluation of dynamical phases, we start from their definition, which is

(21)$$\gamma_{D,n}(t) ={-} \frac{1}{\hbar} \int_{t_0}^{t} \langle\Phi_n(t') | \hat{H}(\hat{q},\hat{p},t')| \Phi_n(t')\rangle dt' + \gamma_{D,n}(t_0).$$

Let us assume that $\gamma _{D,n}(t_0)=0$ similarly to the previous case. Then, the execution of the integration in Eq. (21) through the use of Eqs. (1) and (2) leads to

(22)$$\gamma_{D,n}(t) = \frac{1}{2} \left( n+\frac{1}{2} \right) \Gamma_{D} ,$$

where

(23)$$\Gamma_{D} ={-}\omega[g_1(t)+g_2(t)]- \frac{g_3(t)}{4\omega}.$$

The substitution of Eq. (14) with Eqs. (15)–(17) into Eq. (23) gives

(24)$$\Gamma_{D} = F_D(t)-F_D(t_0),$$

where

(25)$$F_D(\tau) ={-}(A+B)\omega \tau .$$

Now, Eq. (22) can be rewritten as

(26)$$\gamma_{D,n}(t) = \frac{1}{2} \left( n+\frac{1}{2} \right) [F_D(t)-F_D(t_0)] .$$

We see that this is identical to the final formula of the dynamical phases shown in Eq. (8).

Funding

National Research Foundation of Korea (NRF-2021R1F1A1062849).

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392(1802), 45–57 (1984). [CrossRef]

2. A. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58(16), 1593–1596 (1987). [CrossRef]

3. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228(2), 205–268 (1993). [CrossRef]

4. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228(2), 269–340 (1993). [CrossRef]

5. C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8(1), 1061 (2017). [CrossRef]

6. W. J. Tango and J. Davis, “Application of geometric phase techniques to stellar interferometry,” Appl. Opt. 35(4), 621–623 (1996). [CrossRef]

7. A. Capolupo and G. Vitiello, “Geometric phase and its applications to fundamental physics,” Il Nuovo Cimento C 38(5), 171 (2015). [CrossRef]

8. S. Abdel-Khalek, Y. S. El-Saman, I. Mechai, and M. Abdel-Aty, “Geometric phase and entanglement of a three-level atom with and without rotating wave approximation,” Brazilian J. Phys. 48(1), 9–15 (2018). [CrossRef]

9. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond,” Nat. Rev. Phys. 1(7), 437–449 (2019). [CrossRef]

10. J. Y. Zeng and Y. A. Lei, “Connection between the Berry phase and the Lewis phase,” Phys. Lett. A 215(5-6), 239–244 (1996). [CrossRef]

11. J. R. Choi, “Quadrature squeezing and geometric-phase oscillations in nano-optics,” Nanomaterials 10(7), 1391 (2020). [CrossRef]

12. S. N. Biswas and S. K. Soni, “Berry’s phase for coherent states and canonical transformation,” Phys. Rev. A 43(10), 5717–5719 (1991). [CrossRef]

13. J. R. Choi, “On the possible emergence of nonstatic quantum waves in a static environment,” Nonlinear Dyn. 103(3), 2783–2792 (2021). [CrossRef]

14. J. R. Choi, “Characteristics of nonstatic quantum light waves: The principle for wave expansion and collapse,” Photonics 8(5), 158 (2021). [CrossRef]

15. J. H. Hannay, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian,” J. Phys. A: Math. Gen. 18(2), 221–230 (1985). [CrossRef]

16. G. S. Agarwal and R. Simon, “Berry phase, interference of light beams, and the Hannay angle,” Phys. Rev. A 42(11), 6924–6927 (1990). [CrossRef]

17. H. D. Liu, S. L. Wu, and X. X. Yi, “Berry phase and Hannay’s angle in a quantum-classical hybrid system,” Phys. Rev. A 83(6), 062101 (2011). [CrossRef]

18. P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. Göppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, and A. Wallraff, “Observation of Berry’s phase in a solid-state qubit,” Science 318(5858), 1889–1892 (2007). [CrossRef]

19. A. Mostafazadeh, “Quantum adiabatic approximation, quantum action, and Berry’s phase,” Phys. Lett. A 232(6), 395–398 (1997). [CrossRef]

20. M. P. Silveri, J. A. Tuorila, E. V. Thuneberg, and G. S. Paraoanu, “Quantum systems under frequency modulation,” Rep. Prog. Phys. 80(5), 056002 (2017). [CrossRef]

21. M. S. Sarandy, E. I. Duzzioni, and M. H. Y. Moussa, “Dynamical invariants and nonadiabatic geometric phases in open quantum systems,” Phys. Rev. A 76(5), 052112 (2007). [CrossRef]

22. M. V. Berry, “Classical adiabatic angles and quantal adiabatic phase,” J. Phys. A: Math. Gen. 18(1), 15–27 (1985). [CrossRef]

23. X.-B. Wang, L. C. Kwek, and C. H. Oh, “Quantum and classical geometric phase of the time-dependent harmonic oscillator,” Phys. Rev. A 62(3), 032105 (2000). [CrossRef]

24. G. Ghosh and B. Dutta-Roy, “The Berry phase and the Hannay angle,” Phys. Rev. D 37(6), 1709–1711 (1988). [CrossRef]

25. S. Golin, “Existence of the Hannay angle for single-frequency systems,” J. Phys. A: Math. Gen. 21(24), 4535–4547 (1988). [CrossRef]

26. S. Golin, “Can one measure Hannay angles?” J. Phys. A: Math. Gen. 22(21), 4573–4580 (1989). [CrossRef]

27. S. Golin and S. Marmi, “A class of systems with measurable Hannay angles,” Nonlinearity 3(2), 507–518 (1990). [CrossRef]

28. D. Chruściński and A. Jamiołkowski, Geometric Phases in Classical and Quantum Mechanics (Birkhäuser, Berlin, 2004).

29. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1978).

30. B. Yurke, P. Grangier, R. E. Slusher, and M. J. Potasek, “Generating and detecting short-duration pulses of squeezed light,” Phys. Rev. A 35(8), 3586–3589 (1987). [CrossRef]

31. Y. Li and M. Xiao, “Generation and applications of amplitude-squeezed states of light from semiconductor diode lasers,” Opt. Express 2(3), 110–117 (1998). [CrossRef]

32. Y. Han, X. Wen, J. Liu, J. He, and J. Wang, “Generation of polarization squeezed light with an optical parametric amplifier at 795 nm,” Opt. Commun. 416, 1–4 (2018). [CrossRef]

33. P. Neveu, J. Delpy, S. Liu, C. Banerjee, J. Lugani, F. Bretenaker, E. Brion, and F. Goldfarb, “Generation of squeezed light vacuum enabled by coherent population trapping,” Opt. Express 29(7), 10471–10479 (2021). [CrossRef]

34. L. Ma, H. Guo, H. Sun, K. Liu, B. Su, and J. Gao, “Generation of squeezed states of light in arbitrary complex amplitude transverse distribution,” Photonics Res. 8(9), 1422–1427 (2020). [CrossRef]

35. M. Ericsson, D. Achilles, J. T. Barreiro, D. Branning, N. A. Peters, and P. G. Kwiat, “Measurement of geometric phase for mixed states using single photon interferometry,” Phys. Rev. Lett. 94(5), 050401 (2005). [CrossRef]

36. A. Hannonen, H. Partanen, A. Leinonen, J. Heikkinen, T. K. Hakala, A. T. Friberg, and T. Setälä, “Measurement of the Pancharatnam-Berry phase in two-beam interference,” Optica 7(10), 1435–1439 (2020). [CrossRef]

37. T. Malhotra, R. Gutiérrez-Cuevas, J. Hassett, M. R. Dennis, A. N. Vamivakas, and M. A. Alonso, “Measuring geometric phase without interferometry,” Phys. Rev. Lett. 120(23), 233602 (2018). [CrossRef]

38. J. Liu, B. Hu, and B. Li, “Nonadiabatic geometric phase for the cyclic evolution of a time-dependent Hamiltonian system,” Phys. Rev. A 58(5), 3448–3456 (1998). [CrossRef]

39. Y. C. Ge and M. S. Child, “Nonadiabatic geometrical phase during cyclic evolution of a Gaussian wave packet,” Phys. Rev. Lett. 78(13), 2507–2510 (1997). [CrossRef]

40. J. R. Choi, “The decay properties of a single-photon in linear media,” Chinese J. Phys. 41(3), 257–266 (2003).

41. S. K. Suslov, “An analogue of the Berry phase for simple harmonic oscillators,” Phys. Scr. 87(3), 038118 (2013). [CrossRef]

42. H. K. Kim and S. P. Kim, “Dynamical squeezing in the stable, damped oscillator model for biophoton emission,” J. Korean Phys. Soc. 47(1), 42–46 (2005).

43. K. H. Yeon, C. I. Um, and T. F. George, “Coherent states for the damped harmonic oscillator,” Phys. Rev. A 36(11), 5287–5291 (1987). [CrossRef]

44. M. B. Farías, F. C. Lombardo, A. Soba, P. I. Villar, and R. S. Decca, “Towards detecting traces of non-contact quantum friction in the corrections of the accumulated geometric phase,” npj Quantum Inf. 6(1), 25 (2020). [CrossRef]

45. D. M. Tong, E. Sjöqvist, L. C. Kwek, and C. H. Oh, “Kinematic approach to the mixed state geometric phase in nonunitary evolution,” Phys. Rev. Lett. 93(8), 080405 (2004). [CrossRef]

Effects of light-wave nonstaticity on accompanying geometric-phase evolutions

Abstract

1. Introduction

2. Description of nonstatic waves

3. Geometric phase

4. Conclusion

5. Methods

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (26)

Optics Express